A new design for observational studies applied to the study of the effects of high school football on cognition late in life

Do the impacts that occur when playing high school football have concussive effects that accelerate cognitive decline late in life? We examine this possibility using newly available cognitive data describing people in 2020 who graduated high school in 1957. Someone who was 18 in 1957 would be 81 in 2020. For this comparison we develop a new design for an observational study, called a triples design, and discuss its advantages and construction. A triples design consists of M blocks of size 3, where a block contains either one treated individual and two controls or two treated individuals and one control. A triples design is the simplest design that uses weights, with just two weights. The “entire number” is {1−e(x)}/e(x), where e(x) is the propensity score at covariate x, so it is the ratio of controls-to-treated expected at x. Unlike a matched pairs design, which can remove the bias from observed covariates when the “entire number” exceeds 1, the triples design can succeed when the entire number exceeds 1/2, reflecting the possibility of matching two treated individuals to the same control. Like full matching, a triples design can match more people than can matched pairs, yet have smaller within-block covariate distances. Unlike full matching, there are no matched pairs. Like matching with multiple controls, a triples design will have a larger design sensitivity than a design which includes matched pairs, under simple models for continuous outcomes; that is, in favorable situations the design is expected to report greater insensitivity to unmeasured biases. Because there are just two weights, it is easy to construct weighted graphics for exploratory displays from triples designs. A heuristic algorithm containing network optimization constructs the design.