We develop a Bayesian highest-density interval (HDI) for use in within-subject designs. This credible interval is based on a standard noninformative Jeffreys prior and a modified posterior distribution that conditions on both the data and point estimates of the subject-specific random effects. Conditioning on the estimated random effects removes between-subject variance and produces intervals that are the Bayesian analogue of the within-subject confidence interval proposed in Loftus and Masson (1994). We show that the latter interval can also be derived as a Bayesian within-subject HDI under a certain improper prior. We argue that the proposed new interval is superior to the original within-subject confidence interval, on the grounds of (a) it being based on a more sensible prior, (b) it having a clear and intuitively appealing interpretation, and (c) because its length is always smaller. A generalization of the new interval that can be applied to heteroscedastic data is also derived, and we show that the resulting interval is numerically equivalent to the standardization method discussed in Franz and Loftus (2012); however, our work provides a Bayesian formulation for the standardization method, and in doing so we identify the associated prior distribution.

Farouk S. Nathoo

Robyn E. Kilshaw

Michael E.J. Masson

Journal of Mathematical Psychology

《数学心理学杂志》包括数学心理学所有领域的文章、专着和评论、笔记和评论以及书评。经验和理论贡献同样受到欢迎。特别感兴趣的领域包括但不限于基本测量和心理过程模型，例如基于神经网络或信息处理概念的模型。涵盖的实质性领域的部分列表包括感觉和感知、心理物理学、学习和记忆、解决问题、判断和决策以及动机。数学心理学杂志隶属于数学心理学会。研究领域包括：• 模型感觉和感知、学习、记忆和思考• 基本测量和缩放• 决策• 神经建模和网络• 心理物理学和信号检测• 神经心理学理论• 心理语言学• 动机动力学• 动物行为• 心理测量学理论

The Journal of Mathematical Psychology publishes research articles, monographs, reviews, notes, and book reviews across all areas of mathematical psychology. Both theoretical and empirical contributions are welcome, with particular interest in fundamental measurement and models of psychological processes, such as those based on neural networks or information processing concepts. Key areas of research covered include sensation and perception, psychophysics, learning and memory, problem solving, judgment and decision making, and motivation. The journal is affiliated with the Society for Mathematical Psychology, and research topics include: • Modeling sensation and perception, learning, memory, and thinking • Fundamental measurement and scaling • Decision making • Neural modeling and networks • Psychophysics and signal detection • Neuropsychological theory • Psycholinguistics • Dynamics of motivation • Animal behavior • Psychometric theory.

《Journal of Mathematical Psychology》涵盖数学心理学各个领域的研究文章、专著、评论、笔记和书评。我们欢迎理论和经验方面的贡献，特别关注以下领域：基本测量和心理过程模型（例如基于神经网络或信息处理的模型）。涵盖的主要研究领域包括：感觉与感知、心理物理学、学习与记忆、问题解决、判断与决策以及动机。该期刊隶属于数学心理学会，研究主题包括：• 模型化感觉与感知、学习、记忆和思考 • 基本测量与缩放 • 决策 • 神经建模与网络 • 心理物理学与信号检测 • 神经心理学理论 • 心理语言学 • 动机动力学 • 动物行为 • 心理测量理论

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心理学, 心理学数学	4
心理学, 社会科学数理方法	4

A better (Bayesian) interval estimate for within-subject designs

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