Run times for PUMA methods and other available software.

For a typical simulated data set with 5000 individuals, 650 K markers and a pre-screening p-value threshold of 0.01, we report the run times, and the number of total and unique models examined by our methods (top) and available methods using standard/default settings (bottom). We list the number of models assessed during a single run of a method where a model is defined by the set of markers with distinct nonzero coefficients and the number of unique models counts the number of sets of distinct markers, where we note that the metrics reported can vary substantially between datasets. Lasso and Adaptive Lasso are convex and have a single tuning parameter, so relatively few models are examined during the search. For convex penalties, each distinct tuning parameter value produces a model, although another tuning parameter value can cause the coefficients to change but still produce the same set of markers with nonzero coefficients. Thus the number of models examined is larger than the number of unique models. MCP, LOG and NEG penalties are non-convex and have two tuning parameters and were applied with 100 marker reorderings, so they produce orders of magnitude more total and unique models. We note that 1D-MCP is faster than 2D-MCP as the former fixes the value of one tuning parameter. We note that HyperLasso [22] can be extremely computationally expensive for large datasets, so that the time we report is based on analysis of the pre-screened dataset where pre-screening step must be implemented separately. Ayers and Cordell [32] do not provide software but proposes an approach using the grpreg package in R.