# Modelling monotonic effects of ordinal predictors in Bayesian regression models

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**Modelling monotonic effects of ordinal predictors in Bayesian regression models.** / Bürkner, Paul Christian; Charpentier, Emmanuel.

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*BRITISH JOURNAL OF MATHEMATICAL AND STATISTICAL PSYCHOLOGY*. https://doi.org/10.1111/bmsp.12195

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*BRITISH JOURNAL OF MATHEMATICAL AND STATISTICAL PSYCHOLOGY*. https://doi.org/10.1111/bmsp.12195

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TY - JOUR

T1 - Modelling monotonic effects of ordinal predictors in Bayesian regression models

AU - Bürkner, Paul Christian

AU - Charpentier, Emmanuel

PY - 2020/1/1

Y1 - 2020/1/1

N2 - Ordinal predictors are commonly used in regression models. They are often incorrectly treated as either nominal or metric, thus under- or overestimating the information contained. Such practices may lead to worse inference and predictions compared to methods which are specifically designed for this purpose. We propose a new method for modelling ordinal predictors that applies in situations in which it is reasonable to assume their effects to be monotonic. The parameterization of such monotonic effects is realized in terms of a scale parameter b representing the direction and size of the effect and a simplex parameter (Formula presented.) modelling the normalized differences between categories. This ensures that predictions increase or decrease monotonically, while changes between adjacent categories may vary across categories. This formulation generalizes to interaction terms as well as multilevel structures. Monotonic effects may be applied not only to ordinal predictors, but also to other discrete variables for which a monotonic relationship is plausible. In simulation studies we show that the model is well calibrated and, if there is monotonicity present, exhibits predictive performance similar to or even better than other approaches designed to handle ordinal predictors. Using Stan, we developed a Bayesian estimation method for monotonic effects which allows us to incorporate prior information and to check the assumption of monotonicity. We have implemented this method in the R package brms, so that fitting monotonic effects in a fully Bayesian framework is now straightforward.

AB - Ordinal predictors are commonly used in regression models. They are often incorrectly treated as either nominal or metric, thus under- or overestimating the information contained. Such practices may lead to worse inference and predictions compared to methods which are specifically designed for this purpose. We propose a new method for modelling ordinal predictors that applies in situations in which it is reasonable to assume their effects to be monotonic. The parameterization of such monotonic effects is realized in terms of a scale parameter b representing the direction and size of the effect and a simplex parameter (Formula presented.) modelling the normalized differences between categories. This ensures that predictions increase or decrease monotonically, while changes between adjacent categories may vary across categories. This formulation generalizes to interaction terms as well as multilevel structures. Monotonic effects may be applied not only to ordinal predictors, but also to other discrete variables for which a monotonic relationship is plausible. In simulation studies we show that the model is well calibrated and, if there is monotonicity present, exhibits predictive performance similar to or even better than other approaches designed to handle ordinal predictors. Using Stan, we developed a Bayesian estimation method for monotonic effects which allows us to incorporate prior information and to check the assumption of monotonicity. We have implemented this method in the R package brms, so that fitting monotonic effects in a fully Bayesian framework is now straightforward.

KW - Bayesian statistics

KW - brms

KW - isotonic regression

KW - ordinal variables

KW - R

KW - Stan

UR - http://www.scopus.com/inward/record.url?scp=85077868836&partnerID=8YFLogxK

U2 - 10.1111/bmsp.12195

DO - 10.1111/bmsp.12195

M3 - Article

AN - SCOPUS:85077868836

JO - BRITISH JOURNAL OF MATHEMATICAL AND STATISTICAL PSYCHOLOGY

JF - BRITISH JOURNAL OF MATHEMATICAL AND STATISTICAL PSYCHOLOGY

SN - 0007-1102

ER -

ID: 40539566