How to Perform a MANOVA in SPSS In this example, we will look at a “multivariate” analysis of variance. The difference between “univariate” and “multivariate” analyses is that a “univariate” analysis has only one dependent variable (with any number of independent variables / predictors). A “multivariate” analysis, on the other hand, has many dependent variables (again, with any number of IVs). The goal of our analysis is to look for an effect of one or more more IVs on several DVs at the same time. Therefore, we’re going to use the familiar “general linear model” command in SPSS, but choose a “ multivariate multivariate”” analysis.
This dataset has information about whether a person is in one of several different experimental groups (“group”), and their scores on three different types of memory tests (recognition, free recall, and cued recall). The difficulty level of each memory task is also rated. We will treat the three different memory tasks as different examples of a single phenomenon (“memory”). Therefore, we will look at all three together as dependent variables in our analysis.
A dialog box will appear. It should look very similar to the dialog box that we’ve been using for the past two weeks (except that it says “ multivariate” up here, instead).
In a multivariate analysis, you have more than one DV, so you can select many DVs and put them into this box together. In this case, we have three different memory tests and we want to test for an effect of the IV on all three of them at once. In the “fixed factor” box, put the predictor that we’re looking at: experimental group membership.
Now, hit the “Options” button to go on.
Again, this “options” dialog is very similar to the used in the univariate analyses.
Click on the “observed power” button to get the statistical power of the multivariate tests.
Hit “Continue,” and then, back in the main dialog box, hit the “Model” button to go on.
Here’s the “model” dialog box, again very similar to the one for univariate analyses.
As usual, you can use the default value of “full factorial,” or you can select just certain variables to be included in the model. Go back to the main dialog box, and hit “OK” to run the analysis.
The output has two segments. The first part gives you the results of the multivariate tests. d
Multivariate Tests Effect Value Intercept
group
F
Hypothesis df
Error df
Sig.
Noncent.
Observed
Parameter
Power
b
Pillai's Trace
.947
249.681
a
3.000
42.000
.000
749.044
1.000
Wilks' Lambda
. 053
249.681
a
3.000
42.000
.000
749.044
1.000
Hotelling's Trace
17.834
249.681
a
3.000
42.000
.000
749.044
1.000
Roy's Largest Root
17.834
249.681a
3.000
42.000
.000
749.044
1.000
Pillai's Trace
.464
2.683
9.000
132.000
.007
24.147
.940
Wilks' Lambda
.592
2.737
9.000
102.368
.007
19.567
.864
Hotelling's Trace
.597
2.698
9.000
122.000
.007
24.282
.941
Roy's Largest Root
.326
c
3.000
44.000
.006
14.330
.874
4.777
a. Exact statistic b. Computed using alpha = .05 c. The statistic is an upper bound on F that yields a lower bound on the significance level. d. Design: Intercept + group
As usual for these F -test results, ignore the section labeled “Intercept.” These four numbers give you the p-values for the four different multivariate tests. These results tell you if there is a significant effect of the IVs on all of the DVs, considered as a group. If you are asked “overall, is there a significant effect of something on some set of variables, as a group,” then you would run a MANOVA and look at these multivariate tests for your conclusion. Remember that there’s no one single multivariate test; there are four different ones. In this case they’re all significant ( p < .05), so we can conclude that Group Membership did have a significant effect on the three different memory variables.
Here’s the second part of the results section. This gives univariate tests for the effects of Group Membership on each of the different DVs. Think of this table as several F -test tables, all stacked on top of one another. For instance, if you look at all of the rows with blue print, you could put them all back together into a single F -table. You could do similar re-arranging to get the individual F -tables for the other two variables.
Tests of Between-Subjects Effects Source
Dependent Variable
Type III Sum of Squares
Corrected Model
df
Mean Square
F
Sig.
Noncent.
Observed
Parameter
Power
b
a
3
8666.743
3.739
.018
11.217
.773
c
3
5712.132
3.343
.028
10.030
.721
8305.583
d
3
2768.528
3.687
.019
11.061
.767
Recognition Test
827662.688
1
827662.688
357.080
.000
357.080
1.000
Cued Recall Test
708831.021
1
708831.021
414.865
.000
414.865
1.000
Free Recall Test
572470.083
1
572470.083
762.408
.000
762.408
1.000
Recognition Test
26000.229
3
8666.743
3.739
.018
11.217
.773
Cued Recall Test
17136.396
3
5712.132
3.343
.028
10.030
.721
Free Recall Test
8305.583
3
2768.528
3.687
.019
11.061
.767
Recognition Test
101986.083
44
2317.866
Cued Recall Test
75177.583
44
1708.581
Free Recall Test
33038.333
44
750.871
Recognition Test
955649.000
48
Cued Recall Test
801145.000
48
Free Recall Test
613814.000
48
Recognition Test
127986.313
47
Cued Recall Test
92313.979
47
Free Recall Test
41343.917
47
Recognition Test
26000.229
Cued Recall Test
17136.396
Free Recall Test
dime
nsion
1
Intercept dime
nsion
1
group
dime
nsion
1
Error
dime
nsion
1
Total
dime
nsion
1
Corrected Total dime
nsion
1
a. R Squared = .203 (Adjusted R Squared = .149) b. Computed using alpha = .05 c. R Squared = .186 (Adjusted R Squared = .130) d. R Squared = .201 (Adjusted R Squared = .146)
These cells (the shaded ones) are the ones we’re most interested in. These p-values tell you that Group Membership had a significant effect on the results of the Recognition Test ( p = .018), the results of the Cued Recall Test ( p = .028), and the results of the Free Recall Test ( p = .019). Please note that the results in this table are not the results of a MANOVA. They are the results of three separate univariate ANOVAs that are done as a “step down analysis” after you ran the MANOVA. This is something like doing post-hoc tests after a significant
one-way F -test. The univariate results together do not add up to the multivariate test. If the multivariate test is really what you are interested in, look at the first table of the output instead. If the univariate tests are really what you’re interested in, skip the MANOVA and just do the three different univariate tests – but you will need to do a correction to your alpha level to control for inflated Type I error when doing multiple tests! MANCOVA in SPSS Now let’s go back to the main dialog box for the multivariate analysis, and add a covariate to the model. The question here is “does the effect of experimental group membership on the three memory variables remain significant after controlling for the item difficulty of the memory tasks?” To do this test, enter “test item difficulty” as a covariate.
Here’s what the “model” dialog looks like. (Again, you can get the same result by leaving the model on its default setting, “full factorial”).
You do have to leave the Sums of Squares setting on “Type III,” in order to get a test of the effects of each variable after controlling for the effects of the others. (But that’s the default setting, so you don’t actually need to do anything here). Hit “Continue,” and then in the main dialog box, hit “OK” to run the test.
Here are the results—again, the first section shows you the results of the multivariate tests (i.e., the effect of the various predictors on “all of the memory tasks”). d
Multivariate Tests Effect Value Intercept
group
difficlt
F
Hypothesis df
Error df
Sig.
Noncent.
Observed
Parameter
Power
b
a
3.000
41.000
.002
18.250
.942
6.083
a
3.000
41.000
.002
18.250
.942
.445
6.083
a
3.000
41.000
.002
18.250
.942
Roy's Largest Root
.445
6.083
a
3.000
41.000
.002
18.250
.942
Pillai's Trace
.472
2.675
9.000
129.000
.007
24.078
.939
Wilks' Lambda
.585
2.732
9.000
99.934
.007
19.523
.862
Hotelling's Trace
.611
2.694
9.000
119.000
.007
24.242
.940
Roy's Largest Root
.334
4.785
c
3.000
43.000
.006
14.354
.873
Pillai's Trace
.016
.224
a
3.000
41.000
.879
.673
.089
Wilks' Lambda
.984
.224
a
3.000
41.000
.879
.673
.089
Hotelling's Trace
.016
.224
a
3.000
41.000
.879
.673
.089
Roy's Largest Root
.016
.224
a
3.000
41.000
.879
.673
.089
Pillai's Trace
.308
6.083
Wilks' Lambda
.692
Hotelling's Trace
a. Exact statistic b. Computed using alpha = .05 c. The statistic is an upper bound on F that yields a lower bound on the significance level. d. Design: Intercept + group + difficlt
You can see that “difficulty” has been included in this model as a covariate. Again, look at the multivariate test results, just in the rows for the IV called “Group”. Again, all four multivariate tests are significant, so we can conclude that the effects of group membership on the three memory tasks are still significant, even after controlling for the effects of test item difficulty on people’s performance on the three memory tasks.
Congratulations! You have now completed a one-way multivariate analysis of covariance, or MANCOVA. We use this term because we have: --one nominal-level predictor (or “fixed factor”) --many I/R-level DVs --and an I/R-level predictor (as a covariate)
Paul F. Cook, University of Colorado Denver, Center for Nursing Research Updated 1/10 with SPSS (PASW) version 18
