A BRIEF INTRODUCTION TOMULTIVARIATE ANALYSIS OF VARIANCE (MANOVA)Like the analysis of variance (ANOVA), the multivariate analysis of variance (MANOVA) hasvariations. For example, the one-way MANOVA contains a single factor (independent variable)distinguishing participants into groups and two or more quantitative dependent variables. Onecould do three separate one-way ANOVAs; however, using MANOVA, you will see how thecombination of the three variables distinguishes the groups, in one analysis. There is a two-wayor two-factor MANOVA that has two independent variables and two or more quantitativedependent variables. A doubly multivariate or mixed MANOVA has a between groupsindependent variable and a repeated measures (within groups) independent variable and two ormore quantitative dependent variables. Mixed MANOVAs are one way to analyze intervention(experimental) studies that have more than one dependent variable. The MANCOVA(Multivariate Analysis of Covariance) is used when you include both nominal (independentvariable) and scale (covariate) variables as predictors of the linear combination of two or morequantitative dependent variables.As with ANOVA, the independent variables for a MANOVA are factors, and each factor has twoor more levels. Unlike ANOVA, MANOVA includes multiple dependent variables rather than asingle dependent variable. MANOVA evaluates whether the population means on a set ofdependent variables vary across the levels of a factor or factors. That is, a one-way MANOVAtests the hypothesis that the population means for the dependent variables are the same for alllevels of a factor (across all groups). If the population means of the dependent variables are equalfor all groups, the population means for any linear combination of these dependent variables arealso equal for all groups. Consequently, a one-way MANOVA evaluates a hypothesis thatincludes not only equality among groups on the dependent variable, but also equality amonggroups on linear combinations of these dependent variables.The multivariate analysis of variance (MANOVA) is a complex statistic similar to ANOVA butwith multiple dependent variables analyzed together. That is, the MANOVA is a multivariateextension of ANOVA. The dependent variables should be related conceptually, and they shouldbe correlated with one another at a low to moderate level. If they are highly correlated, one runsthe risk of multicollinearity. If they are uncorrelated, there is usually no reason to analyze themtogether. The General Linear Model program in SPSS provides you with a multivariate F basedon the linear combination of dependent variables that maximally distinguishes your groups. Thismultivariate result is the MANOVA. SPSS also automatically prints out univariate Fs for theseparate univariate ANOVAs for each dependent variable. Typically, these ANOVA results arenot examined unless the multivariate results (the MANOVA) are significant, and somestatisticians believe that they should not be used at all.SPSS reports a number of statistics to evaluate the MANOVA hypothesis, labeled Wilks’Lambda, Pillai’s Trace, Hotelling’s Trace (T), and Roy’s Largest Root. Each statistic evaluates amultivariate hypothesis that the population means on the multiple dependent variables are equalacross groups. Most statisticians use Wilks’ Lambda, Λ, and as such, it is frequently reported inthe social (and behavioral) sciences literature. Pillai’s Trace is a reasonable alternative to Wilks’Lambda for use with the MANOVA. Measures of effect size are often reported with (partial) etasquared (η2 is reported in SPSS) or eta (η is the square root of eta squared). A multivariate effectsize index is more commonly used (see Multivariate η2 below).MANOVAPAGE 2If the one-way MANOVA is significant, follow-up analyses can assess whether there aredifferences among groups on the population means for certain dependent variables and forparticular linear combinations of dependent variables. A popular follow-up approach is toconduct multiple ANOVAs, one for each dependent variable, and to control for Type I erroracross these multiple tests using one of the Bonferroni approaches (e.g., α / number of dependentvariables). If any of these ANOVAs yield significance and the factor contains more than twolevels (with two levels, a comparison of group means is conducted), additional follow-up testsare performed. These tests typically involve post hoc pairwise comparisons among levels of thefactor, although they may involve more complex comparisons. For example, some statisticiansprefer conducting parameter estimates, with the use of dummy variables devised to distinguishone group from the others. Parameter estimates tell us how the dependent variables are weightedin the equation that maximally distinguishes the groups.Some statisticians have criticized the strategy of conducting follow-up ANOVAs after obtaininga significant MANOVA because the individual ANOVAs do not take into account themultivariate nature of the MANOVA. Conducting follow-up ANOVAs ignores the fact that theMANOVA hypothesis includes sub-hypotheses about linear combinations of dependentvariables. Of course, if we have particular linear combinations of variables of interest, we canevaluate these linear combinations using ANOVA in addition to, or in place of, the ANOVAsconducted on the individual dependent variables. For example, if two of the dependent variablesfor a MANOVA measure the same construct of introversion, then we may wish to represent themby transforming the variables to z scores, adding them together, and evaluating the resultingcombined scores using ANOVA. This ANOVA could be performed in addition to the ANOVAson the remaining dependent variables. If we have no clue as to what linear combinations ofdependent variables to evaluate, we may choose to conduct follow-up analyses to a significantMANOVA using discriminant analysis. Discriminant analysis yields one or more uncorrelatedlinear combinations of dependent variables that maximize differences among groups. Theselinear combinations are empirically determined and may not be interpretable.Assumptions underlying the One-Way