8 Sensitivity to departures from random assignment So far, we have drawn inferences under the assumption that treatments are randomly assigned within matched sets. In an observational study, this assumption lacks support and is typically doubtful if not implausible. We examine sensitivity to bias from nonrandom assignment by assuming that two individuals with the same observed covariates may differ in their odds of treatment by at most a factor of Γ ≥ 1 due to differences in unobserved covariates; see Rosenbaum (2007b; 2017, § 9). This yields hypothesis tests that falsely reject a true null hypothesis with probability at most α when the bias in treatment assignment is at most Γ. Then Γ is varied to display the magnitude of bias that would need to be present to alter the conclusions of a study. How much bias, measured by Γ, would need to be present to lead us to fail to reject the null hypothesis of no effect of football when, in fact, football causes substantial harm? In the current example where there is no evidence of a harmful effect of football, we may ask a parallel question that is related to equivalence testing: how much bias would need to be present to mask a substantial true effect of football on memory? For example, an increase or decrease of a DWR score by at least one word. Aids to interpreting values of Γ are discussed by Rosenbaum and Silber (2009) and Hsu and Small (2013). In particular, in a matched pair with n$_{i}$ = 2, the value Γ = 1 . 25 corresponds with an unobserved covariate that doubles the odds of playing football and doubles the odds of a worse memory score, while Γ = 1 . 5 corresponds with an unobserved covariate that doubles the odds of playing football and quadruples the odds of a worse memory score; see Rosenbaum and Silber (2009) and Rosenbaum (2017, § 9). Proposition 1 applies to the intervals obtained from upper bounds on P -values from sensitivity analyses, providing the bias in treatment assignment is at most Γ. Figure 1 shows the expansion of I c and I v as Γ increases from Γ = 1 for randomization 21