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arXiv · 2404.16709

Understanding Measurement Precision from a Regression Perspective

Yang Liu·Jolynn Pek·Alberto Maydeu-Olivares
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Abstract

We adopt and expand McDonald's (2011) regression framework for measurement precision, integrating two key perspectives: (a) reliability of observed scores and (b) optimal prediction of latent scores. Reliability arises from a measurement decomposition of an observed score into its true score and measurement error. In contrast, proportional reduction in mean squared error (PRMSE) arises from a prediction decomposition of a latent score into its optimal predictor (the observed expected a posteriori [EAP] score) and prediction error. Reliability is the coefficient of determination obtained by two isomorphic regressions: regressing the observed score on its true score or on all the latent variables. Similarly, PRMSE is the coefficient of determination obtained from two isomorphic regressions: regressing the latent score on its observed EAP score or all the manifest variables. A key implication of this regression framework is that both reliability and PRMSE can be estimated using a Monte Carlo (MC) method, which is particularly useful when no analytic formula is available or when the analytic calculation is involved. We illustrate these concepts with a factor analysis model and a two parameter logistic model, in which we compute reliability coefficients for different observed scores and PRMSE for different latent scores. Additionally, we provide a numerical example demonstrating how the MC method can be used to estimate reliability and PRMSE within a two-dimensional item response tree model.

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Yang Liu, Jolynn Pek, Alberto Maydeu-Olivares. 2024-04-25. Understanding Measurement Precision from a Regression Perspective. https://doi.org/10.1037/met0000763

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