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

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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
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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

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

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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
Patients severity of illness increases the risk
of VAP and the poor health conditions leading to
VAP are also indicative of an increased mortality
risk.
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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
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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
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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 illnesst1
Severity of illnesst
VAPt
Mortality
VAPt1
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Time dependent confounding

• Association between infection and mortality is
  disturbed by time-dependent confounders
• severity of illness at time t1 is a confounder
• ? we need to adjust

Severity of illnesst1
Severity of illnesst
VAPt
Mortality
VAPt1
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Time dependent confounding

• Association between infection and mortality is
  disturbed by time-dependent confounders
• Severity of illness at time t1 may also be
  effected by the patients
• infection status at time t (lies on the
  causal path)
• ? we should not adjust

Severity of illnesst1
Severity of illnesst
VAPt
Mortality
VAPt1
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Importance of modelling evolution in severity of
illness
Severity of illness
ICU admission
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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 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

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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

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Its 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, ß2
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.
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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

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Selection of patients compatible the infection
path no infection
ICU admission Day 1
Day 30

• No infection
• Patients who died or were discharged without
  infection

infection
Discharged alive
Died in ICU
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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 ? ? ? ? ?
? ? ? ? ?

• 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

infection
Discharged alive
Died in ICU
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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
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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
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
21 30
t
1 20
wt
w20w20
w1 w20
Observed information
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Selection of patients compatible with getting
infection on day 5
ICU admission Day 1
Day 5
Day 30
0 0 0 0 1 1
0 0 0 0 1 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

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Estimating equation
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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

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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

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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

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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

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Competing risk analysis ignoring time dependent
confounding
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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

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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

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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