Practical Aspects of Modelling Techp.iques in Logistic Regression Procedures of the SAS System

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1 r""'=~~"''''''''''''''''''''''''''''\;'=="'~''''o''''"'"''~ ~c_,,..! Practical Aspects of Modelling Techp.iques in Logistic Regression Procedures of the SAS System Rainer Muche 1, Josef HogeP and Olaf Gefeller 2 lclinical Documentation, University of Ulm, Germany 2Department of Medical Statistics, University of Gottingen, Germany ABSTRACT Practical apllications of logistic regression analysis often involve the implementation of model-building strategies to find an appropriate final model for a comprehensible description of the relationship under study. The statistical software used for logistic regression modelling has thus not only to be evaluated with respect to its features related to analysing one given model but also with respect to its capability to support the modelbuilding process. In this paper, the procedures within the SAS software performing logistic regression modelling are examined to which extent they offer the necessary features for convenient model-building. The deficits found in this investigation are described in combination with a proposal for theit resolution. INTRODUCTION Logistic regression modelling is the methodologic approach frequently applied in epidemiology and clinical research to detect and to evaluate risk factors for dichotomous disease variables. In this context usually information on a multitude of explanatory variables (potential risk factors) and the response variable (the disease) is obtained. A variable-selecting algorithm is then routinely employed to reduce the number of explanatory variables. Variable selection techniques as part of statistical modelling in logistic regression are applied to obtain an appropriate model fitting the data which should (i) lead to stable parameter estimates, (ii) predict accurately future values, and (iii) allow comprehensible interpretation since only few important explanatory variables are selected [1]. In order to meet these requirements variable selection algorithms should cover two statistical strategies: incorporation of the hierarchy principle in selecting interaction terms and coding of categorical variables via dummy variables. 1045

2 i \.. In practice, the choice of a variable selection algorithm to be employed usually depends not only on its statistical properties but also on the availability of the corresponding software. Since SAS is the leading software package for statistical analyses, the statistical features of variable selection techniques implemented in the SAS-procedures for logistic regression will have a major impact on the current practice of variable selection in applications. In this paper, the present situation within the SASI ST ATTM procedures performing logistic regression modelling is investigated, and a proposal for resolving the deficits found in this investigation is provided. STATISTICAL BACKGROUND Logistic Regression Analysis To analyse the influence of explanatory variables on the risk for a disease described by a dichotomous variable usually the logistic regression model is employed [2]. This model can be used to analyse simultaneously explanatory variables and interaction terms of different data types (D: disease, EI,'..,Ek : explanatory variables, h+b'..,1/: interaction terms): 1 P(D = 1 lei = ej,",ek = ek, h+i = ik+b'. " II = i l ) = -----k---i--- -(/30+ L /3i e;+ L /3jij) 1 + e ;=1 j=k+1 Variable Selection Procedures In epidemiology and clinical research a variable-selecting algorithm is routinely employed in order to reduce the number of explanatory variables in the final model [3-7]. Such a procedure attempts to yield a final model which should primarily lead to unbiased stable estimators of the regression coefficients. There are several existing selection procedures in the literature and statistical software (e.g. st.epwise methods, all possible subset methods, tree structure testing strategy). The criterion for selecting variables for the final model is often based on significance testing. Hierarchy - Principle In model-building strategies, the hierarchy principle refers to the rule that no interaction terms should be included in a model without simultaneously incorporating all components (main effects and interactions of minor order) involved in these terms [8-10]. This requirement is essential for the interpretability of estimates in the final model. Applying the principle means to start analysing the data at the highest order interactions. If an interaction stays in the model after variable selection, the involved components are 1046

3 forced into the model. This procedure is then repeated from highest order interactions down to main effects. Dummy Variables Categorical variables are usually included into the model after coding them as a set of dummy variables [11]. In this situation, assessment of their common effect is performed by simultaneous criteria (e.g. simultaneous likelihood-ratio test). Most text books of modelbuilding give the rule that categorical main effects have to be represented in the final model by the whole set of their dummy variables. IMPLEMENTATION OF LOGISTIC REGRESSION IN PROCEDURES OF THE SAS SOFTWARE First, a brief overview on realizations of logistic regression modelling and variable selection techniques as part of the modelling process in procedures of the SAS software is given. Logistic regression is directly implemented in three procedures of SAS/STAT [12]: PROC LOGISTIC, PROC CATMOD, and PROC PROBIT. PROC PROBIT will not be considered in this paper because this is not the usual procedure for performing logistic regression analysis. Automatic variable selection algorithms are only available in PROC LOGISTIC. Syntax and relevant features for model-building of PROC LOGISTIC [A] and PROC CATMOD [B] are as follows: [A] PROC LOGISTIC DATA=data ; MODEL d = el... ek i k+1 il / LINK=logit SELECTION= INCLUDE= ; PROC LOGISTIC gives maximum-likelihood estimators and a likelihood-ratio test to assess the effects of the variables included in the model statement. Four automatic variable selection algorithms are implemented in the procedure: forward, backward, stepwise selection, and an all possible subset selection method based on the score statistic [13]. Interaction terms have to be coded explicitly in the DATA step as a product of the main effects entered into the model. The dummy variables corresponding to a categorical or ordinal exploratory variable also have to be constructed explicitly in the DATA step. Hence, using this procedure, dummy variables are treated as independent variables in model-building without recognizing their interrelationship. There is no possibility of testing the common effect of several dummy variables. The hierarchy-principle can only be applied manually by using the INCLUDE= option. With this option all variables except 1047

4 the highest order interactions can be fixed. In this context, the variable selection algorithm will only give the relevant variables among these highest order interactions. However, for each level of interaction you have to start model-building again. [B] PROC CATMOD DATA=data ; DIRECT ; MODEL d = el... ek ik+i il / ML; PROC CATMOD offers an alternative possibility to perform logistic regression analysis. Maximum-likelihood estimators for the model parameters are calculated using the ML option. However, there is no automatic variable selection algorithm implemented in this procedure and, consequently, variable selection has to be carried out "by hand" using differences of the model likelihood values. Proceeding like this, the hierarchy principle can be applied, but in a very time consuming way: each new model resulting by removing a "non-explanatory" variable has to be restarted again. The main advantage in using PROC CATMOD is that categorical variables not listed in the DIRECT statement are coded automatically as dummy variables and a test of the common effect of these dummy variables (simultaneous likelihood-ratio test) is performed. Table 1 summarizes the statistical features relevant for model-building purposes of both procedures. Table 1: Features of the SAS/STAT procedures PROC LOGISTIC and PROC CATMOD for logistic regression model-building Features Name of automatic automatic SAS procedure variable selection hierarchy-principle dummy-coding PROC LOGISTIC x - - PROC CATMOD - - x. DEMAND FOR IMPLEMENTATION OF HIERARCHY-PRINCIPLE AND DUMMY-CODING IN VARIABLE SELECTION ALGORITHMS As already mentioned, implementation of the hierarchy-principle and simultaneous analysis of dummy variables is necessary for using statistical software in model-building. The results of SAS software-ballots of the last years demonstrate that a substantial proportion of SAS users miss also these tools within the SAS software. Table 2 gives the corresponding items of the ballot together with the number of votes and ranking. 1048

5 Table 2: SAS software-ballot results ( ) for topics connected with modelbuilding in logistic regression procedures SAS procedure SUGI-Ballot Rank Votes Item PROC provide a mechanism for variables to enter and LOGISTIC leave the model as a group during model selection add a CLASS statement allowing all resulting dummy variables for a variable to enter and leave the model together during model selection add a CLASS statement allowing all resulting dummy variables for a variable to enter and leave the model together during model selection provide a TEST statement to test linear combinations of parameters PROC implement stepwise selection of variables CATMOD implement stepwise selection of variables allow automatic model selection It should not be a problem to implement such analytical model-building strategies in PROC LOGISTIC: Modelling of interactions could be included into the model statement allowing the same notation as in PROC GLM multiplying the main effects, using an asterisk (*). In doing so, the procedure will know which variables are interactions and of what order they are. As a consequence, implementation of the hierarchy-principle in automatic variable selection algorithms then will not present any problem. Activation of the hierarchy-principle could be implemented by a separate option (like HIERARCHY). The following syntactical approach can handle this problem: PROC LOGISTIC DATA=data ; MODEL d = el... ek el * e2 el * e3... ek-l * ek jlink=logit SELECTION=.. HIERARCHY; By implementing an extra statement (like CLASS) the identification of categorical or ordinal variables becomes possible. Like in PROC CATMOD, automatic dummy coding and simultaneous assessment of the common effects can be applied. \ \, ;'_ -_;" -, ;,_., ~~'_'r' _ r, _.;.- _.,' _.", ~ ".:;~:..._

6 REFERENCES [1] Kron, M., Gefeller, O. (1992). A critical appraisal of variable selection methods in regression models. In: A. Westlake, R. Blanks, C. Payne, T. Orchards (Eds.): Survey and statistical computing. North-Holland, Amsterdam, [2] Hosmer, D.W., Lemeshow, S. (1989). Applied Logistic Regression. John Wiley, New York [3] Aldrich, J.H.,Nelson, F.D. (1984). Linear probability, logit, and probit models. SAGE Publications, Beverly Hills [4] Cox, D.R., Snell, E.J. (1989). Analysis of binary data. Chapman and Hall, London [5] Greenland, S. (1989). Modelling and variable selection in epidemiologic analysis. Am. J. Public Health 79, [6] Harrell, F.E., Lee, K.L., Califf, R.M., Pryor, D.B., Rosati, R.A. (1984). Regression modelling strategies for improved prognostic prediction. Statistics in Medicine 3, [7] Miller, A.J. (1984). Selection of subsets of regression variables (with discussion). J.R. Statist. Soc. A 147, [8] Kleinbaum, D.G., Kupper, L.L., Morgenstern, H. (1982). Epidemiologic Research: Principles and quantitative methods. Lifetime Learning Publications, Belmont/California [9] Kleinbaum, D.G., Kupper, L.L., Chambless, L.E. (1982). Logistic regression analysis of epidemiologic data: Theory and practice. Commun. Stat. - Theor. Meth. 11, John Wiley, New York [10] Bishop, Y.M.M., Fienberg, S.E., Holland, P.W. (1975). Discrete multivariate analysis. MIT Press, Cambridge [11] Cohen, A. (1991). Dummy variables in stepwise regression. The American Statistician 45, [12] SAS Institute Inc. (1990). SAS/STAT User's Guide, 4th Edition. SAS Institute Inc., Cary, NC. [13] Gefeller, O., Muche, R. (1993). Variable Selection Techniques implemented in Procedures of the SAS software. Proceedings of SEUGI 93, Address for correspondence: Dipl.-Stat. Rainer Muche Klinische Dokumentation, Universitat Ulm D VIm, Germany SAS and SAS/STAT are registered trademarks of SAS Institute Inc., Cary, NC, USA. \, 1050