In this paper we study the impact of exposure misclassification when cluster size is potentially informative (i.e., related to outcomes) and when misclassification is differential by cluster size. First, we show that misclassification in an exposure related to cluster size can induce informativeness when cluster size would otherwise be non-informative. Second, we show that misclassification that is differential by informative cluster size can not only attenuate estimates of exposure effects but even inflate or reverse the sign of estimates. To correct for bias in estimating marginal parameters, we propose two frameworks: (i) an observed likelihood approach for joint marginalized models of cluster size and outcomes and (ii) an expected estimating equations approach. Although we focus on estimating marginal parameters, a corollary is that the observed likelihood approach permits valid inference for conditional parameters as well. Using data from the Nurses Health Study II, we compare the results of the proposed correction methods when applied to motivating data on the multigenerational effect of in-utero diethylstilbestrol exposure on attention-deficit/hyperactivity disorder in 106,198 children of 47,450 nurses.
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