In studying the effects of a treatment — here, high school football — it is typically in- appropriate to adjust for events subsequent to the start of treatment, as this may introduce bias even where none existed prior to adjustments, because part of the treatment effect may be removed (Rosenbaum 1984). However, there are certain adult health outcomes that may be different between football players and controls due to disparities in unmea- sured baseline health characteristics rather than an effect of playing football. This may threaten the validity of our study if these baseline health characteristics also play a role in later-life cognitive health. Comparing these health outcomes may can be used, at least partially, to assess the comparability of the baseline health of the comparison groups. We checked on the health status of football players and matched controls at age 65 using the Mantel-Haenszel procedure, failing to find a difference significant at the 0.05 level for “ever had high blood pressure,” “ever had diabetes,” and “ever had heart problems”. Football players were more likely to report that they had “ever had a stroke,” with a P -value of 0.03, and a 95% confidence interval for the odds ratio of [1 . 09 , 3 . 21]. Extensive comparisons of this kind are reported in Deshpande et al. (2017). If there were a single version of treatment or control, then individual ij would have two potential delayed word recall scores, Y$_{ij}$ (1) if he played football and Y$_{ij}$ (0) if he did not, where we observe only one of these, namely Y$_{ij}$ = Z$_{ij}$ Y$_{ij}$ (1)+(1 − Z$_{ij}$ ) Y$_{ij}$ (0), and the effect caused by playing football, namely δ$_{ij}$ = Y$_{ij}$ (1) − Y$_{ij}$ (0), is not observed for any individual; see Neyman (1923) and Rubin (1974). Fisher’s (1935) sharp null hypothesis of no effect says H$_{0}$ : Y$_{ij}$ (1) = Y$_{ij}$ (0), i = 1 , . . . , I , j = 1 , . . . , n$_{i}$ , which we henceforth abbreviate as H$_{0}$ : Y$_{ij}$ (1) = Y$_{ij}$ (0), ∀ i, j or as H$_{0}$ : δ$_{ij}$ = 0, ∀ i, j . The treatment has an additive constant effect if there exists some constant τ such that δ$_{ij}$ = Y$_{ij}$ (1) − r$_{Cij}$ = τ , ∀ i, j . The hypothesis 4 Review of randomization inference without versions of treatment 9