Introduction: The Basics of Multivariate Analysis of Variance

Multivariate analysis of variance (MANOVA) with one between-subjects factor is appropriate when an analysis involves the following:

• a single predictor variable measured on a nominal scale;

• multiple criterion variables, each of which is measured on an interval or ratio scale.

MANOVA is similar to ANOVA in that it tests for significant differences between two or more groups of participants. There is an important difference between MANOVA and ANOVA, however: ANOVA is appropriate when the study involves just one criterion variable whereas MANOVA is appropriate when the study involves more than one criterion variable. With MANOVA, you can perform a single computation that determines whether there is a significant difference between treatment groups when compared simultaneously on all criterion variables.

The Aggression Study

To illustrate one possible use of MANOVA, consider the study on aggression that was introduced in Chapter 9, “One-Way ANOVA with One Between-Groups Factor.” In that study, each of 60 children was assigned to one of the following treatment groups:

• a group that consumed 0 grams of sugar at lunch;

• a group that consumed 20 grams of sugar at lunch;

• a group that consumed 40 grams of sugar at lunch.

A pair of observers watched each child after lunch and recorded the number of aggressive acts displayed by that child. The total number of aggressive acts performed by a given child over a two-week period served as that child’s score on the “aggression” dependent variable.

It was appropriate to analyze data from the preceding study with ANOVA because there was a single dependent variable: level of aggression. Now, consider how the study could be modified so that it would, instead, analyze the data using a multivariate procedure: MANOVA. Imagine that, as the researcher, you wanted to have more than one measure of aggression. After reviewing the literature, you believe that there are at least four different types of aggression that children might display:

• aggressive acts directed at children of the same sex;

• aggressive acts directed at children of the opposite sex;

• aggressive acts directed at teachers;

• aggressive acts directed at parents.

Assume that you now want to replicate your earlier study; this time, your observers will note the number of times each child displays an aggressive act in each of the four preceding categories. At the end of the two-week period, you will have scores for each child on each of the dependent variables listed above.

You now have a number of options as to how you will analyze your data. One option is to simply perform four ANOVAs (as described in Chapter 9). In each ANOVA, the independent variable would again be “amount of sugar consumed.” In the first ANOVA, the dependent variable would be “number of aggressive acts directed at children of the same sex,” in the second ANOVA, the dependent variable would be “number of aggressive acts directed at children of the opposite sex,” and so forth.

However, a more appropriate alternative would be to perform a single computation that allows you to assess the effect of your independent variable on all four of the dependent variables simultaneously. This is what MANOVA allows. Performing a MANOVA will allow you to test the following null hypothesis:

In the population, there is no difference across the various treatment groups when they are compared simultaneously on the dependent variables.

Here is another way of stating this null hypothesis:

In the population, all treatment groups are equal on all dependent variables.

MANOVA produces a single statistic that allows you to test this null hypothesis. If the null hypothesis is rejected, it means that at least two of the treatment groups are significantly different with respect to at least one dependent variable. You can then perform follow-up tests to identify the pairs of groups that are significantly different, and the specific dependent variables on which they are different. In doing these follow-up tests, you might find that the groups differ on some dependent variables (e.g., “aggressive acts directed toward children of the same sex”) but not on other dependent variables (e.g., “aggressive acts directed toward teachers”).

A Multivariate Measure of Association

Chapter 9 introduced the R² statistic, a measure of association that is often computed when performing ANOVA. Values of R² can range from 0 to 1, with higher values indicating a stronger relationship between the predictor variable and the criterion variable in the study. If the study is a true experiment, you can view R² as an index of the magnitude of the treatment effect.

The current chapter introduces a multivariate measure of association: one that can be used when there are multiple criterion variables (as in MANOVA). This multivariate measure of association is called Wilks’ lambda. Values of Wilks’ lambda can range from 0 to 1, but the way that you interpret lambda is the opposite of the way that you interpret R². With lambda, small values (near 0) indicate a relatively strong relationship between the predictor variable and the multiple criterion variables (taken as a group), while larger values (near 1) indicate a relatively weak relationship. The F statistic that tests the significance of the relationship between the predictor and the multiple criterion variables is actually based on Wilks’ lambda.

Overview of the Steps Followed in Performing a MANOVA

When you compute a multivariate analysis of variance with PROC GLM, the procedure produces two sets of results. First are the results from the univariate ANOVAs with one univariate ANOVA for each criterion variable in the analysis (here, univariate means “one criterion variable”). Each univariate ANOVA involves the same type of analysis that was described in Chapter 9. For example, with the present study, the predictor variable (“amount of sugar consumed”) is the same in each analysis. However, one ANOVA will have “aggressive acts directed toward children of the same sex” as the criterion variable, one will have “aggressive acts directed toward children of the opposite sex” as the criterion, and so forth.

The second set of results produced when you request a MANOVA are the results of the multivariate analysis of variance (here, multivariate means “multiple criterion variables”). These multivariate results will include Wilks’ lambda and the F statistic derived from Wilks’ lambda.

As the preceding section suggests, there is a specific sequence of steps that you should follow when interpreting these results. First, you should review the multivariate F statistic derived from Wilks’ lambda. If this multivariate F statistic is significant, you can reject the null hypothesis of no overall effect for the predictor variable. In other words, you can reject the null hypothesis that all groups are equal on all criterion variables. At that point, you can proceed to the univariate ANOVAs and interpret them.

When interpreting univariate ANOVAs, you first identify those criterion variables for which the univariate F statistic is significant. If the F statistic is significant for a given criterion variable, you can then proceed to interpret the results of the Tukey multiple comparison test to determine which pairs of groups significantly differ.

However, if the multivariate F statistic that is computed in the MANOVA is nonsignificant, this means that you cannot reject the null hypothesis that all groups have equal means on the criterion variables in the population. In most cases, your analysis should terminate at this point; you generally should not proceed to interpret the univariate ANOVAs, even if one or more of them displays a significant F statistic.

Similarly, even if the multivariate F statistic is significant, you should not interpret the results for any specific criterion variable that did not display a significant univariate F statistic. This is consistent with the general guidelines for univariate ANOVA presented in Chapter 9.