The CATMOD Procedure

Suppose the dependent variable A has three levels and is the only response-effect in the MODEL statement. The following table shows the proportions upon which the response functions are defined:

Note that . The following table shows the response functions generated for each population:

Now, suppose the dependent variables A and B each have three levels (valued 1, 2, and 3 each) and the response-effect in the MODEL statement is A*B. The following table shows the proportions upon which the response functions are defined:

The marginal totals for the preceding table are defined as follows:

where . The following table shows the response functions generated for each population:

The READ and transformation function specifications are not shown in the preceding table. For these two situations, there is not a general response function; the response functions that are generated depend on what you specify.

Another important aspect of the function specification is the number of response functions generated per population, q. Let represent the number of levels for the ith dependent variable in the MODEL statement, and let d represent the number of dependent variables in the MODEL statement. Then, if the function specification is ALOGITS, CLOGITS, LOGITS, or MARGINALS, the number of response functions is

If the function specification is JOINT or the default (generalized logits), the number of response functions per population is

where r is the number of response profiles. If every possible cross-classification of the dependent variables is observed in the samples, then

Otherwise, r is the number of cross-classifications actually observed.

If the function specification is MEANS, the number of response functions per population is .

Some example response statements are shown in the following table:

If you specify a transformation, it is applied to the vector that contains the sample proportions in each population. The transformation can be any combination of the following four operations:

If more than one operation is specified, then PROC CATMOD applies the operations consecutively from right to left.

A matrix literal is a matrix of numbers with each row of the matrix separated from the next by a comma. If you specify a linear combination, in most cases the is not needed. The following statement defines the response function . The is needed to separate the two matrix literals '1' and '1 0'.

response + 1 * 1 0;

The of a vector transforms each element of the vector into its natural logarithm; the of a vector transforms each element into its exponential function (antilogarithm).

In order to specify a linear response function for data that have r = 3 response categories, you can specify either of the following RESPONSE statements:

response  * 1 0 0 , 0 1 0;
response    1 0 0 , 0 1 0;

The matrix literal in the preceding statements specifies a matrix, which is applied to each population as follows:

where , , and are sample proportions for the three response categories in a population, and and are the two response functions computed for that population. Therefore, this response function sets and in each population.

As another example of the linear response function, suppose you have two dependent variables corresponding to two observers who evaluate the same subjects. If the observers grade on the same three-point scale and if all nine possible responses are observed, then the following RESPONSE statement would compute the probability that the observers agree on their assessments:

response 1 0 0 0 1 0 0 0 1;

This response function is then computed as

where denotes the probability that a subject gets a grade of i from the first observer and j from the second observer.

If the function is a compound function, requiring more than one operation to specify it, then the operations should be listed so that the first operation to be applied is on the right and the last operation to be applied is on the left. For example, if there are two response levels, you can have the following response function:

response 1 -1 log;

This is equivalent to the matrix expression

which is the logit response function since when there are only two response levels.

The following statement specifies another example of a compound response function:

response exp 1 -1 * 1 0 0 1, 0 1 1 0 log;

This is equivalent to the matrix expression

where is the vector of sample proportions for some population,

If the four responses are based on two dependent variables, each with two levels, then the function can also be written as

which is the odds (crossproduct) ratio for a table.

If no RESPONSE statement is specified, PROC CATMOD computes the standard response functions, which contrast the log of each response probability with the log of the probability for the last response category. If there are r response categories, then there are standard response functions. For example, if there are four response categories, using no RESPONSE statement is equivalent to specifying the following:

response  1 0 0 -1,
          0 1 0 -1,
          0 0 1 -1  log;

This results in three response functions:

If there are only two response levels, the resulting response function would be a logit, which is why the standard response functions are called generalized logits. They are useful in dealing with the log-linear model:

If denotes the matrix in the preceding RESPONSE statement, then because of the restriction that the probabilities sum to 1, it follows that an equivalent model is

But is simply the vector of standard response functions. Thus, fitting a log-linear model on the cell probabilities is equivalent to fitting a linear model on the generalized logits.