Analyzing Ordinal Repeated Measures Data Using SAS (r)
This paper provides a brief review of commonly used statistical methods for analyses of ordinal response data. Generalized CMH Score Tests of Marginal Homogeneity, GEE, and random intercepts logistic regression ordinal model for analysis of repeated ordinal response data will be particularly discussed. SAS procedures, Proc NLMIXED, Proc GENMOD, Proc IML, and Proc FREQ for categorical ordinal analysis, are described and illustrated with data from a clinical trial. A SAS macro to produce estimated marginal Probabilities will be presented. INTRODUCTION Ordinal variables are common in clinical research studies. Ordinal variables have a hierarchical ordering, Such as Severity score (none, mild, moderate, severe) Repeated measures refer to multiple measurements taken from the experimental unit over time. For example, Many clinical studies require patients to return to the clinic at several times and observe the response variable at each visit. The layout for the sample data with two treatments is shown in the following: PATINET TREATMENT VISIT Y1 Y2 Y3 1000 A 1 111 y 112 y 113 y 6 161 y 162 y 163 y 1001 B 1 211 y 212 y 213 y 6 261 y 262 y 263 y These longitudinally repeated measures can be used to characterize a response profile over time. Repeated measures within a subject are usually positively correlated. The correlation structure of the longitudinal data can be described by random subject effects. Random subject effects indicate the degree of subject variation that exists in the population of subjects. Data from studies with repeated measurement in general are incomplete due to drop out. We will use terminology of little and Rubin (1987, Chapter 6) for the missing value process. A non response process is said to be missing completely a random (MCAR) if the missing is independent of both unobserved and observed data and missing at random (MAR) if, conditional on the observed data, the missing is independent of the unobserved measurements. When either of these is plausible, with a likelihood based analysis it is not necessary to model the missingness mechanism (Agresti A. 2002) For ordinal response data, we can compare mean response score between treatments by assigning numerically score which are the ranks of the categories when ordered from smallest to largest. This continuous model does not take into account the ceiling and floor effects of the ordinal outcome. The result can be biased when the ordinal variable is highly skewed. Here we first use a generalized CMH test for testing marginal homogeneity at the end of the study. Next, we use the GEE method implemented in the GENMOD procedure to fit a marginal model. The mean response of the marginal model depends only on covariates variable, and not on random effects. GEE uses quasi likelihood estimation and assumes that the missing data are MCAR. To accommodate random effects, we describe a mixed effects proportional odds model using the NLMIXED procedure. The NLMIXED procedure relies on approximating the marginal log likelihood by integral approximation through Gaussian quadrature. The objective function for the NLMIXED procedure is the marginal log likelihood obtained by integrating out the random effects from the joint distribution of responses and random effects using quadrature techniques. Although these are very accurate, the number of random effects that can be practically managed is limited (SAS/STAT) Since the estimation of model parameters is based on Maximum likelihood approach, the missing data are assumed to be MAR.