Comparison of CATMOD, GENMOD, LOGISTIC, and PROBIT Procedures

The CATMOD, GENMOD, LOGISTIC, and PROBIT procedures can all be used for statistical modeling of categorical data. The CATMOD procedure provides maximum likelihood estimation for logistic regression, including the analysis of logits for dichotomous outcomes and the analysis of generalized logits for polychotomous outcomes . It provides weighted least squares estimation of many other response functions, such as means, cumulative logits, and proportions, and you can also compute and analyze other response functions that can be formed from the proportions corresponding to the rows of a contingency table. In addition, a user can input and analyze a set of response functions and user-supplied covariance matrix with weighted least squares. With the CATMOD procedure, by default, all explanatory (independent) variables are treated as classification variables .

The GENMOD procedure is also a general statistical modeling tool which fits generalized linear models to data: it fits several useful models to categorical data including logistic regression, the proportional odds model, and Poisson regression. The GENMOD procedure also provides a facility for fitting generalized estimating equations to correlated response data that are categorical, such as repeated dichotomous outcomes. The GENMOD procedure fits models using maximum likelihood estimation, and you include classification variables in your models with a CLASS statement. PROC GENMOD can perform type I and type III tests, and it provides predicted values and residuals.

The LOGISTIC procedure is specifically designed for logistic regression. It performs the usual logistic regression analysis for dichotomous outcomes and it fits the proportional odds model and the generalized logit model for ordinal and nominal outcomes, respectively, by the method of maximum likelihood. With the CLASS statement, you can include independent CLASS variables in the model. This procedure has capabilities for a variety of model-building techniques, including stepwise, forward, and backward selection. It computes predicted values, the receiver operating characteristics (ROC) curve and the area beneath the curve, and a number of regression diagnostics. It can create output data sets containing these values and other statistics. PROC LOGISTIC can perform a conditional logistic regression analysis (matched-set and case-controlled) for binary response data. For small data sets, PROC LOGISTIC can perform the exact conditional logistic analysis of Hirji, Mehta, and Patel (1987) and Mehta, Patel, and Senchaudhuri (1992).

The PROBIT procedure is designed for quantal assay or other discrete event data. In additional to performing the logistic regression analysis, it can estimate the threshold response rate. PROC PROBIT can also estimate the values of independent variables that yield a desired response. With the CLASS statement, you can include CLASS variables in the model. PROC PROBIT allows only the less-than -full-rank parameterization for the CLASS variables.

Stokes, Davis, and Koch (2000) provide substantial discussion of these procedures, particularly the use of the FREQ, LOGISTIC, GENMOD, and CATMOD procedures for statistical modeling.

Logistic Regression

Dichotomous Response

You have many choices of performing logistic regression in the SAS System. The CATMOD, GENMOD, LOGISTIC, and PROBIT procedures fit the usual logistic regression model.

PROC LOGISTIC provides the capability of model-building, and performs conditional logistic regression analysis for case-control studies and exact conditional logistic regression analysis. You may choose to use it for these reasons.

PROC CATMOD may not be efficient when there are continous independent variables with large numbers of different values. For a continuous variable with a very limited number of values, PROC CATMOD may be useful. You list the continuous variables in the DIRECT statement.

The LOGISTIC, GENMOD, and PROBIT procedures can analyze summarized data by enabling you to input the numbers of events and trials; the ratio of events to trials must be between 0 and 1. PROC PROBIT enables you to estimate the natural response rate and compute fiducial limits for the dose variable.

Ordinal Response

PROC LOGISTIC fits the proportional odds model to the ordinal response data by default. PROC PROBIT fits this model if you specify the logistic distribution, and PROC GENMOD fits the same model if you specify the CLOGIT link and the multinomial distribution.

Nominal Response

When the response variable is nominal, there is no concept of ordering of the response values. PROC CATMOD fits a logistic model to response functions called generalized logits . PROC LOGISTIC fits the generalized logit model if you specify the GLOGIT link.

Parameterization

There are some differences in the way that models are parameterized, which means that you might get different parameter estimates if you were to perform logistic regression in each of these procedures.

• Parameter estimates from the procedures may differ in sign, depending on the ordering of response levels, which you can change if you want.

• The parameter estimates associated with a categorical independent variable may differ among the procedures, since the estimates depend on the coding of the indicator variables in the design matrix. By default, the design matrix column produced by PROC CATMOD for a binary independent variable is coded using the values 1 and ˆ’ 1. The same column produced by the CLASS statement of PROC PROBIT is coded using 1 and 0. PROC CATMOD uses the deviation from the mean coding, which is a full-rank parameterization, and PROC PROBIT uses the less-than-full-rank coding. As a result, the parameter estimate printed by PROC CATMOD is one-half of the estimate produced by PROC PROBIT. Both PROC GENMOD and PROC LOGISTIC allow either a full-rank parameterization or the less-than-full-rank parameterization. See the Details sections in the chapters on the CATMOD, GENMOD, LOGISTIC, and PROBIT procedures for more information on the generation of the design matrices used by these procedures.

• The maximum-likelihood algorithm used differs among the procedures. PROC LOGISTIC uses the Fisher s scoring method by default, while PROC PROBIT, PROC GENMOD, and PROC CATMOD use the Newton-Raphson method. The parameter estimates should be the same for all three procedures, and the standard errors should be the same for the logistic model. For the normal and extreme-value (Gompertz) distributions in PROC PROBIT, which correspond to the probit and cloglog links, respectively, in PROC GENMOD and PROC LOGISTIC, the standard errors may differ. In general, tests computed using the standard errors from the Newton-Raphson method will be more conservative.

• The LOGISTIC, GENMOD, and PROBIT procedures can be used to fit a cumulative regression model for ordinal response data using maximum-likelihood estimation. PROC LOGISTIC and PROC GENMOD use a different parameterization from that of PROC PROBIT, which results in different intercept parameters. Estimates of the slope parameters, however, should be the same for both procedures. The estimated standard errors of the slope estimates are slightly different between the two procedures because of the different computational algorithms used as default.