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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)?

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ilanman
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  • 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

2 Answers2

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The two approaches for handling either continuous or ordinal responses make the same amount of assumptions. I have a detailed case study in the 2nd edition of Regression Modeling Strategies for which course notes may be obtained at http://biostat.mc.vanderbilt.edu/rms . You'll see that the fit of the PH vs. the PO model was dramatically different for my example, and that the PH model (log-log link) was a nearly perfect fit.

Frank Harrell
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  • Was it obvious why the PH model was a better fit than PO before performing model fit? i.e. log-log link was more suitable than logit link? – ilanman Dec 13 '16 at 19:57
  • 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
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Proportional hazards is a survival model. So this represents a different way to think about your data. More specifically, you need to consider the nature of your response variable. Sometimes study units just have a particular value on a variable that is ordinal in nature (or that isn't only ordinal, but where you don't want to have to rely on distributional assumptions, etc.). Other times, an ordinal variable constitutes a sequence through which units have to pass. An example of the former might be ratings of accident severity by ER doctors. An example of the latter might be educational attainment. Setting aside weird exceptions, you do not move to the next educational level until you've finished the previous one.

The nature of the jump between high school and college might be very different than between college and graduate school. However, the proportional odds model assumes that moves between all levels are the same; the levels can be closer together or further apart, but that's it. On the other hand, because the baseline hazard is left unspecified, whether or not those jumps have the same slope on the latent variable is irrelevant. All that you need is that the hazard for one group is a constant multiple of the hazard for the other group in every step.

gung - Reinstate Monica
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  • Right - difference between PO and PH makes sense to me. But interpreting the hazard function between levels still isn't clear to me. What does the hazard for one group mean, vis-a-vis the hazard for another group? – ilanman Dec 13 '16 at 19:58
  • Survival models can be thought of as modeling the cumulative distribution function, but the actual distribution is unspecified in a Cox PH model. Instead, all you are saying is that, whatever the hazard rate is in the baseline group, the hazard in the treatment group is a fixed multiple of that. In a proportional odds model, you are modeling a latent variable as a straight line (it has the same slope at every point). That line is cut into increments, & the cutpoints can be closer or further apart, but the slope of that line is the same no matter which segment you're in. – gung - Reinstate Monica Dec 14 '16 at 00:19