Analysis of multivariate reliability structures and the induced bias in linear model estimation

Least squares provides consistent estimates of the regression coefficients beta in the model E[Y [symbol: see text] x] = beta x when fully accurate measurements of x are available. However, in biomedical studies one must frequently substitute unreliable measurements X in place of x. This induces bias in the least squares coefficient estimates. In the univariate case, the bias manifests itself as a shrinkage toward zero, but this result does not generalize. When x is multivariate, then there are no predictable relationships between the signs or magnitudes of actual and estimated regression coefficients. In this article, we characterize the estimation bias, and review a relatively simple adjustment procedure to correct it. We also show that several natural conjectures about the bias are false. We present three definitions of reliability coefficient matrices that generalize the univariate case, and we illustrate their application to dietary intake data from a cancer prevention study.