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:
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.
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)?
Best Answer
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.
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