An approximate likelihood ratio test for a normal mean vector with nonnegative components with application to clinical trials

A new multivariate statistic, designed to be sensitive to the alternative hypothesis that a mean vector lies in the positive orthant, rather than to all alternatives or to alternatives on a line (O'Brien, 1984), is proposed. The statistic is shown to approximate the likelihood ratio test statistic for such alternatives (Kudô, 1963; Perlman, 1969). Its null distribution is a special case of the chi-bar-squared distribution, which allows critical values to be obtained. Such a statistic may be used in two-armed clinical trials where we are interested in showing that one treatment is better than another with respect to several responses. As an example, a clinical trial is reanalyzed, with stronger conclusions than had been drawn originally.