Efficiency comparisons of rank and permutation tests based on summary statistics computed from repeated measures data

Abstract

A popular method of using repeated measures data to compare treatment groups in a clinical trial is to summarize each individual's outcomes with a scalar summary statistic, and then to perform a two-group comparison of the resulting statistics using a rank or permutation test. Many different types of summary statistics are used in practice, including discrete and continuous functions of the underlying repeated measures data. When the repeated measures processes of the comparison groups differ by a location shift at each time point, the asymptotic relative efficiency of (continuous) summary statistics that are linear functions of the repeated measures has been determined and used to compare tests in this class. However, little is known about the non-null behaviour of discrete summary statistics, about continuous summary statistics when the groups differ in more complex ways than location shifts or where the summary statistics are not linear functions of the repeated measures. Indeed, even simple distributional structures on the repeated measures variables can lead to complex differences between the distribution of common summary statistics of the comparison groups. The presence of left censoring of the repeated measures, which can arise when these are laboratory markers with lower limits of detection, further complicates the distribution of, and hence the ability to compare, summary statistics. This paper uses recent theoretical results for the non-null behaviour of rank and permutation tests to examine the asymptotic relative efficiencies of several popular summary statistics, both discrete and continuous, under a variety of common settings. We assume a flexible linear growth curve model to describe the repeated measures responses and focus on the types of settings that commonly arise in HIV/AIDS and other diseases.