hypothesis test two types of errors

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• 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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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

Type 2 Error Overview & Example

By Jim Frost 3 Comments

What is a Type 2 Error?

A type 2 error (AKA Type II error) occurs when you fail to reject a false null hypothesis in a hypothesis test. In other words, a statistically non-significant test result indicates that a population effect does not exist when it actually does. A type 2 error is a false negative because the effect exists in the population, but the test doesn’t detect it in the sample .

By failing to reject a false null hypothesis, you incorrectly conclude that the effect does not exist when it does. Unfortunately, you’re unaware of this error at the time. You’re simply interpreting the results of your hypothesis test.

Type 2 errors can have profound implications. For example, a false negative in medical testing might mean overlooking an effective treatment. Recognizing and controlling these errors is crucial for sound statistical findings.

Related post : Hypothesis Testing Overview

Type 2 Error Example

Let’s illustrate this concept with an example of a type 2 error in practice. For our scenario, we’ll assume the effect exists — a detail typically unknown in real-world situations, hence the need for the study.

Imagine we’re testing a new drug that genuinely is effective. We conduct a study, gather data, and carry out the hypothesis test.

The hypotheses for this study are:

• Null : The drug has no effect in the population.
• Alternative : The drug is effective in the population.

Our analysis yields a p-value of 0.08, above our alpha level of 0.05. The study is not statistically significant . Consequently, we fail to reject the null and conclude the drug is ineffective.

Regrettably, this conclusion is incorrect because the drug is effective. The non-significant results lead us to believe the medication doesn’t work even though it is effective. It’s a false negative. A type 2 error has occurred, and we’re none the wiser!

Learn more about the Null Hypothesis .

Why Do They Occur?

Hypothesis tests employ sample data to make inferences about populations. Using random samples is beneficial as examining entire populations is often impractical.

However, relying on samples can introduce issues, including Type 2 errors. While random samples usually represent the population accurately, they can sometimes give a misleading picture and produce false negatives.

Consider flipping a coin. Occasionally, by sheer chance, you might get fewer heads than expected. Similarly, randomness can yield atypical samples that do not accurately portray the population.

However, unlike Type I errors , which primarily arise from random sampling error, Type 2 errors stem from various factors . These include sampling error but also small effects, small samples, and high data variability.

These conditions make it more difficult for a hypothesis test to use a sample to detect a population effect when one truly exists.

Learn more about Representative Samples and Random Sampling .

Probability of a Type 2 Error

While it’s impossible to identify when studies yield false negative results, we can estimate their rate of occurrence rate. Statisticians denote the probability of making a Type 2 error using the Greek letter beta (β). By designing your study effectively, you minimize the false negative rate.

The Type 2 error rate is the probability of a false negative. Therefore, 1 – β is the probability of correctly detecting an effect that exists. Statisticians call this the power of a hypothesis test. Analysts typically estimate power rather than beta itself.

Unlike Type I errors, you can’t set the Type 2 error rate for your analysis. Instead, analysts estimate the properties of the alternative hypothesis and enter them into statistical software to approximate statistical power. This process is known as power analysis.

A crucial benefit of hypothesis testing is that when the null hypothesis is false because an effect exists in the population, researchers can design a study with a low false negative rate and high statistical power. This process lends credibility to the results because the study has a low probability of producing a false negative.

Related post : What is Power in Statistics?

Minimizing False Negatives

Analysts can’t wholly avoid Type 2 errors, but increasing statistical power can lessen their likelihood. However, augmenting power usually requires spending more time and resources on the study. It’s a matter of balancing false negatives with the resources available for the analysis.

Reduced variability and larger effect sizes can lower the Type 2 error rate. Unfortunately, these aspects are frequently challenging for researchers to control because they are properties inherent to the population under study.

Generally, the aspect researchers can influence the most is sample size, making it the primary factor in regulating false negatives. Keeping all other aspects constant, increasing the sample size leads to a lower Type 2 error rate (β) and, correspondingly, higher statistical power (1 – β). Learn how to calculate the sample size for statistical power .

In hypothesis testing, understanding Type 2 errors is essential. They represent a false negative, where we fail to detect a significant effect that genuinely exists. By thoughtfully designing our studies, we can reduce the risk of these errors and make more informed statistical decisions.

Compare and contrast Type I vs. Type II Errors .

Acosta, Griselda; Smith, Eric; and Kreinovich, Vladik, “ Why Area Under the Curve in Hypothesis Testing? ” (2019). Departmental Technical Reports (CS) . 1360.

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Reader Interactions

July 2, 2024 at 8:17 pm

Hi Jim, I have a question about Type 2 error computation.

In type 2 error computation, why do we compute the standard deviation of the proportion using the value assumed under the null hypothesis even if we provide the alternative hypothesis as possible true value?

Let’s say we have following problem:

An airline claims that 92% of its flights leave on schedule, but an FAA investigator believes the true figure is lower. He decides that 125 flights will be checked at the 5% significance level. What is the probability of a Type II error if the true percentage is 90%?

STD using H0 = √((.92)(.08)/125) = 0.0243. With a = .05 the critical z-score is 1.645, and the critical proportion is .92 – 1.645(0.0243) = .880. If the true proportion of on-schedule flights is .90, then the z-score of .880 is (.880-.90)/0.0243 = 0.82, and B = 5 + .2939 = .7939.

My question is why we use STD based on H0 instead of alternative hypothesis i.e. the true percentage .90? Because if the alternative hypothesis were used to calculate the standard deviation, it would lead to a different value, which would not represent the probability of a type 2 error under the null hypothesis?

Thank you so much for your time!

July 2, 2024 at 10:35 pm

When calculating the power of a hypothesis test, you primarily use the standard deviation under the null hypothesis to determine the critical values. Then, you use the standard deviation under the alternative hypothesis to calculate the probability that the test statistic falls within the non-rejection region when the alternative hypothesis is true.

I haven’t done the manual calculations myself, but I plugged the values into my statistical software and got a different answer than you. For N = 125, a hypothesized value of 0.9, and a comparison proportion of 0.92, alpha = 0.05, I get a power of 0.09, which equates to a Type 2 error rate of 0.91.

I hope that helps!

July 3, 2024 at 3:17 am

Thank you Jim for detailed explanation 🙏 Really appreciate it.

About β, I got 79.4% using the following R script.

# SE under H0: 0.0242652 # z critical: 1.644854 # critical proportion: 0.8800873 # z for true proportion -0.820628 # Type II error (Beta): 0.7940709

# Define parameters n <- 125 # sample size p0 <- 0.92 # assumed proportion under H0 p1 <- 0.90 # true proportion alpha <- 0.05 # significance level

# Calculate standard error under H0 se0 <- sqrt(p0 * (1 – p0) / n) cat("SE under H0:", se0, "\n")

# Find critical z-value and rejection region boundary z_crit <- qnorm(alpha, lower.tail = FALSE) cat("z critical:", z_crit, "\n") p_hat_crit <- p0 – z_crit * se0 cat("critical proportion: ", p_hat_crit, "\n")

# Calculate z-value for the true proportion z1 <- (p_hat_crit – p1) / se0 cat("z for true proportion", z1, "\n")

# Calculate the probability of Type II error beta <- pnorm(z1, lower.tail = FALSE)

# Print the results cat("Type II error (Beta):", beta, "\n")

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• Knowledge Base

Hypothesis Testing | A Step-by-Step Guide with Easy Examples

Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.

There are 5 main steps in hypothesis testing:

• State your research hypothesis as a null hypothesis and alternate hypothesis (H o ) and (H a  or H 1 ).
• Collect data in a way designed to test the hypothesis.
• Perform an appropriate statistical test .
• Decide whether to reject or fail to reject your null hypothesis.
• Present the findings in your results and discussion section.

Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.

Table of contents

Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.

After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.

The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.

• H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.

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For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.

There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).

If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.

Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.

Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .

• an estimate of the difference in average height between the two groups.
• a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.

Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.

In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.

In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).

The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .

In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.

In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.

However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.

If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”

These are superficial differences; you can see that they mean the same thing.

You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.

If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .

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

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

Methodology

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

Research bias

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

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

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).

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.

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Bevans, R. (2023, June 22). Hypothesis Testing | A Step-by-Step Guide with Easy Examples. Scribbr. Retrieved September 3, 2024, from https://www.scribbr.com/statistics/hypothesis-testing/

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9.2 Outcomes and the Type I and Type II Errors

When you perform a hypothesis test, there are four possible outcomes depending on the actual truth, or falseness, of the null hypothesis H 0 and the decision to reject or not. The outcomes are summarized in the following table:

The four possible outcomes in the table are as follows:

• The decision is not to reject H 0 when H 0 is true (correct decision).
• The decision is to reject H 0 when, in fact, H 0 is true (incorrect decision known as a Type I error ).
• The decision is not to reject H 0 when, in fact, H 0 is false (incorrect decision known as a Type II error ).
• The decision is to reject H 0 when H 0 is false (correct decision whose probability is called the Power of the Test ).

Each of the errors occurs with a particular probability. The Greek letters α and β represent the probabilities.

α = probability of a Type I error = P (Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.

β = probability of a Type II error = P (Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.

α and β should be as small as possible because they are probabilities of errors. They are rarely zero.

The Power of the Test is 1 – β . Ideally, we want a high power that is as close to one as possible. Increasing the sample size can increase the Power of the Test.

The following are examples of Type I and Type II errors.

Example 9.5

Suppose the null hypothesis, H 0 , is: Frank's rock climbing equipment is safe.

Type I error: Frank does not go rock climbing because he considers that the equipment is not safe, when in fact, the equipment is really safe. Frank is making the mistake of rejecting the null hypothesis, when the equipment is actually safe!

Type II error: Frank goes climbing, thinking that his equipment is safe, but this is a mistake, and he painfully realizes that his equipment is not as safe as it should have been. Frank assumed that the null hypothesis was true, when it was not.

α = probability that Frank thinks his rock climbing equipment may not be safe when, in fact, it really is safe. β = probability that Frank thinks his rock climbing equipment may be safe when, in fact, it is not safe.

Notice that, in this case, the error with the greater consequence is the Type II error. (If Frank thinks his rock climbing equipment is safe, he will go ahead and use it.)

Suppose the null hypothesis, H 0 , is: the blood cultures contain no traces of pathogen X . State the Type I and Type II errors.

Example 9.6

Suppose the null hypothesis, H 0 , is: a tomato plant is alive when a class visits the school garden.

Type I error: The null hypothesis claims that the tomato plant is alive, and it is true, but the students make the mistake of thinking that the plant is already dead.

Type II error: The tomato plant is already dead (the null hypothesis is false), but the students do not notice it, and believe that the tomato plant is alive.

α = probability that the class thinks the tomato plant is dead when, in fact, it is alive = P (Type I error). β = probability that the class thinks the tomato plant is alive when, in fact, it is dead = P (Type II error).

The error with the greater consequence is the Type I error. (If the class thinks the plant is dead, they will not water it.)

Suppose the null hypothesis, H 0 , is: a patient is not sick. Which type of error has the greater consequence, Type I or Type II?

Example 9.7

It’s a Boy Genetic Labs, a genetics company, claims to be able to increase the likelihood that a pregnancy will result in a boy being born. Statisticians want to test the claim. Suppose that the null hypothesis, H 0 , is: It’s a Boy Genetic Labs has no effect on gender outcome.

Type I error : This error results when a true null hypothesis is rejected. In the context of this scenario, we would state that we believe that It’s a Boy Genetic Labs influences the gender outcome, when in fact it has no effect. The probability of this error occurring is denoted by the Greek letter alpha, α .

Type II error : This error results when we fail to reject a false null hypothesis. In context, we would state that It’s a Boy Genetic Labs does not influence the gender outcome of a pregnancy when, in fact, it does. The probability of this error occurring is denoted by the Greek letter beta, β .

The error with the greater consequence would be the Type I error since couples would use the It’s a Boy Genetic Labs product in hopes of increasing the chances of having a boy.

Red tide is a bloom of poison-producing algae—a few different species of a class of plankton called dinoflagellates. When the weather and water conditions cause these blooms, shellfish such as clams living in the area develop dangerous levels of a paralysis-inducing toxin. In Massachusetts, the Division of Marine Fisheries montors levels of the toxin in shellfish by regular sampling of shellfish along the coastline. If the mean level of toxin in clams exceeds 800 μg (micrograms) of toxin per kilogram of clam meat in any area, clam harvesting is banned there until the bloom is over and levels of toxin in clams subside. Describe both a Type I and a Type II error in this context, and state which error has the greater consequence.

Example 9.8

A certain experimental drug claims a cure rate of at least 75 percent for males with a disease. Describe both the Type I and Type II errors in context. Which error is the more serious?

Type I : A patient believes the cure rate for the drug is less than 75 percent when it actually is at least 75 percent.

Type II : A patient believes the experimental drug has at least a 75 percent cure rate when it has a cure rate that is less than 75 percent.

In this scenario, the Type II error contains the more severe consequence. If a patient believes the drug works at least 75 percent of the time, this most likely will influence the patient’s (and doctor’s) choice about whether to use the drug as a treatment option.

Determine both Type I and Type II errors for the following scenario:

Assume a null hypothesis, H 0 , that states the percentage of adults with jobs is at least 88 percent.

Identify the Type I and Type II errors from these four possible choices.

• Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88 percent when that percentage is actually less than 88 percent
• Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88 percent when the percentage is actually at least 88 percent
• Reject the null hypothesis that the percentage of adults who have jobs is at least 88 percent when the percentage is actually at least 88 percent
• Reject the null hypothesis that the percentage of adults who have jobs is at least 88 percent when that percentage is actually less than 88 percent

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What are type I and type II errors?

The probability of rejecting the null hypothesis when it is false is equal to 1–β. This value is the power of the test.

Example of type I and type II error

To understand the interrelationship between type I and type II error, and to determine which error has more severe consequences for your situation, consider the following example.

Null hypothesis (H 0 ): μ 1 = μ 2

The two medications are equally effective.

Alternative hypothesis (H 1 ): μ 1 ≠ μ 2

The two medications are not equally effective.

A type I error occurs if the researcher rejects the null hypothesis and concludes that the two medications are different when, in fact, they are not. If the medications have the same effectiveness, the researcher may not consider this error too severe because the patients still benefit from the same level of effectiveness regardless of which medicine they take. However, if a type II error occurs, the researcher fails to reject the null hypothesis when it should be rejected. That is, the researcher concludes that the medications are the same when, in fact, they are different. This error is potentially life-threatening if the less-effective medication is sold to the public instead of the more effective one.

As you conduct your hypothesis tests, consider the risks of making type I and type II errors. If the consequences of making one type of error are more severe or costly than making the other type of error, then choose a level of significance and a power for the test that will reflect the relative severity of those consequences.

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The Difference Between Type I and Type II Errors in Hypothesis Testing

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The statistical practice of hypothesis testing is widespread not only in statistics but also throughout the natural and social sciences. When we conduct a hypothesis test there a couple of things that could go wrong. There are two kinds of errors, which by design cannot be avoided, and we must be aware that these errors exist. The errors are given the quite pedestrian names of type I and type II errors. What are type I and type II errors, and how we distinguish between them? Briefly:

• Type I errors happen when we reject a true null hypothesis
• Type II errors happen when we fail to reject a false null hypothesis

We will explore more background behind these types of errors with the goal of understanding these statements.

Hypothesis Testing

The process of hypothesis testing can seem to be quite varied with a multitude of test statistics. But the general process is the same. Hypothesis testing involves the statement of a null hypothesis and the selection of a level of significance . The null hypothesis is either true or false and represents the default claim for a treatment or procedure. For example, when examining the effectiveness of a drug, the null hypothesis would be that the drug has no effect on a disease.

After formulating the null hypothesis and choosing a level of significance, we acquire data through observation. Statistical calculations tell us whether or not we should reject the null hypothesis.

In an ideal world, we would always reject the null hypothesis when it is false, and we would not reject the null hypothesis when it is indeed true. But there are two other scenarios that are possible, each of which will result in an error.

Type I Error

The first kind of error that is possible involves the rejection of a null hypothesis that is actually true. This kind of error is called a type I error and is sometimes called an error of the first kind.

Type I errors are equivalent to false positives. Let’s go back to the example of a drug being used to treat a disease. If we reject the null hypothesis in this situation, then our claim is that the drug does, in fact, have some effect on a disease. But if the null hypothesis is true, then, in reality, the drug does not combat the disease at all. The drug is falsely claimed to have a positive effect on a disease.

Type I errors can be controlled. The value of alpha, which is related to the level of significance that we selected has a direct bearing on type I errors. Alpha is the maximum probability that we have a type I error. For a 95% confidence level, the value of alpha is 0.05. This means that there is a 5% probability that we will reject a true null hypothesis. In the long run, one out of every twenty hypothesis tests that we perform at this level will result in a type I error.

Type II Error

The other kind of error that is possible occurs when we do not reject a null hypothesis that is false. This sort of error is called a type II error and is also referred to as an error of the second kind.

Type II errors are equivalent to false negatives. If we think back again to the scenario in which we are testing a drug, what would a type II error look like? A type II error would occur if we accepted that the drug had no effect on a disease, but in reality, it did.

The probability of a type II error is given by the Greek letter beta. This number is related to the power or sensitivity of the hypothesis test, denoted by 1 – beta.

How to Avoid Errors

Type I and type II errors are part of the process of hypothesis testing. Although the errors cannot be completely eliminated, we can minimize one type of error.

Typically when we try to decrease the probability one type of error, the probability for the other type increases. We could decrease the value of alpha from 0.05 to 0.01, corresponding to a 99% level of confidence . However, if everything else remains the same, then the probability of a type II error will nearly always increase.

Many times the real world application of our hypothesis test will determine if we are more accepting of type I or type II errors. This will then be used when we design our statistical experiment.

• Type I and Type II Errors in Statistics
• What Level of Alpha Determines Statistical Significance?
• What Is the Difference Between Alpha and P-Values?
• The Runs Test for Random Sequences
• What 'Fail to Reject' Means in a Hypothesis Test
• How to Construct a Confidence Interval for a Population Proportion
• How to Find Critical Values with a Chi-Square Table
• Null Hypothesis and Alternative Hypothesis
• An Example of a Hypothesis Test
• What Is ANOVA?
• Degrees of Freedom for Independence of Variables in Two-Way Table
• How to Find Degrees of Freedom in Statistics
• Confidence Interval for the Difference of Two Population Proportions
• An Example of Chi-Square Test for a Multinomial Experiment
• Example of a Permutation Test
• How to Calculate the Margin of Error

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Hypothesis testing, type I and type II errors

Amitav banerjee.

Department of Community Medicine, D. Y. Patil Medical College, Pune, India

U. B. Chitnis

S. l. jadhav, j. s. bhawalkar, s. chaudhury.

1 Department of Psychiatry, RINPAS, Kanke, Ranchi, India

Hypothesis testing is an important activity of empirical research and evidence-based medicine. A well worked up hypothesis is half the answer to the research question. For this, both knowledge of the subject derived from extensive review of the literature and working knowledge of basic statistical concepts are desirable. The present paper discusses the methods of working up a good hypothesis and statistical concepts of hypothesis testing.

Karl Popper is probably the most influential philosopher of science in the 20 th century (Wulff et al ., 1986). Many scientists, even those who do not usually read books on philosophy, are acquainted with the basic principles of his views on science. The popularity of Popper’s philosophy is due partly to the fact that it has been well explained in simple terms by, among others, the Nobel Prize winner Peter Medawar (Medawar, 1969). Popper makes the very important point that empirical scientists (those who stress on observations only as the starting point of research) put the cart in front of the horse when they claim that science proceeds from observation to theory, since there is no such thing as a pure observation which does not depend on theory. Popper states, “… the belief that we can start with pure observation alone, without anything in the nature of a theory, is absurd: As may be illustrated by the story of the man who dedicated his life to natural science, wrote down everything he could observe, and bequeathed his ‘priceless’ collection of observations to the Royal Society to be used as inductive (empirical) evidence.

STARTING POINT OF RESEARCH: HYPOTHESIS OR OBSERVATION?

The first step in the scientific process is not observation but the generation of a hypothesis which may then be tested critically by observations and experiments. Popper also makes the important claim that the goal of the scientist’s efforts is not the verification but the falsification of the initial hypothesis. It is logically impossible to verify the truth of a general law by repeated observations, but, at least in principle, it is possible to falsify such a law by a single observation. Repeated observations of white swans did not prove that all swans are white, but the observation of a single black swan sufficed to falsify that general statement (Popper, 1976).

CHARACTERISTICS OF A GOOD HYPOTHESIS

A good hypothesis must be based on a good research question. It should be simple, specific and stated in advance (Hulley et al ., 2001).

Hypothesis should be simple

A simple hypothesis contains one predictor and one outcome variable, e.g. positive family history of schizophrenia increases the risk of developing the condition in first-degree relatives. Here the single predictor variable is positive family history of schizophrenia and the outcome variable is schizophrenia. A complex hypothesis contains more than one predictor variable or more than one outcome variable, e.g., a positive family history and stressful life events are associated with an increased incidence of Alzheimer’s disease. Here there are 2 predictor variables, i.e., positive family history and stressful life events, while one outcome variable, i.e., Alzheimer’s disease. Complex hypothesis like this cannot be easily tested with a single statistical test and should always be separated into 2 or more simple hypotheses.

Hypothesis should be specific

A specific hypothesis leaves no ambiguity about the subjects and variables, or about how the test of statistical significance will be applied. It uses concise operational definitions that summarize the nature and source of the subjects and the approach to measuring variables (History of medication with tranquilizers, as measured by review of medical store records and physicians’ prescriptions in the past year, is more common in patients who attempted suicides than in controls hospitalized for other conditions). This is a long-winded sentence, but it explicitly states the nature of predictor and outcome variables, how they will be measured and the research hypothesis. Often these details may be included in the study proposal and may not be stated in the research hypothesis. However, they should be clear in the mind of the investigator while conceptualizing the study.

Hypothesis should be stated in advance

The hypothesis must be stated in writing during the proposal state. This will help to keep the research effort focused on the primary objective and create a stronger basis for interpreting the study’s results as compared to a hypothesis that emerges as a result of inspecting the data. The habit of post hoc hypothesis testing (common among researchers) is nothing but using third-degree methods on the data (data dredging), to yield at least something significant. This leads to overrating the occasional chance associations in the study.

TYPES OF HYPOTHESES

For the purpose of testing statistical significance, hypotheses are classified by the way they describe the expected difference between the study groups.

Null and alternative hypotheses

The null hypothesis states that there is no association between the predictor and outcome variables in the population (There is no difference between tranquilizer habits of patients with attempted suicides and those of age- and sex- matched “control” patients hospitalized for other diagnoses). The null hypothesis is the formal basis for testing statistical significance. By starting with the proposition that there is no association, statistical tests can estimate the probability that an observed association could be due to chance.

The proposition that there is an association — that patients with attempted suicides will report different tranquilizer habits from those of the controls — is called the alternative hypothesis. The alternative hypothesis cannot be tested directly; it is accepted by exclusion if the test of statistical significance rejects the null hypothesis.

One- and two-tailed alternative hypotheses

A one-tailed (or one-sided) hypothesis specifies the direction of the association between the predictor and outcome variables. The prediction that patients of attempted suicides will have a higher rate of use of tranquilizers than control patients is a one-tailed hypothesis. A two-tailed hypothesis states only that an association exists; it does not specify the direction. The prediction that patients with attempted suicides will have a different rate of tranquilizer use — either higher or lower than control patients — is a two-tailed hypothesis. (The word tails refers to the tail ends of the statistical distribution such as the familiar bell-shaped normal curve that is used to test a hypothesis. One tail represents a positive effect or association; the other, a negative effect.) A one-tailed hypothesis has the statistical advantage of permitting a smaller sample size as compared to that permissible by a two-tailed hypothesis. Unfortunately, one-tailed hypotheses are not always appropriate; in fact, some investigators believe that they should never be used. However, they are appropriate when only one direction for the association is important or biologically meaningful. An example is the one-sided hypothesis that a drug has a greater frequency of side effects than a placebo; the possibility that the drug has fewer side effects than the placebo is not worth testing. Whatever strategy is used, it should be stated in advance; otherwise, it would lack statistical rigor. Data dredging after it has been collected and post hoc deciding to change over to one-tailed hypothesis testing to reduce the sample size and P value are indicative of lack of scientific integrity.

STATISTICAL PRINCIPLES OF HYPOTHESIS TESTING

A hypothesis (for example, Tamiflu [oseltamivir], drug of choice in H1N1 influenza, is associated with an increased incidence of acute psychotic manifestations) is either true or false in the real world. Because the investigator cannot study all people who are at risk, he must test the hypothesis in a sample of that target population. No matter how many data a researcher collects, he can never absolutely prove (or disprove) his hypothesis. There will always be a need to draw inferences about phenomena in the population from events observed in the sample (Hulley et al ., 2001). In some ways, the investigator’s problem is similar to that faced by a judge judging a defendant [ Table 1 ]. The absolute truth whether the defendant committed the crime cannot be determined. Instead, the judge begins by presuming innocence — the defendant did not commit the crime. The judge must decide whether there is sufficient evidence to reject the presumed innocence of the defendant; the standard is known as beyond a reasonable doubt. A judge can err, however, by convicting a defendant who is innocent, or by failing to convict one who is actually guilty. In similar fashion, the investigator starts by presuming the null hypothesis, or no association between the predictor and outcome variables in the population. Based on the data collected in his sample, the investigator uses statistical tests to determine whether there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis that there is an association in the population. The standard for these tests is shown as the level of statistical significance.

The analogy between judge’s decisions and statistical tests

TYPE I (ALSO KNOWN AS ‘α’) AND TYPE II (ALSO KNOWN AS ‘β’)ERRORS

Just like a judge’s conclusion, an investigator’s conclusion may be wrong. Sometimes, by chance alone, a sample is not representative of the population. Thus the results in the sample do not reflect reality in the population, and the random error leads to an erroneous inference. A type I error (false-positive) occurs if an investigator rejects a null hypothesis that is actually true in the population; a type II error (false-negative) occurs if the investigator fails to reject a null hypothesis that is actually false in the population. Although type I and type II errors can never be avoided entirely, the investigator can reduce their likelihood by increasing the sample size (the larger the sample, the lesser is the likelihood that it will differ substantially from the population).

False-positive and false-negative results can also occur because of bias (observer, instrument, recall, etc.). (Errors due to bias, however, are not referred to as type I and type II errors.) Such errors are troublesome, since they may be difficult to detect and cannot usually be quantified.

EFFECT SIZE

The likelihood that a study will be able to detect an association between a predictor variable and an outcome variable depends, of course, on the actual magnitude of that association in the target population. If it is large (such as 90% increase in the incidence of psychosis in people who are on Tamiflu), it will be easy to detect in the sample. Conversely, if the size of the association is small (such as 2% increase in psychosis), it will be difficult to detect in the sample. Unfortunately, the investigator often does not know the actual magnitude of the association — one of the purposes of the study is to estimate it. Instead, the investigator must choose the size of the association that he would like to be able to detect in the sample. This quantity is known as the effect size. Selecting an appropriate effect size is the most difficult aspect of sample size planning. Sometimes, the investigator can use data from other studies or pilot tests to make an informed guess about a reasonable effect size. When there are no data with which to estimate it, he can choose the smallest effect size that would be clinically meaningful, for example, a 10% increase in the incidence of psychosis. Of course, from the public health point of view, even a 1% increase in psychosis incidence would be important. Thus the choice of the effect size is always somewhat arbitrary, and considerations of feasibility are often paramount. When the number of available subjects is limited, the investigator may have to work backward to determine whether the effect size that his study will be able to detect with that number of subjects is reasonable.

α,β,AND POWER

After a study is completed, the investigator uses statistical tests to try to reject the null hypothesis in favor of its alternative (much in the same way that a prosecuting attorney tries to convince a judge to reject innocence in favor of guilt). Depending on whether the null hypothesis is true or false in the target population, and assuming that the study is free of bias, 4 situations are possible, as shown in Table 2 below. In 2 of these, the findings in the sample and reality in the population are concordant, and the investigator’s inference will be correct. In the other 2 situations, either a type I (α) or a type II (β) error has been made, and the inference will be incorrect.

Truth in the population versus the results in the study sample: The four possibilities

The investigator establishes the maximum chance of making type I and type II errors in advance of the study. The probability of committing a type I error (rejecting the null hypothesis when it is actually true) is called α (alpha) the other name for this is the level of statistical significance.

If a study of Tamiflu and psychosis is designed with α = 0.05, for example, then the investigator has set 5% as the maximum chance of incorrectly rejecting the null hypothesis (and erroneously inferring that use of Tamiflu and psychosis incidence are associated in the population). This is the level of reasonable doubt that the investigator is willing to accept when he uses statistical tests to analyze the data after the study is completed.

The probability of making a type II error (failing to reject the null hypothesis when it is actually false) is called β (beta). The quantity (1 - β) is called power, the probability of observing an effect in the sample (if one), of a specified effect size or greater exists in the population.

If β is set at 0.10, then the investigator has decided that he is willing to accept a 10% chance of missing an association of a given effect size between Tamiflu and psychosis. This represents a power of 0.90, i.e., a 90% chance of finding an association of that size. For example, suppose that there really would be a 30% increase in psychosis incidence if the entire population took Tamiflu. Then 90 times out of 100, the investigator would observe an effect of that size or larger in his study. This does not mean, however, that the investigator will be absolutely unable to detect a smaller effect; just that he will have less than 90% likelihood of doing so.

Ideally alpha and beta errors would be set at zero, eliminating the possibility of false-positive and false-negative results. In practice they are made as small as possible. Reducing them, however, usually requires increasing the sample size. Sample size planning aims at choosing a sufficient number of subjects to keep alpha and beta at acceptably low levels without making the study unnecessarily expensive or difficult.

Many studies s et al pha at 0.05 and beta at 0.20 (a power of 0.80). These are somewhat arbitrary values, and others are sometimes used; the conventional range for alpha is between 0.01 and 0.10; and for beta, between 0.05 and 0.20. In general the investigator should choose a low value of alpha when the research question makes it particularly important to avoid a type I (false-positive) error, and he should choose a low value of beta when it is especially important to avoid a type II error.

The null hypothesis acts like a punching bag: It is assumed to be true in order to shadowbox it into false with a statistical test. When the data are analyzed, such tests determine the P value, the probability of obtaining the study results by chance if the null hypothesis is true. The null hypothesis is rejected in favor of the alternative hypothesis if the P value is less than alpha, the predetermined level of statistical significance (Daniel, 2000). “Nonsignificant” results — those with P value greater than alpha — do not imply that there is no association in the population; they only mean that the association observed in the sample is small compared with what could have occurred by chance alone. For example, an investigator might find that men with family history of mental illness were twice as likely to develop schizophrenia as those with no family history, but with a P value of 0.09. This means that even if family history and schizophrenia were not associated in the population, there was a 9% chance of finding such an association due to random error in the sample. If the investigator had set the significance level at 0.05, he would have to conclude that the association in the sample was “not statistically significant.” It might be tempting for the investigator to change his mind about the level of statistical significance ex post facto and report the results “showed statistical significance at P < 10”. A better choice would be to report that the “results, although suggestive of an association, did not achieve statistical significance ( P = .09)”. This solution acknowledges that statistical significance is not an “all or none” situation.

Hypothesis testing is the sheet anchor of empirical research and in the rapidly emerging practice of evidence-based medicine. However, empirical research and, ipso facto, hypothesis testing have their limits. The empirical approach to research cannot eliminate uncertainty completely. At the best, it can quantify uncertainty. This uncertainty can be of 2 types: Type I error (falsely rejecting a null hypothesis) and type II error (falsely accepting a null hypothesis). The acceptable magnitudes of type I and type II errors are set in advance and are important for sample size calculations. Another important point to remember is that we cannot ‘prove’ or ‘disprove’ anything by hypothesis testing and statistical tests. We can only knock down or reject the null hypothesis and by default accept the alternative hypothesis. If we fail to reject the null hypothesis, we accept it by default.

Source of Support: Nil

Conflict of Interest: None declared.

• Daniel W. W. In: Biostatistics. 7th ed. New York: John Wiley and Sons, Inc; 2002. Hypothesis testing; pp. 204–294. [ Google Scholar ]
• Hulley S. B, Cummings S. R, Browner W. S, Grady D, Hearst N, Newman T. B. 2nd ed. Philadelphia: Lippincott Williams and Wilkins; 2001. Getting ready to estimate sample size: Hypothesis and underlying principles In: Designing Clinical Research-An epidemiologic approach; pp. 51–63. [ Google Scholar ]
• Medawar P. B. Philadelphia: American Philosophical Society; 1969. Induction and intuition in scientific thought. [ Google Scholar ]
• Popper K. Unended Quest. An Intellectual Autobiography. Fontana Collins; p. 42. [ Google Scholar ]
• Wulff H. R, Pedersen S. A, Rosenberg R. Oxford: Blackwell Scientific Publicatons; Empirism and Realism: A philosophical problem. In: Philosophy of Medicine. [ Google Scholar ]

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5.6 Hypothesis Tests in Depth

Establishing the parameter of interest, type of distribution to use, the test statistic, and p -value can help you figure out how to go about a hypothesis test. However, there are several other factors you should consider when interpreting the results.

Rare Events

Suppose you make an assumption about a property of the population (this assumption is the null hypothesis). Then you gather sample data randomly. If the sample has properties that would be very unlikely to occur if the assumption is true, then you would conclude that your assumption about the population is probably incorrect. Remember that your assumption is just an assumption; it is not a fact, and it may or may not be true. But your sample data are real and are showing you a fact that seems to contradict your assumption.

Errors in Hypothesis Tests

When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis H 0 and the decision to reject or not. The outcomes are summarized in the following table:

The four possible outcomes in the table are:

• The decision is not to reject H 0 when H 0 is true (correct decision).
• The decision is to reject H 0 when H 0 is true (incorrect decision known as a type I error ).
• The decision is not to reject H 0 when, in fact, H 0 is false (incorrect decision known as a type II error ).
• The decision is to reject H 0 when H 0 is false (correct decision whose probability is called the power of the test).

Each of the errors occurs with a particular probability. The Greek letters α and β represent the probabilities.

α = probability of a type I error = P (type I error) = probability of rejecting the null hypothesis when the null hypothesis is true. These are also known as false positives. We know that α is often determined in advance, and α = 0.05 is often widely accepted. In that case, you are saying, “We are OK making this type of error in 5% of samples.” In fact, the p -value is the exact probability of a type I error based on what you observed.

β = probability of a type II error = P (type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false. These are also known as false negatives.

The power of a test is 1 – β .

Ideally, α and β should be as small as possible because they are probabilities of errors but are rarely zero. We want a high power that is as close to one as well. Increasing the sample size can help us achieve these by reducing both α and β and therefore increasing the power of the test.

Suppose the null hypothesis, H 0 , is that Frank’s rock climbing equipment is safe.

Type I error: Frank thinks that his rock climbing equipment may not be safe when, in fact, it really is safe. Type II error: Frank thinks that his rock climbing equipment may be safe when, in fact, it is not safe.

α = probability that Frank thinks his rock climbing equipment may not be safe when, in fact, it really is safe. β = probability that Frank thinks his rock climbing equipment may be safe when, in fact, it is not safe.

Notice that, in this case, the error with the greater consequence is the type II error, in which Frank thinks his rock climbing equipment is safe, so he goes ahead and uses it.

Suppose the null hypothesis, H 0 , is that the blood cultures contain no traces of pathogen X . State the type I and type II errors.

Statistical Significance vs. Practical Significance

When the sample size becomes larger, point estimates become more precise and any real differences in the mean and null value become easier to detect and recognize. Even a very small difference would likely be detected if we took a large enough sample. Sometimes, researchers will take such large samples that even the slightest difference is detected, even differences where there is no practical value. In such cases, we still say the difference is statistically significant , but it is not practically significant.

For example, an online experiment might identify that placing additional ads on a movie review website statistically significantly increases viewership of a TV show by 0.001%, but this increase might not have any practical value.

One role of a data scientist in conducting a study often includes planning the size of the study. The data scientist might first consult experts or scientific literature to learn what would be the smallest meaningful difference from the null value. She also would obtain other information, such as a very rough estimate of the true proportion p , so that she could roughly estimate the standard error. From here, she could suggest a sample size that is sufficiently large enough to detect the real difference if it is meaningful. While larger sample sizes may still be used, these calculations are especially helpful when considering costs or potential risks, such as possible health impacts to volunteers in a medical study.

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The decision is to reject the null hypothesis when, in fact, the null hypothesis is true

Erroneously rejecting a true null hypothesis or erroneously failing to reject a false null hypothesis

The probability of failing to reject a true hypothesis

Finding sufficient evidence that the observed effect is not just due to variability, often from rejecting the null hypothesis

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Type I and Type II Errors

Type I and Type II Errors are central for hypothesis testing in general, which subsequently impacts various aspects of science including but not limited to statistical analysis. False discovery refers to a Type I error where a true Null Hypothesis is incorrectly rejected. On the other end of the spectrum, Type II errors occur when a true null hypothesis fails to get rejected.

In this article, we will discuss Type I and Type II Errors in detail, including examples and differences.

Table of Content

Type I and Type II Error in Statistics

What is error, what is type i error (false positive), what is type ii error (false negative), type i and type ii errors – table, type i and type ii errors examples, examples of type i error, examples of type ii error, factors affecting type i and type ii errors, how to minimize type i and type ii errors, difference between type i and type ii errors.

In statistics , Type I and Type II errors represent two kinds of errors that can occur when making a decision about a hypothesis based on sample data. Understanding these errors is crucial for interpreting the results of hypothesis tests.

In the statistics and hypothesis testing , an error refers to the emergence of discrepancies between the result value based on observation or calculation and the actual value or expected value.

The failures may happen in different factors, such as turbulent sampling, unclear implementation, or faulty assumptions. Errors can be of many types, such as

• Measurement Error
• Calculation Error
• Human Error
• Systematic Error
• Random Error

In hypothesis testing, it is often clear which kind of error is the problem, either a Type I error or a Type II one.

Type I error, also known as a false positive , occurs in statistical hypothesis testing when a null hypothesis that is actually true is rejected. In other words, it’s the error of incorrectly concluding that there is a significant effect or difference when there isn’t one in reality.

In hypothesis testing, there are two competing hypotheses:

• Null Hypothesis (H 0 ): This hypothesis represents a default assumption that there is no effect, no difference, or no relationship in the population being studied.
• Alternative Hypothesis (H 1 ): This hypothesis represents the opposite of the null hypothesis. It suggests that there is a significant effect, difference, or relationship in the population.

A Type I error occurs when the null hypothesis is rejected based on the sample data, even though it is actually true in the population.

Type II error, also known as a false negative , occurs in statistical hypothesis testing when a null hypothesis that is actually false is not rejected. In other words, it’s the error of failing to detect a significant effect or difference when one exists in reality.

A Type II error occurs when the null hypothesis is not rejected based on the sample data, even though it is actually false in the population. In other words, it’s a failure to recognize a real effect or difference.

Suppose a medical researcher is testing a new drug to see if it’s effective in treating a certain condition. The null hypothesis (H 0 ) states that the drug has no effect, while the alternative hypothesis (H 1 ) suggests that the drug is effective. If the researcher conducts a statistical test and fails to reject the null hypothesis (H 0 ), concluding that the drug is not effective, when in fact it does have an effect, this would be a Type II error.

The table given below shows the relationship between True and False:

Some of examples of type I error include:

• Medical Testing : Suppose a medical test is designed to diagnose a particular disease. The null hypothesis ( H 0 ) is that the person does not have the disease, and the alternative hypothesis ( H 1 ) is that the person does have the disease. A Type I error occurs if the test incorrectly indicates that a person has the disease (rejects the null hypothesis) when they do not actually have it.
• Legal System : In a criminal trial, the null hypothesis ( H 0 ) is that the defendant is innocent, while the alternative hypothesis ( H 1 ) is that the defendant is guilty. A Type I error occurs if the jury convicts the defendant (rejects the null hypothesis) when they are actually innocent.
• Quality Control : In manufacturing, quality control inspectors may test products to ensure they meet certain specifications. The null hypothesis ( H 0 ) is that the product meets the required standard, while the alternative hypothesis ( H 1 ) is that the product does not meet the standard. A Type I error occurs if a product is rejected (null hypothesis is rejected) as defective when it actually meets the required standard.

Using the same H 0 and H 1 , some examples of type II error include:

• Medical Testing : In a medical test designed to diagnose a disease, a Type II error occurs if the test incorrectly indicates that a person does not have the disease (fails to reject the null hypothesis) when they actually do have it.
• Legal System : In a criminal trial, a Type II error occurs if the jury acquits the defendant (fails to reject the null hypothesis) when they are actually guilty.
• Quality Control : In manufacturing, a Type II error occurs if a defective product is accepted (fails to reject the null hypothesis) as meeting the required standard.

Some of the common factors affecting errors are:

• Sample Size: In statistical hypothesis testing, larger sample sizes generally reduce the probability of both Type I and Type II errors. With larger samples, the estimates tend to be more precise, resulting in more accurate conclusions.
• Significance Level: The significance level (α) in hypothesis testing determines the probability of committing a Type I error. Choosing a lower significance level reduces the risk of Type I error but increases the risk of Type II error, and vice versa.
• Effect Size: The magnitude of the effect or difference being tested influences the probability of Type II error. Smaller effect sizes are more challenging to detect, increasing the likelihood of failing to reject the null hypothesis when it’s false.
• Statistical Power: The power of Statistics (1 – β) dictates that the opportunity for rejecting a wrong null hypothesis is based on the inverse of the chance of committing a Type II error. The power level of the test rises, thus a chance of the Type II error dropping.

To minimize Type I and Type II errors in hypothesis testing, there are several strategies that can be employed based on the information from the sources provided:

• By setting a lower significance level, the chances of incorrectly rejecting the null hypothesis decrease, thus minimizing Type I errors.
• Increasing the sample size reduces the variability of the statistic, making it less likely to fall in the non-rejection region when it should be rejected, thus minimizing Type II errors.

Some of the key differences between Type I and Type II Errors are listed in the following table:

Conclusion – Type I and Type II Errors

In conclusion, type I errors occur when we mistakenly reject a true null hypothesis, while Type II errors happen when we fail to reject a false null hypothesis. Being aware of these errors helps us make more informed decisions, minimizing the risks of false conclusions.

People Also Read:

Difference between Null and Alternate Hypothesis Z-Score Table

Type I and Type II Errors – FAQs

What is type i error.

Type I Error occurs when a null hypothesis is incorrectly rejected, indicating a false positive result, concluding that there is an effect or difference when there isn’t one.

What is an Example of a Type 1 Error?

An example of Type I Error is that convicting an innocent person (null hypothesis: innocence) based on insufficient evidence, incorrectly rejecting the null hypothesis of innocence.

What is Type II Error?

Type II Error happens when a null hypothesis is incorrectly accepted, failing to detect a true effect or difference when one actually exists.

What is an Example of a Type 2 Error?

An example of type 2 error is that failing to diagnose a disease in a patient (null hypothesis: absence of disease) despite them actually having the disease, incorrectly failing to reject the null hypothesis.

What is the difference between Type 1 and Type 2 Errors?

Type I error involves incorrectly rejecting a true null hypothesis, while Type II error involves failing to reject a false null hypothesis. In simpler terms, Type I error is a false positive, while Type II error is a false negative.

What is Type 3 Error?

Type 3 Error is not a standard statistical term. It’s sometimes informally used to describe situations where the researcher correctly rejects the null hypothesis but for the wrong reason, often due to a flaw in the experimental design or analysis.

How are Type I and Type II Errors related to hypothesis testing?

In hypothesis testing, Type I Error relates to the significance level (α), which represents the probability of rejecting a true null hypothesis. Type II Error relates to the power of the test (β), which represents the probability of failing to reject a false null hypothesis.

What are some examples of Type I and Type II Errors?

Type I Error: Rejecting a null hypothesis that a new drug has no side effects when it actually does (false positive). Type II Error: Failing to reject a null hypothesis that a new drug has no effect when it actually does (false negative).

How can one minimize Type I and Type II Errors?

Type I Error can be minimized by choosing a lower significance level (α) for hypothesis testing. Type II Error can be minimized by increasing the sample size or improving the sensitivity of the test.

What is the relationship between Type I and Type II Errors?

There is often a trade-off between Type I and Type II Errors. Decreasing the probability of one type of error typically increases the probability of the other.

How do Type I and Type II Errors impact decision-making?

Type I Errors can lead to false conclusions, such as mistakenly believing a treatment is effective when it’s not. Type II Errors can result in missed opportunities, such as failing to identify an effective treatment.

In which fields are Type I and Type II Errors commonly encountered?

Type I and Type II Errors are encountered in various fields, including medical research, quality control, criminal justice, and market research.

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Exclusive Hypothesis Testing for Cox’s Proportional Hazards Model

• Published: 30 August 2024
• Volume 37 , pages 2157–2172, ( 2024 )

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• Qiang Wu 1 ,
• Xingwei Tong 1 &
• Xiaogang Duan 1  

Exclusive hypothesis testing is a new and special class of hypothesis testing. This kind of testing can be applied in survival analysis to understand the association between genomics information and clinical information about the survival time. Besides, it is well known that Cox’s proportional hazards model is the most commonly used model for regression analysis of failure time. In this paper, the authors consider doing the exclusive hypothesis testing for Cox’s proportional hazards model with right-censored data. The authors propose the comprehensive test statistics to make decision, and show that the corresponding decision rule can control the asymptotic Type I errors and have good powers in theory. The numerical studies indicate that the proposed approach works well for practical situations and it is applied to a set of real data arising from Rotterdam Breast Cancer Data study that motivated this study.

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Qiang Wu, Xingwei Tong & Xiaogang Duan

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This research was supported by the National Natural Science Foundation of China under Grant Nos. 11971064, 12371262, and 12171374.

This paper was recommended for publication by Editor SUN Liuquan.

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Wu, Q., Tong, X. & Duan, X. Exclusive Hypothesis Testing for Cox’s Proportional Hazards Model. J Syst Sci Complex 37 , 2157–2172 (2024). https://doi.org/10.1007/s11424-024-3283-0

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Received : 24 July 2023

Revised : 25 September 2023

Published : 30 August 2024

Issue Date : October 2024

DOI : https://doi.org/10.1007/s11424-024-3283-0

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