A family of ordered logistic regression models fit by data expansion.

Sirs—One of us (SRC) coauthored a paper in the International Journal of Epidemiology that detailed a use of data expansion to fit the continuation-ratio model1 as sketched by Armstrong and Slone,2 as well as a paper in the Annals of Epidemiology that detailed a related use of data expansion to fit proportional-odds models.3 In fact, a family of ordered logistic regression models, including continuation-ratio, proportional-odds and adjacent category models, can be fit using data expansion as detailed briefly below. Consider an ordered response Yi = 1, 2, ..., K levels for i = 1, 2, ..., N participants. We can write a general ordered regression model as g [Yij] = x j for j = 1, 2, ..., (K 1) thresholds, where g [ ] is a link function. Setting g [ ] to log [Pr(Y Yj)/Pr(Y = Yj)], log [Pr(Y Yj) / Pr(Y Yj)], or log [Pr(Y = Yj 1)/Pr(Y = Yj)] results in the continuation-ratio, proportional-odds or adjacent category models, respectively. The size of the expanded data set is given by i j I [j y], i j I [1 0], and i j I [j y j 1] for the continuation-ratio, proportional-odds and adjacent category models, respectively, where I [ ] is the indicator function. SAS code (SAS Institute, Cary NC) to expand an observed data set, where K = 4, into the three expanded data sets needed to fit these three members of this family of ordered logistic regression models is provided in the Appendix. Implementation of analysis using any of the members of this family may follow references.1,3 To the best of our knowledge, use of data expansion to fit the adjacent-category ordered logistic regression model has not been previously shown.4 Data expansion is an extremely flexible quantitative tool that has been applied in varied settings. Specifically, in addition to ordered regression data expansion has been employed when a participant has multiple person-time contributions,5 events,6 competing risks,7 or informants.8