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I am using ordinal R-package to fit a cumulative link mixed model to an ordered, categorical outcome (5 levels) using logit function as the link function. The model is a random intercepts only model. I was thinking that since the probability of a response $Y_i$ is conditionally multinomial on estimated conditional modes of random effects, i.e.

$$\begin{align} \operatorname {logit} [P(Y_{ijk} \le j \mid x, b_k)] = \alpha_j - x_{i}\beta + b_k, \end{align}$$

where $i$ indexes observations, $j$ indexes response category and $k$ indexes participants, I should (somehow) incorporate estimated conditional modes $b_k$ into a visual diagnostic* of the proportional odds assumption. If the assumption is violated, I would analyse the data with a non-proportional odds model. What would be an appropriate way to incorporate estimated conditional modes of random effects into the visual plot? If this is not a good approach, is there an another way to check the proportional odds assumption?

*Please see the second figure from this answer for reference.

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Lack of fit is a trickier concept than is apparent. That's because assessing lack of fit increases model uncertainty (and true but not nominal confidence interval widths) and especially because fixes for lack of fit (e.g., relaxing the proportional odds assumption with a partial proportional odds model) can be worse than the lack of fit. So it is more important to assess the impact of the assumptions and of altering the model to account for lack of fit. See this article and the R rms package impactPO function.

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