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

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

Case study: Estimation of attributable mortality of ventilator associated pneumonia

MotivationAttributable 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:

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

Informative censoringThe 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 interestDischarge from the ICU competing eventModels based on the hazard associated with the CIF are used in the ICU setting

Causal inferenceConfounding:

Infected and non-infected patients are not comparable because they differ in terms of factors other than their infection status

Infection Mortality

Severity of illness

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.

Assumption of no unmeasured confounders

Severity of illness

Unmeasured confounders

VAP Mortality

No unmeasured confounding

Information that leads to acquiring VAP is completely contained within

the measured confounders

Non causal paths between VAP and mortality

Unmeasured confounders

VAP MortalityCausal path

Severity of illness

In a non-randomized setting at a single time point, we can adjust for confounding variables

by including them in a regression model

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

VAPt

Severity of illnesst Severity of illnesst+1

VAPt+1 Mortality

Time dependent confounding

Association between infection and mortality is disturbed by time-dependent confounders:

severity of illness at time t+1 is a confounderwe need to adjust

VAPt

Severity of illnesst Severity of illnesst+1

VAPt+1 Mortality

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

VAPt

Severity of illnesst Severity of illnesst+1

VAPt+1 Mortality

Seve

rity

of i

llnes

s

ICU admission Time of infection Time of dead

Died with VAP

Died from VAP

Importance of modelling evolution in severity of illness

Marginal structural survival model in the presence of competing risks

Notation:Let At and Dt be 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 patientwould, 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

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:

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.

1 1

It’s a marginal model because we do not condition on time varying confounders because theyare themselves affected by early infections !!

Estimation principleHow to fit this model:1.

Select those patients whose observed data are compatible with the given infection path

2.

Perform a competing risk analysis on those data, using inverse probability weighting to account for the selective nature of that subset

Selection of patients compatible the infection path no infection

No infection:Patients who died or were discharged without infection

ICU admissionDay 1 Day 30

= infection Discharged aliveDied in ICU

= infection Discharged aliveDied in ICU

Discharge without infection

Patients who are discharged by time t stay in the risk setSurvival time of infinity (30 days)We need to expand the data setSeveral possible infection paths after discharge

ICU admissionDay 1 Day 30Day 20

0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   ?   ?   ?   ?   ?   ?   ?   ?   ?   ?  At

ICU admissionDay 1 Day 30

0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   ?   ?   ?   ?   ?   ?   ?   ?   ?   ?  

Discharge without infection

At

Day 20

0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   ?   ?   ?   ?   ?   ?   ?   ?   ?   ? Yt

Observed information

1   ………………………………………………………………………………

20   ?   ?   ?   ?   ?   ?   ?   ?   ?   ? 

w1

………………………………………………………………………………w20  

?   ?   ?   ?   ?   ?   ?   ?   ?   ? 

twt

Data expansion

ICU admissionDay 1 Day 30

0   0   0   0   0   0   0   0   0  00   0   0   0   0   0   0   0   0  10   0   0   0   0   0   0   0   1  10   0   0   0   0   0   0   1   1  10   0   0   0   0   0   1   1   1  10   0   0   0   0   1   1   1   1  10   0   0   0   1   1   1   1   1  10   0   0   1   1   1   1   1   1  10   0   1   1   1   1   1   1   1  10   1   1   1   1   1   1   1   1  11   1   1   1   1   1   1   1   1  1

Discharge without infectionDay 20

(30 -

time of discharge) +1

possible infection paths

0   0   0  0   0   0   0   0   0  0 21  …………………………………30

w20

…………………………………w20

0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0At

0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0Yt

Observed information

1   ………………………………………………………………………………

20

w1

………………………………………………………………………………w20

twt

Data expansion

Selection of patients compatible with getting infection on day 5

Infection on day 5:Patients who died before day 5Patients who acquired infection on day 5 and died in the ICU within 30 daysPatients who were discharged after day 5 with an infection acquired on day 5 Patients who were discharged before day 5

ICU admissionDay 1 Day 30Day 5

0 0 0 0

1 1 ……

0 0 0 0 1 1 ……

Estimating equation

Estimating equation

Calculation of the patient specific time dependent weights :

Estimate using a logistic regressionFor patients who are discharged =1Calculate the weights as:

where K = discharge time

Weights

Data analysisData 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

Confounders included in the analysisBaseline 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

Preliminary resultCrude 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

Competing risk analysis ignoring time dependent confounding

1. Separated analysis per potential infection path

We selected patients compatible with a given infection pathAnalyse the data with a weighted pooled logistic regression model with a flexible time trend.Plot the cumulative incidence function

2. Results after solving the weighted estimating equation

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

Discussion

and future

workWhen ignoring the informative censoring we get biased resultsIn 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 modelWork in progress:

Calculation of sandwich estimators of the standard errorWe will develop semi-parametric estimators for the time-evolution in severity of illnessUsing the COSARA data set we will be able to account for a lot more time dependent confoundersCheck results with simulation studies