Multiple Comparisons - Methods

Methods

Multiple testing correction refers to re-calculating probabilities obtained from a statistical test which was repeated multiple times. In order to retain a prescribed familywise error rate α in an analysis involving more than one comparison, the error rate for each comparison must be more stringent than α. Boole's inequality implies that if each test is performed to have type I error rate α/n, the total error rate will not exceed α. This is called the Bonferroni correction, and is one of the most commonly used approaches for multiple comparisons.

In some situations, the Bonferroni correction is substantially conservative, i.e., the actual familywise error rate is much less than the prescribed level α. This occurs when the test statistics are highly dependent (in the extreme case where the tests are perfectly dependent, the familywise error rate with no multiple comparisons adjustment and the per-test error rates are identical). For example, in fMRI analysis, tests are done on over 100000 voxels in the brain. The Bonferroni method would require p-values to be smaller than .05/100000 to declare significance. Since adjacent voxels tend to be highly correlated, this threshold is generally too stringent.

Because simple techniques such as the Bonferroni method can be too conservative, there has been a great deal of attention paid to developing better techniques, such that the overall rate of false positives can be maintained without inflating the rate of false negatives unnecessarily. Such methods can be divided into general categories:

• Methods where total alpha can be proved to never exceed 0.05 (or some other chosen value) under any conditions. These methods provide "strong" control against Type I error, in all conditions including a partially correct null hypothesis.
• Methods where total alpha can be proved not to exceed 0.05 except under certain defined conditions.
• Methods which rely on an omnibus test before proceeding to multiple comparisons. Typically these methods require a significant ANOVA/Tukey's range test before proceeding to multiple comparisons. These methods have "weak" control of Type I error.
• Empirical methods, which control the proportion of Type I errors adaptively, utilizing correlation and distribution characteristics of the observed data.

The advent of computerized resampling methods, such as bootstrapping and Monte Carlo simulations, has given rise to many techniques in the latter category. In some cases where exhaustive permutation resampling is performed, these tests provide exact, strong control of Type I error rates; in other cases, such as bootstrap sampling, they provide only approximate control.