The SURVEYLOGISTIC Procedure |

If the response categories of the response variable can be restricted to a number of ordinal values, you can fit cumulative probabilities of the response categories with a cumulative logit model, a complementary log-log model, or a probit model. Details of cumulative logit models (or proportional odds models) can be found in McCullagh and Nelder (1989). If the response categories of are nominal responses without natural ordering, you can fit the response probabilities with a generalized logit model. Formulation of the generalized logit models for nominal response variables can be found in Agresti (2002). For each model, the procedure estimates the model parameter by using a pseudo-log-likelihood function. The procedure obtains the pseudo-maximum likelihood estimator by using iterations described in the section Iterative Algorithms for Model Fitting and estimates its variance described in the section Variance Estimation.

A cumulative logit model uses the **logit** function

as the link function.

Denote the cumulative sum of the expected proportions for the first categories of variable by

for Then the cumulative logit model can be written as

with the model parameters

A complementary log-log model uses the **complementary log-log** function

as the link function. Denote the cumulative sum of the expected proportions for the first categories of variable by

for Then the complementary log-log model can be written as

with the model parameters

A probit model uses the **probit** (or normit) function, which is the inverse of the cumulative standard normal distribution function,

as the link function, where

Denote the cumulative sum of the expected proportions for the first categories of variable by

for Then the probit model can be written as

with the model parameters

For nominal response, a generalized logit model is to fit the ratio of the expected proportion for each response category over the expected proportion of a reference category with a logit link function.

Without loss of generality, let category be the reference category for the response variable . Denote the expected proportion for the th category by as in the section Notation. Then the generalized logit model can be written as

for with the model parameters

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