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Test the Equality of Independent Covariance Matrices

PURPOSE:

Test equality of two or more independent covariance matrices under the assumption of normality.

REQUIREMENTS:

Base SAS and SAS/STAT, Version 7 or later. This example can be run in previous releases by removing the ODS statements.

DETAILS:

This example uses the POOL=TEST option in PROC DISCRIM to request Bartlett's modification of the likelihood ratio test of the homogeneity of the within-group covariance matrices. The test is unbiased. However, it is not robust to nonnormality.

SEE ALSO:

As described in this Frequently-Asked Question, tests of equal variances can be accomplished in PROC GLM using the HOVTEST= option. Variances can also be modeled simultaneously with the mean. For only two independent groups, a test of equal variances is provided in PROC TTEST.

REFERENCES:

Anderson, T.W. (1984), An Introduction to Multivariate Statistical Analysis, Second Edition, New York: John Wiley & Sons, Inc.

Morrison, D.F. (1976), Multivariate Statistical Methods, New York: McGraw-Hill.

Perlman, M.D. (1980), "Unbiasedness of the Likelihood Ratio Tests for Equality of Several Covariance Matrices and Equality of Several Multivariate Normal Populations," Annals of Statistics, 8, 247 -263.

data iris;
   input sepallen sepalwid petallen petalwid spec_no @@;
   if spec_no=1 then species='Setosa    ';
   else if spec_no=2 then species='Versicolor';
   else species='Virginica ';
   drop spec_no;
   datalines;
50 33 14 02 1 64 28 56 22 3 65 28 46 15 2
67 31 56 24 3 63 28 51 15 3 46 34 14 03 1
69 31 51 23 3 62 22 45 15 2 59 32 48 18 2
46 36 10 02 1 61 30 46 14 2 60 27 51 16 2
65 30 52 20 3 56 25 39 11 2 65 30 55 18 3
58 27 51 19 3 68 32 59 23 3 51 33 17 05 1
57 28 45 13 2 62 34 54 23 3 77 38 67 22 3
63 33 47 16 2 67 33 57 25 3 76 30 66 21 3
49 25 45 17 3 55 35 13 02 1 67 30 52 23 3
70 32 47 14 2 64 32 45 15 2 61 28 40 13 2
48 31 16 02 1 59 30 51 18 3 55 24 38 11 2
63 25 50 19 3 64 32 53 23 3 52 34 14 02 1
49 36 14 01 1 54 30 45 15 2 79 38 64 20 3
44 32 13 02 1 67 33 57 21 3 50 35 16 06 1
58 26 40 12 2 44 30 13 02 1 77 28 67 20 3
63 27 49 18 3 47 32 16 02 1 55 26 44 12 2
50 23 33 10 2 72 32 60 18 3 48 30 14 03 1
51 38 16 02 1 61 30 49 18 3 48 34 19 02 1
50 30 16 02 1 50 32 12 02 1 61 26 56 14 3
64 28 56 21 3 43 30 11 01 1 58 40 12 02 1
51 38 19 04 1 67 31 44 14 2 62 28 48 18 3
49 30 14 02 1 51 35 14 02 1 56 30 45 15 2
58 27 41 10 2 50 34 16 04 1 46 32 14 02 1
60 29 45 15 2 57 26 35 10 2 57 44 15 04 1
50 36 14 02 1 77 30 61 23 3 63 34 56 24 3
58 27 51 19 3 57 29 42 13 2 72 30 58 16 3
54 34 15 04 1 52 41 15 01 1 71 30 59 21 3
64 31 55 18 3 60 30 48 18 3 63 29 56 18 3
49 24 33 10 2 56 27 42 13 2 57 30 42 12 2
55 42 14 02 1 49 31 15 02 1 77 26 69 23 3
60 22 50 15 3 54 39 17 04 1 66 29 46 13 2
52 27 39 14 2 60 34 45 16 2 50 34 15 02 1
44 29 14 02 1 50 20 35 10 2 55 24 37 10 2
58 27 39 12 2 47 32 13 02 1 46 31 15 02 1
69 32 57 23 3 62 29 43 13 2 74 28 61 19 3
59 30 42 15 2 51 34 15 02 1 50 35 13 03 1
56 28 49 20 3 60 22 40 10 2 73 29 63 18 3
67 25 58 18 3 49 31 15 01 1 67 31 47 15 2
63 23 44 13 2 54 37 15 02 1 56 30 41 13 2
63 25 49 15 2 61 28 47 12 2 64 29 43 13 2
51 25 30 11 2 57 28 41 13 2 65 30 58 22 3
69 31 54 21 3 54 39 13 04 1 51 35 14 03 1
72 36 61 25 3 65 32 51 20 3 61 29 47 14 2
56 29 36 13 2 69 31 49 15 2 64 27 53 19 3
68 30 55 21 3 55 25 40 13 2 48 34 16 02 1
48 30 14 01 1 45 23 13 03 1 57 25 50 20 3
57 38 17 03 1 51 38 15 03 1 55 23 40 13 2
66 30 44 14 2 68 28 48 14 2 54 34 17 02 1
51 37 15 04 1 52 35 15 02 1 58 28 51 24 3
67 30 50 17 2 63 33 60 25 3 53 37 15 02 1
;
/* Testing equality of the covariance matrices
===================================================================*/
ods select chisq;
proc discrim pool=test data=iris;
 class species;
 run;


/* Creating a TYPE=COV data set containing the covariance matrices 
===================================================================*/
proc sort data=iris out=sortiris; 
 by species;
 run;
proc corr data=sortiris out=covmxs(type=cov) nocorr cov noprint;
 by species;
 run;
proc print;
 run;

/* Example DATA step that creates a TYPE=COV data set containing the
   covariance matrices above.  This method can be useful if you only
   have the matrices and not the original data.
===================================================================*/
data incovs (type=cov);
input
SPECIES $10. _TYPE_ $   _NAME_ $   SEPALLEN    SEPALWID    PETALLEN    PETALWID;
datalines;
Setosa         COV      SEPALLEN     12.4249      9.9216      1.6355      1.0331
Setosa         COV      SEPALWID      9.9216     14.3690      1.1698      0.9298
Setosa         COV      PETALLEN      1.6355      1.1698      3.0159      0.6069
Setosa         COV      PETALWID      1.0331      0.9298      0.6069      1.1106
Setosa         MEAN         .        50.0600     34.2800     14.6200      2.4600
Setosa         STD          .         3.5249      3.7906      1.7366      1.0539
Setosa         N            .        50.0000     50.0000     50.0000     50.0000
Versicolor     COV      SEPALLEN     26.6433      8.5184     18.2898      5.5780
Versicolor     COV      SEPALWID      8.5184      9.8469      8.2653      4.1204
Versicolor     COV      PETALLEN     18.2898      8.2653     22.0816      7.3102
Versicolor     COV      PETALWID      5.5780      4.1204      7.3102      3.9106
Versicolor     MEAN         .        59.3600     27.7000     42.6000     13.2600
Versicolor     STD          .         5.1617      3.1380      4.6991      1.9775
Versicolor     N            .        50.0000     50.0000     50.0000     50.0000
Virginica      COV      SEPALLEN     40.4343      9.3763     30.3290      4.9094
Virginica      COV      SEPALWID      9.3763     10.4004      7.1380      4.7629
Virginica      COV      PETALLEN     30.3290      7.1380     30.4588      4.8824
Virginica      COV      PETALWID      4.9094      4.7629      4.8824      7.5433
Virginica      MEAN         .        65.8800     29.7400     55.5200     20.2600
Virginica      STD          .         6.3588      3.2250      5.5189      2.7465
Virginica      N            .        50.0000     50.0000     50.0000     50.0000
;

/* Testing equality of the covariance matrices.  The slight numerical
   difference in the chi-square statistic is due to inputting the 
   covariance matrices only to four decimal places.
===================================================================*/
ods select chisq;
proc discrim pool=test data=incovs;
 class species;
 run;

Below is the single table selected by the ODS statement for display. The significant chi-square statistic indicates that the covariance matrices are unequal, assuming normality.

                      The DISCRIM Procedure
        Test of Homogeneity of Within Covariance Matrices

                 Chi-Square        DF    Pr > ChiSq

                 140.943050        20        <.0001

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