− 2 − 1 0 1 2 Words Remembered τ Versus all controls Versus no sport Versus other sport Constant effect Versions Γ = 1 Versus all controls Versus no sport Versus other sport Constant effect Versions Γ = 1.25 Versus all controls Versus no sport Versus other sport Constant effect Versions Γ = 1.5 Figure 1: Comparison of interval estimates for the effect of high school football on the delayed word recall score. The intervals for Γ = 1 assume that there is no bias from unmeasured covariates, while Γ > 1 permits unmeasured biases of unknown form but limited magnitude. The top three intervals for “all controls”, “no sport”, and “other sport”, are conventional confidence intervals lacking simultaneous coverage. The bottom two intervals are I c and I $_{v}$. Notice that the “all controls” interval equals I c and the union of the first three intervals equals I $_{v}$. Figure 1: Comparison of interval estimates for the effect of high school football on the delayed word recall score. The intervals for Γ = 1 assume that there is no bias from unmeasured covariates, while Γ > 1 permits unmeasured biases of unknown form but limited magnitude. The top three intervals for “all controls”, “no sport”, and “other sport” are conventional confidence intervals lacking simultaneous coverage. The bottom two intervals are I c and I $_{v}$. Noitce that the “all controls” interval equals I c and the union of the first three intervals equals I $_{v}$. 1 6 Comparison to conventional approaches to multiple versions: F-tests and Bonferroni correction The method described in § 5.2 makes an important trade-off: it prioritizes the primary comparison against all controls under the assumption of a constant treatment effect, al- lowing us to report the corresponding confidence interval with no correction, in exchange for the ability to distinguish between versions if they do, in fact, exists. In other words, our method emphasizes detection of non-zero, constant treatment effects over detecting different versions of the effect. Conventional approaches to multiple comparisons, are of- 17