The SURVEYLOGISTIC procedure fits a logistic regression model and estimates the corresponding regression parameters. Each model uses the link function you specified in the LINK= option in the MODEL statement. There are four types of model you can use with the procedure: cumulative logit model, complementary log-log model, probit model, and generalized logit model.
Let Y be the response variable with categories . The p covariates are denoted by a p-dimension row vector .
For a stratified clustered sample design, each observation is represented by a row vector, , where
is the stratum index
is the cluster index within stratum h
is the unit index within cluster i of stratum h
denotes the sampling weight
is a D-dimensional column vector whose elements are indicator variables for the first D categories for variable Y. If the response of the jth unit of the ith cluster in stratum h falls in category d, the dth element of the vector is one, and the remaining elements of the vector are zero, where .
is the indicator variable for the category of variable Y
denotes the k-dimensional row vector of explanatory variables for the jth unit of the ith cluster in stratum h. If there is an intercept, then .
is the total number of clusters in the sample
is the total sample size
The following notations are also used:
denotes the sampling rate for stratum h
is the expected vector of the response variable:
Note that , where 1 is a D-dimensional column vector whose elements are 1.
If the response categories of the response variable Y 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 Y 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 d categories of variable Y 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 d categories of variable Y 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 d categories of variable Y 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 Y. Denote the expected proportion for the dth category by as in the section Notation. Then the generalized logit model can be written as
for with the model parameters
Let be a link function such that
where is a column vector for regression coefficients. The pseudo-log likelihood is
Denote the pseudo-estimator as , which is a solution to the estimating equations:
where is the matrix of partial derivatives of the link function with respect to .
To obtain the pseudo-estimator , the procedure uses iterations with a starting value for . See the section Iterative Algorithms for Model Fitting for more details.