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 Introduction to Categorical Data Analysis Procedures

## 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 -1 . The same column produced by the CLASS statement of PROC GENMOD and PROC PROBIT is coded 1 and 0. PROC CATMOD uses fullrank parameterization using differential effects. As a result, the parameter estimate printed by PROC CATMOD is one-half of the estimate produced by the others. PROC LOGISTIC does not automatically create indicator variables for categorical independent variables. So, the parameterization depends on how you code the indicator variables (1,0 versus -1,1). See the "Details" sections in the chapters on the CATMOD, GENMOD, 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 Fisher's scoring method while PROC PROBIT, PROC GENMOD, and PROC CATMOD use the Newton-Raphson method (the PROC PROBIT algorithm is ridge stabilized and is a modified Newton-Raphson algorithm.) 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 (handled by the PROBIT, GENMOD, and LOGISTIC procedures), 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 fit logistic regression models 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.

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