Publication Description
In many studies, a K degree of freedom large sample χ2 test is used to assess the effect of treatment on a multivariate response, such as an omnibus T2‐like test of a difference between two treatment groups in any of K repeated measures. Alternately, a K df χ2 test may be used to test the equality of K+1 groups in a single outcome measure. Jennison and Turnbull (Biometrika 1991; 78: 133–141) describe group sequential χ2 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 χ2 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. Copyright © 2003 John Wiley & Sons, Ltd.