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I have an ordered regression model as described in ?polr:

The ordered factor which is observed is which bin Y_i falls into with breakpoints

zeta_0 = -Inf < zeta_1 < … < zeta_K = Inf

This leads to the model

logit P(Y <= k | x) = zeta_k - eta

(?polr documents a logit, but in my particular example I'm using a probit.)

I don't understand how the cutpoints zeta are identified in that model. It seems like you should be able to stretch them out as much as you like. (And the coefficients beta would follow suit.) How are these parameters identified mathematically? I know that in some models linear constraints are specified, such as setting one of the coefficients to 0, or stating that they sum to 1. Is that the technique used in ordered probit regression?

Part 2: How does the R package polr in particular identify these parameters? It doesn't seem like any of them is set to 0, and they don't sum to 1. I don't see any answers in ?polr.

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Whether using polr or the R rms package's lrm and orm functions, there is one cutpoint for every unique value of $Y$ except for the first. This is automatic. Printed output of lrm and orm fit objects uses a notation that is a bit more clear. The intercepts are constrained to be in order automatically, and if you compute the correct individual probabilities (see predict.lrm in rms) they will sum to one. But we usually deal with exceedance probabilities.

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  • $\begingroup$ Ok! Those packages are very nice - thanks for the pointer. My current approach to thinking about this is that three constraints are used for ordinal probit by orm and such packages: (1) $\tau_j$'s constrained to be in order. (2) The scale of the $\tau_j$'s is set by the equivalent constraint of $V(\epsilon)=1$. (3) The location of the $\tau_j$'s is set by the equivalent constraint of $\beta_0 = 0$. Am I understanding correctly for all three of these? $\endgroup$ – Hatshepsut May 15 '16 at 21:50
  • $\begingroup$ I don't know about those last 2 items. The constraints are automatic, and if all the $\beta$s were exactly zero, these intercepts are just logit or normal inverse transformations of all the cumulative probabilities of $Y$ values. The is no variance consideration here, nor do these models contain error terms. $\endgroup$ – Frank Harrell May 15 '16 at 22:33
  • $\begingroup$ That's helpful, thanks. I think the approach I described (specifically #2 and #3) is consistent with the latent variable interpretation set up as follows: $$y=j \iff \tau_{j-1}<y^*<\tau_j$$ where $$y^*=X\beta + \epsilon$$ and $E[\epsilon] = 0$, $V[\epsilon] = 1$ which holds in the ordered probit model, where $p(\epsilon) = \phi(\epsilon)$, the standard normal density. The probability that $y_i$ falls into some bin $j$ is equal to the area under the density curve of $y_i^*$ between the cutpoints $\tau_j$ and $\tau_{j+1}$ bounding that bin. $\endgroup$ – Hatshepsut May 15 '16 at 22:55
  • $\begingroup$ Yes - I just find it most helpful to think only about observables. $\endgroup$ – Frank Harrell May 15 '16 at 22:56

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