unscientific hypothesis example

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scientific hypothesis

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• National Center for Biotechnology Information - PubMed Central - On the scope of scientific hypotheses
• LiveScience - What is a scientific hypothesis?
• The Royal Society - Open Science - On the scope of scientific hypotheses

scientific hypothesis , an idea that proposes a tentative explanation about a phenomenon or a narrow set of phenomena observed in the natural world. The two primary features of a scientific hypothesis are falsifiability and testability, which are reflected in an “If…then” statement summarizing the idea and in the ability to be supported or refuted through observation and experimentation. The notion of the scientific hypothesis as both falsifiable and testable was advanced in the mid-20th century by Austrian-born British philosopher Karl Popper .

The formulation and testing of a hypothesis is part of the scientific method , the approach scientists use when attempting to understand and test ideas about natural phenomena. The generation of a hypothesis frequently is described as a creative process and is based on existing scientific knowledge, intuition , or experience. Therefore, although scientific hypotheses commonly are described as educated guesses, they actually are more informed than a guess. In addition, scientists generally strive to develop simple hypotheses, since these are easier to test relative to hypotheses that involve many different variables and potential outcomes. Such complex hypotheses may be developed as scientific models ( see scientific modeling ).

Depending on the results of scientific evaluation, a hypothesis typically is either rejected as false or accepted as true. However, because a hypothesis inherently is falsifiable, even hypotheses supported by scientific evidence and accepted as true are susceptible to rejection later, when new evidence has become available. In some instances, rather than rejecting a hypothesis because it has been falsified by new evidence, scientists simply adapt the existing idea to accommodate the new information. In this sense a hypothesis is never incorrect but only incomplete.

The investigation of scientific hypotheses is an important component in the development of scientific theory . Hence, hypotheses differ fundamentally from theories; whereas the former is a specific tentative explanation and serves as the main tool by which scientists gather data, the latter is a broad general explanation that incorporates data from many different scientific investigations undertaken to explore hypotheses.

Countless hypotheses have been developed and tested throughout the history of science . Several examples include the idea that living organisms develop from nonliving matter, which formed the basis of spontaneous generation , a hypothesis that ultimately was disproved (first in 1668, with the experiments of Italian physician Francesco Redi , and later in 1859, with the experiments of French chemist and microbiologist Louis Pasteur ); the concept proposed in the late 19th century that microorganisms cause certain diseases (now known as germ theory ); and the notion that oceanic crust forms along submarine mountain zones and spreads laterally away from them ( seafloor spreading hypothesis ).

The Unfalsifiable Hypothesis Paradox

What is the unfalsifiable hypothesis paradox.

Imagine someone tells you a story about a dragon that breathes not fire, but invisible, heatless fire. You grab a thermometer to test the claim but no matter what, you can’t prove it’s not true because you can’t measure something that’s invisible and has no heat. This is what we call an ‘unfalsifiable hypothesis’—it’s a claim that’s made in such a way that it can’t be proven wrong, no matter what.

Now, the paradox is this: in science, being able to prove or disprove a claim makes it strong and believable. If nobody could ever prove a hypothesis wrong, you’d think it’s completely reliable, right? But actually, in science, that makes it weak! If we can’t test a claim, then it’s not really playing by the rules of science. So, the paradox is that not being able to prove something wrong can make a claim scientifically useless—even though it seems like it would be the ultimate truth.

Key Arguments

• An unfalsifiable hypothesis is a claim that can’t be proven wrong, but just because we can’t disprove it, that doesn’t make it automatically true.
• Science grows and improves through testing ideas; if we can’t test a claim, we can’t know if it’s really valid.
• Being able to show that an idea could be wrong is a fundamental part of scientific thinking. Without this testability, a claim is more like a personal belief or a philosophical idea than a scientific one.
• An unfalsifiable hypothesis might look like it’s scientific, but it’s misleading since it doesn’t stick to the strict rules of testing and evidence that science needs.
• Using unfalsifiable claims can block our paths to understanding since they stop us from asking questions and looking for verifiable answers.
• The dragon with invisible, heatless fire: This is an example of an unfalsifiable hypothesis because no test or observation could ever show that the dragon’s fire isn’t real, since it can’t be detected in any way.
• Saying a celestial teapot orbits the Sun between Earth and Mars: This teapot is said to be small and far enough away that no telescope could spot it. Because it’s undetectable, we can’t disprove its existence.
• A theory that angels are responsible for keeping us gravitationally bound to Earth: Since we can’t test for the presence or actions of angels, we can’t refute the claim, making it unfalsifiable.
• The statement that the world’s sorrow is caused by invisible spirits: It sounds serious, but if we can’t measure or observe these spirits, we can’t possibly prove this idea right or wrong.

Answer or Resolution

Dealing with the Unfalsifiable Hypothesis Paradox means finding a balance. We can’t just ignore all ideas that can’t be tested because some might lead to real scientific breakthroughs one day. On the other side, we can’t treat untestable claims as true science. It’s about being open to possibilities but also clear about what counts as scientific evidence.

Some people might say we should only focus on what can be proven wrong. Others think even wild ideas have their place at the starting line of science—they inspire us and can evolve into something testable later on.

Major Criticism

Some people criticize the idea of rejecting all unfalsifiable ideas because that could block new ways of thinking. Sometimes a wild guess can turn into a real scientific discovery. Plus, falsifiability is just one part of what makes a theory scientific. We shouldn’t throw away potentially good ideas just because they don’t fit one rule, especially when they’re still in the early stages and shouldn’t be held too tightly to any rules at all.

Another point is that some important ideas have been unfalsifiable at first but later became testable. So, we have to recognize that science itself can change and grow.

Practical Applications

You might wonder, “Why does this matter to me?” Well, knowing about the Unfalsifiable Hypothesis Paradox actually affects a lot of real-world situations, like how we learn things in school, the kinds of products we buy, and even the rules and laws that are made.

• Education: By learning what makes science solid, students can tell the difference between real science and just a bunch of fancy words that sound scientific but aren’t based on testable ideas.
• Consumer Protection: Sometimes companies try to sell things by using science-sounding claims that can’t be proven wrong—and that’s where knowing about unfalsifiable hypotheses helps protect us from buying into false promises.
• Legal and Policy Making: For people who make laws or guide big decisions, understanding this concept helps them judge if a study or report is really based on solid science.

Related Topics

The Unfalsifiable Hypothesis Paradox is linked with a couple of other important ideas you might hear about:

• Scientific Method: This is the set of steps scientists use to learn about the world. Part of the process is making sure ideas can be tested.
• Pseudoscience: These are beliefs or practices that try to appear scientific but don’t follow the scientific method properly, often using unfalsifiable claims.
• Empiricism : This big word just means learning by observation and experiment—the backbone of science and everything opposite of unfalsifiable concepts.

Wrapping up, the Unfalsifiable Hypothesis Paradox shows us that science isn’t just about coming up with ideas—it’s about being able to test them, too. Untestable claims may be interesting, but they can’t help us understand the world in a scientific way. But remember, just because an idea is unfalsifiable now doesn’t mean it will be forever. The best approach is using that creative spark but always grounding it in what we can observe and prove. This balance keeps our imaginations soaring but our facts checked, forming a bridge between our wildest ideas and the world we can measure and know.

15 Hypothesis Examples

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Dr. Chris Drew is the founder of the Helpful Professor. He holds a PhD in education and has published over 20 articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education. [Image Descriptor: Photo of Chris]

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A hypothesis is defined as a testable prediction , and is used primarily in scientific experiments as a potential or predicted outcome that scientists attempt to prove or disprove (Atkinson et al., 2021; Tan, 2022).

In my types of hypothesis article, I outlined 13 different hypotheses, including the directional hypothesis (which makes a prediction about an effect of a treatment will be positive or negative) and the associative hypothesis (which makes a prediction about the association between two variables).

This article will dive into some interesting examples of hypotheses and examine potential ways you might test each one.

Hypothesis Examples

1. “inadequate sleep decreases memory retention”.

Field: Psychology

Type: Causal Hypothesis A causal hypothesis explores the effect of one variable on another. This example posits that a lack of adequate sleep causes decreased memory retention. In other words, if you are not getting enough sleep, your ability to remember and recall information may suffer.

How to Test:

To test this hypothesis, you might devise an experiment whereby your participants are divided into two groups: one receives an average of 8 hours of sleep per night for a week, while the other gets less than the recommended sleep amount.

During this time, all participants would daily study and recall new, specific information. You’d then measure memory retention of this information for both groups using standard memory tests and compare the results.

Should the group with less sleep have statistically significant poorer memory scores, the hypothesis would be supported.

Ensuring the integrity of the experiment requires taking into account factors such as individual health differences, stress levels, and daily nutrition.

Relevant Study: Sleep loss, learning capacity and academic performance (Curcio, Ferrara & De Gennaro, 2006)

2. “Increase in Temperature Leads to Increase in Kinetic Energy”

Field: Physics

Type: Deductive Hypothesis The deductive hypothesis applies the logic of deductive reasoning – it moves from a general premise to a more specific conclusion. This specific hypothesis assumes that as temperature increases, the kinetic energy of particles also increases – that is, when you heat something up, its particles move around more rapidly.

This hypothesis could be examined by heating a gas in a controlled environment and capturing the movement of its particles as a function of temperature.

You’d gradually increase the temperature and measure the kinetic energy of the gas particles with each increment. If the kinetic energy consistently rises with the temperature, your hypothesis gets supporting evidence.

Variables such as pressure and volume of the gas would need to be held constant to ensure validity of results.

3. “Children Raised in Bilingual Homes Develop Better Cognitive Skills”

Field: Psychology/Linguistics

Type: Comparative Hypothesis The comparative hypothesis posits a difference between two or more groups based on certain variables. In this context, you might propose that children raised in bilingual homes have superior cognitive skills compared to those raised in monolingual homes.

Testing this hypothesis could involve identifying two groups of children: those raised in bilingual homes, and those raised in monolingual homes.

Cognitive skills in both groups would be evaluated using a standard cognitive ability test at different stages of development. The examination would be repeated over a significant time period for consistency.

If the group raised in bilingual homes persistently scores higher than the other, the hypothesis would thereby be supported.

The challenge for the researcher would be controlling for other variables that could impact cognitive development, such as socio-economic status, education level of parents, and parenting styles.

Relevant Study: The cognitive benefits of being bilingual (Marian & Shook, 2012)

4. “High-Fiber Diet Leads to Lower Incidences of Cardiovascular Diseases”

Field: Medicine/Nutrition

Type: Alternative Hypothesis The alternative hypothesis suggests an alternative to a null hypothesis. In this context, the implied null hypothesis could be that diet has no effect on cardiovascular health, which the alternative hypothesis contradicts by suggesting that a high-fiber diet leads to fewer instances of cardiovascular diseases.

To test this hypothesis, a longitudinal study could be conducted on two groups of participants; one adheres to a high-fiber diet, while the other follows a diet low in fiber.

After a fixed period, the cardiovascular health of participants in both groups could be analyzed and compared. If the group following a high-fiber diet has a lower number of recorded cases of cardiovascular diseases, it would provide evidence supporting the hypothesis.

Control measures should be implemented to exclude the influence of other lifestyle and genetic factors that contribute to cardiovascular health.

Relevant Study: Dietary fiber, inflammation, and cardiovascular disease (King, 2005)

5. “Gravity Influences the Directional Growth of Plants”

Field: Agronomy / Botany

Type: Explanatory Hypothesis An explanatory hypothesis attempts to explain a phenomenon. In this case, the hypothesis proposes that gravity affects how plants direct their growth – both above-ground (toward sunlight) and below-ground (towards water and other resources).

The testing could be conducted by growing plants in a rotating cylinder to create artificial gravity.

Observations on the direction of growth, over a specified period, can provide insights into the influencing factors. If plants consistently direct their growth in a manner that indicates the influence of gravitational pull, the hypothesis is substantiated.

It is crucial to ensure that other growth-influencing factors, such as light and water, are uniformly distributed so that only gravity influences the directional growth.

6. “The Implementation of Gamified Learning Improves Students’ Motivation”

Field: Education

Type: Relational Hypothesis The relational hypothesis describes the relation between two variables. Here, the hypothesis is that the implementation of gamified learning has a positive effect on the motivation of students.

To validate this proposition, two sets of classes could be compared: one that implements a learning approach with game-based elements, and another that follows a traditional learning approach.

The students’ motivation levels could be gauged by monitoring their engagement, performance, and feedback over a considerable timeframe.

If the students engaged in the gamified learning context present higher levels of motivation and achievement, the hypothesis would be supported.

Control measures ought to be put into place to account for individual differences, including prior knowledge and attitudes towards learning.

Relevant Study: Does educational gamification improve students’ motivation? (Chapman & Rich, 2018)

7. “Mathematics Anxiety Negatively Affects Performance”

Field: Educational Psychology

Type: Research Hypothesis The research hypothesis involves making a prediction that will be tested. In this case, the hypothesis proposes that a student’s anxiety about math can negatively influence their performance in math-related tasks.

To assess this hypothesis, researchers must first measure the mathematics anxiety levels of a sample of students using a validated instrument, such as the Mathematics Anxiety Rating Scale.

Then, the students’ performance in mathematics would be evaluated through standard testing. If there’s a negative correlation between the levels of math anxiety and math performance (meaning as anxiety increases, performance decreases), the hypothesis would be supported.

It would be crucial to control for relevant factors such as overall academic performance and previous mathematical achievement.

8. “Disruption of Natural Sleep Cycle Impairs Worker Productivity”

Field: Organizational Psychology

Type: Operational Hypothesis The operational hypothesis involves defining the variables in measurable terms. In this example, the hypothesis posits that disrupting the natural sleep cycle, for instance through shift work or irregular working hours, can lessen productivity among workers.

To test this hypothesis, you could collect data from workers who maintain regular working hours and those with irregular schedules.

Measuring productivity could involve examining the worker’s ability to complete tasks, the quality of their work, and their efficiency.

If workers with interrupted sleep cycles demonstrate lower productivity compared to those with regular sleep patterns, it would lend support to the hypothesis.

Consideration should be given to potential confounding variables such as job type, worker age, and overall health.

9. “Regular Physical Activity Reduces the Risk of Depression”

Field: Health Psychology

Type: Predictive Hypothesis A predictive hypothesis involves making a prediction about the outcome of a study based on the observed relationship between variables. In this case, it is hypothesized that individuals who engage in regular physical activity are less likely to suffer from depression.

Longitudinal studies would suit to test this hypothesis, tracking participants’ levels of physical activity and their mental health status over time.

The level of physical activity could be self-reported or monitored, while mental health status could be assessed using standard diagnostic tools or surveys.

If data analysis shows that participants maintaining regular physical activity have a lower incidence of depression, this would endorse the hypothesis.

However, care should be taken to control other lifestyle and behavioral factors that could intervene with the results.

Relevant Study: Regular physical exercise and its association with depression (Kim, 2022)

10. “Regular Meditation Enhances Emotional Stability”

Type: Empirical Hypothesis In the empirical hypothesis, predictions are based on amassed empirical evidence . This particular hypothesis theorizes that frequent meditation leads to improved emotional stability, resonating with numerous studies linking meditation to a variety of psychological benefits.

Earlier studies reported some correlations, but to test this hypothesis directly, you’d organize an experiment where one group meditates regularly over a set period while a control group doesn’t.

Both groups’ emotional stability levels would be measured at the start and end of the experiment using a validated emotional stability assessment.

If regular meditators display noticeable improvements in emotional stability compared to the control group, the hypothesis gains credit.

You’d have to ensure a similar emotional baseline for all participants at the start to avoid skewed results.

11. “Children Exposed to Reading at an Early Age Show Superior Academic Progress”

Type: Directional Hypothesis The directional hypothesis predicts the direction of an expected relationship between variables. Here, the hypothesis anticipates that early exposure to reading positively affects a child’s academic advancement.

A longitudinal study tracking children’s reading habits from an early age and their consequent academic performance could validate this hypothesis.

Parents could report their children’s exposure to reading at home, while standardized school exam results would provide a measure of academic achievement.

If the children exposed to early reading consistently perform better acadically, it gives weight to the hypothesis.

However, it would be important to control for variables that might impact academic performance, such as socioeconomic background, parental education level, and school quality.

12. “Adopting Energy-efficient Technologies Reduces Carbon Footprint of Industries”

Field: Environmental Science

Type: Descriptive Hypothesis A descriptive hypothesis predicts the existence of an association or pattern related to variables. In this scenario, the hypothesis suggests that industries adopting energy-efficient technologies will resultantly show a reduced carbon footprint.

Global industries making use of energy-efficient technologies could track their carbon emissions over time. At the same time, others not implementing such technologies continue their regular tracking.

After a defined time, the carbon emission data of both groups could be compared. If industries that adopted energy-efficient technologies demonstrate a notable reduction in their carbon footprints, the hypothesis would hold strong.

In the experiment, you would exclude variations brought by factors such as industry type, size, and location.

13. “Reduced Screen Time Improves Sleep Quality”

Type: Simple Hypothesis The simple hypothesis is a prediction about the relationship between two variables, excluding any other variables from consideration. This example posits that by reducing time spent on devices like smartphones and computers, an individual should experience improved sleep quality.

A sample group would need to reduce their daily screen time for a pre-determined period. Sleep quality before and after the reduction could be measured using self-report sleep diaries and objective measures like actigraphy, monitoring movement and wakefulness during sleep.

If the data shows that sleep quality improved post the screen time reduction, the hypothesis would be validated.

Other aspects affecting sleep quality, like caffeine intake, should be controlled during the experiment.

Relevant Study: Screen time use impacts low‐income preschool children’s sleep quality, tiredness, and ability to fall asleep (Waller et al., 2021)

14. Engaging in Brain-Training Games Improves Cognitive Functioning in Elderly

Field: Gerontology

Type: Inductive Hypothesis Inductive hypotheses are based on observations leading to broader generalizations and theories. In this context, the hypothesis deduces from observed instances that engaging in brain-training games can help improve cognitive functioning in the elderly.

A longitudinal study could be conducted where an experimental group of elderly people partakes in regular brain-training games.

Their cognitive functioning could be assessed at the start of the study and at regular intervals using standard neuropsychological tests.

If the group engaging in brain-training games shows better cognitive functioning scores over time compared to a control group not playing these games, the hypothesis would be supported.

15. Farming Practices Influence Soil Erosion Rates

Type: Null Hypothesis A null hypothesis is a negative statement assuming no relationship or difference between variables. The hypothesis in this context asserts there’s no effect of different farming practices on the rates of soil erosion.

Comparing soil erosion rates in areas with different farming practices over a considerable timeframe could help test this hypothesis.

If, statistically, the farming practices do not lead to differences in soil erosion rates, the null hypothesis is accepted.

However, if marked variation appears, the null hypothesis is rejected, meaning farming practices do influence soil erosion rates. It would be crucial to control for external factors like weather, soil type, and natural vegetation.

The variety of hypotheses mentioned above underscores the diversity of research constructs inherent in different fields, each with its unique purpose and way of testing.

While researchers may develop hypotheses primarily as tools to define and narrow the focus of the study, these hypotheses also serve as valuable guiding forces for the data collection and analysis procedures, making the research process more efficient and direction-focused.

Hypotheses serve as a compass for any form of academic research. The diverse examples provided, from Psychology to Educational Studies, Environmental Science to Gerontology, clearly demonstrate how certain hypotheses suit specific fields more aptly than others.

It is important to underline that although these varied hypotheses differ in their structure and methods of testing, each endorses the fundamental value of empiricism in research. Evidence-based decision making remains at the heart of scholarly inquiry, regardless of the research field, thus aligning all hypotheses to the core purpose of scientific investigation.

Testing hypotheses is an essential part of the scientific method . By doing so, researchers can either confirm their predictions, giving further validity to an existing theory, or they might uncover new insights that could potentially shift the field’s understanding of a particular phenomenon. In either case, hypotheses serve as the stepping stones for scientific exploration and discovery.

Atkinson, P., Delamont, S., Cernat, A., Sakshaug, J. W., & Williams, R. A. (2021).  SAGE research methods foundations . SAGE Publications Ltd.

Curcio, G., Ferrara, M., & De Gennaro, L. (2006). Sleep loss, learning capacity and academic performance.  Sleep medicine reviews ,  10 (5), 323-337.

Kim, J. H. (2022). Regular physical exercise and its association with depression: A population-based study short title: Exercise and depression.  Psychiatry Research ,  309 , 114406.

King, D. E. (2005). Dietary fiber, inflammation, and cardiovascular disease.  Molecular nutrition & food research ,  49 (6), 594-600.

Marian, V., & Shook, A. (2012, September). The cognitive benefits of being bilingual. In Cerebrum: the Dana forum on brain science (Vol. 2012). Dana Foundation.

Tan, W. C. K. (2022). Research Methods: A Practical Guide For Students And Researchers (Second Edition) . World Scientific Publishing Company.

Waller, N. A., Zhang, N., Cocci, A. H., D’Agostino, C., Wesolek‐Greenson, S., Wheelock, K., … & Resnicow, K. (2021). Screen time use impacts low‐income preschool children’s sleep quality, tiredness, and ability to fall asleep. Child: care, health and development, 47 (5), 618-626.

• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd-2/ 10 Reasons you’re Perpetually Single
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd-2/ 20 Montessori Toddler Bedrooms (Design Inspiration)
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd-2/ 21 Montessori Homeschool Setups
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd-2/ 101 Hidden Talents Examples

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Hypothesis Examples

A hypothesis is a prediction of the outcome of a test. It forms the basis for designing an experiment in the scientific method . A good hypothesis is testable, meaning it makes a prediction you can check with observation or experimentation. Here are different hypothesis examples.

Null Hypothesis Examples

The null hypothesis (H 0 ) is also known as the zero-difference or no-difference hypothesis. It predicts that changing one variable ( independent variable ) will have no effect on the variable being measured ( dependent variable ). Here are null hypothesis examples:

• Plant growth is unaffected by temperature.
• If you increase temperature, then solubility of salt will increase.
• Incidence of skin cancer is unrelated to ultraviolet light exposure.
• All brands of light bulb last equally long.
• Cats have no preference for the color of cat food.
• All daisies have the same number of petals.

Sometimes the null hypothesis shows there is a suspected correlation between two variables. For example, if you think plant growth is affected by temperature, you state the null hypothesis: “Plant growth is not affected by temperature.” Why do you do this, rather than say “If you change temperature, plant growth will be affected”? The answer is because it’s easier applying a statistical test that shows, with a high level of confidence, a null hypothesis is correct or incorrect.

Research Hypothesis Examples

A research hypothesis (H 1 ) is a type of hypothesis used to design an experiment. This type of hypothesis is often written as an if-then statement because it’s easy identifying the independent and dependent variables and seeing how one affects the other. If-then statements explore cause and effect. In other cases, the hypothesis shows a correlation between two variables. Here are some research hypothesis examples:

• If you leave the lights on, then it takes longer for people to fall asleep.
• If you refrigerate apples, they last longer before going bad.
• If you keep the curtains closed, then you need less electricity to heat or cool the house (the electric bill is lower).
• If you leave a bucket of water uncovered, then it evaporates more quickly.
• Goldfish lose their color if they are not exposed to light.
• Workers who take vacations are more productive than those who never take time off.

Is It Okay to Disprove a Hypothesis?

Yes! You may even choose to write your hypothesis in such a way that it can be disproved because it’s easier to prove a statement is wrong than to prove it is right. In other cases, if your prediction is incorrect, that doesn’t mean the science is bad. Revising a hypothesis is common. It demonstrates you learned something you did not know before you conducted the experiment.

Test yourself with a Scientific Method Quiz .

• Mellenbergh, G.J. (2008). Chapter 8: Research designs: Testing of research hypotheses. In H.J. Adèr & G.J. Mellenbergh (eds.), Advising on Research Methods: A Consultant’s Companion . Huizen, The Netherlands: Johannes van Kessel Publishing.
• Popper, Karl R. (1959). The Logic of Scientific Discovery . Hutchinson & Co. ISBN 3-1614-8410-X.
• Schick, Theodore; Vaughn, Lewis (2002). How to think about weird things: critical thinking for a New Age . Boston: McGraw-Hill Higher Education. ISBN 0-7674-2048-9.
• Tobi, Hilde; Kampen, Jarl K. (2018). “Research design: the methodology for interdisciplinary research framework”. Quality & Quantity . 52 (3): 1209–1225. doi: 10.1007/s11135-017-0513-8

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How to Write a Great Hypothesis

Hypothesis Definition, Format, Examples, and Tips

Verywell / Alex Dos Diaz

• The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis.

• Operationalization

Hypothesis Types

Hypotheses examples.

• Collecting Data

A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study. It is a preliminary answer to your question that helps guide the research process.

Consider a study designed to examine the relationship between sleep deprivation and test performance. The hypothesis might be: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

At a Glance

A hypothesis is crucial to scientific research because it offers a clear direction for what the researchers are looking to find. This allows them to design experiments to test their predictions and add to our scientific knowledge about the world. This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

• Forming a question
• Performing background research
• Creating a hypothesis
• Designing an experiment
• Collecting data
• Analyzing the results
• Drawing conclusions
• Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. At this point, researchers then begin to develop a testable hypothesis.

Unless you are creating an exploratory study, your hypothesis should always explain what you  expect  to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore numerous factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment  do not  support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk adage that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

• Is your hypothesis based on your research on a topic?
• Can your hypothesis be tested?
• Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the  journal articles you read . Many authors will suggest questions that still need to be explored.

How to Formulate a Good Hypothesis

To form a hypothesis, you should take these steps:

• Collect as many observations about a topic or problem as you can.
• Evaluate these observations and look for possible causes of the problem.
• Create a list of possible explanations that you might want to explore.
• After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method ,  falsifiability is an important part of any valid hypothesis. In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that  if  something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

The Importance of Operational Definitions

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

Operational definitions are specific definitions for all relevant factors in a study. This process helps make vague or ambiguous concepts detailed and measurable.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in various ways. Clearly defining these variables and how they are measured helps ensure that other researchers can replicate your results.

Replicability

One of the basic principles of any type of scientific research is that the results must be replicable.

Replication means repeating an experiment in the same way to produce the same results. By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. For example, how would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

To measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming others. The researcher might utilize a simulated task to measure aggressiveness in this situation.

Hypothesis Checklist

• Does your hypothesis focus on something that you can actually test?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate the variables?
• Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

• Simple hypothesis : This type of hypothesis suggests there is a relationship between one independent variable and one dependent variable.
• Complex hypothesis : This type suggests a relationship between three or more variables, such as two independent and dependent variables.
• Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
• Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
• Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative population sample and then generalizes the findings to the larger group.
• Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the  dependent variable  if you change the  independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

A few examples of simple hypotheses:

• "Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
• "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."​
• "Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."
• "Children who receive a new reading intervention will have higher reading scores than students who do not receive the intervention."

Examples of a complex hypothesis include:

• "People with high-sugar diets and sedentary activity levels are more likely to develop depression."
• "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

Examples of a null hypothesis include:

• "There is no difference in anxiety levels between people who take St. John's wort supplements and those who do not."
• "There is no difference in scores on a memory recall task between children and adults."
• "There is no difference in aggression levels between children who play first-person shooter games and those who do not."

Examples of an alternative hypothesis:

• "People who take St. John's wort supplements will have less anxiety than those who do not."
• "Adults will perform better on a memory task than children."
• "Children who play first-person shooter games will show higher levels of aggression than children who do not." 

Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

Descriptive Research Methods

Descriptive research such as  case studies ,  naturalistic observations , and surveys are often used when  conducting an experiment is difficult or impossible. These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a  correlational study  can examine how the variables are related. This research method might be used to investigate a hypothesis that is difficult to test experimentally.

Experimental Research Methods

Experimental methods  are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually  cause  another to change.

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Thompson WH, Skau S. On the scope of scientific hypotheses .  R Soc Open Sci . 2023;10(8):230607. doi:10.1098/rsos.230607

Taran S, Adhikari NKJ, Fan E. Falsifiability in medicine: what clinicians can learn from Karl Popper [published correction appears in Intensive Care Med. 2021 Jun 17;:].  Intensive Care Med . 2021;47(9):1054-1056. doi:10.1007/s00134-021-06432-z

Eyler AA. Research Methods for Public Health . 1st ed. Springer Publishing Company; 2020. doi:10.1891/9780826182067.0004

Nosek BA, Errington TM. What is replication ?  PLoS Biol . 2020;18(3):e3000691. doi:10.1371/journal.pbio.3000691

Aggarwal R, Ranganathan P. Study designs: Part 2 - Descriptive studies .  Perspect Clin Res . 2019;10(1):34-36. doi:10.4103/picr.PICR_154_18

Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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Falsifiability

Karl popper's basic scientific principle, karl popper's basic scientific principle.

Falsifiability, according to the philosopher Karl Popper, defines the inherent testability of any scientific hypothesis.

This article is a part of the guide:

• Inductive Reasoning
• Deductive Reasoning
• Hypothetico-Deductive Method
• Scientific Reasoning
• Testability

Browse Full Outline

• 1 Scientific Reasoning
• 2.1 Falsifiability
• 2.2 Verification Error
• 2.3 Testability
• 2.4 Post Hoc Reasoning
• 3 Deductive Reasoning
• 4.1 Raven Paradox
• 5 Causal Reasoning
• 6 Abductive Reasoning
• 7 Defeasible Reasoning

Science and philosophy have always worked together to try to uncover truths about the universe we live in. Indeed, ancient philosophy can be understood as the originator of many of the separate fields of study we have today, including psychology, medicine, law, astronomy, art and even theology.

Scientists design experiments and try to obtain results verifying or disproving a hypothesis, but philosophers are interested in understanding what factors determine the validity of scientific endeavors in the first place.

Whilst most scientists work within established paradigms, philosophers question the paradigms themselves and try to explore our underlying assumptions and definitions behind the logic of how we seek knowledge. Thus there is a feedback relationship between science and philosophy - and sometimes plenty of tension!

One of the tenets behind the scientific method is that any scientific hypothesis and resultant experimental design must be inherently falsifiable. Although falsifiability is not universally accepted, it is still the foundation of the majority of scientific experiments. Most scientists accept and work with this tenet, but it has its roots in philosophy and the deeper questions of truth and our access to it.

What is Falsifiability?

Falsifiability is the assertion that for any hypothesis to have credence, it must be inherently disprovable before it can become accepted as a scientific hypothesis or theory.

For example, someone might claim "the earth is younger than many scientists state, and in fact was created to appear as though it was older through deceptive fossils etc.” This is a claim that is unfalsifiable because it is a theory that can never be shown to be false. If you were to present such a person with fossils, geological data or arguments about the nature of compounds in the ozone, they could refute the argument by saying that your evidence was fabricated to appeared that way, and isn’t valid.

Importantly, falsifiability doesn’t mean that there are currently arguments against a theory, only that it is possible to imagine some kind of argument which would invalidate it. Falsifiability says nothing about an argument's inherent validity or correctness. It is only the minimum trait required of a claim that allows it to be engaged with in a scientific manner – a dividing line between what is considered science and what isn’t. Another important point is that falsifiability is not any claim that has yet to be proven true. After all, a conjecture that hasn’t been proven yet is just a hypothesis.

The idea is that no theory is completely correct , but if it can be shown both to be falsifiable  and supported with evidence that shows it's true, it can be accepted as truth.

For example, Newton's Theory of Gravity was accepted as truth for centuries, because objects do not randomly float away from the earth. It appeared to fit the data obtained by experimentation and research , but was always subject to testing.

However, Einstein's theory makes falsifiable predictions that are different from predictions made by Newton's theory, for example concerning the precession of the orbit of Mercury, and gravitational lensing of light. In non-extreme situations Einstein's and Newton's theories make the same predictions, so they are both correct. But Einstein's theory holds true in a superset of the conditions in which Newton's theory holds, so according to the principle of Occam's Razor , Einstein's theory is preferred. On the other hand, Newtonian calculations are simpler, so Newton's theory is useful for almost any engineering project, including some space projects. But for GPS we need Einstein's theory. Scientists would not have arrived at either of these theories, or a compromise between both of them, without the use of testable, falsifiable experiments. 

Popper saw falsifiability as a black and white definition; that if a theory is falsifiable, it is scientific , and if not, then it is unscientific. Whilst some "pure" sciences do adhere to this strict criterion, many fall somewhere between the two extremes, with  pseudo-sciences  falling at the extreme end of being unfalsifiable. 

Pseudoscience

According to Popper, many branches of applied science, especially social science, are not truly scientific because they have no potential for falsification.

Anthropology and sociology, for example, often use case studies to observe people in their natural environment without actually testing any specific hypotheses or theories.

While such studies and ideas are not falsifiable, most would agree that they are scientific because they significantly advance human knowledge.

Popper had and still has his fair share of critics, and the question of how to demarcate legitimate scientific enquiry can get very convoluted. Some statements are logically falsifiable but not practically falsifiable – consider the famous example of “it will rain at this location in a million years' time.” You could absolutely conceive of a way to test this claim, but carrying it out is a different story.

Thus, falsifiability is not a simple black and white matter. The Raven Paradox shows the inherent danger of relying on falsifiability, because very few scientific experiments can measure all of the data, and necessarily rely upon generalization . Technologies change along with our aims and comprehension of the phenomena we study, and so the falsifiability criterion for good science is subject to shifting.

For many sciences, the idea of falsifiability is a useful tool for generating theories that are testable and realistic. Testability is a crucial starting point around which to design solid experiments that have a chance of telling us something useful about the phenomena in question. If a falsifiable theory is tested and the results are significant , then it can become accepted as a scientific truth.

The advantage of Popper's idea is that such truths can be falsified when more knowledge and resources are available. Even long accepted theories such as Gravity, Relativity and Evolution are increasingly challenged and adapted.

The major disadvantage of falsifiability is that it is very strict in its definitions and does not take into account the contributions of sciences that are observational and descriptive .

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Martyn Shuttleworth , Lyndsay T Wilson (Sep 21, 2008). Falsifiability. Retrieved Sep 03, 2024 from Explorable.com: https://explorable.com/falsifiability

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How to Write a Research Hypothesis: Good & Bad Examples

What is a research hypothesis?

A research hypothesis is an attempt at explaining a phenomenon or the relationships between phenomena/variables in the real world. Hypotheses are sometimes called “educated guesses”, but they are in fact (or let’s say they should be) based on previous observations, existing theories, scientific evidence, and logic. A research hypothesis is also not a prediction—rather, predictions are ( should be) based on clearly formulated hypotheses. For example, “We tested the hypothesis that KLF2 knockout mice would show deficiencies in heart development” is an assumption or prediction, not a hypothesis. 

The research hypothesis at the basis of this prediction is “the product of the KLF2 gene is involved in the development of the cardiovascular system in mice”—and this hypothesis is probably (hopefully) based on a clear observation, such as that mice with low levels of Kruppel-like factor 2 (which KLF2 codes for) seem to have heart problems. From this hypothesis, you can derive the idea that a mouse in which this particular gene does not function cannot develop a normal cardiovascular system, and then make the prediction that we started with. 

What is the difference between a hypothesis and a prediction?

You might think that these are very subtle differences, and you will certainly come across many publications that do not contain an actual hypothesis or do not make these distinctions correctly. But considering that the formulation and testing of hypotheses is an integral part of the scientific method, it is good to be aware of the concepts underlying this approach. The two hallmarks of a scientific hypothesis are falsifiability (an evaluation standard that was introduced by the philosopher of science Karl Popper in 1934) and testability —if you cannot use experiments or data to decide whether an idea is true or false, then it is not a hypothesis (or at least a very bad one).

So, in a nutshell, you (1) look at existing evidence/theories, (2) come up with a hypothesis, (3) make a prediction that allows you to (4) design an experiment or data analysis to test it, and (5) come to a conclusion. Of course, not all studies have hypotheses (there is also exploratory or hypothesis-generating research), and you do not necessarily have to state your hypothesis as such in your paper. 

But for the sake of understanding the principles of the scientific method, let’s first take a closer look at the different types of hypotheses that research articles refer to and then give you a step-by-step guide for how to formulate a strong hypothesis for your own paper.

Types of Research Hypotheses

Hypotheses can be simple , which means they describe the relationship between one single independent variable (the one you observe variations in or plan to manipulate) and one single dependent variable (the one you expect to be affected by the variations/manipulation). If there are more variables on either side, you are dealing with a complex hypothesis. You can also distinguish hypotheses according to the kind of relationship between the variables you are interested in (e.g., causal or associative ). But apart from these variations, we are usually interested in what is called the “alternative hypothesis” and, in contrast to that, the “null hypothesis”. If you think these two should be listed the other way round, then you are right, logically speaking—the alternative should surely come second. However, since this is the hypothesis we (as researchers) are usually interested in, let’s start from there.

Alternative Hypothesis

If you predict a relationship between two variables in your study, then the research hypothesis that you formulate to describe that relationship is your alternative hypothesis (usually H1 in statistical terms). The goal of your hypothesis testing is thus to demonstrate that there is sufficient evidence that supports the alternative hypothesis, rather than evidence for the possibility that there is no such relationship. The alternative hypothesis is usually the research hypothesis of a study and is based on the literature, previous observations, and widely known theories. 

Null Hypothesis

The hypothesis that describes the other possible outcome, that is, that your variables are not related, is the null hypothesis ( H0 ). Based on your findings, you choose between the two hypotheses—usually that means that if your prediction was correct, you reject the null hypothesis and accept the alternative. Make sure, however, that you are not getting lost at this step of the thinking process: If your prediction is that there will be no difference or change, then you are trying to find support for the null hypothesis and reject H1. 

Directional Hypothesis

While the null hypothesis is obviously “static”, the alternative hypothesis can specify a direction for the observed relationship between variables—for example, that mice with higher expression levels of a certain protein are more active than those with lower levels. This is then called a one-tailed hypothesis. 

Another example for a directional one-tailed alternative hypothesis would be that 

H1: Attending private classes before important exams has a positive effect on performance. 

Your null hypothesis would then be that

H0: Attending private classes before important exams has no/a negative effect on performance.

Nondirectional Hypothesis

A nondirectional hypothesis does not specify the direction of the potentially observed effect, only that there is a relationship between the studied variables—this is called a two-tailed hypothesis. For instance, if you are studying a new drug that has shown some effects on pathways involved in a certain condition (e.g., anxiety) in vitro in the lab, but you can’t say for sure whether it will have the same effects in an animal model or maybe induce other/side effects that you can’t predict and potentially increase anxiety levels instead, you could state the two hypotheses like this:

H1: The only lab-tested drug (somehow) affects anxiety levels in an anxiety mouse model.

You then test this nondirectional alternative hypothesis against the null hypothesis:

H0: The only lab-tested drug has no effect on anxiety levels in an anxiety mouse model.

How to Write a Hypothesis for a Research Paper

Now that we understand the important distinctions between different kinds of research hypotheses, let’s look at a simple process of how to write a hypothesis.

Writing a Hypothesis Step:1

Ask a question, based on earlier research. Research always starts with a question, but one that takes into account what is already known about a topic or phenomenon. For example, if you are interested in whether people who have pets are happier than those who don’t, do a literature search and find out what has already been demonstrated. You will probably realize that yes, there is quite a bit of research that shows a relationship between happiness and owning a pet—and even studies that show that owning a dog is more beneficial than owning a cat ! Let’s say you are so intrigued by this finding that you wonder: 

What is it that makes dog owners even happier than cat owners? 

Let’s move on to Step 2 and find an answer to that question.

Writing a Hypothesis Step 2:

Formulate a strong hypothesis by answering your own question. Again, you don’t want to make things up, take unicorns into account, or repeat/ignore what has already been done. Looking at the dog-vs-cat papers your literature search returned, you see that most studies are based on self-report questionnaires on personality traits, mental health, and life satisfaction. What you don’t find is any data on actual (mental or physical) health measures, and no experiments. You therefore decide to make a bold claim come up with the carefully thought-through hypothesis that it’s maybe the lifestyle of the dog owners, which includes walking their dog several times per day, engaging in fun and healthy activities such as agility competitions, and taking them on trips, that gives them that extra boost in happiness. You could therefore answer your question in the following way:

Dog owners are happier than cat owners because of the dog-related activities they engage in.

Now you have to verify that your hypothesis fulfills the two requirements we introduced at the beginning of this resource article: falsifiability and testability . If it can’t be wrong and can’t be tested, it’s not a hypothesis. We are lucky, however, because yes, we can test whether owning a dog but not engaging in any of those activities leads to lower levels of happiness or well-being than owning a dog and playing and running around with them or taking them on trips.  

Writing a Hypothesis Step 3:

Make your predictions and define your variables. We have verified that we can test our hypothesis, but now we have to define all the relevant variables, design our experiment or data analysis, and make precise predictions. You could, for example, decide to study dog owners (not surprising at this point), let them fill in questionnaires about their lifestyle as well as their life satisfaction (as other studies did), and then compare two groups of active and inactive dog owners. Alternatively, if you want to go beyond the data that earlier studies produced and analyzed and directly manipulate the activity level of your dog owners to study the effect of that manipulation, you could invite them to your lab, select groups of participants with similar lifestyles, make them change their lifestyle (e.g., couch potato dog owners start agility classes, very active ones have to refrain from any fun activities for a certain period of time) and assess their happiness levels before and after the intervention. In both cases, your independent variable would be “ level of engagement in fun activities with dog” and your dependent variable would be happiness or well-being . 

Examples of a Good and Bad Hypothesis

Let’s look at a few examples of good and bad hypotheses to get you started.

Good Hypothesis Examples

Bad Hypothesis Examples

Tips for Writing a Research Hypothesis

If you understood the distinction between a hypothesis and a prediction we made at the beginning of this article, then you will have no problem formulating your hypotheses and predictions correctly. To refresh your memory: We have to (1) look at existing evidence, (2) come up with a hypothesis, (3) make a prediction, and (4) design an experiment. For example, you could summarize your dog/happiness study like this:

(1) While research suggests that dog owners are happier than cat owners, there are no reports on what factors drive this difference. (2) We hypothesized that it is the fun activities that many dog owners (but very few cat owners) engage in with their pets that increases their happiness levels. (3) We thus predicted that preventing very active dog owners from engaging in such activities for some time and making very inactive dog owners take up such activities would lead to an increase and decrease in their overall self-ratings of happiness, respectively. (4) To test this, we invited dog owners into our lab, assessed their mental and emotional well-being through questionnaires, and then assigned them to an “active” and an “inactive” group, depending on… 

Note that you use “we hypothesize” only for your hypothesis, not for your experimental prediction, and “would” or “if – then” only for your prediction, not your hypothesis. A hypothesis that states that something “would” affect something else sounds as if you don’t have enough confidence to make a clear statement—in which case you can’t expect your readers to believe in your research either. Write in the present tense, don’t use modal verbs that express varying degrees of certainty (such as may, might, or could ), and remember that you are not drawing a conclusion while trying not to exaggerate but making a clear statement that you then, in a way, try to disprove . And if that happens, that is not something to fear but an important part of the scientific process.

Similarly, don’t use “we hypothesize” when you explain the implications of your research or make predictions in the conclusion section of your manuscript, since these are clearly not hypotheses in the true sense of the word. As we said earlier, you will find that many authors of academic articles do not seem to care too much about these rather subtle distinctions, but thinking very clearly about your own research will not only help you write better but also ensure that even that infamous Reviewer 2 will find fewer reasons to nitpick about your manuscript. 

Perfect Your Manuscript With Professional Editing

Now that you know how to write a strong research hypothesis for your research paper, you might be interested in our free AI Proofreader , Wordvice AI, which finds and fixes errors in grammar, punctuation, and word choice in academic texts. Or if you are interested in human proofreading , check out our English editing services , including research paper editing and manuscript editing .

On the Wordvice academic resources website , you can also find many more articles and other resources that can help you with writing the other parts of your research paper , with making a research paper outline before you put everything together, or with writing an effective cover letter once you are ready to submit.

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• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1. ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

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• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

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A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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• B.A., Physics and Mathematics, Hastings College

A hypothesis is an educated guess about what you think will happen in a scientific experiment, based on your observations. Before conducting the experiment, you propose a hypothesis so that you can determine if your prediction is supported.

There are several ways you can state a hypothesis, but the best hypotheses are ones you can test and easily refute. Why would you want to disprove or discard your own hypothesis? Well, it is the easiest way to demonstrate that two factors are related. Here are some good scientific hypothesis examples:

• Hypothesis: All forks have three tines. This would be disproven if you find any fork with a different number of tines.
• Hypothesis: There is no relationship between smoking and lung cancer. While it is difficult to establish cause and effect in health issues, you can apply statistics to data to discredit or support this hypothesis.
• Hypothesis: Plants require liquid water to survive. This would be disproven if you find a plant that doesn't need it.
• Hypothesis: Cats do not show a paw preference (equivalent to being right- or left-handed). You could gather data around the number of times cats bat at a toy with either paw and analyze the data to determine whether cats, on the whole, favor one paw over the other. Be careful here, because individual cats, like people, might (or might not) express a preference. A large sample size would be helpful.
• Hypothesis: If plants are watered with a 10% detergent solution, their growth will be negatively affected. Some people prefer to state a hypothesis in an "If, then" format. An alternate hypothesis might be: Plant growth will be unaffected by water with a 10% detergent solution.
• What Are Examples of a Hypothesis?
• What Is a Hypothesis? (Science)
• What Is a Testable Hypothesis?
• Null Hypothesis Examples
• What Are the Elements of a Good Hypothesis?
• Scientific Method Flow Chart
• Six Steps of the Scientific Method
• Scientific Method Vocabulary Terms
• Understanding Simple vs Controlled Experiments
• Scientific Variable
• What Is an Experimental Constant?
• The Role of a Controlled Variable in an Experiment
• What Is the Difference Between a Control Variable and Control Group?
• Random Error vs. Systematic Error
• DRY MIX Experiment Variables Acronym
• What Is a Controlled Experiment?

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Is it "unscientific" to be sceptical without offering alternative explanations?

Alice has made some anecdotal observations. Through a process of elimination, she proposes a hypothesis to explain the phenomenon, as well as an experiment to validate (or otherwise) her hypothesis. The experiment has yet to be performed.

Bob largely agrees with Alice's deductions, however he advises that there may be other factors at play. When challenged on this by Alice, Bob mentions things that (unbeknownst to him) Alice has considered in her elimination process. Regardless, Bob still believes that it is reasonable be open-minded, even though he cannot immediately offer an alternative hypothesis.

Is Bob flouting the scientific method? I believe that playing the role of sceptic is both valuable and necessary -- please correct me if I'm wrong -- but does the fact that he cannot provide a tangible alternative relegate him to quackery and pseudoscience?

• philosophy-of-science
• scientific-method

• Comments have been moved to chat ; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments . Comments that do not request clarification or suggest improvements usually belong as an answer , on Philosophy Meta , or in Philosophy Chat . Comments continuing discussion may be removed. –  Philip Klöcking ♦ Commented Dec 8, 2023 at 22:52

19 Answers 19

Alice has made some anecdotal observations. Through a process of elimination, she proposes a hypothesis to explain the phenomenon, as well as an experiment to validate (or otherwise) her hypothesis .

You didn't mention if Alice was successful with her experiments. Assuming that she was, as long as her hypothesis remains unfalsified experiment after experiment, Alice's hypothesis would remain as the best working explanation, until Bob is able to come up with a different experiment that falsifies it. If Bob additionally believes there are other factors that Alice's hypothesis is probably overlooking, the burden of proof rests on him to elucidate these factors, construct a more robust hypothesis that is testable, and design a set of experiments that would corroborate his hypothesis's predictions while simultaneously falsifying Alice's.

Updated answer (taking comment section feedback into account)

Bob thinks there might be other factors at play. The question is which factors?

Your story claims the following:

Bob largely agrees with Alice's deductions, however he advises that there may be other factors at play. When challenged on this by Alice, Bob mentions things that (unbeknownst to him) Alice has considered in her elimination process .

Thus, we can split the factors into two groups:

• Group 1: Factors that are challenged by Alice's elimination process
• Group 2: Factors that are not challenged by Alice's elimination process

At the same time, we can think of different goals that Bob might have, each one entailing a different burden of proof:

Goal 1 : Bob wants to convince Alice that her elimination process is not (necessarily) sound, so factors she believes she has deductively ruled out are not necessarily ruled out.

In this case Bob would need to engage Alice's elimination process, and it would suffice for him to refute it at a purely logical level. He would need to expose logical fallacies in Alice's elimination process, and show how her elimination process fails to rule out certain factors she claims to be ruling out. Perhaps Alice's deductive arguments are logically invalid, or they are valid but rely on questionable premises that might turn out to be false. Now, if Alice's elimination process relies on other kinds of reasoning as well (e.g. inductive , abductive ), then those by their very nature are questionable too (they do not logically entail their conclusions, so there is always epistemic room for the possibility that they might be false).

Goal 2 : Bob wants to convince Alice that one or more factors in Group 1 must be considered in a revised version of Alice's hypothesis.

In this case Bob not only has to convince Alice that she is unjustified in ruling out one or more factors in Group 1 (i.e., Goal 1), but he also has to make a positive case for the need of including specific factors into a revised version of Alice's hypothesis. To make this case, I see two options for Bob:

Design an experiment where some of these factors play a crucial role, test Alice's hypothesis and show that it fails to make accurate predictions (i.e., falsify it).

Offer a better hypothesis that takes these factors into account, and design a set of experiments that confirm this hypothesis, while simultaneously falsifying Alice's.

Goal 3 : Bob wants to convince Alice that one or more factors in Group 2 must be considered in a revised version of Alice's hypothesis.

In this case, Bob would need to show Alice how her elimination process doesn't rule out these factors (it should be relatively straightforward since Alice's elimination process says nothing about them), and then he would need to proceed exactly as he would for Goal 2.

Goal 4 : Bob just wants to convince Alice that she cannot justify not being epistemically open to the possibility that perhaps (for all they know) factors in Group 2 might turn out relevant (eventually).

I think this is the most modest of all goals, and it would be sufficient for Bob to simply explain to Alice that she might be committing the Holmesian fallacy , as Mark Foskey's answer correctly points out.

It all depends on the epistemic goals he sets himself to accomplish, and whether he manages to meet the corresponding burden of proof. If Bob just wants to defend the claim that Alice is unjustified in her rejection of certain factors, he just needs to show how Alice fails to have justification for this rejection, which can be done as modestly as simply appealing to Alice's lack of omniscience. If, on the other hand, Bob wants to go one step further and defend the claim that some factors need to be considered, he cannot be as modest: he would need to design experiments that verify that these factors are clearly playing a role in the outcomes, falsifying Alice's hypothesis in the process. In this case it would help a lot (in terms of progress of the scientific knowledge) if Bob also comes up with a better hypothesis of his own, that can account for the role these factors seem to be playing in the outcomes of the experiments.

• 9 I don't think we should require that Bob constructs a new hypothesis. It is entirely above board to falsify a hypothesis (presumably through experiment) without having an alternative ready. –  Arthur Commented Dec 6, 2023 at 10:51
• 3 @Mark I think it's enough for him to say "I think you forgot to take X into account, here is an experiment I designed where I think X greatly affects the result". The hypothesis "X affects the result noticeably compared to Alice's hypothesis" is enough in my mind, he shouldn't have to hypothesize anything about how X affects the result in order to invalidate Alice's assumption that it doesn't. –  Arthur Commented Dec 6, 2023 at 11:25
• 4 I don't even think, in every situation like this, Bob has to construct a better hypothesis than Alice. I mean, suppose Alice's hypothesis is that the reason people lose their socks all the time is that tiny little gnomes are running around peoples' houses. Does Bob really have a "burden of proof" here? Does he have to have an explicitly better explanation for the process of losing a sock than tiny little gnomes, in order for his rejection of her idea to be considered scientifically allowable? I don't think so. I think he can just say "That's a preposterous idea" and leave it at that. –  TKoL Commented Dec 6, 2023 at 16:42
• 4 I almost agree with you, @TKoL. I agree that Bob doesn't need to provide an alternative explanation to reject Alice's hypothesis, and I certainly don't think he needs to come up with an experiment (theoretical physicist here ^_^) showing Alice's hypothesis is insufficient, but in order for me to consider his rejection scientific, he would need to offer some reasoning (even if purely theoretical) as to why Alice's hypothesis should be considered preposterous. –  jecado Commented Dec 6, 2023 at 16:59
• 2 @TKoL Agreed and agreed! I think the only part that makes me uncomfortable is that skeptics often seem to think they are being scientific when they dismiss or ignore "bad" ideas, and I think that is bad. –  jecado Commented Dec 6, 2023 at 17:55

Definitely not. To say that it is unscientific is to fall into what is sometimes called the Sherlock Holmes fallacy. Alice seems to be saying that her explanation must be right because she has ruled out all other possible explanations. But she and Bob both might have missed some.

After all, Alice does still plan to perform an experiment to validate her theory, so it seems like she herself is still a little skeptical. Why can't Bob be as well?

• Planning an experiment to test a theory is not “skepticism” about a theory. It is a required and critical part of the scientific method. –  Jagerber48 Commented Dec 7, 2023 at 15:03
• 2 I think you are interpreting the word "skepticism" a little differently from me. My point is that Bob is merely enunciating why you always want to test a theory. He is not dismissing Alice's theory out of hand. –  Mark Foskey Commented Dec 7, 2023 at 16:59
• yeah, upon re-reading the question I see that Bob was not as dismissive of Alice's theory as I thought on my original reading. I guess, on my reading, Bob is being slightly pigheaded in that he has is arguing against Alice's theory but isn't raising any real objections other than the generic objection "there might be other factors" that can be levied against ANY theory. –  Jagerber48 Commented Dec 7, 2023 at 23:37
• "Alice seems to be saying that her explanation must be right..." This is not evidenced in the question. –  Jagerber48 Commented Dec 8, 2023 at 9:54

she proposes a hypothesis to explain the phenomenon, as well as an experiment to validate (or otherwise) her hypothesis... Bob still believes that it is reasonable be open-minded, even though he cannot immediately offer an alternative hypothesis. Is Bob flaunting the scientific method?

It depends on the example.

A patient's signs and symptoms match every standard indicator of a common and well-understood infection, except for double-checking the microbes' presence in blood, which is what Alice wants to do next. But Bob's sceptical just as an epistemological purist. After all, what if a different microbe that does the same things hasn't been discovered yet? (If he had a rare known infection in mind, that's something different .)

Take this with a grain of salt as I'm not a medical doctor, but I think Bob is being silly.

Alice wonders why we (i) see no signs of alien life and (ii) are very early in the trillions of years when stars shine, which you might not expect of a random civilization. She likes the grabby aliens model, according to which interstellar civilizations seize resources that would have otherwise evolved into other civilizations, so they only form for a few billion years after civilizations first emerge. The model is compatible after fitting some parameters with very rare civilizations that control much of the Universe, but nothing within (say) a billion light years of us. This predicts our descendants, if extant long enough, will meet aliens in 200 million to 2 billion years, so that's technically testable. (If you're interested in such data fitting, see here .)

Bob, worried about positing unseen aliens to explain something (especially when that includes their being unseen ), thinks alien civilizations might just not be feasible around the longest-lived stars, but he can't articulate why exactly. Sure, M-type stars' habitability has been critiqued , but ideally he'd need K-type stars not to work either.

Partial as I am to the grabby aliens idea, I think Bob is being sensible.

Let's suppose the answer to your question was yes, it is unscientific to doubt theories without alternative explanations. Bob would have to say 'No need to perform your experiment, Alice; I can't think of any alternative hypothesis, so there is no reason to doubt yours'.

I would say that Bob is not "flouting" the scientific method, but he's not yet doing very good science. Alice is doing better science than him so far.

I would say that Bob pointing out "there may be other factors at play" is a correct statement, but it's more in the realm of philosophy of science than actual science. That is, "there may be other factors at play" is very related to the Quine-Duhem thesis which is an interesting bit of philosophy of science, but a bit boring from the perspective of ACTUAL science unless you are making theoretical suggestions or experimental investigations about what those factors may be.

Alice on the other hand is doing good science. She has proposed a hypothesis AND an experiment to test that hypothesis. This is the best a theorist can do, the next steps are to run the experiment.

So I would say that Alice is being a good scientist. Bob is not necessarily being a bad scientist but he's not really being a good one. Depending on Bob's tone when he is expressing his skepticism he may be being a good or bad colleague (but that is not what the question is about).

Above is my general reaction. However, there is a degree to which the answer to this question becomes a matter of scale rather than kind. If Alice has a pretty good theory and she has considered and refuted all of Bob's doubts and Bob still continues to parrot "there may be other factors at play" without giving a competing theory then he could he easily be venturing into pseudoscience. Again, from a strict philosophy of science point of view, he's not technically wrong, but from a "doing good science" point of view he's not being helpful, and, in fact, he's being counter-productive by wasting the time of a good scientist after she's already addressed all of his points.

I think this very last point is very important when it comes to the relationship between science and pseudoscientists (in fact I wonder if this question is really about the demarcation problem between science and pseudoscience). Both scientists and psuedoscientists are skeptical. So what is the difference? It can be a tough question. But Lakatos talks about comparing theories based on their simplicity (Occam's Razor), explanatory power and experimental corroboration. I think the difference between the scientist and the pseudoscientist is that the scientist follows Lakatos's suggestions about how to evaluate theories. Instead, I think the pseudoscientists is MORE motivated by knock-on ramifications of a particular theory.

For example,

• The young earth creationist might reject the theory of evolution because it poses a threat to their religious views. They are doing pseudoscience if they value the preservation of their religious views to the point that they neglect the simplicity, explanatory power, or experimental corroboration of evolution.
• The flat earther might reject globe earth theory because they are generally contrarian and mistrustful of authority. They also stand to "lose a lot of face" if they admit they were wrong. Again, they are doing pseudoscience if they value these things over the simplicity, explanatory power, or experimental corroboration of globe earth theory.
• Perhaps there is a financial stake in one theory over another. If a scientist is swayed to promote one theory because of the financial stake at the expense of evaluating the theory based on its simplicity, explanatory power, and experimental corroboration, then they are doing pseudoscience.

Not only is it not unscientific, it's not uncommon either. Scientists often offer up alternative explanations for things, but they didn't arrive at those conclusions instantly. Einstein's general relativity proposed a new explanation for phenomena that had been covered by Newtonian mechanics for over a century. He knew something wasn't right, but it took many years of research before he could develop something cohesive enough to publish as an alternative explanation. That work was bona fide scientific research. Some scientists spend their entire careers doing research into something that will only be distilled into a complete explanation by a future generation.

The scientific process is about a method for gathering and evaluating knowledge. Bob's skepticism isn't flouting the scientific method at all, he's actually working through the first stages of it. The first step in the scientific method is to determine the question that you want to answer. In Bob's case, the question he's asking is "is Alice's theory correct?". He suspects the answer is "no". Like any other scientific endeavor, the next step would be to refine the question and design an experiment to test his hypothesis, or to find a logical flaw that invalidates Alice's reasoning.

• Your answer could be improved with additional supporting information. Please edit to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center . –  Community Bot Commented Dec 6, 2023 at 8:16

We should always be skeptical of scientific hypotheses and results. The scientific method never proves that a hypothesis is absolutely true, it can only disprove things (when results are inconsistent with the hypothesis and we can't find a flaw in the process). Throughout history many accepted theories have been disproven (the aether model of space) or modified (the theories of relativity's refinements to Newton's laws of motion and gravity).

There's no way to know when there's something missing in a theory. Even if experiments support the hypothesis, there are potentially unknown factors that may require changes to the model. Since these are unknown, it would be unreasonable to expect the skeptic to enumerate the problems.

Of course, this doesn't mean we shouldn't proceed with the experiment and accept the results. Newton's laws are close enough to serve us well in almost all circumstances -- relativity only comes into play at extreme speeds or gravity, or when extreme precision is needed (e.g. GPS makes adjustments due to the difference in gravity between the surface of the Earth and the satellite's orbit). Theories are always provisional -- if we confirm the hypothesis adequately, we can treat it as a law until a future experiment forces us to change the model. As with Newton vs. Einstein, these will mostly be minor tweaks, but occasionally we do get paradigm changes (e.g. geocentric to heliocentric models of the Solar System, the germ theory of disease).

Is the method reasonable?

You mentioned Alice using some method to arrive at her explanation. Whether this method is reasonable is distinct from whether any other explanation exists, and one can question that.

You mentioned the process of elimination, which is generally reasonable, but it's not perfect, and any reliability it may have depends heavily on which explanations you start your process of elimination with, and how you decide on which explanations to eliminate. If your initial set of explanations excludes better or comparable explanations, that'll be an issue. It also doesn't account for observations that we simply don't have enough information to explain - sometimes we just don't know. As for experiments, those are good, but we should also consider whether the results of the experiment (if successful) would actually support the given explanation.

Is the explanation reasonable?

An explanation could also contradict itself, or be inconsistent with what we know about reality, or it may involve claims you can arbitrarily swap out with completely different claims and it would still have the same explanatory power and evidentiary support (which would mean we have no reason to favour that explanation above any other). You can point that out without offering an alternative explanation.

To explain why a door slammed shut, I could say an alien named Steve on the planet Stevington on the other side of the galaxy fired a beam of gamma rays directly at it. One could reject that explanation given that there's no clear evidence of the existence of such an alien, that even if there are aliens elsewhere, that most likely isn't that specific one, and they most likely don't know about our existence, never mind having the level of knowledge and ability to influence what happens on Earth down to that much detail. Those would all be reasons to reject the explanation, and if you don't have an alternative explanation, one might just be left with "I don't know".

So no, you do not need to offer an alternative explanation to question or reject one explanation.

One might also posit that you'd likely only find alternative explanations if you're skeptical of some given explanation, which is part of why skeptics and scientists advocate for (responsibly and consistently) questioning explanations and beliefs.

Although one could also take the other approach of considering which explanation best explains the data.

The go-to case to keep in mind for questions like this is Ignaz Semmelweis. You may have heard of him. Back in the day, he achieved great success at the clinic where he worked in eliminating the dreaded "childbed fever" that had been killing new mothers for a long time, by requiring all surgeons to wash their hands with a harsh chemical after performing autopsies and before delivering babies.

He theorized that there was some "cadaverous particle" in dead bodies that caused the deadly fever, and that the cleaning chemical, which was known to remove the stench of death and decay, broke down cadaverous particles.

Despite his resounding success in practice, Semmelweis had an astonishing amount of difficulty convincing doctors in general to adopt his handwashing protocol. They took it personally, and felt offended by the accusation that they were responsible for patient deaths by being physically unclean in some way, and because Semmelweis could not produce any proper hypothesis for what a "cadaverous particle" actually was or how it worked, they scoffed and called the theory unscientific and the man mad.

Not long after Semmelweis' death, Louis Pasteur's theory of germs provided the scientific underpinning needed to make sense of this infection and why a strong disinfecting agent would prevent transmission. Semmelweis was right all along, but doctors used "that's unscientific" as an excuse to not take his theory seriously, and ended up with the blood of far too many women on their hands.

Beware, beware, beware conflating "scientific" with "valid"!

• 3 Fantastic historical example. Although, in the end, the only "unscientific" thing that happened was the rejection of hand-washing despite clear statistical evidence that hand-washing worked. At the end of the day, if you have statistical proof that a certain technique works, then even if you don't have an explanation for WHY the technique works, the effectiveness of the technique itself is still "scientific". –  TKoL Commented Dec 7, 2023 at 9:18

First, just a reminder that hypotheses stand to be disproved, not proved. Karl Popper explained this. The phrase in brackets, unbeknownst to him, causes a problem with your question. We cannot know what we do not know. Simply listening to a hypothesis and stating that other factors may be at play is valid. It happens all the time! It takes into account anything that we do not know.

It is my opinion that this is the way science works a lot of the time. An observation is made of some phenomena. Lots of scientists work on this until eventually one comes up with a theory for the phenomena. They then create a mathematical model that describes it. They use this model to predict some future observation. This observation is duly made and within the error bars of the current technology the theory appears to be correct. The theory is declared to be the truth and everybody else gives up looking at the problem. Eventually maybe hundreds of years later more powerful technology shows that the models predictions lie outside the updated error bars and everybody starts to panic because so much science was based on the theory being the truth.

Not necessarily unscientific.

If I tell you that my anti-gravity device can save humans in 100mph concrete wall crash, and I have designed an experiment to try with a volunteer.. to test this hypothesis, then it is unscientific if you happily agree with me.

Now this example is extreme. I think it's alright if one finds a hypothesis fishy or ridiculous. In the presence of solid proof though, that would be different.

On the other hand "solid proof" is sometimes not easily defined.

At its core, the scientific method boils down to:

• Make an observation.
• Come up with a theory that can explain the observation.
• Perform experiments whose results can disprove the theory.

Eventually, after multiple experiments fail to disprove the theory, you accept it as the best approximation of the truth you have come up with so far.

Now, in your example, the experiments have not been carried out. Therefore, accepting the theory is "unscientific". Bob is absolutely correct and would have been very wrong to accept a theory that has not been tested just because it sounds reasonable or he hasn't come up with a better one. Not only is he not flouting the scientific method, he, unlike Alice, is following it. Alice is indeed flouting it since she wants to accept her hypothesis before testing it.

As a working scientist, the first thing I will do if someone brings me a neat new theory is try to poke holes in it. The better the theory, the more elegant, the more true and correct it feels, the more important it is to not accept it until I have tried to disprove it as hard as I possibly can. That is the very basis of the scientific method!

So no, your Bob is the one who is following the scientific method and it is your Alice who is flouting it. Accepting an untested theory is quackery and pseudoscience. Refusing to accept it is correct, and whether one can come up with a better alternative or not is irrelevant. A single experiment can be enough to disprove a theory, and if that happens, the theory is abandoned irrespective of whether another one exists.

Or, to think of it another way, the logical conclusion of your suggestion would be that we must continue to accept clearly disproven theories just because we haven't come up with an alternative. That isn't how science works. If a theory is disproven, then we stop using it; we don't continue to treat disproven theories as a decent approximation of the truth just because we have no better theories. If we have no better theories, then we simply do not have a theory to explain the observed phenomenon and go back to the drawing board and try and find one.

• In the question Alice does not "want to accept her hypothesis before testing it"... –  Jagerber48 Commented Dec 7, 2023 at 23:35
• @Jagerber48 "The experiment has yet to be performed." –  terdon Commented Dec 8, 2023 at 9:03
• nowhere in the question does it say that Alice "accepts" her hypothesis or thinks Bob should "accept" her hypothesis. It says she has proposed a hypothesis and an experimental test for it. She's doing a perfect job as a scientist so far so it's strange that you're criticizing her. –  Jagerber48 Commented Dec 8, 2023 at 9:18
• @Jagerber48 "Alice has made anecdotal observations [...] proposes an experiment [which has] yet to be performed". So the premise here is that Alice has come up with a clever theory but has not yet tested it. Bob isn't accepting the untested theory. He cannot offer another theory, but isn't willing to accept one without testing it either. You're right that it isn't explicitly stated that Alice wants to accept the theory, but I don't see how else we can read this. The OP seems to suggest that in the absence of an alternative, one must accept a theory even when untested which is just silly. –  terdon Commented Dec 8, 2023 at 9:23
• "...I don't see how else we can read this." I think it can be read without assuming Alice accepts or wants Bob to accept her theory. On this reading parts of your answer are strange. In this case I read it as: Alice has done good work but Bob is still skeptical. Is Bob being unscientific? The answer to that, for me, depends on how stubborn Bob is being in his skepticism. I think there are healthy and unhealthy levels of skepticism and there's not really enough info in the question to determine if Bob's skepticism is healthy or unhealthy (I think it can be read both ways). –  Jagerber48 Commented Dec 8, 2023 at 9:27

A lot of the answers seem to be missing the bigger picture.

The objective of science is to extend our understanding of the nature of reality. And to that end we try to weave our observations of reality into a narrative that explains them. What we thus end up with is some sort of toy model or simulation of a (usually) simplified universe. Which has the neat advantage that if we can describe the universe with such a hypothesis, this also allows us to make predictions for how it should behave in unexplored cases if the theory were to be true. Which we can then test.

That being said we almost certainly know that the theory is wrong, because usually it's a massive simplification and based on observations that already came with a margin of error, rather than being precise.

And so the scientific process is a continuous loop of observing, explaining and testing. Where each test is the opportunity to either extend the model to more applications or to have it break and find new properties of the universe that you haven't yet accounted for.

So distrusting the theory is the default and you're almost bound to be correct with that. So BOB IS NOT FLOUTING SCIENCE when he argues to stay open minded. If you're engaged in pure science that is precisely what you ought to do. Never trust the theory, always trying to poke holes in it even if it is all to human to like your own ideas if they are beautiful, elegant, powerful, ...

The problem is: Not all science is pure. As soon as you enter the domain of "science adjacent"-disciplines idk you either drop the uncertainty or you drop reality. Like idk math and logic often times drop reality and get lost in their own models, where they accept a few axioms and then build an entire universe upon the assertions "Assuming XYZ is true" (regardless of the question whether XYZ is ACTUALLY true).

While on the other hand the more applied sciences like idk medicine, engineering, etc, are usually not interested in the purity of models explaining how the world works and much more interested in whether they are good enough approximations so that they "reliably" work in reality. So the explanation might as well be bullshit... as long as it works...

Which brings us to the crucial question that is likely hiding behind yourquestion, namely: "What should we do with scientific theories".

And to the apparent surprise of many people that is not a scientific question but a political one.

Science is only interested in broadening the understanding of the universe and it likely will remain a never ending work in progress, so it cannot provide and doesn't claim to provide eternal truths.

So following the advice of a scientific theory is a gamble upon a(n educated) guess. And ultimately gambling is a decision (or if it involves more than 1 person a collective decision aka political). So science might tell you it's likely a 90:10 coin flip, but whether you test your luck and go for the 10% probability or whether you go for the 90% is a decision that you need to make.

For example if your 90% option is certain death, you might speculate on whatever the 10% option is. While in a different scenario you'd plan with the more likely outcome. All with the asterisk that even the 90:10 probability is a guess.

Though "science" as a whole usually is the collective of all the data and provides you with models trying to explain the data, so you have both some empirical hints as to what is to be expected and you've models that allow you to plan for certain scenarios (under the assumption that these models hold water), which is usually much better than blind guesses or basing your decision making on way fewer data and more unreasonable explanations. But the outcome of the experiment is decided by reality not by science and it's science that needs to adapt not reality.

So it's not unscientific to be skeptical, on the contrary. Though usually science expects you to engage with the data or to contribute your own. The problem being that pure skepticism and epistemic nihilism are pretty unproductive. Like if you reject the very ability to acquire any knowledge and reject science for being likely incorrect than you end up with a total mist and no direction and poking in that mist is usually way more likely to be incorrect then to rely on all the observations (and ideas how to put these pieces of information together) that we've acquired so far.

So at some point you're likely forced to make a decision an usually an educated guess is still better than a blind guess.

Also last thing worth noting. The reliability of science is also to some extend a matter of time. Like Alice is one person making experiments on a novel subject, so chances are she's not seeing the whole picture but just a fragment of it. So with time more people might find her work interesting and research in that direction revealing more parts of the picture which might extend Alice's work or put it into a different context. Maybe it was all just an outlier, maybe it was a textbook example of something, aso. So the reliability often relies on the data that is being gathered so with respect to novel findings there's usually a lot of hype, but usually the actual scientists try to moderate and argue what it could be not what it is, because it's still very likely that they are "wrong" or perhaps it's better expressed as incomplete .

Like with morality, the key to this is the motivation for the doubt. Bob might remain unconvinced of pregnancy being caused by sexuality because Bob prefers magical thinking. Bob might remain unconvinced that earth is spherical rather than flat because Bob likes to be anti-establishment, and Anti-Main-Stream.

It is unscientific to remain unconvinced of some things but not others based on speficic biases and personal preferences. A scientist would seek a consistent approach of accepting a best explaining theory as working model of what to believe, and not cherry pick convictions to stick with feel-good beliefs and reject inconvenient truths.

So the example in the question lacks the most important aspect to decide, the motivation of Bob and their standard in life for selecting what to believe.

I think we can say no, nothing about Bob's attitude "relegate him to quackery and pseudoscience" and we can use a historical example to demonstrate this.

Until Einstein proposed the theory of special relativity, the main theory for how light was transmitted was the existence of the Luminiferous aether . There were no other generally accepted hypotheses. The problem was no one could demonstrate its existence (because it was an incorrect theory.) Now imagine Bob is someone during that time expressing doubts about the aether's existence and Alice's hypothesis is based on its existence. He's not Einstein, though, and doesn't know the correct answer, he just doesn't believe the in the Aether despite knowing all the arguments for it. Would you say Bob was a quack or psuedo-scientist knowing what we know now? If you think that he was, then what you are ultimately claiming is that believing incorrect ideas is an essential part of science. I find that notion to be preposterous.

There are harms related to this kind of attitude. Similar to Mason Wheeler's example, many people (mostly women) who are having medical issues are regularly diagnosed by doctors with 'anxiety' and sent away. No investigation into what else might be a factor. Often, later, it is found that these patients were sick with cancer or some condition that was not well understood. Many suffer and/or die unnecessarily due to this. This pattern is essentially caused by what you are describing. One hypothesis is deemed acceptable and doubting it is considered quackery or psuedo-science.

That's not science. That's people abusing their status as scientists (or doctors) to assert their beliefs in lieu of actual science. The idea that we must accept the only known hypothesis boils down to an argument from ignorance. Similar to a false dilemma, the possibility of an unknown alternatives is ignored, and one option is falsely presented as the only possible choice.

First let us agree that science is a social endeavor- the dialogue and collaboration are key aspects of it. The pseudo-problem here arises since Alice and Bob never properly discussed the matter. Bob is trying to investigate Alice's hypothesis because he believes she must have missed something- i.e., it is his personal opinion that there must be something else at play. Alice discusses things with Bob but does not clarify she herself tested the hypothesis against the same alternate explanations Bob had- so far , they are both applying the scientific method but are at different steps in its classic three part course- Alice is at the hypothesis validation phase, preparing the experiment (but has done a few observations that seem to confirm her hypothesis) while Bob only knows about her initial thesis and he disagrees.

So far the only problem is lack of proper scientific communication.

The scientific method as many of you above said, is based on observation,then formulating a simplified model of reality that aims to explain the observation, necessarily followed by its validation- repeated application on available data, and various experiments. I do agree that all of our models are inherently flawed but not necessarily wrong- approximations and simplifications exist, but sometimes they are made to ease the burden on the scientists and avoiding unneeded complexity. One does not design a bridge by solving quantum mechanics problems- but indeed such phenomena might ultimately explain properties of some materials and will play a part in material choice for example. So Alice has a theory and wants to prove it by experiment . Great! Third step in the scientific method. She trusts her theory, 'because she applied a process of elimination' - indeed here there are possible things that she might have missed and Bob can make the remark in good faith. So far nothing wrong with either. The logical step is for Bob to review the experimental results and further ponder if he can design an experiment that can invalidate Alice's results. If not, then he must keep working- nothing wrong here, and any individual opinion is just that. Bob would make a mistake if he were to simply state Alice is wrong because she might have missed something without mentioning what and how. But as long as he states his opinion - it's just that , an opinion and outside proper scientific method. Of course she MIGHT be wrong, or she MIGHT have missed something (else entirely, even if not his initial guessed factors) - but until there is evidence that she did, or the experiment is not done to invalidate her theory we have no facts.Bob is now in the first stage of the scientific method. His theory is that Alice is wrong. The burden of proof is on him. But as long as he continues to work and arrives at experimental validation phase and then presents his findings - he is not wrong, he just has an opinion. That being said- opinions are outside the scientific method. He cannot present his opinions AS FACTS. If Bob thinks he is right and Alice is wrong, then he should prove that and then the theory proves true or false. The potentiality that one MIGHT be right or wrong is not science-fact it is an opinion and a possibility. Quite valid if Alice's theory is high stakes or untested or in a completely new field, i.e.fundamental research.But still not fact.In science we can believe things just like anyone else does- but as researchers it is our duty to try to explain why we believe what we do and make experiments to prove (or disprove) our opinions. Then and only then we can state them as facts...either true ,verified, or false, verified again. Anyhow,there was this famous quote that said, to paraphrase, intelligent people are happy when they discover the truth while the fools are happy when they discover the false. Scientific method is the way of intelligent people. Negative results are of course good too- but only to tell us to change the way we considered modelling a certain thing.

• Your answer could be improved with additional supporting information. Please edit to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center . –  Community Bot Commented Dec 8, 2023 at 17:36

No. That is not a logical need or obligation.

However. Providing the right answer, is always better than skepticism alone.

I strongly disagree with the first answer that says that the burden of proof rests on Bob.

My view is that "science" is just a name for the rules of the game called prover-skeptic: the prover tries to prove something and the skeptic emits objections. In this case, if Bob finds himself in the position of not finding any other objections, he has therefore lost the game, and Alice has won. Whether Bob is happy or not with the result of the game has nothing to do with being unscientific. If, in the future, he finds a good objection, he can ask Alice to play again, and Alice may lose.

If I lose against Rafael Nadal at tennis, and upon leaving the court I keep thinking "I may have won", am I being "untennistic"?

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Scientific Hypotheses: Writing, Promoting, and Predicting Implications

Armen yuri gasparyan.

1 Departments of Rheumatology and Research and Development, Dudley Group NHS Foundation Trust (Teaching Trust of the University of Birmingham, UK), Russells Hall Hospital, Dudley, West Midlands, UK.

Lilit Ayvazyan

2 Department of Medical Chemistry, Yerevan State Medical University, Yerevan, Armenia.

Ulzhan Mukanova

3 Department of Surgical Disciplines, South Kazakhstan Medical Academy, Shymkent, Kazakhstan.

Marlen Yessirkepov

4 Department of Biology and Biochemistry, South Kazakhstan Medical Academy, Shymkent, Kazakhstan.

George D. Kitas

5 Arthritis Research UK Epidemiology Unit, University of Manchester, Manchester, UK.

Scientific hypotheses are essential for progress in rapidly developing academic disciplines. Proposing new ideas and hypotheses require thorough analyses of evidence-based data and predictions of the implications. One of the main concerns relates to the ethical implications of the generated hypotheses. The authors may need to outline potential benefits and limitations of their suggestions and target widely visible publication outlets to ignite discussion by experts and start testing the hypotheses. Not many publication outlets are currently welcoming hypotheses and unconventional ideas that may open gates to criticism and conservative remarks. A few scholarly journals guide the authors on how to structure hypotheses. Reflecting on general and specific issues around the subject matter is often recommended for drafting a well-structured hypothesis article. An analysis of influential hypotheses, presented in this article, particularly Strachan's hygiene hypothesis with global implications in the field of immunology and allergy, points to the need for properly interpreting and testing new suggestions. Envisaging the ethical implications of the hypotheses should be considered both by authors and journal editors during the writing and publishing process.

INTRODUCTION

We live in times of digitization that radically changes scientific research, reporting, and publishing strategies. Researchers all over the world are overwhelmed with processing large volumes of information and searching through numerous online platforms, all of which make the whole process of scholarly analysis and synthesis complex and sophisticated.

Current research activities are diversifying to combine scientific observations with analysis of facts recorded by scholars from various professional backgrounds. 1 Citation analyses and networking on social media are also becoming essential for shaping research and publishing strategies globally. 2 Learning specifics of increasingly interdisciplinary research studies and acquiring information facilitation skills aid researchers in formulating innovative ideas and predicting developments in interrelated scientific fields.

Arguably, researchers are currently offered more opportunities than in the past for generating new ideas by performing their routine laboratory activities, observing individual cases and unusual developments, and critically analyzing published scientific facts. What they need at the start of their research is to formulate a scientific hypothesis that revisits conventional theories, real-world processes, and related evidence to propose new studies and test ideas in an ethical way. 3 Such a hypothesis can be of most benefit if published in an ethical journal with wide visibility and exposure to relevant online databases and promotion platforms.

Although hypotheses are crucially important for the scientific progress, only few highly skilled researchers formulate and eventually publish their innovative ideas per se . Understandably, in an increasingly competitive research environment, most authors would prefer to prioritize their ideas by discussing and conducting tests in their own laboratories or clinical departments, and publishing research reports afterwards. However, there are instances when simple observations and research studies in a single center are not capable of explaining and testing new groundbreaking ideas. Formulating hypothesis articles first and calling for multicenter and interdisciplinary research can be a solution in such instances, potentially launching influential scientific directions, if not academic disciplines.

The aim of this article is to overview the importance and implications of infrequently published scientific hypotheses that may open new avenues of thinking and research.

Despite the seemingly established views on innovative ideas and hypotheses as essential research tools, no structured definition exists to tag the term and systematically track related articles. In 1973, the Medical Subject Heading (MeSH) of the U.S. National Library of Medicine introduced “Research Design” as a structured keyword that referred to the importance of collecting data and properly testing hypotheses, and indirectly linked the term to ethics, methods and standards, among many other subheadings.

One of the experts in the field defines “hypothesis” as a well-argued analysis of available evidence to provide a realistic (scientific) explanation of existing facts, fill gaps in public understanding of sophisticated processes, and propose a new theory or a test. 4 A hypothesis can be proven wrong partially or entirely. However, even such an erroneous hypothesis may influence progress in science by initiating professional debates that help generate more realistic ideas. The main ethical requirement for hypothesis authors is to be honest about the limitations of their suggestions. 5

EXAMPLES OF INFLUENTIAL SCIENTIFIC HYPOTHESES

Daily routine in a research laboratory may lead to groundbreaking discoveries provided the daily accounts are comprehensively analyzed and reproduced by peers. The discovery of penicillin by Sir Alexander Fleming (1928) can be viewed as a prime example of such discoveries that introduced therapies to treat staphylococcal and streptococcal infections and modulate blood coagulation. 6 , 7 Penicillin got worldwide recognition due to the inventor's seminal works published by highly prestigious and widely visible British journals, effective ‘real-world’ antibiotic therapy of pneumonia and wounds during World War II, and euphoric media coverage. 8 In 1945, Fleming, Florey and Chain got a much deserved Nobel Prize in Physiology or Medicine for the discovery that led to the mass production of the wonder drug in the U.S. and ‘real-world practice’ that tested the use of penicillin. What remained globally unnoticed is that Zinaida Yermolyeva, the outstanding Soviet microbiologist, created the Soviet penicillin, which turned out to be more effective than the Anglo-American penicillin and entered mass production in 1943; that year marked the turning of the tide of the Great Patriotic War. 9 One of the reasons of the widely unnoticed discovery of Zinaida Yermolyeva is that her works were published exclusively by local Russian (Soviet) journals.

The past decades have been marked by an unprecedented growth of multicenter and global research studies involving hundreds and thousands of human subjects. This trend is shaped by an increasing number of reports on clinical trials and large cohort studies that create a strong evidence base for practice recommendations. Mega-studies may help generate and test large-scale hypotheses aiming to solve health issues globally. Properly designed epidemiological studies, for example, may introduce clarity to the hygiene hypothesis that was originally proposed by David Strachan in 1989. 10 David Strachan studied the epidemiology of hay fever in a cohort of 17,414 British children and concluded that declining family size and improved personal hygiene had reduced the chances of cross infections in families, resulting in epidemics of atopic disease in post-industrial Britain. Over the past four decades, several related hypotheses have been proposed to expand the potential role of symbiotic microorganisms and parasites in the development of human physiological immune responses early in life and protection from allergic and autoimmune diseases later on. 11 , 12 Given the popularity and the scientific importance of the hygiene hypothesis, it was introduced as a MeSH term in 2012. 13

Hypotheses can be proposed based on an analysis of recorded historic events that resulted in mass migrations and spreading of certain genetic diseases. As a prime example, familial Mediterranean fever (FMF), the prototype periodic fever syndrome, is believed to spread from Mesopotamia to the Mediterranean region and all over Europe due to migrations and religious prosecutions millennia ago. 14 Genetic mutations spearing mild clinical forms of FMF are hypothesized to emerge and persist in the Mediterranean region as protective factors against more serious infectious diseases, particularly tuberculosis, historically common in that part of the world. 15 The speculations over the advantages of carrying the MEditerranean FeVer (MEFV) gene are further strengthened by recorded low mortality rates from tuberculosis among FMF patients of different nationalities living in Tunisia in the first half of the 20th century. 16

Diagnostic hypotheses shedding light on peculiarities of diseases throughout the history of mankind can be formulated using artefacts, particularly historic paintings. 17 Such paintings may reveal joint deformities and disfigurements due to rheumatic diseases in individual subjects. A series of paintings with similar signs of pathological conditions interpreted in a historic context may uncover mysteries of epidemics of certain diseases, which is the case with Ruben's paintings depicting signs of rheumatic hands and making some doctors to believe that rheumatoid arthritis was common in Europe in the 16th and 17th century. 18

WRITING SCIENTIFIC HYPOTHESES

There are author instructions of a few journals that specifically guide how to structure, format, and make submissions categorized as hypotheses attractive. One of the examples is presented by Med Hypotheses , the flagship journal in its field with more than four decades of publishing and influencing hypothesis authors globally. However, such guidance is not based on widely discussed, implemented, and approved reporting standards, which are becoming mandatory for all scholarly journals.

Generating new ideas and scientific hypotheses is a sophisticated task since not all researchers and authors are skilled to plan, conduct, and interpret various research studies. Some experience with formulating focused research questions and strong working hypotheses of original research studies is definitely helpful for advancing critical appraisal skills. However, aspiring authors of scientific hypotheses may need something different, which is more related to discerning scientific facts, pooling homogenous data from primary research works, and synthesizing new information in a systematic way by analyzing similar sets of articles. To some extent, this activity is reminiscent of writing narrative and systematic reviews. As in the case of reviews, scientific hypotheses need to be formulated on the basis of comprehensive search strategies to retrieve all available studies on the topics of interest and then synthesize new information selectively referring to the most relevant items. One of the main differences between scientific hypothesis and review articles relates to the volume of supportive literature sources ( Table 1 ). In fact, hypothesis is usually formulated by referring to a few scientific facts or compelling evidence derived from a handful of literature sources. 19 By contrast, reviews require analyses of a large number of published documents retrieved from several well-organized and evidence-based databases in accordance with predefined search strategies. 20 , 21 , 22

The format of hypotheses, especially the implications part, may vary widely across disciplines. Clinicians may limit their suggestions to the clinical manifestations of diseases, outcomes, and management strategies. Basic and laboratory scientists analysing genetic, molecular, and biochemical mechanisms may need to view beyond the frames of their narrow fields and predict social and population-based implications of the proposed ideas. 23

Advanced writing skills are essential for presenting an interesting theoretical article which appeals to the global readership. Merely listing opposing facts and ideas, without proper interpretation and analysis, may distract the experienced readers. The essence of a great hypothesis is a story behind the scientific facts and evidence-based data.

ETHICAL IMPLICATIONS

The authors of hypotheses substantiate their arguments by referring to and discerning rational points from published articles that might be overlooked by others. Their arguments may contradict the established theories and practices, and pose global ethical issues, particularly when more or less efficient medical technologies and public health interventions are devalued. The ethical issues may arise primarily because of the careless references to articles with low priorities, inadequate and apparently unethical methodologies, and concealed reporting of negative results. 24 , 25

Misinterpretation and misunderstanding of the published ideas and scientific hypotheses may complicate the issue further. For example, Alexander Fleming, whose innovative ideas of penicillin use to kill susceptible bacteria saved millions of lives, warned of the consequences of uncontrolled prescription of the drug. The issue of antibiotic resistance had emerged within the first ten years of penicillin use on a global scale due to the overprescription that affected the efficacy of antibiotic therapies, with undesirable consequences for millions. 26

The misunderstanding of the hygiene hypothesis that primarily aimed to shed light on the role of the microbiome in allergic and autoimmune diseases resulted in decline of public confidence in hygiene with dire societal implications, forcing some experts to abandon the original idea. 27 , 28 Although that hypothesis is unrelated to the issue of vaccinations, the public misunderstanding has resulted in decline of vaccinations at a time of upsurge of old and new infections.

A number of ethical issues are posed by the denial of the viral (human immunodeficiency viruses; HIV) hypothesis of acquired Immune deficiency Syndrome (AIDS) by Peter Duesberg, who overviewed the links between illicit recreational drugs and antiretroviral therapies with AIDS and refuted the etiological role of HIV. 29 That controversial hypothesis was rejected by several journals, but was eventually published without external peer review at Med Hypotheses in 2010. The publication itself raised concerns of the unconventional editorial policy of the journal, causing major perturbations and more scrutinized publishing policies by journals processing hypotheses.

WHERE TO PUBLISH HYPOTHESES

Although scientific authors are currently well informed and equipped with search tools to draft evidence-based hypotheses, there are still limited quality publication outlets calling for related articles. The journal editors may be hesitant to publish articles that do not adhere to any research reporting guidelines and open gates for harsh criticism of unconventional and untested ideas. Occasionally, the editors opting for open-access publishing and upgrading their ethics regulations launch a section to selectively publish scientific hypotheses attractive to the experienced readers. 30 However, the absence of approved standards for this article type, particularly no mandate for outlining potential ethical implications, may lead to publication of potentially harmful ideas in an attractive format.

A suggestion of simultaneously publishing multiple or alternative hypotheses to balance the reader views and feedback is a potential solution for the mainstream scholarly journals. 31 However, that option alone is hardly applicable to emerging journals with unconventional quality checks and peer review, accumulating papers with multiple rejections by established journals.

A large group of experts view hypotheses with improbable and controversial ideas publishable after formal editorial (in-house) checks to preserve the authors' genuine ideas and avoid conservative amendments imposed by external peer reviewers. 32 That approach may be acceptable for established publishers with large teams of experienced editors. However, the same approach can lead to dire consequences if employed by nonselective start-up, open-access journals processing all types of articles and primarily accepting those with charged publication fees. 33 In fact, pseudoscientific ideas arguing Newton's and Einstein's seminal works or those denying climate change that are hardly testable have already found their niche in substandard electronic journals with soft or nonexistent peer review. 34

CITATIONS AND SOCIAL MEDIA ATTENTION

The available preliminary evidence points to the attractiveness of hypothesis articles for readers, particularly those from research-intensive countries who actively download related documents. 35 However, citations of such articles are disproportionately low. Only a small proportion of top-downloaded hypotheses (13%) in the highly prestigious Med Hypotheses receive on average 5 citations per article within a two-year window. 36

With the exception of a few historic papers, the vast majority of hypotheses attract relatively small number of citations in a long term. 36 Plausible explanations are that these articles often contain a single or only a few citable points and that suggested research studies to test hypotheses are rarely conducted and reported, limiting chances of citing and crediting authors of genuine research ideas.

A snapshot analysis of citation activity of hypothesis articles may reveal interest of the global scientific community towards their implications across various disciplines and countries. As a prime example, Strachan's hygiene hypothesis, published in 1989, 10 is still attracting numerous citations on Scopus, the largest bibliographic database. As of August 28, 2019, the number of the linked citations in the database is 3,201. Of the citing articles, 160 are cited at least 160 times ( h -index of this research topic = 160). The first three citations are recorded in 1992 and followed by a rapid annual increase in citation activity and a peak of 212 in 2015 ( Fig. 1 ). The top 5 sources of the citations are Clin Exp Allergy (n = 136), J Allergy Clin Immunol (n = 119), Allergy (n = 81), Pediatr Allergy Immunol (n = 69), and PLOS One (n = 44). The top 5 citing authors are leading experts in pediatrics and allergology Erika von Mutius (Munich, Germany, number of publications with the index citation = 30), Erika Isolauri (Turku, Finland, n = 27), Patrick G Holt (Subiaco, Australia, n = 25), David P. Strachan (London, UK, n = 23), and Bengt Björksten (Stockholm, Sweden, n = 22). The U.S. is the leading country in terms of citation activity with 809 related documents, followed by the UK (n = 494), Germany (n = 314), Australia (n = 211), and the Netherlands (n = 177). The largest proportion of citing documents are articles (n = 1,726, 54%), followed by reviews (n = 950, 29.7%), and book chapters (n = 213, 6.7%). The main subject areas of the citing items are medicine (n = 2,581, 51.7%), immunology and microbiology (n = 1,179, 23.6%), and biochemistry, genetics and molecular biology (n = 415, 8.3%).

Interestingly, a recent analysis of 111 publications related to Strachan's hygiene hypothesis, stating that the lack of exposure to infections in early life increases the risk of rhinitis, revealed a selection bias of 5,551 citations on Web of Science. 37 The articles supportive of the hypothesis were cited more than nonsupportive ones (odds ratio adjusted for study design, 2.2; 95% confidence interval, 1.6–3.1). A similar conclusion pointing to a citation bias distorting bibliometrics of hypotheses was reached by an earlier analysis of a citation network linked to the idea that β-amyloid, which is involved in the pathogenesis of Alzheimer disease, is produced by skeletal muscle of patients with inclusion body myositis. 38 The results of both studies are in line with the notion that ‘positive’ citations are more frequent in the field of biomedicine than ‘negative’ ones, and that citations to articles with proven hypotheses are too common. 39

Social media channels are playing an increasingly active role in the generation and evaluation of scientific hypotheses. In fact, publicly discussing research questions on platforms of news outlets, such as Reddit, may shape hypotheses on health-related issues of global importance, such as obesity. 40 Analyzing Twitter comments, researchers may reveal both potentially valuable ideas and unfounded claims that surround groundbreaking research ideas. 41 Social media activities, however, are unevenly distributed across different research topics, journals and countries, and these are not always objective professional reflections of the breakthroughs in science. 2 , 42

Scientific hypotheses are essential for progress in science and advances in healthcare. Innovative ideas should be based on a critical overview of related scientific facts and evidence-based data, often overlooked by others. To generate realistic hypothetical theories, the authors should comprehensively analyze the literature and suggest relevant and ethically sound design for future studies. They should also consider their hypotheses in the context of research and publication ethics norms acceptable for their target journals. The journal editors aiming to diversify their portfolio by maintaining and introducing hypotheses section are in a position to upgrade guidelines for related articles by pointing to general and specific analyses of the subject, preferred study designs to test hypotheses, and ethical implications. The latter is closely related to specifics of hypotheses. For example, editorial recommendations to outline benefits and risks of a new laboratory test or therapy may result in a more balanced article and minimize associated risks afterwards.

Not all scientific hypotheses have immediate positive effects. Some, if not most, are never tested in properly designed research studies and never cited in credible and indexed publication outlets. Hypotheses in specialized scientific fields, particularly those hardly understandable for nonexperts, lose their attractiveness for increasingly interdisciplinary audience. The authors' honest analysis of the benefits and limitations of their hypotheses and concerted efforts of all stakeholders in science communication to initiate public discussion on widely visible platforms and social media may reveal rational points and caveats of the new ideas.

Disclosure: The authors have no potential conflicts of interest to disclose.

Author Contributions:

• Conceptualization: Gasparyan AY, Yessirkepov M, Kitas GD.
• Methodology: Gasparyan AY, Mukanova U, Ayvazyan L.
• Writing - original draft: Gasparyan AY, Ayvazyan L, Yessirkepov M.
• Writing - review & editing: Gasparyan AY, Yessirkepov M, Mukanova U, Kitas GD.

Scientific Hypothesis — Definition & Examples - Expii

From the Editors

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How we edit science part 1: the scientific method

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We take science seriously at The Conversation and we work hard to report it accurately. This series of five posts is adapted from an internal presentation on how to understand and edit science by our Australian Science & Technology Editor, Tim Dean. We thought you might also find it useful.

Introduction

If I told you that science was a truth-seeking endeavour that uses a single robust method to prove scientific facts about the world, steadily and inexorably driving towards objective truth, would you believe me?

Many would. But you shouldn’t.

The public perception of science is often at odds with how science actually works. Science is often seen to be a separate domain of knowledge, framed to be superior to other forms of knowledge by virtue of its objectivity, which is sometimes referred to as it having a “ view from nowhere ”.

But science is actually far messier than this - and far more interesting. It is not without its limitations and flaws, but it’s still the most effective tool we have to understand the workings of the natural world around us.

In order to report or edit science effectively - or to consume it as a reader - it’s important to understand what science is, how the scientific method (or methods) work, and also some of the common pitfalls in practising science and interpreting its results.

This guide will give a short overview of what science is and how it works, with a more detailed treatment of both these topics in the final post in the series.

What is science?

Science is special, not because it claims to provide us with access to the truth, but because it admits it can’t provide truth .

Other means of producing knowledge, such as pure reason, intuition or revelation, might be appealing because they give the impression of certainty , but when this knowledge is applied to make predictions about the world around us, reality often finds them wanting.

Rather, science consists of a bunch of methods that enable us to accumulate evidence to test our ideas about how the world is, and why it works the way it does. Science works precisely because it enables us to make predictions that are borne out by experience.

Science is not a body of knowledge. Facts are facts, it’s just that some are known with a higher degree of certainty than others. What we often call “scientific facts” are just facts that are backed by the rigours of the scientific method, but they are not intrinsically different from other facts about the world.

What makes science so powerful is that it’s intensely self-critical. In order for a hypothesis to pass muster and enter a textbook, it must survive a battery of tests designed specifically to show that it could be wrong. If it passes, it has cleared a high bar.

The scientific method(s)

Despite what some philosophers have stated , there is a method for conducting science. In fact, there are many. And not all revolve around performing experiments.

One method involves simple observation, description and classification, such as in taxonomy. (Some physicists look down on this – and every other – kind of science, but they’re only greasing a slippery slope .)

However, when most of us think of The Scientific Method, we’re thinking of a particular kind of experimental method for testing hypotheses.

This begins with observing phenomena in the world around us, and then moves on to positing hypotheses for why those phenomena happen the way they do. A hypothesis is just an explanation, usually in the form of a causal mechanism: X causes Y. An example would be: gravitation causes the ball to fall back to the ground.

A scientific theory is just a collection of well-tested hypotheses that hang together to explain a great deal of stuff.

Crucially, a scientific hypothesis needs to be testable and falsifiable .

An untestable hypothesis would be something like “the ball falls to the ground because mischievous invisible unicorns want it to”. If these unicorns are not detectable by any scientific instrument, then the hypothesis that they’re responsible for gravity is not scientific.

An unfalsifiable hypothesis is one where no amount of testing can prove it wrong. An example might be the psychic who claims the experiment to test their powers of ESP failed because the scientific instruments were interfering with their abilities.

(Caveat: there are some hypotheses that are untestable because we choose not to test them. That doesn’t make them unscientific in principle, it’s just that they’ve been denied by an ethics committee or other regulation.)

Experimentation

There are often many hypotheses that could explain any particular phenomenon. Does the rock fall to the ground because an invisible force pulls on the rock? Or is it because the mass of the Earth warps spacetime , and the rock follows the lowest-energy path, thus colliding with the ground? Or is it that all substances have a natural tendency to fall towards the centre of the Universe , which happens to be at the centre of the Earth?

The trick is figuring out which hypothesis is the right one. That’s where experimentation comes in.

A scientist will take their hypothesis and use that to make a prediction, and they will construct an experiment to see if that prediction holds. But any observation that confirms one hypothesis will likely confirm several others as well. If I lift and drop a rock, it supports all three of the hypotheses on gravity above.

Furthermore, you can keep accumulating evidence to confirm a hypothesis, and it will never prove it to be absolutely true. This is because you can’t rule out the possibility of another similar hypothesis being correct, or of making some new observation that shows your hypothesis to be false. But if one day you drop a rock and it shoots off into space, that ought to cast doubt on all of the above hypotheses.

So while you can never prove a hypothesis true simply by making more confirmatory observations, you only one need one solid contrary observation to prove a hypothesis false. This notion is at the core of the hypothetico-deductive model of science.

This is why a great deal of science is focused on testing hypotheses, pushing them to their limits and attempting to break them through experimentation. If the hypothesis survives repeated testing, our confidence in it grows.

So even crazy-sounding theories like general relativity and quantum mechanics can become well accepted, because both enable very precise predictions, and these have been exhaustively tested and come through unscathed.

The next post will cover hypothesis testing in greater detail.

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Three Famous Hypotheses and How They Were Tested

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Key Takeaways

• Ivan Pavlov's experiment demonstrated conditioned responses in dogs.
• Pavlov's work exemplifies the scientific method, starting with a hypothesis about conditioned responses and testing it through controlled experiments.
• Pavlov's findings not only advanced an understanding of animal physiology but also laid foundational principles for behaviorism, a major school of thought in psychology that emphasizes the study of observable behaviors.

Coho salmon ( Oncorhynchus kisutch ) are amazing fish. Indigenous to the Pacific Northwest, they begin their lives in freshwater streams and then relocate to the open ocean. But when a Coho salmon reaches breeding age, it'll return to the waterway of its birth , sometimes traveling 400 miles (644 kilometers) to get there.

Enter the late Arthur Davis Hasler. While an ecologist and biologist at the University of Wisconsin, he was intrigued by the question of how these creatures find their home streams. And in 1960, he used a Hypothesis-Presentation.pdf">basic tenet of science — the hypothesis — to find out.

So what is a hypothesis? A hypothesis is a tentative, testable explanation for an observed phenomenon in nature. Hypotheses are narrow in scope — unlike theories , which cover a broad range of observable phenomena and draw from many different lines of evidence. Meanwhile, a prediction is a result you'd expect to get if your hypothesis or theory is accurate.

So back to 1960 and Hasler and those salmon. One unverified idea was that Coho salmon used eyesight to locate their home streams. Hasler set out to test this notion (or hypothesis). First, he rounded up several fish who'd already returned to their native streams. Next, he blindfolded some of the captives — but not all of them — before dumping his salmon into a faraway stretch of water. If the eyesight hypothesis was correct, then Hasler could expect fewer of the blindfolded fish to return to their home streams.

Things didn't work out that way. The fish without blindfolds came back at the same rate as their blindfolded counterparts. (Other experiments demonstrated that smell, and not sight, is the key to the species' homing ability.)

Although Hasler's blindfold hypothesis was disproven, others have fared better. Today, we're looking at three of the best-known experiments in history — and the hypotheses they tested.

Ivan Pavlov and His Dogs (1903-1935)

Isaac newton's radiant prisms (1665), robert paine's revealing starfish (1963-1969).

The Hypothesis : If dogs are susceptible to conditioned responses (drooling), then a dog who is regularly exposed to the same neutral stimulus (metronome/bell) before it receives food will associate this neutral stimulus with the act of eating. Eventually, the dog should begin to drool at a predictable rate when it encounters said stimulus — even before any actual food is offered.

The Experiment : A Nobel Prize-winner and outspoken critic of Soviet communism, Ivan Pavlov is synonymous with man's best friend . In 1903, the Russian-born scientist kicked off a decades-long series of experiments involving dogs and conditioned responses .

Offer a plate of food to a hungry dog and it'll salivate. In this context, the stimulus (the food) will automatically trigger a particular response (the drooling). The latter is an innate, unlearned reaction to the former.

By contrast, the rhythmic sound of a metronome or bell is a neutral stimulus. To a dog, the noise has no inherent meaning and if the animal has never heard it before, the sound won't provoke an instinctive reaction. But the sight of food sure will .

So when Pavlov and his lab assistants played the sound of the metronome/bell before feeding sessions, the researchers conditioned test dogs to mentally link metronomes/bells with mealtime. Due to repeated exposure, the noise alone started to make the dogs' mouths water before they were given food.

According to " Ivan Pavlov: A Russian Life in Science " by biographer Daniel P. Todes, Pavlov's big innovation here was his discovery that he could quantify the reaction of each pooch by measuring the amount of saliva it generated. Every canine predictably drooled at its own consistent rate when he or she encountered a personalized (and artificial) food-related cue.

Pavlov and his assistants used conditioned responses to look at other hypotheses about animal physiology, as well. In one notable experiment, a dog was tested on its ability to tell time . This particular pooch always received food when it heard a metronome click at the rate of 60 strokes per minute. But it never got any food after listening to a slower, 40-strokes-per-minute beat. Lo and behold, Pavlov's animal began to salivate in response to the faster rhythm — but not the slower one . So clearly, it could tell the two rhythmic beats apart.

The Verdict : With the right conditioning — and lots of patience — you can make a hungry dog respond to neutral stimuli by salivating on cue in a way that's both predictable and scientifically quantifiable.

The Hypothesis : If white sunlight is a mixture of all the colors in the visible spectrum — and these travel at varying wavelengths — then each color will refract at a different angle when a beam of sunlight passes through a glass prism.

The Experiments : Color was a scientific mystery before Isaac Newton came along. During the summer of 1665, he started experimenting with glass prisms from the safety of a darkened room in Cambridge, England.

He cut a quarter-inch (0.63-centimeter) circular hole into one of the window shutters, allowing a single beam of sunlight to enter the place. When Newton held up a prism to this ray, an oblong patch of multicolored light was projected onto the opposite wall.

This contained segregated layers of red, orange, yellow, green, blue, indigo and violet light. From top to bottom, this patch measured 13.5 inches (33.65 centimeters) tall, yet it was only 2.6 inches (6.6 centimeters) across.

Newton deduced that these vibrant colors had been hiding within the sunlight itself, but the prism bent (or "refracted") them at different angles, which separated the colors out.

Still, he wasn't 100 percent sure. So Newton replicated the experiment with one small change. This time, he took a second prism and had it intercept the rainbow-like patch of light. Once the refracted colors entered the new prism, they recombined into a circular white sunbeam. In other words, Newton took a ray of white light, broke it apart into a bunch of different colors and then reassembled it. What a neat party trick!

The Verdict : Sunlight really is a blend of all the colors in the rainbow — and yes, these can be individually separated via light refraction.

The Hypothesis : If predators limit the populations of the organisms they attack, then we'd expect the prey species to become more common after the eradication of a major predator.

The Experiment : Meet Pisaster ochraceus , also known as the purple sea star (or the purple starfish if you prefer).

Using an extendable stomach , the creature feeds on mussels, limpets, barnacles, snails and other hapless victims. On some seaside rocks (and tidal pools) along the coast of Washington state, this starfish is the apex predator.

The animal made Robert Paine a scientific celebrity. An ecologist by trade, Paine was fascinated by the environmental roles of top predators. In June 1963, he kicked off an ambitious experiment along Washington state's Mukkaw Bay. For years on end, Paine kept a rocky section of this shoreline completely starfish-free.

It was hard work. Paine had to regularly pry wayward sea stars off "his" outcrop — sometimes with a crowbar. Then he'd chuck them into the ocean.

Before the experiment, Paine observed 15 different species of animals and algae inhabiting the area he decided to test. By June 1964 — one year after his starfish purge started — that number had dropped to eight .

Unchecked by purple sea stars, the barnacle population skyrocketed. Subsequently, these were replaced by California mussels , which came to dominate the terrain. By anchoring themselves to rocks in great numbers, the mussels edged out other life-forms. That made the outcrop uninhabitable to most former residents: Even sponges, anemones and algae — organisms that Pisaster ochraceus doesn't eat — were largely evicted.

All those species continued to thrive on another piece of shoreline that Paine left untouched. Later experiments convinced him that Pisaster ochraceus is a " keystone species ," a creature who exerts disproportionate influence over its environment. Eliminate the keystone and the whole system gets disheveled.

The Verdict : Apex predators don't just affect the animals that they hunt. Removing a top predator sets off a chain reaction that can fundamentally transform an entire ecosystem.

Contrary to popular belief, Pavlov almost never used bells in his dog experiments. Instead, he preferred metronomes, buzzers, harmoniums and electric shocks.

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A Good check on the Bayes factor

• Original Manuscript
• Open access
• Published: 04 September 2024

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• Nikola Sekulovski   ORCID: orcid.org/0000-0001-7032-1684 1 ,
• Maarten Marsman   ORCID: orcid.org/0000-0001-5309-7502 1 &
• Eric-Jan Wagenmakers   ORCID: orcid.org/0000-0003-1596-1034 1  

Bayes factor hypothesis testing provides a powerful framework for assessing the evidence in favor of competing hypotheses. To obtain Bayes factors, statisticians often require advanced, non-standard tools, making it important to confirm that the methodology is computationally sound. This paper seeks to validate Bayes factor calculations by applying two theorems attributed to Alan Turing and Jack Good. The procedure entails simulating data sets under two hypotheses, calculating Bayes factors, and assessing whether their expected values align with theoretical expectations. We illustrate this method with an ANOVA example and a network psychometrics application, demonstrating its efficacy in detecting calculation errors and confirming the computational correctness of the Bayes factor results. This structured validation approach aims to provide researchers with a tool to enhance the credibility of Bayes factor hypothesis testing, fostering more robust and trustworthy scientific inferences.

Avoid common mistakes on your manuscript.

Introduction

The Bayes factor (Kass & Raftery, 1995 ; Jeffreys, 1935 ) serves as a valuable tool for testing scientific hypotheses by comparing the relative predictive adequacy of two competing statistical models. In recent decades, there has been a surge in the adoption of Bayes factors as a tool for hypothesis testing (e.g., in psychology, Heck et al., 2023 ; van de Schoot et al., 2017 ). This increasing trend towards Bayesian hypothesis testing and model comparison has been catalyzed by a growing critique of traditional frequentist null hypothesis significance testing methods (e.g., Wasserstein and Lazar, 2016 ; Wagenmakers, 2007 ; Cohen, 1994 ; Wagenmakers et al., 2018b ; Benjamin et al., 2018 ; for an early critique see Edwards et al., 1963 ). In addition, the emergence of user-friendly software packages (e.g., JASP Team, 2023 ; Morey and Rouder, 2022 ; Gu et al., 2021 ) and associated tutorial articles have played a crucial role in making the benefits of the Bayesian framework more accessible to applied researchers (e.g., van Doorn et al., 2021 ; Rouder et al., 2012 ; Hoijtink et al., 2019 ; Marsman and Wagenmakers, 2017 ; Wagenmakers et al., 2018b ; Wagenmakers et al., 2018a ). Overall, this upswing in Bayesian methodology has ushered in a new era of statistical analysis, offering researchers valuable alternatives to traditional approaches.

Although Bayes factors have gained popularity in scientific practice, calculating them can be challenging, especially when comparing the relative likelihood of two complex models, such as hierarchical or nonlinear models with a large number of parameters. In such cases, Bayes factors often need to be approximated using various numerical (sampling) techniques such as bridge sampling (Gronau et al., 2017 ) or path sampling (Zhou et al., 2012 ); for a general introduction to stochastic sampling in Bayesian inference see Gamerman and Lopes ( 2006 ). These techniques often require the user to specify proposal distributions or tune certain parameters within the sampler, which may lead to inaccuracies. There are also state-of-the-art sampling methods designed to obtain joint posterior probabilities over many models; some notable examples of these transdimensional methods are Reversible Jump MCMC (Green, 1995 ), MCMC with mixtures of mutually singular distributions (Gottardo & Raftery, 2008 ) and the product space method (Lodewyckx et al., 2011 ; Carlin & Chib, 1995 ). These methods, even though very powerful, are quite complex to implement in software and therefore error-prone. Therefore, despite their utility, the use of these numerical techniques can introduce errors, such as the one highlighted by Tsukamura and Okada ( 2023 ), who pointed out a common coding error when computing Bayes factors in certain settings in the Stan programming language (Stan Development Team, 2023 ). Recently, Schad and Vasishth ( 2024 ) showed that Bayes factor estimates can be biased in some commonly used factorial designs.

In addition to the potential inaccuracies of existing approac-hes, ongoing research is constantly advancing the methods used to compute Bayes factors; a recent development by Kim and Rockova ( 2023 ) introduces a deep learning estimator as an addition to the toolkit of techniques available for computing Bayes factors. While the diversity of computational approaches is crucial, it is important to note that the complexity of these tools can lead to inaccuracies in Bayes factor calculations in applied research contexts. Thus, the development of appropriate controls and checks becomes imperative.

Schad et al. ( 2022 ) highlight five key considerations that warrant attention when computing Bayes factors, two of which are (i) the Bayes factor estimates for complex statistical models can be unstable, and (ii) the Bayes factor estimates can be biased. Therefore, Schad et al. ( 2022 ) propose a structured approach based on simulation-based calibration, which was originally developed as a method to validate the computational correctness of applied Bayesian inference more generally, and use it to verify the accuracy of Bayes factor calculations (see Talts et al., 2018 ; Cook et al., 2006 ; Geweke, 2004 ). Their method is based on the idea that that the marginal expected posterior model probability is equal to the prior model probability. We provide a more detailed description of the method proposed by Schad et al. ( 2022 ) in one of the following sections.

Before proposing another formal Bayes factor check in the spirit of the one by Schad et al. ( 2022 ), we would like to mention two other methods that, while not explicitly described as Bayes factor checks, can be used for this purpose. For the first method, suppose a researcher is interested in computing the Bayes factor for the relative adequacy of two complex (possibly non-nested) models, \(\mathcal {H}_1\) and \(\mathcal {H}_{\text {2}}\) , and has already chosen a numerical method implemented in some software for computing \(\text {BF}_{12}\) . To check that the calculation has been carried out correctly, they can construct nested versions of each of the models by selecting a single parameter and setting it to its maximum likelihood estimate (MLE) value, which would act as a surrogate oracle null model. They can then use the Savage–Dickey density ratio (Dickey & Lientz, 1970 ; Wagenmakers et al., 2010 ) to compute \(\text {BF}_{\text {ou}}\) – the Bayes factor in favor of the oracle null over the unconstrained model – for both \(\mathcal {H}_1\) and \(\mathcal {H}_{\text {2}}\) . When both models have Savage–Dickey \(\text {BF}_{\text {ou}}\) ’s that match the \(\text {BF}_{\text {ou}}\) ’s obtained from the method under scrutiny, then this gives the researcher reason to believe that \(\text {BF}_{12}\) has been computed correctly. A similar approach has been implemented by Gronau et al. ( 2020 ) for computing the marginal likelihood in evidence accumulation models, achieved by introducing a Warp-III bridge sampling algorithm. A second method to check the Bayes factor is pragmatic and can be used whenever multiple computational methods are available for a specific application. The idea is that one can use all methods – if they agree, they will mutually reinforce the conclusion and provide evidence that the Bayes factor has been calculated correctly. Furthermore, a Bayes factor can be computed for this agreement. Given that the probability of two correct methods yielding the same outcome is 1, the Bayes factor is calculated as 1 divided by the probability of a chance agreement between two methods, assuming at least one is incorrect. Since the probability of two methods converging on the same wrong value is very small, the Bayes factor provides very strong evidence that both methods are correct.

In this paper, we draw attention to two theorems by Alan Turing and Jack Good (e.g., Good, 1950 , 1985 , 1994 ), which they proposed could be used to verify the computation of Bayes factors. We introduce a structured approach to perform this verification, aiming to revive and highlight an idea that, until now, has not received the attention it deserves.

The remainder of this paper is structured as follows. In the next section, we provide an overview of the material in Good ( 1985 ), where we discuss the theorems, introduce key concepts, and establish notation. Following this, we present a simple binomial model to illustrate the conditions under which these theorems apply. Next, we outline the workflow for the Bayes factor check tool and offer two numerical examples to demonstrate its application-one employing an ANOVA design and the other utilizing a complex psychometric network model. We conclude the paper by comparing the strengths and limitations of this method, as well as highlighting potential avenues for improvement.

Theoretical background

The weight of evidence.

Good ( 1985 ) points out that the concept of weight of evidence , which is used in many areas (e.g., in science, medicine, law, and daily life), is a function of the probabilities of the data under two hypotheses (see also Good, 1950 , 1965 , 1979 , 1994 , 1995 ). Formally, this relation takes the form

where \(\mathcal {W}(\mathcal {H}_1:\text {data})\) denotes the weight of evidence in favor of the hypothesis \(\mathcal {H}_1\) provided by the evidence (data), while \(p(\text {data} \mid \mathcal {H}_\cdot )\) denote the probabilities of the data under each of the hypotheses (i.e., what is usually called the marginal likelihood of the data). Good ( 1985 ) further points out that this function should be mathematically independent of \(p(\mathcal {H}_\cdot )\) , known as the prior probability of a hypothesis, but that \(p(\mathcal {H}_\cdot \mid \text {data})\) (i.e., the posterior probability) should depend both on the weight of evidence and the prior probability. This relationship can therefore be expressed as

Thus, the Bayes factor can be interpreted as the factor by which the initial odds are multiplied to give the final odds, or as the ratio of the posterior odds for \(\mathcal {H}_1\) to its prior odds. When \(\mathcal {H}_1\) and \(\mathcal {H}_2\) are simple (point) hypotheses the Bayes factor is equal to the likelihood ratio (Royall, 2017 ). Good defined the weight of evidence as the logarithm of the Bayes factor (Good, 1950 , 1985 , 1994 ), because it is additive and symmetric (e.g., \(\log (\text {BF} = 10) = 2.3\) and \(\log (\text {BF} = {1}/{10}) = -2.3\) , the average of which is 0). In contrast, the Bayes factor scale is not symmetric – the average of a Bayes factor of 10 and 1/10 is larger than 1. In writing about an appropriate metric for the weight of evidence, Good ( 1985 ) draws attention to a counterintuitive theorem about the Bayes factor, and suggested it may be used to check whether a particular procedure computes Bayes factors correctly. The theorem states that “the expected (Bayes) factor in favor of the false hypothesis is 1” . Good attributed this paradoxical insight to Alan Turing, whose team at Bletchley Park decrypted German naval messages during World War II (cf. Zabell, 2023 ).

In the following subsection, we first introduce Turing’s theorem. We then present another related theorem proposed by Good, which shows the relationship between higher-order moments of Bayes factors.

Moments of the Bayes factor

Theorem 1: the expected (bayes) factor in favor of the false hypothesis equals 1. – alan turing.

Suppose the possible outcomes of an experiment are \(E_1, E_2,...,E_M\) , where \(\mathcal {H}_t\) is the true hypothesis and \(\mathcal {H}_f\) is the false hypothesis. Footnote 1 Taking the expectation of the Bayes factor in favor of one of the hypotheses simply means calculating the weighted average of that Bayes factor where the weights are provided by the probability of the evidence given the true hypothesis (i.e., \(p(\text {E} \mid \mathcal {H}_t)\) ). Then the expected Bayes factor in favor of \(\mathcal {H}_f\) is given by

\(\square \)

The theorem states that the expected Bayes factor against the truth is 1, regardless of sample size. For example, consider a binomial experiment with \(n = 2\) trials and k successes, where \(\mathcal {H}_0\text {: } \theta = {1}/{2}\) and \(\mathcal {H}_1\text {: } \theta \sim \text {Beta}(\alpha =1\text {, }\beta =1)\) . There are three possible outcomes for this experiment, \(E_1\text {: } k = 0\) , \(E_2\text {: } k = 1\) , and \(E_3\text {: } k = 2\) . It follows from the beta-binomial distribution that the probability is the same for each possible outcome under \(\mathcal {H}_1\) , which in this case is 1/3 \(\forall \ E_i\) . Under \(\mathcal {H}_0\) the probability of \(E_1\) and \(E_3\) is 1/4 and for \(E_2\) is 1/2. Assuming that \(\mathcal {H}_1\) is the correct hypotheses we have

As a Bayes factor of 1 indicates the complete absence of evidence, this theorem is paradoxical; intuition suggests that – especially for large sample sizes – the average Bayes factor against the truth should be much smaller than 1. As mentioned in the previous subsection, unlike the weight of evidence, the Bayes factor is not symmetric. For example, the mean of \(\text {BF}_{10} = {1}/{10}\) and \(\text {BF}_{10} = 10\) is 5.05 and not 1, whereas the mean of \(\log ({1}/{10})\) and \(\log (10)\) is 0. This theorem implies that the sampling distribution of the Bayes factor is skewed to the right. Therefore, Good ( 1985 ) suggests that the Bayes factor is likely to have a (roughly) log-normal distribution while the weight of evidence has a (roughly) normal distribution (see also, Good, 1994 ). Finally, Good ( 1985 ) shows that the expected weight of evidence in favor of the truth (i.e., \(\mathcal {W}(\mathcal {H}_t: \text {data})\) ) is non-negative and vanishes when the weight of evidence is 0. This again illustrates that the weight of evidence is additive and its expected value is more meaningful than that of the Bayes factor.

Until now, Theorem 1 has been used almost exclusively to establish the universal bound on obtaining misleading evidence (e.g., Royall, 2000 ; Sanborn and Hills, 2014 ). The universal bound states that the probability of obtaining a Bayes factor greater than or equal to \({1}/{\alpha }\) in favor of the false hypothesis is less than or equal to some threshold \(\alpha \) . For example, the probability of obtaining a Bayes factor of 100 in favor of the false hypothesis is less than or equal to \(1\%\) . This is related to the fact that a Bayes factor in favor of the false hypothesis is related to a non-negative test martingale where the expected value of the martingale at any point t is 1. Footnote 2 That is, the test martingale measures the evidence against a hypothesis \(\mathcal {H}\) , and its inverse at some point t is a Bayes factor in favor of \(\mathcal {H}\) (see e.g., Shafer et al., 2011 ; Grünwald et al., 2020 ). Footnote 3 These properties have also been used independently in sequential analysis by Abraham Wald (Wald, 1945 ). Since the concept of a martingale (Ville, 1939 ) predates the work of Good and Turing, this suggests that they were not the first to be (at least implicitly) aware of this theorem. However, Jack Good was apparently the first to propose that the theorem may be used to verify the computation of the Bayes factor (Good, 1985 , p. 255). This paper implements Good’s idea.

Theorem 1 shows that the first moment of the Bayes factor under the false hypothesis is equal to 1. This is the main result; however, Good ( 1985 ) shows that Theorem 1 is a special case of another theorem which shows the equivalence between higher-order moments of Bayes factors; we turn to this theorem next.

Theorem 2: Equivalence of moments for Bayes factors under \(\mathcal {H}_1\) and \(\mathcal {H}_2 \) . – Jack Good

The second theorem generalizes the first and states that

The theorem can be expressed as

Using the product law of exponents, the right-hand side of the equation above can be rewritten as

which immediately proves the result.

This theorem states that the \(k^{th}\) moment of the Bayes factor in favor of \(\mathcal {H}_1\) about the origin, given that \(\mathcal {H}_1\) is true is equal to the \((k+1)^{st}\) moment of the Bayes factor in favor of \(\mathcal {H}_1\) given that \(\mathcal {H}_2\) is true. Here we refer to the raw moments, that is the moments about the origin and not to the central moments (such as the variance, which is the second moment about the mean). When \(k = 0\) , this result reduces to that of the first theorem.

Considering the binomial example from earlier with \(n = 2\) and hypotheses \(\mathcal {H}_0\text {: } \theta = {1}/{2}\) and \(\mathcal {H}_1\text {: } \theta \sim \text {Beta}(\alpha = 1\text {, } \beta = 1)\) one can see that

Numerical illustrations

Consider a sequence of n coin tosses that forms the basis of a test of the null hypothesis \(\mathcal {H}_0\text {: } \theta = {1}/{2}\) against the alternative hypothesis \(\mathcal {H}_1\text {: } \theta \sim \text {Uniform}(0,1)\) , where \(\theta \) represents the probability of the coin landing heads. Footnote 4 Additionally, in the last part of this section, we consider a restricted (directional) hypothesis \(\mathcal {H}_{\text {r}}\text {: } \theta > {1}/{2}\) . We simulated a total of \(m = 100{,}000\) data sets either under \(\mathcal {H}_0\) , \(\mathcal {H}_1\) or \(\mathcal {H}_{\text {r}}\) for sample sizes of \(n = \{10, 50, 100\}\) . For each simulation setting, we averaged the \(m = 2, \dots , 100{,}000\) Bayes factors in favor of the wrong hypothesis. The code to reproduce the examples in this paper is publicly available in an OSF repository at https://osf.io/438vy/ .

The average Bayes factor in favor of the null hypothesis quickly converges to 1 for synthetic data sets generated under the alternative hypothesis. The figure depicts the average \(\text {BF}_{01}\) as a function of the number of synthetic data sets m generated under \(\mathcal {H}_1\) , for \(n = 10, 50, 100\) ; the black solid line is for \(n = 10\) , the red dashed line is for \(n = 50\) , and the green dotted line is for \(n = 100\) . The left panel plots the cumulative mean across \(m = 100{,}000\) data sets; the right panel zooms in on the first \(m = 1{,}000\) iterations

The average Bayes factor in favor of the alternative hypothesis does not converge to 1 as n increases for the synthetic data sets generated under the null hypothesis. The figure depicts the average \(\text {BF}_{10}\) as a function of the number of synthetic data sets m generated under \(\mathcal {H}_0\) , for \(n = 10, 50, 100\) ; the black solid line is for \(n = 10\) , the red dashed line is for \(n = 50\) , and the green dotted line is for \(n = 100\)

Illustration of Theorem 1

Figure 1 illustrates the situation where \(\mathcal {H}_1\) is true and plots the mean Bayes factor in favor of \(\mathcal {H}_0\) , that is, the average \(\text {BF}_{01}\) . For all three values of n , the average \(\text {BF}_{01}\) quickly stabilizes towards 1. There is a slightly larger instability in the mean for larger sample sizes n ; however, the results quickly converge as m increases.

Figure 2 illustrates the situation where \(\mathcal {H}_0\) is true and plots the mean Bayes factor in favor of \(\mathcal {H}_1\) , that is, the average \(\text {BF}_{10}\) calculated for the data sets simulated under \(\mathcal {H}_0\) . It is immediately evident that for larger sample size n , the mean \(\text {BF}_{10}\) becomes unstable and moves away from 1. As m increases, the average appears to stabilize on values different from 1. This observation suggests that under \(\mathcal {H}_0\) , with a large sample size, a very large number of iterations would be necessary to obtain a mean \(\text {BF}_{10}\) that approaches 1. This phenomenon arises because, under \(\mathcal {H}_0\) , there exist rare outcomes that produce extreme \(\text {BF}_{10}\) values, a situation that does not occur with \(\text {BF}_{01}\) when \(\mathcal {H}_1\) is the true hypothesis. The chance of encountering these extreme results under \(\mathcal {H}_0\) , which in turn yields extreme \(\text {BF}_{10}\) values, becomes less probable as the sample size n increases. Consequently, in this scenario the mean \(\text {BF}_{10}\) does not quickly converge to 1. We conclude that the Turing–Good theorems exhibit more robust performance in practice when the true hypothesis is not a point null hypothesis (i.e., when the more complicated hypothesis is true).

When the encompassing hypothesis is true, the average Bayes factor in favor of the restricted hypothesis rapidly converges to 1, whereas for when the restricted hypothesis is true the average Bayes factor in favor of the encompassing hypothesis does not converge to 1 when the sample size is large. The left panel shows the average \(\text {BF}_{\text {re}}\) as a function of the number of synthetic data sets m generated under \(\mathcal {H}_{\text {e}}\) , for \(n = 10, 50, 100\) ; the black solid line is for \(n = 10\) , the red dashed line is for \(n = 50\) , and the green dotted line is for \(n = 100\) . The right panel shows the average \(\text {BF}_{\text {er}}\) as a function of the number of synthetic data sets m generated under \(\mathcal {H}_{\text {r}}\) , for \(n = 10, 50, 100\)

Illustration of Theorem 2

To illustrate the second theorem, we compare the first moment of the Bayes factor in favor of the true hypothesis with the second raw moment in favor of the false hypothesis. We first calculated these moments analytically for \(n = \{10, 50, 100\}\) with \(\mathcal {H}_0\text {: } \theta = {1}/{2}\) and \(\mathcal {H}_1\text {: } \theta \sim \text {Uniform}(0,1)\) . We then calculated the same moments for the Bayes factors based on the synthetic data. We calculated the second raw moments for the Bayes factors using the following formula:

The results are summarized in Table 1 .

The eighth column of Table 1 shows that, on average, the evidence for \(\mathcal {H}_0\) increases with the sample size n . Comparing the seventh and eighth columns (shaded in gray) confirms that the mean of \(\text {BF}_{01}\) when \(\mathcal {H}_0\) is true is approximately equal to the second raw moment of \(\text {BF}_{01}\) when \(\mathcal {H}_1\) is true, regardless of sample size.

When the restricted hypothesis is true, the Bayes factor in favor of the null hypothesis rapidly converges to 1, whereas when the null hypothesis is true the Bayes factor in favor of the restricted hypothesis does not converge to 1 when the sample size is large. The left panel shows the average \(\text {BF}_{\text {0r}}\) as a function of the number of synthetic data sets m generated under \(\mathcal {H}_{\text {r}}\) , for \(n = 10, 50, 100\) ; the black solid line is for \(n = 10\) , the red dashed line is for \(n = 50\) , and the green dotted line is for \(n = 100\) . The right panel shows the average \(\text {BF}_{\text {r0}}\) as a function of the number of synthetic data sets m generated under \(\mathcal {H}_0\) , for \(n = 10, 50, 100\)

The sixth column of Table 1 shows that the expected evidence in favor of \(\mathcal {H}_1\) becomes extreme as n increases; contrasting this with the second moment of \(\text {BF}_{10}\) when \(\mathcal {H}_0\) shows that the values are equal for \(n = 10\) , but as n increases these values diverge. These instabilities are due to the same reasons highlighted in the previous subsection. Note, however, that the theorems still hold in this situation, and for a very large number of iterations m the moments are expected to eventually converge. This is supported by the analytical solutions presented in columns 2 through 6. However, the results computed from the synthetic data suggest that in practice, when dealing with a point null hypothesis, one should compute the first moment from the data generated under \(\mathcal {H}_0\) and compare it with the second raw moment computed from the data generated under \(\mathcal {H}_1\) .

It is also possible to compare, for example, the second and third raw moments. In the results from the simulation, the second raw moments of \(\text {BF}_{01}\) for the data sets generated under \(\mathcal {H}_0\) are 4.28, 19, and 37.32, for \(n = 10, 50\) , and 100, respectively. And the third raw moments of \(\text {BF}_{01}\) for the data sets generated under \(\mathcal {H}_1\) are 4.3, 18.8 and 37.1. These results illustrate that the second theorem holds for higher-order moments in general.

Directional hypotheses

In this subsection, we examine how the Bayes factor behaves when one of the hypotheses under consideration is a directional (i.e., inequality constrained or restricted) hypothesis. Hypotheses that consist of a combination of inequality and equality constraints among the parameters are known as informative hypotheses (Hoijtink, 2011 ). Informative hypotheses allow researchers to express their substantive theory and expectations and have become popular in recent years; therefore, it is important to also consider how inequality constrained hypotheses perform under the two theorems.

We make use of the restricted hypothesis \(\mathcal {H}_{\text {r}}: \theta > {1}/{2}\) , which we specify as \(\mathcal {H}_{\text {r}}: \theta \sim \text {Uniform}(0.5, 1)\) . This is equivalent to setting a truncated Beta distribution from 0.5 to 1 for the probability \(\theta \) . We then compare \(\mathcal {H}_{\text {r}}\) with the alternative hypothesis ( \(\mathcal {H}_1\) ) and the null hypothesis ( \(\mathcal {H}_0\) ) from the previous subsections. In line with previous literature (e.g., Klugkist et al., 2005 ), we rename the alternative hypothesis ( \(\mathcal {H}_1\) ) to the encompassing hypothesis and denote it as \(\mathcal {H}_{\text {e}}\) , as both \(\mathcal {H}_0\) and \(\mathcal {H}_{\text {r}}\) are nested under this encompassing hypothesis.

Figure 3 illustrates the situation of comparing \(\mathcal {H}_{\text {e}}\) and \(\mathcal {H}_{\text {r}}\) . In the left plot, the average \(\text {BF}_{\text {re}}\) when \(\mathcal {H}_{\text {e}}\) is the true hypothesis quickly stabilizes towards 1 for all three sample size values. Note also that the initial fluctuations are all greater than 1; this is because half of the outcomes expected under \(\mathcal {H}_{\text {e}}\) are also plausible under \(\mathcal {H}_{\text {r}}\) . The right panel of Fig. 3 illustrates the reverse situation, where \(\mathcal {H}_{\text {r}}\) is the true hypothesis. As can be seen, the Bayes factor now does not quickly converge to 1 for larger sample sizes, because under \(\mathcal {H}_{\text {r}}\) , outcomes that produce large \(\text {BF}_{\text {er}}\) ’s are highly improbable; similar to the case when considering \(\text {BF}_{10}\) when \(\mathcal {H}_0\) is true (cf. Figure 2 ).

Figure 4 illustrates the situation of comparing \(\mathcal {H}_0\) with \(\mathcal {H}_{\text {r}}\) . In the left panel, the average \(\text {BF}_{0r}\) when \(\mathcal {H}_{\text {r}}\) is the true hypothesis approaches 1 for all three sample size values; note, however, that for \(n = 100\) it takes a considerable number of iterations for the average \(\text {BF}_{\text {0r}}\) to converge to 1. The right panel of Fig. 4 illustrates the situation when \(\mathcal {H}_0\) is the true hypothesis; as was the case in Fig. 2 , when the point (null) hypothesis is the true hypothesis, for a finite number of iterations, the average Bayes factor in favor of the false hypothesis does not converge to 1 as the sample size increases. Again, this is due to the fact that under \(\mathcal {H}_0\) very few outcomes produce large \(\text {BF}_{\text {r0}}\) .

Examining the third and fourth columns of Table 2 , we see that the second raw moment of \(\text {BF}_{\text {re}}\) when \(\mathcal {H}_{\text {e}}\) is true is equal to the mean of \(\text {BF}_{\text {re}}\) when \(\mathcal {H}_{\text {r}}\) is true. A similar observation can be made when comparing the sixth and ninth columns. This illustrates that the second theorem also holds for inequality-constrained hypotheses. However, if we compare the mean of \(\text {BF}_{\text {er}}\) when \(\mathcal {H}_{\text {e}}\) is true with the second moment of \(\text {BF}_{\text {er}}\) when \(\mathcal {H}_0\) is true, we observe that these values diverge, especially as the sample size increases. The same divergence occurs when we compare the mean of \(\text {BF}_{12}\) when \(\mathcal {H}_1\) is true with the second moment of \(\text {BF}_{12}\) when \(\mathcal {H}_0\) is true (cf. Table 1 ).

These results illustrate that both theorems are applicable to directional hypotheses and can be used as a general method for checking Bayes factors. Furthermore, generalizing from all the examples, the first theorem shows more robust performance when the more general (encompassing) hypothesis is true. For the second theorem, the (more) specific hypothesis should be set to true, and the average Bayes factor in favor of the more specific hypothesis should be compared with the second moment of the Bayes factor in favor of the more specific hypothesis when the more general hypothesis is true.

An exception to the rule

In the philosophy of science, a universal generalization is a hypothesis stating that a parameter or characteristic is true for the entire population without exceptions (e.g., all ravens are black). So for the binomial example, this would be equivalent to \(\mathcal {H}_0\text {: } \theta = 1\) . The two theorems do not hold in this situation, since they require that the true hypothesis (in this case \(\mathcal {H}_0\) ) must assign a non-zero prior mass to all events that are considered plausible under the false hypothesis. In other words, both hypotheses must assign non-zero mass to the same sample space.

A formal approach for checking the Bayes factor calculation

In their method for checking the calculation of the Bayes factor, Schad et al. ( 2022 ) recommend simulating multiple data sets from statistical models (with predefined prior model probabilities) and then obtaining Bayes factors and posterior model probabilities using the same method that is to be used to calculate the Bayes factor(s) on the empirical data. This method represents a structured approach based on simulation-based calibration (Geweke, 2004 ; Cook et al., 2006 ). The idea is based on the fact that the expected posterior model probability should equal the prior model probability (see e.g., Skyrms, 1997 ; Goldstein, 1983 ; Huttegger, 2017 ). Therefore, if the average posterior model probability across the simulated data sets is equal to the prior model probability, then the calculation of the Bayes factor (and the posterior model probability) should be considered accurate.

For the correctly specified calculation the average \(\text {BF}_{01}\) rapidly converges to 1, whereas for the misspecified calculation, it does not. The figure depicts average \(\text {BF}_{01}\) calculated for the data generated under \(\mathcal {H}_1\) as a function of the number of synthetic data sets m . The Bayes factor is calculated using two different values for the scale of the scaled inverse chi-squared distribution

In this paper, we follow the approach by Schad et al. ( 2022 ) and propose a new method for checking the Bayes factor, based on Turing and Good’s theorems described in the previous sections. The check (steps 1-4) assumes that if the calculation of the Bayes factor is executed correctly and if all the assumptions are met, then its expected value in favor of the wrong hypothesis should be (approximately) equal to 1. Additionally, it is possible to extend this check by comparing higher-order moments (steps 5-6). After collecting the data and selecting the appropriate analysis, the proposed methodology can be summarized as follows:

Specify two rival models; since the prior can be seen as an integral part of the model (e.g., Vanpaemel, 2010 ; Vanpaemel and Lee, 2012 ), this step includes the assignment of prior distributions to the model parameters.

Calculate the Bayes factor based on the observed data using the computational methodology of interest.

Select one of the models to generate simulated data from – we strongly recommend this to be the more complex model; in nested models, one should therefore simulate from the alternative hypothesis and not from the null hypothesis.

Sample data from the prior predictive distribution. This could, for example, be done by selecting a parameter (vector) from the joint prior distribution and use this to generate a synthetic data set of the same length as the observed data (although it could be any length in principle).

Compute the Bayes factor in favor of the false hypothesis over the true hypothesis for the synthetic data set, using the same computational technique used for the observed data (step 2).

Repeat steps b-c m times, yielding m Bayes factors in favor of the false hypothesis.

Calculate the average Bayes factor in favor of the false hypothesis across the m Bayes factors obtained in the previous step. If this mean value is close to 1 for a sufficiently large number of simulations m , this provides strong evidence that the Bayes factor calculation has been executed correctly. Then one can confidently report the value obtained in step 2.

Additionally, simulate data as described in step 3, but this time set the other hypothesis under consideration (e.g., \(\mathcal {H}_0\) ) to true. Calculate the Bayes factor in favor of the true hypothesis. Repeat this step m times and calculate the average Bayes factor in favor of the true hypothesis.

Compare the mean Bayes factor from step 5 with the second moment of the Bayes factor in favor of the wrong hypothesis based on the data generated in step 3. If these two values are approximately equal, this provides additional evidence that the Bayes factor calculation was performed correctly.

This step-by-step approach helps validate the Bayes factor calculations and ensures that the results obtained are reliable. More specifically, if the Bayes factor calculation is done correctly, we should be confident that there were no issues with the calculation of the Bayes factor. In the following two subsections, we illustrate these steps with two concrete examples.

Note that the purpose of the following examples-one using a simple Bayes factor for an intervention effect in an ANOVA design, and another using a transdimensional Bayes factor for the inclusion of an edge in a graphical model-is to demonstrate how to perform the proposed check. A comprehensive review of the performance of various software packages in calculating Bayes factors is beyond the scope of this paper.

Example 1: A Bayes factor test for an intervention effect in one-way ANOVA

Consider a one-way ANOVA model where the standard alternative hypothesis ( \(\mathcal {H}_1\) ), which states that not all means between the 3 groups are equal, is tested against the null hypothesis ( \(\mathcal {H}_0\) ), which states that the means are equal. The model can be expressed as

where \(y_i\) is the value of the dependent variable for participant i , \(\alpha \) is the intercept, \(x_i\) is the factor variable denoting the group membership, \(\beta \) is the parameter representing the effect of the experimental manipulation, and \(\epsilon _i\) is the residual term normally distributed around 0 with variance \(\sigma ^2\) . To calculate the Bayes factor on the empirical data one can use the default settings in the R package BayesFactor (Morey & Rouder, 2022 ). The function anovaBF assigns Jeffreys priors to the intercept and residual variance, and a normal prior to the main effect

where g is given an independent scaled inverse-chi-squared hyperprior with 1 degree of freedom. The interested reader is referred to Rouder et al. ( 2012 ) for the details of the prior specifications. We now illustrate how the check can be performed for the current example.

Suppose we have collected data from 150 participants (50 participants in each of the 3 groups) and we wish to test \(\mathcal {H}_1\) versus \(\mathcal {H}_0\) . We simulate \(m = 200{,}000\) data sets under \(\mathcal {H}_1\) by sampling the parameter \(\beta \) from its prior distribution, employing the same default specification as used in the package (i.e., applying a scaled inverse-chi-squared hyperprior for g with a scale of 1/2 and Jeffreys priors on \(\alpha \) and \(\sigma ^2\) with a value of \(\sigma ^2 = 0.5\) ). Additionally, we generate m datasets under \(\mathcal {H}_0\) by setting \(\beta = 0\) . In both cases, we calculate the Bayes factors using the default settings as described above. To illustrate what happens when the Bayes factor calculation is misspecified, we re-calculate the Bayes factor for the data generated under \(\mathcal {H}_1\) by altering the default value for the scale of the inverse chi-squared distribution. Specifically, we change the scale from medium to ultrawide , corresponding to values of 1/2 and 1, respectively. For the Bayes factors calculated on the data sets where \(\mathcal {H}_1\) is true, approximately 0.28% of the Bayes factors calculations failed due to computational difficulties.

Figure 5 depicts the cumulative mean for \(\text {BF}_{01}\) when \(\mathcal {H}_1\) is true. Notably, for the Bayes factors calculated using the default settings of the package, which precisely mirror how the data was generated, the average \(\text {BF}_{01}\) rapidly converges to 1. However, when there is a discrepancy between the data and the Bayes factor calculation, which for the purpose of this example was achieved by altering the scale of the inverse chi-squared hyperprior from 1/2 to 1, we notice that the average Bayes factor deviates significantly from 1. It eventually stabilizes at a value of approximately 3.16, illustrating the sensitivity of the Bayes factor when its calculation is misspecified.

For the second set of synthetic data generated under \(\mathcal {H}_0\) , we calculate the average \(\text {BF}_{01}\) , which yields a value of 8.18, which we can compare with the second raw moment of \(\text {BF}_{01}\) from the data sets where the alternative hypothesis is true, which yields a value of 8.15. This result provides additional proof that the calculation of the Bayes factor was done correctly.

Example 2: A Bayes factor test for conditional independence in a Markov random field model

Network psychometrics is a relatively new subdiscipline in which psychological constructs (e.g., intelligence, mental disorders) are conceptualized as complex systems of behavioral and cognitive factors (Marsman & Rhemtulla, 2022 ; Borsboom & Cramer, 2013 ). Psychometric network analysis is then used to infer the structure of such systems from multivariate psychological data (Borsboom et al., 2021 ). These analyses use graphical models known as Markov Random Fields (MRFs, Kindermann and Snell, 1980 ; Rozanov, 1982 ) in which psychological variables assume the role of the network nodes. The edges of the network express the direct influence of one variable on another given the remaining network variables, that is, that they are conditionally dependent , and the absence of an edge implies that the two variables are conditionally independent (Lauritzen, 2004 ). The Bayesian approach to analyzing these graphical models (Mohammadi & Wit, 2015 ; Marsman et al., 2015 ; Marsman, 2022 ; Marsman et al., 2023 ; Williams, 2021 ; Williams & Mulder, 2020 ) allows researchers to quantify the evidence in the data for the presence or absence of edges, and thus to formally test for conditional (in)dependence (see Sekulovski et al., 2024 , for an overview of three Bayesian methods for testing conditional independence).

Sekulovski et al. ( 2024 ) discuss two types of Bayes factor tests for conditional independence. In one test, the predictive success of a particular network structure with the relationship of interest is compared against the same network structure with the relationship of interest removed. One problem with testing for conditional independence in this way is that even for relatively small networks, there are many possible structures to consider, and as Sekulovski et al. ( 2024 ) have shown, Bayes factor tests for conditional independence can be highly sensitive to the choice of that network structure. In the second Bayes factor test, we use Bayesian model averaging (BMA, Hoeting et al., 1999 ; Hinne et al., 2020 ) and contrast the predictive success of all structures with the relationship of interest against the predictive success of all structures without that relationship. This is known as the inclusion Bayes factor (Marsman, 2022 ; Marsman et al., 2023 ). Sekulovski et al. ( 2024 ) showed that the inclusion Bayes factor is robust to variations in the structures underlying the rest of the network. However, the BMA methods for psychometric network analysis required to estimate the inclusion Bayes factor are much more complex and thus more prone to the computational problems identified above. For an accessible introduction to BMA with a specific example on network models, see Hinne et al. ( 2020 ) and for an accessible introduction to BMA analysis of psychometric network models, see Huth et al. ( 2023 ) and Sekulovski et al. ( 2024 ).

The average \(\text {BF}_{01}\) converges to 1. The figure depicts the average inclusion \(\text {BF}_{01}\) calculated for the data generated under \(\mathcal {H}_1\) as a function of the number of synthetic data sets m

In this paper, we scrutinize the Bayesian edge selection method developed by Marsman et al. ( 2023 ) for analyzing MRF models for binary and ordinal data, and which can be used to estimate the inclusion Bayes factor. This method, implemented in the R package bgms (Marsman et al., 2023 ), stipulates a discrete spike and slab prior distribution on the edge weights of the MRF, and models the inclusion and exclusion of pairwise relations in the model with an edge indicator ( \(\gamma \) ), which when present designates the corresponding edge weight a diffuse prior and when absent sets it to 0. That is, for a single edge weight \(\theta _{ij}\) , between variables i and j , the prior distribution can be expressed as

The transdimensional Markov chain Monte Carlo method proposed by Gottardo and Raftery ( 2008 ) is used to simulate from the multivariate posterior distribution of the MRF’s parameters and edge indicators. The output of this approach can be used to compute the inclusion Bayes factor which is defined as

Since the inclusion Bayes factor is an extension of the classical Bayes factor presented in Eq.  1 and involves a much more complex calculation, we wish to verify that its computation is performed correctly using the newly proposed methodology. Therefore, we simulated \(m = 30{,}000\) datasets with \(p = 5\) binary variables and \(N = 500\) observations each. We focus on testing whether the first two variables are conditionally independent, that is, we compare \(\mathcal {H}_0\text {: } \theta _{12} = 0\) with \(\mathcal {H}_1\text {: } \theta _{12} \ne 0\) . For the case where \(\mathcal {H}_1\) is true, we simulated data where all ten possible edges have an edge weight value of \(\theta _{ij} = 0.5\) . Additionally, for the case where \(\mathcal {H}_0\) is true, we simulated a second set of data by setting the focal edge weight parameter \(\theta _{12}\) to 0 and leaving the values of the nine remaining edge weights unchanged. We estimated the graphical model for each simulated data set using the R package bgms . We used a unit information prior for \(f_{slab}\) ; a Dirac measure at 0 for \(f_{spike}\) , and an independent Bernoulli distribution for each \(\gamma _{ij}\) with a prior inclusion probability of 1/2 (see Sekulovski et al., 2024 , for a detailed analysis of the prior distributions for these models). Under this prior specification, the prior inclusion odds are equal to 1. In cases where the posterior inclusion probability was equal to 1, we obtained undefined values for the inclusion Bayes factor (i.e., 1/0). For the data sets where \(\mathcal {H}_1\) was true, there were 9,345 Bayes factors with undefined values (31%), and for the data sets where \(\mathcal {H}_0\) was true, there were 53 undefined values (0.2%). To work around this problem, we set all undefined values to 1 + the highest observed finite value of the inclusion Bayes factor.

Figure 6 shows the cumulative mean of the inclusion \(\text {BF}_{01}\) when \(\mathcal {H}_1\) is true (i.e., there is an edge between variables 1 and 2). As the number of simulations increases, the mean inclusion \(\text {BF}_{01}\) stabilizes around 1 (1.01 at the last iteration), indicating that the inclusion Bayes factor obtained with this approach was computed correctly. In addition, we computed the mean \(\text {BF}_{01}\) when \(\mathcal {H}_0\) is true, which was 11.5, and compared it to the second moment of \(\text {BF}_{01}\) when \(\mathcal {H}_1\) is true, which was 9.96. These values are not equal. However, we suspect that the reason for this is twofold: first, the sample size N in each of the simulated data sets was quite large, and second, since the calculation of this Bayes factor is more involved, it probably takes many more iterations m to be sure that the moments are equal. Estimating these models takes much more time than estimating other more standard statistical models, so it was not computationally feasible to do more than \(m = 30{,}000\) repetitions under each of the hypotheses. In addition, we must consider the sampling variability of the simulated data sets. In other words, due to variability, not all of the simulated data sets will show support for the hypothesis under which they were simulated, further reducing the number of “effective” data sets. These reasons also justify the choice to recode the undefined inclusion Bayes factor values as we did, rather than omitting them altogether.

This paper presents a structured approach to checking the accuracy of Bayes factor calculations based on the theorems of Turing and Good. The approach provides researchers with a general and practical method for confirming that their Bayes factor results are reliable. Application to two concrete examples demonstrated the effectiveness of this approach in verifying the correctness of Bayes factor calculations. In particular, if the method of calculating the Bayes factor is consistent with the data generation process, the mean Bayes factor in favor of the false hypothesis converges to approximately 1, in accordance with the first theorem. Furthermore, comparing the first and second moments of the Bayes factors under different hypotheses provides additional evidence for correct calculations. However, as we have seen in the second example when dealing with more complex models, the second theorem requires many more iterations. Due to the variability of the second moment, one can only be sure that the second theorem approximately holds for a finite number of simulations. Therefore, we recommend that researchers focus primarily on the first theorem and perform the additional check based on the second theorem whenever practically possible. This would also make the check less computationally expensive since it would only require simulating data under one of the hypotheses.

Finally, we have demonstrated that for practical applications of the first theorem, it is best to simulate under the more general hypothesis and take the average Bayes factor in favor of the more specific hypothesis. For the second theorem, the optimal approach can be summarized as follows. First, compute the mean Bayes factor in favor of the more specific hypothesis for data where that hypothesis is true. Second, compare this to the second raw moment in favor of the more specific hypothesis computed on data simulated under the more general hypothesis.

Limitations & Possible extensions

While the proposed approach provides a practical way to validate Bayes factor calculations, it is not without limitations. In cases with large sample sizes, or when dealing with highly complex models, the convergence of the values for the higher-order moments may require a significant number of iterations. In such cases, as we have seen, the second moments may not match very closely. In situations where Bayes factors are used for comparing highly complex models, different methods of checking their calculation might be more appropriate, such as the method proposed by Schad et al. ( 2022 ).

However, for certain Bayes factors, particularly those based on Bayesian model averaging (BMA), such as the inclusion Bayes factor for including an edge in a graphical model or a predictor in linear regression, the method proposed in this paper can be straightforwardly applied to verify these calculations. This is because the other two methods are more suitable for checking classical (i.e., non-BMA) Bayes factors, which compare two competing statistical models (see, Sekulovski et al., 2024 , for a discussion of the difference between these two Bayes factors)

One of the reviewers of the paper suggested that the check proposed in this paper could be incorporated as an additional step within the approach proposed by Schad et al. ( 2022 ). This would mean that at the start of the simulation exercise, we would have to (a) assign prior probabilities to two competing models and then randomly select one of those models, (b) simulate synthetic data under the sampled model, (c) compute the Bayes factor and the posterior model probability, and then repeat these steps m times. Then, step 4 would be split into two, where we filter out the data sets generated by only one of the models, and filter out the associated Bayes factors. For each resulting set of Bayes factors, we would compute the mean in favor of the false hypothesis, where we expect both means to be approximately equal to one.

Providing a structured and systematic way to evaluate Bayes factor calculations helps to increase the credibility and rigor of Bayesian hypothesis testing in applied research. The proposed methods serve as a valuable tool for researchers working with Bayes factors, providing a means to validate their results and ensure the robustness of their statistical inferences. We encourage researchers to consider this approach when using Bayes factors in their analyses, thereby fostering greater confidence in the validity of their conclusions.

Code Availability

The data and materials for all simulation examples are available at the OSF repository https://osf.io/438vy/ .

Availability of data and materials

Not applicable.

We use E and ‘data’ interchangeably.

A martingale is a sequence of random variables where the conditional expected value of the next value, given all prior values, equals the current value. For instance, consider a coin flip game where a player starts with 100 euros, winning one euro for heads and losing one euro for tails; in this scenario, the expected amount of money after each flip remains equal to the player’s current amount. Consequently, regardless of the number of flips, the expected future value is always equal to the present value, exemplifying the martingale property.

It should be noted that the marginal Bayes factor (i.e., a Bayes factor not conditioned on the false hypothesis) is not a martingale.

Note that the specification of \(\mathcal {H}_0\) and \(\mathcal {H}_1\) is the same as in the binomial example from the previous section with \(n = 2\) .

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Acknowledgements

The authors would like to thank Wolf Vanpaemel for providing the idea and example script for merging the Good check with the check proposed by Schad et al. ( 2022 ), as well as two other reviewers and the Associate Editor for their comments on earlier versions of the manuscript.

NS and MM were supported by the European Union (ERC, BAYESIAN P-NETS, #101040876). Views and opinions expressed are, however, those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

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Sekulovski, N., Marsman, M. & Wagenmakers, EJ. A Good check on the Bayes factor. Behav Res (2024). https://doi.org/10.3758/s13428-024-02491-4

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DOI : https://doi.org/10.3758/s13428-024-02491-4

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