an example of alternative hypothesis

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• Null and Alternative Hypotheses | Definitions & Examples

Null and Alternative Hypotheses | Definitions & Examples

Published on 5 October 2022 by Shaun Turney . Revised on 6 December 2022.

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 ): There’s an effect in the population.

The effect is usually the effect of the independent variable on the dependent variable .

Table of contents

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

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.” On the other hand, 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.

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.

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 ≤).

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.

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.

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

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.

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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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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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

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Methodology

• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

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

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

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

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

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

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

Variables in hypotheses

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

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

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

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

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Step 1. ask a question.

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

Step 2. Do some preliminary research

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

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

Step 3. Formulate your hypothesis

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

4. Refine your hypothesis

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

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

5. Phrase your hypothesis in three ways

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

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

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

6. Write a null hypothesis

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

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

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

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

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

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

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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.

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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

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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.

• Alternative vs Null Hypothesis: Pros, Cons, Uses & Examples

All research starts with a problem that needs to be solved. From this problem, hypotheses are developed to provide the researcher with a clear statement of the problem.

To understand alternative hypotheses also known as alternate hypotheses, you must first understand what the hypothesis is .

When you hear the word hypothesis it means the accurate explanations in relation to a set of facts that can be analyzed when studied, using some specific method of research.

There are primarily two types of hypothesis which are null hypothesis and alternative hypothesis.

When you think about the word “null” what should come to mind is something that can not change, what you expect is what you get, unlike alternate hypotheses which can change.

Now, the research problems or questions which could be in the form of null hypothesis or alternative hypothesis are expressed as the relationship that exists between two or more variables. The process for this states that the questions should be what expresses the relationship between two variables that can be measured.

Both null hypotheses and alternative hypotheses are used by statisticians and researchers to conduct research in various industries or fields such as mathematics, psychology, science, medicine, and technology.

We are going to discuss alternative hypotheses and null hypotheses in this post and how they work in research.

What is an Alternative Hypothesis?

Alternative hypothesis simply put is another viable option to the null hypothesis. It means looking for a substantial change or option that can allow you to reject the null hypothesis.

It is an opposing theory to a null hypothesis.

If you develop a null hypothesis, you make an informed guess on whether a thing is true or whether there is a relationship between that thing and another variable. An alternate hypothesis will always take an opposite stand against a null hypothesis. So if according to a null hypothesis something is correct to an alternate hypothesis that same thing will be incorrect.

For example, let’s assume that you develop a null hypothesis that states “I”m going to be $500 richer” the alternate hypothesis will be “I’m going to get $500 or be richer”

When you are trying to disprove a null hypothesis, that is when you test an alternate hypothesis. If there is enough data to back up the alternative hypothesis then you can dispose of the null hypothesis. 

Get Answers: What is Empirical Research Study? [Examples & Method]

What is a Null Hypothesis?

The null hypothesis is best explained as the statement showing that no relationship exists between two variables that are being considered or that two groups are not related. As we have earlier established, a hypothesis is an assumed statement that has not been proven with sufficient data that could serve as a piece of evidence. 

The null hypothesis is now the statement that a researcher or an investigator wants to disprove. The null hypothesis is capable of being tested, being verifiable, and also capable of being rejected.

For example, if you want to conduct a study that will compare the relationship between project A and project B if the study is based on the assumption that both projects are of equal standard, the assumption is referred to as the null hypothesis.

This is because the null hypothesis should be specific at all times.

Learn: Hypothesis Testing in Research: Definition, Procedure, Uses, Limitations + Examples

Advantages of the Alternative Hypothesis 

• Alternative hypothesis gives a researcher specific clarifications on the research questions or problems .
• It provides a study with the direction that can be used to collect data and obtain results of interest by the researcher.
• An alternative hypothesis is always selected before commencing the studies which gives the researcher the opportunity to prove that the restatement is backed up by evidence and not just from the researcher’s ideas or values.
• Another good thing about alternative hypotheses is that it provides 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 .
• An alternate hypothesis is also useful to prove that there is a relationship between two selected variables and the outcomes of the conducted study are relevant.

Principles of the Alternative Hypothesis

• Alternative hypotheses will be accepted if the amount of data that is gone is insignificant within the significance level. This means that the null hypothesis will be rejected.
• Another principle of the alternative hypothesis is that the data gathered from random samples go through a statistical tool that analyzes the effect of the amount of data leaving the null hypothesis.

For the curious: Sampling Bias: Definition, Types + [Examples]

Purpose of the Null Hypothesis 

Here are the purposes of the null hypothesis in an experiment or study:

• The primary purpose of a null hypothesis is to disprove an assumption.
• Null hypotheses can help to further progress a theory in some scientific cases.
• You can also use a null hypothesis to ascertain how consistent the outcomes of multiple studies are.

Principle of the Null Hypothesis 

Now, these are the principles of the null hypothesis:

1. The primary principle of the null hypothesis is to prove that the assumed statement is true. This is done by collecting data and analyzing in the study , what chance the collected data has in the random sample.

2. If the collected data does not meet the expectation of the null hypothesis, it is determined that the data lacks sufficient evidence to back up the null hypothesis therefore the null hypothesis statement is rejected.

Just as in the case of the alternative hypothesis the collected data in a null hypothesis is analyzed using some statistical tools that are made to measure the extent to which data left the null hypothesis.

The process will determine whether the data that left the null hypothesis is larger than a set value. If the data collected from the random sample is enough to serve as evidence to prove the null hypothesis then the null hypothesis will be accepted as true. And also defined that it has no relationship with other variables .

Learn About: Research Reports: Definition, Types + [Writing Guide]

Types of Alternative Hypothesis (Advantages of Each and When to Use)

There are four types of alternative hypotheses, and we will briefly discuss them below.

• One-tailed directional: For one of the sampling distributions one tail, this type of alternative hypothesis focuses on the rejected part only.
• Two-tailed directional: In an alternative hypothesis, a two-tailed directional focus on the two parts or directions that were rejected in the sampling distribution.
• Point: Point is another alternate hypothesis. It occurs in hypothesis testing when the sample population in the alternate hypothesis has been completely defined in a distribution. If there are no known parameters, the hypothesis will serve no interest. They are, however, important to the foundation of the statistical inferences.
• Non-directional: In an alternate hypothesis, a non-directional does not focus on the two directions of rejection. The only focus of the nondirectional alternative hypothesis is to prove that the null hypothesis is incorrect.

Read: Type I vs Type II Errors: Definition, Examples & Prevention

Difference between Null Hypothesis and Alternative Hypothesis 

We are going to look at the differences between the alternate hypothesis and the null hypothesis based on these six factors which are:

• Mathematical expression
• Observation
• Acceptance criteria
• The difference in Mathematical expression

Null hypothesis is followed by an ‘equals to’ (=) sign. While the Alternative hypothesis is followed by these three signs; 

• The difference in how they are observed

In the null hypothesis, it is believed that the results that are observed are as a result of chance. While In the alternative hypothesis, it is believed that the observed results are the outcome of some real causes.

• Differences in results

The result of the null hypothesis always shows that there have been no changes in statements or opinions. While the result of the alternative hypothesis shows that there have been significant changes in statements and opinions.

• Differences in Acceptance criteria

If the p-value in a null hypothesis is greater than the significance level, then the null hypothesis is accepted.

If the p-value in an alternate hypothesis is smaller than the significance level, then the alternative hypothesis is accepted.

• Differences in importance

The null hypothesis accepts true existing theories and also if there has been consistency in multiple experiments of similar hypotheses.

The alternative hypothesis establishes whether a relationship exists between two variables, and the result will then lead to new improved theories.

Read: T-testing: Definition, Formula & Interpretations

Examples of an Alternative Hypothesis and Null Hypothesis

Here are some examples of the alternative hypothesis:.

A researcher assumes that a bridge’s bearing capacity is over 10 tons, the researcher will then develop an hypothesis to support this study. The hypothesis will be:

For the null hypothesis H0: µ= 10 tons

For the alternate hypothesis Ha: µ>10 tons

In another study being conducted, the researcher wants to find out whether there is a noticeable difference or change in a patient’s heart arrest medicine and the patient’s heart condition.

For the alternate hypothesis: The hypothesis is that there might indeed be a relationship between the new medicine and the frequency or chances of heart arrest in a patient.

Here are the examples of the null hypothesis

The hypothesis from example 2 in the alternate hypothesis implies that the use of one specific medicine can reduce the frequency and chances of heart arrest.

For the null hypothesis: The hypothesis will be that the use of that particular medicine cannot reduce the chance and frequency of heart arrest in a patient.

An alternate hypothesis states that the random exam scores are collected from both men and women. But are the scores of the two groups (men and women) the same or are they different?

For the null hypothesis: The hypothesis will state that the calculated mean of the men’s exam score is equal to the exam score of the women.

This is represented as

H0= The null hypothesis

µ1= The calculated mean score of men

µ2= The calculated mean score of women

Read: What is Empirical Research Study? [Examples & Method]

• Can you reject an alternative hypothesis?

It is quite inappropriate to say or report that an alternate hypothesis was rejected. It is much better to use the phrase “the alternate hypothesis was rather not supported”.

The reason behind this use of words is that only the null hypothesis is designed to be rejected in a study. The alternative hypothesis is designed to prove the null hypothesis incorrect, to introduce new facts that can disprove the null hypothesis but it is not designed to be rejected.

It can either be accepted or not supported.

• How do you identify alternative hypotheses?

A researcher can use this formula to identify the alternate hypothesis in a study or experiment.

H0 and Ha are in contrast.

Therefore, if Ho has:

Equal to (=)

Greater than or equal to (≥)

Less than or equal to (≤)

And then Ha has:

Not equal (≠) 

Greater than (>) or less than (

Less than ( )

If in a study, α ≤ p-value, then the researcher should not reject H0.

If in a study, α > p-value, then the researcher should reject H0.

α is preconceived. The value of α is determined even before the hypothesis test is conducted. While the p-value is derived from the calculation in the data.

• Which is better in formulating hypotheses of your study alternative or null?

The study a researcher wants to conduct will determine what hypothesis should be developed. However, the researcher should keep in mind what the purpose of the null and alternative two hypotheses are while developing the study hypothesis. So while the null hypothesis will accept existing theories that it found to be true or correct, and measure the consistency of multiple experiments, alternative hypotheses will find the relationship that exists (if any) between two phenomena and may lead to the development of a new and improved theory.

In this article, it has been clearly defined the relationship that exists between the null hypothesis and the alternative hypothesis. While the null hypothesis is always an assumption that needs to be proven with evidence for it to be accepted, the alternative hypothesis puts in all the effort to make sure the null hypothesis is disproved. 

Researchers should note that for every null hypothesis, one or more alternate hypotheses can be developed.

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You may also like:

Type I vs Type II Errors: Causes, Examples & Prevention

This article will discuss the two different types of errors in hypothesis testing and how you can prevent them from occurring in your research

Extrapolation in Statistical Research: Definition, Examples, Types, Applications

In this article we’ll look at the different types and characteristics of extrapolation, plus how it contrasts to interpolation.

Acceptance Sampling: Meaning, Examples, When to Use

In this post, we will discuss extensively what acceptance sampling is and when it is applied.

Hypothesis Testing: Definition, Uses, Limitations + Examples

The process of research validation involves testing and it is in this context that we will explore hypothesis testing.

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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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Research Hypothesis In Psychology: Types, & Examples

Saul McLeod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul McLeod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

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On This Page:

A research hypothesis, in its plural form “hypotheses,” is a specific, testable prediction about the anticipated results of a study, established at its outset. It is a key component of the scientific method .

Hypotheses connect theory to data and guide the research process towards expanding scientific understanding

Some key points about hypotheses:

• A hypothesis expresses an expected pattern or relationship. It connects the variables under investigation.
• It is stated in clear, precise terms before any data collection or analysis occurs. This makes the hypothesis testable.
• A hypothesis must be falsifiable. It should be possible, even if unlikely in practice, to collect data that disconfirms rather than supports the hypothesis.
• Hypotheses guide research. Scientists design studies to explicitly evaluate hypotheses about how nature works.
• For a hypothesis to be valid, it must be testable against empirical evidence. The evidence can then confirm or disprove the testable predictions.
• Hypotheses are informed by background knowledge and observation, but go beyond what is already known to propose an explanation of how or why something occurs.

Predictions typically arise from a thorough knowledge of the research literature, curiosity about real-world problems or implications, and integrating this to advance theory. They build on existing literature while providing new insight.

Types of Research Hypotheses

Alternative hypothesis.

The research hypothesis is often called the alternative or experimental hypothesis in experimental research.

It typically suggests a potential relationship between two key variables: the independent variable, which the researcher manipulates, and the dependent variable, which is measured based on those changes.

The alternative hypothesis states a relationship exists between the two variables being studied (one variable affects the other).

A hypothesis is a testable statement or prediction about the relationship between two or more variables. It is a key component of the scientific method. Some key points about hypotheses:

• Important hypotheses lead to predictions that can be tested empirically. The evidence can then confirm or disprove the testable predictions.

In summary, a hypothesis is a precise, testable statement of what researchers expect to happen in a study and why. Hypotheses connect theory to data and guide the research process towards expanding scientific understanding.

An experimental hypothesis predicts what change(s) will occur in the dependent variable when the independent variable is manipulated.

It states that the results are not due to chance and are significant in supporting the theory being investigated.

The alternative hypothesis can be directional, indicating a specific direction of the effect, or non-directional, suggesting a difference without specifying its nature. It’s what researchers aim to support or demonstrate through their study.

Null Hypothesis

The null hypothesis states no relationship exists between the two variables being studied (one variable does not affect the other). There will be no changes in the dependent variable due to manipulating the independent variable.

It states results are due to chance and are not significant in supporting the idea being investigated.

The null hypothesis, positing no effect or relationship, is a foundational contrast to the research hypothesis in scientific inquiry. It establishes a baseline for statistical testing, promoting objectivity by initiating research from a neutral stance.

Many statistical methods are tailored to test the null hypothesis, determining the likelihood of observed results if no true effect exists.

This dual-hypothesis approach provides clarity, ensuring that research intentions are explicit, and fosters consistency across scientific studies, enhancing the standardization and interpretability of research outcomes.

Nondirectional Hypothesis

A non-directional hypothesis, also known as a two-tailed hypothesis, predicts that there is a difference or relationship between two variables but does not specify the direction of this relationship.

It merely indicates that a change or effect will occur without predicting which group will have higher or lower values.

For example, “There is a difference in performance between Group A and Group B” is a non-directional hypothesis.

Directional Hypothesis

A directional (one-tailed) hypothesis predicts the nature of the effect of the independent variable on the dependent variable. It predicts in which direction the change will take place. (i.e., greater, smaller, less, more)

It specifies whether one variable is greater, lesser, or different from another, rather than just indicating that there’s a difference without specifying its nature.

For example, “Exercise increases weight loss” is a directional hypothesis.

Falsifiability

The Falsification Principle, proposed by Karl Popper , is a way of demarcating science from non-science. It suggests that for a theory or hypothesis to be considered scientific, it must be testable and irrefutable.

Falsifiability emphasizes that scientific claims shouldn’t just be confirmable but should also have the potential to be proven wrong.

It means that there should exist some potential evidence or experiment that could prove the proposition false.

However many confirming instances exist for a theory, it only takes one counter observation to falsify it. For example, the hypothesis that “all swans are white,” can be falsified by observing a black swan.

For Popper, science should attempt to disprove a theory rather than attempt to continually provide evidence to support a research hypothesis.

Can a Hypothesis be Proven?

Hypotheses make probabilistic predictions. They state the expected outcome if a particular relationship exists. However, a study result supporting a hypothesis does not definitively prove it is true.

All studies have limitations. There may be unknown confounding factors or issues that limit the certainty of conclusions. Additional studies may yield different results.

In science, hypotheses can realistically only be supported with some degree of confidence, not proven. The process of science is to incrementally accumulate evidence for and against hypothesized relationships in an ongoing pursuit of better models and explanations that best fit the empirical data. But hypotheses remain open to revision and rejection if that is where the evidence leads.

• Disproving a hypothesis is definitive. Solid disconfirmatory evidence will falsify a hypothesis and require altering or discarding it based on the evidence.
• However, confirming evidence is always open to revision. Other explanations may account for the same results, and additional or contradictory evidence may emerge over time.

We can never 100% prove the alternative hypothesis. Instead, we see if we can disprove, or reject the null hypothesis.

If we reject the null hypothesis, this doesn’t mean that our alternative hypothesis is correct but does support the alternative/experimental hypothesis.

Upon analysis of the results, an alternative hypothesis can be rejected or supported, but it can never be proven to be correct. We must avoid any reference to results proving a theory as this implies 100% certainty, and there is always a chance that evidence may exist which could refute a theory.

How to Write a Hypothesis

• Identify variables . The researcher manipulates the independent variable and the dependent variable is the measured outcome.
• Operationalized the variables being investigated . Operationalization of a hypothesis refers to the process of making the variables physically measurable or testable, e.g. if you are about to study aggression, you might count the number of punches given by participants.
• Decide on a direction for your prediction . If there is evidence in the literature to support a specific effect of the independent variable on the dependent variable, write a directional (one-tailed) hypothesis. If there are limited or ambiguous findings in the literature regarding the effect of the independent variable on the dependent variable, write a non-directional (two-tailed) hypothesis.
• Make it Testable : Ensure your hypothesis can be tested through experimentation or observation. It should be possible to prove it false (principle of falsifiability).
• Clear & concise language . A strong hypothesis is concise (typically one to two sentences long), and formulated using clear and straightforward language, ensuring it’s easily understood and testable.

Consider a hypothesis many teachers might subscribe to: students work better on Monday morning than on Friday afternoon (IV=Day, DV= Standard of work).

Now, if we decide to study this by giving the same group of students a lesson on a Monday morning and a Friday afternoon and then measuring their immediate recall of the material covered in each session, we would end up with the following:

• The alternative hypothesis states that students will recall significantly more information on a Monday morning than on a Friday afternoon.
• The null hypothesis states that there will be no significant difference in the amount recalled on a Monday morning compared to a Friday afternoon. Any difference will be due to chance or confounding factors.

More Examples

• Memory : Participants exposed to classical music during study sessions will recall more items from a list than those who studied in silence.
• Social Psychology : Individuals who frequently engage in social media use will report higher levels of perceived social isolation compared to those who use it infrequently.
• Developmental Psychology : Children who engage in regular imaginative play have better problem-solving skills than those who don’t.
• Clinical Psychology : Cognitive-behavioral therapy will be more effective in reducing symptoms of anxiety over a 6-month period compared to traditional talk therapy.
• Cognitive Psychology : Individuals who multitask between various electronic devices will have shorter attention spans on focused tasks than those who single-task.
• Health Psychology : Patients who practice mindfulness meditation will experience lower levels of chronic pain compared to those who don’t meditate.
• Organizational Psychology : Employees in open-plan offices will report higher levels of stress than those in private offices.
• Behavioral Psychology : Rats rewarded with food after pressing a lever will press it more frequently than rats who receive no reward.

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• Open access
• Published: 03 September 2024

World economies’ progress in decoupling from CO 2 emissions

• Jaume Freire-González 1 ,
• Emilio Padilla Rosa 2 &
• Josep Ll. Raymond 3  

Scientific Reports volume  14 , Article number:  20480 ( 2024 ) Cite this article

Metrics details

• Climate-change mitigation
• Climate-change policy
• Environmental economics
• Environmental impact
• Sustainability

The relationship between economic growth and CO 2 emissions has been analyzed testing the environmental Kuznets curve hypothesis, but traditional econometric methods may be flawed. An alternative method is proposed using segmented-sample regressions and implemented in 164 countries (98.34% of world population) over different periods from 1822 to 2018. Results suggest that while the association between GDP per capita and CO 2 emissions per capita is weakening over time, it remains positive globally, with only some high-income countries showing a reversed association in recent years. While 49 countries have decoupled emissions from economic growth, 115 have not. Most African, American, and Asian countries have not decoupled, whereas most European and Oceanians have. These findings highlight the urgency for effective climate policies because decoupling remains unachieved on a global scale, and we are moving away from, rather than approaching, the Paris Agreement goal of limiting temperature increase to 1.5 °C above preindustrial levels.

Introduction

The relationship between economic growth and environmental impacts has been extensively studied in the economic literature. In the last decades the debate has mostly focused on the environmental Kuznets curve (EKC) 1 , 2 , an optimistic hypothesis on the relationship between environmental impacts and gross domestic product (GDP) per capita. The EKC posits that in the early stages of development, economic growth leads to an increase in environmental degradation, but after achieving a certain GDP per capita threshold or turning point, environmental degradation decreases with economic growth 3 . That is, it hypothesizes an inverted-U or bell-shaped relationship between GDP per capita and environmental degradation. The EKC hypothesis was initially proposed by Grossman and Krueger 3 , while Panayotou 4 was the first to coin the term EKC due to its similarity to the relationship that Kuznets 5 suggested between income inequality and per capita income. Some empirical studies have supported the hypothesis for specific pollutants, resources use, or impacts in some countries or groups of countries, but there is also much evidence rejecting it for several environmental degradation indicators, countries, and time frames, so the EKC cannot be considered a general pattern 6 . Empirical studies have also identified other patterns, like linear relationships 7 , U-shaped relationships 8 , or patterns that include two turning points starting to rise or decrease again after a specific GDP per capita level. These can adopt an N-shaped form 9 , 10 , 11 or an inverted N-shaped form 12 . Most of the studies on the EKC hypothesis are based on atmospheric indicators, whereas there is more limited evidence on other pollutants or resources, like land oceans, seas, coasts and biodiversity indicators, and freshwater indicators 13 .

In relation to climate change, there is some controversy on the form of the relationship between GDP per capita and greenhouse gas emissions. Studies have used different methodologies to empirically test the shape of the relationship, particularly for carbon dioxide (CO 2 ) emissions. Some studies focus on single country analysis 7 , 8 , 14 , 15 , 16 , 17 , 18 , while some of them direct their analyses on groups of countries 11 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , with a diversity of methods and results. Some recent literature reviews summarize well this literature. Shahbaz and Sinha 29 found that the literature on the EKC for CO 2 emissions is inconclusive, which may be caused by the different contexts, time periods, explanatory variables, and methods employed. They suggest that studies should focus on new explanatory variables and refining the data set. Sarkodie and Strezov 13 , using meta-analysis methods, found heterogeneity among the GDP per capita level of the turning point in studies that validate the EKC hypothesis due to differences in the period of study and econometric methods used in model estimation.

Our paper aims to determine whether there is a decoupling between GDP and CO 2 emissions, while avoiding the methodological issues of previous studies. This decoupling is the focus of the EKC literature. Although there are other approaches like convergence or inequality analysis 30 , 31 , 32 , 33 that also evaluate the evolution between GDP and CO 2 , they address different research questions than ours and the EKC literature, which specifically examines this decoupling.

Most studies that empirically test the EKC hypothesis suffer from shortcomings that can affect the conclusions drawn from literature reviews. As pointed out by Jardon et al. 21 , the empirical validity of EKC studies has been questioned due to the sensitivity of the results to variations in model specifications, lack of diagnosis of the stationarity properties of the variables, assumption of cross-sectional independence (for panel data), and possible presence of structural breaks in the long-run relationship implied by the EKC hypothesis (for details see Refs. 34 , 35 , 36 , 37 ). Furthermore, numerous studies are constrained by limited time frames, potentially capturing only short-term or time-specific effects, or are limited to specific geographic areas, failing to account for transboundary effects, for instance. Not all studies conduct all needed statistical verifications to provide robust results. Another critical limitation of the EKC literature is that studies analyzing panels or cross-section data of countries impose the same functional form and parameters for the entire set of countries in estimations. This assumption has been rejected in some studies when tested 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 . In fact, the relation estimated in the literature for panels of countries may be the result of the juxtaposition of different trajectories of various countries 25 . Therefore, even if some studies find evidence supporting the EKC for a panel of countries, it may not provide valid conclusions on the expected path for the different countries. Individual analysis is essential to understand the dynamics and characteristics of each country 7 , 43 .

Another important shortcoming, not usually acknowledged in literature, is the potential existence of multicollinearity in estimated regressions 44 . It tends to appear when a regressor is specified along with its squared and/or cube form in a multiple regression model. It does not necessarily reduce the predictive power of the model as a whole but affects the estimations of individual parameters. As individual values assigned to different (GDP per capita) parameters are key in determining the functional form of the relationship between GDP per capita and pollution, multicollinearity may lead to estimate wrong curves. Some authors have considered it and provided alternative estimation methods as a solution. In this sense, Narayan and Narayan 24 compare short- and long-run elasticities of GDP on CO 2 emissions for some developing countries. They assume that if short-run estimates are larger than long-run ones, more income will lead to fewer CO 2 emissions, and the other way round. Al-Mulali et al. 15 also used this approach to avoid multicollinearity for estimates for Kenya. However, this method does not allow to track the functional form for a whole period, and it is not well suited to examine decoupling, which does not depend on the differences between long- and short-term elasticities but on the existence of a negative long-term elasticity.

This study aims to examinate the relationship between CO 2 emissions and economic growth across a comprehensive set of 164 countries over different time periods, with a focus on the 1822–2018 period. The countries included in the sample accounted for 98.34% of the total world population in 2018, representing 7534 million people, with only a few small countries discarded from the sample due to data limitations. The study covers the most extensive time frame in studies of this nature to date. Most important, we address the limitations of previous studies and avoid the methodological flaws that may have led to inferring incorrect patterns. The motivation of the study is to contribute to the EKC debate by: (1) correcting errors that previous studies may have committed by not considering the serious problems multicollinearity cause in estimated parameters; (2) proposing a new econometric approach, which avoids the mentioned problem from previous methods in estimating the relationship between CO 2 emissions and economic growth; (3) providing estimates for most of the countries of the world with this method, in a homogenous framework.

First, we apply an exhaustive statistical and verification analysis to provide robust results, which includes diagnosis of multicollinearity and diagnosis of the stationarity and cointegration properties of variables. Second, we apply segmented-sample regressions for individual countries, an alternative estimation method consisting in splitting the samples in smaller periods for each country and regressing GDP per capita on CO 2 per capita. A total of 932 regressions have been conducted. This way we are able to find elasticities for each country and period, and therefore, we build sequentially the curve for each country without imposing a priori assumptions on the functional form, such as including a quadratic form of GDP per capita and allowing the data to speak by themselves. The large data set used, along with the proposed methodological approach, allows us to provide robust evidence and overcome the shortcomings incurred in previous studies.

We find that the relationship between CO 2 emissions per capita and GDP per capita is still positive on average. However, there is a global tendency toward the weakening of this relationship. Therefore, decoupling of economic growth and CO 2 has not yet been achieved on a global scale. The article is structured as follows: section " Methods " describes the flaws of the methods used in literature, as well as the methods used in this research; section " Results " shows the results obtained in different ways, considering global and individual results. Section " Discussion " discusses the main results in relation to the state-of-art of the topic, and section " Conclusion " contains the main conclusions of this research.

The first part of the methodology consists in performing multicollinearity analysis between regressors as well as unit root and cointegration tests to the series to avoid spurious regressions and test the existence of a long-run relationship between the relevant variables for the analysis. The second part of the methodology consists in performing multiple individual regressions, for each country in different periods (segmented-sample regressions), to test the sign and intensity of the relationship between CO 2 emissions and economic growth. The tested specification is a simple model relating CO 2 emissions per capita with GDP per capita: \(CO{2}_{cap}=f(GD{P}_{cap})\) . Unlike some previous studies, we have not included other control variables often found in literature like urban population, trade openness, and energy use 45 . The exclusion of the former two is due to a potential problem of overparameterization of the model because we have a limited number of observations, as we will detail later; the latter, in addition to the issue of overparameterization, is also for the consideration that CO 2 and energy use are coupled variables 21 , as for the years covered in this analysis, most of the primary energy comes from fossil sources. This happens when a variable directly or indirectly contains the whole or components of another variable, leading to invalid conclusions if they are included in the same regression equation (Archie Jr. 46 ). In addition, the use of a reduced form model captures all the direct and indirect relationship between CO 2 and GDP per capita, including effects associated to omitted variables that may be correlated with both economic activity and time 47 , so including additional variables could potentially distort the analysis 38 . Although this is the appropriate analysis to study apparent elasticities between the variables, it does not allow us to assess the causes of the relationship or the determinants of emissions 25 .

We have obtained a data set for 164 countries for the period 1822–2018 for three variables: GDP, production-based CO 2 emissions, and population. However, data are not available for all the years of the period in all countries. GDP is measured in international dollars using 2011 prices to adjust for inflation and price differences between countries. It has been obtained from the Maddison Project Database 48 . Production-based CO 2 emissions are measured in million tons and include all emissions from energy production (coal, oil, natural gas and flaring) as well as industrial emissions from cement and steel production. They do not include emissions from land use change. This variable is obtained from the Global Carbon Project 49 . Population is used to calculate per capita GDP and CO 2 emissions. From 1800 to 1949 data on population comes from historical estimates by Gapminder v7. From 1950 to 2018 population records are by the United Nations Population Division (2018). Appendix I contains the main descriptive statistics of the two main variables included in the regressions for all the countries: production-based CO 2 emissions per capita and GDP per capita.

• Multicollinearity

Unreliable individual EKC estimators with multicollinearity

Classical model specifications of the EKC are as follows:

where \(CO{2}_{cap}\) is CO 2 emissions per capita, and \(\mathit{GD}{P}_{\mathit{cap}}\) is GDP per capita. Additionally, specifications include \({(GD{P}_{cap})}_{it}^{2}\) to capture a quadratic relationship and the turning point in the curve and, in some cases, also \({(GD{P}_{cap})}_{it}^{3}\) to capture more complex polynomial functions. The signs of parameters determine the functional form of the relationship 10 , 12 : if \({\beta }_{1}={\beta }_{2}={\beta }_{3}=0\) , there is either a flat pattern or no relationship between CO 2 and GDP; if \({\beta }_{1}>0\) and \({\beta }_{2}={\beta }_{3}=0\) , there is a monotonic increasing relationship between both variables; if \({\beta }_{1}<0\) and \({\beta }_{2}={\beta }_{3}=0\) , there is a monotonic decreasing relationship; if \({\beta }_{1}>0\) and \({\beta }_{2}<0\) and \({\beta }_{3}=0\) there is an inverted U-shaped relationship, that is, an EKC; if \({\beta }_{1}<0\) and \({\beta }_{2}>0\) and \({\beta }_{3}=0\) , there is a U-shaped curve; if \({\beta }_{1}>0\) and \({\beta }_{2}<0\) and \({\beta }_{3}>0\) , there is an N-shaped relationship; if \({\beta }_{1}<0\) and \({\beta }_{2}>0\) and \({\beta }_{3}<0\) , there is an inverted N-shaped relationship.

Multicollinearity occurs when there is a high correlation between explanatory variables. The extreme multicollinearity that this formulation introduces can result in unreliable parameter estimates, invalidating the conclusions on individual estimators. To test multicollinearity, we use the variance inflation factor (VIF) as a measure of its degree. In our sample of countries, using a quadratic form of the GDP per capita as explanatory variables, frequent VIF values exceeding 800 were observed. These results suggest the presence of extreme multicollinearity, indicating that the estimated individual coefficients are unreliable in all cases.

Because multicollinearity affects estimated values of individual coefficients, we have developed an alternative approach instead of testing the specifications by including one turning point fitting a quadratic form (U-shaped or inverted U-shaped curves) or two turning points fitting a cubic form (testing the N-shaped or the inverted N-shaped form).

Problems with previous methods in the literature to avoid multicollinearity in estimations testing the EKC hypothesis

Some previous studies have proposed alternative methods for testing the EKC hypothesis to avoid the multicollinearity problem. Particularly, Narayan and Narayan 24 proposed an alternative way of judging if countries had reduced CO 2 emissions over time with growth in incomes by comparing the short- with the long-run income elasticities. They argue that if the latter is smaller than the former, GDP per capita growth will eventually lead to less CO 2 emissions. However, this is not necessarily correct because emissions per capita will increase in the long run if the long-run elasticity between emissions and GDP per capita is greater than zero and will only decrease if this long-run elasticity is lower than zero. This method is neither suited to test a weak decoupling—a decrease in the emissions per unit of GDP—because it needs a long-run elasticity lower than one independently on the difference between short- and long-run elasticities.

Consider a general dynamic model of the type:

where \(\alpha \left(L\right)\) and \(\beta \left(L\right)\) are polynomials in the lag operator L , \(\mathit{ln}{{CO2}_{cap}}_{t}\) is the logarithm of CO 2 emissions per capita, and \(\mathit{ln}{{Y}_{cap}}_{t}\) is the logarithm of GDP per capita. Given that the model can be expressed as follows:

Short-term elasticity is \({\beta }_{0}\) , whereas \(\frac{{\beta }_{0}+{\beta }_{1}\cdot \cdot \cdot +{\beta }_{s}}{1-{\alpha }_{1}\cdot \cdot \cdot -{\alpha }_{r}}=k\) is the long-term emissions–GDP per capita elasticity.

If an economy grows at a stable rate of \(\dot{Y}_{cap}\) , CO 2 emissions per capita will also grow at a stable rate of \(k\dot{Y}_{cap}\) . If k  > 1, emissions per unit of GDP will tend to increase regardless of whether \({\beta }_{0}>k\) or \({\beta }_{0}<k\) is verified.

Therefore, the key factor determining the increase or decrease in CO 2 emissions per unit of GDP is whether the long-run elasticity is greater or smaller than 1. Moreover, CO 2 emissions per capita only decrease with an increase in GDP per capita if the long-run elasticity is negative. This is the requisite for decoupling, which does not depend on the difference between short- and long-run elasticities but on the sign of long-run elasticity.

Stationarity and cointegration

To ensure that regression models are not spurious, it is necessary to conduct a unit root test analysis of the random disturbance term, testing its stationarity. If the variables are not stationary but are cointegrated, it can be concluded that they have a long-term relationship. In this section, we first conduct panel unit root tests for all the countries of the analysis for CO 2 per capita and GDP per capita. Then, we also conduct different cointegration tests.

We specifically use six panel unit root tests to evaluate the stationarity of the series: the Levin–Lin–Chu (LLC) test 50 , Breitung t-statistic 51 , Hadri Z-statistic 52 , Im–Pesaran–Shin W-statistic (IPS) 53 , Fisher-augmented Dickey–Fuller (ADF–Fisher); (Maddala and Wu 54 ), and Fisher– Phillips and Perron (PP-Fisher) 55 . The LLC, Hadri, and Breitung assume a common unit root process, whereas the IPS, ADF–Fisher and PP–Fisher assume an individual root process. Differences between tests in the context of the EKC literature are detailed in Jardón et al 21 . We have conducted all tests in two ways: with constant (intercept) and with constant plus trend, except Breitung, which is only allowed with trend and intercept. Results of the tests are shown in Table 1 .

Based on the different tests we conducted, we can conclude that, in general, CO 2 per capita and GDP per capita are non-stationary at levels, having a unit root. Therefore, we can consider them as integrated processes of order 1.

We investigate the existence of a long-term relationship between the two variables through cointegration tests. We employ three different panel cointegration tests: Pedroni 56 , 57 , Kao 58 , and a Fisher-type test using an underlying Johansen methodology 54 . Following Pedroni 56 , we estimate the different statistics of his test including a constant and a trend because they are more reliable than just including a constant. The Kao test follows the same approach as the Pedroni but is based on the assumption of homogeneity across the panel. Kao 58 derives two (DF and ADF) types of panel cointegration tests. The results of the cointegration tests are presented in Table 2 .

Results of cointegration tests in Table 2 show that mostly both variables are cointegrated, so they both have a long-term relationship.

Segmented-sample regressions of the relationship between CO 2 emissions per capita and GDP per capita

The application of unit root panel data tests of integration and cointegration is suitable if the relationships that links the different variables are relatively similar among countries. But if this is not the case, a simple technique, the segmented-sample regression approach, is more flexible to encompass the individual specificities of the countries included in the sample, which is our case. Applying rolling regressions 59 would lead to similar conclusions but will require a much larger number of regressions. So, the third step of the methodology consists in performing this method to estimate the relationship between CO 2 emissions per capita and GDP per capita. For each country, we estimate successive regressions for periods of 15 years ( T ), starting at 2018 and going backward, taking the first year of the previous regression as the last year of the current one. We choose this time frame to have at least two regressions (periods) for all countries and be able to compare them because not all series of countries have the same total length. Harrell 60 assumes that it is necessary to have at least 10–20 observations per parameter estimated to be able to detect effects with a reasonable statistical power. Periods with fewer than 10 observations have not been regressed. Therefore, for each period T , we choose this simple functional form with just one regressor. This avoids both multicollinearity existent in polynomial functional forms, and the issue of overparameterization in regression models with few observations:

where \({\beta }_{1}\) represents the income elasticity of CO 2 of emissions for country i in year t . The objective for each country is to obtain different elasticities for different periods and observe their changes to determine the form of the relationship. We estimate them by ordinary least squares using the Newey–West estimator 61 , also known as HAC (heteroskedasticity and autocorrelation consistent), to estimate the standard errors of the coefficients. We have performed a total of 932 regressions for different periods for the 164 countries. Table 3 summarizes the regressions carried out.

Different possibilities can arise with the different results of the income-elasticities: if \({\beta }_{iT}>0\) , there is a positive relationship between CO 2 per capita and GDP per capita in period T , whereas if \({\beta }_{iT}<0\) , there is a negative relationship between both variables. The inverted U-shaped relationship (EKC) will appear if \({\beta }_{iT}>0\) and \({\beta }_{iT+1}<0\) ; an N-shaped relationship will happen if \({\beta }_{iT}>0\) , \({\beta }_{iT+1}<0\) , and \({\beta }_{iT+2}>0\) . If \({\beta }_{iT}>{\beta }_{iT+1}>0\) , there is a growing relationship between both variables along the whole period, although they grow less in successive periods.

The results of the regressions can be summarized in several ways, given the vast number of elasticities obtained for different countries and time periods. The 932 elasticities are grouped to present global results and individual results. For detailed results in all countries and periods, see Appendix II , which contains a table with the results of estimates for each country and period, providing a total of 932 elasticities. Furthermore, Appendix III provides visual representations of the estimated elasticity curves for each one of the 164 countries.

Global results

Figure  1 illustrates the temporal evolution of the average income elasticity between CO 2 emissions per capita and GDP per capita over the entire period of analysis. The trend reveals a consistently positive relationship between the two variables, although it becomes weaker over time. Specifically, in the period 2004–2018 CO 2 per capita grew along with GDP per capita but at a slower rate than in previous periods. Because the average elasticity in this last period is between zero and one, CO 2 emissions per capita grow less than GDP per capita when the latter grows, but they still have a positive relationship. This is sometimes known in the literature as “weak” decoupling. Although the trend does not indicate an inverted U-shape in average, it is important to interpret this figure with caution because the sample sizes differ across time frames and because we plot average elasticities from very different economies.

Average income elasticity of CO 2 of the 164 countries analyzed in each period.

The signs of elasticities play a pivotal role in determining the functional form of the relationships between CO 2 and GDP per capita for each country. Positive elasticities mean a positive association between economic growth and emissions, reflected in the positive slopes of the curves, whereas negative elasticities indicate a negative relationship, evidenced by the negative slopes of curves. Focusing on the signs, Fig.  2 disaggregates the number of positive and negative elasticities in each period regardless of their value, showing a convergence between the two kinds of elasticities along the period. However, there are still more countries with positive elasticities in 2004–2018 than those with negative ones.

Percentage of the number of positive and negative elasticities in each period for the 164 countries.

We have also grouped countries according to their GDP per capita in 2018 into 10 deciles, so we have all countries ordered from higher to lower income per capita. Then we have calculated the average elasticity within each group during the periods analyzed. The results of the analysis are presented in Fig.  3 . Notably, only the richest countries achieve negative elasticities on average in the most recent period of analysis (2004–2018) and partially in the previous period. However, the general trend is toward reducing average elasticities. Some peculiar patterns appear before the period 1962–1976. This is attributable to a reduced number of observations available as we go further back through the different periods. Furthermore, potential biases may arise due to less reliable data for older periods and developing countries. Additionally, because we use 2018 GDP per capita, some countries would be assigned to different deciles in relation to 2018 classification as we go backward in time. We observe, for instance, that there is a negative elasticity in decile 4 for the period 1920–1934. This is because it is based on only two observations in that period (Philippines and India), so the results obtained for those countries carried more weight than observations for other countries in other deciles/periods.

Average income elasticity of CO 2 by period and GDP per capita decile. ID means income decile. ID 1 represents the lowest GDP per capita, whereas ID 10 is the highest.

When we classify them by quartiles of GDP per capita, some noise disappears, and we observe that the wealthiest countries, and only in the last period (2004–2018), have a negative average elasticity. However, we observe the same tendency as in the analysis by deciles toward the reduction of elasticities in all quartiles over the period (Fig.  4 ).

Average income elasticity of CO 2 by period and GDP per capita quartile. IQ means income quartile. IQ 1 represents the lowest GDP per capita, whereas IQ 4 is the highest.

Individual results

We have classified all countries into six different groups based on the shape of the estimated relationship (see Table 4 ). The first three groups reveal “decoupling shapes,” where the last period or periods of analysis have an inverse relationship between CO 2 per capita and GDP per capita. These shapes are a degrowing, an inverted U-shape (EKC curve) and an inverted N-shape. In contrast, we label the last three groups of countries as “non-decoupling shapes” because they show a positive relationship between the two variables in the last period(s) under analysis. Among the countries analyzed, 49 countries exhibit decoupling shapes, representing the 29.88% of the countries, whereas 115 have non-decoupling ones, which represents the 70.12%.

Most developed countries exhibit favorable or improving patterns in the relationship between emissions and economic growth. The average 2018 GDP per capita for these countries is 32,227 in 2011 international dollars, compared to the average of 12,936 dollars of GDP per capita for countries with non-decoupling shapes of the relationship. The total GDP in 2018 for decoupling shapes amount 47,700 billion dollars, whereas that of non-decoupling shapes is 65,573 billion dollars, the two being the 42.11% and the 57.89% of the total GDP, respectively. The total population for countries with decoupling shapes is 1,490 million people, which accounts for 19.79% of the sample population, whereas the population of countries with non-decoupling shapes is 6,043 million people, accounting for 80.21% of the sample. This information leads to a pessimistic outlook for a near and effective decoupling of CO 2 emissions in world economies. Table 4 shows the behavior of different groups of countries regarding whether the shape of the curves of their members are decoupling or non-decoupling.

To better understand the distribution of decoupling and non-decoupling shapes of the relationship between emissions and economic growth across regions and territories, we have grouped the different countries in different ways. First, Fig.  5 displays the percentage of countries with decoupling and non-decoupling shapes in each continent, ranked in descending order based on the proportion of countries with decoupling shapes. We observe that Oceania and Europe have mostly countries with a decoupling shape, whereas in Asia, America, and Africa, most countries have non-decoupling shapes. The sample of Oceania only includes two out of fourteen countries: Australia and New Zealand.

Percentage of decoupling and non-decoupling curve shapes in each continent, ordered from more to less percentage of countries with decoupling curves in each continent. Classification based on the sample of 164 countries. The sample of Oceania only includes two out of fourteen countries: Australia and New Zealand.

We have also classified the different countries into groups based on their membership in international organizations, groups, or internationally accepted classifications. Figure  6 displays the percentage of countries in each group using a multiple pie chart for the 38 countries of the Organisation for Economic Co-operation and Development (OECD), the 27 countries of the European Union (EU) countries, the 5 countries of the emerging economies (BRICS), the 13 countries of the Organization of the Petroleum Exporting Countries (OPEC), the 21 members of the Asia–Pacific Economic Cooperation (APEC), the 10 members of the Economic Cooperation Organization (ECO), the 57 members of the Organization of Islamic Cooperation (OIC), the 9 member states and associate states of the Southern Common Market (MERCOSUR), the 33 members of the Community of Latin American and Caribbean States (CELAC), and the G8 countries and G20 countries. Most countries in the OECD, EU, and G8 show decoupling evolutions of the relationship between emissions and growth. In contrast, BRICS and OPEC countries mostly have non-decoupling trajectories in the relationship. Finally, the G20 has an equal number of countries with proper and non-proper evolutions of the curve.

Percentage of the decoupling and non-decoupling shapes in each group of countries. Ordered from more to less percentage of countries with decoupling shapes in each group. EU as a member of G20 has been included in the “decoupling shapes” group for that category of countries. MERCOSUR includes member states and associate states. We have excluded some countries from some groups due to lack of data: Brunei (APEC, OIC), Papua New Guinea (APEC), Somalia (OIC), Sudan (OIC), Maldives (OIC), Guyana (OIC, MERCOSUR, CELAC), Suriname (OIC, MERCOSUR, CELAC), Antigua and Barbuda (CELAC), Bahamas (CELAC), Belize (CELAC), Grenada (CELAC), Saint Kitts and Nevis (CELAC), and Saint Vincent and the Grenadines (CELAC).

To provide an intuitive and visual representation of the dynamic trends observed in our study, we have summarized the evolution of the elasticities in all countries in Fig.  7 , which shows a colored world map for each period. Red represents positive values of elasticities (undesirable situation—positive slope of the relationship), and green shows negative values (desirable situation—negative slope of the relationship). In both cases, the more intense the color, the higher the absolute value.

Maps of the evolution of income elasticities in the 164 countries of the sample. The income elasticities of CO 2 are represented in maps. Red indicates positive elasticities, whereas green indicates negative elasticities between emissions per capita and GDP per capita.

We observe a transition toward greener maps as we approach more contemporary periods. The intensity of the color also changes, from intense red toward pale tones and from pale greens to more intense ones, although there is no clear pattern, and each country has its own history, as showed in Table 4 . We see some countries that transition from red to green but then again to red, like Bolivia, for instance, which shows a N-shaped curve.

We now focus on the last periods of the study. There are 159 country estimates for the period 1976–1990, as shown in Table 3 , and we have the full sample of 164 countries from 1990 onward. Furthermore, during these three periods (1976–2018), societies have experienced a growing concern over energy availability, climate change, and the environment in general, demanding stronger environmental policies. Renewable energy sources have become more widespread, triggering energy and sustainable transitions facilitated by supportive legislative measures and policies and increasing environmental awareness. To visualize this transition more clearly, Fig.  8 shows, individually, the change experienced by each country from the period 1976–1990 to the period 1990–2004, and from the period 1990–2004 to the period 2004–2018. Green arrows show a reduction of the elasticity of the country between periods, whereas red arrows indicate an increase of the elasticity between the two periods. Countries are ordered by the lowest value in the first period. Arrows in Fig.  8 show the direction of the evolution of the income elasticity of CO 2 between periods.

Change in elasticities by country between time frames. Periods 1976–1990 to 1990–2004 and from 1990–2004 to 2004–2018.

Our analysis offers interesting insights in light of the data and methodology used. Regarding the methodology used, unlike previous studies we avoid problems that have been leading to unreliable parameters, as demonstrated in section " Methods ". It is interesting, however, to compare our results with those of other recent studies. The most recent and exhaustive study in terms of the number of countries included is Wang et al. 27 which estimates the EKC for 29 years (1990–2018) using data from 208 countries or regions worldwide, considering aspects like the role of trade openness, human capital, renewable energy and natural resource rent. This study finds an inverted U-shaped curve at the global level using a classical specification and generalized method of moments (GMM) for dynamic panel data. In 2018, 120 countries did not reach the EKC turning point and 72 countries reached the EKC turning point for per capita income, being this a similar proportion than the results obtained in our study in terms of countries that have reached decoupling versus countries that still have not decoupled their emissions. Similarly, Wang et al. 62 in another recent research estimate the EKC for 147 countries for the period from 1995 to 2018. They globally substantiate the validity of the EKC hypothesis, but do not provide country level estimates. A more recent study 22 explores the EKC using a panel data of 38 countries for the period 2002–2020 including other variables like geopolitical risk, natural resource rents, corruption governance, and energy intensity on carbon emissions. They globally find support to the EKC for the selected countries, but no individual results are reported, and their specifications include other factors, so it is not possible to compare our results with theirs. Mohammed et al. 63 found validity of the EKC for EU-27 countries between 1990 and 2019. However, they use the classical specification with the squared independent variable. On the other side, while we find that 63% of European countries have decoupling shapes, they found a lower percentage (38%).

The studies referred above do not provide estimates of the relationship for individual countries, since they impose in their estimates the same functional form and parameters for the entire set of countries. Golpîra et al. 20 studied the EKC hypothesis for 37 countries of the OECD over the period from 1960 to 2019. Both the number of countries and the time span are less than in our study, but it is useful to compare their results for individual countries with the results we obtained for the same countries. In their study about 57% of the OECD countries show decoupling shapes while 43% exhibit non-decoupling shapes. Our research, on the other hand, shows that 71% of OECD countries are decoupling versus 29% that have non-decoupling shapes. There are also differences between individual results by country. The existence of multicollinearity due to the use of classical specifications in their estimates lead to unreliable parameter estimates, as demonstrated in section " Unreliable individual EKC estimators with multicollinearity ", and it is the main driver of these differences. For BRICS countries, Hasan et al. 64 found validity of the EKC hypothesis overall and for all the 5 countries analyzed using the classical specification that included a squared independent variable, while we have only found it valid for South Africa. Focusing on studies for individual countries, another recent study 20 found an inverted U-shape relation between carbon emissions and real GDP in long run for France and Germany for the period 1995–2015. We also find this shape for these countries but using a larger time span with the improved method we propose for correcting multicollinearity. Mahmood et al. 65 found recently that most EKC studies for China validate the EKC hypothesis while we find a positive slope in the relationship when correcting by multicollinearity. Uche et al. 66 found an EKC for India for the period 1980–2018 employing the multiple threshold nonlinear ARDL procedure with a squared exogenous variable, while we found an N-shaped curve for this country.

Focusing on the results obtained in this study, it has worrying implications for both regions and individual countries regarding the climate policies carried out over the last decades, considering the urgency required to mitigate climate change. Only in wealthy regions have a majority of countries already decoupled GDP per capita from CO 2 emissions. However, this represents a small part of the world, and economic growth in most regions will still be associated with higher emissions, leading to greater emissions at the global level. So, the first and most important general implication of these results is that countries must effectively implement, accelerate or rethink this kind of policies, as they may not be working as fast as needed. Given the global nature of the effect of carbon emissions, individual countries have lower incentives than regions or the whole world —many countries have decoupling shapes—. This is why is so important a coordinated and effective action.

A simple but effective method that eludes multicollinearity and a priori imposing of a specific functional form is proposed in this study to estimate the relationship between CO 2 emissions and economic growth: segmented-sample regressions. We find that the relationship between CO 2 per capita and GDP per capita is, on average, still positive, meaning that both variables grow at the same time, although there is a global tendency toward the weakening of this relationship. Only a few countries present a negative elasticity between both variables in any period. When grouping different countries according to income level and taking average elasticities, only the group of wealthiest countries (top 20%) have reversed this association and just during the last period of the analysis (2004–2018). Most countries in Africa (83%), America (82%), and Asia (80%) have non-decoupling shapes, whereas those in Europe and Oceania only represent 38% and 0%, respectively, taking into account that we only consider 2 countries from Oceania (the biggest ones though).

For most countries, the evidence shows that economic growth is still associated with more carbon emissions, which indicates that the decoupling of economic growth and CO 2 has not yet been achieved on a global scale. Even assuming that the general observed trend of decreasing elasticities over time could eventually lead to turning points in the future in which emissions would start decreasing with economic growth, there would still be positive emissions for most countries. This makes current climate targets, like the Paris Agreement’s target of limiting temperature increase to 1.5ºC or 2ºC above pre-industrial levels, difficult to achieve.

Although the causes and drivers of these tendencies are out of the scope of this research, some important aspects to understand these results are as follows. (1) The tendency to outsource some environmental impacts from developed countries to developing countries due to the stronger environmental regulations in the former 67 and the tertiarization of economies. Consumption-based emissions can offset this bias if they account for all the impacts along the international supply chain of products and services. (2) The growth of renewables, particularly in some developed countries may be one of the main causes of the appearance of a negative elasticity between both variables, which is positive to reduce CO 2 emissions 68 . (3) Emissions of international aviation and shipping are not included in any country or region because there is no international agreement on how these emissions should be allocated, representing around 3.45% of total global emissions in 2018 49 . Considering that most commercial aviation is caused by the demand of developed countries and shipping activity feeds in greater measure developed regions, their CO 2 emissions may be underestimated.

Our proposed method can be applied to future studies to test the EKC hypothesis and inform evidence-based policymaking. In this sense, our study arises some policy implications. The most important one is the need of global efforts to mitigate carbon emissions, not just local/national ones. Territorial emissions in many developed countries appear to be decoupling from CO 2 , but countries need to make efforts not just to reduce their emissions but also to help reducing emissions in other countries, like developing ones, with a fair and global justice perspective. Climate change is a global problem with global consequences and, in terms of decoupling, our research suggests a gap between developed and developing countries, or sometimes different velocities in decoupling. The articulation of the different potential mechanisms to achieve decoupling is out of the scope of this research, but dialogue, common understanding and cooperation between countries suppose the first steps in solving complex global issues. National efforts are important, but climate policies need to have a global perspective to be effective, given that the climate and the atmosphere have characteristics of a pure public good. A second policy implication is that policies should be oriented to global decoupling but also, and most important, towards achieving global absolute reductions of CO 2 emissions. Decoupling of some countries or groups of countries in not enough and highlights the importance of reading the results of this research with a global perspective. One country may reach the wrong conclusion from this research that no more climate policy efforts are needed if they are already decoupling. All the contrary, their territorial emissions may be decoupling from GDP but, at the same time, causing other territories to emit more CO 2 due to other complex economic mechanisms mentioned in this article. From a public policy perspective there is no relieve in one territory from being decoupling if globally the other countries are not. Although several countries are decoupling, CO 2 emissions are still globally growing. The relevant aspect in climate policy is the achievement of global absolute reductions of CO 2 emissions. Following the last assessment report of the IPCC 69 , they need to be reduced fast to avoid the worst consequences of climate change.

Further research should focus not just on monitoring this relationship in the future but also on estimating this relationship for other environmental impacts, not just CO 2 emissions and climate change, understanding the links between impacts in an energy/sustainable transition, and studying its relationship with economic growth. It is important to consider that economic growth may be the cause of other environmental problems, beyond CO 2 emissions, not considered in this study. Some studies, for instance, link the deployment of renewable energy sources with the use of more other natural resources, some of them critical materials 70 . Our results are concerning and emphasize the need to accelerate changes toward a low-carbon economy and implement effective and global policies that incentivize emissions reductions.

Data availability

Data is available upon reasonable request. Please contact the corresponding author.

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Acknowledgements

Jaume Freire-González acknowledges financial support through the grant PID2021-124256OB-I00 funded by MCIN/AEI/ https://doi.org/10.13039/501100011033 and by ERDF A way of making Europe, the Spanish Agencia Estatal de Investigación (AEI), through the Severo Ochoa Program for Centers of Excellence in R&D (Barcelona School of Economics CEX2019-000915-S), and AGAUR-Generalitat de Catalunya (2021-SGR-416). Emilio Padilla acknowledges support from the Spanish Ministry of Science and Innovation (project PID2021-126295OB-I00) and AGAUR (2021-SGR- 01502).

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Jaume Freire-González conceptualized the study, conducted data curation, contributed to the methodology, performed formal analyses and wrote the original draft; Emilio Padilla Rosa contributed to the methodology, conducted formal analyses and wrote sections of the document. Josep Ll. Raymond developed the methodology and wrote sections of the document.

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Freire-González, J., Padilla Rosa, E. & Raymond, J.L. World economies’ progress in decoupling from CO 2 emissions. Sci Rep 14 , 20480 (2024). https://doi.org/10.1038/s41598-024-71101-2

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