Characteristics of Ordinal Data in Trend Odds Model

Ordinal data widely appear in many scientific areas, are often assumed with proportional odds in ordinal logistic regression. When the ordinal data have monotone manner across cut-points, proportional odds assumption become less appropriate. The lack of proportionality may be possible captured using trend odds model which have constrained structural relationship between the odds and the cut-points. This paper describes briefly cumulative odds which are the foundation of trend odds model and demonstrates algebraically the trend odds model related to latent logistic and uniform distribution to obtain the explicit form of slope of the trend. A data set is used to illustrate the interpretation of trend odds model, we apply this model to cigarettes consumption which has monotone manner across cut-points that seems the proportional assumption is more likely not appropriate. However, characteristic of ordinal data that fit trend odds model properly must be specified because only monotone manner is insufficient. We perform simulation using PROC NLMIXED in SAS system and find that ordinal data must significantly monotone for unexposed group and the ratio between covariates is higher for higher level cut-points.