Restriction as a method for reducing bias in the estimation of direct effects

The direct effect of a treatment on some outcome is that part of the treatment's effect not referred through a specified covariate intermediate on the pathway between treatment and outcome. Such direct effects are often of primary interest in a data analysis. Unfortunately, standard methods of analysis (for example, stratification or modelling) do not, in general, produce consistent estimates of direct effects whether or not the covariate is 'controlled'. Robins and co-authors have proposed two methods for estimation of direct effects applicable when reliable information is available on the covariate. We propose a third approach for reducing bias: data restriction. By restricting the analysis to strata of the data in which the effect of treatment on the covariate is small, we can (under certain assumptions) reduce bias in estimating treatment's direct effect. We discuss these points with reference to difference and ratio measures of treatment effect. The approach will sometimes be applicable even with an unmeasured or poorly measured covariate. We illustrate these points with data from an observational study of the effect of hormone replacement therapy on breast cancer.