Measurement Error in the Response in the General Linear Model

Problems where there is measurement error in the response variable in a general linear model are considered. With Y denoting the true value and U the observed/surrogate value, a rich class of measurement error models is considered in which (a) the mean of U given Y = y can be either linear or nonlinear in y, (b) the variance of U given Y = y can depend on y, and (c) the parameters in the model for U given Y can change across observations in the study. This could occur, for example, when using a radioimmunoassay to measure the outcome variable with recalibration for each batch of reagents. The parameters in the measurement error models are estimated using independent “calibration” data. Full and pseudo—maximum likelihood estimators (MLE's) and their asymptotic properties are described under distributional assumptions. The pseudo-MLE's and naive estimates using imputed values are special cases of a general class of estimators. The asymptotic properties of such estimators are given and then used to derive detailed results about the performance of analyses based on imputed values. This includes an assessment of bias in estimation of the coefficients and the variance and the development of an asymptotic covariance matrix of the estimated coefficients and ways to estimate it. This approach leads to modification of the standard errors obtained using imputed values.