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 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.

中文翻译：

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贝叶斯回归模型中序数预测变量的单调效应建模。

序数预测变量通常用于回归模型。它们经常被错误地视为名义或公制，从而低估或高估所包含的信息。与专门为此目的设计的方法相比，这种做法可能会导致更糟糕的推理和预测。我们提出了一种新方法来对有序预测变量进行建模，该方法适用于假设它们的影响是单调的情况是合理的。这种单调效应的参数化是通过一个表示效应方向和大小的尺度参数b和一个单纯形参数来实现的对类别之间的标准化差异进行建模。这确保预测单调增加或减少，而相邻类别之间的变化可能因类别而异。该公式可推广到交互项以及多级结构。单调效应不仅可以应用于序数预测变量，还可以应用于其他可能存在单调关系的离散变量。在模拟研究中，我们表明该模型经过良好校准，并且如果存在单调性，则其预测性能与旨在处理有序预测变量的其他方法相似甚至更好。使用 Stan，我们开发了一种用于单调效应的贝叶斯估计方法，它允许我们合并先验信息并检查单调性假设。brms，以便在完全贝叶斯框架中拟合单调效应现在很简单。