Analysis of covariance is an effective method for addressing two considerations for randomized clinical trials. One is reduction of variance for estimates of treatment effects and thereby the production of narrower confidence intervals and more powerful statistical tests. The other is the clarification of the magnitude of treatment effects through adjustment of corresponding estimates for any random imbalances between the treatment groups with respect to the covariables. The statistical basis of covariance analysis can be either non-parametric, with reliance only on the randomization in the study design, or parametric through a statistical model for a postulated sampling process. For non-parametric methods, there are no formal assumptions for how a response variable is related to the covariables, but strong correlation between response and covariables is necessary for variance reduction. Computations for these methods are straightforward through the application of weighted least squares to fit linear models to the differences between treatment groups for the means of the response variable and the covariables jointly with a specification that has null values for the differences that correspond to the covariables. Moreover, such analysis is similarly applicable to dichotomous indicators, ranks or integers for ordered categories, and continuous measurements. Since non-parametric covariance analysis can have many forms, the ones which are planned for a clinical trial need careful specification in its protocol. A limitation of non-parametric analysis is that it does not directly address the magnitude of treatment effects within subgroups based on the covariables or the homogeneity of such effects. For this purpose, a statistical model is needed. When the response criterion is dichotomous or has ordered categories, such a model may have a non-linear nature which determines how covariance adjustment modifies results for treatment effects. Insight concerning such modifications can be gained through their evaluation relative to non-parametric counterparts. Such evaluation usually indicates that alternative ways to compare treatments for a response criterion with adjustment for a set of covariables mutually support the same conclusion about the strength of treatment effects. This robustness is noteworthy since the alternative methods for covariance analysis have substantially different rationales and assumptions. Since findings can differ in important ways across alternative choices for covariables (as opposed to methods for covariance adjustment), the critical consideration for studies with covariance analyses planned as the primary method for comparing treatments is the specification of the covariables in the protocol (or in an amendment or formal plan prior to any unmasking of the study.
A non-parametric strategy for the analysis of ordinal data from cross-over studies with two treatment sequences and d(> or = 2) periods is examined through Mann-Whitney rank measures of association. For each period, these statistics estimate the probability of larger response for a randomly selected patient in one group relative to a randomly selected patient in the other group. Such estimates are as well formed for comparisons between groups for u pairs of periods with the same treatment. Methods for U-statistics are used to produce a consistent estimate of the covariance matrix for the (d + u) Mann-Whitney estimates. The effects of periods and treatments on the respective Mann-Whitney estimates are evaluated through linear (or log-linear) models. For estimation of the parameters in these models, a modified weighted least squares method is applied through a (2d - 1) < or = (d + u) dimensional basis which effectively addresses potentially near singularities in the estimated covariance matrix of the Mann-Whitney estimates. The proposed methods are applicable to response variables with an interval or an ordered categorical scale. Their scope additionally has capabilities for controlling strata in the design of a cross-over study or concomitant variables for which covariance adjustment is of interest for reduction of variance. Applications of the methods are illustrated through three cross-over studies with different specifications for the two sequences of two treatments during two to four periods.
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