Lachin J M
Department of Statistics/Computer and Information Systems, George Washington University, Rockville, MD 20852.
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.
多变量观测最常见的情况是随时间重复测量的情形。用于两组K次重复测量分析的两种最广泛使用的方法是K自由度（d.f.）的T2多变量方差分析F检验和受试者内1自由度方差分析F检验。两者都要求从正态分布总体中获取完整样本。在本文中，我描述了替代的K和1自由度无分布程序，这些程序允许随机缺失观测值。这些方法包括均值的大样本分析、带有曼-惠特尼参数估计的魏和拉钦多变量威尔科克森检验，以及基于魏和约翰逊多变量U统计量的多变量霍奇斯-莱曼位置偏移估计量。这些方法中的每一种都提供了组间差异大小的无分布K变量估计，可作为组间差异总体检验的基础。这些检验包括K自由度的综合T2类检验、1自由度的受限假设检验，如魏-拉钦多变量随机排序单侧检验，以及基于平均组间差异的最小方差广义最小二乘（GLS）估计的一般关联检验。然后我描述了协变量分层调整的GLS估计和组间差异检验。这种方法还提供了受试者内和受试者间效应的同质性（交互作用）检验。我通过对两组按性别分层的患者重复胆固醇测量的分析来说明这些分析。此类分析提供了一个总体无分布汇总估计，并检验了通过合并时间（重复测量）和分层上的组间差异所获得的治疗效果。
