Analysis of cross-over designs via growth curve models

Abstract

Models for repeated measures cross-over designs are defined in terms of growth curve models. In the paper two specific cross-over designs, called the AB:BA and ABAB:BABA design, are studied. The maximum-likelihood (ML) estimators for the parameters are derived by utilizing the theory for growth curve models. A model with a structured dispersion matrix is defined for the AB:BA design, and a specific linear transformation is used to derive estimators in a convenient way. To illustrate numerically, ML estimates are calculated for an ABAB:BABA study.

Introduction

The simplest example of a cross-over design is the two-sequence two-period cross-over design for comparing two treatments. That is, if the two treatments involved are denoted $A$ and $B$, then half of the subjects receive treatment $A$ in the first period followed by treatment $B$ in the second period, whereas the remaining subjects receive the treatments in the reverse order. This design is denoted AB:BA, where the colon is used to separate the treatment sequences. Higher-order cross-over designs are obtained by including more than two treatments or two periods. Typically, in a standard cross-over design, a response variable is measured once at the end of each period for each subject. A repeated measures cross-over design is a cross-over design, where a sequence of measurements is collected within each period. For example, to compare two test drugs, the concentration of the drug in the blood might be measured every 30 min for each subject during 3 h after administration of the drug.

The aim of this paper is to show that statistical models for a repeated measures cross-over design in a useful way can be defined through the growth curve model (GCM). The GCM (other names of the model are generalized linear model, GMANOVA, multivariate linear normal model, etc.) was introduced by Potthoff and Roy (1964) as a method for analysis of growth curve experiments. An extended version of the GCM was presented by von Rosen (1989). In general, the model applies to longitudinal data in which subjects are followed for a period of time. To date the GCM has only been applied to cross-over designs by Banken (1984), in a canonical version, to study hypothesis testing problems. Putt and Chinchilli (1999) developed a mixed effects model and gave conditions under which explicit estimators of the variance components exist as well as hypothesis test of treatment differences. However, explicit maximum-likelihood (ML) estimators of the fixed effects are not obtained, unless independent random errors are assumed, in which case the estimators coincide with the ordinary least-squares estimators. The main advantages of using the GCM is that explicit ML estimators of the parameters are obtained.

Two specific cross-over designs, called the AB:BA and ABAB:BABA design, are studied. Models for these designs are described and the ML estimators for the parameters are derived by utilizing the theory for the GCM. Two models are defined for each of the design, with respect to inclusion or exclusion of a so-called carry-over effect. When the carry-over effect is omitted, the extended GCM is applied. Furthermore, a model with a structured dispersion matrix is defined for the AB:BA design, and a specific linear transformation is used to derive estimators in a convenient way. To illustrate numerically, ML estimates are calculated for an ABAB:BABA study published by Ciminera and Wolfe (1953) and later re-analysed by Putt and Chinchilli (1999).