We introduce a pathwise algorithm for the Cox proportional hazards model, regularized by convex combinations of ℓ1 and ℓ2 penalties (elastic net). Our algorithm fits via cyclical coordinate descent, and employs warm starts to find a solution along a regularization path. We demonstrate the efficacy of our algorithm on real and simulated data sets, and find considerable speedup between our algorithm and competing methods.

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Females have generally more robust immune responses than males for reasons that are not well-understood. Here we used a systems analysis to investigate these differences by analyzing the neutralizing antibody response to a trivalent inactivated seasonal influenza vaccine (TIV) and a large number of immune system components, including serum cytokines and chemokines, blood cell subset frequencies, genome-wide gene expression, and cellular responses to diverse in vitro stimuli, in 53 females and 34 males of different ages. We found elevated antibody responses to TIV and expression of inflammatory cytokines in the serum of females compared with males regardless of age. This inflammatory profile correlated with the levels of phosphorylated STAT3 proteins in monocytes but not with the serological response to the vaccine. In contrast, using a machine learning approach, we identified a cluster of genes involved in lipid biosynthesis and previously shown to be up-regulated by testosterone that correlated with poor virus-neutralizing activity in men. Moreover, men with elevated serum testosterone levels and associated gene signatures exhibited the lowest antibody responses to TIV. These results demonstrate a strong association between androgens and genes involved in lipid metabolism, suggesting that these could be important drivers of the differences in immune responses between males and females.

Summary We consider rules for discarding predictors in lasso regression and related problems, for computational efficiency. El Ghaoui and his colleagues have propose ‘SAFE’ rules, based on univariate inner products between each predictor and the outcome, which guarantee that a coefficient will be 0 in the solution vector. This provides a reduction in the number of variables that need to be entered into the optimization. We propose strong rules that are very simple and yet screen out far more predictors than the SAFE rules. This great practical improvement comes at a price: the strong rules are not foolproof and can mistakenly discard active predictors, i.e. predictors that have non-zero coefficients in the solution. We therefore combine them with simple checks of the Karush–Kuhn–Tucker conditions to ensure that the exact solution to the convex problem is delivered. Of course, any (approximate) screening method can be combined with the Karush–Kuhn–Tucker, conditions to ensure the exact solution; the strength of the strong rules lies in the fact that, in practice, they discard a very large number of the inactive predictors and almost never commit mistakes. We also derive conditions under which they are foolproof. Strong rules provide substantial savings in computational time for a variety of statistical optimization problems.

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