Multivariate Analysis of Variance (MANOVA) is a procedure used for determining if the expected value of a given random vector is different in different groups. The random vector is composed of several random variables, which are called dependent variables. The different groups are observations with different values of certain categorical factors, which are called independent variables. There can be one or more independent variables. If there is one independent variable, whose levels define the groups, then the analysis is called one-way MANOVA. Two independent variables is the case of two-way MANOVA, three variables is the case of there-way MANOVA, and so on.
Univariate analysis of variance (ANOVA) is the degenerate case of MANOVA having only one dependent variable. Further on, the degenerate case of ANOVA comparing the means between only two groups is algebraically equivalent to t-test. The conclusions from the t-test and the one-way two-group ANOVA are exactly the same.
The explanation that follows gives a conceptual feel for what multivariate analysis of variance is attempting to accomplish. Suppose we are asked to determine if the mean scores on several types of quizzes are different between five ethnic groups of students: White, Black, Native, Asian and Hispanic. This is a case of one-way MANOVA. The independent variable here is the ethnicity. You can see that it is categorical (not continuous), having five distinct levels. The dependent variables are quiz_1, quiz_2, ... , quiz_p, which record performance of the students on p different quizzes. These variables are combined into one random vector quiz = (quiz_1, quiz_2, ... , quiz_p). The sample means of random vector quiz in different ethnic groups are compared with each other: Natives with Asians, Natives with Blacks, Natives with Whites, Natives with Hispanics, Asians with Blacks, Asians with Whites, Asians with Hispanics, Blacks with Whites, Blacks with Hispanics and Whites with Hispanics. MANOVA will generate a significance value indicating whether there are significant differences within the comparisons being made. If the differences are significant, it means that we accept the hypothesis saying that there is one or more quizzes such that, for each quiz, there are two or more ethnicities with different expected performance on the quiz. MANOVA does not tell us which ethnicity performs better. It does not tell us which quizzes are sensitive to the ethnicity issue. All we are told is that there are differences among the groups. We are told that people are no equal... We will have to follow up with more tests to determine where exactly the differences lie. To determine which quizzes present differences, we run univariate ANOVA tests. To determine which groups are different, we run the so-called "post-hoc" tests: Tukey test, Scheffe test, least-significant difference test (LSD) or Bonferroni test.
MANOVA REFERENCES
Weerahandi, S. (2004). Generalized Inference in Repeated Measures : Exact Methods in MANOVA and Mixed Models. Wiley-Interscience, Hoboken, New Jersey.
Huberty, C. J., & Olejnik, S. (2006). Applied MANOVA and Discriminant Analysis (2nd ed). Wiley-Interscience, Hoboken, New Jersey.
D'Agostino, R., Sullivan, L., & Beiser, A. (2005). Introductory Applied Biostatistics. Cengage Learning.
Teller, G. R. (1999). Mathematical Statistics: A Unified Introduction. New York: Springer.
Dean, A. M., & Voss, D. (2000). Design and Analysis of Experiments. Springer-Verlag New York.
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