From Alan Agresti:

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Response to reviewers criticizing use of ordinal models:

The proportional odds ordinal logistic model accomodates the strange statistical distribution (and possible floor and ceiling effects) of variables such as angina frequency.  It involves no categorization and is statistically very efficient while only using the rank order of frequency across patients.  The commonly used Wilcoxon-Mann-Whitney 2-sample rank-sum test is a special case of this ordinal logistic model when there is only one covariate and it is binary.  Even if the response variable were normally distributed, the proportional odds model has efficiency of 3/pi or about 0.95.

In the text add a footnote or parenthetical text the first time you mention an odds ratio for the ordinal model results:

This odds ratio does not come from a dichotomization of angina frequency but from the proportional odds model and involves the ratio of odds of a frequency > f for two groups, for any non-zero f.


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The primary efficacy hypothesis will be tested with a mixed effects proportional odds ordinal logistic model using the ordinal package in R.  The fixed effects time trend in log odds of the WHO scale exceeding level j (for all j) will be modeled using a quadratic time effect, and all available data (until subject dropout) will be used in the analysis.  The model's random effects corresponds to subjects.  Because of known occasional convergence problems with frequentist mixed effects ordinal models, if this frequentist approach is not robust we will use a Bayesian hierarchical mixed effects proportional odds model using the R brms package, and will report the exact posterior probability of efficacy.  The prior distribution on the log odds ratio for treatment would be a normal distribution with standard deviation computped so that the prior probability of a treatment effect that yields a hazard ratio < 0.5 is only 0.05.  Note that the treatment hazard ratio in the ordinal logistic model does not use cutoffs in analyzing the WHO scale, but the hazard ratios correspond to exceedance probabilities, i.e. P(WHO >= j) for all j other than the first level.  Because the frequentist and Bayesian ordinal mixed effects models are full likelihood models, they only assume missing at random if the longitudinal records (not missing completely at random).  Hence they do not require imputation of outcomes for missed visits.

------------- Alan Agresti 2020-03-24
Thanks for your email.  This is an interesting modeling problem, and as usual your creative ideas seem very sensible to me.  I can't recall seeing a paper dealing with this in the ordinal-modelling literature, but I've not kept up with it since I revised my book on ordinal categorical data in 2010.

In your separate email to David and Scott, you mentioned having a student doing a dissertation on a Bayesian approach to the proportional odds model.  I imagine you've seen that Hoff's Bayesian book has a good chapter on this.  I guess quite a bit has been done on such models since the Albert and Chib (1993) latent variable approach with the probit link, so it will be interesting to see what your student comes up with.  How about other ordinal models with proportional odds structure, such as adjacent-categories logit or sequential logit?  Or your own partial proportional odds for any such model?

Well, what a nightmare having Covid-19 to motivate statistical analyses.  I now have three statistician friends in Italy who have it, one who has also lost both parents.  Best wishes, David, in your work.

Alan