Estimation of marginal structural survival models in the presence of competing risks

1. Estimation of marginal structural survival models in the presenceof competingrisks Case study: Estimation of attributable mortality of ventilator associated pneumonia Maarten Bekaert and Stijn Vansteelandt Department of Applied Mathematics and Computer Science, Ghent University, Ghent, Belgium Karl Mertens Epidemiology Unit, Scientic Institute of Public Health, Brussels, Belgium

2. Motivation • Attributable mortality of ventilator associated pneumonia (VAP) on 30-day ICU-mortality • A nosocomial pneumonia associated with mechanical ventilation that develops within 48 hours or more after hospital admission • Controversial results in ICU-literature due to:

3. Main question: “To what extent does pneumonia itself, rather than underlying comorbidity, contribute to mortality in critically ill patients.”

4. Informative censoring • The decision to discharge patients is closely related to their health status • Patients are typically discharged alive because they have a lower risk of death. • These patients are therefore not comparable with those who stayed within the hospital. • Competing risk analysis: • ICU-death  event of interest • Discharge from the ICU  competing event • Models based on the hazard associated with the CIF are used in the ICU setting

5. Causal inference • Confounding: • Infected and non-infected patients are not comparable because they differ in terms of factors other than their infection status Severity of illness Infection Mortality Patient’s severity of illness increases the risk of VAP and the poor health conditions leading to VAP are also indicative of an increased mortality risk.

6. Assumption of no unmeasured confounders Information that leads to acquiring VAP is completely contained within the measured confounders Severity of illness VAP Mortality No unmeasured confounding Unmeasured confounders

7. Non causal paths between VAP and mortality In a non-randomized setting at a single time point, we can adjust for confounding variables by including them in a regression model Severity of illness VAP Mortality Causal path Unmeasured confounders

8. Time dependent confounding • Confounders are time-dependent: • They are also intermediate on the causal path from infection to mortality because infection makes an increase in severity of illness more likely Severity of illnesst+1 Severity of illnesst VAPt Mortality VAPt+1

9. Time dependent confounding • Association between infection and mortality is disturbed by time-dependent confounders: • severity of illness at time t+1 is a confounder  we need to adjust Severity of illnesst+1 Severity of illnesst VAPt Mortality VAPt+1

10. Time dependent confounding • Association between infection and mortality is disturbed by time-dependent confounders: • Severity of illness at time t+1 may also be effected by the patients infection status at time t (lies on the causal path)  we should not adjust Severity of illnesst+1 Severity of illnesst VAPt Mortality VAPt+1

11. Importance of modellingevolution in severity of illness Died with VAP Died from VAP Severity of illness Time of infection Time of dead ICU admission

12. Marginal structural survival model in the presence of competing risks • Notation: • Let At and Dtbe two counting processes that respectively indicates 1 for ICU-acquired infection or ICU discharge at or prior to time t and 0 otherwise. • Under infection path = ( 0,0,0,0,1,1,1,1,1,1,… ) we would infect all ICU-patients 5 days after admission • expresses the counterfactual survival time, which an ICU patient would, possibly contrary to fact, have had under a given infection path • represents the counterfactual event status at time t (0 = still alive in ICU, 1 = dead, 2 = discharged alive from ICU) • For an event of type k (k = 1, 2) we define: = which is equal to the time until event k occurs or infinity when the competing event occurs

13. Marginal structural survival model in the presence of competing risks • The counterfactual cumulative incidence function: = which is the probability that, under an infection path , an event of type k occurs at or before time t. • Discrete time setting  pooled logistic regression model for the subdistribution hazard of death: 1 1 It’s a marginal model because we do not condition on time varying confounders because they are themselves affected by early infections !! For patients who have not died in the ICU, β2 describes the effect on the hazard of ICU- death of acquiring infection on a given day t, versus not acquiring infection up to that day.

14. Estimation principle • How to fit this model: • Select those patients whose observed data are compatible with the given infection path • Perform a competing risk analysis on those data, using inverse probability weighting to account for the selective nature of that subset

15. Selection of patients compatible the infection path no infection ICU admission Day 1 • No infection: • Patients who died or were discharged without infection Day 30 = infection Died in ICU Discharged alive

16. Discharge without infection ICU admission Day 1 • Patients who are discharged by time t stay in the risk set • Survival time of infinity (30 days) • We need to expand the data set • Several possible infection paths after discharge Day 20 Day 30 At 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ? ? ? ? ? ? ? ? ? ? = infection Died in ICU Discharged alive

17. Discharge without infection ICU admission Day 1 Day 20 Day 30 At 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ? ? ? ? ? ? ? ? ? ? Yt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ? ? ? ? ? ? ? ? ? ? t 1 ……………………………………………………………………………… 20 ? ? ? ? ? ? ? ? ? ? wt w1 ………………………………………………………………………………w20 ? ? ? ? ? ? ? ? ? ? Observed information Data expansion

18. Discharge without infection ICU admission Day 1 Day 20 Day 30 At 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 (30 - time of discharge) +1 possible infection paths 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Yt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t 21 …………………………………30 1 ……………………………………………………………………………… 20 wt w20…………………………………w20 w1 ………………………………………………………………………………w20 Observed information Data expansion

19. Selection of patients compatible with getting infection on day 5 ICU admission Day 1 • Infection on day 5: • Patients who died before day 5 • Patients who acquired infection on day 5 and died in the ICU within 30 days • Patients who were discharged after day 5 with an infection acquired on day 5 • Patients who were discharged before day 5 Day 5 Day 30 0 0 0 0 1 1 …… 0 0 0 0 1 1 ……

20. Estimating equation

21. Estimating equation Weights • Calculation of the patient specific time dependent weights : • Estimate using a logistic regression • For patients who are discharged =1 • Calculate the weights as: where K = discharge time

22. Data analysis • Data set: • Data from the National Surveillance Study of Nosocomial Infections in ICU's (Belgium). • A total of 16868 ICU patients were analyzed. • Of the 939 (5,6%) patients who acquired VAP in ICU and stayed more than 3 days, 186 (19,8%) died in the ICU, as compared to 1353(8,4%) deaths among the 15929 patients who remained VAP-free in ICU

23. Confounders included in the analysis • Baseline confounders: • age, gender, reason for ICU admission, acute coronary care, multiple trauma, presence and type of infections upon ICU admission, prior surgery, baseline antibiotic use and the SAPS score • Time dependent confounders: • Invasive therapeutic treatment indicators collected on day t: • indicators of exposure to mechanical ventilation, central vascular catheter, parenteral feeding, presence and/or feeding through naso- or oro-intestinal tube, tracheotomy intubation, nasal intubation, oral intubation, stoma feeding and surgery

24. Preliminary result • Crude analysis: • Ignoring informative censoring: pooled logistic regression • When not take into account time dependent confounding, the OR associated with infection is equal to 0,67 with 95% CI (0,57 ; 0,79) • Including time dependent confounders as covariates in the model the OR equals 0,75 with 95% CI (0,63 ; 0,89)  infected patients have a significant decreased mortality

26. Competing risk analysisignoring time dependentconfounding

27. 1. Separated analysis per potential infection path • We selected patients compatible with a given infection path • Analyse the data with a weighted pooled logistic regression model with a flexible time trend. • Plot the cumulative incidence function

29. 2. Resultsaftersolving the weightedestimatingequation • We defined a simple model for the effect of infection and a quadratic time trend without taking into acount the baseline confounders • OR equals 1,15 (no estimation of SE available yet) • Still working on models with a more complex impact of infection

31. Discussion and future work • When ignoring the informative censoring we get biased results • In order to get insight into the problem of time dependent confounding we will do a competing risk analysis by including the confounders as time dependent covariates in the model • Work in progress: • Calculation of sandwich estimators of the standard error • We will develop semi-parametric estimators for the time-evolution in severity of illness • Using the COSARA data set we will be able to account for a lot more time dependent confounders • Check results with simulation studies