# Chapter 31

*Permutation Tests in Clinical Trials*

*David M. Zucker*

# 31.1 Randomization Inference—Introduction

Randomization—allocation of study treatments to subjects in a random fashion—is a fundamental pillar of the modern controlled clinical trial. Randomization bolsters the internal validity of a trial in three major respects:

**1.** It prevents possible investigator bias (which may otherwise exist even unintentionally) in the allocation of subjects to treatments.

**2.** It generates study groups that, on average, are balanced with respect to both known and unknown factors.

**3.** It provides a probability structure whereby the study results may be evaluated statistically without exogenous statistical modeling assumptions.

The purpose of this article is to elaborate on the last of these three points.

To begin, we note that many, probably most, statistical analyses in empirical research, especially in nonexperimental studies, involve some statistical modeling assumptions. For example, to take a simple case, the classical two-sample *t*-test for comparing two groups makes the following assumptions:

**1.** Each observation in each group is equal to the sum of a fixed group-specific population mean value plus a mean zero random error term.

**2.** All random error terms, both within and between groups, are statistically independent.

**3.** The random error terms have a normal distribution.

**4.** Within each group, the error terms all have the same variance. (On occasion it is also assumed that the variance is ...