On the minimum strength of (unobserved) covariates to overturn an insignificant result

We study conditions under which the addition of variables to a regression equation can turn a previously statistically insignificant result into a significant one. Specifically, we characterize the minimum strength of association required for these variables--both with the dependent and independent variables, or with the dependent variable alone--to elevate the observed t-statistic above a specified significance threshold. Interestingly, we show that it is considerably difficult to overturn a statistically insignificant result solely by reducing the standard error. Instead, included variables must also alter the point estimate to achieve such reversals in practice. Our results can be used for sensitivity analysis and for bounding the extent of p-hacking, and may also offer algebraic explanations for patterns of reversals seen in empirical research, such as those documented by Lenz and Sahn (2021).