Version info: Code for this page was tested in Stata 12.

MANOVA is used to model two or more dependent variables that are
continuous with one or more categorical predictor variables.

Please note: The purpose of this page is to show how to use various data
analysis commands.  It does not cover all aspects of the research process which
researchers are expected to do.  In particular, it does not cover data
cleaning and checking, verification of assumptions, model diagnostics or
potential follow-up analyses.

Examples of one-way multivariate analysis of variance

Example 1.

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.  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.

Example 2.  A clinical psychologist recruits 100 people who suffer from
panic disorder into his study.  Each subject receives one of four types of
treatment for eight weeks.  At the end of treatment, each subject
participates in a structured interview, during which the clinical psychologist
makes three ratings:  physiological, emotional and cognitive.  The
clinical psychologist wants to know which type of treatment most reduces the
symptoms of the panic disorder as measured on the physiological, emotional and
cognitive scales.  (This example was adapted from Grimm and Yarnold, 1995,
page 246.)

Description of the data

Let’s pursue Example 1 from above.

We have a data file, manova.dta, with 33 observations on three response
variables.  The response variables are ratings called useful, difficulty and importance.  Level 1 of the group variable is the treatment group, level 2 is control group 1 and
level 3 is control group 2.

Let’s look at the data.  It is always a good idea to start with descriptive
statistics.

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

 summarize difficulty useful 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

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

correlate useful difficulty importance
(obs=33)

             |   useful diffic~y import~e
-------------+---------------------------
      useful |   1.0000
  difficulty |   0.0978   1.0000
  importance |  -0.3411   0.1978   1.0000

Analysis methods you might consider

Below is a list of some analysis methods you may have
encountered.  Some of the methods listed are quite reasonable, while others
have either fallen out of favor or have limitations.

One-way MANOVA

We will start by running the 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

Stata provides four multivariate tests by default.  Each of these tests
is statistically significant.  For more information on these tests, please
see our Stata Annotated Output: MANOVA
page.

The overall multivariate test is significant, which means that differences
between the levels of the variable group exist.  To find where the
differences lie, we
will follow up with several post-hoc tests.  We will begin with the multivariate test of group 1 versus the
average of groups 2 and 3.  First, we will use the
manova, showorder command to determine the order of the elements in the
design matrix.  Knowing the order of the elements in the design matrix is
necessary to run the post-hoc tests.  (Note that the order of the elements
in the design matrix changed in Stata 11.)

manovatest, showorder

 Order of columns in the design matrix
      1: (group==1)
      2: (group==2)
      3: (group==3)
      4: _cons

We will begin by comparing the treatment group (group 1) to an average of the
control groups (groups 2 and 3).  This tests the hypothesis that the mean
control groups equals the treatment group.  The
output above indicates that the fourth element in the matrix is the constant, so
in the matrix command below, we will set it to 0.  Once we have
created a matrix (which we call c1), we can use the manovatest
command to test c1.

matrix c1=(2,-1,-1,0)
manovatest, test(c1)

 Test constraint
 (1)    2*1.group - 2.group - 3.group = 0

                           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
              -----------+--------------------------------------------------
              manovatest | W   0.5290      1     3.0    28.0     8.31 0.0004 e
                         | P   0.4710            3.0    28.0     8.31 0.0004 e
                         | L   0.8904            3.0    28.0     8.31 0.0004 e
                         | R   0.8904            3.0    28.0     8.31 0.0004 e
                         |--------------------------------------------------
                Residual |                30
              --------------------------------------------------------------
                           e = exact, a = approximate, u = upper bound on F

These results indicate that group 1 is statistically significantly different
from the average of groups 2 and 3.

Now we will compare control group 1 (group 2) to control group 2 (group 3).  Again, we need to create a
matrix (called c2 in this example) to do this comparison, and then use
that matrix in the manovatest command.

matrix c2=(0,1,-1,0)
manovatest, test(c2)

 Test constraint
 (1)    2.group - 3.group = 0

                           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
              -----------+--------------------------------------------------
              manovatest | W   0.9932      1     3.0    28.0     0.06 0.9785 e
                         | P   0.0068            3.0    28.0     0.06 0.9785 e
                         | L   0.0068            3.0    28.0     0.06 0.9785 e
                         | R   0.0068            3.0    28.0     0.06 0.9785 e
                         |--------------------------------------------------
                Residual |                30
              --------------------------------------------------------------
                           e = exact, a = approximate, u = upper bound on F

The results indicate that control group 1 is not statistically significantly
different from control group 2.

We can use the margins command to obtain adjusted predicted values for
each of the groups.  In the first example below, we get the predicted means
for the dependent variable difficulty.  In the next two examples, we
get the predicted means for the dependent variables useful and
importance.  These values can be helpful in seeing where differences
between levels of the predictor variable are and describing the model.

margins group, predict(equation(difficulty))

Adjusted predictions                              Number of obs   =         33

Expression   : Linear prediction:  difficulty, predict(equation(difficulty))

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       group |
          1  |   6.190909   .6186184    10.01   0.000     4.978439    7.403379
          2  |   5.581818   .6186184     9.02   0.000     4.369349    6.794288
          3  |   5.372727   .6186184     8.69   0.000     4.160257    6.585197
------------------------------------------------------------------------------

margins group, predict(equation(useful))

Adjusted predictions                              Number of obs   =         33

Expression   : Linear prediction:  useful, predict(equation(useful))

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       group |
          1  |   18.11818   .9438243    19.20   0.000     16.26832    19.96804
          2  |   15.52727   .9438243    16.45   0.000     13.67741    17.37713
          3  |   15.34545   .9438243    16.26   0.000     13.49559    17.19532
------------------------------------------------------------------------------

margins group, predict(equation(importance))

Adjusted predictions                              Number of obs   =         33

Expression   : Linear prediction:  importance, predict(equation(importance))

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       group |
          1  |   8.681818   1.136676     7.64   0.000     6.453973    10.90966
          2  |   5.109091   1.136676     4.49   0.000     2.881246    7.336936
          3  |   5.636364   1.136676     4.96   0.000     3.408519    7.864208
------------------------------------------------------------------------------

In each of the three outputs above, we see that the predicted means for
groups 2 and 3 are very similar; the predicted mean for group 1 is higher than
those for groups 2 and 3.

In the examples below, we obtain the differences in the means for each of the
dependent variables for each of the control groups (groups 2 and 3) compared to
the treatment group (group1).  With respect to the dependent variable
difficulty, the difference between the means for control group 1 versus the
treatment group is approximately -0.61 (5.58 – 6.19).  The difference
between the means for control group 2 versus the treatment group is
approximately -0.82 (5.37 – 6.19). 

margins, dydx(group) predict(equation(difficulty))

Conditional marginal effects                      Number of obs   =         33

Expression   : Linear prediction:  difficulty, predict(equation(difficulty))
dy/dx w.r.t. : 2.group 3.group

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       group |
          2  |  -.6090908   .8748585    -0.70   0.486    -2.323782      1.1056
          3  |  -.8181818   .8748585    -0.94   0.350    -2.532873    .8965094
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

margins, dydx(group) predict(equation(useful))

Conditional marginal effects                      Number of obs   =         33

Expression   : Linear prediction:  useful, predict(equation(useful))
dy/dx w.r.t. : 2.group 3.group

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       group |
          2  |  -2.590909   1.334769    -1.94   0.052    -5.207008    .0251907
          3  |  -2.772727   1.334769    -2.08   0.038    -5.388827   -.1566278
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

margins, dydx(group) predict(equation(importance))

Conditional marginal effects                      Number of obs   =         33

Expression   : Linear prediction:  importance, predict(equation(importance))
dy/dx w.r.t. : 2.group 3.group

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       group |
          2  |  -3.572727   1.607503    -2.22   0.026    -6.723375   -.4220792
          3  |  -3.045454   1.607503    -1.89   0.058    -6.196103    .1051936
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

Finally, let’s run separate univariate
ANOVAs.  We will use a foreach loop to run the ANOVA for each
dependent variable.

foreach vname in difficulty useful importance {
  anova `vname' group
}
              /* useful */
                           Number of obs =      33     R-squared     =  0.1526
                           Root MSE      = 3.13031     Adj R-squared =  0.0961

                  Source |  Partial SS    df       MS           F     Prob > F
              -----------+----------------------------------------------------
                   Model |  52.9242378     2  26.4621189       2.70     0.0835
                         |
                   group |  52.9242378     2  26.4621189       2.70     0.0835
                         |
                Residual |  293.965442    30  9.79884808   
              -----------+----------------------------------------------------
                   Total |   346.88968    32  10.8403025   
                   
              /* difficulty */
                           Number of obs =      33     R-squared     =  0.0305
                           Root MSE      = 2.05173     Adj R-squared = -0.0341

                  Source |  Partial SS    df       MS           F     Prob > F
              -----------+----------------------------------------------------
                   Model |  3.97515121     2   1.9875756       0.47     0.6282
                         |
                   group |  3.97515121     2   1.9875756       0.47     0.6282
                         |
                Residual |  126.287277    30  4.20957589   
              -----------+----------------------------------------------------
                   Total |  130.262428    32  4.07070087   
                   
              /* importance */
                           Number of obs =      33     R-squared     =  0.1610
                           Root MSE      = 3.76993     Adj R-squared =  0.1051

                  Source |  Partial SS    df       MS           F     Prob > F
              -----------+----------------------------------------------------
                   Model |  81.8296936     2  40.9148468       2.88     0.0718
                         |
                   group |  81.8296936     2  40.9148468       2.88     0.0718
                         |
                Residual |  426.370896    30  14.2123632   
              -----------+----------------------------------------------------
                   Total |   508.20059    32  15.8812684   

While none of the three ANOVAs were statistically significant at the alpha = .05 level,
in particular, the F-ratio for difficulty was less than 1.

Things to consider

See also

References