The following provides further interpretation of the model parameters in the example titled "Repeated Measures, 4 Response Levels, 1 Population" in the PROC CATMOD documentation.

Since the RESPONSE MARGINALS statement is specified, and the RIGHT and LEFT variables each have four possible levels, there are three marginal probabilities modeled for RIGHT and for LEFT. These are the six response functions modeled by this analysis. They are listed below. Since RIGHT was specified first in the MODEL statement, its marginal probabilities are first.

The MODEL and REPEATED statements specify a model with one categorical predictor, SIDE, having two levels (RIGHT and LEFT). Since there are six response functions, there are six parameters to be estimated — three intercepts and three SIDE parameters. The first three columns of the design matrix correspond to the three intercepts. The last three columns correspond to the three SIDE parameters.

The first row of the design matrix means that the probability that RIGHT=1 is estimated by the first intercept plus the first SIDE parameter. This is the value in the Response Function column (any slight difference is due to rounding).

Pr(RIGHT=1) = 0.2597 + 0.00461 = 0.2643

The fourth row of the design matrix means that the probability that LEFT=1 is estimated by the first intercept minus the first SIDE parameter.

Pr(LEFT=1) = 0.2597 - 0.00461 = 0.2551

The difference of these two equations shows that the first SIDE parameter is half the estimated difference in the marginal probabilities of level 1 in the two eyes.

Pr(RIGHT=1) - Pr(LEFT=1) = 2*0.00461

By similar argument, the second and third SIDE parameters are found to be half the difference in the marginal probabilities of level 2 and of level 3 in the two eyes.

If you add, rather than subtract, the equations from the first and fourth rows of the design matrix, you find that the first intercept estimates the average of the marginal probabilities of level 1 in the two eyes. Similarly, the other two intercepts estimate the average probabilities for levels 2 and 3.

Since the model is saturated (that is, all available degrees of freedom are used in the model), the model fit is perfect. As a result, the estimates of the above probabilities and difference in probabilities exactly match the observed values. Notice that the above values match the observed row and column percentages shown here using PROC FREQ (slight differences due to rounding).

      proc freq data=vision; 
        table Right*Left / norow nocol; 
        weight count; 
        run;

In order to obtain a parameter and test comparing the eyes on the fourth response level it is necessary to use a CONTRAST statement. Under the default full-rank parameterization (effects coding) used to fit the model, the SIDE parameters are constrained to sum to zero. Therefore, the fourth parameter is constrained to equal the negative sum of the other three parameters. The following CONTRAST statement produces the fourth parameter estimate and test.

contrast 'Level 4' all_parms 0 0 0 -1 -1 -1 / estimate=parm;

The results show that the parameter comparing the eyes at response level 4 is -0.00348 . Consequently, there is evidence to conclude that the two eyes differ on levels 1 (p =0.0174) and 4 (p =0.0407) of the response, but not on levels 2 (p =0.3726) and 3 (p =0.1757).

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