Modeling Monotonic Effects of Ordinal Predictors in Bayesian Regression Models

Abstract

Ordinal predictors are commonly used in regression models. They are often incorrectly treated as either nominal or metric, thus under- or overestimating the contained information. 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 modeling 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 $\zeta$ modeling 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 not only be applied 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, in case of monotonicity, has similar or even better predictive performance than other approaches designed to handle ordinal predictors. Using Stan, we developed a Bayesian estimation method for monotonic effects, which allows 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.

Intonation does aid serial recall after all

“…The main fixed effects were response modality (spoken, keyboard, grid) Contour A and B sequences, vertical broken lines mark group boundaries (intonational phrases), whereas in the grouped-by-pause sequence, double vertical broken lines mark silent intervals (pauses) between groups and condition (Intonation A, Intonation B, pause, control). In addition, we included a fixed effect for "position within triplet," which was added as a monotonic variable (see Bürkner & Charpentier, 2018). This variable codes for the first, second, and third position within each triplet (1, 2, 3 versus 4, 5, 6 versus 7, 8, 9).…”

Prestige Does Not Affect the Cultural Transmission of Novel Controversial Arguments in an Online Transmission Chain Experiment

“…This was a change from the preregistered script (https://osf.io/dt2uq/), in which we did the analyses with dummy data with the package lme4 (Bates, Maechler, Bolker, & Walker, 2015). The change was due to the advantage of brms in handling ordinal predictors (Bürkner & Charpentier, 2018). This change did not affect the qualitative conclusions derived from the results (see SM2 and SM3).…”

“…We first ran a generation model with generation (one to four) as the sole predictor of recall (number of correctly recalled propositions). Generation was treated as a monotonic variable (Bürkner & Charpentier, 2018) as recall tends to decrease across generations but the amount of the decrease varies between adjacent generations. This model was compared with an opinion model, which included opinion ('pro-tablets' vs 'anti-tablets') and generation as predictors.…”

mcp: An R Package for Regression With Multiple Change Points

“…mcp supports a scaled and shifted Dirichlet prior on the difference(s) between adjacent change points -an approach that is also used to model monotonic effects in brms (Bürkner and Charpentier, 2018). The Dirichlet is itself a simplex: ζ k > 0 ensures ordering and while the support is ζ k ∈ (0, 1) the property that K−1 k=1 ζ k = 1 renders it easy to shift and scale the support to the observed range of x ([min(x), max(x)]).…”