Properties of preliminary test estimators and shrinkage estimators for evaluating multiple exposures—application to questionnaire data from the ‘Study of nevi in children’

Summary.  Epidemiology studies increasingly examine multiple exposures in relation to disease by selecting the exposures of interest in a thematic manner. For example, sun exposure, sunburn and sun protection behaviour could be themes for an investigation of sun‐related exposures. Several studies now use predefined linear combinations of the exposures pertaining to the themes to estimate the effects of the individual exposures. Such analyses may improve the precision of the exposure effects, but they can lead to inflated bias and type I errors when the linear combinations are inaccurate. We investigate preliminary test estimators and empirical Bayes‐type shrinkage estimators as alternative approaches when it is desirable to exploit the thematic choice of exposures, but the accuracy of the predefined linear combinations is unknown. We show that the two types of estimator are intimately related under certain assumptions. The shrinkage estimator that is derived under the assumption of an exchangeable prior distribution gives precise estimates and is robust to misspecifications of the user‐defined linear combinations. The precision gains and robustness of the shrinkage estimation approach are illustrated by using data from the `Study of nevi in children', where the exposures are the individual questionnaire items and the outcome is log(total back naevus count).