Accounting for Estimation Uncertainty and Shrinkage in Bayesian Within-Subject Intervals: A Comment on Nathoo, Kilshaw, and Masson (2018)

To facilitate the interpretation of systematic mean differences in within-subject designs, Nathoo, Kilshaw, and Masson (2018, Journal of Mathematical Psychology, 86, 1-9) proposed a Bayesian within-subject highest-density interval (HDI). However, their approach rests on independent maximum-likelihood estimates for the random effects which do not take estimation uncertainty and shrinkage into account. I propose an extension of Nathoo et al.'s method using a fully Bayesian, two-step approach. First, posterior samples are drawn for the linear mixed model. Second, the within-subject HDI is computed repeatedly based on the posterior samples, thereby accounting for estimation uncertainty and shrinkage. After marginalizing over the posterior distribution, the two-step approach results in a Bayesian within-subject HDI with a width similar to that of the classical within-subject confidence interval proposed by Loftus and Masson (1994, Psychonomic Bulletin & Review, 1, 476-490).

为便于解析受试者内设计的系统平均差异，Nathoo、Kilshaw和Masson（2018年，《数学心理学杂志》，第86卷，第1-9页）提出了一种贝叶斯受试者内最高密度区间（HDI）。然而，他们的方法基于对随机效应的独立最大似然估计，该估计未考虑估计不确定性和收缩。本文提出了一种对Nathoo等人方法的扩展，采用了一种完全贝叶斯的分两步走的方法。首先，从线性混合模型中抽取后验样本。其次，基于后验样本反复计算受试者内HDI，从而考虑估计不确定性和收缩。在对方差分布进行边缘化之后，两步法最终得到一个贝叶斯受试者内HDI，其宽度与Loftus和Masson（1994年，《心理科学通报与评论》，第1卷，第476-490页）提出的经典受试者内置信区间相似。