Multivariate analysis of variance.

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Abstract
We provide an expository presentation of multivariate analysis of variance (MANOVA) for both consumers of research and investigators by capitalizing on its relation to univariate analysis of variance models. We address several questions: (a) Why should one use MANOVA? (b) What is the structure of MANOVA? (c) How are MANOVA test statistics obtained and interpreted? (d) How are MANOVA follow-up tests obtained and interpreted? (e) How is strength of association assessed in MANOVA? (f) How should the results of MANOVA be presented? (g) Are there any alternatives to MANOVA? We use an example data set throughout the article to illustrate these points. Readers of psychological journals have probably observed that the analysis of variance (ANOVA) is the most prevalent statistical model for the analysis of data in the discipline. ANOVA models are popular because of their rich develop- mental history, because they include a vast array of designs that can address most typical research situations, and because of the wide availability of computer programs that can be used to perform the calculations. One can get a sense of how longer confronted with the tedium of extensive matrix calcu- lations, a major impediment to the undertaking of these more complex analyses. In this article, we capitalize on the reader's understanding of the univariate ANOVA to develop the multi- variate analog, MANOVA. Our purpose is not to present a comprehensive discussion of MANOVA but to present a descrip- tion of it in sufficient detail to point out the utility, even necessity, of such models in counseling psychology research and to encourage the reader to undertake a study of the topic and to use the technique more routinely in his or her research. To achieve the goals outlined above, we address several common questions about MANOVA models that might arise. In addressing these questions, we hope to provide an overview of the method that will capture the reader's interest and spur further study of the topic. The questions we address are (a) Why should one use MANOVA? (b) What is the structure of MANOVA? (c) How are MANOVA test statistics obtained and interpreted? (d) How does one interpret the follow-up tests in MANOVA? (e) How is strength of association assessed in MAN- OVA? (f) How should the results of MANOVA be presented? (g) Are there any alternatives to MANOVA in multiple-dependent- variable research?

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