A bayesian approach to logistic regression models having measurement error following a mixture distribution

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Abstract
To estimate the parameters in a logistic regression model when the predictors are subject to random or systematic measurement error, we take a Bayesian approach and average the true logistic probability over the conditional posterior distribution of the true value of the predictor given its observed value. We allow this posterior distribution to consist of a mixture when the measurement error distribution changes form with observed exposure. We apply the method to study the risk of alcohol consumption on breast cancer using the Nurses Health Study data. We estimate measurement error from a small subsample where we compare true with reported consumption. Some of the self‐reported non‐drinkers truly do not drink. The resulting risk estimates differ sharply from those computed by standard logistic regression that ignores measurement error.