Lab 4 -- MANOVA on SPSS

A Multivariate Analysis of Variance (MANOVA) is often used as a multivariate analog of the ANOVA, but when there are multiple dependent variables.  (Note that an ANOVA can handle multiple independent variables, but only one dependent variable at a time.)   I will have much to say in lecture about the limited appropriateness of MANOVA, however you should know of its existence since it is used often.  These are the SPSS instructions and an example of a simple MANOVA.

For a simple oneway MANOVA, the data set should have one independent variable (grouping variable) and at least two dependent variables.  In the example herein I use gender as the IV and the GRE verbal and quantitative (grev and greq) scores as DVs.  The data set used herein (DESC.sav) is included in this MANOVA folder.

Click Analyze/General Linear Model/Multivariate.   Place the dependent variables in the box labeled such and the independent variable in the "Fixed Factors" box.  Open "Options" and check the boxes as denoted in the screen shot showing all of this below.

Click "Continue" in the Options window and "OK" in the "Multivariate" window and your job will run.   The output for this run is:

General Linear Model

Between-Subjects Factors

Value Label N
GENDER 1.00 female 193
2.00 male 45

Descriptive Statistics

GENDER Mean Std. Deviation N
GRE- VERBAL female 493.8860 90.35767 193
male 470.8889 93.07432 45
Total 489.5378 91.12631 238
GRE-QUANTITATIVE female 485.1813 95.87912 193
male 525.7778 118.92490 45
Total 492.8571 101.62098 238

Box's Test of Equality of Covariance Matrices(a)
Box's M 3.722
F 1.219
df1 3
df2 86913.177
Sig. .301
Tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups.
a Design: Intercept+GENDER

Multivariate Tests(c)
Effect Value F Hypothesis df Error df Sig. Partial Eta Squared Noncent. Parameter Observed Power(a)
Intercept Pillai's Trace .959 2771.204(b) 2.000 235.000 .000 .959 5542.409 1.000
Wilks' Lambda .041 2771.204(b) 2.000 235.000 .000 .959 5542.409 1.000
Hotelling's Trace 23.585 2771.204(b) 2.000 235.000 .000 .959 5542.409 1.000
Roy's Largest Root 23.585 2771.204(b) 2.000 235.000 .000 .959 5542.409 1.000
GENDER Pillai's Trace .054 6.707(b) 2.000 235.000 .001 .054 13.415 .913
Wilks' Lambda .946 6.707(b) 2.000 235.000 .001 .054 13.415 .913
Hotelling's Trace .057 6.707(b) 2.000 235.000 .001 .054 13.415 .913
Roy's Largest Root .057 6.707(b) 2.000 235.000 .001 .054 13.415 .913
a Computed using alpha = .05
b Exact statistic
c Design: Intercept+GENDER

Levene's Test of Equality of Error Variances(a)

F df1 df2 Sig.
GRE- VERBAL .001 1 236 .976
GRE-QUANTITATIVE 3.443 1 236 .065
Tests the null hypothesis that the error variance of the dependent variable is equal across groups.
a Design: Intercept+GENDER

Tests of Between-Subjects Effects
Source Dependent Variable Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Noncent. Parameter Observed Power(a)
Corrected Model GRE- VERBAL 19299.223(b) 1 19299.223 2.337 .128 .010 2.337 .331
GRE-QUANTITATIVE 60140.712(c) 1 60140.712 5.945 .015 .025 5.945 .680
Intercept GRE- VERBAL 33966035.357 1 33966035.357 4113.398 .000 .946 4113.398 1.000
GRE-QUANTITATIVE 37295811.300 1 37295811.300 3686.906 .000 .940 3686.906 1.000
GENDER GRE- VERBAL 19299.223 1 19299.223 2.337 .128 .010 2.337 .331
GRE-QUANTITATIVE 60140.712 1 60140.712 5.945 .015 .025 5.945 .680
Error GRE- VERBAL 1948749.937 236 8257.415




GRE-QUANTITATIVE 2387316.431 236 10115.748




Total GRE- VERBAL 59004100.000 238





GRE-QUANTITATIVE 60259600.000 238





Corrected Total GRE- VERBAL 1968049.160 237





GRE-QUANTITATIVE 2447457.143 237





a Computed using alpha = .05
b R Squared = .010 (Adjusted R Squared = .006)
c R Squared = .025 (Adjusted R Squared = .020)

Interpretation

An assumption of the MANOVA is that the covariance matrices of the dependent variables are the same across groups (determined by levels of the independent variable) in the population.  This is the multivariate analog of the assumption of equal variances for the ANOVA.  Box's M tests that assumption.  In the case at hand the p value of  .301 suggests that the hypothesis of equal covariance matrices can not be rejected.  So we have not violated an assumption of MANOVA, and may feel confident in continuing (at least in respect to this assumption).

The Multivariate Tests (Pillai's, Wilks', Hotelling's, and Roy's) all test the MANOVA null hypothesis -- that the mean on the composite variable is the same across groups.   In the multivariate case, these tests can, in general, provide different results.  In our present simple example contrasting across two groups, they are necessarily the same.  Thus we find the multivariate hypothesis that the mean on the composite is the same across groups rejected.  Remember that this is a test of the equality of a composite of the means (optimized to yield the maximum possible F-ratio) across groups. 

Almost all MANOVA programs provide univariate tests for each of the dependent variables used in the MANOVA.  This is probably done for a bad reason, as the practice has been to only pursue univariate tests if the multivariate test is significant (in an incorrect attempt to protect against a Type I error). 

For this reason, we have the standard Levene's test of the assumption of equal variances for each of our dependent variables as this is an assumption of the ANOVA.  For both grev and greq, the test produces an nonsignificant p value, so the null hypotheses regarding equal variances can not be rejected for either dependent variable, thus ANOVA is fine.

We can, however, consider these univariate tests if we wish (although we should realize that they are not directly related to the multivariate test), as long as we treat the error rate appropriately.  A simple (although not necessarily optimal) way to adjust the error rate is to use the Bonferroni inequality, thus we test each of our two null hypotheses regarding each of our two dependent variables at the α/2 level.  For the sake of demonstration, let α=.05, thus the adjusted error rate is .025.  We see (under the Tests of Between-Subjects Effects") that, using this modified α, the null hypothesis regarding greq would be rejected (and looking at the means we see that the males were superior), but that the null for grev would not be rejected.