# Proportional hazards vs proportional odds for modeling ordinal data

When working with ordinal response data, we can use a proportional odds model to calculate the log odds in favor of one category over another. Often we use a logit link yielding the following model:

$${\rm logit}\big(P(Y \leq j|\bf{x})\big) = \alpha_j + \bf{x'\beta}$$

for $j = 1, ..., J$ responses.

Alternatively, I've seen it formulated as a proportional hazards model, which is basically the same model, except we use an Extreme Value Function ($F(x) = 1 - e^{-e^{-x}}$) as the link:

$$\log(-\log(1 - P(Y \leq j|{\bf x})) = \alpha_j + \bf{x'\beta}$$

From empirical analysis, I see that my coefficients and resulting fits are similar in both cases.

Question:

1. Can someone intuitively explain the connection between this proportional hazards model for calculating the odds of an ordinal response, and the proportional hazards model for determining survival probabilities via a hazard function?

Note that I have a mathematical derivation of how a hazard function fits in the proportional hazard model, but the connection between the two seemingly different types of problems eludes me.

2. Is there a situation when one model is preferred over another for ordinal response data (other than just that a logit link is more interpretable than the extreme value link)?

• This article by Bender and Bender here has quite an extensive discussion which I think answers your question. If it is behind a pay-wall it does seem to be available elsewhere on the web. – mdewey Dec 11 '16 at 13:40

• Not sure why the measurement of Hba1c has this property but the fit was better by a long shot. See the figure where I stratified by six intervals of $X\hat{\beta}$ and computed the log-log empirical CDF of $Y$ and the logit empirical CDF of $Y$. – Frank Harrell Dec 13 '16 at 20:56