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Regression calibration inference seeks to estimate regression models with measurement error in explanatory variables by replacing the mismeasured variable by its conditional expectation, given a surrogate variable, in an estimation procedure that would have been used if the true variable were available. This study examines the effect of the uncertainty in the estimation of the required conditional expectation on inference about regression parameters, when the true explanatory variable and its surrogate are observed in a calibration dataset and related through a normal linear model. The exact sampling distribution of the regression calibration estimator is derived for normal linear regression when independent calibration data are available. The sampling distribution is skewed and its moments are not defined, but its median is the parameter of interest. It is shown that, when all random variables are normally distributed, the regression calibration estimator is equivalent to maximum likelihood provided a natural estimate of variance is non-negative. A check for this equivalence is useful in practice for judging the suitability of regression calibration. Results about relative efficiency are provided for both external and internal calibration data. In some cases maximum likelihood is substantially more efficient than regression calibration. In general, though, a more important concern when the necessary conditional expectation is uncertain, is that inferences based on approximate normality and estimated standard errors may be misleading. Bootstrap and likelihood-ratio inferences are preferable. |
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