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• Null and Alternative Hypotheses | Definitions & Examples

Null & Alternative Hypotheses | Definitions, Templates & Examples

Published on May 6, 2022 by Shaun Turney . Revised on June 22, 2023.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

• Null hypothesis ( H 0 ): There’s no effect in the population .
• Alternative hypothesis ( H a or H 1 ) : There’s an effect in the population.

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, similarities and differences between null and alternative hypotheses, how to write null and alternative hypotheses, other interesting articles, frequently asked questions.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”:

• The null hypothesis ( H 0 ) answers “No, there’s no effect in the population.”
• The alternative hypothesis ( H a ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample. It’s critical for your research to write strong hypotheses .

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

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The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept . Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect,” “no difference,” or “no relationship.” When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

You can never know with complete certainty whether there is an effect in the population. Some percentage of the time, your inference about the population will be incorrect. When you incorrectly reject the null hypothesis, it’s called a type I error . When you incorrectly fail to reject it, it’s a type II error.

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis ( H a ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect,” “a difference,” or “a relationship.” When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes < or >). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

• They’re both answers to the research question.
• They both make claims about the population.
• They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

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To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

General template sentences

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

• Null hypothesis ( H 0 ): Independent variable does not affect dependent variable.
• Alternative hypothesis ( H a ): Independent variable affects dependent variable.

Test-specific template sentences

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

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

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

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.

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.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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Module 9: Hypothesis Testing With One Sample

Null and alternative hypotheses, learning outcomes.

• Describe hypothesis testing in general and in practice

The actual test begins by considering two  hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 : The null hypothesis: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

H a : The alternative hypothesis : It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make adecision. There are two options for a  decision . They are “reject H 0 ” if the sample information favors the alternative hypothesis or “do not reject H 0 ” or “decline to reject H 0 ” if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in  H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ 30

H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

H 0 : The drug reduces cholesterol by 25%. p = 0.25

H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:

H 0 : μ = 2.0

H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 66 H a : μ __ 66

• H 0 : μ = 66
• H a : μ ≠ 66

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

H 0 : μ ≥ 5

H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 45 H a : μ __ 45

• H 0 : μ ≥ 45
• H a : μ < 45

In an issue of U.S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.

H 0 : p ≤ 0.066

H a : p > 0.066

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : p __ 0.40 H a : p __ 0.40

• H 0 : p = 0.40
• H a : p > 0.40

Concept Review

In a  hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis , typically denoted with H 0 . The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥) Always write the alternative hypothesis , typically denoted with H a or H 1 , using less than, greater than, or not equals symbols, i.e., (≠, >, or <). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

H 0 and H a are contradictory.

• OpenStax, Statistics, Null and Alternative Hypotheses. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:58/Introductory_Statistics . License : CC BY: Attribution
• Introductory Statistics . Authored by : Barbara Illowski, Susan Dean. Provided by : Open Stax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
• Simple hypothesis testing | Probability and Statistics | Khan Academy. Authored by : Khan Academy. Located at : https://youtu.be/5D1gV37bKXY . License : All Rights Reserved . License Terms : Standard YouTube License

Alternative hypothesis

by Marco Taboga , PhD

In a statistical test, observed data is used to decide whether or not to reject a restriction on the data-generating probability distribution.

The assumption that the restriction is true is called null hypothesis , while the statement that the restriction is not true is called alternative hypothesis.

A correct specification of the alternative hypothesis is essential to decide between one-tailed and two-tailed tests.

Table of contents

Mathematical setting

Choice between one-tailed and two-tailed tests, the critical region, the interpretation of the rejection, the interpretation must be coherent with the alternative hypothesis.

• Power function

Accepting the alternative

More details, keep reading the glossary.

In order to fully understand the concept of alternative hypothesis, we need to remember the essential elements of a statistical inference problem:

we observe a sample drawn from an unknown probability distribution;

in principle, any valid probability distribution could have generated the sample;

however, we usually place some a priori restrictions on the set of possible data-generating distributions;

A couple of simple examples follow.

When we conduct a statistical test, we formulate a null hypothesis as a restriction on the statistical model.

The alternative hypothesis is

The alternative hypothesis is used to decide whether a test should be one-tailed or two-tailed.

The null hypothesis is rejected if the test statistic falls within a critical region that has been chosen by the statistician.

The critical region is a set of values that may comprise:

only the left tail of the distribution or only the right tail (one-tailed test);

both the left and the right tail (two-tailed test).

The choice of the critical region depends on the alternative hypothesis. Let us see why.

The interpretation is different depending on the tail of the distribution in which the test statistic falls.

The choice between a one-tailed or a two-tailed test needs to be done in such a way that the interpretation of a rejection is always coherent with the alternative hypothesis.

When we deal with the power function of a test, the term "alternative hypothesis" has a special meaning.

We conclude with a caveat about the interpretation of the outcome of a test of hypothesis.

The interpretation of a rejection of the null is controversial.

According to some statisticians, rejecting the null is equivalent to accepting the alternative.

However, others deem that rejecting the null does not necessarily imply accepting the alternative. In fact, it is possible to think of situations in which both hypotheses can be rejected. Let us see why.

According to the conceptual framework illustrated by the images above, there are three possibilities:

the null is true;

the alternative is true;

neither the null nor the alternative is true because the true data-generating distribution has been excluded from the statistical model (we say that the model is mis-specified).

If we are in case 3, accepting the alternative after a rejection of the null is an incorrect decision. Moreover, a second test in which the alternative becomes the new null may lead us to another rejection.

You can find more details about the alternative hypothesis in the lecture on Hypothesis testing .

Previous entry: Almost sure

Next entry: Binomial coefficient

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Please cite as:

Taboga, Marco (2021). "Alternative hypothesis", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/alternative-hypothesis.

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Statistics By Jim

Making statistics intuitive

Alternative hypothesis

By Jim Frost

The alternative hypothesis is one of two mutually exclusive hypotheses in a hypothesis test. The alternative hypothesis states that a population parameter does not equal a specified value. Typically, this value is the null hypothesis value associated with no effect , such as zero. If your sample contains sufficient evidence, you can reject the null hypothesis and favor the alternative hypothesis. The alternative hypothesis is often denoted as H 1 or H A .

If you are performing a two-tailed hypothesis test, the alternative hypothesis states that the population parameter does not equal the null hypothesis value. For example, when the alternative hypothesis is H A : μ ≠ 0, the test can detect differences both greater than and less than the null value.

A one-tailed alternative hypothesis can test for a difference only in one direction. For example, H A : μ > 0 can only test for differences that are greater than zero.

• How Hypothesis Tests Work: Significance Levels (Alpha) and P values
• How to Identify the Distribution of Your Data
• When Can I Use One-Tailed Hypothesis Tests?
• Examples of Hypothesis Tests: Busting Myths about the Battle of the Sexes
• Failing to Reject the Null Hypothesis

• Math Article

Alternative Hypothesis

Alternative hypothesis defines there is a statistically important relationship between two variables. Whereas null hypothesis states there is no statistical relationship between the two variables. In statistics, we usually come across various kinds of hypotheses. A statistical hypothesis is supposed to be a working statement which is assumed to be logical with given data. It should be noticed that a hypothesis is neither considered true nor false.

The alternative hypothesis is a statement used in statistical inference experiment. It is contradictory to the null hypothesis and denoted by H a or H 1 . We can also say that it is simply an alternative to the null. In hypothesis testing, an alternative theory is a statement which a researcher is testing. This statement is true from the researcher’s point of view and ultimately proves to reject the null to replace it with an alternative assumption. In this hypothesis, the difference between two or more variables is predicted by the researchers, such that the pattern of data observed in the test is not due to chance.

To check the water quality of a river for one year, the researchers are doing the observation. As per the null hypothesis, there is no change in water quality in the first half of the year as compared to the second half. But in the alternative hypothesis, the quality of water is poor in the second half when observed.

Difference Between Null and Alternative Hypothesis

Basically, there are three types of the alternative hypothesis, they are;

Left-Tailed : Here, it is expected that the sample proportion (π) is less than a specified value which is denoted by π 0 , such that;

H 1 : π < π 0

Right-Tailed: It represents that the sample proportion (π) is greater than some value, denoted by π 0 .

H 1 : π > π 0

Two-Tailed: According to this hypothesis, the sample proportion (denoted by π) is not equal to a specific value which is represented by π 0 .

H 1 : π ≠ π 0

Note: The null hypothesis for all the three alternative hypotheses, would be H 1 : π = π 0 .

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Once you have developed a clear and focused research question or set of research questions, you’ll be ready to conduct further research, a literature review, on the topic to help you make an educated guess about the answer to your question(s). This educated guess is called a hypothesis.

In research, there are two types of hypotheses: null and alternative. They work as a complementary pair, each stating that the other is wrong.

• Null Hypothesis (H 0 ) – This can be thought of as the implied hypothesis. “Null” meaning “nothing.”  This hypothesis states that there is no difference between groups or no relationship between variables. The null hypothesis is a presumption of status quo or no change.
• Alternative Hypothesis (H a ) – This is also known as the claim. This hypothesis should state what you expect the data to show, based on your research on the topic. This is your answer to your research question.

Null Hypothesis:   H 0 : There is no difference in the salary of factory workers based on gender. Alternative Hypothesis :  H a : Male factory workers have a higher salary than female factory workers.

Null Hypothesis :  H 0 : There is no relationship between height and shoe size. Alternative Hypothesis :  H a : There is a positive relationship between height and shoe size.

Null Hypothesis :  H 0 : Experience on the job has no impact on the quality of a brick mason’s work. Alternative Hypothesis :  H a : The quality of a brick mason’s work is influenced by on-the-job experience.

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13 Different Types of Hypothesis

Chris Drew (PhD)

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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There are 13 different types of hypothesis. These include simple, complex, null, alternative, composite, directional, non-directional, logical, empirical, statistical, associative, exact, and inexact.

A hypothesis can be categorized into one or more of these types. However, some are mutually exclusive and opposites. Simple and complex hypotheses are mutually exclusive, as are direction and non-direction, and null and alternative hypotheses.

Below I explain each hypothesis in simple terms for absolute beginners. These definitions may be too simple for some, but they’re designed to be clear introductions to the terms to help people wrap their heads around the concepts early on in their education about research methods .

Types of Hypothesis

Before you Proceed: Dependent vs Independent Variables

A research study and its hypotheses generally examine the relationships between independent and dependent variables – so you need to know these two concepts:

• The independent variable is the variable that is causing a change.
• The dependent variable is the variable the is affected by the change. This is the variable being tested.

Read my full article on dependent vs independent variables for more examples.

Example: Eating carrots (independent variable) improves eyesight (dependent variable).

1. Simple Hypothesis

A simple hypothesis is a hypothesis that predicts a correlation between two test variables: an independent and a dependent variable.

This is the easiest and most straightforward type of hypothesis. You simply need to state an expected correlation between the dependant variable and the independent variable.

You do not need to predict causation (see: directional hypothesis). All you would need to do is prove that the two variables are linked.

Simple Hypothesis Examples

2. Complex Hypothesis

A complex hypothesis is a hypothesis that contains multiple variables, making the hypothesis more specific but also harder to prove.

You can have multiple independent and dependant variables in this hypothesis.

Complex Hypothesis Example

In the above example, we have multiple independent and dependent variables:

• Independent variables: Age and weight.
• Dependent variables: diabetes and heart disease.

Because there are multiple variables, this study is a lot more complex than a simple hypothesis. It quickly gets much more difficult to prove these hypotheses. This is why undergraduate and first-time researchers are usually encouraged to use simple hypotheses.

3. Null Hypothesis

A null hypothesis will predict that there will be no significant relationship between the two test variables.

For example, you can say that “The study will show that there is no correlation between marriage and happiness.”

A good way to think about a null hypothesis is to think of it in the same way as “innocent until proven guilty”[1]. Unless you can come up with evidence otherwise, your null hypothesis will stand.

A null hypothesis may also highlight that a correlation will be inconclusive . This means that you can predict that the study will not be able to confirm your results one way or the other. For example, you can say “It is predicted that the study will be unable to confirm a correlation between the two variables due to foreseeable interference by a third variable .”

Beware that an inconclusive null hypothesis may be questioned by your teacher. Why would you conduct a test that you predict will not provide a clear result? Perhaps you should take a closer look at your methodology and re-examine it. Nevertheless, inconclusive null hypotheses can sometimes have merit.

Null Hypothesis Examples

4. Alternative Hypothesis

An alternative hypothesis is a hypothesis that is anything other than the null hypothesis. It will disprove the null hypothesis.

We use the symbol H A or H 1 to denote an alternative hypothesis.

The null and alternative hypotheses are usually used together. We will say the null hypothesis is the case where a relationship between two variables is non-existent. The alternative hypothesis is the case where there is a relationship between those two variables.

The following statement is always true: H 0 ≠ H A .

Let’s take the example of the hypothesis: “Does eating oatmeal before an exam impact test scores?”

We can have two hypotheses here:

• Null hypothesis (H 0 ): “Eating oatmeal before an exam does not impact test scores.”
• Alternative hypothesis (H A ): “Eating oatmeal before an exam does impact test scores.”

For the alternative hypothesis to be true, all we have to do is disprove the null hypothesis for the alternative hypothesis to be true. We do not need an exact prediction of how much oatmeal will impact the test scores or even if the impact is positive or negative. So long as the null hypothesis is proven to be false, then the alternative hypothesis is proven to be true.

5. Composite Hypothesis

A composite hypothesis is a hypothesis that does not predict the exact parameters, distribution, or range of the dependent variable.

Often, we would predict an exact outcome. For example: “23 year old men are on average 189cm tall.” Here, we are giving an exact parameter. So, the hypothesis is not composite.

But, often, we cannot exactly hypothesize something. We assume that something will happen, but we’re not exactly sure what. In these cases, we might say: “23 year old men are not on average 189cm tall.”

We haven’t set a distribution range or exact parameters of the average height of 23 year old men. So, we’ve introduced a composite hypothesis as opposed to an exact hypothesis.

Generally, an alternative hypothesis (discussed above) is composite because it is defined as anything except the null hypothesis. This ‘anything except’ does not define parameters or distribution, and therefore it’s an example of a composite hypothesis.

6. Directional Hypothesis

A directional hypothesis makes a prediction about the positivity or negativity of the effect of an intervention prior to the test being conducted.

Instead of being agnostic about whether the effect will be positive or negative, it nominates the effect’s directionality.

We often call this a one-tailed hypothesis (in contrast to a two-tailed or non-directional hypothesis) because, looking at a distribution graph, we’re hypothesizing that the results will lean toward one particular tail on the graph – either the positive or negative.

Directional Hypothesis Examples

7. Non-Directional Hypothesis

A non-directional hypothesis does not specify the predicted direction (e.g. positivity or negativity) of the effect of the independent variable on the dependent variable.

These hypotheses predict an effect, but stop short of saying what that effect will be.

A non-directional hypothesis is similar to composite and alternative hypotheses. All three types of hypothesis tend to make predictions without defining a direction. In a composite hypothesis, a specific prediction is not made (although a general direction may be indicated, so the overlap is not complete). For an alternative hypothesis, you often predict that the even will be anything but the null hypothesis, which means it could be more or less than H 0 (or in other words, non-directional).

Let’s turn the above directional hypotheses into non-directional hypotheses.

Non-Directional Hypothesis Examples

8. Logical Hypothesis

A logical hypothesis is a hypothesis that cannot be tested, but has some logical basis underpinning our assumptions.

These are most commonly used in philosophy because philosophical questions are often untestable and therefore we must rely on our logic to formulate logical theories.

Usually, we would want to turn a logical hypothesis into an empirical one through testing if we got the chance. Unfortunately, we don’t always have this opportunity because the test is too complex, expensive, or simply unrealistic.

Here are some examples:

• Before the 1980s, it was hypothesized that the Titanic came to its resting place at 41° N and 49° W, based on the time the ship sank and the ship’s presumed path across the Atlantic Ocean. However, due to the depth of the ocean, it was impossible to test. Thus, the hypothesis was simply a logical hypothesis.
• Dinosaurs closely related to Aligators probably had green scales because Aligators have green scales. However, as they are all extinct, we can only rely on logic and not empirical data.

9. Empirical Hypothesis

An empirical hypothesis is the opposite of a logical hypothesis. It is a hypothesis that is currently being tested using scientific analysis. We can also call this a ‘working hypothesis’.

We can to separate research into two types: theoretical and empirical. Theoretical research relies on logic and thought experiments. Empirical research relies on tests that can be verified by observation and measurement.

So, an empirical hypothesis is a hypothesis that can and will be tested.

• Raising the wage of restaurant servers increases staff retention.
• Adding 1 lb of corn per day to cows’ diets decreases their lifespan.
• Mushrooms grow faster at 22 degrees Celsius than 27 degrees Celsius.

Each of the above hypotheses can be tested, making them empirical rather than just logical (aka theoretical).

10. Statistical Hypothesis

A statistical hypothesis utilizes representative statistical models to draw conclusions about broader populations.

It requires the use of datasets or carefully selected representative samples so that statistical inference can be drawn across a larger dataset.

This type of research is necessary when it is impossible to assess every single possible case. Imagine, for example, if you wanted to determine if men are taller than women. You would be unable to measure the height of every man and woman on the planet. But, by conducting sufficient random samples, you would be able to predict with high probability that the results of your study would remain stable across the whole population.

You would be right in guessing that almost all quantitative research studies conducted in academic settings today involve statistical hypotheses.

Statistical Hypothesis Examples

• Human Sex Ratio. The most famous statistical hypothesis example is that of John Arbuthnot’s sex at birth case study in 1710. Arbuthnot used birth data to determine with high statistical probability that there are more male births than female births. He called this divine providence, and to this day, his findings remain true: more men are born than women.
• Lady Testing Tea. A 1935 study by Ronald Fisher involved testing a woman who believed she could tell whether milk was added before or after water to a cup of tea. Fisher gave her 4 cups in which one randomly had milk placed before the tea. He repeated the test 8 times. The lady was correct each time. Fisher found that she had a 1 in 70 chance of getting all 8 test correct, which is a statistically significant result.

11. Associative Hypothesis

An associative hypothesis predicts that two variables are linked but does not explore whether one variable directly impacts upon the other variable.

We commonly refer to this as “ correlation does not mean causation ”. Just because there are a lot of sick people in a hospital, it doesn’t mean that the hospital made the people sick. There is something going on there that’s causing the issue (sick people are flocking to the hospital).

So, in an associative hypothesis, you note correlation between an independent and dependent variable but do not make a prediction about how the two interact. You stop short of saying one thing causes another thing.

Associative Hypothesis Examples

• Sick people in hospital. You could conduct a study hypothesizing that hospitals have more sick people in them than other institutions in society. However, you don’t hypothesize that the hospitals caused the sickness.
• Lice make you healthy. In the Middle Ages, it was observed that sick people didn’t tend to have lice in their hair. The inaccurate conclusion was that lice was not only a sign of health, but that they made people healthy. In reality, there was an association here, but not causation. The fact was that lice were sensitive to body temperature and fled bodies that had fevers.

12. Causal Hypothesis

A causal hypothesis predicts that two variables are not only associated, but that changes in one variable will cause changes in another.

A causal hypothesis is harder to prove than an associative hypothesis because the cause needs to be definitively proven. This will often require repeating tests in controlled environments with the researchers making manipulations to the independent variable, or the use of control groups and placebo effects .

If we were to take the above example of lice in the hair of sick people, researchers would have to put lice in sick people’s hair and see if it made those people healthier. Researchers would likely observe that the lice would flee the hair, but the sickness would remain, leading to a finding of association but not causation.

Causal Hypothesis Examples

13. Exact vs. Inexact Hypothesis

For brevity’s sake, I have paired these two hypotheses into the one point. The reality is that we’ve already seen both of these types of hypotheses at play already.

An exact hypothesis (also known as a point hypothesis) specifies a specific prediction whereas an inexact hypothesis assumes a range of possible values without giving an exact outcome. As Helwig [2] argues:

“An “exact” hypothesis specifies the exact value(s) of the parameter(s) of interest, whereas an “inexact” hypothesis specifies a range of possible values for the parameter(s) of interest.”

Generally, a null hypothesis is an exact hypothesis whereas alternative, composite, directional, and non-directional hypotheses are all inexact.

See Next: 15 Hypothesis Examples

This is introductory information that is basic and indeed quite simplified for absolute beginners. It’s worth doing further independent research to get deeper knowledge of research methods and how to conduct an effective research study. And if you’re in education studies, don’t miss out on my list of the best education studies dissertation ideas .

[1] https://jnnp.bmj.com/content/91/6/571.abstract

[2] http://users.stat.umn.edu/~helwig/notes/SignificanceTesting.pdf

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2 thoughts on “13 Different Types of Hypothesis”

Wow! This introductionary materials are very helpful. I teach the begginers in research for the first time in my career. The given tips and materials are very helpful. Chris, thank you so much! Excellent materials!

You’re more than welcome! If you want a pdf version of this article to provide for your students to use as a weekly reading on in-class discussion prompt for seminars, just drop me an email in the Contact form and I’ll get one sent out to you.

When I’ve taught this seminar, I’ve put my students into groups, cut these definitions into strips, and handed them out to the groups. Then I get them to try to come up with hypotheses that fit into each ‘type’. You can either just rotate hypothesis types so they get a chance at creating a hypothesis of each type, or get them to “teach” their hypothesis type and examples to the class at the end of the seminar.

Cheers, Chris

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10.1 - setting the hypotheses: examples.

A significance test examines whether the null hypothesis provides a plausible explanation of the data. The null hypothesis itself does not involve the data. It is a statement about a parameter (a numerical characteristic of the population). These population values might be proportions or means or differences between means or proportions or correlations or odds ratios or any other numerical summary of the population. The alternative hypothesis is typically the research hypothesis of interest. Here are some examples.

Example 10.2: Hypotheses with One Sample of One Categorical Variable Section  

About 10% of the human population is left-handed. Suppose a researcher at Penn State speculates that students in the College of Arts and Architecture are more likely to be left-handed than people found in the general population. We only have one sample since we will be comparing a population proportion based on a sample value to a known population value.

• Research Question : Are artists more likely to be left-handed than people found in the general population?
• Response Variable : Classification of the student as either right-handed or left-handed

State Null and Alternative Hypotheses

• Null Hypothesis : Students in the College of Arts and Architecture are no more likely to be left-handed than people in the general population (population percent of left-handed students in the College of Art and Architecture = 10% or p = .10).
• Alternative Hypothesis : Students in the College of Arts and Architecture are more likely to be left-handed than people in the general population (population percent of left-handed students in the College of Arts and Architecture > 10% or p > .10). This is a one-sided alternative hypothesis.

Example 10.3: Hypotheses with One Sample of One Measurement Variable Section  

A generic brand of the anti-histamine Diphenhydramine markets a capsule with a 50 milligram dose. The manufacturer is worried that the machine that fills the capsules has come out of calibration and is no longer creating capsules with the appropriate dosage.

• Research Question : Does the data suggest that the population mean dosage of this brand is different than 50 mg?
• Response Variable : dosage of the active ingredient found by a chemical assay.
• Null Hypothesis : On the average, the dosage sold under this brand is 50 mg (population mean dosage = 50 mg).
• Alternative Hypothesis : On the average, the dosage sold under this brand is not 50 mg (population mean dosage ≠ 50 mg). This is a two-sided alternative hypothesis.

Example 10.4: Hypotheses with Two Samples of One Categorical Variable Section  

Many people are starting to prefer vegetarian meals on a regular basis. Specifically, a researcher believes that females are more likely than males to eat vegetarian meals on a regular basis.

• Research Question : Does the data suggest that females are more likely than males to eat vegetarian meals on a regular basis?
• Response Variable : Classification of whether or not a person eats vegetarian meals on a regular basis
• Explanatory (Grouping) Variable: Sex
• Null Hypothesis : There is no sex effect regarding those who eat vegetarian meals on a regular basis (population percent of females who eat vegetarian meals on a regular basis = population percent of males who eat vegetarian meals on a regular basis or p females = p males ).
• Alternative Hypothesis : Females are more likely than males to eat vegetarian meals on a regular basis (population percent of females who eat vegetarian meals on a regular basis > population percent of males who eat vegetarian meals on a regular basis or p females > p males ). This is a one-sided alternative hypothesis.

Example 10.5: Hypotheses with Two Samples of One Measurement Variable Section  

Obesity is a major health problem today. Research is starting to show that people may be able to lose more weight on a low carbohydrate diet than on a low fat diet.

• Research Question : Does the data suggest that, on the average, people are able to lose more weight on a low carbohydrate diet than on a low fat diet?
• Response Variable : Weight loss (pounds)
• Explanatory (Grouping) Variable : Type of diet
• Null Hypothesis : There is no difference in the mean amount of weight loss when comparing a low carbohydrate diet with a low fat diet (population mean weight loss on a low carbohydrate diet = population mean weight loss on a low fat diet).
• Alternative Hypothesis : The mean weight loss should be greater for those on a low carbohydrate diet when compared with those on a low fat diet (population mean weight loss on a low carbohydrate diet > population mean weight loss on a low fat diet). This is a one-sided alternative hypothesis.

Example 10.6: Hypotheses about the relationship between Two Categorical Variables Section  

• Research Question : Do the odds of having a stroke increase if you inhale second hand smoke ? A case-control study of non-smoking stroke patients and controls of the same age and occupation are asked if someone in their household smokes.
• Variables : There are two different categorical variables (Stroke patient vs control and whether the subject lives in the same household as a smoker). Living with a smoker (or not) is the natural explanatory variable and having a stroke (or not) is the natural response variable in this situation.
• Null Hypothesis : There is no relationship between whether or not a person has a stroke and whether or not a person lives with a smoker (odds ratio between stroke and second-hand smoke situation is = 1).
• Alternative Hypothesis : There is a relationship between whether or not a person has a stroke and whether or not a person lives with a smoker (odds ratio between stroke and second-hand smoke situation is > 1). This is a one-tailed alternative.

This research question might also be addressed like example 11.4 by making the hypotheses about comparing the proportion of stroke patients that live with smokers to the proportion of controls that live with smokers.

Example 10.7: Hypotheses about the relationship between Two Measurement Variables Section  

• Research Question : A financial analyst believes there might be a positive association between the change in a stock's price and the amount of the stock purchased by non-management employees the previous day (stock trading by management being under "insider-trading" regulatory restrictions).
• Variables : Daily price change information (the response variable) and previous day stock purchases by non-management employees (explanatory variable). These are two different measurement variables.
• Null Hypothesis : The correlation between the daily stock price change (\$) and the daily stock purchases by non-management employees (\$) = 0.
• Alternative Hypothesis : The correlation between the daily stock price change (\$) and the daily stock purchases by non-management employees (\$) > 0. This is a one-sided alternative hypothesis.

Example 10.8: Hypotheses about comparing the relationship between Two Measurement Variables in Two Samples Section  

• Research Question : Is there a linear relationship between the amount of the bill (\$) at a restaurant and the tip (\$) that was left. Is the strength of this association different for family restaurants than for fine dining restaurants?
• Variables : There are two different measurement variables. The size of the tip would depend on the size of the bill so the amount of the bill would be the explanatory variable and the size of the tip would be the response variable.
• Null Hypothesis : The correlation between the amount of the bill (\$) at a restaurant and the tip (\$) that was left is the same at family restaurants as it is at fine dining restaurants.
• Alternative Hypothesis : The correlation between the amount of the bill (\$) at a restaurant and the tip (\$) that was left is the difference at family restaurants then it is at fine dining restaurants. This is a two-sided alternative hypothesis.

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Alternative Hypothesis: Definition, Types and Examples

In statistical hypothesis testing, the alternative hypothesis is an important proposition in the hypothesis test. The goal of the hypothesis test is to demonstrate that in the given condition, there is sufficient evidence supporting the credibility of the alternative hypothesis instead of the default assumption made by the null hypothesis.

Alternative Hypotheses

Both hypotheses include statements with the same purpose of providing the researcher with a basic guideline. The researcher uses the statement from each hypothesis to guide their research. In statistics, alternative hypothesis is often denoted as H a or H 1 .

Table of Content

What is a Hypothesis?

Alternative hypothesis, types of alternative hypothesis, difference between null and alternative hypothesis, formulating an alternative hypothesis, example of alternative hypothesis, application of alternative hypothesis.

“A hypothesis is a statement of a relationship between two or more variables.” It is a working statement or theory that is based on insufficient evidence.

While experimenting, researchers often make a claim, that they can test. These claims are often based on the relationship between two or more variables. “What causes what?” and “Up to what extent?” are a few of the questions that a hypothesis focuses on answering. The hypothesis can be true or false, based on complete evidence.

While there are different hypotheses, we discuss only null and alternate hypotheses. The null hypothesis, denoted H o , is the default position where variables do not have a relation with each other. That means the null hypothesis is assumed true until evidence indicates otherwise. The alternative hypothesis, denoted H 1 , on the other hand, opposes the null hypothesis. It assumes a relation between the variables and serves as evidence to reject the null hypothesis.

Example of Hypothesis:

Mean age of all college students is 20.4 years. (simple hypothesis).

An Alternative Hypothesis is a claim or a complement to the null hypothesis. If the null hypothesis predicts a statement to be true, the Alternative Hypothesis predicts it to be false. Let’s say the null hypothesis states there is no difference between height and shoe size then the alternative hypothesis will oppose the claim by stating that there is a relation.

We see that the null hypothesis assumes no relationship between the variables whereas an alternative hypothesis proposes a significant relation between variables. An alternative theory is the one tested by the researcher and if the researcher gathers enough data to support it, then the alternative hypothesis replaces the null hypothesis.

Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

There are a few types of alternative hypothesis that we will see:

1. One-tailed test H 1 : A one-tailed alternative hypothesis focuses on only one region of rejection of the sampling distribution. The region of rejection can be upper or lower.

• Upper-tailed test H 1 : Population characteristic > Hypothesized value
• Lower-tailed test H 1 : Population characteristic < Hypothesized value

2. Two-tailed test H 1 : A two-tailed alternative hypothesis is concerned with both regions of rejection of the sampling distribution.

3. Non-directional test H 1 : A non-directional alternative hypothesis is not concerned with either region of rejection; rather, it is only concerned that null hypothesis is not true.

4. Point test H 1 : Point alternative hypotheses occur when the hypothesis test is framed so that the population distribution under the alternative hypothesis is a fully defined distribution, with no unknown parameters; such hypotheses are usually of no practical interest but are fundamental to theoretical considerations of statistical inference and are the basis of the Neyman–Pearson lemma.

the differences between Null Hypothesis and Alternative Hypothesis is explained in the table below:

Formulating an alternative hypothesis means identifying the relationships, effects or condition being studied. Based on the data we conclude that there is a different inference from the null-hypothesis being considered.

• Understand the null hypothesis.
• Consider the alternate hypothesis
• Choose the type of alternate hypothesis (one-tailed or two-tailed)

Alternative hypothesis must be true when the null hypothesis is false. When trying to identify the information need for alternate hypothesis statement, look for the following phrases:

• “Is it reasonable to conclude…”
• “Is there enough evidence to substantiate…”
• “Does the evidence suggest…”
• “Has there been a significant…”

When alternative hypotheses in mathematical terms, they always include an inequality ( usually ≠, but sometimes < or >) . When writing the alternate hypothesis, make sure it never includes an “=” symbol.

To help you write your hypotheses, you can use the template sentences below.

Does independent variable affect dependent variable?

• Null Hypothesis (H 0 ): Independent variable does not affect dependent variable.
• Alternative Hypothesis (H a ): Independent variable affects dependent variable.

Various examples of Alternative Hypothesis includes:

Two-Tailed Example

• Research Question : Do home games affect a team’s performance?
• Null-Hypothesis: Home games do not affect a team’s performance.
• Alternative Hypothesis: Home games have an effect on team’s performance.
• Research Question: Does sleeping less lead to depression?
• Null-Hypothesis: Sleeping less does not have an effect on depression.
• Alternative Hypothesis : Sleeping less has an effect on depression.

One-Tailed Example

• Research Question: Are candidates with experience likely to get a job?
• Null-Hypothesis: Experience does not matter in getting a job.
• Alternative Hypothesis: Candidates with work experience are more likely to receive an interview.
• Alternative Hypothesis : Teams with home advantage are more likely to win a match.

Some applications of Alternative Hypothesis includes:

• Rejecting Null-Hypothesis : A researcher performs additional research to find flaws in the null hypothesis. Following the research, which uses the alternative hypothesis as a guide, they may decide whether they have enough evidence to reject the null hypothesis.
• Guideline for Research : An alternative and null hypothesis include statements with the same purpose of providing the researcher with a basic guideline. The researcher uses the statement from each hypothesis to guide their research.
• New Theories : Alternative hypotheses can provide the opportunity to discover new theories that a researcher can use to disprove an existing theory that may not have been backed up by evidence.

We defined the relationship that exist between null-hypothesis and alternative hypothesis. While the null hypothesis is always a default assumption about our test data, the alternative hypothesis puts in all the effort to make sure the null hypothesis is disproved.

Null-hypothesis always explores new relationships between the independent variables to find potential outcomes from our test data. We should note that for every null hypothesis, one or more alternate hypotheses can be developed.

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FAQs on Alternative Hypothesis

What is hypothesis.

A hypothesis is a statement of a relationship between two or more variables.” It is a working statement or theory that is based on insufficient evidence.

What is an Alternative Hypothesis?

Alternative hypothesis, denoted by H 1 , opposes the null-hypothesis. It assumes a relation between the variables and serves as an evidence to reject the null-hypothesis.

What is the Difference between Null-Hypothesis and Alternative Hypothesis?

Null hypothesis is the default claim that assumes no relationship between variables while alternative hypothesis is the opposite claim which considers statistical significance between the variables.

What is Alternative and Experimental Hypothesis?

Null hypothesis (H 0 ) states there is no effect or difference, while the alternative hypothesis (H 1 or H a ) asserts the presence of an effect, difference, or relationship between variables. In hypothesis testing, we seek evidence to either reject the null hypothesis in favor of the alternative hypothesis or fail to do so.

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Null hypothesis and Alternative Hypothesis

A hypothesis is a proposed explanation for a phenomenon, based on observation, reasoning, or scientific theory, awaiting verification or falsification through experimentation and data analysis. It serves as a starting point for investigation, guiding the research process by suggesting what outcomes to expect. In the realm of statistics and scientific research, hypotheses are crucial for designing experiments, analyzing results, and advancing knowledge.

The null hypothesis and alternative hypothesis are required to be fragmented properly before the data collection and interpretation phase in the research. Well fragmented hypotheses indicate that the researcher has adequate knowledge in that particular area and is thus able to take the investigation further because they can use a much more systematic system. It gives direction to the researcher on his/her collection and interpretation of data.

The null hypothesis and alternative hypothesis are useful only if they state the expected relationship between the variables or if they are consistent with the existing body of knowledge. They should be expressed as simply and concisely as possible. They are useful if they have explanatory power.

The purpose and importance of the null hypothesis and alternative hypothesis are that they provide an approximate description of the phenomena. The purpose is to provide the researcher or an investigator with a relational statement that is directly tested in a research study. The purpose is to provide the framework for reporting the inferences of the study. The purpose is to behave as a working instrument of the theory. The purpose is to prove whether or not the test is supported, which is separated from the investigator’s own values and decisions. They also provide direction to the research.

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The null hypothesis is generally denoted as H0. It states the exact opposite of what an investigator or an experimenter predicts or expects. It basically defines the statement which states that there is no exact or actual relationship between the variables.

The alternative hypothesis is generally denoted as H1. It makes a statement that suggests or advises a potential result or an outcome that an investigator or the researcher may expect. It has been categorized into two categories: directional alternative hypothesis and non directional alternative hypothesis.

The directional hypothesis is a kind that explains the direction of the expected findings. Sometimes this type of alternative hypothesis is developed to examine the relationship among the variables rather than a comparison between the groups.

The non directional hypothesis is a kind that has no definite direction of the expected findings being specified.

Related Pages:

Hypothesis Testing

Research Hypotheses

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Null Hypothesis and Alternative Hypothesis

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Hypothesis testing involves the careful construction of two statements: the null hypothesis and the alternative hypothesis. These hypotheses can look very similar but are actually different.

How do we know which hypothesis is the null and which one is the alternative? We will see that there are a few ways to tell the difference.

The Null Hypothesis

The null hypothesis reflects that there will be no observed effect in our experiment. In a mathematical formulation of the null hypothesis, there will typically be an equal sign. This hypothesis is denoted by H 0 .

The null hypothesis is what we attempt to find evidence against in our hypothesis test. We hope to obtain a small enough p-value that it is lower than our level of significance alpha and we are justified in rejecting the null hypothesis. If our p-value is greater than alpha, then we fail to reject the null hypothesis.

If the null hypothesis is not rejected, then we must be careful to say what this means. The thinking on this is similar to a legal verdict. Just because a person has been declared "not guilty", it does not mean that he is innocent. In the same way, just because we failed to reject a null hypothesis it does not mean that the statement is true.

For example, we may want to investigate the claim that despite what convention has told us, the mean adult body temperature is not the accepted value of 98.6 degrees Fahrenheit . The null hypothesis for an experiment to investigate this is “The mean adult body temperature for healthy individuals is 98.6 degrees Fahrenheit.” If we fail to reject the null hypothesis, then our working hypothesis remains that the average adult who is healthy has a temperature of 98.6 degrees. We do not prove that this is true.

If we are studying a new treatment, the null hypothesis is that our treatment will not change our subjects in any meaningful way. In other words, the treatment will not produce any effect in our subjects.

The Alternative Hypothesis

The alternative or experimental hypothesis reflects that there will be an observed effect for our experiment. In a mathematical formulation of the alternative hypothesis, there will typically be an inequality, or not equal to symbol. This hypothesis is denoted by either H a or by H 1 .

The alternative hypothesis is what we are attempting to demonstrate in an indirect way by the use of our hypothesis test. If the null hypothesis is rejected, then we accept the alternative hypothesis. If the null hypothesis is not rejected, then we do not accept the alternative hypothesis. Going back to the above example of mean human body temperature, the alternative hypothesis is “The average adult human body temperature is not 98.6 degrees Fahrenheit.”

If we are studying a new treatment, then the alternative hypothesis is that our treatment does, in fact, change our subjects in a meaningful and measurable way.

The following set of negations may help when you are forming your null and alternative hypotheses. Most technical papers rely on just the first formulation, even though you may see some of the others in a statistics textbook.

• Null hypothesis: “ x is equal to y .” Alternative hypothesis “ x is not equal to y .”
• Null hypothesis: “ x is at least y .” Alternative hypothesis “ x is less than y .”
• Null hypothesis: “ x is at most y .” Alternative hypothesis “ x is greater than y .”
• What 'Fail to Reject' Means in a Hypothesis Test
• Type I and Type II Errors in Statistics
• An Example of a Hypothesis Test
• The Runs Test for Random Sequences
• An Example of Chi-Square Test for a Multinomial Experiment
• The Difference Between Type I and Type II Errors in Hypothesis Testing
• What Level of Alpha Determines Statistical Significance?
• What Is the Difference Between Alpha and P-Values?
• What Is ANOVA?
• How to Find Critical Values with a Chi-Square Table
• Example of a Permutation Test
• Degrees of Freedom for Independence of Variables in Two-Way Table
• Example of an ANOVA Calculation
• How to Find Degrees of Freedom in Statistics
• How to Construct a Confidence Interval for a Population Proportion
• Degrees of Freedom in Statistics and Mathematics

9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 : The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

H a : The alternative hypothesis: It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 . This is usually what the researcher is trying to prove.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject H 0 " if the sample information favors the alternative hypothesis or "do not reject H 0 " or "decline to reject H 0 " if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ .30 H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

In an issue of U. S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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• Key Differences

Know the Differences & Comparisons

Difference Between Null and Alternative Hypothesis

Null hypothesis implies a statement that expects no difference or effect. On the contrary, an alternative hypothesis is one that expects some difference or effect. Null hypothesis This article excerpt shed light on the fundamental differences between null and alternative hypothesis.

Content: Null Hypothesis Vs Alternative Hypothesis

Comparison chart.

Definition of Null Hypothesis

A null hypothesis is a statistical hypothesis in which there is no significant difference exist between the set of variables. It is the original or default statement, with no effect, often represented by H 0 (H-zero). It is always the hypothesis that is tested. It denotes the certain value of population parameter such as µ, s, p. A null hypothesis can be rejected, but it cannot be accepted just on the basis of a single test.

Definition of Alternative Hypothesis

A statistical hypothesis used in hypothesis testing, which states that there is a significant difference between the set of variables. It is often referred to as the hypothesis other than the null hypothesis, often denoted by H 1 (H-one). It is what the researcher seeks to prove in an indirect way, by using the test. It refers to a certain value of sample statistic, e.g., x¯, s, p

The acceptance of alternative hypothesis depends on the rejection of the null hypothesis i.e. until and unless null hypothesis is rejected, an alternative hypothesis cannot be accepted.

Key Differences Between Null and Alternative Hypothesis

The important points of differences between null and alternative hypothesis are explained as under:

• A null hypothesis is a statement, in which there is no relationship between two variables. An alternative hypothesis is a statement; that is simply the inverse of the null hypothesis, i.e. there is some statistical significance between two measured phenomenon.
• A null hypothesis is what, the researcher tries to disprove whereas an alternative hypothesis is what the researcher wants to prove.
• A null hypothesis represents, no observed effect whereas an alternative hypothesis reflects, some observed effect.
• If the null hypothesis is accepted, no changes will be made in the opinions or actions. Conversely, if the alternative hypothesis is accepted, it will result in the changes in the opinions or actions.
• As null hypothesis refers to population parameter, the testing is indirect and implicit. On the other hand, the alternative hypothesis indicates sample statistic, wherein, the testing is direct and explicit.
• A null hypothesis is labelled as H 0 (H-zero) while an alternative hypothesis is represented by H 1 (H-one).
• The mathematical formulation of a null hypothesis is an equal sign but for an alternative hypothesis is not equal to sign.
• In null hypothesis, the observations are the outcome of chance whereas, in the case of the alternative hypothesis, the observations are an outcome of real effect.

There are two outcomes of a statistical test, i.e. first, a null hypothesis is rejected and alternative hypothesis is accepted, second, null hypothesis is accepted, on the basis of the evidence. In simple terms, a null hypothesis is just opposite of alternative hypothesis.

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Zipporah Thuo says

February 22, 2018 at 6:06 pm

The comparisons between the two hypothesis i.e Null hypothesis and the Alternative hypothesis are the best.Thank you.

Getu Gamo says

March 4, 2019 at 3:42 am

Thank you so much for the detail explanation on two hypotheses. Now I understood both very well, including their differences.

Jyoti Bhardwaj says

May 28, 2019 at 6:26 am

Thanks, Surbhi! Appreciate the clarity and precision of this content.

January 9, 2020 at 6:16 am

John Jenstad says

July 20, 2020 at 2:52 am

Thanks very much, Surbhi, for your clear explanation!!

Navita says

July 2, 2021 at 11:48 am

Thanks for the Comparison chart! it clears much of my doubt.

GURU UPPALA says

July 21, 2022 at 8:36 pm

Thanks for the Comparison chart!

Enock kipkoech says

September 22, 2022 at 1:57 pm

What are the examples of null hypothesis and substantive hypothesis

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8 Different Types of Hypotheses (Plus Essential Facts)

The hypothesis is an idea or a premise used as a jumping off the ground for further investigation. It’s essential to scientific research because it serves as a compass for scientists or researchers in carrying out their experiments or studies.

There are different types of hypotheses but crafting a good hypothesis can be tricky. A sound hypothesis should be logical, affirmative, clear, precise, quantifiable, or can be tested, and has a cause and effect factor.

Types 

Alternative hypothesis.

Also known as a maintained hypothesis or a research hypothesis, an alternative hypothesis is the exact opposite of a null hypothesis, and it is often used in statistical hypothesis testing. There are four main types of alternative hypothesis:

• Point alternative hypothesis . This hypothesis occurs when the population distribution in the hypothesis test is fully defined and has no unknown parameters. It usually has no practical interest, but it is considered important in other statistical activities.
• Non-directional alternative hypothesis. These hypotheses have nothing to do with the either region of rejection (i.e., one-tailed or two-tailed directional hypotheses) but instead, only that the null hypothesis is untrue.
• One-tailed directional hypothesis. This hypothesis is only concerned with the region of direction for one tail of a sampling distribution, not both of them.
• Two-tailed directional hypothesis. This hypothesis is concerned with both regions of rejection of a particular sampling distribution

Known by the symbol H1, this type of hypothesis proclaims the expected relationship between the variables in the theory.

Associative and Causal Hypothesis

Associative hypotheses simply state that there is a relationship between two variables, whereas causal hypotheses state that any difference in the type or amount of one particular variable is going to directly affect the difference in the type or amount of the next variable in the equation.

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These hypotheses are often used in the field of psychology. A causal hypothesis looks at how manipulation affects events in the future, while an associative hypothesis looks at how specific events co-occur.

A good example of its practical use occurs when discussing the psychological aspects of eyewitness testimonies, and they generally affect four areas of this phenomenon: emotion and memory, system variables in the line-up, estimation of the duration of the event, and own-race bias.

Complex Hypothesis

In a complex hypothesis, a relationship exists between the variables . In these hypotheses, there are more than two independent and dependent variables, as demonstrated in the following hypotheses:

• Taking drugs and smoking cigarettes leads to respiratory problems, increased tension, and cancer.
• The people who are older and living in rural areas are happier than people who are younger and who live in the city or suburbs.
• If you eat a high-fat diet and a few vegetables, you are more likely to suffer from hypertension and high cholesterol than someone who eats a lot of vegetables and sticks to a low-fat diet.

Directional Hypothesis

A directional hypothesis is one regarding either a positive or negative difference or change in the two variables involved. Typically based on aspects such as accepted theory, literature printed on the topic at hand, past research, and even accepted theory, researchers normally develop this type of hypothesis from research questions, and they use statistical methods to check its validity.

Words you often hear in hypotheses that are directional in nature include more, less, increase, decrease, positive, negative, higher, and lower. Directional hypotheses specify the direction or nature of the relationship between two or more independent variables and two or more dependent variables.

Non-Directional Hypothesis

This hypothesis states that there is a distinct relationship between two variables; however, it does not predict the exact nature or direction of that particular relationship.

Null Hypothesis

Indicated by the symbol Ho, a null hypothesis predicts that the variables in a certain hypothesis have no relationship to one another and that the hypothesis is normally subjected to some type of statistical analysis. It essentially states that the data and variables being investigated do not actually exist.

A perfect example of this comes when looking at scientific medical studies, where you have both an experimental and control group, and you are hypothesizing that there will be no difference in the results of these two groups.

Simple Hypothesis

This hypothesis consists of two variables, an independent variable or cause, and a dependent variable or cause. Simple hypotheses contain a relationship between these two variables. For example, the following are examples of simple hypotheses:

• The more you chew tobacco, the more likely you are to develop mouth cancer.
• The more money you make, the less likely you are to be involved in criminal activity.
• The more educated you are, the more likely you are to have a well-paying job.

Statistical Hypothesis

This is just a hypothesis that is able to be verified through statistics. It can be either logical or illogical, but if you can use statistics to verify it, it is called a statistical hypothesis.

Facts about Hypotheses

Difference Between Simple and Complex Hypotheses

In a simple hypothesis, there is a dependent and an independent variable, as well as a relationship between the two. The independent variable is the cause and comes first when they’re in chronological order, and the dependent variable describes the effect. In a complex hypothesis, the relationship is between two or more independent variables and two or more dependent variables.

Difference Between Non-Directional and Directional Hypotheses

In a directional research hypothesis, the direction of the relationship is predicted. The advantages of this type of hypothesis include one-tailed statistical tests, theoretical propositions that can be tested in a more precise manner, and the fact that the researcher’s expectations are very clear right from the start.

In a non-directional research hypothesis, the relationship between the variables is predicted but not the direction of that relationship. Reasons to use this type of research hypothesis include when your previous research findings contradict one another and when there is no theory on which to base your predictions.

Difference Between a Hypothesis and a Theory

There are many different differences between a theory and a hypothesis, including the following:

• A hypothesis is a suggestion of what might happen when you test out a theory. It is a prediction of a possible correlation between various phenomena. On the other hand, a theory has been tested and is well-substantiated. If a hypothesis succeeds in proving a certain point, it can then be called a theory.
• The data for a hypothesis is most often very limited, whereas the data relating to theory has been tested under numerous circumstances.
• A hypothesis offers a very specific instance; that is, it is limited to just one observation. On the other hand, a theory is more generalized and is put through a multitude of experiments and tests, which can then apply to various specific instances.
• The purposes of these two items are different as well. A hypothesis starts with a possibility that is uncertain but can be studied further via observations and experiments. A theory is used to explain why large sets of observations are continuously made.
• Hypotheses are based on various suggestions and possibilities but have uncertain results, while theories have a steady and reliable consensus among scientists and other professionals.
• Both theories and hypotheses are testable and falsifiable, but unlike theories, hypotheses are neither well-tested nor well-substantiated.

What is the Interaction Effect?

This effect describes the two variables’ relationship to one another.

When Writing the Hypothesis, There is a Certain Format to Follow

This includes three aspects:

• The correlational statement
• The comparative statement
• A statistical analysis

How are Hypotheses Used to Test Theories?

• Do not test the entire theory, just the proposition
• It can never be either proved or disproved

When Formulating a Hypothesis, There are Things to Consider

These include:

• You have to write it in the present tense
• It has to be empirically testable
• You have to write it in a declarative sentence
• It has to contain all of the variables
• It must contain three parts: the purpose statement, the problem statement, and the research question
• It has to contain the population

What is the Best Definition of a Scientific Hypothesis?

It is essentially an educated guess; however, that guess will lose its credibility if it is falsifiable.

How to Use Research Questions

There are two ways to include research questions when testing a theory. The first is in addition to a hypothesis related to the topic’s other areas of interest, and the second is in place of the actual hypothesis, which occurs in some instances.

Tips to Keep in Mind When Developing a Hypothesis

• Use language that is very precise. Your language should be concise, simple, and clean. This is not a time when you want to be vague, because everything needs to be spelled out in great detail.
• Be as logical as possible. If you believe in something, you want to prove it, and remaining logical at all times is a great start.
• Use research and experimentation to determine whether your hypothesis is testable. All hypotheses need to be proven. You have to know that proving your theory is going to work, even if you find out different in the end.

What is the Number-One Purpose of a Scientific Method?

Scientific methods are there to provide a structured way to get the appropriate evidence in order to either refute or prove a scientific hypothesis.

Glossary of Terms Related to Hypotheses

Bivariate Data: This is data that includes two distinct variables, which are random and usually graphed via a scatter plot.

Categorical Data: These data fit into a tiny number of very discrete categories. They are usually either nominal, or non-ordered, which can include things such as age or country; or they can be ordinal, or ordered, which includes aspects such as hot or cold temperature.

Correlation: This is a measure of how closely two variables are to one another. It measures whether a change in one random variable corresponds to a change in the other random variable. For example, the correlation between smoking and getting lung cancer has been widely studied.

Data: These are the results found from conducting a survey or experiment, or even an observation study of some type.

Dependent Event: If the happening of one event affects the probability of another event occurring also, they are said to be dependent events.

Distribution: The way the probability of a random variable taking a certain value is described is called its distribution. Possible distribution functions include the cumulative, probability density, or probability mass function.

Element: This refers to an object in a certain set, and that object is an element of that set.

Empirical Probability: This refers to the likelihood of an outcome happening, and it is determined by the repeat performance of a particular experiment.  You can do this by dividing the number of times that event took place by the number of times you conducted the experiment.

Equality of Sets: If two sets contain the exact same elements, they are considered equal sets. In order to determine if this is so, it can be advantageous to show that each set is contained in the other set.

Equally Likely Outcomes: Refers to outcomes that have the same probability; for example, if you toss a coin there are only two likely outcomes.

Event: This term refers to the subset of a sample space.

Expected Value: This demonstrates the average value of a quantity that is random and which has been observed numerous times in order to duplicate the same results of previous experiments.

Experiment: A scientific process that results in a set of outcomes that is observable. Even selecting a toy from a box of toys can be considered an experiment in this instance.

Experimental Probability: When you estimate how likely something is to occur, this is an experimental probability example. To get this probability, you divide the number of trials that were successful by the total number of trials that were performed.

Finite Sample Space: These sample spaces have a finite number of outcomes that could possibly occur.

Frequency: The frequency is the number of times a certain value occurs when you observe an experiment’s results.

Frequency Distribution: This refers to the data that describes possible groups or values and the frequencies that correspond to those groups or values.

Histogram: A histogram, or frequency histogram, is a bar graph that demonstrates how frequently data points occur.

Independent Event: If two events occur, and one event’s outcome has no effect on the other’s outcome, this is known as an independent event.

Infinite Sample Space: This refers to a sample space that consists of outcomes with an infinite number of possibilities.

Mutually Exclusive: Events are mutually exclusive if their outcomes have absolutely nothing in common.

Notations: Notations are operations or quantities described by symbols instead of numbers.

Observational Study: Like the name implies, these are studies that allow you to collect data through basic observation.

Odds: This is a way to express the likelihood that a certain event will happen. If you see odds of m:n, it means it is expected that a certain event will happen m times for every n times it does not happen.

One-Variable Data: Data that have related behaviors usually associated in some important way.

Outcome: The outcome is simply the result of a particular experiment. If you consider a set of all of the possible outcomes, this is called the sample space.

Probability: A probability is merely the likelihood that a certain event will take place, and it is expressed on a scale of 0 to one, with 0 meaning it is impossible that it will happen and one being a certainty that it will happen. Probability can also be expressed as a percentage, starting with 0 and ending at 100%.

Random Experiment: A random experiment is one whereby the outcome can’t be predicted with any amount of certainty, at least not before the experiment actually takes place.

Random Variable: Random variables take on different numerical values, based on the results of a particular experiment.

Replacement: Replacement is the act of returning or replacing an item back into a sample space, which takes place after an event and allows the item to be chosen more than one time.

Sample Space: This term refers to all of the possible outcomes that could result from a probability experiment.

Set: A collection of objects that is well-defined is called a set.

Simple Event: When an event is a single element of the sample space, it is known as a simple event.

Simulation: A simulation is a type of experiment that mimics a real-life event.

Single-Variable Data: These are data that use only one unknown variable.

Statistics: This is the branch of mathematics that deals with the study of quantitative data. If you analyze certain events that are governed by probability, this is called statistics.

Theoretical Probability: This probability describes the ratio of the number of outcomes in a specific event to the number of outcomes found in the sample space. It is based on the presumption that all outcomes are equally liable.

Union: Usually described by the symbol ∪, or the cup symbol, a union describes the combination of two or more sets and their elements.

Variable: A variable is a quantity that varies and is almost always represented by letters.

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• Open access
• Published: 26 August 2024

Larger colony sizes favoured the evolution of more worker castes in ants

• Louis Bell-Roberts   ORCID: orcid.org/0000-0002-4035-8168 1 ,
• Juliet F. R. Turner 1 ,
• Gijsbert D. A. Werner 1 , 2 ,
• Philip A. Downing   ORCID: orcid.org/0000-0002-5286-3153 3 ,
• Laura Ross   ORCID: orcid.org/0000-0003-3184-4161 4 &
• Stuart A. West   ORCID: orcid.org/0000-0003-2152-3153 1  

Nature Ecology & Evolution ( 2024 ) Cite this article

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• Behavioural ecology
• Social evolution

The size–complexity hypothesis is a leading explanation for the evolution of complex life on earth. It predicts that in lineages that have undergone a major transition in organismality, larger numbers of lower-level subunits select for increased division of labour. Current data from multicellular organisms and social insects support a positive correlation between the number of cells and number of cell types and between colony size and the number of castes. However, the implication of these results is unclear, because colony size and number of cells are correlated with other variables which may also influence selection for division of labour, and causality could be in either direction. Here, to resolve this problem, we tested multiple causal hypotheses using data from 794 ant species. We found that larger colony sizes favoured the evolution of increased division of labour, resulting in more worker castes and greater variation in worker size. By contrast, our results did not provide consistent support for alternative hypotheses regarding either queen mating frequency or number of queens per colony explaining variation in division of labour. Overall, our results provide strong support for the size–complexity hypothesis.

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Division of labour has played a pivotal role in the evolution of life on earth, by facilitating evolutionary transitions to more complex organisms 1 , 2 , 3 , 4 . In these transitions, individuals cooperate to form a new higher-level individual (organism). For example, cells formed multicellular organisms, with germline and soma, and insects formed complex colonies termed ‘superorganisms’, with queen and worker castes 3 . Division of labour is fundamental to these transitions because specialization allows the individuals that form a higher-level organism to perform more diverse functions and to become more reliant upon each other for reproduction (mutual dependence) 4 . Consequently, a major aim in evolutionary biology is to understand the factors that favour an increased level of division of labour 2 , 5 , 6 .

The size–complexity hypothesis is one of the leading explanations for the evolution of increased division of labour because of the generality of its explanatory power across different levels of life. In lineages that have undergone a major evolutionary transition, such as to obligate multicellularity or superorganismality, the hypothesis predicts that larger numbers of lower-level individuals (subunits) select for increased division of labour 1 , 2 , 6 , 7 , 8 , 9 . This prediction arises because organisms or superorganisms formed by larger numbers of individuals have more tasks that require doing and are better able to maintain the optimal ratio of specialized individuals performing each task 2 , 6 , 10 , 11 . Support for the size–complexity hypothesis has been provided by examining division of labour in both individual multicellular organisms and insect colonies. Across multicellular species, organisms with larger numbers of cells have more cell types 7 , 8 , 9 . Across ants, species with larger colony sizes have more worker castes 12 , 13 , 14 , 15 , 16 , 17 . However, despite these correlations, challenges to the size–complexity hypothesis remain.

First, alternative factors have been suggested to explain variation in the level of division of labour. For example, in ants, variation in both queen mating frequency and number of queens per colony has been predicted to influence the evolution of number of worker castes 6 , 18 , 19 . When colonies are formed by a single monogamous pair, relatedness among colony members will be maximal, which may reduce conflict within the colony and favour the evolution of multiple worker castes 6 , 19 . Alternatively, the opposite prediction could also be made in hymenopteran species, where multiple mating by the queen selects for workers to remove (police) the eggs laid by other workers 20 . Policing could reduce conflict, leading to a positive correlation between the number of castes and queen mating frequency 18 . The problem of multiple hypotheses is exasperated by the fact that colony size can be correlated with queen mating frequency 13 , 14 , 21 . Consequently, the correlation between colony size and number of castes could just be the result of a relationship caused by variation in queen mating frequency.

Second, in superorganismal social insects, the correlation between colony size and number of worker castes is open to alternative causal explanations. For example, while larger colony sizes may favour the evolution of greater division of labour, it could alternatively be that increased division of labour favours the evolution of larger colony sizes. The problem of causal direction becomes even greater when we consider that there are multiple correlated variables. For example, it could be that an increase in colony size favours an increase in queen mating frequency, and it is an increase in queen mating frequency that favours the evolution of a greater number of worker castes. In this case, colony size and the number of castes would be correlated. However, this pattern would lead to the false impression that larger colony size directly favours increased division of labour and misses the involvement of an intermediate factor. Ultimately, distinguishing between the role of different factors requires phylogenetically based analyses examining the causal direction and order of evolutionary changes 22 .

We resolve this problem by performing a series of phylogenetic analyses on the number of physical worker castes in ants. We focus on physical castes which are irreversibly fixed for life and determined during development, rather than behavioural differentiation among workers. This is because extreme specialization, involving mutual dependence, requires differences in size and morphology. Ants provide an excellent opportunity for testing the size–complexity hypothesis because they are a single monophyletic clade in which the number of worker castes varies across species from one to four, and there is considerable information on factors that could influence the evolution of castes 23 , 24 , 25 , 26 , 27 . We analysed data from 794 species to examine the influence of three possible factors: colony size, queen mating frequency and number of queens per colony. We performed phylogenetic regressions, to determine how these three traits are correlated with the number of castes and each other. This step provided an overview that helped to guide further causal analyses and allowed comparison with previous studies. We then examined the likely causal relationships between number of worker castes, queen mating frequency and colony size with three methods: (1) phylogenetic path analysis 28 , (2) transition rate analysis between pairs of traits 29 and (3) ancestral state reconstruction 30 . These analyses allowed us to tease apart the different roles of each variable. Last, to assess the robustness of our results, we also tested whether the same patterns emerged when examining an alternative possible measure of division of labour—variation in worker size 12 , 13 , 18 , 31 .

Summary of ant social traits

Across different ant species, there was considerable variation in the characteristics of a colony. The number of discrete physical worker castes varied from one to four, although 76% of species had only one worker caste (Fig. 1a ). Ancestral state reconstruction suggested that polymorphic workers, with more than one caste, had evolved from monomorphic ancestors at least 27 times (Fig. 2 and Supplementary Table 1 ). Colony sizes varied from 7 to 14,750,000, with a median of 300 (Fig. 1b ). The effective number of males that inseminated each queen varied from 1 to 26, with 31% of species mating with a single male, 36% mating with between 1 and 2 males and 33% mating with more than 2 males. The median value for queen mating frequency was 1.1 mates (Fig. 1c ). The number of queens per colony varied from 1 to 70, with 63% of species possessing a single queen, 18% possessing between 1 and 2 queens and 19% possessing more than 2 queens. The median number of queens per colony was 1 (Fig. 1d ). The dataset was not complete for all traits for each species, and so for each analysis we used the maximum amount of data available. Therefore, sample sizes vary between analyses depending on the variables being analysed, as detailed within each section.

a , Number of worker castes ( n species  = 699). b , Colony size ( n species  = 541). c , Queen mating frequency ( n species  = 126). d , Number of queens per colony ( n species  = 252). e , Variation in worker size quantified using the CV for worker head width ( n species  = 152). f , Ants vary in their number of physical worker castes. Left: Lasius claviger , a species characterized by having just a single physical worker caste. Right: Two distinct physical worker castes of the same species, Eciton burchelli , representing one of the most extreme cases of morphological caste variation (photo by Alex Wild ( https://www.alexanderwild.com/ )). Axes for colony size and number of queens per colony are plotted on a log 10 scale.

The ancestral state at the root of the ant phylogeny is a single physical worker caste. Coloured circles represent evolutionary increases in number of worker castes, with colour corresponding to the number of castes that have evolved. The number of worker castes present in each species is also displayed around the outside of the phylogeny. Branch lengths are not to scale. Ant phylogenetic lineages, of varying levels, are displayed in black text around the outside of the phylogeny. Analyses were repeated over 400 phylogenetic trees, but we display the results of a single representative tree.

Phylogenetic correlations

We used Bayesian phylogenetic mixed effect models (BPMMs) to test for correlations between the different ant traits. We found that colony size and queen mating frequency were both positively correlated with number of worker castes (Fig. 3a,b ; BPMMs: colony size: β  = 0.17, credible interval (CI) = 0.11 to 0.24, n species  = 436; queen mating frequency: β  = 0.47, CI = 0.04 to 0.91, n species  = 104). By contrast, we found no relationship between number of queens per colony and number of worker castes (Fig. 3c ; BPMM: β  = −0.14, CI = −0.56 to 0.30, n species  = 192).

a , b , Species with larger colony sizes ( n species  = 436) ( a ) and higher queen mating frequencies ( n species  = 104) ( b ) had significantly more worker castes. c , d , f , We did not find a significant association between number of queens per colony and either number of worker castes ( n species  = 192) ( c ), queen mating frequency ( n species  = 111) ( d ) or colony size ( n species  = 195) ( f ). e , Species with higher queen mating frequencies had significantly larger colony sizes ( n species  = 109). a – d , Data show mean ± standard error. e , f , Fitted lines are mean regression slopes with 95% CIs from BPMMs using a single phylogenetic tree. Solid regression lines represent significant relationships, while dashed regression lines represent non-significant relationships. Dots represent species averages. Axes for colony size, queen mating frequency and number of queens per colony are plotted on a log 10 scale.

We also found that species with higher queen mating frequencies had larger colony sizes (Fig. 3e ; BPMM: β  = 1.25, CI = 0.50 to 2.03, n species  = 109, R 2  = 0.08). The significant correlations between queen mating frequency, colony size and number of worker castes emphasizes the need to carry out analyses that can examine the underlying causality between these variables. By contrast, we found no relationship between number of queens per colony and colony size (Fig. 3f ; BPMM: β  = 0.07, CI = −0.37 to 0.49, n species  = 195, R 2  = 0.00). However, we did find a negative relationship, which was close to significance, between number of queens per colony and queen mating frequency (Fig. 3d ; BPMM: β  = −0.20, CI = −0.39 to 0.01, n species  = 111, R 2  = 0.00). Based on the results of our correlational analysis, we excluded the number of queens per colony from our causality analysis, except for a potential relationship between number of queens per colony and queen mating frequency, where a negative relationship was also observed previously 25 .

Causality analyses

We then used three different methods to examine the causal relationships that underly these correlations: (1) phylogenetic path analysis, (2) transition rate analysis and (3) ancestral state reconstruction. These forms of analysis required number of worker castes to be modelled as a binary variable, and so we classified species as possessing either a single (monomorphic) or multiple (>1, polymorphic) worker castes.

Phylogenetic path analysis

We used phylogenetic path analysis to test four alternative causal models of the relationships between the following: number of worker castes, colony size, queen mating frequency and number of queens per colony (Extended Data Fig. 1 ). We found strong evidence in favour of the size–complexity hypothesis. The analysis supported a single causal model, where large colony sizes favoured the evolution of multiple worker castes, and size was the only variable that directly did so (Fig. 4 , Extended Data Fig. 2 and Supplementary Tables 2 and 3 ; ω (relative weight of support for a model within a set of competing models) = 0.67, n species  = 94). We found no support for alternative causal models where colony size is influenced by number of worker castes or where number of worker castes is influenced by queen mating frequency (Supplementary Table 2 ; n species  = 94).

Larger colony sizes facilitated the evolution of multiple worker castes. We display the results from the analysis of one MCC consensus tree (Supplementary Tables 2 and 3 ). However, we repeated the analyses on three additional MCC trees and, in each case, found that the conclusions drawn remained unchanged due to the similarity in the presence, direction and magnitude of the causal relationships identified (Extended Data Fig. 2 and Supplementary Tables 2 and 3 ). Values represent standardized path coefficients with 95% confidence intervals (path coefficients describe the strength and direction of the relationship between two variables in terms of their correlated variance, after accounting for the effects of other variables in the model). Arrows indicate the direction of the relationship between variables, with heavier lines indicating larger coefficients. Number of worker castes was modelled as a discrete binary variable, while colony size, queen mating frequency and number of queens per colony were modelled as continuous variables.

Transition rate analysis

We used transition rate analysis as a second method to test the hypothesized causal relationships between number of worker castes and both colony size and queen mating frequency. This method allowed us to test whether the evolution of two traits is correlated, as well as the underlying causal direction, but required all traits to be modelled as discrete binary variables. We classified species as having small or large colony sizes depending upon whether colony size was smaller (≤300) or larger (>300) than the median colony size. We classified species as having low or high mating frequencies depending on whether queens mated more (>2) or less than or equal to (≤2) two times.

We found very strong support for the size–complexity hypothesis, with correlated evolution between colony size and number of worker castes (Supplementary Table 4 ; Bayes factor (BF) = 20.83, n species  = 436; BFs quantify the relative support for two competing models where BF > 2 offers positive evidence, > 5 provides strong evidence and >10 very strong evidence that the more complex model performs better than the simpler model 29 ). Transitions to multiple castes only occurred in species with large colony sizes (Fig. 5 ). In species with only a single worker caste, transition rates from small colony sizes to large colony sizes were slightly elevated in comparison to species with multiple castes (Fig. 5 ). By contrast, we found that variation in queen mating frequency does not influence the evolution of number of worker castes, as queen mating frequency evolved independently of the number of worker castes (Supplementary Table 5 ; queen mating frequency and worker castes: BF = −1.39, n species  = 104).

Multiple worker castes only evolved in lineages with large colony sizes. Values represent the inferred transition rates between number of worker castes and colony size, accompanied by 95% CIs ( n species  = 436). Transition rates were estimated using rjMCMC implemented in BayesTraits. Each possible state is shown as a box, and the direction of transitions between states are shown with arrows, with higher rates of transition indicated by a heavier line.

Ancestral state reconstruction

We used ancestral state reconstruction as a third method to examine the underlying causal relationships and again found support for the size–complexity hypothesis. We found that the ancestors of species that evolved multiple worker castes had colony sizes that were more than 3.5 times larger compared with the ancestors of species that retained one worker caste (Extended Data Fig. 3a ; BPMM: ancestral colony size in species with one worker caste, β  = 281.67, CI = 40.04 to 1,590.90; ancestral colony size in species with multiple worker castes, β  = 1,025.05, CI = 137.17 to 9,064.82; P single vs multiple  < 0.01, n species  = 436). By contrast, queen mating frequency did not differ significantly between ancestors that evolved multiple worker castes and those that did not (Extended Data Fig. 3b ; BPMM: ancestral queen mating frequency in species with one worker caste, β  = 1.99, CI = 0.95 to 3.90; ancestral queen mating frequency in species with multiple worker castes, β  = 2.46, CI = 1.10 to 5.55; P single vs multiple  = 0.16, n species  = 104). The phylogenetic signal present in both colony size and queen mating frequency across ants allowed us to estimate ancestral values relatively accurately in these traits (Supplementary Table 6 ; BPMMs: phylogenetic heritability of colony size = 72.61%, CI = 59.67% to 84.02%; phylogenetic heritability of queen mating frequency = 88.94%, CI = 74.01% to 99.86%).

Division of labour and worker size

Taken together, our analyses on the number of worker castes provide strong support for the size–complexity hypothesis. Larger colony sizes correlate with and appear to lead to the evolution of more worker castes (Figs. 3 – 5 ). However, most ant species do not possess more than one physical worker caste. Despite this, there can be considerable variation in worker size within castes, and there is some evidence that different-sized workers perform different roles even within ant species that have a single physical worker caste 32 . Consequently, an alternative method for testing the size–complexity hypothesis would be to examine the relationship between colony size and variation in worker size, rather than number of castes, as a measure of non-reproductive division of labour 12 , 13 , 18 , 31 .

Using data obtained from AntWeb, we examined variation in the head width measurements of 1,064 workers from 152 species and found mixed support for the size–complexity hypothesis (Extended Data Fig. 4 ). We found a positive correlation between colony size and variation in worker size and that larger colony sizes may favour the evolution of greater variation in worker size (Fig. 6 , Extended Data Fig. 5 and Supplementary Tables 7 and 8 ; BPMM: colony size, β  = 0.05, CI = 0.03 to 0.08, n species  = 122, R 2  = 0.14; path analysis: n species  = 94). However, there was some uncertainty regarding the causal direction of the relationship between colony size and variation in worker size. Among the three supported causal models identified by path analysis, larger colony sizes favoured the evolution of greater variation in worker size in two cases, and greater variation in worker size favoured the evolution of larger colony sizes in the third case (the model fit was compared using the corrected C-statistic Information Criterion (CICc), where models within 2 CICc units of the best model were considered to have equal support; Supplementary Table 7 ; combined weight for the models supporting the hypothesis that variation in worker size is influenced by colony size: ω  = 0.55). Due to the uncertainty regarding the best-supported model, the confidence intervals for the path coefficients in the average model, linking both colony size and queen mating frequency with variation in worker size, were relatively large and overlapped with zero (Fig. 6b ). When examining species with only a single worker caste, we found the contrasting result that greater variation in worker size favours the evolution of larger colony sizes (Fig. 6a , Extended Data Fig. 6 and Supplementary Tables 9 and 10 ; BPMM: β  = 0.04, CI = 0.01 to 0.07, n species  = 84, R 2  = 0.08; path analysis: ω  = 0.75, n species  = 60).

a , Species with larger colony sizes possess significantly greater variation in worker size when analysing either of the following: (1) all species for which data were available (all dots and regression line in black, n species  = 122) or (2) only species which possess a single worker caste (dots and regression line in light blue, n species  = 84). Fitted lines represent mean regression slopes from BPMMs. Dots represent species averages. Colony size is plotted on a log 10 scale, while variation in worker size is on a square root scale. b , Averaged model from the three supported path analysis models (Supplementary Tables 7 and 8 ). Larger colony sizes may favour the evolution of greater variation in worker size, but opposite causation remains possible. Analysis performed using all species where data were available for variation in worker size, colony size, number of queens per colony and queen mating frequency ( n species  = 94). We display the results from the analysis of one MCC consensus tree. We repeated the analyses on three additional MCC trees and, in each case, found that the conclusions drawn remained unchanged due to the similarity in the presence, direction and magnitude of the causal relationships identified (Extended Data Fig. 5 and Supplementary Tables 7 and 8 ). Values represent standardized path coefficients with 95% confidence intervals. Arrows indicate the direction of the relationship between variables, with heavier lines indicating larger coefficients. Variation in worker size, colony size, queen mating frequency and number of queens per colony were modelled as continuous variables.

By contrast, we did not find support for the hypothesis that variation in worker size is influenced by queen mating frequency. This was the case when analysing either all species or just those with a single worker caste (Fig. 6b , Extended Data Fig. 5 – 7 and Supplementary Tables 7 – 11 ).

Sensitivity analyses

We examined whether our results were robust to alternative methods of quantifying variation in queen mating frequency and number of queens per colony as categorical rather than continuous variables 21 , 25 , 33 . Detailed statistics for each analysis are presented in the supplementary material (Extended Data Figs. 8 – 10 and Supplementary Tables 11 – 19 ). In all cases, the size–complexity hypothesis was still supported, but some of the other relationships depended upon how data were categorized. Our conclusions regarding the phylogenetic correlations were robust, except for the finding that the number of worker castes did not significantly differ between singly mated and facultatively multiply mated species (Supplementary Table 19 ). The conclusions of our causal analyses were also robust, except when analysing the number of discrete worker castes using phylogenetic path analysis. We identified a single supported model where the evolution of multiple worker castes was favoured by both larger colony sizes and obligate multiple mating by queens. However, the confidence interval for the path coefficient associated with queen mating frequency overlapped with zero (Extended Data Fig. 8 and Supplementary Tables 12 and 13 ).

We found strong and consistent support for the size–complexity hypothesis, with larger colony sizes appearing to favour the evolution of multiple worker castes (Figs. 3 – 6 ). By contrast, we did not find consistent support for the hypothesis that multiple mating favours the evolution of multiple worker castes (Figs. 3 , 4 and 6b ). We found no evidence to suggest a direct relationship between the evolution of number of worker castes and number of queens per colony (Fig. 3c ). Our conclusions were robust to different analysis methods.

Our results suggest that larger colony sizes favour the evolution of multiple worker castes in ants. Previous studies had highlighted a positive correlation between increased division of labour and both cell number in multicellular organisms and colony size in social insects, but these results were open to multiple explanations 7 , 8 , 9 , 12 , 13 , 14 , 18 , 21 . Our results reveal a distinct evolutionary pattern where larger colony sizes tend to evolve before multiple worker castes (Figs. 4 and 5 ). There are at least two reasons why larger colonies may promote the evolution of multiple castes. First, as colony size increases, the number of tasks that need to be performed also increases 2 . For example, species with larger colonies may experience greater logistical challenges in transporting food or waste within or outside the colony. Second, species with larger colonies are better able to maintain the optimal ratio of the different castes 10 , 34 . Consequently, these species face a lower risk of losing essential worker functions if workers are lost.

Our analysis also suggests that larger colony sizes favoured the evolution of greater variation in worker size (Fig. 6 ). However, there is some uncertainty regarding the direction of this relationship, as causality could potentially be reversed. In addition, we found that this result only applied when analysing across species with variable numbers of physical worker castes. When analysing variation in worker size exclusively in species with a single caste, we found that greater variation in worker size preceded the evolution of larger colony sizes. Consequently, while our analyses of variation in worker size support the size–complexity hypothesis, they suggest that colony size is not responsible for the evolution of variation in worker size in species that lack discrete physical worker castes.

We did not find consistent support for the hypothesis that multiple mating favours the evolution of multiple worker castes. There is a positive correlation between both number of worker castes and variation in worker size with queen mating frequency, regardless of whether it is analysed as a continuous or categorical trait (Fig. 3b ). However, when analysing the causal relationship between these variables, we found that higher mating frequency was not consistently associated with evolutionary transitions from single to multiple worker castes (Figs. 4 and 6b , Extended Data Figs. 8 and 9 and Supplementary Tables 5 and 18 ). This finding is consistent with the interpretation that queen mating frequency is positively correlated with number of worker castes due to both variables being positively associated with colony size (Figs. 3 and 4 ). Colony size and queen mating frequency may be positively correlated due to the advantages of increased genetic diversity that result from multiple mating 21 , 35 , 36 , 37 , 38 , 39 , 40 , 41 . If species with larger colony sizes are at greater risk of infection from pathogens, for example, resulting from increased traffic of foragers into the nest, they may benefit more from the protection offered against disease by increased genetic diversity 40 , 41 , 42 .

Excluding ants, possessing more than one physical worker caste is extremely rare in the social Hymenoptera. Multiple worker castes appear to be entirely absent in vespine wasps, while in bees, the presence of two castes has only been identified in some stingless bee species 31 , 43 . For example, Tetragonisca angustula possesses two castes—soldiers are both approximately 30% heavier than foragers and have a different morphology 43 . This contrasts with the much larger variation in ants—for example, major workers in Atta colombica leafcutter ants can be 60 times heavier than the smallest workers 44 . Why does the causal link between colony size and number of castes that we have identified in ants seemingly not apply in superorganismal species of bees and wasps 31 ? Multiple hypotheses have been proposed, suggesting that factors such as the presence of a powerful sting or body size constraints imposed by having winged workers could have impeded the evolution of multiple worker castes in bees and wasps 6 , 24 , 45 . However, the phylogenetic power available to carry out statistical tests of these hypotheses is low. Other ways to test the size–complexity hypothesis include examining behavioural variation in bees and wasps or the physical caste variation in termites 6 , 31 , 46 , 47 , 48 .

To conclude, our results provide strong support for the size–complexity hypothesis in ants, suggesting that larger colony sizes have favoured the evolution of greater division of labour (more castes and greater variation in size). The generality of this hypothesis could be tested by carrying out similar analyses on other forms of division of labour, such as the number of cell types in multicellular organisms 9 , 49 , 50 or the difference between queens and workers in superorganismal social insects 48 , 51 .

To test whether colony size, queen mating frequency or number of queens per colony explains variation in non-reproductive division of labour in ants, we performed a large-scale phylogenetic comparative analysis. We first tested which factors correlate with number of worker castes and then used three different methods to examine the causal relationships that underly these correlations: phylogenetic path analysis, transition rate analysis and ancestral state reconstruction. We examined causality in these three different ways as it also allowed us to test that our results are robust to different methods of analysis. We also examined whether our results were robust to alternative methods of quantifying variation in queen mating frequency and number of queens per colony. All the species used in the analysis are listed in the supplementary materials (Supplementary Table 20 ), and we follow the taxonomic classification of ants listed in Bolton’s Catalogue available from AntCat.org 52 . When analysed as continuous variables, colony size, queen mating frequency and number of queens per colony were log 10 transformed for all analyses, while variation in worker size was square root transformed.

Data collection

We collected data on five ant traits: (1) the number of discrete physical worker castes, (2) colony size (number of workers in mature colonies), (3) effective queen mating frequency 21 , 53 (estimated number of mates weighted by the proportion of offspring sired by each male; referred to as ‘queen mating frequency’), (4) observed number of queens per colony and (5) variation in worker size. We started by gathering data from major reviews, comparative studies and books and then performed a topic search of the primary literature using Web of Science 24 , 25 , 26 , 27 . We also incorporated data collected by the Global Ant Genomics Alliance consortium 54 . In total, we collected data for 794 species from 160 different genera.

We collected data on queen mating frequency and number of queens per colony by searching papers published between 1 January 2008 and 2 May 2020. Among these papers, we included data from Hughes et al. 25 , which summarizes the available data on these variables in the eusocial Hymenoptera. We assumed that all relevant data published before 1 January 2008 had been captured by Hughes et al. 25 . We used the following keywords: ‘Ant’ AND (monandr* OR monogyn* OR polyandr* OR polygyn* OR effective-mating-frequenc* OR mating-frequenc* OR paternity-frequenc* OR mating-system* OR sociogenetic-structure*). We chose to use estimates of effective mating frequency, rather than number of copulations, because it more accurately reflects the genetic relatedness among workers in a colony, considering the potential influence of biased sperm use by queens that have mated multiple times 21 , 55 . When multiple estimates of queen mating frequency were available, we calculated species-level averages for queen mating frequency using the harmonic mean. We used the harmonic mean because when mating frequency is low, increases in mating frequency will have greater effects on intra-colony relatedness. Therefore, harmonic means, which assign greater weight to small numbers, better represent the genetic effects of multiple mating on colony-wide relatedness than arithmetic means 56 . We collected data on the average observed number of queens per colony, rather than the effective number of queens per colony 57 (the estimated number of reproductive queens weighted by the respective contribution of each queen to the production of offspring) because this was the only available type of continuous data with a sufficient sample size for a large-scale phylogenetic comparative analysis. When multiple estimates of number of queens per colony were available, we calculated species-level averages. For queen mating frequency and number of queens per colony, we also collected categorical data with three factor levels for each trait from the papers identified in our literature search. These factor levels include obligate single mating (monandry), facultative multiple mating (facultative polyandry) where some monandrous queens still occur in the population and obligate multiple mating (obligate polyandry) where mature colonies are never monandrous. Similarly, for number of queens per colony, these levels were obligate single queen (monogyny), facultative multiple queens (facultative polygyny) and obligate multiple queens (obligate polygyny).

Species were classed as having one to four physical worker castes based on the following criteria: Species were classed as having a single worker caste if they showed limited variation in size or monophasic allometric scaling. Species were classified as having more than one caste if there was variation in body size with non-allometric scaling, where the number of castes depended on the number of different scaling relationships between body parts. When morphometric data were absent, the number of castes was determined using direct estimates provided in the literature. We searched for published data up to and including 13 July 2020, and we used the following keywords: ‘Ant’ AND (worker polymorph* OR worker monomorph* OR (morphometric AND worker* AND caste*) OR subcaste* OR sub-caste* OR worker dimorph* OR major-worker* OR minor-worker* OR worker AND allometr*). We then performed two additional searches to collect missing data for number of worker castes. First, we performed a species-specific literature search for instances where data on queen mating frequency or number of queens per colony was available for a given species but number of castes was not. Our search query included the species name along with the terms ‘(worker polymorph* OR worker monomorph* OR (morphometric AND worker* AND caste*) OR subcaste* OR sub-caste* OR worker dimorph* OR major-worker* OR minor-worker* OR worker AND allometr* OR soldier* OR replete*)’. Second, we performed a search at the genus level for genera where no caste data had been collected. The search query used the same terms as mentioned above, in combination with the genus name.

To collect data on colony size, we performed a literature search focussing on species for which we already possessed data on queen mating frequency or number of queens per colony. Using Google Scholar, we searched for the following keywords in combination with the species name ‘colony size OR colony collection OR worker number’. We adopted this approach based on the methodology outlined in Dornhaus et al. 14 . Google Scholar was chosen as it indexes the entire text of research papers. Notably, details regarding colony size are often not the primary focus of papers, resulting in brief mentions within the methods or results sections. Therefore, it is less likely to be recovered by search engines that rely solely on keywords, titles and abstracts to retrieve such information. When multiple measurements existed for a given species, we calculated arithmetic means weighted by sample size.

We collected data on variation in worker size by measuring the width of ant heads 12 , 13 , 18 , 31 . We quantified the relative variation in worker size for each species by calculating the coefficient of variation (CV), which is achieved by dividing the standard deviation of worker head width by its mean value \(({\rm{CV}}_{{\rm{worker}}\; {\rm{size}}}=\frac{{\rm{worker}}\; {\rm{head}}\; {\rm{width}}\; {\rm{standard}}\; {\rm{deviation}}}{{\rm{worker}}\; {\rm{head}}\; {\rm{width}}\; {\rm{mean}}})\) . To obtain images of the species for which we already possessed data on colony size and queen mating frequency, we searched for scaled images on the AntWeb online database (antweb.org) 58 . We downloaded front-view images of the ants’ heads and measured at the widest point excluding the eyes using the image-processing software ImageJ v1.53i 59 . In total, we measured 1,064 worker ant heads (Extended Data Fig. 4 ). The number of workers measured per species ranged from 2 to 33, with a mean of 7.

We excluded 112 species from our dataset for the analysis based on their distinct life history traits. These traits represent evolutionarily derived elaborations of the ancestral full-sibling-colony state at the transition to superorganismal colonies at the root of the ant clade. They included (1) species that formed supercolonies (vast networks of connected nests); (2) social parasites, lacking some or all of the worker castes (although temporary social parasites which only establish new colonies with the assistance of a host species were included in the analysis); (3) those that can reproduce parthenogenetically to produce queens and/or workers; (4) those that reproduce via gamergates (mated workers that reproduce sexually) following the evolutionary loss of the original queen caste; or (5) those that use interlineage hybridization for genetic caste determination (Supplementary Table 21 ). We gathered the information from papers that had summarized the available data on these five traits 60 , 61 , 62 , 63 , 64 . We excluded these species from the analysis as they represent secondary reductions of complexity in social organization and are likely experiencing different selection pressures for either the evolution of worker castes, colony size, queen mating frequency or number of queens per colony (Supplementary Table 22 ). This decision to remove species characterized by evolutionarily derived social systems improved our ability to detect which variables could influence the evolution of the number of worker castes.

To control for shared evolutionary history in our comparative analyses, we used a posterior sample of 400 phylogenetic trees produced by Economo et al. 65 , which included 14,594 species and of which 731 overlapped with the species in our dataset 65 . However, as time-calibrated, molecular phylogenies are only available for ants at the genus level, the species topology within genera varied widely across the sample of 400 trees. Therefore, we took steps to account for phylogenetic uncertainty in each of our analyses (see each section below for details).

Statistical analyses

All analyses were performed in R v4.2.2 apart from transition rate models that were conducted in Bayestraits V4 29 , 66 .

Bayesian phylogenetic models

We fitted BPMMs with Markov chain Monte Carlo (MCMC) estimation using the MCMCglmm package v2.34 67 . Models were run for a minimum of 1,100,000 iterations, with a burn-in of 100,000 and thinning interval of 1,000. However, models were run for as long as necessary to obtain posterior effective sample sizes of at least 300 for all parameters. Overall, the majority of estimates had an effective sample size of at least 1,000. To ensure model convergence, we used the coda package v0.19-4 to calculate the degree of autocorrelation between successive iterations in each chain. We fitted each model independently two times and used Gelman and Rubin’s convergence test to compare within- and between-chain variance 68 , 69 . We modelled the number of worker castes as a discrete variable for our regression analyses and used a Poisson error distribution with a log-link function. We modelled colony size, queen mating frequency and number of queens per colony as Gaussian traits. The prior settings used for each analysis are specified in the supplementary R code. To select priors for random effects, we first checked model convergence using inverse-Wishart priors ( V  = 1, ν  = 0.002). However, in situations where the MCMC chain showed poor mixing properties, particularly in cases involving discrete response variables, we examined two different parameter expanded priors: the Fisher prior ( V  = 1, ν  = 1, α.μ  = 0, α.V  = 1,000) and the χ 2 prior ( V  = 1, ν  = 1,000, α.μ  = 0, α.V  = 1). We specified an inverse-Wishart prior for residual variances ( V  = 1, ν  = 0.002) 70 . For fixed effects, we used the default priors in MCMCglmm. We report parameter estimates from models as posterior modes along with 95% lower and upper CIs. For BPMMs with a Gaussian error distribution, we also calculated marginal R 2 values and report the median value for each analysis from a distribution of R 2 values estimated across a sample of 400 trees 71 . The reported P values, which assess differences between levels, such as species with a single worker caste versus species with multiple worker castes, represent the number of iterations where one level is greater than the other level divided by the total number of iterations.

Correlational analyses

Estimating phylogenetic correlations using mcmcglmm.

We used BPMMs to test for correlations between four traits: number of worker castes, colony size, queen mating frequency and number of queens per colony. In addition, we investigated the correlations between variation in worker size and colony size, queen mating frequency and number of queens per colony. When analysing variation in worker size, our analysis was split into two parts. The first part examined correlations across all species for which worker size data were available, while the second part specifically examined species that have a single worker caste. To account for phylogenetic uncertainty in our BPMMs, we repeated each analysis 400 times, each time with a different tree, and combined the posterior samples produced from each tree before parameter estimation. Model convergence was assessed as described in the section ‘Statistical analyses’.

Causality analyses: examining colony size and levels of polyandry that preceded the evolution of multiple worker castes

Phylogenetic path analysis analysing number of discrete physical worker castes.

To test how the evolution of number of worker castes was influenced by colony size and queen mating frequency, we used phylogenetic path analysis 28 , 72 . This method compares alternative models of the causal relationships between traits, disentangling direct from indirect effects. After removing species with incomplete data for all variables, we analysed the data for a total of 94 species. By comparing four different causal models, we tested between the following possibilities: (1) variation in colony size directly influenced the evolution of number of worker castes, but queen mating frequency did not; (2) variation in number of worker castes influenced the evolution of colony size, which then predicted queen mating frequency; (3) variation in both colony size and queen mating frequency directly influenced the evolution of number of worker castes; and (4) variation in queen mating frequency directly influenced the evolution of number of worker castes, but colony size did not (list of full models in Extended Data Fig. 1 ). In each of the causal models tested, we assumed that the evolution of queen mating frequency was influenced by variation in both the number of queens per colony and colony size. We made these assumptions because (1) it has been suggested that the evolution of queen mating frequency has been influenced by variation in the number of queens per colony 21 , 25 and (2) larger colony sizes may select for multiple mating 40 , 41 . Larger colonies may be at greater risk of infection from pathogens, and multiple mating may offer protection against disease because it increases genetic diversity within colonies 37 , 40 , 41 , 42 . However, multiple mating is a costly trait, and therefore, we hypothesized that larger colony sizes may evolve first and then select for multiple mating if the benefits of multiple mating outweigh the costs 73 . We excluded the relationship between the number of queens per colony and either the number of castes or colony size in our models as we found no correlation between these variables in our regression analysis (section ‘Phylogenetic correlations’).

Based on the causal relationships presented in the alternative path analysis models, we generated conditional independence statements that can be formulated and tested as a set of phylogenetic generalized linear models. For each model, we calculated Fisher’s C statistic and conducted the d-sep test 28 . P values less than 0.05 indicate that a proposed candidate model should be rejected. Next, we compared the fit of different models using CICc scores, where the lowest score represents the best candidate model 74 . Models that have ΔCICc < 2 were considered to have equal support. We used a model-averaging approach to estimate path coefficients that assigned weights to causal links based on the CICc weight ( ω ) of the supported models (ΔCICc < 2). When a path was absent in a model, we assumed its coefficients and variance to be zero for the purpose of model averaging. We transformed the number of castes into a binary variable (single worker caste/multiple worker castes) for the analysis as the phylopath package does not support discrete count data as response variables. We then used a binomial family for binary response variables when modelling number of worker castes. To the best of our knowledge, there is currently no established method for combining results obtained across different trees to account for phylogenetic uncertainty in frequentist analyses. Therefore, we performed our analysis four times, each time using a different tree, from a sample of four maximum clade credibility (MCC) trees produced by Economo et al. 65 . Each MCC tree was constructed from a sample of 100 trees. We then compared the results from each of the four analyses to determine whether our results remained consistent irrespective of the phylogeny that was used. In the main text, we present the results from a single tree as we found that the conclusions drawn across trees remained unchanged due to the similarity in the presence, direction and magnitude of the causal relationships identified. We present the results using the alternative MCC trees in the supplementary material (Extended Data Fig. 2 and Supplementary Tables 2 and 3 ). All phylogenetic path analyses were carried out using phylopath v1.1.3 in R 75 .

Phylogenetic path analysis analysing variation in worker size

We repeated our analysis, examining variation in worker size instead of number of worker castes. We used head width as a measure of worker size and modelled it as continuous variable. Models were constructed using Pagel’s λ 76 for the associated error structure, and we compared the same four causal models that were used in the previous path analysis (Extended Data Fig. 1 ). To investigate whether differences in the number of worker castes were driving the results when analysing worker size, we performed the analysis in two different ways: (1) with the full set of species for which we had data available for all four traits ( n species  = 94) and (2) limiting our analysis to only species that possess a single worker caste (monomorphic; n species  = 60). To account for phylogenetic uncertainty, we repeated each analysis using the same four MCC trees used in the previous path analysis. In the main text, we present the results from a single tree as we found that the conclusions drawn across trees remained unchanged due to the similarity in the presence, direction and magnitude of the causal relationships identified. We present the results for both analyses using the alternative MCC trees in the supplementary material (Extended Data Figs. 5 and 6 and Supplementary Tables 7 – 10 ).

Testing models of dependent versus independent evolution and estimating evolutionary transition rates using BayesTraits

We tested for correlated evolution between number of worker castes and both colony size and queen mating frequency. We used the Discrete module with reverse jump MCMC estimation implemented in BayesTraits V4. This method requires pairs of binary traits, and so we transformed all traits into binary variables. We classified species as having either a single worker caste or multiple worker castes, to focus on the evolution of multiple castes. We classified species as having queens that mated with less than or equal to two males (≤2 males) or more than two males (>2 males), because (1) the influence of larger numbers of matings on relatedness is diminishing; (2) an effective mating frequency value of 2 leads to the average level of relatedness among workers in the colony being midway between the maximal and minimal values of 0.75 and 0.25 that are possible with a single queen; and (3) mating with 2 males is the threshold at which worker policing is favoured 20 . We classified species as having either small or large colonies depending upon whether colony size was smaller (≤300) or larger (>300) than the median colony size. To assess the sensitivity of our results to different colony size classification thresholds, we repeated our analysis, but divided species into classes based on the 40th/60th and 60th/40th quantile boundaries. Across these thresholds, we obtained the same result that large colony size was necessary for the evolution of multiple worker castes. Therefore, we present the results of the 40th/60th and 60th/40th only in the supplementary material (Supplementary Tables 23 and 24 ). In all cases, we tested the fit of the independent model of evolution against the dependent model of evolution using BFs (2 × (log(likelihood of the dependent model) − log(likelihood of independent model))). We estimated the marginal likelihood of both the independent model and the dependent model using a stepping-stone sampler 77 . As recommended in the Bayestraits manual, we scaled tree branch lengths by a factor of 0.001. This helps prevent the estimated transition rates from becoming very small, which can make the values hard to estimate or search for. We accounted for phylogenetic uncertainty in our analysis by resampling each iteration from the posterior distribution of 400 trees to estimate parameters.

To reduce uncertainty over prior selection, we used a hyper prior approach to seed the mean and variance of an exponential prior 29 . The values for the hyper priors were drawn from a uniform distribution ranging from 0 to 100, based on the estimated range of transition rates determined by using analyses with maximum likelihood estimation. In addition, we investigated the sensitivity of our models to the selection of priors by running models with gamma priors initialized using hyper priors. We found that the conclusions drawn between models using different priors remained the same. Therefore, we only present the results from the models using the gamma priors in the supplementary materials (Supplementary Tables 25 and 26 ). All models were run for at least 11,000,000 iterations with 1,000,000 iterations of burn-in and a thinning interval of 5,000. However, some models were run for 110,000,000 with 10,000,000 iterations of burn-in and a thinning interval of 50,000 iterations to improve chain convergence and mixing. Model convergence was assessed using the same approach described in ‘Statistical analyses’.

We identified correlated evolution between colony size and number of worker castes. Therefore, using the transition rates estimated by the model, we examined whether transitions to multiple worker castes were influenced by small or large colony sizes, as well as whether transitions to large colony sizes were influenced by number of worker castes. We found that queen mating frequency and number of worker castes evolved independently. Therefore, we did not explore the potential causal relationship between these traits. To determine the likelihood of transitions occurring, we examined the proportion of models visited by the reverse jump MCMC algorithm in which the rates were set to zero (Supplementary Tables 4 , 5 , 18 and 23 – 26 ).

Ancestral state reconstructions

We used ancestral state reconstruction to examine how colony size and queen mating frequency differed between the ancestors of lineages with a single caste and lineages with multiple castes 30 . First, we reconstructed the ancestral number of worker castes (single/multiple castes) to identify transitions between single and multiple worker castes. Second, we tested whether species that evolved multiple worker castes had greater estimated colony sizes and queen mating frequencies. To account for potential variation in the rate of caste evolution across the ant phylogeny, we used a hidden Markov model technique called hidden rate models (HRMs) to reconstruct the ancestral numbers of castes 78 . This method enabled us to incorporate heterogeneity in the loss and gain rates of our reconstruction. We used the R package corHMM v2.8 to examine HRMs with one to three rate classes and generated both equal rates and all-rates-different models of evolution for each rate class 78 . To determine the best HRM model from the candidate set, we examined corrected Akaike Information Criteria (AICc) values for each model across 400 trees using all available species with data on the number of castes. We selected the equal rates model with two rate categories because it had the lowest AICc value over 57.5% of the 400 trees. We then ran the equal rates model with two rate categories for each analysis, examining the relationship between colony size and number of worker castes and queen mating frequency and number of worker castes. Next, we classified each node in the phylogeny depending on which state (single/multiple castes) had the highest likelihood, resulting in four possible classification categories for each node: (1) single caste with all descendants having a single caste, (2) single caste with at least one descendant having multiple castes, (3) multiple castes with all descendants having multiple castes and (4) multiple castes with at least one descendant having a single caste.

The nodal classifications were entered as an explanatory variable in a BPMM and treated as a fixed factor with four levels. Depending on whether we were analysing the relationship between number of worker castes and either colony size or queen mating frequency, we specified either colony size or queen mating frequency as the response variable and used a phylogenetic covariance matrix that was linked to ancestral nodes as a random effect. We removed the global intercept and estimated the colony size and queen mating frequency values before the evolutionary origin of multiple worker castes (comparison of classification 1 versus 2). To account for phylogenetic uncertainty in our analysis, we repeated this process 400 times, once for each tree from our sample of 400 phylogenetic trees. We then combined the resulting posterior samples across the models to calculate parameter estimates. Accurate estimation of ancestral colony size and queen mating frequency values required phylogenetic signal in these traits. Using BPMMs, we estimated the amount of variation in each trait that can be attributed to shared ancestry between species, calculated as phylogenetic heritability ( \({{{\rm{phylo}}\; h}}^{2}=\left(\frac{{V}_{{\rm{A}}}}{{V}_{{\rm{A}}}+{V}_{{\rm{R}}}}\right)\times 100\) , where V A  = phylogenetic variance, V R  = residual variance). For each trait, we repeated the analysis 400 times, each time with a different tree, and combined the posterior samples of the variance components before phylogenetic heritability was estimated.

In addition, we estimated the ancestral number of discrete worker castes (one to four worker castes) using HRMs. We ran each model over a sample of 400 trees and used the same model selection approach for HRMs as described above. The symmetric model with 2 rate classes performed the best over 58.5% of trees. We then estimated the frequency of transitions between each number of castes, classifying each node based on the state with the highest likelihood estimate.

Sensitivity analysis

To assess the robustness of our findings to different methods of quantifying queen mating frequency and number of queens per colony, we performed a sensitivity analysis. We repeated our phylogenetic correlations analysis, treating queen mating frequency and number of queens per colony as categorical variables with three factor levels. This approach was chosen based on evidence indicating that these categorizations may reflect discrete reproductive strategies that may have evolved for different reasons and align with previous studies 21 , 25 , 33 , 79 , 80 . They were categorized as follows: obligate single mating (monandry), facultative multiple mating (facultative polyandry) and obligate multiple mating (obligate polyandry); and obligate single queen (monogyny), facultative multiple queens (facultative polygyny) and obligate multiple queens (obligate polygyny) 79 , 80 .

We also repeated our phylogenetic path analysis and transition rate analysis. However, we transformed queen mating frequency and number of queens per colony into discrete binary variables, as these analytical methods cannot accept categorical variables with more than two factor levels. Species were classified as being obligately polyandrous or not, and obligately polygynous or not. We combined monandry with facultative polyandry, and monogyny with facultative polygyny to avoid excluding a large proportion of species from our analysis. We combined these categories because species that are obligately polyandrous or polygynous have made an evolutionary transition where mature colonies can no longer possess a monandrous or monogynous queen. By contrast, species that are facultatively polyandrous or polygynous may still show monandry or monogyny. Detailed statistics for each analysis are provided in the supplementary material (Extended Data Figs. 8 – 10 and Supplementary Tables 11 – 19 ).

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Data availability

All data are provided in Supplementary Table 20 and are made available, along with their associated references, via the public repository, Oxford University Research Archive (ORA) at https://ora.ox.ac.uk/objects/uuid:453278a3-61b9-4144-b08a-98d9379d5ce8 (ref. 81 ).

Code availability

R code for analyses is available via GitHub at https://github.com/LouisBell-Roberts/Larger_colony_sizes_favoured_the_evolution_of_more_worker_castes_in_ants .

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Acknowledgements

We thank R. Bonifacii, J. Green, T. Scott, A. Dewar and M. Liu for their helpful comments, M. Liu for producing ant silhouettes, C. Cornwallis for advice on performing ancestral state reconstruction and M. Brindle for advice on performing transition rate analysis. We thank the Global Ant Genomics Alliance consortium for making their data on ant life history traits available to us, and the Biotechnology and Biological Sciences Research Council (BB/M011224/1: L.B.-R.) and European Research Council (834164: S.A.W. and J.F.R.T.) for funding.

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Louis Bell-Roberts, Juliet F. R. Turner, Gijsbert D. A. Werner & Stuart A. West

Netherlands Scientific Council for Government Policy, The Hague, The Netherlands

Gijsbert D. A. Werner

Ecology & Genetics Research Unit, University of Oulu, Oulu, Finland

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Institute of Ecology and Evolution, University of Edinburgh, Edinburgh, UK

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Contributions

Conceptualization: S.A.W., L.R. and G.D.A.W. Data collection: L.B.-R., J.F.R.T. and L.R. Methodology: L.B.-R., J.F.R.T., S.A.W., G.D.A.W. and P.A.D. Investigation: L.B.-R., J.F.R.T., S.A.W. and G.D.A.W. Visualization: L.B.-R., J.F.R.T. and S.A.W. Funding acquisition: S.A.W. Supervision: S.A.W. and G.D.A.W. Writing—original draft: L.B.-R., J.F.R.T. and S.A.W. Writing—review and editing: L.B.-R., J.F.R.T., S.A.W., G.D.A.W., P.A.D. and L.R.

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Extended data

Extended data fig. 1 the full set of potential causal models tested in the phylogenetic path analyses illustrated using directed acyclic graphs (dags)..

In different path analyses, division of labour was modelled either as the number of discrete physical worker castes (Fig. 4 ) or as variation in worker size (Fig. 6b ).

Extended Data Fig. 2 Average of the best-performing models where division of labour analysed as the number of physical worker castes (single caste/multiple castes).

Analysis repeated over four different MCC ant trees from Economo et al. 65 : a ) NC uniform stem b ) NC uniform crown c ) FBD stem d ) FBD crown. Values represent standardised average coefficients. CN = number of worker castes; CS = colony size; MF = queen mating frequency; QN = number of queens per colony.

Extended Data Fig. 3 Ancestral state reconstruction: larger colony sizes favour the evolution of multiple worker castes.

( a ) The posterior distribution from a BPMM estimating colony size in ancestors with a single worker caste that led to descendants with a single caste (single to single) or multiple castes (single to multiple). Colony size is larger in the ancestors of lineages that evolve multiple castes, P single vs multiple  = 0.003 (n species = 436). ( b ) The posterior distribution from a BPMM estimating queen mating frequency in ancestors with a single worker caste that led to descendants with a single caste (single to single) or multiple castes (single to multiple). Queen mating frequency was not significantly different in the ancestors of lineages that evolved multiple castes and the ancestors of lineages that retained a single caste, P single vs multiple  = 0.16 (n species = 104).

Extended Data Fig. 4 Measurement of ant head width.

We downloaded front-view images of ant heads from www.antweb.org and measured at the widest point excluding the eyes (horizontal white line) using the image-processing software ImageJ. © California Academy of Sciences. Clockwise from top left, images by Erin Prado, Erin Prado, Will Ericson, and Z. Lieberman. Specimen codes: CASENT0179923, CASENT0179924, CASENT0280193, CASENT0911107. All images licensed under CC BY 4.0. https://creativecommons.org/licenses/by/4.0/ .

Extended Data Fig. 5 Average of the best-performing models where division of labour analysed as variation in worker size.

Analysis performed over all species for which data on all four traits was available. Analysis repeated over four different MCC ant trees from Economo et al. 65 a ) NC uniform stem b ) NC uniform crown c ) FBD stem d ) FBD crown. Values represent standardised average coefficients. SV = variation in worker size (degree of worker polymorphism); CS = colony size; MF = queen mating frequency; QN = number of queens per colony.

Extended Data Fig. 6 Average of the best-performing models where division of labour analysed as variation in worker size and only using species with a single physical worker caste.

Analysis repeated over four different MCC ant trees from Economo et al. 65 : a ) NC uniform stem b ) NC uniform crown c ) FBD stem d ) FBD crown. Values represent standardised average coefficients. SV = variation in worker size (degree of worker polymorphism); CS = colony size; MF = queen mating frequency; QN = number of queens per colony.

Extended Data Fig. 7 Relationships between queen mating frequency and number of queens per colony with variation in worker size.

Species with higher queen mating frequencies exhibited significantly greater variation in worker size ( a , BPMM: β = 0.12, CI = 0.01 to 0.23, n species  = 105). We did not find a significant association between number of queens per colony and variation in worker size ( b , BPMM: β = −0.03, CI = −0.12 to 0.07, n species  = 106). When reanalysing our data so that we only included species with a single worker caste ( c - d ), we found no association between either queen mating frequency or number of queens per colony with variation in worker size (c, BPMMs: queen mating frequency: β = −0.00, CI = −0.10 to 0.10, n species  = 65; d, queen number: β = −0.02, CI = −0.12 to 0.07, n species  = 68). Fitted lines are mean regression slopes with 95% CIs from BPMMs using a single phylogenetic tree. Solid regression lines represent significant relationships, while dashed regression lines represent non-significant relationships. Dots represent species averages. Axes for queen mating frequency and number of queens per colony are plotted on a log 10 scale. Axes for variation in worker size are on a square root scale.

Extended Data Fig. 8 Average of the best-performing models where division of labour analysed as the number of physical worker castes (single caste/multiple castes).

Queen mating frequency (obligately monandrous & facultatively polyandrous or obligately polyandrous) and number of queens per colony (obligately monogynous & facultatively polygynous or obligately polygynous) were analysed as binary traits. Analysis repeated over four different MCC ant trees from Economo et al. 65 : a ) NC uniform stem b ) NC uniform crown c ) FBD stem d ) FBD crown. However, when using the FBD stem tree for the analysis, the model did not converge, and we exclude this result. Values represent standardised average coefficients. CN = number of worker castes; CS = colony size; MF = queen mating frequency; QN = number of queens per colony.

Extended Data Fig. 9 Average of the best-performing models where division of labour analysed as variation in worker size.

Analysis performed over all species for which data on all four traits was available. Queen mating frequency (obligately monandrous & facultatively polyandrous or obligately polyandrous) and number of queens per colony (obligately monogynous & facultatively polygynous or obligately polygynous) were analysed as binary traits. Analysis repeated over four different MCC ant trees from Economo et al. 65 a ) NC uniform stem b ) NC uniform crown c ) FBD stem d ) FBD crown. Values represent standardised average coefficients. SV = variation in worker size (degree of worker polymorphism); CS = colony size; MF = queen mating frequency; QN = number of queens per colony.

Extended Data Fig. 10 Average of the best-performing models where division of labour analysed as variation in worker size and only using species with a single physical worker caste.

Queen mating frequency (obligately monandrous & facultatively polyandrous or obligately polyandrous) and number of queens per colony (obligately monogynous & facultatively polygynous or obligately polygynous) were analysed as binary traits. Analysis repeated over four different MCC ant trees from Economo et al. 65 : a ) NC uniform stem b ) NC uniform crown c ) FBD stem d ) FBD crown. However, when using the NCuniform stem tree for the analysis, the model did not converge, and we exclude this result. Values represent standardised average coefficients. SV = variation in worker size (degree of worker polymorphism); CS = colony size; MF = queen mating frequency; QN = number of queens per colony.

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Bell-Roberts, L., Turner, J.F.R., Werner, G.D.A. et al. Larger colony sizes favoured the evolution of more worker castes in ants. Nat Ecol Evol (2024). https://doi.org/10.1038/s41559-024-02512-7

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DOI : https://doi.org/10.1038/s41559-024-02512-7

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