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Improving the treatment of trauma, a leading cause of death worldwide, is of great clinical and public health interest. This analysis introduces flexible statistical methods for estimating center-level effects on individual outcomes in the context of highly variable patient populations, such as those of the PRospective, Observational, Multi-center Major Trauma Transfusion study. Ten US level I trauma centers enrolled a total of 1,245 trauma patients who survived at least 30 minutes after admission and received at least one unit of red blood cells. Outcomes included death, multiple organ failure, substantial bleeding, and transfusion of blood products. The centers involved were classified as either large or small-volume based on the number of massive transfusion patients enrolled during the study period. We focused on estimation of parameters inspired by causal inference, specifically estimated impacts on patient outcomes related to the volume of the trauma hospital that treated them. We defined this association as the change in mean outcomes of interest that would be observed if, contrary to fact, subjects from large-volume sites were treated at small-volume sites (the effect of treatment among the treated). We estimated this parameter using three different methods, some of which use data-adaptive machine learning tools to derive the outcome models, minimizing residual confounding by reducing model misspecification. Differences between unadjusted and adjusted estimators sometimes differed dramatically, demonstrating the need to account for differences in patient characteristics in clinic comparisons. In addition, the estimators based on robust adjustment methods showed potential impacts of hospital volume. For instance, we estimated a survival benefit for patients who were treated at large-volume sites, which was not apparent in simpler, unadjusted comparisons. By removing arbitrary modeling decisions from the estimation process and concentrating on parameters that have more direct policy implications, these potentially automated approaches allow methodological standardization across similar comparativeness effectiveness studies.