Multivariate Analysis of Variance (MANOVA)

Multivariate analysis of variance (MANOVA) designs are appropriate when multiple dependent variables are included in the analysis. The dependent variables should represent continuous measures (i.e., interval or ratio data). Dependent variables should be moderately correlated. If there is no correlation at all, MANOVA offers no improvement over an analysis of variance (ANOVA); if the variables are highly correlated, the same variable may be measured more than once. In many MANOVA situations, multiple independent variables, called factors, with multiple levels are included. The independent variables should be categorical (qualitative). Unlike ANOVA procedures that analyze differences across two or more groups on one dependent variable, MANOVA procedures analyze differences across two or more groups on two or more dependent variables. Investigating two or more dependent variables simultaneously is important in various disciplines, ranging from the natural and physical sciences to government and business and to the behavioral and social sciences. Many research questions cannot be answered adequately by an investigation of only one dependent variable because treatments in experimental studies are likely to affect subjects in more than one way. The focus of this entry is on the various types of MANOVA procedures and associated assumptions. The logic of MANOVA and advantages and disadvantages of MANOVA are included.

MANOVA is a special case of the general linear models. MANOVA may be represented in a basic linear equation as Y = Xβ + ∊, where Y represents a vector of dependent variables, X represents [Page 858]a matrix of independent variables, β represents a vector of weighted regression coefficients, and ∊ represents a vector of error terms. Calculations for the multivariate procedures are based on matrix algebra, making hand calculations virtually impossible. For example, the null hypothesis for MANOVA states no difference among the population mean vectors. The form of the omnibus null hypothesis is written as H0 = μ1 = … = μk. It is important to remember that the means displayed in the null hypothesis represent mean vectors for the population, rather than the population means. The complexity of MANO VA calculations requires the use of statistical software for computing.