Statistical analysis
We first computed summary statistics to delineate baseline patient characteristics at ICU admission and compare the characteristics between patients treated with and without PMX-HA both in the derivation and validation cohort.Next, to evaluate the heterogeneity in the treatment effects of PMX-HA, we applied the machine learning-based causal forest model to the derivation cohort using the Based on the constructed causal forest model, we estimated the individual treatment effects (ITEs) of PMX-HA on 28-day survival in all patients from the derivation cohort and ranked them into quintiles of estimated ITEs. We estimated conditional average treatment effects in each quintile group to evaluate the heterogeneity in treatment effects of PMX-HA and compared the characteristics in each group in order to identify the candidate determinants of treatment effects. To investigate the contribution of the candidate determinants to the variation in the treatment effects of PMX-HA, we drew heatmaps of predicted treatment effects, conditioning on the candidate determinants with the rest of the covariates fixed at the median for continuous variables and at the mode for binary variables [Finally, to validate the criteria derived from our causal forest model, we estimated treatment effects of PMX-HA on 28-day survival in the (1) full sample and (2) targeted subpopulation of the validation cohort (i.e., patients who met the derived criteria). We also performed survival analysis for up to 28 days using Kaplan–Meier curves with the log-rank test and the Cox proportional hazard model adjusted for APACHE II and SOFA scores.To address the bias due to the exclusion of all patients who were treated with RRT from the derivation cohort, we estimated treatment effects of PMX-HA on 28-day survival in patients who were on RRT and not on RRT at randomization for PMX-HA in the targeted subpopulation of the validation cohort.A 