Instrumental variables (IVs) are extensively used to estimate treatment effects when the treatment and outcome are confounded by unmeasured confounders; however, weak IVs are often encountered in empirical studies and may cause problems. Many studies have considered building a stronger IV from the original, possibly weak, IV in the design stage of a matched study at the cost of not using some of the samples in the analysis. It is widely accepted that strengthening an IV tends to render nonparametric tests more powerful and will increase the power of sensitivity analyses in large samples. In this article, we re-evaluate this conventional wisdom to bring new insights into this topic. We consider matched observational studies from three perspectives. First, we evaluate the trade-off between IV strength and sample size on nonparametric tests assuming the IV is valid and exhibit conditions under which strengthening an IV increases power and conversely conditions under which it decreases power. Second, we derive a necessary condition for a valid sensitivity analysis model with continuous doses. We show that the $Gamma$ sensitivity analysis model, which has been previously used to come to the conclusion that strengthening an IV increases the power of sensitivity analyses in large samples, does not apply to the continuous IV setting and thus this previously reached conclusion may be invalid. Third, we quantify the bias of the Wald estimator with a possibly invalid IV under an oracle and leverage it to develop a valid sensitivity analysis framework; under this framework, we show that strengthening an IV may amplify or mitigate the bias of the estimator, and may or may not increase the power of sensitivity analyses. We also discuss how to better adjust for the observed covariates when building an IV in matched studies.
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