Example 25.19 Testing Equality of Two Covariance Matrices Using a Multiple-Group Analysis

You can use PROC CALIS to do multiple-group or multiple-sample analysis. The groups in the analysis must be independent. In this example, a relatively simple multiple-group analysis is carried out. The covariance matrices of two independent groups are tested for equality. Hence, individual covariance matrices are actually not structured. Rather, they are constrained to be the same under the null hypothesis. See Example 25.25 for a more sophisticated example of multiple-group analysis.

In this example, a reaction time experiment was conducted on two groups of individuals. One group () was considered to be an expert group with prior training related to the tasks of the experiment. Another group () was a control group without prior training. Three tasks of dexterity were administered to all individuals. These tasks differed by their required complexity levels of body skills. They were labeled as high, medium, and low complexities.

Apparently, the differential performance of the two groups under different task complexities was the primary research objective. In this example, however, you are interested in testing whether the groups have the same covariance matrix for the tasks. Equality of covariance matrices might be an essential assumption in some statistical tests for comparing group means. In this example, you use PROC CALIS to see the tenability of such an assumption. The covariance matrices for the two groups are stored in the data sets Expert and Novice, as shown in the following:

data expert(type=cov);
   input _type_ $ _name_ $ high medium low;
   datalines;
COV   high    5.88     .      .
COV   medium  2.88    7.16    .
COV   low     3.12    4.44   8.14
;

data novice(type=cov);
   input _type_ $ _name_ $ high medium low;
   datalines;
COV   high    6.42     .      .
COV   medium  1.24    8.25    .
COV   low     4.26    2.75   7.99
;

These data sets are read into the analysis through the GROUP statements in the following PROC CALIS specification:

proc calis;
   group 1 / data=expert nobs=20 label="Expert";
   group 2 / data=novice nobs=18 label="Novice";
   model 1 / groups=1,2;
      mstruct 
         var=high medium low;
   fitindex NoIndexType On(only)=[chisq df probchi] 
            chicorrect=eqcovmat;
   ods select ModelingInfo MSTRUCTVariables MSTRUCTCovInit Fit;
   run;

The first GROUP statement defines group 1 for the expert group. The second GROUP statement defines group 2 for the novice group. You use the NOBS= option in both statements to provide the number of observations of these groups. You use the LABEL= option in these statements to provide meaningful group labels.

The MODEL statement defines MODEL 1. In the analysis, this model fits to both groups 1 and 2, as indicated by the GROUPS= option of the statement. This is done to test the null hypothesis of equality of covariance matrices in the two groups. An MSTRUCT model for MODEL 1 is defined immediately afterward. Three variables, high, medium, and low, are specified in the VAR= option of the MSTRUCT statement.

Without further specification about the MSTRUCT model, PROC CALIS assumes all non redundant elements in the covariance matrix are free parameters. This is what is required under the null hypothesis of the equality of covariance matrices in the two groups—the groups have the same covariance matrix, but the covariance matrix itself is unconstrained. Your model under the null hypothesis is now well-defined and ready to run. In addition, you use FITINDEX and ODS SELECT statements to customize or fine tune the analysis.

By using the options in the FITINDEX statement, you can customize the fit summary table and control some analytic options. In the current example, you use the NOINDEXTYPE option to suppress the printing of the index types in the fit summary table. Then, you use the ON(ONLY)= option to specify the fit indices printed in the fit summary table. In this example, you request only the model fit chi-square statistic, degrees of freedom, and the probability value of the chi-square be printed. Finally, you use the CHICORRECT=EQCOVMAT option to request a chi-square correction for the test of equality of covariance matrices. This correction is due to Box (1949) and is implemented in PROC CALIS as a built-in chi-square correction option.

In addition, because you are not interested in all displayed output for the current hypothesized model, you use the ODS SELECT statement to display only those output (or ODS tables) of interest. In this example, you request only the modeling information, the variables involved, the initial covariance matrix specification, and the fit summary table be printed. All output in PROC CALIS are named as an ODS table. To locate a particular output in PROC CALIS, you must know the corresponding ODS table name. See the section ODS Table Names for a listing of ODS tables produced by PROC CALIS.

Output 25.19.1 displays some information regarding the basic model setup.

The modeling information table summarizes some basic information about the two groups. Both of them are fitted by Model 1. The next table shows the variables involved: high, medium, and low. The order of variables in this table is the same as that of the row and column variables of the covariance model matrix, which is shown next in Output 25.19.1. The parameters for the entries in the covariance matrix are shown. The names of parameters are displayed in parentheses. All these parameters are set by default and their names have the prefix _Add. No initial estimates are given as input, as indicated by the missing value '.'.

Output 25.19.2 shows the customized fit summary table, which has been much simplified for the current example due to the uses of some options in the FITINDEX statement.

As shown in Output 25.19.2, the chi-square test statistic is . With six degrees of freedom, the test statistic is not significant at . Therefore, the hypothesized model is supported, which means that the equality of the covariance matrices of the groups is supported.