An Example of Analyzing Ordinal Categories of Response Using Continuation-Ratio Models
Henry S.H. Hsu, Ph.D., MPH; Peter A. Lachenbruch, Ph.D; Rolf. E. Taffs, Ph.D.
1401 Rockville Pike, HFM-215, Rockville, MD 20854
When the categories of response probabilities X1, X2, ..., Xk are ordered, the cumulative response probabilities, K1 = X1, K2 = X1+ X2, ... , Kk = X1+X2+...+Xk = 1, provide a basis to use the logit link function in the linear logistic regression. The parallel-lines regression model has the form:
log{Kj(x) / [1!Kj(x)]} = Ij + JTx, where J is the vector of slope parameters and x is the covariate vector. This proportional-odds model is commonly used in analysis of polychotomous data. However, there are situations such as toxicity or adverse event studies whose ordinal categories of response are formed as the sequence of conditional factors according to a nested or hierarchical scale. The probability of a positive response is Xj/(1!Kj-1), and the probability of a negative response is (1!Kj)/(1!Kj-1). Thus, the log odds corresponding to category j in a continuous-ratio model are log{Xj/(1!Kj)} which use the individual probabilities, rather than their cumulative distribution probabilities. In this presentation, we have used a computational procedure proposed by Berridge and Whitehead (1991) to analyze the neurovirulence test data of oral poliomyelitis vaccine, in which the nature of lesion severity was in ordinal hierarchical categories. We also compared the results based on the continuation-odds model and the proportional-odds model.