Which is Worse, Type I or Type II errors?

The trade-offs between Type I and Type II errors

Type I and Type II errors are statistical concepts that describe the errors that can occur when making decisions based on statistical tests. They are important because they help us to understand the trade-offs between the risks of making incorrect decisions and the benefits of making correct decisions.

What is the Type I error?

A Type I error, also known as a false positive, occurs when a researcher rejects a null hypothesis that is actually true.

This can happen when the sample data appears to support the alternative hypothesis, but in reality, it is just a random fluctuation or chance event (i.e., the sampling error). In the population data, the null hypothesis is supported.

For example, consider a medical study that is testing a new drug to see if it is effective at lowering cholesterol levels. If the null hypothesis is that the drug has no effect, and the researcher rejects this hypothesis based on the data, but in reality, the drug has no effect, then a Type I error has occurred.

How does the Type I error relate to the Significance Level (α)?

We usually talk about the Type I error and the significance level in the context of hypothesis testing.

In hypothesis testing, the null hypothesis represents the default assumption that there is no difference or relationship between the variables being tested. The alternative hypothesis represents the opposite assumption.

The purpose of a hypothesis test is to determine if the sample data provide sufficient evidence to reject the null hypothesis for the entire population.

Making a statistical decision based on the sample data always involves uncertainties due to sampling error, which is unavoidable.

Therefore, before a hypothesis test, researchers need to set the Significant Level (α), the probability of making a Type I error when the null hypothesis is correct.

The typical significance levels are 0.1, 0.05, and 0.01. For example, α = 0.05 means if we repeat the hypothesis tests many times with different sets of sample data, we expect that 5% of the time we incorrectly reject the null hypothesis when it is true, and 95% of the time we won’t reject the null hypothesis.

When we make a Type I error using a given set of sample data, it doesn’t mean that we made a mistake. We acknowledge the risk that we could observe extreme sample statistics due to sampling error. It is just a matter of luck.

What is the Type II error?

A Type II error, also known as a false negative, occurs when a researcher fails to reject a null hypothesis that is actually false.

This can happen when the sample data does not appear to support the alternative hypothesis, but in reality, the population does.

Continuing with the example of the medical study, if the null hypothesis is that the drug has no effect and the researcher fails to reject this hypothesis based on the sample data, but in reality, the drug does have an effect, then a Type II error has occurred.

How does the Type II error relate to the Statistical Power (1-β)?

Making a Type II error means you fail to detect the effect when the effect actually exists. When that happens, your study or analysis doesn’t have enough Statistical Power.

Statistical power (1-β) is a measure of the ability of a statistical test to detect a true difference of a certain size between the null and alternative hypotheses if one exists. It is defined as the probability of correctly rejecting the null hypothesis when it is false.

Therefore, the relationship between Type II error rate (β) and statistical power (1-β) is that a high level of statistical power means a low probability of making a Type II error, and a low level of statistical power means a high probability of making a Type II error.

How to reduce Type I and Type II errors?

Increase sample size: A large size can decrease the variance of the distribution of sample statistics. Therefore it can reduce the chance of making a Type I error rate due to random fluctuation or sampling errors. It can also increase the statistical power of detecting the true effect if one exists, therefore reducing the chance of making a Type II error.
Increase effect size: If the effect size is large, this means that the difference between the null and alternative hypotheses is more pronounced, which can increase the statistical power of the study and reduce the Type II error rate. Moreover, as the effect size increase, we’re less likely to observe the difference between the null and alternative hypotheses due to random fluctuation or sampling errors, therefore, reducing the Type I error.

3. Adjust Significance Level: If we lower the significance level, we have a stricter criterion (e.g., data must be more extreme) for rejecting the null hypothesis, therefore reducing the Type I error rate, but this also decreases the statistical power, therefore increases the Type II error rate.

If we increase the significance level, it means that the data does not have to be as extreme in order to reject the null hypothesis. Therefore, the Type I error rate increases, but the Type II error rate decreases.

We can visualize the trade-off between Type I and Type II errors in the following graph.

For a certain effect size and a given sample data, the Type I and II errors have an inverse relationship.

Therefore, if we want to maintain a given Significance Level (α, e.g., 0.05), Statistical Power (β, e.g., 0.80), and practical effect size, we would need carefully compute the sample size for the hypothesis test.

Under what situations the Type I error is worse than the Type II error?

There may be situations in which the consequences of making a Type I error (false positive) are worse than the consequences of making a Type II error (false negative). Here is an example:

Imagine that a researcher is studying the effectiveness of a new security system at detecting intruders. The null hypothesis is that the security system is not effective, and the alternative hypothesis is that it is effective. If the researcher rejects the null hypothesis based on the data and implements the security system, but in reality the system is not effective, then a Type I error has occurred. In this case, the consequences of the Type I error (implementing an ineffective security system) may be worse than the consequences of a Type II error (failing to implement an effective security system). This is because the cost of implementing an ineffective security system (such as the cost of purchasing and installing the system, as well as the cost of any security breaches that occur) may be higher than the cost of not implementing an effective system.

Under what situations the Type II error is worse than the Type I error?

Sometimes, the consequences of making a Type II error (false negative) are worse than the consequences of making a Type I error (false positive). Here is an example:

Imagine that a researcher is studying the effectiveness of a new drug at reducing the risk of heart attacks. The null hypothesis is that the drug has no effect, and the alternative hypothesis is that it is effective. If the researcher fails to reject the null hypothesis based on the data and does not prescribe the drug to patients, but in reality the drug is effective at reducing the risk of heart attacks, then a Type II error has occurred. In this case, the consequences of the Type II error (not prescribing an effective drug) may be worse than the consequences of a Type I error (prescribing an ineffective drug). This is because the cost of not prescribing an effective drug (such as the cost of additional medical treatment or lost productivity due to heart attacks) may be higher than the cost of prescribing an ineffective drug.

Summary

Type I and Type II errors are statistical errors that can occur when making a decision based on data. A Type I error occurs when a researcher rejects a null hypothesis that is actually true, while a Type II error occurs when a researcher fails to reject a null hypothesis that is actually false. It is important to consider both of these errors when designing a statistical study, as the trade-off between the chances of making these errors and the cost and time required to collect the data can have important consequences.

If you would like to explore more posts related to Statistics, please check out my articles:

Thank you for reading !!!

If you enjoy this article and would like to Buy Me a Coffee, please click here.

You can sign up for a membership to unlock full access to my articles, and have unlimited access to everything on Medium. Please subscribe if you’d like to get an email notification whenever I post a new article.