Pate1 analysed a two-period cross-over design with baseline measurements assuming bivariate normality for the joint distribution of the period responses. In this paper, we propose non-parametric methods for analysing this design, including the use of the Wilcoxon rank sum test to derive the preliminary tests from the baseline measurements. We fit a robust regression line of the treatment response on baseline for each period and compute residuals. We also fit a robust locally weighted regression as an alternative method for computing residuals. Then, following Koch's procedure, we analyse the residuals for testing the significance of the treatment x period interaction and the treatment difference. We provide a numerical example to illustrate the methods.
1. Introduction
The two-period two-sequence cross-over design is useful in clinical trials if used in a proper setting. For example, if the disease conditions are relatively stable and one considers a sufficiently long washout period, the cross-over design should give a valid and efficient estimate of the treatment difference. In this design subjects are randomly assigned to two treatment sequences: subjects assigned to sequence 1 receive treatment 1 in period 1 and treatment 2 in period 2, whereas those assigned to sequence 2 receive these two treatments in the reverse order. Because of lack of power of the carry-over effect test in the context of Grizzle's' model, the use of this design in clinical trials had come under question by researchers, including the Food and Drug Administration. To remedy this and other problems, there have been some modifications in the design and analysis proposed (see for example, Wallenstein,' Armitage and Hills,3 Patel:, Kenward and Jones,6 and Laska' et al.). Matthews' gives an excellent review of the cross-over design.
In this paper we consider a two-period two-sequence design with baseline measurements, obtained just before the start of periods 1 and 2 for each subject. As shown in the following schematic, the design includes a placebo run-in period before the first treatment period and a washout period of adequate length between the two treatment periods. Furthermore, we limit the analysis methods to continuous data.