This paper explores the use of simple summary statistics for analysing repeated measurements in randomized clinical trials with two treatments. Quite often the data for each patient may be effectively summarized by a pre-treatment mean and a post-treatment mean. Analysis of covariance is the method of choice and its superiority over analysis of post-treatment means or analysis of mean changes is quantified, as regards both reduced variance and avoidance of bias, using a simple model for the covariance structure between time points. Quantitative consideration is also given to practical issues in the design of repeated measures studies: the merits of having more than one pre-treatment measurement are demonstrated, and methods for determining sample sizes in repeated measures designs are provided. Several examples from clinical trials are presented, and broad practical recommendations are made. The examples support the value of the compound symmetry assumption as a realistic simplification in quantitative planning of repeated measures trials. The analysis using summary statistics makes no such assumption. However, allowance in design for alternative non-equal correlation structures can and should be made when necessary.

Repeated measures in clinical trials: Analysis using mean summary statistics and its implications for design

The INR system was developed to standardize PT reporting in patients on oral anticoagulants. We prospectively collected blood samples from 29 patients with liver impairment (INR 1.5-3.5). Control patients were on warfarin (n = 31). PT's were measured on an ACL-300 with three thromboplastin reagents. INR's were calculated using instrument specific ISI's. Other tests performed were FDP's, fibrinogen, aPTT, factors II, V, VII and X. The INR's for each patient in the study population using the three thromboplastin reagents were significantly different (p = 0.0001). Those for the control population were not (p = 0.0658). Fibrinogen, factors V, II and X were different at the 5% level of significance between the populations. FDP's were detected in 17 study subjects. The INR system is not valid for comparison of patients with liver impairment because different reagents do not give the same INR for the same sample. It is, however, no less valid than the use of PT with different thromboplastin reagents. Further study is recommended.

Assessment of the validity of the INR system for patients with liver impairment.

Chapter 12 – Missing Data: A Review of the Literature

A common feature of many statistical investigations is the collection of data from groups of experimental units each of which is observed under two or more conditions. Such studies are generally called either split-plot experiments or repeated measurements experiments. This paper is concerned with reviewing general statistical strategies for the analysis of data from these types of research designs. For this purpose, primary attention is directed at two basic dimensions. One of these is the nature of the randomization processes for the data as obtained among and within the experimental units. The other is the level of the measurement scale as either nominal, ordinal, or interval. This framework is then used as the basis of discussion of alternative statistical methods such as repeated measurements analysis of variance, multivariate analysis of variance, and their non-parametric rank and categorical data counterparts in both a general sense and for some specific classes of examples. Finally, a selected bibliography of references for these and related methods is given.

Some Views on Parametric and Non-Parametric Analysis for Repeated Measurements and Selected Bibliography

A variational method is used to obtain sufficient conditions for an optimal two-treatment, p-period design, given a repeated-measures model with arbitrary within-subject covariance. The design minimizes the variance of the estimator of treatment differences over all possible allocations of N subjects to the 2 p possible treatment sequences. The method is applied to a model that includes carryover effects. When subject effects are random with equicorrelated observations, designs are optimal if they are (a) strongly balanced; (b) uniform on the rows; and (c) either uniform on the subjects p is even and greater than 2, or uniform on the first p — 1 periods if p is odd. For two periods the optimal design is AB, BA, AA, BB. The same designs are optimal when baseline measurements are obtained in each period. Baseline observations improve efficiency considerably for p = 2, only slightly for p odd, and are of no help for p even and greater than 2. A second model of errors in which a subject's response ha...

A Variational Approach to Optimal Two-Treatment Crossover Designs: Application to Carryover-Effect Models

SUMMARY Tests of the hypothesis of equality of the elements of the multivariate normal mean vector have been developed under various assumptions on the covariance matrix. The tests include an alternative form of the usual Hotelling-Hsu statistic, and methods based upon covariance matrices of the equal variance, equal covariance, reducible, compound symmetry, and known correlation types. The efficacies of the special methods are measured in terms of the average squared lengths of their simultaneous confidence intervals. The problem of incomplete data is considered, and a test statistic offered for the paired t situation with missing values on one variate and known correlation.

The analysis of a single sample of repeated measurements.

The treatments in a repeated measurements design can frequently be ordered along some natural continuum (e.g. time), and as a consequence the expected values and covariance structure of the responses can be represented as functions of their positions on the scale. We shall consider rules for spacing the treatments so as to maximize the power of the T2 test for equal treatment effects, as well as general two-sample T2 tests for the equality of mean vectors. These results will be obtained under the assumptions of covariance structures of the kinds associated with Wiener and Markov stochastic processes, and alternative hypothesis mean vectors whose elements are linear or quadratic functions of the scale variable.

Donald F. Morrison

Papers

The analysis of a single sample of repeated measurements.

The Optimal Spacing of Repeated Measurements