One Way MANOVA
This is example 5.9 from a text by Morrison on page 240. There are 4 groups of subjects and each subject was scored on 3 scales called A B and C. The data are in a file called table5.7 in four columns. The first column is group number and the remaining 3 are the A B and C scores respectively.
SAS code to read in the data and print it out:
data long;
infile 'table5.7';
input group a b c;
run;
proc print;
run;
The result is
OBS GROUP A B C
1 1 19 20 18
2 1 20 21 19
3 1 19 22 22
4 1 18 19 21
5 1 16 18 20
6 1 17 22 19
7 1 20 19 20
8 1 15 19 19
9 2 12 14 12
10 2 15 15 17
11 2 15 17 15
12 2 13 14 14
13 2 14 16 13
14 3 15 14 17
15 3 13 14 15
16 3 12 15 15
17 3 12 13 13
18 4 8 9 10
19 4 10 10 12
20 4 11 10 10
21 4 11 7 12
A one way MANOVA, testing the hypothesis of no group effect is
proc glm;
class group;
model a b c = group;
manova h=group ;
run;
which produces the output
General Linear Models Procedure
Dependent Variable: A
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 190.438095 63.479365 25.21 0.0001
Error 17 42.800000 2.517647
Corrected Total 20 233.238095
R-Square C.V. Root MSE A Mean
0.816497 10.92489 1.58671 14.5238
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 190.438095 63.479365 25.21 0.0001
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 190.438095 63.479365 25.21 0.0001
Dependent Variable: B
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 340.152381 113.384127 62.58 0.0001
Error 17 30.800000 1.811765
Corrected Total 20 370.952381
R-Square C.V. Root MSE B Mean
0.916970 8.617799 1.34602 15.6190
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 340.152381 113.384127 62.58 0.0001
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 340.152381 113.384127 62.58 0.0001
Dependent Variable: C
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 232.271429 77.423810 34.37 0.0001
Error 17 38.300000 2.252941
Corrected Total 20 270.571429
R-Square C.V. Root MSE C Mean
0.858448 9.465640 1.50098 15.8571
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 232.271429 77.423810 34.37 0.0001
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 232.271429 77.423810 34.37 0.0001
General Linear Models Procedure
Multivariate Analysis of Variance
Characteristic Roots and Vectors of: E Inverse * H, where
H = Type III SS&CP Matrix for GROUP E = Error SS&CP Matrix
Characteristic Percent Characteristic Vector V'EV=1
Root
A B
C
15.3752900 98.30 0.01127992 0.13817484
0.08125989
0.2307260 1.48 -0.04455525 -0.09323432
0.15450684
0.0356937 0.23 -0.17287851 0.09020041
0.04776837
Manova Test Criteria and F Approximations for
the Hypothesis of no Overall GROUP Effect
H = Type III SS&CP Matrix for GROUP E = Error SS&CP Matrix
S=3 M=-0.5 N=6.5
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.04790913 10.1213 9 36.65666 0.0001
Pillai's Trace 1.16086747 3.5768 9 51 0.0016
Hotelling-Lawley Trace 15.64170973 23.7522 9 41 0.0001
Roy's Greatest Root 15.37528995 87.1266 3 17 0.0001
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
The output shows univariate analysis for each of the 3 response variables and
then the one way MANOVA analysis. The data set is actually an unbalanced design.
Here is what happens if we analyze only the data for the first two groups.
data long;
infile 'tab57sh';
input group a b c;
run;
proc print;
run;
proc glm;
class group;
model a b c = group;
manova h=group / printh printe;
run;
The result is
The SAS System 52
08:02 Wednesday, January 28, 1998
OBS GROUP A B C
1 1 19 20 18
2 1 20 21 19
3 1 19 22 22
4 1 18 19 21
5 1 16 18 20
6 1 17 22 19
7 1 20 19 20
8 1 15 19 19
9 2 12 14 12
10 2 15 15 17
11 2 15 17 15
12 2 13 14 14
13 2 14 16 13
General Linear Models Procedure
Dependent Variable: A
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 54.2769231 54.2769231 19.38 0.0011
Error 11 30.8000000 2.8000000
Corrected Total 12 85.0769231
R-Square C.V. Root MSE A Mean
0.637975 10.21275 1.67332 16.3846
Dependent Variable: B
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 70.8923077 70.8923077 34.20 0.0001
Error 11 22.8000000 2.0727273
Corrected Total 12 93.6923077
R-Square C.V. Root MSE B Mean
0.756650 7.930534 1.43970 18.1538
Dependent Variable: C
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 94.7769231 94.7769231 39.64 0.0001
Error 11 26.3000000 2.3909091
Corrected Total 12 121.0769231
R-Square C.V. Root MSE C Mean
0.782783 8.777875 1.54626 17.6154
E = Error SS&CP Matrix
A B C
A 30.8 12.2 10.2
B 12.2 22.8 3.8
C 10.2 3.8 26.3
General Linear Models Procedure
Multivariate Analysis of Variance
Partial Correlation Coefficients from the Error SS&CP Matrix / Prob > |r|
DF = 11 A B C
A 1.000000 0.460381 0.358383
0.0001 0.1320 0.2527
B 0.460381 1.000000 0.155181
0.1320 0.0001 0.6301
C 0.358383 0.155181 1.000000
0.2527 0.6301 0.0001
H = Type III SS&CP Matrix for GROUP
A B C
A 54.276923077 62.030769231 71.723076923
B 62.030769231 70.892307692 81.969230769
C 71.723076923 81.969230769 94.776923077
Characteristic Roots and Vectors of: E Inverse * H, where
H = Type III SS&CP Matrix for GROUP E = Error SS&CP Matrix
Characteristic Percent Characteristic Vector V'EV=1
Root
A B
C
5.81615862 100.00 0.00403434 0.12874606
0.13332232
The SAS System 62
08:02 Wednesday, January 28, 1998
General Linear Models Procedure
Multivariate Analysis of Variance
Characteristic Roots and Vectors of: E Inverse * H, where
H = Type III SS&CP Matrix for GROUP E = Error SS&CP Matrix
Characteristic Percent Characteristic Vector V'EV=1
Root
A B
C
0.00000000 0.00 -0.09464169 -0.10311602
0.16080216
0.00000000 0.00 -0.19278508 0.16868694
0.00000000
General Linear Models Procedure
Multivariate Analysis of Variance
Manova Test Criteria and Exact F Statistics for
the Hypothesis of no Overall GROUP Effect
H = Type III SS&CP Matrix for GROUP E = Error SS&CP Matrix
S=1 M=0.5 N=3.5
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.14671020 17.4485 3 9 0.0004
Pillai's Trace 0.85328980 17.4485 3 9 0.0004
Hotelling-Lawley Trace 5.81615862 17.4485 3 9 0.0004
Roy's Greatest Root 5.81615862 17.4485 3 9 0.0004
General Linear Models Procedure
Multivariate Analysis of Variance
Characteristic Roots and Vectors of: E Inverse * H, where
H = Type III SS&CP Matrix for GROUP E = Error SS&CP Matrix
Characteristic Percent Characteristic Vector V'EV=1
Root
A B
C
5.81615862 100.00 0.00403434 0.12874606
0.13332232
0.00000000 0.00 -0.09464169 -0.10311602
0.16080216
0.00000000 0.00 -0.19278508 0.16868694
0.00000000
Manova Test Criteria and Exact F Statistics for
the Hypothesis of no Overall GROUP Effect
H = Type III SS&CP Matrix for GROUP E = Error SS&CP Matrix
S=1 M=0.5 N=3.5
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.14671020 17.4485 3 9 0.0004
Pillai's Trace 0.85328980 17.4485 3 9 0.0004
Hotelling-Lawley Trace 5.81615862 17.4485 3 9 0.0004
Roy's Greatest Root 5.81615862 17.4485 3 9 0.0004
The quantities labelled "Characteristic Root" are eigenvalues; only one is non-zero in
this problem. The various test statistics all lead to the same F value and the
conclusion that groups 1 and 2 have different multivariate means.
Now we examine univariate approaches to the data. Here we tell SAS that the variables A, B and C are repeated measurements of the same quantity.
proc glm;
class group;
model a b c = group;
repeated scale;
run;
These statements actually do two analyses. The class group statement says that the
numbers 1 to 4 in the column called group are levels of a categorical
factor.
The model statement says that there are 3 response variables and 1 independent variable. It does multivariate analysis of variance tests for the hypotheses of no scale by group interaction (the hypothesis of parallelism) and for the hypothesis of no scale effect. It does a univariate test of no group effect by adding together the 3 scales to get a single total score for each individual.
The results are as follows:
Linear Models Procedure
Class Level Information
Class Levels Values
GROUP 4 1 2 3 4
Number of observations in data set = 21
Dependent Variable: A
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 190.4380952 63.4793651 25.21 0.0001
Error 17 42.8000000 2.5176471
Corrected Total 20 233.2380952
R-Square C.V. Root MSE A Mean
0.816497 10.92489 1.586710 14.52381
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 190.4380952 63.4793651 25.21 0.0001
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 190.4380952 63.4793651 25.21 0.0001
Dependent Variable: B
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 340.1523810 113.3841270 62.58 0.0001
Error 17 30.8000000 1.8117647
Corrected Total 20 370.9523810
R-Square C.V. Root MSE B Mean
0.916970 8.617799 1.346018 15.61905
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 340.1523810 113.3841270 62.58 0.0001
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 340.1523810 113.3841270 62.58 0.0001
Dependent Variable: C
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 232.2714286 77.4238095 34.37 0.0001
Error 17 38.3000000 2.2529412
Corrected Total 20 270.5714286
R-Square C.V. Root MSE C Mean
0.858448 9.465640 1.500980 15.85714
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 232.2714286 77.4238095 34.37 0.0001
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 232.2714286 77.4238095 34.37 0.0001
General Linear Models Procedure
Repeated Measures Analysis of Variance
Repeated Measures Level Information
Dependent Variable A B C
Level of SCALE 1 2 3
Manova Test Criteria and Exact F Statistics for the Hypothesis of no
SCALE Effect
H = Type III SS&CP Matrix for SCALE E = Error SS&CP Matrix
S=1 M=0 N=7
Statistic
Value F Num DF Den DF Pr > F
Wilks' Lambda 0.56372926 6.1912 2 16 0.0102
Pillai's Trace 0.43627074 6.1912 2 16 0.0102
Hotelling-Lawley Trace 0.77390120 6.1912 2 16 0.0102
Roy's Greatest Root 0.77390120 6.1912 2 16 0.0102
Manova Test Criteria and F Approximations for the Hypothesis of no
SCALE*GROUP Effect
H = Type III SS&CP Matrix for SCALE*GROUP E = Error SS&CP Matrix
S=2 M=0 N=7
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.56333190 1.7725 6 32 0.1364
Pillai's Trace 0.48726039 1.8253 6 34 0.1234
Hotelling-Lawley Trace 0.68534343 1.7134 6 30 0.1522
Roy's Greatest Root 0.50884896 2.8835 3 17 0.0662
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.
The repeated statement causes sas to do a univariate anova based analysis
for the model. The results are
General Linear Models Procedure
Repeated Measures Analysis of Variance
Tests of Hypotheses for Between Subjects Effects
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 743.900000 247.966667 70.93 0.0001
Error 17 59.433333 3.496078
General Linear Models Procedure
Repeated Measures Analysis of Variance
Univariate Tests of Hypotheses for Within Subject Effects
Source: SCALE Adj Pr > F
DF Type III SS Mean Square F Value Pr > F G - G H - F
2 16.62424242 8.31212121 5.39 0.0093 0.0101 0.0093 ÿ
Source: SCALE*GROUP
Adj Pr > F
DF Type III SS Mean Square F Value Pr > F G - G H - F
6 18.96190476 3.16031746 2.05 0.0860 0.0889 0.0860
Source: Error(SCALE)
DF Type III SS Mean Square
34 52.46666667 1.54313725
Greenhouse-Geisser Epsilon = 0.9664
Huynh-Feldt Epsilon = 1.2806 The following two files do univariate anova:
The data reordered:
1 1 1 19 1 1 2 20 1 1 3 18 2 1 1 20 2 1 2 21 2 1 3 19 3 1 1 19 3 1 2 22 3 1 3 22 4 1 1 18 4 1 2 19 4 1 3 21 5 1 1 16 5 1 2 18 5 1 3 20 6 1 1 17 6 1 2 22 6 1 3 19 7 1 1 20 7 1 2 19 7 1 3 20 8 1 1 15 8 1 2 19 8 1 3 19 1 2 1 12 1 2 2 14 1 2 3 12 2 2 1 15 2 2 2 15 2 2 3 17 3 2 1 15 3 2 2 17 3 2 3 15 4 2 1 13 4 2 2 14 4 2 3 14 5 2 1 14 5 2 2 16 5 2 3 13 1 3 1 15 1 3 2 14 1 3 3 17 2 3 1 13 2 3 2 14 2 3 3 15 3 3 1 12 3 3 2 15 3 3 3 15 4 3 1 12 4 3 2 13 4 3 3 13 1 4 1 8 1 4 2 9 1 4 3 10 2 4 1 10 2 4 2 10 2 4 3 12 3 4 1 11 3 4 2 10 3 4 3 10 4 4 1 11 4 4 2 7 4 4 3 12The sas commands:
data long;
infile 'table5.7uni';
input subject group scale score;
run;
proc print;
run;
proc glm;
class group;
class scale;
class subject;
model score =group subject(group) scale group*scale;
random subject(group) ;
run;
Some of the output:
General Linear Models Procedure
Dependent Variable: SCORE
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 28 843.5333333 30.1261905 19.52 0.0001
Error 34 52.4666667 1.5431373
Corrected Total 62 896.0000000
R-Square C.V. Root MSE SCORE Mean
0.941443 8.101505 1.242231 15.33333
Source DF Type I SS Mean Square F Value Pr > F
GROUP 3 743.9000000 247.9666667 160.69 0.0001
SUBJECT(GROUP) 17 59.4333333 3.4960784 2.27 0.0208
SCALE 2 21.2380952 10.6190476 6.88 0.0031
GROUP*SCALE 6 18.9619048 3.1603175 2.05 0.0860
Source DF Type III SS Mean Square F Value Pr > F
GROUP 3 743.9000000 247.9666667 160.69 0.0001
SUBJECT(GROUP) 17 59.4333333 3.4960784 2.27 0.0208
SCALE 2 16.6242424 8.3121212 5.39 0.0093
GROUP*SCALE 6 18.9619048 3.1603175 2.05 0.0860
General Linear Models Procedure
Source Type III Expected Mean Square
GROUP Var(Error) + 3 Var(SUBJECT(GROUP)) + Q(GROUP,GROUP*SCALE)
SUBJECT(GROUP) Var(Error) + 3 Var(SUBJECT(GROUP))
SCALE Var(Error) + Q(SCALE,GROUP*SCALE)
GROUP*SCALE Var(Error) + Q(GROUP*SCALE)
The difference between the types of error sums of squares arises because the design is unbalanced. Notice that the hypothesis of no group by scale interaction is acceptable. Under the assumption of no such group by scale interaction the hypothesis of no group effect is tested by dividing group ms by subject(group) ms for which the value is 70.9 on 3,17 degrees of freedom. This is NOT the F value in the table above since the table above is for FIXED effects. Notice that the sums of squares in this table match those produced in the repeated measures ANOVA. This is not accidental.