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 The MULTTEST Procedure

# Overview

The MULTTEST procedure addresses the multiple testing problem. This problem arises when you perform many hypothesis tests on the same data set. Carrying out multiple tests is often reasonable because of the cost of obtaining data, the discovery of new aspects of the data, and the many alternative statistical methods. However, a negative feature of multiple testing is the greatly increased probability of declaring false significances.

For example, suppose you carry out 10 hypothesis tests at the 5% level, and you assume that the distributions of the p-values from these tests are uniform and independent. Then, the probability of declaring a particular test significant under its null hypothesis is 0.05, but the probability of declaring at least 1 of the 10 tests significant is 0.401. If you perform 20 hypothesis tests, the latter probability increases to 0.642. These high chances illustrate the danger in multiple testing.

PROC MULTTEST approaches the multiple testing problem by adjusting the p-values from a family of hypothesis tests. An adjusted p-value is defined as the smallest significance level for which the given hypothesis would be rejected, when the entire family of tests is considered. The decision rule is to reject the null hypothesis when the adjusted p-value is less then ; in most cases, this procedure controls the familywise error rate at or below the level. PROC MULTTEST offers the following p-value adjustments:

• Bonferroni
• Sidak
• stepdown
• Hochberg's method
• Hochberg and Benjamini's method
• bootstrap
• permutation

The Bonferroni and Sidak adjustments are simple functions of the raw p-values. They are computationally quick, but they can be too conservative. Stepdown methods remove some conservativeness, as do the step-up methods of Hochberg (1988). The bootstrap and permutation adjustments resample the data with and without replacement, respectively, to approximate the distribution of the minimum p-value of all tests. This distribution is then used to adjust the individual raw p-values. The bootstrap and permutation methods are computationally intensive but appealing in that, unlike the other methods, correlations and distributional characteristics are incorporated into the adjustments (Westfall and Young 1989, 1993).

PROC MULTTEST handles data arising from a multivariate one-way ANOVA model, possibly stratified, with continuous and discrete response variables; it can also accept raw p-values as input data. You can perform a t-test for the mean for continuous data and the following statistical tests for discrete data:

• Cochran-Armitage (CA) linear trend test
• Freeman-Tukey (FT) double arcsine test
• Peto (PETO) mortality-prevalence (log-rank) test
• Fisher (FISHER) exact test

The CA and PETO tests have exact versions that use permutation distributions and asymptotic versions that use an optional continuity correction. Also, with the exception of the FISHER test, you can use a stratification variable to construct Mantel-Haenszel type tests. All of the previously mentioned tests can be one- or two-sided.

As in the GLM procedure, you can specify linear contrasts that compare means or proportions of the treated groups. The output contains summary statistics and regular and multiplicity-adjusted p-values. You can create output data sets containing raw and adjusted p-values, test statistics and other intermediate calculations, permutation distributions, and resampling information.

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