few in number. Because many young people play high school football, the safety of high school football is an important question apart from the safety of professional football. 3 Full matching of football players and controls We matched the 591 male football players to all 1,290 male controls who did not play football and did not play a contact sport. The match controlled for several factors that may affect later-life cognition, including the student’s IQ score in high school, their high school rank-in-class recorded as a percent, planned years of future education, as well as binary indicators of whether teachers rated him as an exceptional student, and whether his teachers and parents encouraged him to pursue a college education. We also accounted for aspects of family background like parental income and education. The match was a “full match,” meaning that a matched set could contain one football player and one or more controls, or else one control and one or more football players. A full match is the form of an optimal stratification in the sense that people in the same stratum are as similar as possible subject to the requirement that every stratum contain at least one treated subject and one control; see Rosenbaum (1991). Although the proof of this claim requires some attention to detail, the key idea is simple: if a matched set contained two treated subjects and two controls, it could be subdivided into two matched sets that are at least as close on covariates and are typically closer. See Hansen and Klopfer (2006) for an algorithm for optimal full matching, Hansen (2007) for software, and Hansen (2004) and Stuart and Green (2008) for applications. The match was constructed using Hansen’s optmatch package in R with the ratio of controls to treated units constrained between 1:6 and 6:1 to avoid excessively large matched sets. In a full match, there are I matched sets, i = 1 , . . . , I and n$_{i}$ individuals, j = 1 , . . . , n$_{i}$ , in set i . If individual ij played on a football team in high school, write Z$_{ij}$ = 1; otherwise, 7