Bayesian Inference for Correlations in the Presence of Measurement Error and Estimation Uncertainty

Dora Matzke,Alexander Ly,Ravi Selker,Eric-Jan Wagenmakers,Wouter D Weeda,Michael D Lee,Benjamin Scheibehenne,Simine Vazire,Samantha Bouwmeester

doi:10.1525/collabra.78

Abstract

Whenever parameter estimates are uncertain or observations are contaminated by measurement error, the Pearson correlation coefficient can severely underestimate the true strength of an association. Various approaches exist for inferring the correlation in the presence of estimation uncertainty and measurement error, but none are routinely applied in psychological research. Here we focus on a Bayesian hierarchical model proposed by Behseta, Berdyyeva, Olson, and Kass (2009) that allows researchers to infer the underlying correlation between error-contaminated observations. We show that this approach may be also applied to obtain the underlying correlation between uncertain parameter estimates as well as the correlation between uncertain parameter estimates and noisy observations. We illustrate the Bayesian modeling of correlations with two empirical data sets; in each data set, we first infer the posterior distribution of the underlying correlation and then compute Bayes factors to quantify the evidence that the data provide for the presence of an association.

Highlights

Formal mathematical models are useful tools for analyzing data obtained from psychological experiments
We focus on parameter estimates obtained with Bayesian inference because the resulting posterior distributions can be conveniently used to quantify estimation uncertainty
As the cumulative prospect theory (CPT) parameters are assumed to be relatively stable across short periods of time, we examine the correlation between the δ parameters estimated from data collected during the two experimental sessions

Summary

Introduction

Formal mathematical models are useful tools for analyzing data obtained from psychological experiments D. Lee & Wagenmakers, 2013; Levine, 2000; Lewandowsky & Farrell, 2010). These models come in many flavors, ranging from simple statistical distributions to sophisticated cognitive process models. Regardless of its degree of sophistication, the general goal of mathematical modeling is to capture regularities in the data using parameters that represent separate statistical components or distinct psychological variables

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