In a recent paper, Weller, Milton, Eisen, and Spiegelman discuss fitting logistic regression models when a scalar main explanatory variable is measured with error by several surrogates, i.e., a situation with more surrogates than variables measured with error. Using a regression calibration approximate model as if it were exact, they compare two methods of adjusting for measurement error. One is the standard regression calibration approach consisting of substituting an estimated conditional expectation of the true covariate given observed data in the logistic regression. The other is a novel two-stage approach when the logistic regression is fitted to multiple surrogates, and then a linear combination of estimated slopes is formed as the estimate of interest. Applying estimated asymptotic variances for both methods in a single data set with some sensitivity analysis, the authors assert superiority of their two-stage approach. We investigate this claim in some detail. A troubling aspect of the proposed two-stage method is that, unlike standard regression calibration and a natural form of maximum likelihood, the resulting estimates are not invariant to reparameterization of nuisance parameters in the model. We show, however, that, under the regression calibration approximation, the two-stage method is asymptotically equivalent to a maximum likelihood formulation, and is therefore in theory superior to standard regression calibration. However, our extensive finite-sample simulations in the practically important parameter space where the regression calibration model provides a good approximation failed to uncover such superiorityof the two-stage method. We also discuss extensions to different data structures.
Keywords: attenuation, logistic regression, measurement error, reparameterization