Efficiencies and Surrogate Variables in Logistic Regression

Raymond J. Carroll Department of Statistics Texas A&M University College Station, TX 77843 We study logistic regression with response y when the true predictor x is measured with· error and the observable data consist of pairs (y,w), where w is correlated with x. Two approaches to estimation are studied. In the first, integrated likelihood estimates are obtained from the conditional distribution of y given E(x Iw). In the second approach, the calibration curve for (x, w) is used to construct unbiased estimates of x and then errors in variables estimation techniques are employed. When (x, w) has a bivariate normal distribution, the two approaches yield consistent and asymptotically normally distributed parameter estimates. The intergrated likelihood estimates are more efficient here, although the errors in variables estimates are shown to be nearly efficient for many choices of the parameters. Our results also apply to the problem of assessing the loss of efficiency in semiparametric regression models, where in this context it is shown that there is often little potential gain for specifying the distribution of x completely.