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 
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 12
The 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.



Richard Lockhart
Tue Feb 17 22:48:57 PST 1998