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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.16709 |
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| _version_ | 1866913900358795264 |
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| author | Liu, Yang Pek, Jolynn Maydeu-Olivares, Alberto |
| author_facet | Liu, Yang Pek, Jolynn Maydeu-Olivares, Alberto |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_16709 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Understanding Measurement Precision from a Regression Perspective Liu, Yang Pek, Jolynn Maydeu-Olivares, Alberto Methodology 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. |
| title | Understanding Measurement Precision from a Regression Perspective |
| topic | Methodology |
| url | https://arxiv.org/abs/2404.16709 |