This page shows an example of multivariate analysis of variance (MANOVA) in
Stata 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 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.

use https://stats.idre.ucla.edu/stat/stata/dae/manova, clear

We can start by examining the three outcome variables. Note that Stata labels
group 1 as the treatment group, group 2 as control_1, and group 3 as
control_2.

summarize useful difficulty importance
    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
      useful |        33     16.3303    3.292461       11.9       24.3
  difficulty |        33    5.715152    2.017598        2.4      10.25
  importance |        33    6.475758    3.985131         .2       18.8

tabulate group, nolabel     

      group |      Freq.     Percent        Cum.
------------+-----------------------------------
  treatment |         11       33.33       33.33
  control_1 |         11       33.33       66.67
  control_2 |         11       33.33      100.00
------------+-----------------------------------
      Total |         33      100.00
      
tabstat difficulty useful importance, by(group) 

Summary statistics: mean
  by categories of: group 

    group |  diffic~y    useful  import~e
----------+------------------------------
treatment |  6.190909  18.11818  8.681818
control_1 |  5.581818  15.52727  5.109091
control_2 |  5.372727  15.34545  5.636364
----------+------------------------------
    Total |  5.715152   16.3303  6.475758
-----------------------------------------

Next, we can enter our MANOVA command.

manova difficulty useful importance = group
                           Number of obs =      33

                           W = Wilks' lambda      L = Lawley-Hotelling trace
                           P = Pillai's trace     R = Roy's largest root

                  Source |  Statistic     df   F(df1,    df2) =   F   Prob>F
              -----------+--------------------------------------------------
                   group | W   0.5258      2     6.0    56.0     3.54 0.0049 e
                         | P   0.4767            6.0    58.0     3.02 0.0122 a
                         | L   0.8972            6.0    54.0     4.04 0.0021 a
                         | R   0.8920            3.0    29.0     8.62 0.0003 u
                         |--------------------------------------------------
                Residual |                30
              -----------+--------------------------------------------------
                   Total |                32
              --------------------------------------------------------------
                           e = exact, a = approximate, u = upper bound on F

As we look at our results, we will want to refer to the eigenvalues of the
sum-of-squares matrix of the model and the sum-of-squares matrix of the error.
These values will be informative in understanding the MANOVA output. To display
the values, we ask Stata to list the matrix of eigenvalues from the model.

matrix list e(eigvals_m)
           c1         c2
r1   .8919879  .00524207

Eigenvalues^a

           c1         c2
r1   .8919879  .00524207

MANOVA Output^b

                           Number of obs =      33

                           W = Wilks' lambdac     L = Lawley-Hotelling tracee
                           P = Pillai's traced    R = Roy's largest rootf

                  Sourceg|  Statistich    dfi  F(df1,    df2) =   Fj  Prob>Fk
              -----------+--------------------------------------------------
                   group | W   0.5258      2     6.0    56.0     3.54 0.0049 e
                         | P   0.4767            6.0    58.0     3.02 0.0122 a
                         | L   0.8972            6.0    54.0     4.04 0.0021 a
                         | R   0.8920            3.0    29.0     8.62 0.0003 u
                         |--------------------------------------------------
                Residual |                30
              -----------+--------------------------------------------------
                   Total |                32
              --------------------------------------------------------------
                           e = exact, a = approximate, u = upper bound on Fl

a. Eigenvalues – These are the eigenvalues
of the product of the sum-of-squares matrix of the model and the sum-of-squares
matrix of the error.  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 eigenvalues are among the saved results of our manova in Stata.
They are used in the calculation of the multivariate test statistics and are
therefore useful to consider when looking at MANOVA output.

b. MANOVA Output – In Stata, MANOVA output includes four multivariate
test statistics for each predictor variable. The four tests are listed above the
output table. For each of the four test statistics, an F statistic and
associated p-value are also displayed.

c. Wilks’ 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 = 0.5258.

d. Pillai’s trace – This is another multivariate test statistic.  To calculate
Pillai’s trace, divide each eigenvalue by 1 + the characteristic root,
then sum these ratios.  So in this example, you would first calculate 0.8919879/(1+0.8919879)
= 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) = 0.4767.

e. Lawley-Hotelling trace – This is very similar to Pillai’s Trace. It is the sum of the roots of the product of the
sum-of-squares matrix of the model and the sum-of-squares matrix of the error
for the two linear regression functions 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 =
0.8972.

f. Roy’s largest root – This is the largest of the roots of the
product of the sum-of-squares matrix of the model and the sum-of-squares matrix
of the error for the two linear regression functions. Because it is a maximum,
it 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 insignificant.

g. Source – This indicates the predictor variable in question. In our model, we are looking at
group as a source of variability in
the ratings.

h. Statistic – This is the test statistic for the given source
listed in the prior column and the multivariate statistic indicated with the
letter (W, P, L or R). For each independent variable, there are four
multivariate test statistics calculated.  See superscripts c, d, e and f.

i. df – This is the number degrees of freedom. Here, our predictor has
three categories and our dataset has 33 observations, so we have 2 degrees of
freedom for the hypothesis, 30 residual degrees of freedom, and 32 total
degrees of freedom.

j. F(df1, df2), F – The first two columns (df1 and df2) list the
degrees of freedom used in determining the F statistics. The third column
lists the F statistic for the given source and multivariate test.

k. Prob > F – This is the p-value associated with the F statistic 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.

l. e = exact, a = approximate, u = upper bound on F – This indicates
how the F statistic was calculated (whether it was an exact calculation, an approximation, or
an upper bound) for each of the multivariate tests.