Fisher’s iris data are widely used in multivariate statistics. They comprise measurements in millimeters of four flower attributes, the length and width of sepals and petals for 50 specimens from each of three species, Iris setosa, I. versicolor, and I. virginica (Fisher, 1936).
When modeling multiple attributes from the same specimen, correlations among measurements from the same flower must be taken into account. Unstructured covariance matrices are common in this multivariate setting. Species comparisons can focus on comparisons of mean response, but comparisons of the variation and covariation are also of interest. In this example, the equivalence of covariance and correlation matrices among the species are examined.
The iris
data set is available in the Sashelp
library. The following step displays the first 10 observations of the iris
data in multivariate format—that is, each observation contains multiple response variables. The DATA step that follows creates
a data set in univariate form, where each observation corresponds to a single response variable. This is the form needed by
the GLIMMIX procedure.
proc print data=Sashelp.iris(obs=10); run;
Output 41.9.1: Fisher (1936) Iris Data
Obs  Species  SepalLength  SepalWidth  PetalLength  PetalWidth 

1  Setosa  50  33  14  2 
2  Setosa  46  34  14  3 
3  Setosa  46  36  10  2 
4  Setosa  51  33  17  5 
5  Setosa  55  35  13  2 
6  Setosa  48  31  16  2 
7  Setosa  52  34  14  2 
8  Setosa  49  36  14  1 
9  Setosa  44  32  13  2 
10  Setosa  50  35  16  6 
data iris_univ; set sashelp.iris; retain id 0; array y (4) SepalLength SepalWidth PetalLength PetalWidth; id+1; do var=1 to 4; response = y{var}; output; end; drop SepalLength SepalWidth PetalLength PetalWidth:; run;
The following GLIMMIX statements fit a model with separate unstructured covariance matrices for each species:
ods select FitStatistics CovParms CovTests; proc glimmix data=iris_univ; class species var id; model response = species*var; random _residual_ / type=un group=species subject=id; covtest homogeneity; run;
The mean function is modeled as a cellmeans model that allows for different means for each species and outcome variable.
The covariances are modeled directly (Rside) rather than through random effects. The ID
variable identifies the individual plant, so that responses from different plants are independent. The GROUP=SPECIES option varies the parameters of the unstructured covariance matrix by species. Hence, this model has 30 covariance
parameters: 10 unique parameters for a covariance matrix for each of three species.
The COVTEST statement requests a test of homogeneity—that is, it tests whether varying the covariance parameters by the group effect provides a significantly better fit compared to a model in which different groups share the same parameter.
Output 41.9.2: Fit Statistics for Analysis of Fisher’s Iris Data
Fit Statistics  

2 Res Log Likelihood  2812.89 
AIC (smaller is better)  2872.89 
AICC (smaller is better)  2876.23 
BIC (smaller is better)  2963.21 
CAIC (smaller is better)  2993.21 
HQIC (smaller is better)  2909.58 
Generalized ChiSquare  588.00 
Gener. ChiSquare / DF  1.00 
The “Fit Statistics” table shows the –2 restricted (residual) log likelihood in the full model and other fit statistics (Output 41.9.2). The “2 Res Log Likelihood” sets the benchmark against which a model with homogeneity constraint is compared. Output 41.9.3 displays the 30 covariance parameters in this model.
There appear to be substantial differences among the covariance parameters from different groups. For example, the residual variability of the petal length of the three species is 12.4249, 26.6433, and 40.4343, respectively. The homogeneity hypothesis restricts these variances to be equal and similarly for the other covariance parameters. The results from the COVTEST statement are shown in Output 41.9.4.
Output 41.9.3: Covariance Parameters Varied by Species (TYPE=UN)
Covariance Parameter Estimates  

Cov Parm  Subject  Group  Estimate  Standard Error 
UN(1,1)  id  Species Setosa  12.4249  2.5102 
UN(2,1)  id  Species Setosa  9.9216  2.3775 
UN(2,2)  id  Species Setosa  14.3690  2.9030 
UN(3,1)  id  Species Setosa  1.6355  0.9052 
UN(3,2)  id  Species Setosa  1.1698  0.9552 
UN(3,3)  id  Species Setosa  3.0159  0.6093 
UN(4,1)  id  Species Setosa  1.0331  0.5508 
UN(4,2)  id  Species Setosa  0.9298  0.5859 
UN(4,3)  id  Species Setosa  0.6069  0.2755 
UN(4,4)  id  Species Setosa  1.1106  0.2244 
UN(1,1)  id  Species Versicolor  26.6433  5.3828 
UN(2,1)  id  Species Versicolor  8.5184  2.6144 
UN(2,2)  id  Species Versicolor  9.8469  1.9894 
UN(3,1)  id  Species Versicolor  18.2898  4.3398 
UN(3,2)  id  Species Versicolor  8.2653  2.4149 
UN(3,3)  id  Species Versicolor  22.0816  4.4612 
UN(4,1)  id  Species Versicolor  5.5780  1.6617 
UN(4,2)  id  Species Versicolor  4.1204  1.0641 
UN(4,3)  id  Species Versicolor  7.3102  1.6891 
UN(4,4)  id  Species Versicolor  3.9106  0.7901 
UN(1,1)  id  Species Virginica  40.4343  8.1690 
UN(2,1)  id  Species Virginica  9.3763  3.2213 
UN(2,2)  id  Species Virginica  10.4004  2.1012 
UN(3,1)  id  Species Virginica  30.3290  6.6262 
UN(3,2)  id  Species Virginica  7.1380  2.7395 
UN(3,3)  id  Species Virginica  30.4588  6.1536 
UN(4,1)  id  Species Virginica  4.9094  2.5916 
UN(4,2)  id  Species Virginica  4.7629  1.4367 
UN(4,3)  id  Species Virginica  4.8824  2.2750 
UN(4,4)  id  Species Virginica  7.5433  1.5240 
Output 41.9.4: Likelihood Ratio Test of Homogeneity
Tests of Covariance Parameters Based on the Restricted Likelihood 


Label  DF  2 Res Log Like  ChiSq  Pr > ChiSq  Note 
Homogeneity  20  2959.55  146.66  <.0001  DF 
Denote as the covariance matrix for species with elements . In processing the COVTEST hypothesis , the GLIMMIX procedure fits a model that satisfies the constraints










where is the covariance between the ith and jth variable for the kth species. The –2 restricted log likelihood of this restricted model is 2959.55 (Output 41.9.4). The change of 146.66 compared to the full model is highly significant. There is sufficient evidence to reject the notion of equal covariance matrices among the three iris species.
Equality of covariance matrices implies equality of correlation matrices, but the reverse is not true. Fewer constraints are needed to equate correlations because the diagonal entries of the covariance matrices are free to vary. In order to test the equality of the correlation matrices among the three species, you can parameterize the unstructured covariance matrix in terms of the correlations and use a COVTEST statement with general contrasts, as shown in the following statements:
ods select FitStatistics CovParms CovTests; proc glimmix data=iris_univ; class species var id; model response = species*var; random _residual_ / type=unr group=species subject=id; covtest 'Equal Covariance Matrices' homogeneity; covtest 'Equal Correlation Matrices' general 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0, 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0, 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0, 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0, 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0, 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0, 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0, 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0, 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0, 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0, 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1, 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 / estimates; run;
The TYPE=UNR structure is a reparameterization of TYPE=UN. The models provide the same fit, as seen by comparison of the “Fit Statistics” tables in Output 41.9.2 and Output 41.9.5. The covariance parameters are ordered differently, however. In each group, the four variances precede the six correlations (Output 41.9.5). The first COVTEST statement tests the homogeneity hypothesis in terms of the UNR parameterization, and the result is identical to the test in Output 41.9.4. The second COVTEST statement restricts the correlations to be equal across groups. If is the correlation between the ith and jth variable for the kth species, the 12 restrictions are












The ESTIMATES option in the COVTEST statement requests that the GLIMMIX procedure display the covariance parameter estimates in the restricted model (Output 41.9.5).
Output 41.9.5: Fit Statistics, Covariance Parameters (TYPE=UNR), and Likelihood Ratio Tests for Equality of Covariance and Correlation Matrices
Fit Statistics  

2 Res Log Likelihood  2812.89 
AIC (smaller is better)  2872.89 
AICC (smaller is better)  2876.23 
BIC (smaller is better)  2963.21 
CAIC (smaller is better)  2993.21 
HQIC (smaller is better)  2909.58 
Generalized ChiSquare  588.00 
Gener. ChiSquare / DF  1.00 
Covariance Parameter Estimates  

Cov Parm  Subject  Group  Estimate  Standard Error 
Var(1)  id  Species Setosa  12.4249  2.5102 
Var(2)  id  Species Setosa  14.3690  2.9030 
Var(3)  id  Species Setosa  3.0159  0.6093 
Var(4)  id  Species Setosa  1.1106  0.2244 
Corr(2,1)  id  Species Setosa  0.7425  0.06409 
Corr(3,1)  id  Species Setosa  0.2672  0.1327 
Corr(3,2)  id  Species Setosa  0.1777  0.1383 
Corr(4,1)  id  Species Setosa  0.2781  0.1318 
Corr(4,2)  id  Species Setosa  0.2328  0.1351 
Corr(4,3)  id  Species Setosa  0.3316  0.1271 
Var(1)  id  Species Versicolor  26.6433  5.3828 
Var(2)  id  Species Versicolor  9.8469  1.9894 
Var(3)  id  Species Versicolor  22.0816  4.4612 
Var(4)  id  Species Versicolor  3.9106  0.7901 
Corr(2,1)  id  Species Versicolor  0.5259  0.1033 
Corr(3,1)  id  Species Versicolor  0.7540  0.06163 
Corr(3,2)  id  Species Versicolor  0.5605  0.09797 
Corr(4,1)  id  Species Versicolor  0.5465  0.1002 
Corr(4,2)  id  Species Versicolor  0.6640  0.07987 
Corr(4,3)  id  Species Versicolor  0.7867  0.05445 
Var(1)  id  Species Virginica  40.4343  8.1690 
Var(2)  id  Species Virginica  10.4004  2.1012 
Var(3)  id  Species Virginica  30.4588  6.1536 
Var(4)  id  Species Virginica  7.5433  1.5240 
Corr(2,1)  id  Species Virginica  0.4572  0.1130 
Corr(3,1)  id  Species Virginica  0.8642  0.03616 
Corr(3,2)  id  Species Virginica  0.4010  0.1199 
Corr(4,1)  id  Species Virginica  0.2811  0.1316 
Corr(4,2)  id  Species Virginica  0.5377  0.1015 
Corr(4,3)  id  Species Virginica  0.3221  0.1280 
Tests of Covariance Parameters Based on the Restricted Likelihood 


Label  DF  2 Res Log Like  ChiSq  Pr > ChiSq  Estimates H0  Note  
Est1  Est2  Est3  Est4  Est5  Est6  Est7  Est8  Est9  Est10  Est11  Est12  Est13  Est14  Est15  Est16  Est17  Est18  Est19  Est20  Est21  Est22  Est23  Est24  Est25  Est26  Est27  Est28  Est29  Est30  
Equal Covariance Matrices  20  2959.55  146.66  <.0001  26.5004  11.5395  18.5179  4.1883  0.5302  0.7562  0.3779  0.3645  0.4705  0.4845  26.5004  11.5395  18.5179  4.1883  0.5302  0.7562  0.3779  0.3645  0.4705  0.4845  26.5004  11.5395  18.5179  4.1883  0.5302  0.7562  0.3779  0.3645  0.4705  0.4845  DF 
Equal Correlation Matrices  12  2876.38  63.49  <.0001  16.4715  14.8656  4.8427  1.4392  0.5612  0.6827  0.4016  0.3844  0.4976  0.5219  24.4020  9.1566  17.4434  3.0021  0.5612  0.6827  0.4016  0.3844  0.4976  0.5219  35.0544  10.8350  27.3593  8.1395  0.5612  0.6827  0.4016  0.3844  0.4976  0.5219  DF 
The result of the homogeneity test is identical to that in Output 41.9.4. The hypothesis of equality of the correlation matrices is also rejected with a chisquare value of 63.49 and a pvalue of . Notice, however, that the chisquare statistic is smaller than in the test of homogeneity due to the smaller number of restrictions imposed on the full model. The estimate of the common correlation matrix in the restricted model is
