The equivalence of two models for ordinal data

SUMMARY The equivalence of the grouped continuous and continuation ratio models for ordinal data, when the complementary log-log link is used, is demonstrated. Recent experience using two regression models for ordinal data gave essentially identical results when the complementary log-log transform was used for the probabilities. Pregibon (1980) implies that these models are the same; the present note makes the equivalence explicit. , Suppose that Y is an ordinal response variable whose values are labelled 1, ..., k. Let 7rj(x) = pr ( Y = j I x), for an observation with covariate vector x, and yj(x) = 1 (x) + ... + rj(x) = pr (Y K j I x) and set bj(x) = pr (Y = Ij Y > j, x) = 7tj(x)/{I-yj l(x)} (j = 1, ...,k-1). Grouped continuous models fit a linear predictor to some monotonic function, f, of the cumulative probabilities yj(x), namely f {7j(x)} =oj oc x. Continuation ratio models use the probabilities bj(x) instead of the yj's, that is