Testing homogeneity of normal means with a simply ordered alternative and dependent observations

Abstract

When subjects are given a treatment and studied for a period of time, it is often the case that the mean responses are believed to be nondecreasing (or nonincreasing) with time and it is desired to determine if the mean responses remain constant or increase. Procedures have been studied extensively for the case in which the correlations between measurements on the same subject are constant, but this requirement is too restrictive in some applications. Assuming the vectors of measurements on the subjects comprise a random sample from a multivariate normal distribution with unknown positive definite covariance matrix, Perlman (“One-sided problems in multivariate analysis”, Ann. Math. Statist., 40 (1969) 549-67) derived the likelihood ratio statistic, expressed its null distribution in terms of the unknown covariance matrix and found the supremum of the rejection probabilities over all positive definite covariance matrices. While this exact approach is very conservative, it is shown by Monte Carlo simulation that for a variety of correlation structures, evaluating the true null distribution at the sample covariance matrix provides an approximation that performs quite well even for small sample sizes. The exact and approximate tests are illustrated on a numerical example. Similar results are noted for the testing situation in which the trend is the null hypothesis.