More than 1 million students played high school American football in 2014,¹ but, recently, many have questioned the safety of the sport^2,3 or even called for its ban.⁴ These concerns have been driven partially by reports of chronic traumatic encephalopathy,^5,6 increased risks of neurodegenerative disease,⁷ and associations between a history of concussion and cognitive impairment and depression later in life^8-10 among retired professional football players.

Effect modification occurs when the effect of the treatment on an outcome varies according to the level of other covariates and often has important implications in decision-making. When there are tens or hundreds of covariates, it becomes necessary to use the observed data to select a simpler model for effect modification and then make valid statistical inference. We propose a two-stage procedure to solve this problem. First, we use Robinson's transformation to decouple the nuisance parameters from the treatment effect of interest and use machine learning algorithms to estimate the nuisance parameters. Next, after plugging in the estimates of the nuisance parameters, we use the lasso to choose a low-complexity model for effect modification. Compared to a full model consisting of all the covariates, the selected model is much more interpretable. Compared to the univariate subgroup analyses, the selected model greatly reduces the number of false discoveries. We show that the conditional selective inference for the selected model is asymptotically valid given the rate assumptions in classical semiparametric regression. Extensive simulation studies are conducted to verify the asymptotic results and an epidemiological application is used to demonstrate the method.

Background Previous studies on the association of malaria and stunted growth delivered inconsistent results. These conflicting results may be due to different levels of confounding and to considerable difficulties in elucidating a causal relationship. Randomized experiments are impractical and previous observational studies have not fully controlled for potential confounding including nutritional deficiencies, breastfeeding habits, other infectious diseases and socioeconomic status. Methods This study aims to estimate the causal effect between malaria episodes and stunted growth by applying a combination of Mendelian randomization, using the sickle cell trait, and matching. We demonstrate the method on a cohort of children in the Ashanti Region, Ghana. Results We found that the risk of stunting increases by 0.32 (P-value: 0.004, 95% CI: 0.09, 1.0) for every malaria episode. The risk estimate based on Mendelian randomization substantially differs from the multiple regression estimate of 0.02 (P-value: 0.02, 95% CI: 0.003, 0.03). In addition, based on the sensitivity analysis, our results were reasonably insensitive to unmeasured confounders. Conclusions The method applied in this study indicates a causal relationship between malaria and stunting in young children in an area of high endemicity and demonstrates the usefulness of the sickle cell trait as an instrument for the analysis of conditions that might be causally related to malaria.