# Ordinal logistic regression assumption

I aim to build for the first time an ordinal logistic regression between a dependent ordinal categorical variable (temperature sensation vote TSV, Y) coded on a bipolar ordered 6-point scale (from very cold to very hot) and a continuous independent variable, which represents the intensity of human body stress to heat (X).

I checked following the book of Harrel F., 2015 ("Regression modeling strategies: with applications to linear models, logistic and ordinal regression, and survival analysis", chapter 13 )the ordinality assumption by computing the mean of the heat stress indices X by levels of temperature sensation Y. " a basic assumption of all commonly used ordinal regression model is that the response variable behaves in an ordinal fashion with respect to each predictor. Assuming that the predictor X is linearly related to the log of the odds of some appropriate event, a simple way to check for ordinality is to plot the mean of X stratified by levels of Y. These means should be in a consistent order. if for many of the Xs, two adjacent categories of Y do not distinguish the means, that it is the evidence that those levels of Y should be pooled."

It appears that the means of X do not always behave in an ordinal fashion.

Here, is an example of the means of X stratified by levels of Y for a given location:

In my case, I cannot pool the Y as I need to build the best accurate predictive model for different locations and compare them. The question is what happens if this assumption is not fully respected by the data? Can I still apply an ordinal logistic regression? This is not stated in the book.

Answer from Frank Harrel posted on the comments of a deleted answer: "You're interpreted the material in RMS correctly. But you neglected to put sample sizes in the table. When all cell sample sizes are not huge you can't worry too much about a lack of ordinality. Note for others not used to thinking this way: The mean of X | Y is highly related to score residuals from the ordinal logistic model that are used to check the proportional odds assumption when X has a linear effect on log odds. For resources on ordinal models see: fharrell.com/post/rpo and fharrell.com/post/impactpo, the latter showing the harm of not assuming prop. odds. – Frank Harrell, Jan 4 at 10:25 "

• "It appears that the means of X do not always behave in an ordinal fashion." What does this mean? Ordinality is about the meaning of the measurement scale. You can't diagnose from the data whether a variable is ordinal or not. Commented Jan 3, 2023 at 23:40
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– Sycorax
Commented Jan 4, 2023 at 13:55
• Referring to the comments for your deleted answer, the assumption to be checked here is not "ordinality", but proportional odds (and maybe monotonicity). Commented Jan 4, 2023 at 23:03

What you observe isn't that much different from what one often sees in ordinary least squares regression with multiple observations at a set of predictor values. Even with an overall upward trend of outcome as a function of predictor values, you still might occasionally have higher average outcomes at a lower predictor value than at a higher predictor value. Sometimes that's the luck of the draw, particularly with small sample sizes. Sometimes it means the model isn't well specified.

As you seem to have theoretical reasons to consider an ordinal relationship between Y and X, then it makes sense to try to fit the model and then use the methods in Chapters 13 and 14 of Regression Modeling Strategies to document how well the assumptions of the model hold overall. Don't forget that flexible modeling of a continuous predictor, e.g. with splines, can be as helpful in ordinal regression as it is in other types of regression. Even with this limited sample size, it looks like you could envision using up 3 degrees of freedom in fitting results for this location.