This page shows an example of multivariate analysis of variance (MANOVA) in
SAS 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.sas7bdat .

We can start by examining the three outcome variables.

data manova; 
  set "C:tempmanova";
run;

proc means data = manova;
  var useful difficulty importance;
run;

The MEANS Procedure

Variable       N            Mean         Std Dev         Minimum         Maximum
USEFUL        33      16.3303030       3.2924615      11.8999996      24.2999992
DIFFICULTY    33       5.7151515       2.0175978       2.4000001      10.2500000
IMPORTANCE    33       6.4757576       3.9851309       0.2000000      18.7999992

proc freq data = manova;
  table group;
run;

The FREQ Procedure

                                  Cumulative    Cumulative
GROUP    Frequency     Percent     Frequency      Percent
    1          11       33.33            11        33.33
    2          11       33.33            22        66.67
    3          11       33.33            33       100.00

proc sort data = manova;
  by group;
run;
proc means data = manova;
  by group;
  var useful difficulty importance;
run;

GROUP=1
The MEANS Procedure
Variable       N            Mean         Std Dev         Minimum         Maximum
USEFUL        11      18.1181817       3.9037974      13.0000000      24.2999992
DIFFICULTY    11       6.1909091       1.8997129       3.7500000      10.2500000
IMPORTANCE    11       8.6818181       4.8630890       3.3000000      18.7999992

GROUP=2
Variable       N            Mean         Std Dev         Minimum         Maximum
USEFUL        11      15.5272729       2.0756162      12.8000002      19.7000008
DIFFICULTY    11       5.5818183       2.4342631       2.4000001       9.8500004
IMPORTANCE    11       5.1090909       2.5311873       0.2000000       8.5000000


GROUP=3
Variable       N            Mean         Std Dev         Minimum         Maximum
USEFUL        11      15.3454545       3.1382682      11.8999996      19.7999992
DIFFICULTY    11       5.3727273       1.7590287       2.6500001       8.7500000
IMPORTANCE    11       5.6363637       3.5469065       0.7000000      10.3000002

Next, we can enter our MANOVA command. In SAS, MANOVA is an option within
proc glm, the generalized linear model procedure. We use the class statement
to indicate our categorical predictor variable group, then specify our model by
listing our outcome variables to the left of the equal sign and our predictor to
the right. We are only interested in type III sum of squares, which we indicate
with the SS3 option. In the manova statement, we indicate that our
hypothesized effect, represented in SAS as h, is group.

proc glm data = manova;
  class group;
  model useful difficulty importance = group / SS3;
  manova h = group;
run;

The GLM Procedure
   Class Level Information
Class         Levels    Values
GROUP              3    1 2 3

Number of Observations Read          33
Number of Observations Used          33

Dependent Variable: USEFUL
                                        Sum of
Source                      DF         Squares     Mean Square    F Value    Pr > F
Model                        2      52.9242378      26.4621189       2.70    0.0835
Error                       30     293.9654425       9.7988481
Corrected Total             32     346.8896803

R-Square     Coeff Var      Root MSE    USEFUL Mean
0.152568      19.16873      3.130311       16.33030

Source                      DF     Type III SS     Mean Square    F Value    Pr > F
GROUP                        2     52.92423783     26.46211891       2.70    0.0835

Dependent Variable: DIFFICULTY
                                        Sum of
Source                      DF         Squares     Mean Square    F Value    Pr > F
Model                        2       3.9751512       1.9875756       0.47    0.6282
Error                       30     126.2872767       4.2095759
Corrected Total             32     130.2624279


R-Square     Coeff Var      Root MSE    DIFFICULTY Mean
0.030516      35.89975      2.051725           5.715152

Source                      DF     Type III SS     Mean Square    F Value    Pr > F
GROUP                        2      3.97515121      1.98757560       0.47    0.6282

Dependent Variable: IMPORTANCE
                                        Sum of
Source                      DF         Squares     Mean Square    F Value    Pr > F
Model                        2      81.8296936      40.9148468       2.88    0.0718
Error                       30     426.3708962      14.2123632
Corrected Total             32     508.2005898

R-Square     Coeff Var      Root MSE    IMPORTANCE Mean
0.161018      58.21603      3.769929           6.475758

Source                      DF     Type III SS     Mean Square    F Value    Pr > F
GROUP                        2     81.82969356     40.91484678       2.88    0.0718

Multivariate Analysis of Variance

        Characteristic Roots and Vectors of: E Inverse * H, where
                    H = Type III SSCP Matrix for GROUP
                          E = Error SSCP Matrix

Characteristic               Characteristic Vector  V'EV=1
          Root    Percent          USEFUL      DIFFICULTY      IMPORTANCE
    0.89198790      99.42      0.06410227     -0.00186162      0.05375069
    0.00524207       0.58      0.01442655      0.06888878     -0.02620577
    0.00000000       0.00     -0.03149580      0.05943387      0.01270798


MANOVA Test Criteria and F Approximations for the Hypothesis of No Overall GROUP Effect
                          H = Type III SSCP Matrix for GROUP
                                 E = Error SSCP Matrix

                                  S=2    M=0    N=13

Statistic                        Value    F Value    Num DF    Den DF    Pr > F
Wilks' Lambda               0.52578838       3.54         6        56    0.0049
Pillai's Trace              0.47667013       3.02         6        58    0.0122
Hotelling-Lawley Trace      0.89722998       4.12         6     35.61    0.0031
Roy's Greatest Root         0.89198790       8.62         3        29    0.0003

         NOTE: F Statistic for Roy's Greatest Root is an upper bound.
                 NOTE: F Statistic for Wilks' Lambda is exact.

Class Level Information

The GLM Procedure

   Class Level Information
Classa        Levelsb  Valuesc
GROUP              3    1 2 3

Number of Observations Read          33
Number of Observations Used          33

a. Class – This is the categorical predictor variable in the MANOVA.

b. Levels – This is the number of possible values of the specified
predictor.  Our predictor in this example has three levels (group = 1,
group = 2 and group = 3).

c. Values – These are the values of the predictor.

Univariate Output^d

Dependent Variablee: USEFUL
                                        Sum of
Sourcef                     DFg        Squaresh    Mean Squarei   F Valuej   Pr > Fk
Model                        2      52.9242378      26.4621189       2.70    0.0835
Error                       30     293.9654425       9.7988481
Corrected Total             32     346.8896803

R-Squarel    Coeff Varm     Root MSEn   USEFUL Meano
0.152568      19.16873      3.130311       16.33030

Source                      DF     Type III SSp    Mean Square    F Value    Pr > F
GROUP                        2     52.92423783     26.46211891       2.70    0.0835

d. Univariate Output – Within MANOVA, SAS provides both univariate and
multivariate
output. The univariate results are presented separately for each dependent variable.
Here, we see the univariate output for useful (the univariate output for
difficulty and importance have been excluded to increase
readability).
Within each set of output for a dependent variable, there are two sets of
results. The first set of results matches a one-way ANOVA using the MANOVA predictor and the single dependent variable.  The second set of results
presents the type III sum of squares results.

e. Dependent Variable – This is one of the dependent variables from
the MANOVA.

f. Source – This is the source of the variability in the specified dependent
variable.

g. DF – This is the degrees of freedom.  Because our predictor,
group, has 3 levels, the degrees of freedom associated with the model is 2.

h. Sum of Squares – These are the model, error, and total sum of squares.
The model sum of squares is the sum of
the squared differences between the predicted values and the mean of the outcome
variable. The error sum of squares is the sum of the squared differences between
the predicted values and the outcome values. The total sum of squares is the sum
of the model and error sums of squares.

i. Mean Square – This is the sum of squares divided by the degrees of freedom (see g and
h).

j. F Value – This is the F statistic associated with the given source.

k. Pr > F – This is the p-value associated with the F statistic of
a given source.  The null hypothesis that the predictor has no effect on
the outcome variable 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.

l. R-Square – This is the proportion of variability in the dependent
variable (useful) that can be explained by the model.
It is the ratio of the model sum of squares to the total sum of squares.

m. Coeff Var – This is the coefficient of variation expressed as a
percent.  The proportion can be calculated as the ratio of the root mean
squared error to the mean of the outcome variable (see n and o), expressed as a
percent. It describes
the amount of variation in the outcome variable.

n. Root MSE – This is the square root of the Mean Square.

o. USEFUL mean – This is the mean value of the dependent
variable.

p. Type III SS – This is a type of sum-of-squares calculation. Here,
we are looking at the sum of squares of the predictor, group. Because our
model consists of just one predictor, the sum of squares of the predictor is the
same as the model sum of squares. Type III sum of squares are calculated for each predictor as if it is the last predictor added to the model. However, in this example, we only have one predictor, and we can see that the Type III sum of squares matches the sum of squares from the ANOVA.

MANOVA Output

Multivariate Analysis of Variance

        Characteristic Roots and Vectors of: E Inverse * H, where
                    H = Type III SSCP Matrix for GROUP
                          E = Error SSCP Matrix

Characteristic               Characteristic Vectorr V'EV=1
          Rootq   Percent          USEFUL      DIFFICULTY      IMPORTANCE
    0.89198790      99.42      0.06410227     -0.00186162      0.05375069
    0.00524207       0.58      0.01442655      0.06888878     -0.02620577
    0.00000000       0.00     -0.03149580      0.05943387      0.01270798


MANOVA Test Criteria and F Approximations for the Hypothesis of No Overall GROUP Effect
                          H = Type III SSCP Matrix for GROUP
                                 E = Error SSCP Matrix

                                  S=2    M=0    N=13s

Statistict                       Value    F Valuey   Num DFz   Den DFaa   Pr > Fab
Wilks' Lambdau              0.52578838       3.54         6        56    0.0049
Pillai's Tracev             0.47667013       3.02         6        58    0.0122
Hotelling-Lawley Tracew     0.89722998       4.12         6     35.61    0.0031
Roy's Greatest Rootx        0.89198790       8.62         3        29    0.0003

         NOTE: F Statistic for Roy's Greatest Root is an upper bound.
                 NOTE: F Statistic for Wilks' Lambda is exact.

q. Characteristic Root –
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 eigenvectors of the product of the model
sum of squares matrix and the error sum of squares matrix, a 3×3 matrix. The
percents listed next to the characteristic roots indicate the amount of
variability in the outcomes a given root and vector account for. In this
example, the first root and vector account for 99.42% of the variability in the
outcomes and the second for .58% of the variability in the outcomes.

r. Characteristic Vector – These are the eigenvectors of the product
of the sum-of-squares matrix of the model and the sum-of-squares matrix of the
error. The three numbers that compose a vector can be read across a row (one
under useful, one under difficulty, and one under importance).

s. S=2 M=0 N=13 – These are intermediate results that are used in computing the
multivariate test statistics and their associated degrees of freedom. If P is the number of
dependent variables, Q is the hypothesis degrees of freedom, and NE is the residual or
error degrees of freedom, then S = min(P, Q), M = .5(abs(P-Q)-1) and N = .5(NE-P-1).

t. Statistic – MANOVA calculates four multivariate test statistics.
All four are based on the characteristic roots (see superscript q). The null
hypothesis for each of these tests is the same: the independent variable (group)
has no effect on any of the dependent variables (useful, difficulty
and importance).

u. 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
characteristic root, calculate 1/(1 + the characteristic root), then find the
product of these ratios.  So in this example, you would first calculate
1/(1+0.89198790) = 0.5285446, 1/(1+0.00524207) = 0.9947853, and 1/(1+0)=1. Then
multiply 0.5285446 * 0.9947853 * 1 = 0.52578838.

v. Pillai’s Trace – This is another one of
the four multivariate test statistics used in MANOVA.  To calculate
Pillai’s trace, divide each characteristic root by 1 + the characteristic root,
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) = 0.47667013.

w. Hotelling-Lawley 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: 0.89198790 + 0.00524207 + 0 =
0.89723.

x. Roy’s Greatest 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. We can see that the value
of Roy’s Greatest Root is the largest of the characteristic roots (see
superscript q). 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 non-significant. For further information on the
calculations underlying MANOVA results, consult SAS online documentation .

y. F Value – This is the F statistic for the given predictor and test
statistic.

z. Num DF –  This is the number of degrees of freedom in the
model.

aa. Den DF – This is the number of degrees of freedom associated with
the model errors.  Note that there are instances in MANOVA when the degrees
of freedom may be a non-integer (here, the DF associated with Hotelling-Lawley
Trace is a non-integer) because these degrees of freedom are calculated using
the mean squared errors, which are often non-integers.

ab. Pr > 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 useful, difficulty or importance scores at alpha level .05 because the p-values are
all less than .05.