Estimation of the number needed to treat, the number needed to be exposed, and the exposure impact number with instrumental variables

Abstract

Objectives The Number Needed to Treat (NNT) is an efficacy index defined as the average number of patients needed to treat to attain one additional treatment benefit. In observational studies, specifically in epidemiology, the adequacy of the populationwise NNT is questionable since the exposed group characteristics may substantially differ from the unexposed. To address this issue, groupwise efficacy indices were defined: the Exposure Impact Number (EIN) for the exposed group and the Number Needed to be Exposed (NNE) for the unexposed. Each defined index answers a unique research question since it targets a unique sub-population. In observational studies, the group allocation is typically affected by confounders that might be unmeasured. The available estimation methods that rely either on randomization or the sufficiency of the measured covariates for confounding control result in statistically inconsistent estimators of the true EIN, NNE, and NNT. This study presents a theoretical framework for statistically consistent point and interval estimation of the NNE, EIN and NNE in observational studies with unmeasured confounders. Methods Using Rubin’s potential outcomes framework, this study explicitly defines the NNT and its derived indices, EIN and NNE, as causal measures. Then, we use instrumental variables to introduce a novel method to estimate the three aforementioned indices in observational studies where the omission of unmeasured confounders cannot be ruled out. To illustrate the novel methods, we present two analytical examples – double logit and double probit models. Next, a corresponding simulation study and a real-world data example are presented. Results This study provides an explicit causal formulation of the EIN, NNE, and NNT indices and a comprehensive theoretical framework for their point and interval estimation using the G-estimators in observational studies with unmeasured confounders. The analytical proofs and the corresponding simulation study illustrate the improved performance of the new estimation method compared to the available methods in terms of consistency and the confidence intervals empirical coverage rates. Conclusions In observational studies, traditional estimation methods to estimate the EIN, NNE, or NNT result in statistically inconsistent estimators. We introduce a novel estimation method that overcomes this pitfall. The novel method produces consistent estimators and reliable CIs for the true EIN, NNE, and NNT. Such a method may facilitate more accurate clinical decision-making and the development of efficient public health policies.