The number needed to treat is often used to measure the efficacy of a binary outcome in randomized clinical trials. There are three different available measures of the number needed to treat. Two of these measures, Furukawa and Leucht’s and Kraemer and Kupfer’s, focus on converting Cohen’s δ index into the number needed to treat, while Laupacis et al.’s measure deals primarily with the number needed to treat’s estimation rather than with a reformulation. Mathematical and numerical analysis of numbers needed to treat and their estimators was conducted. Three novel number needed to treat estimators were introduced to supplement the numbers needed to treat introduced by Laupacis, Furukawa and Leucht, and Kraemer and Kupfer. The analysis showed that Laupacis et al.’s number needed to treat is intrinsically different from Kraemer and Kupfer’s number needed to treat, and that Furukawa and Leucht’s estimator is appropriate to use only for normally distributed outcomes with equal standard deviations. Based on the numerical analysis, the novel numbers needed to treat outperformed the existing ones under correct model specifications. Asymptotic analysis was used to test three different types of confidence intervals to supplement the numbers needed to treat. An R-package to calculate these numbers needed to treat and their confidence intervals has been developed and made available for users online.

ObjectiveWe aim to explain the unadjusted, adjusted and marginal number needed to treat (NNT) and provide software for clinicians to compute them.MethodsThe NNT is an efficacy index that is commonly used in randomised clinical trials. The NNT is the average number of patients needed to treat to obtain one successful outcome (ie, response) due to treatment. We developed the nntcalc R package for desktop use and extended it to a user-friendly web application. We provided users with a user-friendly step-by-step guide. The application calculates the NNT for various models with and without explanatory variables. The implemented models for the adjusted NNT are linear regression and analysis of variance (ANOVA), logistic regression, Kaplan-Meier and Cox regression. If no explanatory variables are available, one can compute the unadjusted Laupacis et al’s NNT, Kraemer and Kupfer’s NNT and the Furukawa and Leucht’s NNT. All NNT estimators are computed with their associated appropriate 95% confidence intervals. All calculations are in R and are replicable.ResultsThe application provides the user with an easy-to-use web application to compute the NNT in different settings and models. We illustrate the use of the application from examples in schizophrenia research based on the Positive and Negative Syndrome Scale. The application is available from https://nntcalc.iem.technion.ac.il. The output is given in a journal compatible text format, which users can copy and paste or download in a comma-separated values format.ConclusionThis application will help researchers and clinicians assess the efficacy of treatment and consequently improve the quality and accuracy of decisions.

Instrumental variables regression is a tool that is commonly used in the analysis of observational data. The instrumental variables are used to make causal inference about the effect of a certain exposure in the presence of unmeasured confounders. A valid instrumental variable is a variable that is associated with the exposure, affects the outcome only through the exposure (exclusion), and is not confounded with the outcome (exogeneity). Unlike the first assumption, the other two are generally untestable and rely on subject‐matter knowledge. Therefore, a sensitivity analysis is desirable to assess the impact of assumptions' violation on the estimated parameters. In this paper, we propose and demonstrate a new method of sensitivity analysis for G‐estimators in causal linear and non‐linear models. We introduce two novel aspects of sensitivity analysis in instrumental variables studies. The first is a single sensitivity parameter that captures violations of exclusion and exogeneity assumptions. The second is an application of the method to non‐linear models. The introduced framework is theoretically justified and is illustrated via a simulation study. Finally, we illustrate the method by application to real‐world data and provide guidelines on conducting sensitivity analysis.

Abstract-Lifetime data are usually assumed to stem from a continuous distribution supported on [0, b) for some b ≤ ∞. The continuity assumption implies that the support of the distribution does not have atom points, particularly not at 0. Accordingly, it seems reasonable that with an accurate measurement tool all data observations will be positive. This suggests that the true support may be truncated from the left. In this work we investigate the effects of adding a left truncation parameter to a continuous lifetime data statistical model. We consider two main settings: right truncation parametric models with possible left truncation, and exponential family models with possible left truncation. We analyze the performance of some optimal estimators constructed under the assumption of no left truncation when left truncation is present, and vice versa. We investigate both asymptotic and finite-sample behavior of the estimators. We show that when left truncation is not assumed but is, in fact present, the estimators have a constant bias term, and therefore will result in inaccurate and inefficient estimation. We also show that assuming left truncation where actually there is none, typically does not result in substantial inefficiency, and some estimators in this case are asymptotically unbiased and efficient.

The number needed to treat (NNT) is an efficacy index commonly used in randomized clinical trials. The NNT is the average number of treated patients for each undesirable patient outcome, for example, death, prevented by the treatment. We introduce a systematic theoretically-based framework to model and estimate the conditional and the harmonic mean NNT in the presence of explanatory variables, in various models with dichotomous and nondichotomous outcomes. The conditional NNT is illustrated in a series of four primary examples; logistic regression, linear regression, Kaplan-Meier estimation, and Cox regression models. Also, we establish and prove mathematically the exact relationship between the conditional and the harmonic mean NNT in the presence of explanatory variables. We introduce four different methods to calculate asymptotically-correct confidence intervals for both indices. Finally, we implemented a simulation study to provide numerical demonstrations of the aforementioned theoretical results and the four examples. Numerical analysis showed that the parametric estimators of the NNT with nonparametric bootstrap-based confidence intervals outperformed other examined combinations in most settings. An R package and a web application have been developed and made available online to calculate the conditional and the harmonic mean NNTs with their corresponding confidence intervals.