Approximate likelihoods for generalized linear errors-in-variables models

When measurement error is present in covariates, it is well known that naively fitting a generalized linear model results in inconsistent inferences. Several methods have been proposed to adjust for measurement error without making undue distributional assumptions about the unobserved true covariates. Stefanski and Carroll focused on an unbiased estimating function rather than a likelihood approach. Their estimating function, known as the conditional score, exists for logistic regression models but has two problems: a poorly behaved Ward test and multiple solutions. They suggested a heuristic procedure to identify the best solution that works well in practice but has little theoretical support compared with maximum likelihood estimation. To help to resolve these problems, we propose a conditional quasi-likelihood to accompany the conditional score that provides an alternative to Wald's test and successfully identifies the consistent solution in large samples.