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Linear Prediction of a True Score From a Direct Estimate and Several Derived Estimates

Statistical prediction problems often involve both a direct estimate of a true score and covariates of this true score. Given the criterion of mean squared error, this study determines the best linear predictor of the true score given the direct estimate and the covariates. Results yield an extension of Kelley’s formula for estimation of the true score to cases in which covariates are present. The best linear predictor is a weighted average of the direct estimate and of the linear regression of the direct estimate onto the covariates. The weights depend on the reliability of the direct estimate and on the multiple correlation of the true score with the covariates. One application of the best linear predictor is to use essay features provided by computer analysis and an observed holistic score of an essay provided by a human rater to approximate the true score corresponding to the holistic score.