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When the dependent variable in a regression model is ordinal, I know that we often use ordered probit/logit to estimate the model. These have an assumption called the parallel regression assumption. It states that if we fix the order of the outcomes 0, 1, 2, ... , J and partition them into two categories with outcomes 0, 1,... , m in one category (labelled 0) and outcomes m+1, m+2, ... J in another (labelled 1) and fit a binary probit model, then the coefficients associated with the independent variables will be the same regardless of the value of m.

However, what do we do when this assumption is violated? My hunch is to simply run multinomial logit/probit, but that throws out all of the information contained in the ordering of the dependent variable. Is there a better way to approach this?

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It looks to me that you are looking for the "partial" version of proportional odds. Reference:

R. S. Society, “Partial Proportional Odds Models for Ordinal Response Variables,” vol. 39, no. 2, pp. 205–217, 1999.

If in the "standard" ordered logit (or proportional odds), the cumulative probability is modeled as

$$ P(Y > j | X_i) = \frac{1}{1 + \exp(-\alpha_j - X_i \beta)} $$

where $\alpha$ is the vector of thresholds (as many as the number of classes - 1) and $\beta$ is the vector of coefficients. In the partial version of the proportional odds model, the cumulative probability takes instead the more general form $$ P(Y > j | X_i) = \frac{1}{1 + \exp(-\alpha_j - X_i \beta - T_i \gamma_j )} $$ where $T$ is a vector containing the values of observation $i$ on that subset of the explanatory variables for which the proportional odds assumption is either not assumed or not verified, and $\gamma_i$ is a vector of coefficients (to be estimated) associated with the variables in $T$.

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    $\begingroup$ Can you say more about this '"partial" version of proportional odds'? $\endgroup$ – gung Oct 20 '15 at 1:25
  • $\begingroup$ I expanded my answer with a brief description of this model. $\endgroup$ – Fabian Pedregosa Oct 21 '15 at 8:36

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