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The CATMOD Procedure |
ALOGIT ALOGITS | specifies response functions as adjacent-category logits of the marginal probabilities for each of the dependent variables. For each dependent variable, the response functions are a set of linearly independent adjacent-category logits, obtained by taking the logarithms of the ratios of two probabilities. The denominator of the kth ratio is the marginal probability corresponding to the kth level of the variable, and the numerator is the marginal probability corresponding to the (k+1)th level. If a dependent variable has two levels, then the adjacent-category logit is the negative of the generalized logit. |
CLOGIT CLOGITS | specifies that the response functions are cumulative logits of the marginal probabilities for each of the dependent variables. For each dependent variable, the response functions are a set of linearly independent cumulative logits, obtained by taking the logarithms of the ratios of two probabilities. The denominator of the kth ratio is the cumulative probability, c_{k}, corresponding to the kth level of the variable, and the numerator is 1-c_{k} (Agresti 1984, 113 -114). If a dependent variable has two levels, then PROC CATMOD computes its cumulative logit as the negative of its generalized logit. You should use cumulative logits only when the dependent variables are ordinally scaled. |
JOINT | specifies that the response functions are the joint response probabilities. A linearly independent set is created by deleting the last response probability. For the case of one dependent variable, the JOINT and MARGINALS specifications are equivalent. |
LOGIT LOGITS | specifies that the response functions are generalized logits of the marginal probabilities for each of the dependent variables. For each dependent variable, the response functions are a set of linearly independent generalized logits, obtained by taking the logarithms of the ratios of two probabilities. The denominator of each ratio is the marginal probability corresponding to the last observed level of the variable, and the numerators are the marginal probabilities corresponding to each of the other levels. If there is one dependent variable, then specifying LOGIT is equivalent to using the standard response functions. |
MARGINAL MARGINALS | specifies that the response functions are marginal probabilities for each of the dependent variables in the MODEL statement. For each dependent variable, the response functions are a set of linearly independent marginals, obtained by deleting the marginal probability corresponding to the last level. |
MEAN MEANS | specifies that the response functions are the means of the dependent variables in the MODEL statement. This specification requires that all of the dependent variables be numeric. |
READ variables | specifies that the response functions and their covariance matrix are to be read directly from the input data set with one response function for each variable named. See the section "Inputting Response Functions and Covariances Directly" for more information. |
transformation | specifies response functions that can be expressed by using successive applications of the four operations: LOG, EXP, * matrix literal, or + matrix literal. The operations are described in detail in the "Using a Transformation to Specify Response Functions" section. |
Value of A: | 1 | 2 | 3 |
proportions: | p_{1} | p_{2} | p_{3} |
Function | Value | |
Specification | of q | Response Function |
none^{*} | 2 | ln( [(p_{1})/(p_{3})] ), ln( [(p_{2})/(p_{3})] ) |
ALOGITS | 2 | ln( [(p_{2})/(p_{1})] ), ln( [(p_{3})/(p_{2})] ) |
CLOGITS | 2 | ln( [(1-p_{1})/(p_{1})] ), ln( [(1-(p_{1}+p_{2}))/(p_{1}+p_{2})] ) |
JOINT | 2 | p_{1}, p_{2} |
LOGITS | 2 | ln( [(p_{1})/(p_{3})] ), ln( [(p_{2})/(p_{3})] ) |
MARGINAL | 2 | p_{1}, p_{2} |
MEAN | 1 | 1 p_{1} + 2 p_{2} + 3 p_{3} |
^{*}Without
a function specification, the default response
functions are generalized logits. |
Value of A: | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 |
Value of B: | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 |
proportions: | p_{1} | p_{2} | p_{3} | p_{4} | p_{5} | p_{6} | p_{7} | p_{8} | p_{9} |
Function | Value | |
Specification | of q | Response Function |
none^{*} | 8 | ln( [(p_{1})/(p_{9})] ), ln( [(p_{2})/(p_{9})] ), ln( [(p_{3})/(p_{9})] ), ... , ln( [(p_{8})/(p_{9})] ) |
ALOGITS | 4 | ln( [(p_{2 ·})/(p_{1 ·})] ), ln( [(p_{3 ·})/(p_{2 ·})] ), ln( [(p_{·2})/(p_{·1})] ), ln( [(p_{·3})/(p_{·2})] ) |
CLOGITS | 4 | ln( [(1-p_{1 ·})/(p_{1 ·})] ), ln( [(1-(p_{1 ·}+p_{2 ·}))/(p_{1 ·}+p_{2 ·})] ), ln( [(1-p_{·1})/(p_{·1})] ), ln( [(1-(p_{·1}+p_{·2}))/(p_{·1}+p_{·2})] ) |
JOINT | 8 | p_{1}, p_{2}, p_{3}, p_{4}, p_{5}, p_{6}, p_{7}, p_{8} |
LOGITS | 4 | ln( [(p_{1 ·})/(p_{3 ·})] ), ln( [(p_{2 ·})/(p_{3 ·})] ), ln( [(p_{·1})/(p_{·3})] ), ln( [(p_{·2})/(p_{·3})] ) |
MARGINAL | 4 | p_{1 ·}, p_{2 ·}, p_{·1}, p_{·2} |
MEAN | 2 | 1 p_{1 ·} + 2 p_{2 ·} + 3 p_{3 ·}, 1 p_{·1} + 2 p_{·2} + 3 p_{·3} |
^{*} Without a function specification, the default response functions are generalized logits. |
Example | Result |
response marginals; | marginals for each dependent variable |
response means; | the mean of each dependent variable |
response logits; | generalized logits of the marginal probabilities |
response clogits; | cumulative logits of the marginal probabilities |
response alogits; | adjacent-category logits of the marginal probabilities |
response joint; | the joint probabilities |
response 1 -1 log; | the logit |
response; | generalized logits |
response 1 2 3; | the mean score, with scores of 1, 2, and 3 corresponding to the three response levels |
response read b1-b4; | four response functions and their covariance matrix, read directly from the input data set |
Operation | Specification |
linear combination | * matrix literal |
matrix literal | |
logarithm | LOG |
exponential | EXP |
adding constant | + matrix literal |
response + 1 * 1 0;
response * 1 0 0 , 0 1 0; response 1 0 0 , 0 1 0;
response 1 0 0 0 1 0 0 0 1;
response 1 -1 log;
response exp 1 -1 * 1 0 0 1, 0 1 1 0 log;which is equivalent to the matrix expression
response 1 0 0 -1, 0 1 0 -1, 0 0 1 -1 log;This results in three response functions:
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