Some large-sample distribution-free estimators and tests for multivariate partially incomplete data from two populations

Stat Med. 1992 Jun 30;11(9):1151-70. doi: 10.1002/sim.4780110903.


The most common instance of multivariate observations is the case of repeated measures over time. The two most widely used methods for the analysis of K repeated measures for two groups are the K degrees of freedom (d.f.) T2 MANOVA F-test and the within-subjects 1 degree of freedom ANOVA F-test. Both require complete samples from normally distributed populations. In this paper, I describe alternative K and 1 d.f. distribution-free procedures which allow for randomly missing observations. These include a large-sample analysis of means, the Wei and Lachin multivariate Wilcoxon test with estimates of the Mann-Whitney parameter, and a multivariate Hodges-Lehmann location shift estimator based on the multivariate U-statistic of Wei and Johnson. Each of these methods provides a distribution-free K-variate estimate of the magnitude of group differences which can be used as the basis for an overall test of group differences. These tests include the K d.f. omnibus T2-like test, 1 d.f. tests of restricted hypotheses, such as the Wei-Lachin multivariate one-sided test of stochastic ordering, and the test of general association based on a minimum variance generalized least squares (GLS) estimate of the average group difference. I then describe covariate stratified-adjusted GLS estimates and tests of group differences. This approach also provides tests of homogeneity (interaction) for within-subjects and between-subjects effects. I illustrate these analyses with an analysis of repeated cholesterol measurements in two groups of patients, stratified by sex. Such analyses provide an overall distribution-free summary estimate and test of the treatment effect obtained by combining the group differences over both time (repeated measures) and strata.

Publication types

  • Research Support, U.S. Gov't, P.H.S.

MeSH terms

  • Chenodeoxycholic Acid / therapeutic use
  • Cholelithiasis / chemistry
  • Cholelithiasis / drug therapy
  • Cholelithiasis / epidemiology
  • Cholesterol / chemistry
  • Female
  • Humans
  • Least-Squares Analysis
  • Male
  • Multivariate Analysis*
  • Sex Factors
  • Stochastic Processes


  • Chenodeoxycholic Acid
  • Cholesterol