One-Way Multivariate Analysis of Variance

Question 1

Multivariate analysis of variance (MANOVA) is a multivariate extension of analysis of variance. As with ANOVA, the independent variables for a MANOVA are factors, and each factor has two or more levels. Unlike ANOVA, MANOVA includes multiple dependent variables rather than a single dependent variable. MANOVA evaluates whether the population means on a set of dependent variables vary across levels of a factor or factors. Here, we will discuss only a MANOVA with a single factor, that is, a ­one-way MANOVA. Each case in the SPSS data file for a one-way MANOVA contains a factor distinguishing participants into groups and two or more quantitative dependent variables.

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Question 4

One-way MANOVAs can analyze data from different types of studies.

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Experimental studies
Quasi-experimental studies
Field studies

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Question 8

SPSS reports a number of statistics to evaluate the MANOVA hypothesis, labeled Wilks’s lambda, Pillai’s trace, Hotelling’s trace, and Roy’s largest root. (Please note that the APA style manual prefers Wilks’s lambda to Wilks’ lambda.) Each statistic evaluates a multivariate hypothesis that the population means on the multiple dependent variables are equal across groups. We will use Wilks’s lambda because it is frequently reported in the social science literature. Pillai’s trace is a reasonable alternative to Wilks’s lambda.

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Question 11

If the one-way MANOVA is significant, follow-up analyses can assess whether there are differences among groups on the population means for certain dependent variables and for particular linear combinations of dependent variables. A popular follow-up approach is to conduct multiple ANOVAs, one for each dependent variable, and to control for Type I error across these multiple tests by using one of the Bonferroni approaches. If any of these ANOVAs yield significance and the factor contains more than two levels, additional follow-up tests are performed. These tests typically involve post hoc pairwise comparisons among levels of the factor, although they may involve more complex comparisons. We will illustrate follow-up tests using this strategy.

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Question 12

The strategy that we have described (i.e., use of MANOVA, multiple ANOVAs, and pairwise comparisons) has been criticized on a number of grounds, although it is frequently used in practice. First, MANOVA is not necessary to control for Type I errors across the multiple ANOVAs and from this perspective could be skipped in applying this strategy. Second, the decision to use MANOVA and multiple ANOVAs in a sequential process is inconsistent. More specifically, conducting follow-up ANOVAs ignores the fact that the MANOVA hypothesis includes subhypotheses about linear combinations of dependent variables. Of course, if we have particular linear combinations of variables of interest, we can evaluate these linear combinations by using ANOVA in addition to, or in place of, the ANOVAs conducted on the individual dependent variables. For example, if two of the dependent variables for a MANOVA measure the same construct of introversion, then we may wish to represent them by transforming the variables to z scores, adding them together, and evaluating the resulting combined scores by using ANOVA. This ANOVA could be performed in addition to the ANOVAs on the remaining dependent variables.

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Question 13

If we have no clue as to what linear combinations of dependent variables to evaluate, we may choose to conduct follow-up analyses to a significant MANOVA with the use of discriminant analysis. Discriminant analysis (see Lesson 35) yields one or more uncorrelated linear combinations of dependent variables that maximize differences among the groups. These linear combinations are empirically determined and may not be interpretable.

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Question 14

Assumption 1:

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The Dependent Variables Are Multivariately Normally Distributed for Each Population, with the Different Populations Being Defined by the Levels of the Factor

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Assumption 2:

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The Population Variances and Covariances among the Dependent Variables Are the Same across All Levels of the Factor

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Assumption 3:

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The Participants Are Randomly Sampled, and the Score on a Variable for Any One Participant Is Independent from the Scores on This Variable for All Other Participants (MANOVA should not be ­conducted if the independence assumption is violated.)

Question 17

Effect Size Statistics for a One-Way MANOVA
The multivariate General Linear Model procedure computes a multivariate effect size index. The multivariate effect size associated with Wilks’s lambda (Λ) is the multivariate eta square:

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Effect Size Statistics for a One-Way MANOVA
The multivariate General Linear Model procedure computes a multivariate effect size index. The multivariate effect size associated with Wilks’s lambda (Λ) is the multivariate eta square: