<table><caption>S1 Table. Complete description of the confounders for the main analysis.</caption><thead><tr><th></th><th>Missing</th><th>Overall</th><th>Cristalloids only</th><th>Cristalloids + Albumin</th><th>P-Value</th></tr></thead><tbody><tr><td>n</td><td></td><td>18421</td><td>14862</td><td>3559</td><td></td></tr><tr><td>Glycopeptide, n (%)</td><td></td><td>9492 (51.5)</td><td>7650 (51.5)</td><td>1842 (51.8)</td><td></td></tr><tr><td>Beta-lactams, n (%)</td><td></td><td>5761 (31.3)</td><td>5271 (35.5)</td><td>490 (13.8)</td><td></td></tr><tr><td>Carbapenems, n (%)</td><td></td><td>727 (3.9)</td><td>636 (4.3)</td><td>91 (2.6)</td><td></td></tr><tr><td>Aminoglycosides, n (%)</td><td></td><td>314 (1.7)</td><td>290 (2.0)</td><td>24 (0.7)</td><td></td></tr><tr><td>suspected_infection_blood, n (%)</td><td></td><td>170 (0.9)</td><td>149 (1.0)</td><td>21 (0.6)</td><td></td></tr><tr><td>RRT, n (%)</td><td></td><td>229 (1.2)</td><td>205 (1.4)</td><td>24 (0.7)</td><td></td></tr><tr><td>ventilation, n (%)</td><td></td><td>16376 (88.9)</td><td>12931 (87.0)</td><td>3445 (96.8)</td><td></td></tr><tr><td>vasopressors, n (%)</td><td></td><td>9058 (49.2)</td><td>6204 (41.7)</td><td>2854 (80.2)</td><td></td></tr><tr><td>Female, n (%)</td><td></td><td>7653 (41.5)</td><td>6322 (42.5)</td><td>1331 (37.4)</td><td></td></tr><tr><td>White, n (%)</td><td></td><td>12366 (67.1)</td><td>9808 (66.0)</td><td>2558 (71.9)</td><td></td></tr><tr><td>Emergency admission, n (%)</td><td></td><td>9605 (52.1)</td><td>8512 (57.3)</td><td>1093 (30.7)</td><td></td></tr><tr><td>Insurance, Medicare, n (%)</td><td></td><td>9727 (52.8)</td><td>7958 (53.5)</td><td>1769 (49.7)</td><td></td></tr><tr><td>myocardial_infarct, n (%)</td><td></td><td>3135 (17.0)</td><td>2492 (16.8)</td><td>643 (18.1)</td><td></td></tr><tr><td>malignant_cancer, n (%)</td><td></td><td>2465 (13.4)</td><td>2128 (14.3)</td><td>337 (9.5)</td><td></td></tr><tr><td>diabetes_with_cc, n (%)</td><td></td><td>1633 (8.9)</td><td>1362 (9.2)</td><td>271 (7.6)</td><td></td></tr><tr><td>diabetesWithout_cc, n (%)</td><td></td><td>4369 (23.7)</td><td>3532 (23.8)</td><td>837 (23.5)</td><td></td></tr><tr><td>metastatic_solid_tumor, n (%)</td><td></td><td>1127 (6.1)</td><td>1016 (6.8)</td><td>111 (3.1)</td><td></td></tr><tr><td>severe_liver_disease, n (%)</td><td></td><td>1289 (7.0)</td><td>880 (5.9)</td><td>409 (11.5)</td><td></td></tr><tr><td>renal_disease, n (%)</td><td></td><td>3765 (20.4)</td><td>3159 (21.3)</td><td>606 (17.0)</td><td></td></tr><tr><td>aki_stage_0.0, n (%)</td><td></td><td>7368 (40.0)</td><td>6284 (42.3)</td><td>1084 (30.5)</td><td></td></tr><tr><td>aki_stage_1.0, n (%)</td><td></td><td>4019 (21.8)</td><td>3222 (21.7)</td><td>797 (22.4)</td><td></td></tr><tr><td>aki_stage_2.0, n (%)</td><td></td><td>6087 (33.0)</td><td>4605 (31.0)</td><td>1482 (41.6)</td><td></td></tr><tr><td>aki_stage_3.0, n (%)</td><td></td><td>947 (5.1)</td><td>751 (5.1)</td><td>196 (5.5)</td><td></td></tr><tr><td>SOFA, mean (SD)</td><td>0</td><td>6.0 (3.5)</td><td>5.7 (3.4)</td><td>6.9 (3.6)</td><td>&lt;0.001</td></tr><tr><td>SAPSII, mean (SD)</td><td>0</td><td>40.3 (14.1)</td><td>39.8 (14.1)</td><td>42.8 (13.6)</td><td>&lt;0.001</td></tr><tr><td>Weight, mean (SD)</td><td>97</td><td>83.3 (23.7)</td><td>82.5 (24.2)</td><td>86.4 (21.2)</td><td>&lt;0.001</td></tr><tr><td>temperature, mean (SD)</td><td>966</td><td>36.9 (0.6)</td><td>36.9 (0.6)</td><td>36.8 (0.6)</td><td>&lt;0.001</td></tr><tr><td>mbp, mean (SD)</td><td>0</td><td>75.6 (10.2)</td><td>76.3 (10.7)</td><td>72.4 (7.2)</td><td>&lt;0.001</td></tr><tr><td>resp_rate, mean (SD)</td><td>9</td><td>19.3 (4.3)</td><td>19.6 (4.4)</td><td>18.0 (3.8)</td><td>&lt;0.001</td></tr><tr><td>heart_rate, mean (SD)</td><td>0</td><td>86.2 (16.3)</td><td>86.2 (16.8)</td><td>86.5 (14.3)</td><td>0.197</td></tr><tr><td>spo2, mean (SD)</td><td>4</td><td>97.4 (2.2)</td><td>97.3 (2.3)</td><td>98.0 (2.1)</td><td>&lt;0.001</td></tr><tr><td>lactate, mean (SD)</td><td>4616</td><td>3.0 (2.5)</td><td>2.8 (2.4)</td><td>3.7 (2.6)</td><td>&lt;0.001</td></tr><tr><td>urineoutput, mean (SD)</td><td>301</td><td>24.0 (52.7)</td><td>24.7 (58.2)</td><td>21.1 (16.6)</td><td>&lt;0.001</td></tr><tr><td>admission_age, mean (SD)</td><td>0</td><td>66.3 (16.2)</td><td>66.1 (16.8)</td><td>67.3 (13.1)</td><td>&lt;0.001</td></tr><tr><td>delta mortality to inclusion, mean (SD)</td><td>11121</td><td>316.9 (640.2)</td><td>309.6 (628.8)</td><td>365.0 (708.9)</td><td>0.022</td></tr><tr><td>delta intervention to inclusion, mean (SD)</td><td>14862</td><td>0.3 (0.2)</td><td>nan (nan)</td><td>0.3 (0.2)</td><td>nan</td></tr><tr><td>delta inclusion to intime, mean (SD)</td><td>0</td><td>0.1 (0.2)</td><td>0.1 (0.2)</td><td>0.1 (0.1)</td><td>0.041</td></tr><tr><td>delta ICU intime to hospital admission, mean (SD)</td><td>0</td><td>1.1 (3.7)</td><td>1.0 (3.7)</td><td>1.6 (3.4)</td><td>&lt;0.001</td></tr><tr><td>los_hospital, mean (SD)</td><td>0</td><td>12.6 (12.5)</td><td>12.6 (12.5)</td><td>12.9 (12.4)</td><td>0.189</td></tr><tr><td>los_icu, mean (SD)</td><td>0</td><td>5.5 (6.7)</td><td>5.5 (6.5)</td><td>5.5 (7.2)</td><td>0.605</td></tr></tbody></table>

<caption>**Table 6.** Characteristics of the trial population measured on the first 24 hours of ICU stay. Risk scores (AKI, SOFA, SAPSII) and lactates have been summarized as the maximum value during the 24 hour period for each stay. Total cumulative urine output has been computed. Other variables have been aggregated by taking mean during the 24 hour period.</caption>
S6 Fig. Directed Acyclic Graph.

The expert DAG in figure depicts the known causal links between these variables.

<figure><img src="image_16.png" /><figcaption>**Fig 11.** *Causal graph for the Albumin vs crystalloids emulated trial* – The green arrow indicates the effect studied. Black arrows show causal links known to medical expertise. Dotted red arrows highlight confounders not directly observed. For readability, we draw only the most important edges from an expert point of view. All white nodes correspond to variables included in our study.</figcaption></figure>
S7 Fig. Complete results for the main analysis.

Compared to Figure 3, we also report in Figure 12 the estimates for Causal forest estimators and other choices of feature aggregation (first and last).

<figure><img src="image_17.png" /><figcaption>Fig 12. Full sensitivity analysis: The estimators with forest nuisances point to no effect for almost every causal estimator consistently with the RCT gold standard. Only matching with forest yields an unconvincingly high estimate. Linear nuisance used with doubly robust methods suggest a reduced mortality risk for albumin. The choices of aggregation only marginally modify the results expect for propensity score matching. The green diamonds depict the mean effect and the bar are the 95% confidence intervals obtained by 50 bootstrap repetitions.</figcaption></figure>
S8 Fig. Complete results for the Immortal time bias.

Compared to Figure 3a, we also report in Figure 13 the estimates for Double Machine Learning, Inverse Propensity Weighting for both Random Forest and Ridge Regression. Feature aggregation was concatenation of first and last for all estimates.

<figure><img src="image_18.png" /><figcaption>**Fig 13.** Sensitivity analysis for immortal time bias: Every choice of estimates show an improvement of the albumin treatment when increasing the observation period, thus increasing the blank period between inclusion and administration of albumin. Aggregation was concatenation of first and last features. The green diamonds depict the mean effect and the bar are the 95% confidence intervals obtained by 50 bootstrap repetitions.</figcaption></figure>
**S6 Appendix. Deviating from expert ignorability – Impact of smaller confounders sets.**

We conducted a dedicated vibration analysis on the different choices of confounders. We created three confounder subsets in addition to all confounders (24 variables): all confounders without antibiotics (Glycopeptides, Beta-lactams, Carbapenems, Aminoglycosides), all confounders without any measurement (weight, lactate, heart rate, spo2, mbp, urine output, temperature, AKI stage, SAPSII, respiratory rate, SOFA), only socio-demographics (admission age, female, emergency admission, insurance–medicare, race).

Figure 3b shows that small deviation from the ignorability assumptions is tolerable: for example, removing antibiotics does not impact the estimate. However, the larger the deviation from graph S6 Fig, the larger the bias compared to the gold-standard. Adjusting only for socio-demographics features is the closest from an unadjusted risk difference, indicating that we lack important confounders on the patient health state. This stability of the treatment effect estimator once sufficient confounders have been included has already been described and suggested as a confounder selection method [36].
S9 Fig. Vibration analysis for aggregation.

We conducted a dedicated vibration analysis on the different choices of features aggregation, studying the impact on the estimated ATE. We also studied if some choices of aggregation led to substantially poorer overlap.

We assessed overlap with two different methods. As recommended by [44], we did a graphical assessment by plotting the distribution of the estimated. The treatment model hyper-parameters were chosen by random search, then predicted propensity scores were obtained by refitting this estimator with cross-fitting on the full dataset.

As shown in Figure 14, we did not find substantial differences between methods when plotting graphically the distribution of the estimated propensity score.

<figure><img src="image_19.png" /><figcaption>Fig 14. Different choices of aggregation yield qualitatively close distributions of the propensity score: Figure 14a)a) shows a concatenation of first, last and median measures whereas Figure 14b)b) shows an aggregation by taking the first measure only. The underlying treatment effect estimator is a random forest.</figcaption></figure>

We also used normalized total variation (NTV) as a summary statistic of the estimated propensity score to measure the distance between treated and control population [73]. This statistic varies between 0 – perfect overlap – and 1 – no overlap at all. Fig 15 shows no marked differences in overlap as measured by NTV between aggregation choices, comforting us in our expert-driven choice of the aggregation: a concatenation of first and last feature observed before inclusion time.
<figure><img src="image_20.png" /><figcaption>Fig 15. Vibration analysis dedicated to the aggregation choices. The choices of aggregation only marginally modify the results. When assessed with Normalized Total Variation, the overlap assumption is respected for all our choices of aggregation. The green diamonds depict the mean effect and the bar are the 95% confidence intervals obtained by 50 bootstrap repetitions.</figcaption></figure>
# S7 Appendix. Details on treatment heterogeneity analysis. Detailed estimation procedure

The estimation of heterogeneous effect based on Double Machine Learning adds another step after the computation, regressing the residuals of the outcome nuisance $\tilde{Y} - \mu(X)$ against the residuals of the treatment nuisance $\tilde{A} = A - e(X)$ with the heterogeneity features $X_{CATE}$. Noting the final CATE model $\theta$, Double ML solves:

$$ \arg\min_{\theta} \mathbb{E}_n [(\tilde{Y} - \tau(X_{CATE}) \cdot \tilde{A})^2] $$

Where $\tilde{Y} = Y - \hat{m}(X)$ and $\tilde{A} = A - \hat{e}(X)$

To avoid the over-fitting of this last regression model, we split the dataset of the main analysis into a train set (size=0.8) where the causal estimator and the final model are learned, and a test set (size=0.2) on which we report the predicted Conditional Average Treatment Effects.

## Known heterogeneity of treatment for the emulated trial

[82] observed statistical differences in the post-hoc subgroup analysis between patient with and without septic shock at inclusion. They found increasing treatment effect measured as relative risk for patients with septic shock (RR=0.87; 95% CI, 0.77 to 0.99 vs 1.13;95% CI, 0.92 to 1.39).

[108] conducted a post-hoc subgroup analysis of patients with or without brain injury –defined as Glasgow Coma Scale between 3 to 8–. The initial population was patients with traumatic brain injury (defined as history or evidence on A CT scan of head trauma, and a GCS score <= 13). They found higher mortality rate at 24 months in the albumin group for patients with severe head injuries.

[83] conducted a subgroup analysis on age (<60 vs >60), septic shock and sex. They conclude for increasing treatment effect measured as Restricted Mean Survival Time for Sepsis vs septic shock (3.47 vs. 2.58), for age >=60 (3.75 vs 2.44), for Male (3.4 vs 2.69). None of these differences were statistically significant.

## Vibration analysis

The choice of the final model for the CATE estimation should also be informed by statistical and clinical rationals. Figure 16 shows the distribution of the individual effects of a final random forest estimator, yielding CATE estimates that are not consistent with the main ATE analysis. Figure 17 shows that the choice of this final model imposes an inductive bias on the form of the heterogeneity and different sources of noise depending of the nature of the model. A random forest is noisier than a linear model. Figure 17 shows the difference of modelization on the subpopulation of non-white male patients without septic shock. One can see that the decreasing linear trend is reflected by the random forest model only for patients aged between 55 and 80.
<figure><img src="image_21.png" /><figcaption>**Fig 16.** Distribution of Conditional Average Treatment effects on sex, age, race and pre-treatment septic shock estimated with a final forest estimator. The CATE are positive for each subgroups, which is not consistent with the null treatment effect obtained in the main analysis. The boxes contain between the 25th and 75th percentiles of the CATE distributions with the median indicated by a vertical line. The whiskers extends to 1.5 the inter-quartile range of the distribution.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Fig 17.** Distribution of Conditional Average Treatment effects on sex, age, race and pre-treatment septic shock plotted for different ages. On the top the final estimator is a linear model; on the bottom, it is a random forest. The forest-based CATE displays more noisy trends than the linear-based CATE. This suggest that the flexibility of the random forest might be underfitting the data.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>**Fig 18.** Figure 17 on the subpopulation of white male patients without septic shock. Contrary to the ridge regression (on top) inducing a nicely interpretable trend, using random forests as the final estimator failed to recover CATE on ages: the predicted estimates do not exhibit any trend and display inconsistently large effect sizes, suggesting data underfitting.</figcaption></figure>