Abstract

A Bayes factor is proposed for testing whether the effect of a key predictor variable on a dependent variable is linear or nonlinear, possibly while controlling for certain covariates. The test can be used (i) in substantive research for assessing the nature of the relationship between certain variables based on scientific expectations, and (ii) for statistical model building to infer whether a (transformed) variable should be added as a linear or nonlinear predictor in a regression model. Under the nonlinear model, a Gaussian process prior is employed using a parameterization similar to Zellner’s g prior resulting in a scale-invariant test. Unlike existing p-values, the proposed Bayes factor can be used for quantifying the relative evidence in the data in favor of linearity. Furthermore the Bayes factor does not overestimate the evidence against the linear null model resulting in more parsimonious models. An extension is proposed for Bayesian one-sided testing of whether a nonlinear effect is consistently positive, consistently negative, or neither. Applications are provided from various fields including social network research and education.

摘要

贝叶斯因子被提议用于测试关键预测变量对因变量的影响是线性的还是非线性的，可能同时控制某些协变量。该测试可用于 (i) 在实质性研究中根据科学预期评估某些变量之间关系的性质，以及 (ii) 用于统计模型构建以推断（转换后的）变量是否应添加为线性或非线性变量回归模型中的预测变量。在非线性模型下，使用类似于 Zellner 的g先验的参数化来采用高斯过程先验，从而进行尺度不变测试。不同于现有的p-values，所提出的贝叶斯因子可用于量化数据中有利于线性的相关证据。此外，贝叶斯因子不会高估导致更简约模型的线性零模型的证据。针对非线性效应是否始终为正、始终为负或两者都不是的贝叶斯单边测试提出了扩展。应用程序来自各个领域，包括社交网络研究和教育。