Type I and II Errors

The post covers the Type I and Type II Errors examples.

Hypothesis testing helps us to determine whether the results are statistically significant or occurred by chance. Hypothesis testing is based on probability, therefore, there is always a chance of making the wrong decision about the null hypothesis (a hypothesis about population). It means that there are two types of errors (Type I and Type II errors) that can be made when drawing a conclusion or decision.

Errors in Statistical Decision-Making

To understand the errors in statistical decision-making, we first need to see the step-by-step process of hypothesis testing:

1. State the null hypothesis and the alternative hypothesis.
2. Choose a level of significance (also called type-I error).
3. Compute the required test statistics
4. Find the critical value or p-value
5. Reject or fail to reject the null hypothesis.

When you decide to reject or fail to reject the null hypothesis, there are four possible outcomes–two represent correct choices, and two represent errors. You can:
• Reject the null hypothesis when it is actually true (Type-I error)
• Reject the null hypothesis when it is actually false (Correct)
• Fail to reject the null hypothesis when it is actually true (Correct)
• Fail to reject the null hypothesis when it is actually false (Type-II error)

These four possibilities can be presented in the truth table.

Type I and Type II Errors Examples: Clinical Trial

To understand Type I and Type II errors, consider the example from clinical trials. In clinical trials, Hypothesis tests are often used to determine whether a new medicine leads to better outcomes in patients. Imagine you are a data professional and working in a pharmaceutical company. The company invents a new medicine to treat the common cold. The company tests a random sample of 200 people with cold symptoms. Without medicine, the typical person experiences cold symptoms for 7.5 days. The average recovery time for people who take the medicine is 6.2 days.

You conduct a hypothesis test to determine if the effect of the medicine on recovery time is statistically significant, or due to chance.

In this case:

• Your null hypothesis ($H_0$) is that the medicine has no effect.
• Your alternative hypothesis ($H_a$) is that the medicine is effective.

Type I Error

A Type-I error (also known as a false positive) occurs when a true null hypothesis is rejected. In other words, one can conclude that the result is statistically significant when in fact the results occurred by chance. To understand this, let in your clinical trial, the results indicate that the null hypothesis is true, which means that the medicine has no effect. In case, you make a Type-I error and reject the null hypothesis, it means that you incorrectly conclude that the medicine relieves cold symptoms while the medicine was (actually) ineffective.

The probability of making a Type I error is represented by $\alpha$ (the level of significance. Typically, a 0.05 (or 5%) significance level is used. A significance level of 5% means you are willing to accept a 5% chance you are wrong when you reject the null hypothesis.

Reduce the risk of Type I error

To reduce your chances of making Type I errors, it is advised to choose a lower significance level. For example, one can choose the significance level of 1% instead of the standard 5%. It will reduce the chances of making a Type I error from 5% to 1%.

Type II Error

A Type II error occurs when we fail to reject a null hypothesis when it is false. In other words, one may conclude that the result occurred by chance, however, in fact, it didn’t. For example, in a clinical study, if the null hypothesis is false, it means that the medicine is effective. In case you make a Type II Error and fail to reject the null hypothesis, it means that you incorrectly conclude that the medicine is ineffective while in reality, the medicine relieves cold symptoms.

The probability of making a Type II error is represented by $\beta$ and it is related to the power of a hypothesis test (power = $1- \beta$). Power refers to the likelihood that a test can correctly detect a real effect when there is one.

Note that reducing the risk of making a Type I error means that it is more likely to make a Type II error or false negative.

Reduce your risk of making Type II Error

One can reduce the risk of making a Type II error by ensuring that the test has enough power. In data work, power is usually set at 0.80 or 80%. The higher the statistical power, the lower the probability of making a Type II error. To increase power, you can increase your sample size or your significance level.

Potential Risks of Type I and Type II Errors

As a data professional, it is important to be aware of the potential risks involved in making the two types of errors.

• A Type I error means rejecting a true null hypothesis. In general, making a Type I error often leads to implementing changes that are unnecessary and ineffective, and which waste valuable time and resources.
  For example, if you make a Type I error in your clinical trial, the new medicine will be considered effective even though it is ineffective. Based on this incorrect conclusion, ineffective medication may be prescribed to a large number of people. While other treatment options may be rejected in favor of the new medicine.

• A Type II error means failing to reject a false null hypothesis. In general, making a Type II error may result in missed opportunities for positive change and innovation. A lack of innovation can be costly for people and organizations.
  For example, if you make a Type II error in your clinical trial, the new medicine will be considered ineffective even though it’s effective. This means that a useful medication may not reach a large number of people who could benefit from it.

In summary, as a data professional, it helps to be aware of the potential errors built into hypothesis testing and how they can affect the final decisions. Depending on the certain situation, one may choose to minimize the risk of either a Type I or Type II error. Ultimately, it is the responsibility of a data professional to determine which type of error is riskier based on the goals of your analysis.

R Language Quick Reference

In hypothesis testing, two types of errors can be made: Type I and Type II errors.

Type I and Type II Errors

• A Type I error occurs when you reject a true null hypothesis (remember that when the null hypothesis is true you hope to retain it). Type-I error is a false positive error.
  α=P(type I error)=P(Rejecting the null hypothesis when it is true)
  Type I error is more serious than type II error and therefore more important to avoid than a type II error.
• A Type II error occurs when you fail to reject a false null hypothesis (remember that when the null hypothesis is false you hope to reject it). Type II error is a false negative error.
  $\beta$=P(type II error) = P(accepting null hypothesis when alternative hypothesis is true)
• The best way to allow yourself to set a low alpha level (i.e., to have a small chance of making a Type I error) and to have a good chance of rejecting the null when it is false (i.e., to have a small chance of making a Type II error) is to increase the sample size.
• The key to hypothesis testing is to use a large sample in your research study rather than a small sample!

If you do reject your null hypothesis, then it is also essential that you determine whether the size of the relationship is practically significant.
Therefore, the hypothesis test procedure is adjusted so that there is a guaranteed “low” probability of rejecting the null hypothesis wrongly; this probability is never zero.

Therefore, for type I and Type II errors remember that falsely rejecting the null hypothesis results in an error called Type-I error and falsely accepting the null hypothesis results in Type-II Error.

Read more about Level of significance in Statistics

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Significance Level

The significance level in statistics is the level of probability at which it is agreed that the null hypothesis will be rejected. In academic research, usually, a 0.05 level of significance (level of significance) is used. The level of significance is also called a level of risk. Typical values for the selection of significance level range from 0.01 to 0.05, but other values can also be used depending on the context and the desired level of risk. A lower significance level means a lower probability of rejecting a true null hypothesis.

Significance Level in Statistics

The level of significance of an event (such as a statistical test) is the probability that the event will occur by chance. If the level is quite low then the probability of occurring that event by chance will be quite small. One can say that the event is significant as its occurrence is very small.

The significance level is the probability of rejecting the null hypothesis when it is true. In other words, the significance level is the probability of making a Type-I error, which is the error of incorrectly rejecting a true null hypothesis.

Type I Error

It has become part of the statistical hypothesis-testing culture.

• It is a longstanding convention.
• It reflects a concern over making type I errors (i.e., wanting to avoid the situation where you reject the null when it is true, that is, wanting to avoid “false positive” errors).
• If you set the level of significance at 0.05, then you will only reject a true null hypothesis 5% of the time (i.e., you will only make a type-I error 5% of the time) in the long run.

The trade-off between Type-I and type-II Error

The choice of significance level is a trade-off between Type-I and Type-II errors. A smaller/ lower level of significance reduces the likelihood (probability) of Type-I errors (false positives) but increases the likelihood of Type-II errors (false negatives). In other words, the chance of type-I error increases for a higher significance level but decreases the chance of type-II error.

### Factors Affecting Significance Level:

• Type of test: Different statistical tests have different formulas for calculating p-values, which are used to determine significance.
• Sample size: Larger sample sizes generally lead to more powerful tests and lower p-values, making it easier to reject the null hypothesis.
• Effect size: The magnitude of the difference between the null and alternative hypotheses can also influence the p-value.

In conclusion, the level of significance is a powerful tool that helps us to navigate the uncertainties in data analysis. By understanding the role of the significance level, one can make more wise informed decisions about the validity of research findings. In summary, the significance level is a crucial stage in the hypothesis testing procedure that helps the researchers make decisions about whether to accept or reject a null hypothesis based on the observed data. By carefully considering the significance level, researchers can balance the risk of making a Type-I error with the power of their tests.

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