In a logistic analysis, the response functions are the logits of the dependent variable.
PROC CATMOD can compute the three following types of logits with the use of keywords in the RESPONSE statement. Note that other types of response functions can be generated by specifying appropriate transformations in the RESPONSE statement.
Generalized logits are used primarily for nominally scaled dependent variables, but they can also be used for ordinal data modeling. Maximum likelihood estimation is available for the analysis of these logits.
Cumulative logits are used for ordinally scaled dependent variables. Except for dependent variables with two response levels, only weighted least squares estimation is available for the analysis of these logits.
Adjacent-category logits are equivalent to generalized logits, but they have some advantages for ordinal data analysis because they automatically incorporate integer scores for the levels of the dependent variable. Except for dependent variables with two response levels, only weighted least squares estimation is available for the analysis of these logits.
If the dependent variable has only two responses, then the cumulative logit and the adjacent-category logit are the negative of the generalized logit, as computed by PROC CATMOD. Consequently, parameter estimates obtained by using these logits are the negative of those obtained from using generalized logits. A simple logistic analysis of variance uses statements like the following:
proc catmod; model r=a|b; run;
If the independent variables are treated quantitatively (like continuous variables), then a logistic analysis is known as a logistic regression. If you want PROC CATMOD to treat the independent variables as quantitative variables, specify them in both the DIRECT and MODEL statements, as follows:
proc catmod; direct x1 x2 x3; model r=x1 x2 x3; run;
Since the preceding statements do not include a RESPONSE statement, generalized logits are computed. See Example 32.3 for another example.
The parameter estimates from the CATMOD procedure are the same as those from a logistic regression program such as PROC LOGISTIC (see Chapter 58: The LOGISTIC Procedure). The chi-square statistics and the predicted values are also identical. In the binary response case, PROC CATMOD can be made to model the probability of the maximum value by either (1) organizing the input data so that the maximum value occurs first and specifying ORDER=DATA in the PROC CATMOD statement or (2) specifying cumulative logits (CLOGITS) in the RESPONSE statement.
Caution: Computational difficulties might occur if you use a continuous variable with a large number of unique values in a DIRECT statement. See the section Continuous Variables for more details.
If your dependent variable is ordinally scaled, you can specify the analysis of cumulative logits that take into account the ordinal nature of the dependent variable:
proc catmod; response clogits; direct x; model r=a x; run;
The preceding statements correspond to a simple analysis that addresses the question of existence of an association between the independent variables and the ordinal dependent variable. However, there are some commonly used models for the analysis of ordinal data (Agresti, 1984) that address the structure of association (in terms of odds ratios), as well as its existence.
If the independent variables are classification variables, a typical analysis for such a model uses the following statements:
proc catmod; weight wt; response clogits; model r=_response_ a b; run;
On the other hand, if the independent variables are ordinally scaled, you might specify numeric scores in variables x1
and x2
, and use the following statements:
proc catmod; weight wt; direct x1 x2; response clogits; model r=_response_ x1 x2; run;
See Agresti (1984) for additional details of estimation, testing, and interpretation.
Computational difficulties might occur if you have a continuous variable with a large number of unique values and you use this variable in a DIRECT statement, since an observation often represents a separate population of size one. At this extreme of sparseness, the weighted least squares method is inappropriate since there are too many zero frequencies. Therefore, you should use the maximum likelihood method. PROC CATMOD is not designed optimally for continuous variables; therefore, it might be less efficient and unable to allocate sufficient memory to handle this problem, as compared with a procedure designed specifically to handle continuous data. In these situations, consider using the LOGISTIC or GENMOD procedure to analyze your data.