This page shows an example of multivariate analysis of variance (manova) in
SPSS with footnotes explaining the output. The data used in this example are from
the following experiment.

A researcher randomly assigns 33 subjects to one of three groups. The first
group receives technical dietary information interactively from an on-line
website. Group 2 receives the same information from a nurse practitioner, while
group 3 receives the information from a video tape made by the same nurse
practitioner. Each subject then made three ratings: difficulty, usefulness, and importance
of the information in the presentation. The researcher looks at three different ratings of the
presentation (difficulty, usefulness and importance) to determine if there is a
difference in the modes of presentation. In particular, the researcher is
interested in whether the interactive website is superior because that is the
most cost-effective way of delivering the information. In the dataset, the
ratings are presented in the variables useful, difficulty
and importance. The variable group indicates the group to which a
subject was assigned.

We are interested in how the variability in the three ratings can be explained by
a subject’s group.  Group is a categorical
variable with three possible values: 1, 2 or 3.  Because we have multiple dependent variables that
cannot be combined, we will choose to use manova.  Our null hypothesis in
this analysis is that a subject’s group has no effect on any of the three
different ratings, and we can test this hypothesis on the dataset,
manova.sav.

GET FILE='C:tempmanova.sav'.

We can start by examining the three outcome variables.

DESCRIPTIVES VARIABLES=useful difficulty importance.

FREQUENCIES VARIABLES=group.

MEANS TABLES=useful difficulty importance BY group.

Next, we can enter our manova command. In SPSS, manova can be conducted
through the generalized linear model function, GLM. In the manova command, we
first list the outcome variables, then indicate any categorical factors after
“by” and any covariates after “with”.  Here, group is a categorical
factor.  We must also indicate the lowest and highest values found in
group. We are also asking SPSS to print the eigenvalues generated.
These will be useful in seeing how the test statistics are calculated.

manova useful difficulty importance by group(1,3) 
/print=sig(eigen).

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
The default error term in MANOVA has been changed from WITHIN CELLS to
WITHIN+RESIDUAL.  Note that these are the same for all full factorial
designs.



* * * * * * A n a l y s i s   o f   V a r i a n c e * * * * * *


        33 cases accepted.
         0 cases rejected because of out-of-range factor values.
         0 cases rejected because of missing data.
         3 non-empty cells.

         1 design will be processed.

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -



* * * * * * A n a l y s i s   o f   V a r i a n c e -- design   1 * * * * * *

 EFFECT .. GROUP
 Multivariate Tests of Significance (S = 2, M = 0, N = 13 )

 Test Name       Value  Approx. F Hypoth. DF   Error DF  Sig. of F

 Pillais          .47667    3.02483       6.00      58.00       .012
 Hotellings       .89723    4.03753       6.00      54.00       .002
 Wilks            .52579    3.53823       6.00      56.00       .005
 Roys             .47146
 Note.. F statistic for WILKS' Lambda is exact.

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 Eigenvalues and Canonical Correlations

 Root No.    Eigenvalue        Pct.   Cum. Pct.  Canon Cor.

        1          .892      99.416      99.416        .687
        2          .005        .584     100.000        .072

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 EFFECT .. GROUP (Cont.)
 Univariate F-tests with (2,30) D. F.

 Variable   Hypoth. SS   Error SS Hypoth. MS   Error MS          F  Sig. of F

 USEFUL       52.92424  293.96544   26.46212    9.79885    2.70053       .083
 DIFFICUL      3.97515  126.28728    1.98758    4.20958     .47216       .628
 IMPORTAN     81.82969  426.37090   40.91485   14.21236    2.87882       .072

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Abbreviated  Extended
Name         Name

DIFFICUL     DIFFICULTY
IMPORTAN     IMPORTANCE

Manova Output

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
The default error term in MANOVA has been changed from WITHIN CELLS to
WITHIN+RESIDUAL.  Note that these are the same for all full factorial
designs.



* * * * * * A n a l y s i s   o f   V a r i a n c e * * * * * *


        33 cases accepted.
         0 cases rejected because of out-of-range factor values.
         0 cases rejected because of missing data.
         3 non-empty cells.

         1 design will be processed.a

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -



* * * * * * A n a l y s i s   o f   V a r i a n c e -- design   1 * * * * * *

 EFFECTb .. GROUP
 Multivariate Tests of Significance (S = 2, M = 0, N = 13 )

 Test Name         Valuec    Approx. Fd Hypoth. DFe Error DFf  Sig. of Fg

 Pillaish          .47667    3.02483       6.00      58.00       .012
 Hotellingsi       .89723    4.03753       6.00      54.00       .002
 Wilksj            .52579    3.53823       6.00      56.00       .005
 Roysk             .47146
 Note.. F statistic for WILKS' Lambda is exact.l

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 Eigenvalues and Canonical Correlationsm

 Root No.    Eigenvalue        Pct.   Cum. Pct.  Canon Cor.

        1          .892      99.416      99.416        .687
        2          .005        .584     100.000        .072

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 EFFECT .. GROUP (Cont.)
 Univariate F-tests with (2,30) D. F.n

 Variable   Hypoth. SS  Error SS   Hypoth. MS Error MS     F       Sig. of F

 USEFUL       52.92424  293.96544   26.46212    9.79885    2.70053       .083
 DIFFICUL      3.97515  126.28728    1.98758    4.20958     .47216       .628
 IMPORTAN     81.82969  426.37090   40.91485   14.21236    2.87882       .072

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Abbreviated  Extended
Name         Name

DIFFICUL     DIFFICULTY
IMPORTAN     IMPORTANCE

a.
Case summary
– This provides counts of the observations to be included in the manova and the
counts of observations to be dropped due to missing data or data that falls
out-of-range. For example, a record where the value for group is 4, after
we have specified that the maximum value for group is 3, would be
considered out-of-range.

b. Effect – This indicates the predictor variable in
question. In our model, we are looking at the effect of group.

c. Value – This is the test statistic for the given effect and
multivariate statistic listed in the prior column.  For each predictor
variable, SPSS calculates four test statistics.  All of these test
statistics are calculated using the eigenvalues of the model (see superscript
m).  See superscripts h, i, j and k for explanations of each of the tests.

d. Approx. F – This is the approximate F statistic for the given
effect and test statistic.

e. Hypoth. DF – This is the number of degrees of freedom in the model.

f. Error DF – This is the number of degrees of freedom associated with
the model errors.  There are instances in manova when the degrees
of freedom may be a non-integer.

g. Sig.of F – This is the p-value associated with the F
statistic and the hypothesis and error degrees of freedom of a given effect and
test statistic.  The null hypothesis that a given predictor has no effect
on either of the outcomes is evaluated with regard to this p-value. For a given
alpha level, if the p-value is less than alpha, the null hypothesis is rejected.
If not, then we fail to reject the null hypothesis. In this example, we reject
the null hypothesis that
group has no effect on the three different ratings at alpha level .05
because the p-values are all less than .05.

h. Pillais – This is Pillai’s Trace, one of the four multivariate criteria test statistics used in manova.
We can calculate Pillai’s trace using the generated eigenvalues (see superscript
m). Divide each eigenvalue by (1 + the
eigenvalue), then sum these ratios.  So in this example, you would first
calculate 0.89198790/(1+0.89198790) = 0.471455394, 0.00524207/(1+0.00524207) =
0.005214734, and 0/(1+0)=0.  When these are added, we arrive at Pillai’s trace:
(0.471455394 + 0.005214734 + 0) = .47667.

i. Hotellings – This is Lawley-Hotelling’s Trace. It is very similar to Pillai’s
Trace. It is the sum of the eigenvalues
(see superscript m) and is a direct generalization of the F
statistic in ANOVA. We can calculate the Hotelling-Lawley Trace by summing the
characteristic roots listed in the output: 0.8919879 + 0.00524207 + 0 = 0.89723.

j. Wilks – This is Wilk’s Lambda. This can be interpreted as the proportion of the
variance in the outcomes that is not explained by an effect. To calculate Wilks’ Lambda, for each
eigenvalue, calculate 1/(1 + the eigenvalue), then find the
product of these ratios. So in this example, you would first calculate
1/(1+0.8919879) = 0.5285446, 1/(1+0.00524207) = 0.9947853, and 1/(1+0)=1. Then
multiply 0.5285446 * 0.9947853 * 1 = 0.52579.

k. Roys – This is Roy’s Largest Root. Note there are two definitions of Roy’s Largest Root depending on whether you use SPSS MANOVA or SPSS GLM. In MANOVA, the calculation is 0.8919879/(1+0.8919879) =  0.4714544. In GLM, Roy’s Largest root is defined as the largest eigenvalue, which is 0.8919879. Based on this definition, Roy’s Largest Root can behave differently from the other three test statistics.  In instances where the other three are not significant and Roy’s is significant, the effect should be considered not significant. You can reference the following article for a detailed description of this discrepancy:

Kuhfeld, W. F. (1986). A note on Roy’s largest root. Psychometrika, 51(3), 479-481.

l. Note – This indicates
that the F statistic for Wilk’s Lambda was calculated exactly. For the other
test statistics, the F values are approximate (as indicated by the column
heading).

m. Eigenvalues and Canonical Correlations – This section of output
provides the eigenvalues from the product of the sum-of-squares matrix of the model and the sum-of-squares
matrix of the errors. There is
one eigenvalue for each of the three eigenvectors of the product of the model
sum of squares matrix and the error sum of squares matrix, a 3×3 matrix.
Because only two are listed here, we can assume the third eigenvalue is zero.
These values can be used to calculate the four multivariate test
statistics.

n. Univariate F-tests – The manova procedure provides both univariate
and multivariate output. This section of output provides summarized output from
a one-way anova for each of the outcomes in the manova. Each row corresponds to
a different one-way anova, one for each dependent variable in the manova.  While the
manova tested a single hypothesis,
each line in this output corresponds to a test of a different hypothesis.  Generally, if
your manova suggests that an effect is significant, you would expect at least
one of these one-way anova tests to indicate that the effect is significant on a
single outcome.