Neil H. Timm University of Pittsburgh
Most students in psychology, education, and the social sciences are familiar with the application of the basic principles of univariate analysis of variance (ANOVA) and covariance (ANCOVA) analysis in the design of experiments; that is, reaching valid conclusions about population parameters through tests of hypotheses and the establishment of simultaneous confidence intervals. When introduced to the univariate principles of ANOVA and ANCOVA, students may have learned the concepts via "partitioning the observed total sum of squares" following Fisher ( 1925); the overparameterized less than full rank model following Scheffé ( 1959) and more recently Searle ( 1971, 1987) and Rao ( 1973); the full rank linear model popularized by Hocking and Speed ( 1975), Timm and Carlson ( 1975), and Hocking ( 1985); or the coordinate-free geometric approach discussed by Herr ( 1980). Although all of these approaches have natural generalizations to multivariate analysis of variance (MANOVA) and covariance (MANCOVA) analysis, the multivariate generalizations are complicated by the number of hypotheses that may be tested, the difficulty of choosing among numerous test statistics, and the fact that a computer software program must be available to the researcher to perform the mathematical calculations. Nonorthogonal designs, the analysis of repeated measurements, designs in which the errors form a stationary spacial process, multiresponse designs, and designs in which repetitions of a basic design leads to independent multivariate stationary time series add additional complications to MANOVA and MANCOVA methodologies.
This chapter reviews and integrates the various approaches to the analysis of variance: full rank models; reparameterized less than full rank models; the geom-