# Continuation ratio and Cumulative proportional odds model

I seek the equivalence in terms of parameters of the two models. Consider and i.i.d data set $(x_1,y_1),...,(x_n,y_n)$. Now lets assume first that $Y$ is categorical with say 4 levels. Then a cumulative logistic regression model assuming proportional odds is given $$\mbox{logit}(P(Y\leq j|X)=\alpha_j+\beta X.$$ The forward continuation ratio model is given by $$\mbox{log}(P(Y=j|Y\geq j,X)=\gamma_j+\theta X.$$ Can some prove to me the relationship between the $\alpha_j,\gamma_j,\theta$ and $\beta$?

The equivalence of these two models with the complementary log-log link has been established I just wish to see a similar establishment for the logit link.

@ARTICLE{laa85equ,
author = {L{\"a}{\"a}r{\"a}, E. and Matthews, J. N. S.},
year = 1985,
title = {The equivalence of two models for ordinal data},
journal = "Biometrika",
volume = 72,
pages = {206-7},
annote = {ordinal logistic model; continuation ratio; complementary log-log}}


established an equivalence of the forward continuation ration model and the complementary log-log cumulative probability ordinal model. I have yet to fully understand that, and have run one test and did not see the equivalence but there may have been a notational issue related to "this category vs. the next category". But there is no relationship to my knowledge with the proportional odds model.

The notation you are using, while common, does not make the proportional odds model consistent with the binary logistic model; the coefficients are negated. That's why I prefer $Y \geq j$ notation.

Note that the forward continuation ratio model is a model for the discrete hazard function and anti-logged $\beta$s are discrete hazard ratios. This connects with the Cox model continuous-time proportional hazards model, which generates an interpretation of $\beta$s as differences in log-log cumulative survival probability ($Pr[Y > t]$).

• Thanks Frank, the equivalence proven in the paper you quoted is using the logarithm property: $\mbox{log}(a-c)=\mbox{log}a+\mbox{log}(1-\frac{c}{a})$ for a$>$c. I used this property and proved it, and the effect sizes ($\beta$) are the same but with a different intercept. With the proportional odds model, with a logit link, I found out that the forward continuation ratio model is the sum of the proportional odds model and backward continuation model. – Chamberlain Foncha Apr 3 '14 at 15:03
• Very interesting. That particular equivalence does not aid in interpretation, I think. – Frank Harrell Apr 3 '14 at 16:05