Using the Proportional Odds Model for Health-Related Outcomes: Why, When, and How with Various SAS® Procedures

Health-related outcomes often possess an intrinsic ordering but fail to meet the assumptions usually needed to perform an ordinary least-squares (OLS) regression. When the distribution of scores is highly non-normal, as occurs when the majority of respondents score at the very bottom or top of the scale, ordinal regression can be more valid, and sometimes more informative, than OLS regression. In SAS/STAT® version 9, PROC LOGISTIC allows one to fit a proportional odds model and to test the validity of interpreting significant predictors as affecting the outcome regardless of where the ordinal outcome might be divided into “higher” vs. “lower” scores. An example is provided based on an analysis of patient scores on the Sheehan Disability Scale, an oft-used outcome measure in mental health research. The model includes various mental disorder diagnoses as primary predictors and several sociodemographic variables as covariates. Results from the proportional odds model are compared to analogous results from an OLS regression obtained with PROC GLM. Other SAS/STAT procedures, such as PROC GENMOD and PROC PROBIT, can also be used to fit proportional odds models, and the differences in assumptions, modeling details, and available output will be described. INTRODUCTION The purpose of this paper is to explain the basic features and implementation of the proportional odds model (POM) in SAS. During the past few decades, several methods of ordinal logistic regression have been formalized. In these methods, the ordinality of a response variable having more than two levels can be incorporated into a logistic model. Many outcome variables are difficult, if not impossible, to measure on an interval scale. In biomedical research, for instance, constructs such as self-perceived health can be measured plausibly on an ordinal scale (“very unhealthy,” “unhealthy”, “healthy”, “very healthy”), but numeric values assigned to these levels are at best arbitrary, and may lead to erroneous conclusions when analyzed as if they were equally-spaced points on a continuum (as in ordinary least-squares regression). When faced with this problem, one option is to dichotomize the ordinal outcome and run a binary logistic regression. However, the loss of information and decrease in statistical power are often too high a price to pay. Moreover, the resulting odds ratios may depend on the cut point chosen to dichotomize the outcome, and this choice is often arbitrary. Ordinal logistic regression overcomes some of these problems. Like in binary and multinomial logistic regression, predictors may be categorical and/or continuous, and the computation of crude or adjusted odds ratios is the typical goal. The unique feature of the POM is that the odds ratio for each predictor is taken to be constant across all possible collapsings of the response variable. When a testable assumption is met, odds ratios in a POM are interpreted as the odds of being “lower” or “higher” on the outcome variable across the entire range of the outcome. The wide applicability and intuitive interpretation of the POM are two reasons for its being considered the most popular model for ordinal logistic regression. This paper will be most useful to SAS users already familiar with binary and/or multinomial logistic regression as implemented in the LOGISTIC and GENMOD procedures. After a brief description of the conceptual basis of the POM, the syntax required to implement a POM will be described. This will be followed by an example from a study conducted in a primary care clinic, where strategies for building a predictive model of mental health-related disability were not clear-cut. SAS/STAT version 9 has been used for all the examples. THE PROPORTIONAL ODDS MODEL The proportional odds model (POM) described by McCullagh (1980) is the most popular model for ordinal logistic regression (Bender & Grouven, 1998). The POM is sometimes referred to as the cumulative logit model, however the latter is actually a more general term. In SAS, three types of cumulative logit models are available: the POM (available in the LOGISTIC, PROBIT, and GENMOD procedures), the partial proportional odds model (available in the GENMOD procedure), and the nonproportional odds model (available in the CATMOD procedure). The hallmark of the POM is that the odds ratio for a predictor can be interpreted as a summary of the odds ratios obtained from separate binary logistic regressions using all possible cut points of the ordinal outcome (Scott et al., 1997). Whereas a binary logistic regression models a single logit, the POM models several cumulative logits. Therefore, if the ordinal outcome has four levels (1, 2, 3, and 4), three logits will be modeled, one for each of the following cut points: 1 vs. 2,3,4; 1,2 vs. 3,4; and 1,2,3 vs. 4. Because the various logits in a single POM are constrained to be equal, the POM has also been referred to as the constrained cumulative logit model (Hosmer & Lemeshow, 2000). Details on the statistical theory behind the POM can be found in several books and articles (see References). Agresti (1996) and Hosmer and Lemeshow (2000) are two sources that offer particularly clear explanations, and a SAS-specific approach can be found in sources such as Allison (2001). Statistics and Data Analysis SUGI 30