Surprising effects of measurement error on an aggregate data estimator

In a generalised linear model with a single, normally distributed covariate, for the most part the effect of normally distributed additive measurement error is attenuation, i.e. asymptotic bias towards the null. Prentice & Sheppard (1995) suggested a marginalised random effects approach to combining the results of different studies on binary outcomes. We show that, in probit regression, when the number of observations per study is large, under the stated normality assumptions attenuation never occurs. In fact, the asymptotic bias is away from the null. This appears to be the first known case under reasonable distributional assumptions that the effect of measurement error is reverse-attenuation.