Fig. 3

Different models’ handling of days alive without life support (DAWOLS) using COVID STEROID 2 trial data [15]. Horizontal axes: number or proportion of days; vertical axes: number of patients. The linear regression models the mean value of the distribution. No limits are imposed; thus, predictions outside the valid value space in both directions may occur (indicated by the arrows). The hurdle-negative binomial regression models the proportion of patients with exactly 0 days (red) in a logistic regression sub-model and the mean counts for all patients with ≥ 1 day (blue) using a (zero-truncated) negative binomial sub-model. Predictions lower than the valid value space are thus not possible, while predictions above the maximum valid value may occur (indicated by the arrow). The zero–one-inflated beta-model consists of three sub-models and models the proportion of DAWOLS. Two logistic regression models estimate the probabilities of having either a proportion of 0 or 1 (0 or 100%, red and green), and the probabilities of a proportion of 1 (100%, green) conditional on having either 0 or 1. A beta regression models the proportion of DAWOLS for patients with > 0 and < 1 (> 0% and < 100%, blue) proportion of DAWOLS. The combined model has lower and upper limits corresponding to the valid parameter space; thus, proportions < 0 or > 1 cannot be predicted. The cumulative logistic regression model separately models the probabilities of all distinct values in the dataset as ordinal categories under the proportional odds assumption (Table S1 in Additional file 1). Thus, only values occurring in the dataset will be predicted and specific clinical events (e.g., death) may be included as separate categories, for example, as a category worse than all other values (here -1, black, with all other values visualised using unique colours), although this may complicate prediction on the absolute scale