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• Type I & Type II Errors | Differences, Examples, Visualizations

Type I & Type II Errors | Differences, Examples, Visualizations

Published on January 18, 2021 by Pritha Bhandari . Revised on June 22, 2023.

In statistics , a Type I error is a false positive conclusion, while a Type II error is a false negative conclusion.

Making a statistical decision always involves uncertainties, so the risks of making these errors are unavoidable in hypothesis testing .

The probability of making a Type I error is the significance level , or alpha (α), while the probability of making a Type II error is beta (β). These risks can be minimized through careful planning in your study design.

• Type I error (false positive) : the test result says you have coronavirus, but you actually don’t.
• Type II error (false negative) : the test result says you don’t have coronavirus, but you actually do.

Table of contents

Error in statistical decision-making, type i error, type ii error, trade-off between type i and type ii errors, is a type i or type ii error worse, other interesting articles, frequently asked questions about type i and ii errors.

Using hypothesis testing, you can make decisions about whether your data support or refute your research predictions with null and alternative hypotheses .

Hypothesis testing starts with the assumption of no difference between groups or no relationship between variables in the population—this is the null hypothesis . It’s always paired with an alternative hypothesis , which is your research prediction of an actual difference between groups or a true relationship between variables .

In this case:

• The null hypothesis (H 0 ) is that the new drug has no effect on symptoms of the disease.
• The alternative hypothesis (H 1 ) is that the drug is effective for alleviating symptoms of the disease.

Then , you decide whether the null hypothesis can be rejected based on your data and the results of a statistical test . Since these decisions are based on probabilities, there is always a risk of making the wrong conclusion.

• If your results show statistical significance , that means they are very unlikely to occur if the null hypothesis is true. In this case, you would reject your null hypothesis. But sometimes, this may actually be a Type I error.
• If your findings do not show statistical significance, they have a high chance of occurring if the null hypothesis is true. Therefore, you fail to reject your null hypothesis. But sometimes, this may be a Type II error.

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A Type I error means rejecting the null hypothesis when it’s actually true. It means concluding that results are statistically significant when, in reality, they came about purely by chance or because of unrelated factors.

The risk of committing this error is the significance level (alpha or α) you choose. That’s a value that you set at the beginning of your study to assess the statistical probability of obtaining your results ( p value).

The significance level is usually set at 0.05 or 5%. This means that your results only have a 5% chance of occurring, or less, if the null hypothesis is actually true.

If the p value of your test is lower than the significance level, it means your results are statistically significant and consistent with the alternative hypothesis. If your p value is higher than the significance level, then your results are considered statistically non-significant.

To reduce the Type I error probability, you can simply set a lower significance level.

Type I error rate

The null hypothesis distribution curve below shows the probabilities of obtaining all possible results if the study were repeated with new samples and the null hypothesis were true in the population .

At the tail end, the shaded area represents alpha. It’s also called a critical region in statistics.

If your results fall in the critical region of this curve, they are considered statistically significant and the null hypothesis is rejected. However, this is a false positive conclusion, because the null hypothesis is actually true in this case!

A Type II error means not rejecting the null hypothesis when it’s actually false. This is not quite the same as “accepting” the null hypothesis, because hypothesis testing can only tell you whether to reject the null hypothesis.

Instead, a Type II error means failing to conclude there was an effect when there actually was. In reality, your study may not have had enough statistical power to detect an effect of a certain size.

Power is the extent to which a test can correctly detect a real effect when there is one. A power level of 80% or higher is usually considered acceptable.

The risk of a Type II error is inversely related to the statistical power of a study. The higher the statistical power, the lower the probability of making a Type II error.

Statistical power is determined by:

• Size of the effect : Larger effects are more easily detected.
• Measurement error : Systematic and random errors in recorded data reduce power.
• Sample size : Larger samples reduce sampling error and increase power.
• Significance level : Increasing the significance level increases power.

To (indirectly) reduce the risk of a Type II error, you can increase the sample size or the significance level.

Type II error rate

The alternative hypothesis distribution curve below shows the probabilities of obtaining all possible results if the study were repeated with new samples and the alternative hypothesis were true in the population .

The Type II error rate is beta (β), represented by the shaded area on the left side. The remaining area under the curve represents statistical power, which is 1 – β.

Increasing the statistical power of your test directly decreases the risk of making a Type II error.

The Type I and Type II error rates influence each other. That’s because the significance level (the Type I error rate) affects statistical power, which is inversely related to the Type II error rate.

This means there’s an important tradeoff between Type I and Type II errors:

• Setting a lower significance level decreases a Type I error risk, but increases a Type II error risk.
• Increasing the power of a test decreases a Type II error risk, but increases a Type I error risk.

This trade-off is visualized in the graph below. It shows two curves:

• The null hypothesis distribution shows all possible results you’d obtain if the null hypothesis is true. The correct conclusion for any point on this distribution means not rejecting the null hypothesis.
• The alternative hypothesis distribution shows all possible results you’d obtain if the alternative hypothesis is true. The correct conclusion for any point on this distribution means rejecting the null hypothesis.

Type I and Type II errors occur where these two distributions overlap. The blue shaded area represents alpha, the Type I error rate, and the green shaded area represents beta, the Type II error rate.

By setting the Type I error rate, you indirectly influence the size of the Type II error rate as well.

It’s important to strike a balance between the risks of making Type I and Type II errors. Reducing the alpha always comes at the cost of increasing beta, and vice versa .

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For statisticians, a Type I error is usually worse. In practical terms, however, either type of error could be worse depending on your research context.

A Type I error means mistakenly going against the main statistical assumption of a null hypothesis. This may lead to new policies, practices or treatments that are inadequate or a waste of resources.

In contrast, a Type II error means failing to reject a null hypothesis. It may only result in missed opportunities to innovate, but these can also have important practical consequences.

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

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

Methodology

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

Research bias

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

In statistics, a Type I error means rejecting the null hypothesis when it’s actually true, while a Type II error means failing to reject the null hypothesis when it’s actually false.

The risk of making a Type I error is the significance level (or alpha) that you choose. That’s a value that you set at the beginning of your study to assess the statistical probability of obtaining your results ( p value ).

To reduce the Type I error probability, you can set a lower significance level.

The risk of making a Type II error is inversely related to the statistical power of a test. Power is the extent to which a test can correctly detect a real effect when there is one.

To (indirectly) reduce the risk of a Type II error, you can increase the sample size or the significance level to increase statistical power.

Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.

Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .

When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.

In statistics, power refers to the likelihood of a hypothesis test detecting a true effect if there is one. A statistically powerful test is more likely to reject a false negative (a Type II error).

If you don’t ensure enough power in your study, you may not be able to detect a statistically significant result even when it has practical significance. Your study might not have the ability to answer your research question.

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Bhandari, P. (2023, June 22). Type I & Type II Errors | Differences, Examples, Visualizations. Scribbr. Retrieved September 4, 2024, from https://www.scribbr.com/statistics/type-i-and-type-ii-errors/

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Type 1 and Type 2 Errors in Statistics

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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A statistically significant result cannot prove that a research hypothesis is correct (which implies 100% certainty). Because a p -value is based on probabilities, there is always a chance of making an incorrect conclusion regarding accepting or rejecting the null hypothesis ( H 0 ).

Anytime we make a decision using statistics, there are four possible outcomes, with two representing correct decisions and two representing errors.

The chances of committing these two types of errors are inversely proportional: that is, decreasing type I error rate increases type II error rate and vice versa.

As the significance level (α) increases, it becomes easier to reject the null hypothesis, decreasing the chance of missing a real effect (Type II error, β). If the significance level (α) goes down, it becomes harder to reject the null hypothesis , increasing the chance of missing an effect while reducing the risk of falsely finding one (Type I error).

Type I error 

A type 1 error is also known as a false positive and occurs when a researcher incorrectly rejects a true null hypothesis. Simply put, it’s a false alarm.

This means that you report that your findings are significant when they have occurred by chance.

The probability of making a type 1 error is represented by your alpha level (α), the p- value below which you reject the null hypothesis.

A p -value of 0.05 indicates that you are willing to accept a 5% chance of getting the observed data (or something more extreme) when the null hypothesis is true.

You can reduce your risk of committing a type 1 error by setting a lower alpha level (like α = 0.01). For example, a p-value of 0.01 would mean there is a 1% chance of committing a Type I error.

However, using a lower value for alpha means that you will be less likely to detect a true difference if one really exists (thus risking a type II error).

Scenario: Drug Efficacy Study

Imagine a pharmaceutical company is testing a new drug, named “MediCure”, to determine if it’s more effective than a placebo at reducing fever. They experimented with two groups: one receives MediCure, and the other received a placebo.

• Null Hypothesis (H0) : MediCure is no more effective at reducing fever than the placebo.
• Alternative Hypothesis (H1) : MediCure is more effective at reducing fever than the placebo.

After conducting the study and analyzing the results, the researchers found a p-value of 0.04.

If they use an alpha (α) level of 0.05, this p-value is considered statistically significant, leading them to reject the null hypothesis and conclude that MediCure is more effective than the placebo.

However, MediCure has no actual effect, and the observed difference was due to random variation or some other confounding factor. In this case, the researchers have incorrectly rejected a true null hypothesis.

Error : The researchers have made a Type 1 error by concluding that MediCure is more effective when it isn’t.

Implications

Resource Allocation : Making a Type I error can lead to wastage of resources. If a business believes a new strategy is effective when it’s not (based on a Type I error), they might allocate significant financial and human resources toward that ineffective strategy.

Unnecessary Interventions : In medical trials, a Type I error might lead to the belief that a new treatment is effective when it isn’t. As a result, patients might undergo unnecessary treatments, risking potential side effects without any benefit.

Reputation and Credibility : For researchers, making repeated Type I errors can harm their professional reputation. If they frequently claim groundbreaking results that are later refuted, their credibility in the scientific community might diminish.

Type II error

A type 2 error (or false negative) happens when you accept the null hypothesis when it should actually be rejected.

Here, a researcher concludes there is not a significant effect when actually there really is.

The probability of making a type II error is called Beta (β), which is related to the power of the statistical test (power = 1- β). You can decrease your risk of committing a type II error by ensuring your test has enough power.

You can do this by ensuring your sample size is large enough to detect a practical difference when one truly exists.

Scenario: Efficacy of a New Teaching Method

Educational psychologists are investigating the potential benefits of a new interactive teaching method, named “EduInteract”, which utilizes virtual reality (VR) technology to teach history to middle school students.

They hypothesize that this method will lead to better retention and understanding compared to the traditional textbook-based approach.

• Null Hypothesis (H0) : The EduInteract VR teaching method does not result in significantly better retention and understanding of history content than the traditional textbook method.
• Alternative Hypothesis (H1) : The EduInteract VR teaching method results in significantly better retention and understanding of history content than the traditional textbook method.

The researchers designed an experiment where one group of students learns a history module using the EduInteract VR method, while a control group learns the same module using a traditional textbook.

After a week, the student’s retention and understanding are tested using a standardized assessment.

Upon analyzing the results, the psychologists found a p-value of 0.06. Using an alpha (α) level of 0.05, this p-value isn’t statistically significant.

Therefore, they fail to reject the null hypothesis and conclude that the EduInteract VR method isn’t more effective than the traditional textbook approach.

However, let’s assume that in the real world, the EduInteract VR truly enhances retention and understanding, but the study failed to detect this benefit due to reasons like small sample size, variability in students’ prior knowledge, or perhaps the assessment wasn’t sensitive enough to detect the nuances of VR-based learning.

Error : By concluding that the EduInteract VR method isn’t more effective than the traditional method when it is, the researchers have made a Type 2 error.

This could prevent schools from adopting a potentially superior teaching method that might benefit students’ learning experiences.

Missed Opportunities : A Type II error can lead to missed opportunities for improvement or innovation. For example, in education, if a more effective teaching method is overlooked because of a Type II error, students might miss out on a better learning experience.

Potential Risks : In healthcare, a Type II error might mean overlooking a harmful side effect of a medication because the research didn’t detect its harmful impacts. As a result, patients might continue using a harmful treatment.

Stagnation : In the business world, making a Type II error can result in continued investment in outdated or less efficient methods. This can lead to stagnation and the inability to compete effectively in the marketplace.

How do Type I and Type II errors relate to psychological research and experiments?

Type I errors are like false alarms, while Type II errors are like missed opportunities. Both errors can impact the validity and reliability of psychological findings, so researchers strive to minimize them to draw accurate conclusions from their studies.

How does sample size influence the likelihood of Type I and Type II errors in psychological research?

Sample size in psychological research influences the likelihood of Type I and Type II errors. A larger sample size reduces the chances of Type I errors, which means researchers are less likely to mistakenly find a significant effect when there isn’t one.

A larger sample size also increases the chances of detecting true effects, reducing the likelihood of Type II errors.

Are there any ethical implications associated with Type I and Type II errors in psychological research?

Yes, there are ethical implications associated with Type I and Type II errors in psychological research.

Type I errors may lead to false positive findings, resulting in misleading conclusions and potentially wasting resources on ineffective interventions. This can harm individuals who are falsely diagnosed or receive unnecessary treatments.

Type II errors, on the other hand, may result in missed opportunities to identify important effects or relationships, leading to a lack of appropriate interventions or support. This can also have negative consequences for individuals who genuinely require assistance.

Therefore, minimizing these errors is crucial for ethical research and ensuring the well-being of participants.

Further Information

• Publication manual of the American Psychological Association
• Statistics for Psychology Book Download

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• Knowledge Base
• Type I & Type II Errors | Differences, Examples, Visualizations

Type I & Type II Errors | Differences, Examples, Visualizations

Published on 18 January 2021 by Pritha Bhandari . Revised on 2 February 2023.

In statistics , a Type I error is a false positive conclusion, while a Type II error is a false negative conclusion.

Making a statistical decision always involves uncertainties, so the risks of making these errors are unavoidable in hypothesis testing .

The probability of making a Type I error is the significance level , or alpha (α), while the probability of making a Type II error is beta (β). These risks can be minimized through careful planning in your study design.

• Type I error (false positive) : the test result says you have coronavirus, but you actually don’t.
• Type II error (false negative) : the test result says you don’t have coronavirus, but you actually do.

Table of contents

Error in statistical decision-making, type i error, type ii error, trade-off between type i and type ii errors, is a type i or type ii error worse, frequently asked questions about type i and ii errors.

Using hypothesis testing, you can make decisions about whether your data support or refute your research predictions with null and alternative hypotheses .

Hypothesis testing starts with the assumption of no difference between groups or no relationship between variables in the population—this is the null hypothesis . It’s always paired with an alternative hypothesis , which is your research prediction of an actual difference between groups or a true relationship between variables .

In this case:

• The null hypothesis (H 0 ) is that the new drug has no effect on symptoms of the disease.
• The alternative hypothesis (H 1 ) is that the drug is effective for alleviating symptoms of the disease.

Then , you decide whether the null hypothesis can be rejected based on your data and the results of a statistical test . Since these decisions are based on probabilities, there is always a risk of making the wrong conclusion.

• If your results show statistical significance , that means they are very unlikely to occur if the null hypothesis is true. In this case, you would reject your null hypothesis. But sometimes, this may actually be a Type I error.
• If your findings do not show statistical significance, they have a high chance of occurring if the null hypothesis is true. Therefore, you fail to reject your null hypothesis. But sometimes, this may be a Type II error.

A Type I error means rejecting the null hypothesis when it’s actually true. It means concluding that results are statistically significant when, in reality, they came about purely by chance or because of unrelated factors.

The risk of committing this error is the significance level (alpha or α) you choose. That’s a value that you set at the beginning of your study to assess the statistical probability of obtaining your results ( p value).

The significance level is usually set at 0.05 or 5%. This means that your results only have a 5% chance of occurring, or less, if the null hypothesis is actually true.

If the p value of your test is lower than the significance level, it means your results are statistically significant and consistent with the alternative hypothesis. If your p value is higher than the significance level, then your results are considered statistically non-significant.

To reduce the Type I error probability, you can simply set a lower significance level.

Type I error rate

The null hypothesis distribution curve below shows the probabilities of obtaining all possible results if the study were repeated with new samples and the null hypothesis were true in the population .

At the tail end, the shaded area represents alpha. It’s also called a critical region in statistics.

If your results fall in the critical region of this curve, they are considered statistically significant and the null hypothesis is rejected. However, this is a false positive conclusion, because the null hypothesis is actually true in this case!

A Type II error means not rejecting the null hypothesis when it’s actually false. This is not quite the same as “accepting” the null hypothesis, because hypothesis testing can only tell you whether to reject the null hypothesis.

Instead, a Type II error means failing to conclude there was an effect when there actually was. In reality, your study may not have had enough statistical power to detect an effect of a certain size.

Power is the extent to which a test can correctly detect a real effect when there is one. A power level of 80% or higher is usually considered acceptable.

The risk of a Type II error is inversely related to the statistical power of a study. The higher the statistical power, the lower the probability of making a Type II error.

Statistical power is determined by:

• Size of the effect : Larger effects are more easily detected.
• Measurement error : Systematic and random errors in recorded data reduce power.
• Sample size : Larger samples reduce sampling error and increase power.
• Significance level : Increasing the significance level increases power.

To (indirectly) reduce the risk of a Type II error, you can increase the sample size or the significance level.

Type II error rate

The alternative hypothesis distribution curve below shows the probabilities of obtaining all possible results if the study were repeated with new samples and the alternative hypothesis were true in the population .

The Type II error rate is beta (β), represented by the shaded area on the left side. The remaining area under the curve represents statistical power, which is 1 – β.

Increasing the statistical power of your test directly decreases the risk of making a Type II error.

The Type I and Type II error rates influence each other. That’s because the significance level (the Type I error rate) affects statistical power, which is inversely related to the Type II error rate.

This means there’s an important tradeoff between Type I and Type II errors:

• Setting a lower significance level decreases a Type I error risk, but increases a Type II error risk.
• Increasing the power of a test decreases a Type II error risk, but increases a Type I error risk.

This trade-off is visualized in the graph below. It shows two curves:

• The null hypothesis distribution shows all possible results you’d obtain if the null hypothesis is true. The correct conclusion for any point on this distribution means not rejecting the null hypothesis.
• The alternative hypothesis distribution shows all possible results you’d obtain if the alternative hypothesis is true. The correct conclusion for any point on this distribution means rejecting the null hypothesis.

Type I and Type II errors occur where these two distributions overlap. The blue shaded area represents alpha, the Type I error rate, and the green shaded area represents beta, the Type II error rate.

By setting the Type I error rate, you indirectly influence the size of the Type II error rate as well.

It’s important to strike a balance between the risks of making Type I and Type II errors. Reducing the alpha always comes at the cost of increasing beta, and vice versa .

For statisticians, a Type I error is usually worse. In practical terms, however, either type of error could be worse depending on your research context.

A Type I error means mistakenly going against the main statistical assumption of a null hypothesis. This may lead to new policies, practices or treatments that are inadequate or a waste of resources.

In contrast, a Type II error means failing to reject a null hypothesis. It may only result in missed opportunities to innovate, but these can also have important practical consequences.

In statistics, a Type I error means rejecting the null hypothesis when it’s actually true, while a Type II error means failing to reject the null hypothesis when it’s actually false.

The risk of making a Type I error is the significance level (or alpha) that you choose. That’s a value that you set at the beginning of your study to assess the statistical probability of obtaining your results ( p value ).

To reduce the Type I error probability, you can set a lower significance level.

The risk of making a Type II error is inversely related to the statistical power of a test. Power is the extent to which a test can correctly detect a real effect when there is one.

To (indirectly) reduce the risk of a Type II error, you can increase the sample size or the significance level to increase statistical power.

Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.

Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .

When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.

In statistics, power refers to the likelihood of a hypothesis test detecting a true effect if there is one. A statistically powerful test is more likely to reject a false negative (a Type II error).

If you don’t ensure enough power in your study, you may not be able to detect a statistically significant result even when it has practical significance. Your study might not have the ability to answer your research question.

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Type I & Type II Errors

• Medicine and Healthcare
• Medical Statistics & Computing
• Type I & Type II Errors

Statistical significance, effect size and statistical power

Louis Cohen, Lawrence Manion, Keith Morrison in Research Methods in Education , 2017

In hypothesis testing one has to avoid Type I and Type II errors (see also Chapter 12). A Type I error occurs when one does not support the null hypothesis when it is in fact true (a false positive). This is a particular problem as the sample size increases, as the chances of finding a statistically significant association increases; to overcome this one can set a higher alpha (α) limit (e.g. 0.01 or 0.001) for statistical significance to be achieved. A Type II error occurs when one supports the null hypothesis when it is in fact not true (a false negative), which is often the case if the levels of significance are set too stringently; to overcome this the researcher can set a lower alpha level (α) (e.g. 0.1 or 0.2). Type I and Type II errors are represented in Table 39.1.

Preliminaries: Welcome to the Statistical Inference Club. Some Basic Concepts in Experimental Decision Making

Albert Vexler, Alan D. Hutson, Xiwei Chen in Statistical Testing Strategies in the Health Sciences , 2017

The Type II error rate (β) is defined as β = Pr(L < C|H1) given that we assume that L is the test statistic based on the observed data, C is a threshold, and the decision rule is to reject H0 when L > C. Type II errors may occur when the sample size is relatively small compared to the design objectives; that is, the signal-to-noise ratio is small or the test is biased (Freiman et al. 1978). Essentially, it is the dichotomization of the study results into the categories “significant” or “not significant” that leads to Type I and Type II errors. Although errors resulting from an incorrect classification of the study results would seem to be unnecessary and avoidable, the Neyman–Pearson (dichotomous) hypothesis testing approach is ingrained in scientific research due to the apparent objectivity and definitiveness of the pronouncement of significance (Rothman et al. 2008). The Type II error rate may be reduced by increasing the sample size. However, it is not practical to have large sample sizes in many settings. Hence, in statistical design there is a trade-off between error control and feasibility. In some scientific disciplines, hypothesis testing has been virtually eliminated in exchange for a confidence interval approach. However, like all scientific methods, the strengths and limitations of the methodology need to be understood well by the end user.

Concluding Remarks

Song S. Qian, Mark R. DuFour, Ibrahim Alameddine in Bayesian Applications in Environmental and Ecological Studies with R and Stan , 2023

Before observing data, we want a test to be able to reject a false null hypothesis with a high probability and accept a true null hypothesis most of the time (not necessarily with a high probability). As a result, pre-data criteria for assessing a test are its probabilities of making the type I and type II errors. If we step away from the long-run frequency description of these error probabilities, we can see that the low type I error probability is designed to ensure that we do not make the most relevant type of error easily. Only when the data showed overwhelming evidence against the null could we conclude that the null should be rejected. Once we have observed the data and rejected the null hypothesis, simply stating that the result is significant seems to be unsatisfactory at best. We want to propose a specific alternative hypothesis and subject it to the same level of scrutiny. In other words, the existing hypothesis testing process explicitly defines a rigorous pre-data procedure that would reject the null hypothesis based on overwhelming evidence. Once we have the data, we need the same post-data procedure to defend any specific alternative hypothesis with strong evidence. This is the topic of “severe testing” of Mayo [2018].

Controlling the error probabilities of model selection information criteria using bootstrapping

Published in Journal of Applied Statistics , 2020

Michael Cullan, Scott Lidgard, Beckett Sterner

These innovations mark an important practical advance for explanatory modeling using hypothesis testing [33]. As scientists have expanded their range of available models, it has become harder to identify a uniquely suitable null hypothesis. Models of evolutionary tempo and mode in fossil traits, for instance, can all represent scientifically interesting evolutionary processes. Similarly, a precursor to our method [42] describes a comparison between two models of bacterial growth: one smooth model describing exponential growth and one segmented regression model. These non-nested models do not give a natural choice of null, and to specify either as the null model entails making an explicit choice that can alter the results of the test. Having to choose one model as the null also effectively removes it from consideration in its own right, since concluding the null is true when a test fails to reject it commits a ‘fallacy of rejection’ [35]. This consequence is especially problematic when the null hypothesis is difficult to reject, such as the unbiased random walk model for fossil time series when compared to Gaussian noise around a fixed mean or a biased random walk [20]. In these cases, it would be helpful to formally distinguish cases where the evidence is inconclusive for the null hypotheses versus strongly supporting it [30]. ECIC enables us draw this distinction in an error statistical framework through generalizing Type-I and Type-II errors.

Influence of inter-observer delineation variability on radiomics stability in different tumor sites

Published in Acta Oncologica , 2018

Matea Pavic, Marta Bogowicz, Xaver Würms, Stefan Glatz, Tobias Finazzi, Oliver Riesterer, Johannes Roesch, Leonie Rudofsky, Martina Friess, Patrick Veit-Haibach, Martin Huellner, Isabelle Opitz, Walter Weder, Thomas Frauenfelder, Matthias Guckenberger, Stephanie Tanadini-Lang

The inter-observer variability of radiomic features was studied using the two-way mixed single measures intraclass correlation coefficient (ICC(3,1)). The ICC is a statistical indicator that gives information about the consistency of feature measurements [19]. A value of 0 indicates no reliability whereas a value of 1 means that the measurements are highly stable. An ICC >0.8 is accepted to indicate a good reproducibility. This threshold was chosen based on a guideline where an ICC > 0.75 is indicative of a good reliability [20]. We selected a slightly stricter threshold to better match with previous published work, which have often chosen either 0.8 or 0.9 and to control for type I and type II errors in light of relatively small sample sizes, as already stated above. For the stable radiomic features, an average linkage hierarchical clustering was performed to identify classes of uncorrelated features with a cutoff value of 0.5. The statistical analysis was performed in R (version 3.2.3).

Is self-regulation key in reducing running-related injuries and chronic fatigue? A randomized controlled trial among long-distance runners

Published in Journal of Applied Sport Psychology , 2022

Luuk P. van Iperen, Jan de Jonge, Josette M. P. Gevers, Steven B. Vos, Luiz Hespanhol

To test Hypothesis 3 (i.e., app intervention effects across risk profiles), we (re-)calculated latent profiles in Mplus (version 8.5; Muthén & Muthén, 1998−2017). To establish boundary conditions across profiles, we tested profiles in relation to group assignment, intervention use, and outcomes (i.e., RRIs and chronic fatigue). We employed the stepwise BCH and DCAT approaches as advocated by Asparouhov and Muthén (2020). To test whether the app effects differed across these profiles, we used model constraints and Wald tests (see Asparouhov & Muthén, 2018). The latter entails a subgroup analysis in evaluating a randomized controlled trial. If faced with potential power issues, as may be the case in the current study, this analysis can have increased rates of type I and type II errors and a limited precision of estimates (Keller, 2019). To address these limitations, we corrected for multiple testing using the conservative Bonferroni correction in deciding upon significance (Armstrong, 2014), and we explicitly note that this analysis is post-hoc and exploratory. Still, this risk profile analysis is important to answer the “what works for whom” question (Keller, 2019), which is considered relevant because it may provide directions for future studies and provide important insights into how psychological aspects can affect intervention effects.

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6.1 - type i and type ii errors.

When conducting a hypothesis test there are two possible decisions: reject the null hypothesis or fail to reject the null hypothesis. You should remember though, hypothesis testing uses data from a sample to make an inference about a population. When conducting a hypothesis test we do not know the population parameters. In most cases, we don't know if our inference is correct or incorrect.

When we reject the null hypothesis there are two possibilities. There could really be a difference in the population, in which case we made a correct decision. Or, it is possible that there is not a difference in the population (i.e., \(H_0\) is true) but our sample was different from the hypothesized value due to random sampling variation. In that case we made an error. This is known as a Type I error.

When we fail to reject the null hypothesis there are also two possibilities. If the null hypothesis is really true, and there is not a difference in the population, then we made the correct decision. If there is a difference in the population, and we failed to reject it, then we made a Type II error.

Rejecting \(H_0\) when \(H_0\) is really true, denoted by \(\alpha\) ("alpha") and commonly set at .05

     \(\alpha=P(Type\;I\;error)\)

Failing to reject \(H_0\) when \(H_0\) is really false, denoted by \(\beta\) ("beta")

     \(\beta=P(Type\;II\;error)\)

Example: Trial Section  

A man goes to trial where he is being tried for the murder of his wife.

We can put it in a hypothesis testing framework. The hypotheses being tested are:

• \(H_0\) : Not Guilty
• \(H_a\) : Guilty

Type I error  is committed if we reject \(H_0\) when it is true. In other words, did not kill his wife but was found guilty and is punished for a crime he did not really commit.

Type II error  is committed if we fail to reject \(H_0\) when it is false. In other words, if the man did kill his wife but was found not guilty and was not punished.

Example: Culinary Arts Study Section  

A group of culinary arts students is comparing two methods for preparing asparagus: traditional steaming and a new frying method. They want to know if patrons of their school restaurant prefer their new frying method over the traditional steaming method. A sample of patrons are given asparagus prepared using each method and asked to select their preference. A statistical analysis is performed to determine if more than 50% of participants prefer the new frying method:

• \(H_{0}: p = .50\)
• \(H_{a}: p>.50\)

Type I error  occurs if they reject the null hypothesis and conclude that their new frying method is preferred when in reality is it not. This may occur if, by random sampling error, they happen to get a sample that prefers the new frying method more than the overall population does. If this does occur, the consequence is that the students will have an incorrect belief that their new method of frying asparagus is superior to the traditional method of steaming.

Type II error  occurs if they fail to reject the null hypothesis and conclude that their new method is not superior when in reality it is. If this does occur, the consequence is that the students will have an incorrect belief that their new method is not superior to the traditional method when in reality it is.

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Statistics By Jim

Making statistics intuitive

Types I & Type II Errors in Hypothesis Testing

By Jim Frost 8 Comments

In hypothesis testing, a Type I error is a false positive while a Type II error is a false negative. In this blog post, you will learn about these two types of errors, their causes, and how to manage them.

Hypothesis tests use sample data to make inferences about the properties of a population . You gain tremendous benefits by working with random samples because it is usually impossible to measure the entire population.

However, there are tradeoffs when you use samples. The samples we use are typically a minuscule percentage of the entire population. Consequently, they occasionally misrepresent the population severely enough to cause hypothesis tests to make Type I and Type II errors.

Potential Outcomes in Hypothesis Testing

Hypothesis testing  is a procedure in inferential statistics that assesses two mutually exclusive theories about the properties of a population. For a generic hypothesis test, the two hypotheses are as follows:

• Null hypothesis : There is no effect
• Alternative hypothesis : There is an effect.

The sample data must provide sufficient evidence to reject the null hypothesis and conclude that the effect exists in the population. Ideally, a hypothesis test fails to reject the null hypothesis when the effect is not present in the population, and it rejects the null hypothesis when the effect exists.

Statisticians define two types of errors in hypothesis testing. Creatively, they call these errors Type I and Type II errors. Both types of error relate to incorrect conclusions about the null hypothesis.

The table summarizes the four possible outcomes for a hypothesis test.

Related post : How Hypothesis Tests Work: P-values and the Significance Level

Fire alarm analogy for the types of errors

Using hypothesis tests correctly improves your chances of drawing trustworthy conclusions. However, errors are bound to occur.

Unlike the fire alarm analogy, there is no sure way to determine whether an error occurred after you perform a hypothesis test. Typically, a clearer picture develops over time as other researchers conduct similar studies and an overall pattern of results appears. Seeing how your results fit in with similar studies is a crucial step in assessing your study’s findings.

Now, let’s take a look at each type of error in more depth.

Type I Error: False Positives

When you see a p-value that is less than your significance level , you get excited because your results are statistically significant. However, it could be a type I error . The supposed effect might not exist in the population. Again, there is usually no warning when this occurs.

Why do these errors occur? It comes down to sample error. Your random sample has overestimated the effect by chance. It was the luck of the draw. This type of error doesn’t indicate that the researchers did anything wrong. The experimental design, data collection, data validity , and statistical analysis can all be correct, and yet this type of error still occurs.

Even though we don’t know for sure which studies have false positive results, we do know their rate of occurrence. The rate of occurrence for Type I errors equals the significance level of the hypothesis test, which is also known as alpha (α).

The significance level is an evidentiary standard that you set to determine whether your sample data are strong enough to reject the null hypothesis. Hypothesis tests define that standard using the probability of rejecting a null hypothesis that is actually true. You set this value based on your willingness to risk a false positive.

Related post : How to Interpret P-values Correctly

Using the significance level to set the Type I error rate

When the significance level is 0.05 and the null hypothesis is true, there is a 5% chance that the test will reject the null hypothesis incorrectly. If you set alpha to 0.01, there is a 1% of a false positive. If 5% is good, then 1% seems even better, right? As you’ll see, there is a tradeoff between Type I and Type II errors. If you hold everything else constant, as you reduce the chance for a false positive, you increase the opportunity for a false negative.

Type I errors are relatively straightforward. The math is beyond the scope of this article, but statisticians designed hypothesis tests to incorporate everything that affects this error rate so that you can specify it for your studies. As long as your experimental design is sound, you collect valid data, and the data satisfy the assumptions of the hypothesis test, the Type I error rate equals the significance level that you specify. However, if there is a problem in one of those areas, it can affect the false positive rate.

Warning about a potential misinterpretation of Type I errors and the Significance Level

When the null hypothesis is correct for the population, the probability that a test produces a false positive equals the significance level. However, when you look at a statistically significant test result, you cannot state that there is a 5% chance that it represents a false positive.

Why is that the case? Imagine that we perform 100 studies on a population where the null hypothesis is true. If we use a significance level of 0.05, we’d expect that five of the studies will produce statistically significant results—false positives. Afterward, when we go to look at those significant studies, what is the probability that each one is a false positive? Not 5 percent but 100%!

That scenario also illustrates a point that I made earlier. The true picture becomes more evident after repeated experimentation. Given the pattern of results that are predominantly not significant, it is unlikely that an effect exists in the population.

Type II Error: False Negatives

When you perform a hypothesis test and your p-value is greater than your significance level, your results are not statistically significant. That’s disappointing because your sample provides insufficient evidence for concluding that the effect you’re studying exists in the population. However, there is a chance that the effect is present in the population even though the test results don’t support it. If that’s the case, you’ve just experienced a Type II error . The probability of making a Type II error is known as beta (β).

What causes Type II errors? Whereas Type I errors are caused by one thing, sample error, there are a host of possible reasons for Type II errors—small effect sizes, small sample sizes, and high data variability. Furthermore, unlike Type I errors, you can’t set the Type II error rate for your analysis. Instead, the best that you can do is estimate it before you begin your study by approximating properties of the alternative hypothesis that you’re studying. When you do this type of estimation, it’s called power analysis.

To estimate the Type II error rate, you create a hypothetical probability distribution that represents the properties of a true alternative hypothesis. However, when you’re performing a hypothesis test, you typically don’t know which hypothesis is true, much less the specific properties of the distribution for the alternative hypothesis. Consequently, the true Type II error rate is usually unknown!

Type II errors and the power of the analysis

The Type II error rate (beta) is the probability of a false negative. Therefore, the inverse of Type II errors is the probability of correctly detecting an effect. Statisticians refer to this concept as the power of a hypothesis test. Consequently, 1 – β = the statistical power. Analysts typically estimate power rather than beta directly.

If you read my post about power and sample size analysis , you know that the three factors that affect power are sample size, variability in the population, and the effect size. As you design your experiment, you can enter estimates of these three factors into statistical software and it calculates the estimated power for your test.

Suppose you perform a power analysis for an upcoming study and calculate an estimated power of 90%. For this study, the estimated Type II error rate is 10% (1 – 0.9). Keep in mind that variability and effect size are based on estimates and guesses. Consequently, power and the Type II error rate are just estimates rather than something you set directly. These estimates are only as good as the inputs into your power analysis.

Low variability and larger effect sizes decrease the Type II error rate, which increases the statistical power. However, researchers usually have less control over those aspects of a hypothesis test. Typically, researchers have the most control over sample size, making it the critical way to manage your Type II error rate. Holding everything else constant, increasing the sample size reduces the Type II error rate and increases power.

Learn more about Power in Statistics .

Graphing Type I and Type II Errors

The graph below illustrates the two types of errors using two sampling distributions. The critical region line represents the point at which you reject or fail to reject the null hypothesis. Of course, when you perform the hypothesis test, you don’t know which hypothesis is correct. And, the properties of the distribution for the alternative hypothesis are usually unknown. However, use this graph to understand the general nature of these errors and how they are related.

The distribution on the left represents the null hypothesis. If the null hypothesis is true, you only need to worry about Type I errors, which is the shaded portion of the null hypothesis distribution. The rest of the null distribution represents the correct decision of failing to reject the null.

On the other hand, if the alternative hypothesis is true, you need to worry about Type II errors. The shaded region on the alternative hypothesis distribution represents the Type II error rate. The rest of the alternative distribution represents the probability of correctly detecting an effect—power.

Moving the critical value line is equivalent to changing the significance level. If you move the line to the left, you’re increasing the significance level (e.g., α 0.05 to 0.10). Holding everything else constant, this adjustment increases the Type I error rate while reducing the Type II error rate. Moving the line to the right reduces the significance level (e.g., α 0.05 to 0.01), which decreases the Type I error rate but increases the type II error rate.

Is One Error Worse Than the Other?

As you’ve seen, the nature of the two types of error, their causes, and the certainty of their rates of occurrence are all very different.

A common question is whether one type of error is worse than the other? Statisticians designed hypothesis tests to control Type I errors while Type II errors are much less defined. Consequently, many statisticians state that it is better to fail to detect an effect when it exists than it is to conclude an effect exists when it doesn’t. That is to say, there is a tendency to assume that Type I errors are worse.

However, reality is more complex than that. You should carefully consider the consequences of each type of error for your specific test.

Suppose you are assessing the strength of a new jet engine part that is under consideration. Peoples lives are riding on the part’s strength. A false negative in this scenario merely means that the part is strong enough but the test fails to detect it. This situation does not put anyone’s life at risk. On the other hand, Type I errors are worse in this situation because they indicate the part is strong enough when it is not.

Now suppose that the jet engine part is already in use but there are concerns about it failing. In this case, you want the test to be more sensitive to detecting problems even at the risk of false positives. Type II errors are worse in this scenario because the test fails to recognize the problem and leaves these problematic parts in use for longer.

Using hypothesis tests effectively requires that you understand their error rates. By setting the significance level and estimating your test’s power, you can manage both error rates so they meet your requirements.

The error rates in this post are all for individual tests. If you need to perform multiple comparisons, such as comparing group means in ANOVA, you’ll need to use post hoc tests to control the experiment-wise error rate  or use the Bonferroni correction .

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Reader Interactions

June 4, 2024 at 2:04 pm

Very informative.

June 9, 2023 at 9:54 am

Hi Jim- I just signed up for your newsletter and this is my first question to you. I am not a statistician but work with them in my professional life as a QC consultant in biopharmaceutical development. I have a question about Type I and Type II errors in the realm of equivalence testing using two one sided difference testing (TOST). In a recent 2020 publication that I co-authored with a statistician, we stated that the probability of concluding non-equivalence when that is the truth, (which is the opposite of power, the probability of concluding equivalence when it is correct) is 1-2*alpha. This made sense to me because one uses a 90% confidence interval on a mean to evaluate whether the result is within established equivalence bounds with an alpha set to 0.05. However, it appears that specificity (1-alpha) is always the case as is power always being 1-beta. For equivalence testing the latter is 1-2*beta/2 but for specificity it stays as 1-alpha because only one of the null hypotheses in a two-sided test can fail at one time. I still see 1-2*alpha as making more sense as we show in Figure 3 of our paper which shows the white space under the distribution of the alternative hypothesis as 1-2 alpha. The paper can be downloaded as open access here if that would make my question more clear. https://bioprocessingjournal.com/index.php/article-downloads/890-vol-19-open-access-2020-defining-therapeutic-window-for-viral-vectors-a-statistical-framework-to-improve-consistency-in-assigning-product-dose-values I have consulted with other statistical colleagues and cannot get consensus so I would love your opinion and explanation! Thanks in advance!

June 10, 2023 at 1:00 am

Let me preface my response by saying that I’m not an expert in equivalence testing. But here’s my best guess about your question.

The alpha is for each of the hypothesis tests. Each one has a type I error rate of 0.05. Or, as you say, a specificity of 1-alpha. However, there are two tests so we need to consider the family-wise error rate. The formula is the following:

FWER = 1 – (1 – α)^N

Where N is the number of hypothesis tests.

For two tests, there’s a family-wise error rate of 0.0975. Or a family-wise specificity of 0.9025.

However, I believe they use 90% CI for a different reason (although it’s a very close match to the family-wise error rate). The 90% CI provides consistent results with the two one-side 95% tests. In other words, if the 90% CI is within the equivalency bounds, then the two tests will be significant. If the CI extends above the upper bound, the corresponding test won’t be significant. Etc.

However, using either rational, I’d say the overall type I error rate is about 0.1.

I hope that answers your question. And, again, I’m not an expert in this particular test.

July 18, 2022 at 5:15 am

Thank you for your valuable content. I have a question regarding correcting for multiple tests. My question is: for exactly how many tests should I correct in the scenario below?

Background: I’m testing for differences between groups A (patient group) and B (control group) in variable X. Variable X is a biological variable present in the body’s left and right side. Variable Y is a questionnaire for group A.

Step 1. Is there a significant difference within groups in the weight of left and right variable X? (I will conduct two paired sample t-tests)


If I find a significant difference in step 1, then I will conduct steps 2A and 2B. However, if I don’t find a significant difference in step 1, then I will only conduct step 2C.

Step 2A. Is there a significant difference between groups in left variable X? (I will conduct one independent sample t-test) Step 2B. Is there a significant difference between groups in right variable X? (I will conduct one independent sample t-test)

Step 2C. Is there a significant difference between groups in total variable X (left + right variable X)? (I will conduct one independent sample t-test)

If I find a significant difference in step 1, then I will conduct with steps 3A and 3B. However, if I don’t find a significant difference in step 1, then I will only conduct step 3C.

Step 3A. Is there a significant correlation between left variable X in group A and variable Y? (I will conduct Pearson correlation) Step 3B. Is there a significant correlation between right variable X in group A and variable Y? (I will conduct Pearson correlation)

Step 3C. Is there a significant correlation between total variable X in group A and variable Y? (I will conduct a Pearson correlation)

Regards, De

January 2, 2021 at 1:57 pm

I should say that being a budding statistician, this site seems to be pretty reliable. I have few doubts in here. It would be great if you can clarify it:

“A significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference. ”

My understanding : When we say that the significance level is 0.05 then it means we are taking 5% risk to support alternate hypothesis even though there is no difference ?( I think i am not allowed to say Null is true, because null is assumed to be true/ Right)

January 2, 2021 at 6:48 pm

The sentence as I write it is correct. Here’s a simple way to understand it. Imagine you’re conducting a computer simulation where you control the population parameters and have the computer draw random samples from the populations that you define. Now, imagine you draw samples from two populations where the means and standard deviations are equal. You know this for a fact because you set the parameters yourself. Then you conduct a series of 2-sample t-tests.

In this example, you know the null hypothesis is correct. However, thanks to random sampling error, some proportion of the t-tests will have statistically significant results (i.e., false positives or Type I errors). The proportion of false positives will equal your significance level over the long run.

Of course, in real-world experiments, you never know for sure whether the null is true or not. However, given the properties of the hypothesis, you do know what proportion of tests will give you a false positive IF the null is true–and that’s the significance level.

I’m thinking through the wording of how you wrote it and I believe it is equivalent to what I wrote. If there is no difference (the null is true), then you have a 5% chance of incorrectly supporting the alternative. And, again, you’re correct that in the real world you don’t know for sure whether the null is true. But, you can still know the false positive (Type I) error rate. For more information about that property, read my post about how hypothesis tests work .

July 9, 2018 at 11:43 am

I like to use the analogy of a trial. The null hypothesis is that the defendant is innocent. A type I error would be convicting an innocent person and a type II error would be acquitting a guilty one. I like to think that our system makes a type I error very unlikely with the trade off being that a type II error is greater.

July 9, 2018 at 12:03 pm

Hi Doug, I think that is an excellent analogy on multiple levels. As you mention, a trial would set a high bar for the significance level by choosing a very low value for alpha. This helps prevent innocent people from being convicted (Type I error) but does increase the probability of allowing the guilty to go free (Type II error). I often refer to the significant level as a evidentiary standard with this legalistic analogy in mind.

Additionally, in the justice system in the U.S., there is a presumption of innocence and the prosecutor must present sufficient evidence to prove that the defendant is guilty. That’s just like in a hypothesis test where the assumption is that the null hypothesis is true and your sample must contain sufficient evidence to be able to reject the null hypothesis and suggest that the effect exists in the population.

This analogy even works for the similarities behind the phrases “Not guilty” and “Fail to reject the null hypothesis.” In both cases, you aren’t proving innocence or that the null hypothesis is true. When a defendant is “not guilty” it might be that the evidence was insufficient to convince the jury. In a hypothesis test, when you fail to reject the null hypothesis, it’s possible that an effect exists in the population but you have insufficient evidence to detect it. Perhaps the effect exists but the sample size or effect size is too small, or the variability might be too high.

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Type II Error

"False negative" error

What is a Type II Error?

In statistical hypothesis testing, a type II error is a situation wherein a hypothesis test fails to reject the null hypothesis that is false. In other words, it causes the user to erroneously not reject the false null hypothesis because the test lacks the statistical power to detect sufficient evidence for the alternative hypothesis. The type II error is also known as a false negative.

The type II error has an inverse relationship with the power of a statistical test. This means that the higher power of a statistical test, the lower the probability of committing a type II error. The rate of a type II error (i.e., the probability of a type II error) is measured by beta (β) while the statistical power is measured by 1- β.

How to Avoid the Type II Error?

Similar to the type I error, it is not possible to completely eliminate the type II error from a hypothesis test . The only available option is to minimize the probability of committing this type of statistical error. Since a type II error is closely related to the power of a statistical test, the probability of the occurrence of the error can be minimized by increasing the power of the test.

1. Increase the sample size

One of the simplest methods to increase the power of the test is to increase the sample size used in a test. The sample size primarily determines the amount of sampling error, which translates into the ability to detect the differences in a hypothesis test. A larger sample size increases the chances to capture the differences in the statistical tests, as well as raises the power of a test.

2. Increase the significance level

Another method is to choose a higher level of significance . For instance, a researcher may choose a significance level of 0.10 instead of the commonly acceptable 0.05 level. The higher significance level implies a higher probability of rejecting the null hypothesis when it is true.

The larger probability of rejecting the null hypothesis decreases the probability of committing a type II error while the probability of committing a type I error increases. Thus, the user should always assess the impact of type I and type II errors on their decision and determine the appropriate level of statistical significance.

Practical Example

Sam is a financial analyst . He runs a hypothesis test to discover whether there is a difference in the average price changes for large-cap and small-cap stocks .

In the test, Sam assumes as the null hypothesis that there is no difference in the average price changes between large-cap and small-cap stocks. Thus, his alternative hypothesis states that a difference between the average price changes does exist.

For the significance level, Sam chooses 5%. This means that there is a 5% probability that his test will reject the null hypothesis when it is actually true.

If Sam’s test incurs a type II error, then the results of the test will indicate that there is no difference in the average price changes between large-cap and small-cap stocks. However, in reality, a difference in the average price changes does exist.

More Resources

CFI is the official provider of the global Business Intelligence & Data Analyst (BIDA)®  certification program, designed to help anyone become a world-class financial analyst. To keep learning and advancing your career, the additional CFI resources below will be useful:

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• Framing Bias
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Curbing type I and type II errors

Kenneth j. rothman.

RTI Health Solutions, Research Triangle Park, NC USA

The statistical education of scientists emphasizes a flawed approach to data analysis that should have been discarded long ago. This defective method is statistical significance testing. It degrades quantitative findings into a qualitative decision about the data. Its underlying statistic, the P -value, conflates two important but distinct aspects of the data, effect size and precision [ 1 ]. It has produced countless misinterpretations of data that are often amusing for their folly, but also hair-raising in view of the serious consequences.

Significance testing maintains its hold through brilliant marketing tactics—the appeal of having a “significant” result is nearly irresistible—and through a herd mentality. Novices quickly learn that significant findings are the key to publication and promotion, and that statistical significance is the mantra of many senior scientists who will judge their efforts. Stang et al. [ 2 ], in this issue of the journal, liken the grip of statistical significance testing on the biomedical sciences to tyranny, as did Loftus in the social sciences two decades ago [ 3 ]. The tyranny depends on collaborators to maintain its stranglehold. Some collude because they do not know better. Others do so because they lack the backbone to swim against the tide.

Students of significance testing are warned about two types of errors, type I and II, also known as alpha and beta errors. A type I error is a false positive, rejecting a null hypothesis that is correct. A type II error is a false negative, a failure to reject a null hypothesis that is false. A large literature, much of it devoted to the topic of multiple comparisons, subgroup analysis, pre-specification of hypotheses, and related topics, are aimed at reducing type I errors [ 4 ]. This lopsided emphasis on type I errors comes at the expense of type II errors. The type I error, the false positive, is only possible if the null hypothesis is true. If the null hypothesis is false, a type I error is impossible, but a type II error, the false negative, can occur.

Type I and type II errors are the product of forcing the results of a quantitative analysis into the mold of a decision, which is whether to reject or not to reject the null hypothesis. Reducing interpretations to a dichotomy, however, seriously degrades the information. The consequence is often a misinterpretation of study results, stemming from a failure to separate effect size from precision. Both effect size and precision need to be assessed, but they need to be assessed separately, rather than blended into the P -value, which is then degraded into a dichotomous decision about statistical significance.

As an example of what can happen when significance testing is exalted beyond reason, consider the case of the Wall Street Journal investigative reporter who broke the news of a scandal about a medical device maker, Boston Scientific, having supposedly distorted study results [ 5 ]. Boston Scientific reported to the FDA that a new device was better than a competing device. They based their conclusion in part on results from a randomized trial in which the significance test showing the superiority of their device had a P -value of 0.049, just under the criterion of 0.05 that the FDA used statistical significance. The reporter found, however, that the P -value was not significant when calculated using 16 other test procedures that he tried. The P -values from those procedures averaged 0.051. According to the news story, that small difference between the reported P -value of 0.049 and the journalist’s recalculated P -value of 0.051 was “the difference between success and failure” [ 5 ]. Regardless of what the “correct” P -value is for the data in question, it should be obvious that it is absurd to classify the success or failure of this new device according to whether or not the P -value falls barely on one side or the other of an arbitrary line, especially when the discussion revolves around the third decimal place of the P -value. No sensible interpretation of the data from the study should be affected by the news in this newspaper report. Unfortunately, the arbitrary standard imposed by regulatory agencies, which foster that focus on the P -value, reduces the prospects for more sensible evaluations.

In their article, Stang et al. [ 2 ] not only describe the problems with significance testing, but also allude to the solution, which is to rely on estimation using confidence intervals. Sadly, although the use of confidence intervals is increasing, for many readers and authors they are used only as surrogate tests of statistical significance [ 6 ], to note whether the null hypothesis value falls inside the interval or not. This dichotomy is equivalent to the dichotomous interpretation that results from significance testing. When confidence intervals are misused in this way, the entire conclusion can depend on whether the boundary of the interval is located precisely on one side or the other of an artificial criterion point. This is just the kind of mistake that tripped up the Wall Street Journal reporter. Using a confidence interval as a significance test is an opportunity lost.

How should a confidence interval be interpreted? It should be approached in the spirit of a quantitative estimate. A confidence interval allows a measurement of both effect size and precision, the two aspects of study data that are conflated in a P -value. A properly interpreted confidence interval allows these two aspects of the results to be inferred separately and quantitatively. The effect size is measured directly by the point estimate, which, if not given explicitly, can be calculated from the two confidence limits. For a difference measure, the point estimate is the arithmetic mean of the two limits, and for a ratio measure, it is the geometric mean. Precision is measured by the narrowness of the confidence interval. Thus, the two limits of a confidence interval convey information on both effect size and precision. The single number that is the P -value, even without degrading it into categories of “significant” and “not significant”, cannot measure two distinct things. Instead the P -value mixes effect size and precision in a way that by itself reveals little about either.

Scientists who wish to avoid type I or type II errors at all costs may have chosen the wrong profession, because making and correcting mistakes are inherent to science. There is a way, however, to minimize both type I and type II errors. All that is needed is simply to abandon significance testing. If one does not impose an artificial and potentially misleading dichotomous interpretation upon the data, one can reduce all type I and type II errors to zero. Instead of significance testing, one can rely on confidence intervals, interpreted quantitatively, not simply as surrogate significance tests. Only then would the analyses be truly quantitative.

Finally, here is a gratuitous bit of advice for testers and estimators alike: both P -values and confidence intervals are calculated and all too often interpreted as if the study they came from were free of bias. In reality, every study is biased to some extent. Even those who wisely eschew significance testing should keep in mind that if any study were increased in size, its precision would improve and thus all its confidence intervals would shrink, but as they do, they would eventually converge around incorrect values as a result of bias. The final interpretation should measure effect size and precision separately, while considering bias and even correcting for it [ 7 ].

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What are Type I and Type II Errors?

Posted on 21st April 2017 by Priscilla Wittkopf

When conducting a hypothesis test, we could:

• Reject the null hypothesis when there is a genuine effect in the population;
• Fail to reject the null hypothesis when there isn’t a genuine effect in the population.

However, as we are inferring results from samples and using probabilities to do so, we are never working with 100% certainty of the presence or absence of an effect. There are two other possible outcomes of a hypothesis test.

• Reject the null hypothesis when there isn’t a genuine effect – we have a false positive result and this is called Type I error .
• Fail to reject the null hypothesis when there is a genuine effect – we have a false negative result and this is called Type II error .

So in simple terms, a type I error is erroneously detecting an effect that is not present , while a type II error is the failure to detect an effect that is present.

Type I error

This error occurs when we reject the null hypothesis when we should have retained it. That means that we believe we found a genuine effect when in reality there isn’t one. The probability of a type I error occurring is represented by α and as a convention the threshold is set at 0.05 (also known as significance level). When setting a threshold at 0.05 we are accepting that there is a 5% probability of identifying an effect when actually there isn’t one.

Type II error

This error occurs when we fail to reject the null hypothesis. In other words, we believe that there isn’t a genuine effect when actually there is one. The probability of a Type II error is represented as β and this is related to the power of the test (power = 1- β). Cohen (1998) proposed that the maximum accepted probability of a Type II error should be 20% (β = 0.2).

When designing and planning a study the researcher should decide the values of α and β, bearing in mind that inferential statistics involve a balance between Type I and Type II errors. If α is set at a very small value the researcher is more rigorous with the standards of rejection of the null hypothesis. For example, if α = 0.01 the researcher is accepting a probability of 1% of erroneously rejecting the null hypothesis, but there is an increase in the probability of a Type II error.

In summary, we can see on the table the possible outcomes of a hypothesis test:

Have this table in mind when designing, analysing and reading studies, it will help when interpreting findings.

COHEN, J. 1990. Things I have learned (so far). American psychologist, 45 , 1304.

COHEN, J. 1998. Statistical Power Analysis for the Behavioral Sciences , Lawrence Erlbaum Associates.

FIELD, A. 2013. Discovering statistics using IBM SPSS statistics , Sage.

Priscilla Wittkopf

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Type I Error and Type II Error

Experimental errors in research.

While you might not have heard of Type I error or Type II error, you’re probably familiar with the terms “false positive” and “false negative.”

This article is a part of the guide:

• Null Hypothesis
• Research Hypothesis
• Defining a Research Problem
• Selecting Method

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A common medical example is a patient who takes an HIV test which promises a 99.9% accuracy rate. This means that in 0.1% of cases, or 1 in every 1000, the test gives a 'false positive,' informing a patient that they have the virus when they do not.

On the other hand, the test could also show a false negative reading, giving a person who is actually HIV positive the all-clear. This is why most medical tests require duplicate samples, to stack the odds in our favor. A 1 in 1000 chance of a false positive becomes a 1 in 1 000 000 chance of two false positives, if two tests are taken.

With any scientific process , there is no such thing as total proof or total rejection, whether of test results or of a null hypothesis . Researchers must work instead with probabilities. So even if the probabilities are lowered to 1 in 1000 000, there is still the chance that the results may be wrong.

How Does This Translate to Science?

Type i error.

A Type I error is often referred to as a “false positive” and is the incorrect rejection of the true null hypothesis in favor of the alternative.

In the example above, the null hypothesis refers to the natural state of things or the absence of the tested effect or phenomenon, i.e. stating that the patient is HIV negative. The alternative hypothesis states that the patient is HIV positive. Many medical tests will have the disease they are testing for as the alternative hypothesis and the lack of that disease as the null hypothesis.

A Type I error would thus occur when the patient doesn’t have the virus but the test shows that they do. In other words, the test incorrectly rejects the true null hypothesis that the patient is HIV negative.

Type II Error

A Type II error is the inverse of a Type I error and is the false acceptance of a null hypothesis that is not actually true, i.e. a false negative. A Type II error would entail the test telling the patient they are free of HIV when they are not.

Considering this HIV example, which error type do you think is more acceptable? In other words, would you rather have a test that was more prone to Type I or Type II error? With HIV, it’s likely that the momentary stress of a false positive is better than feeling relieved at a false negative and then failing to take steps to treat the disease. Pregnancy tests, blood tests and any diagnostic tool that has serious consequences for the health of a patient are usually overly sensitive for this reason – it is much better for them to err on the side of a false positive.

But in most fields of science, Type II errors are seen as less serious than Type I errors. With the Type II error, a chance to reject the null hypothesis was lost, and no conclusion is inferred from a non-rejected null. But the Type I error is more serious, because you have wrongly rejected the null hypothesis and ultimately made a claim that is not true. In science, finding a phenomenon where there is none is more egregious than failing to find a phenomenon where there is. Therefore in most research designs, effort is made to err on the side of a false negative.

Replication

This is the key reason why scientific experiments must be replicable.

Even if the highest level of proof is reached, where P < 0.01 ( probability is less than 1%), out of every 100 experiments, there will still be one false result. To a certain extent, duplicate or triplicate samples reduce the chance of error , but may still mask chance if the error -causing variable is present in all samples.

But if other researchers, using the same equipment, replicate the experiment and find that the results are the same, the chances of 5 or 10 experiments giving false results is unbelievably small. This is how science regulates and minimizes the potential for both Type I and Type II errors.

Of course, in certain experiments and medical diagnoses, replication is not always possible, so the possibility of Type I and II errors is always a factor.

One area that is guilty of forgetting about Type I and II errors is in the legal system, where a jury is seldom told that fingerprint and DNA tests may produce false results. There have been many documented failures of justice involving such tests. Today courts will no longer accept these tests alone as proof of guilt, and require other evidence to reduce the possibility of error to acceptable levels.

Type III Errors

Some statisticians are now adopting a third type of error, Type III, which is where the null hypothesis was correctly rejected …but for the wrong reason.

In an experiment, a researcher might postulate a hypothesis and perform research. After analyzing the results statistically, the null hypothesis is rejected.

The problem is that there may indeed be some relationship between the variables , but it’s not the one stated in the hypothesis. There is no error in rejecting the null here, but the error lies in accepting an incorrect alternative hypothesis. Hence a still unknown process may underlie the relationship, and the researchers are none the wiser.

As an example, researchers may be interested to see if there is any difference in two group means, and find that there is one. So they reject the null hypothesis but don’t notice that the difference is actually in the opposite direction to what their results found. Perhaps random chance led them to collect low scores from the group that is in reality higher and high scores from the group that is in reality lower. This is a curious way of being both correct and incorrect at the same time! As you can imagine, Type III errors are rare.

Economist Howard Raiffa gives a different definition for Type III error, one that others have called Type 0: getting the correct answer to an incorrect question.

Additionally, a Type IV error has been defined as incorrectly interpreting a null hypothesis that has been correctly rejected. Type IV error comes down to faulty analysis, bias or fumbling with the data to arrive at incorrect conclusions.

Errors of all types should be taken into account by scientists when conducting research.

Whilst replication can minimize the chances of an inaccurate result, it is no substitute for clear and logical research design, and careful analysis of results.

Many scientists do not accept quasi-experiments , because they are difficult to replicate and analyze, and therefore have a higher risk of being affected by error.

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An Adaptive Differential Evolution Algorithm Based on Data Preprocessing Method and a New Mutation Strategy to Solve Dynamic Economic Dispatch Considering Generator Constraints

• Published: 08 September 2024

Cite this article

• Ruxin Zhao   ORCID: orcid.org/0000-0002-6810-2631 1 ,
• Wei Wang 1 ,
• Tingting Zhang 1 ,
• Chang Liu 2 ,
• Lixiang Fu 1 ,
• Jiajie Kang 1 ,
• Hongtan Zhang 1 ,
• Yang Shi 1 &
• Chao Jiang 1  

Differential evolution (DE) algorithm is a classical natural-inspired optimization algorithm which has a good. However, with the deepening of research, some researchers found that the quality of the candidate solution of the population in the differential evolution algorithm is poor and its global search ability is not enough when solving the global optimization problem. Therefore, in order to solve the above problems, we proposed an adaptive differential evolution algorithm based on the data processing method and a new mutation strategy (ADEDPMS). In this paper, the data preprocessing method is implemented by k -means clustering algorithm, which is used to divide the initial population into multiple clusters according to the average value of fitness, and select candidate solutions in each cluster according to different proportions. This method improves the quality of candidate solutions of the population to a certain extent. In addition, in order to solve the problem of insufficient global search ability in differential evolution algorithm, we also proposed a new mutation strategy, which is called “DE/current-to- \({p}_{1}\) best& \({p}_{2}\) best”. This strategy guides the search direction of the differential evolution algorithm by selecting individuals with good fitness, so that its search range is in the most promising candidate solution region, and indirectly increases the population diversity of the algorithm. We also proposed an adaptive parameter control method, which can effectively balance the relationship between the exploration process and the exploitation process to achieve the best performance. In order to verify the effectiveness of the proposed algorithm, the ADEDPMS is compared with five optimization algorithms of the same type in the past three years, which are AAGSA, DFPSO, HGASSO, HHO and VAGWO. In the simulation experiment, 6 benchmark test functions and 4 engineering example problems are used, and the convergence accuracy, convergence speed and stability are fully compared. We used ADEDPMS to solve the dynamic economic dispatch (ED) problem with generator constraints. It is compared with the optimization algorithms used to solve the ED problem in the last three years which are AEFA, AVOA, OOA, SCA and TLBO. The experimental results show that compared with the five latest optimization algorithms proposed in the past three years to solve benchmark functions, engineering example problems and the ED problem, the proposed algorithm has strong competitiveness in each test index.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.

This work is supported by the National Natural Science Foundation of China (with number 61906164), by the Natural Science Foundation of Jiangsu Province of China (with number BK20190875).

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Ruxin Zhao, Wei Wang, Tingting Zhang, Lixiang Fu, Jiajie Kang, Hongtan Zhang, Yang Shi & Chao Jiang

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Zhao, R., Wang, W., Zhang, T. et al. An Adaptive Differential Evolution Algorithm Based on Data Preprocessing Method and a New Mutation Strategy to Solve Dynamic Economic Dispatch Considering Generator Constraints. Comput Econ (2024). https://doi.org/10.1007/s10614-024-10705-2

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