9.4.18 Type I Error, Type II Error, Power and Sample Size

Knows What is Meant by: Type I Error, Type II Error, Power and Sample Size

When conducting a hypothesis test, there are four possible outcomes, which are true positive, false positive, true negative, and false negative. Type I error and Type II errors are two types of errors that can occur in a hypothesis test. Power and sample size are two concepts that are related to these errors.

Type I error:

Type I error is an error that occurs when a null hypothesis is rejected, even though it is true. The probability of making a Type I error is denoted by α, which is typically set at 0.05 or 0.01. For example, if we conduct a study and reject the null hypothesis, but the null hypothesis is actually true, we have made a Type I error.

Mnemonic: “False Alarm”

• Example: Convicting an innocent person.
• Key Idea: Think of it as a “false alarm”—you think there is a signal (effect) when there is none.
• Visual Aid: Imagine a fire alarm going off when there is no fire.

Type II error:

Type II error is an error that occurs when a null hypothesis is not rejected, even though it is false. The probability of making a Type II error is denoted by β. The power of a study is 1-β. For example, if we conduct a study and fail to reject the null hypothesis, but the null hypothesis is actually false, we have made a Type II error.

Mnemonic: “Missed Opportunity”

• Example: Letting a guilty person go free.
• Key Idea: Think of it as a “missed opportunity”—you missed detecting the signal (effect) that was actually there.
• Visual Aid: Imagine overlooking a fire that is starting to spread.

Power:

Statistical power is the probability that a study will detect an effect when there is an effect to be detected. It is a crucial concept in hypothesis testing, as it reflects the study’s ability to avoid Type 2 errors (false negatives).

Power is the probability of correctly rejecting a false null hypothesis. 1 − β = probability of a “true positive”, i.e., correctly rejecting the null hypothesis. A higher power indicates that the study is more likely to detect a true difference if one exists. Power is affected by several factors, including the sample size, effect size, and significance level.

Typical Power Level:

• Common Standard: A power of 0.80 (or 80%) is commonly used as a standard in many fields. This means there is an 80% chance of detecting an effect if there is one.

Influences on Power:

The power of a study is influenced by several factors:

1. Sample Size (n): Larger sample sizes increase power.
2. Effect Size (d): Larger effect sizes increase power.
3. Significance Level (α): Lower significance levels (e.g., 0.01 vs. 0.05) decrease power.
4. Variance: Lower variability within the data increases power.
5. Test Type: One-tailed tests have more power than two-tailed tests if the direction of the effect is correctly specified.

Power Calculation Formula: While the exact formula for calculating power depends on the specific statistical test being used (e.g., t-test, ANOVA), it generally involves the non-centrality parameter (NCP) and the cumulative distribution function (CDF) of the test statistic under the alternative hypothesis.

Sample Size:

The sample size is the number of observations in a study. It is an important factor that affects the power of a study. A larger sample size generally leads to a higher power and a lower chance of making a Type II error.

Clinical Examples:

Suppose a researcher wants to conduct a study to evaluate the effectiveness of a new medication for treating hypertension. The null hypothesis is that the new medication is no more effective than a placebo, while the alternative hypothesis is that the new medication is more effective than a placebo.

To conduct the study, the researcher needs to determine the appropriate sample size to achieve the desired power. The researcher can use power analysis to determine the sample size required to detect a difference of a certain magnitude with a desired level of power and significance level.

Suppose the researcher decides to use a significance level of 0.05 and a power of 0.80. The researcher estimates that a difference of 5 mmHg in mean systolic blood pressure between the two groups is clinically meaningful. Using power analysis, the researcher determines that a sample size of 80 participants per group is required to achieve the desired power.

In summary, Type I and Type II errors, power, and sample size are important concepts in hypothesis testing. Type I error occurs when a null hypothesis is rejected, even though it is true, while Type II error occurs when a null hypothesis is not rejected, even though it is false. Power is the probability of correctly rejecting a false null hypothesis, and the sample size is the number of observations in a study. These concepts are important for researchers and clinicians to understand when interpreting study results and designing new studies.

References:

1. Armitage P, Berry G, Matthews JNS. Statistical methods in medical research. John Wiley & Sons; 2002.
2. Altman DG. Practical statistics for medical research. Chapman and Hall; 1991.