# The MULTTEST Procedure

## Overview: MULTTEST Procedure

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 disadvantage 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 of 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 than . For most methods, this decision rule controls the familywise error rate at or below the level. However, the false discovery rate controlling procedures control the false discovery rate at or below the level.

PROC MULTTEST provides the following p-value adjustments:

• Bonferroni

• Šidák

• step-down methods

• Hochberg

• Hommel

• Fisher and Stouffer combination

• bootstrap

• permutation

• false discovery rate

• positive FDR

The Bonferroni and Šidák adjustments are simple functions of the raw p-values. They are computationally quick, but they can be too conservative. Step-down methods remove some conservativeness, as do the step-up methods of Hochberg (1988), and the adaptive methods. 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; Westfall et al. 1999).

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 with or without a homogeneity assumption, and the following statistical tests for discrete data:

• Cochran-Armitage linear trend test

• Freeman-Tukey double arcsine test

• Peto mortality-prevalence (log-rank) test

• Fisher exact test

The Cochran-Armitage 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 exact 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.

The MULTTEST procedure uses ODS Graphics to create graphs as part of its output. For general information about ODS Graphics, see Chapter 21: Statistical Graphics Using ODS.

The GLIMMIX , GLM , MIXED , and LIFETEST procedures, and other procedures that implement the ESTIMATE , LSMEANS , LSMESTIMATE , and SLICE statements, also adjust their results for multiple tests. For more information, see the documentation for these procedures and statements, and Westfall et al. (1999).