Suppose I have an ordinal variable $Y$ that takes values $K=\{1,\dots,k\}$ (e.g., a Likert scale, or a score from 1 to 10) and predictors $\boldsymbol{X}$. I fit two ordinal regression models containing parameters $\boldsymbol{\beta}$ and thresholds $\boldsymbol{\theta}$.

  1. $P(Y\leq i|\boldsymbol{X})= f_1(\boldsymbol{X};\boldsymbol{\beta}_1;\boldsymbol{\theta}_1)$ for $i\in K$, where $f_1$ is the complementary log-log inverse link function (link function $f^{-1}_1=log[-log(1-\gamma)]$).

  2. $P(rev(Y)\leq i^\prime|\boldsymbol{X})= f_2(\boldsymbol{X};\boldsymbol{\beta}_2;\boldsymbol{\theta}_2)$ for $i^\prime\in K^\prime$, where $f_2$ is the (negative*) log-log inverse link function (link function $f^{-1}_2=-log[-log(\gamma)]$), $rev(Y)$ indicates that the levels of $Y$ are reversed (the $k$th level becomes level $1$, $k-1$th level becomes level $2$, etc) and $K^\prime=rev(K)$ (meaning $i^\prime=k-i+1$).

In both, $\gamma$ denotes the probability that the event in $P(\cdot)$ occurs.

My intuition is that these two models will be equivalent, or at least very closely related, for two (admittedly imprecise) reasons:

  1. $P(Y\leq i|\boldsymbol{X})=P(rev(Y)> i^\prime|\boldsymbol{X})^\ddagger=1-P(rev(Y)\leq i^\prime|\boldsymbol{X})^\ddagger$ and $f_1$ models $1-\gamma$ while $f_2$ models $\gamma$. (In this case, the two models would generate complementary probabilities without reversing variable $Y$, and would generate identical probabilities when one models $rev(Y)$.)

  2. I have been told that the complementary log-log link is typically used when higher categories are more probable (the distribution of $Y$ is left skewed), and the (negative) log-log link is typically used when lower categories are more probable (the distribution of $Y$ is right skewed).

How close is my intuition here? Can anyone provide some mathematical clarification as to what form the relationship between the two models does (or does not) take?

And if there is a relationship, how are the coefficients $\{\boldsymbol{\beta}_1, \boldsymbol{\beta}_2\}$ and the thresholds $\{\boldsymbol{\theta}_1, \boldsymbol{\theta}_2\}$ related between models?

*I have seen some places where the negative log-log link given above is simply called the log-log link (e.g., the R package ordinal). I'm not sure what the convention is for naming this link function.

$^\ddagger$ I'm having trouble determining which of these two inequalities should have the "or equal to." Brain fog.

  • $\begingroup$ Why do you call Y factor? In what sense factor? Y is a dependent variable, while by "factor" we usually mean a categorical independent variable. $\endgroup$ – ttnphns Sep 10 at 4:49
  • $\begingroup$ Sorry if my language was imprecise. I mean "factor" in the R sense: a categorical variable that takes a limited set of values. I meant "ordinal variable". For example, a Likert scale, or a score from 1 to 10. I have edited my post to reflect this language. $\endgroup$ – mdawgig Sep 10 at 12:49

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