Group sequential large sample T2‐like χ2 tests for multivariate observations

In many studies, a K degree of freedom large sample X 2 test is used to assess the effect of treatment on a multivariate response, such as an omnibus T 2 -like test of a difference between two treatment groups in any of K repeated measures. Alternately, a K df X 2 test may be used to test the equality of K + I groups in a single outcome measure. Jennison and Turnbull (Biometrika 1991; 78:133-141) describe group sequential X 2 and F-tests for normal errors linear models, and Proschan, Follmann and Geller (Statist. Med. 1994; 13:1441-1452) describe group sequential tests for K+1 group comparisons. These methods apply to sequences of statistics that can be characterized as having an independent increments variance-covariance structure, thus simplifying the computation of the sequential variance-covariance matrix and the resulting sequential test boundaries. However, many commonly used statistics do not share this structure, including a Liang-Zeger (Biometrika 1986; 73:13-22) GEE longitudinal analysis with an independence working correlation structure and a Wei-Lachin (J. Amer. Statist. Assoc. 1984; 79:653-661) multivariate Wilcoxon rank test, among others. For such analyses, this paper describes the computation of group sequential boundaries for the interim analysis of emerging results using K df tests that are expressed as quadratic forms in a statistics vector that is distributed as multivariate normal, at least asymptotically. We derive the elements of the covariance matrix of multiple successive K df X 2 statistics based on established theorems on the distribution of quadratic forms. This covariance matrix is estimated by augmenting the data from the successive interim analyses into a single analysis from which the component sequential tests and their variance-covariance matrix can then be extracted. Boundary values for the sequential statistics can then be computed using the method of Slud and Wei (J. Amer. Statist. Assoc. 1982; 77:862-868) or using the α-spending function of Lan and DeMets (Biometrika 1983; 70:659-663) with a surrogate measure of information. An example is presented using the analysis of repeated cholesterol measurements in a clinical trial.