The Importance of Assessing Measurement Reliability in Multivariate Regression

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Abstract
In many contexts involving multivariate linear regression models, some or all of the independent (predictor) variables are measured with error. It is argued that if the goal is to assess the relationship of the dependent variables to the true predictor variables, then it is important to determine the reliability matrix Λ of the measurement X of the vector of true predictors. If Λ is singular, then the slope matrix B is not identifiable. If Λ is nearly singular, then B cannot be accurately estimated. A two-step estimation procedure is proposed in which Λ is estimated from data on the measured predictors X and on prior information from reliability studies on these predictors and the parameters of the (latent) linear relationship are estimated by classical linear regression methods. This approach not only allows the use of available software but also lends itself to traditional diagnostic methods. Large-sample properties of the estimators resulting from this approach are derived. It is shown how a canonical analysis based on the eigenvalues and eigenvectors of Λ (or its estimate ) can be used to assess the influence of measurement error on the accuracy of estimation. Finally, using replications of observed predictors is suggested as a way of estimating Λ and increasing reliability of measurement.

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