Question

True/false: If you are making 4 comparisons between means, then based on the Bonferroni correction, you should use an alpha level of .01 for each test.

Short answer

False, the adjusted alpha level should be 0.0125, not 0.01.

Step-by-step solution

Understand the Bonferroni Correction

The Bonferroni correction is used when conducting multiple statistical tests to reduce the chances of obtaining false positives (Type I errors). It adjusts the significance level (alpha) for each individual test to control the overall Type I error rate.

Calculate the Adjusted Alpha Level

According to the Bonferroni correction, the adjusted alpha level for each test is calculated by dividing the original alpha level by the number of tests. If the original alpha level is 0.05 and there are 4 comparisons, calculate it as follows: \[ \text{Adjusted alpha} = \frac{0.05}{4} = 0.0125 \].

Compare with the Stated Alpha Level

The problem states an alpha level of 0.01 for each test. Compare the calculated adjusted alpha (0.0125) with the stated alpha level (0.01) to determine if they match. Since 0.0125 is greater than 0.01, the provided statement does not hold true.

Key concepts

Type I error

A Type I error occurs in statistical hypothesis testing when we incorrectly reject a true null hypothesis. In simpler terms, it's a false alarm. We think there's an effect or a difference when there actually isn't one. This type of error is a major concern in experiments and studies, as it can lead researchers to draw incorrect conclusions. To prevent such mistakes, statisticians use a significance level (alpha), often set at 0.05, which indicates that there's a 5% risk of making a Type I error. This means, if the data suggests that a finding is statistically significant, there's a 5% chance that this finding could be simply due to random chance rather than a true effect. This becomes particularly important in studies with multiple comparisons, because with each test, the likelihood of encountering at least one Type I error increases.

Adjusted Alpha Level

The adjusted alpha level is a recalculated significance threshold used in multiple testing to control the overall risk of Type I errors. When making multiple comparisons, the probability of making at least one Type I error across all tests rises. This is where methods like the Bonferroni correction come into play.

Significance Level Adjustment

The Bonferroni correction adjusts the alpha level (\( ext{original alpha} \), typically 0.05) by dividing it by the number of comparisons (tests) being conducted.For instance, if you have four comparisons, the adjusted alpha is calculated as:\[ \text{Adjusted alpha} = rac{0.05}{4} = 0.0125 \]Using this adjusted alpha helps maintain the overall risk of Type I errors across all tests at the desired level, ensuring that any individual test result remains statistically reliable.

Multiple Comparisons

In research, it is common to conduct several comparisons at once, which is referred to as multiple comparisons. These arise when researchers want to test different hypotheses simultaneously or compare more than two groups. While beneficial for gathering comprehensive insights, multiple comparisons increase the risk of encountering Type I errors.

Risk Management

Without adjusting the alpha level, the overall error rate would skyrocket with each additional test, making it very likely to find a false positive somewhere among the results. By using methods like the Bonferroni correction, researchers manage this risk by reducing the alpha level applied to each test. This adjustment helps to ensure that the collective statistical findings remain robust and reliable while keeping a lid on Type I errors. For example, if you have 4 tests and you use a typical alpha of 0.05 for each, you'd be almost guaranteed to mistakenly reject a true null hypothesis at least once. Adjusting the alpha level addresses this issue by ensuring that the likelihood of such an error stays within an acceptable range across the study.