Sensitivity analysis for stratified comparisons in an observational study of the effect of smoking on homocysteine levels

Abstract

Sensitivity bounds for randomization inferences exist in several important cases, such as matched pairs with any type of outcome or binary outcomes with any type of stratification, but computationally feasible bounds for any outcome in any stratification are not currently available. For instance, with 20 strata, some large, others small, there is no currently available, computationally feasible sensitivity bound testing the null hypothesis of no treatment effect in the presence of a bias from nonrandom treatment assignment of a specific magnitude. The current paper solves the general problem; it uses an inequality formed by taking a one-step Taylor approximation from a near extreme solution, known as the separable approximation, where the concavity of the underlying function ensures that the Taylor approximation is, at worst, conservative. In practice, the separable approximation and the one-step movement away from it provide computationally feasible lower and upper bounds, thereby providing both a usable, perhaps slightly conservative statement, together with a check that the conservative statement is not unduly conservative. In every example that I have tried, the upper and lower bounds barely differ, although with some effort one can construct examples in which the separable approximation gives a $P$-value of 0.0499 and the Taylor approximation gives 0.0501. The new inequality holds in finite samples, so it strengthens certain existing asymptotic results, additionally simplifying the proof of those results. The method is discussed in the context of an observational study of the effects of smoking on homocysteine levels, a possible risk factor for several diseases including cardiovascular disease, thrombosis and Alzheimer’s disease. This study contains two evidence factors, the comparison of smokers and nonsmokers and the comparison of smokers to one another in terms of recent nicotine exposure. A new $\mathtt{R}$ package, $\mathtt{senstrat}$, implements the procedure and illustrates it with the example from the current paper.