Non-linear association of SHR at admission with outcomes

We evaluated the non-linear association of SHR as a continuous exposure with a given outcome. First, we developed entropy balancing weights via weights optimization to achieve an exact balance of covariates moments [25], accounting for age, sex, baseline systolic BP, baseline eGFR, baseline NT-proBNP, admission department (Department of Cardiology /others), CCI, with or without ischemic heart disease at baseline, whether the use of insulin at baseline (Yes vs no), and whether the use of venous loop diuretics at baseline (Yes vs no). Additional file 1: Fig. S1 presented the correlation of SHR with each covariate before and after applying entropy balancing. For each outcome, a generalized logistic regression model with entropy balancing weights was used to obtain the odds ratio (OR) of each SHR (referent, SHR of 0.8) and fit the non-linear model. The corresponding 95% confidence intervals (CIs) were estimated using the percentile bootstrapping method [26, 27]. The SHR at admission was modeled with restricted cubic splines, and three knots located at the 25th, 50th, and 75th percentiles of SHR.