A Tutorial on Logistic Regression

Many procedures in SAS/STAT can be used to perform logistic regression analysis: CATMOD, GENMOD,LOGISTIC, and PROBIT. Each procedure has special features that make it useful for certain applications. For most applications, PROC LOGISTIC is the preferred choice. It fits binary response or proportional odds models, provides various model-selection methods to identify important prognostic variables from a large number of candidate variables, and computes regression diagnostic statistics. This tutorial discusses some of the problems users encountered when they used the LOGISTIC procedure. INTRODUCTION PROC LOGISTIC can be used to analyze binary response as well as ordinal response data. Binary Response The response, Y, of a subject can take one of two possible values, denoted by 1 and 2 (for example, Y=1 if a disease is present; otherwise Y=2). Let x = (x1; : : :; xk)0 be the vector of explanatory variables. The logistic regression model is used to explain the effects of the explanatory variables on the binary response. logitfPr(Y = 1jx)g = log Pr(Y = 1jx) 1 Pr(Y = 1jx) = 0+x 0 where 0 is the intercept parameter, and is the vector of slope parameters (Hosmer and Lameshow, 1989). Ordinal Response The response, Y, of a subject can take one of m ordinal values, denoted by 1;2; : : :;m. PROC LOGISTIC fits the following cumulative logit model: logitfPr(Y rjx)g = r + x0 1 r < m where 1; : : :; m 1 are (m-1) intercept parameters. This model is also called the proportional odds model because the odds of making response r are exp( 0(x1 x2)) times higher at x = x1 than at x = x2 (Agresti, 1990). This ordinal model is especially appropriate if the ordinal nature of the response is due to methodological limitations in collecting the data in which the researchers are forced to logit of the cumulative probabilities lump together and identify various portions of an otherwise continuous variable. Let T be the underlying continuous variable and suppose that