I am performing ordinal regression, I have 5 response categories and several predictors both continuous and categorical. I would like to add a predictor which is categorical but ordered (1, 2, 3, 4). I don't think it would be appropriate to apply the usual dummy coding for unordered categorical predictors, but when I searched for how to code this I did not find much information. In Steyerberg (2009) "linear coding" or "assuming linearity of the predictor effect" is mentioned, but without further details. Does it mean I just use my ordered values as they are, i.e. use them as a continuous variable?
You could check out Gertheiss & Tutz, Penalized Regression with Ordinal Predictors, & their R package ordPens. They say:–
Rather than estimating the parameters by simple maximum likelihood methods we propose to penalize differences between coefficients of adjacent categories in the estimation procedure. The rationale behind is as follows: the response $y$ is assumed to change slowly between two adjacent categories of the independent variable. In other words, we try to avoid high jumps and prefer a smoother coefficient vector.