Controlling for Time Dependent Confounding Using Marginal Structural Models in the Case of a Continuous Treatment

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Controlling for Time Dependent Confounding Using
Marginal Structural Models in the Case of a
Continuous Treatment

• O Wang1, T McMullan2
• 1Amgen, Thousand Oaks, CA 2inVentiv Clinical,
  Collegeville,PA

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Background

• The management of anemia in end stage renal
  disease is a continuous process over time
• Physicians monitor patient characteristics such
  as hemoglobin levels, iron, other co-morbid
  conditions regularly over time and adjust their
  Epogen dosing behavior accordingly
• Modeling Epogen over time provides a more
  realistic measure of its impact on clinical
  outcomes

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Confounding by Indication

• Hemoglobin measured over time is a good predictor
  of both Epogen dose and patient outcome a
  confounder
• Hemoglobin is also impacted by previous Epogen
  dose time-dependent confounding
• Standard time dependent Cox PH models will
  produce biased parameter estimates need another
  approach!

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Marginal Structural Models (MSM)

• Censoring weights are created similarly
• This weighting creates a pseudo-population,
  without confounding between A and L
• Stabilized vs. non-stabilized weights
• History-adjusted MSM

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MSM Illustrated
Severely Sick Patients
Mildly Sick Patients
High dose
Low dose
High dose
Low dose
IPTW
High dose
Low dose
High dose
Low dose
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MSM weight estimates

• Binary treatment Logistic model that estimates
  probability of on/off treatment
• Ordinal treatment categories Ordinal regression
  that estimates probability of receiving current
  treatment category
• Continuous treatment probability density at
  current treatment could be very small!

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MSM weights
For patient i at time point K

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MSM for continuous treatment?

• Theoretically MSM models can be applied to a
  continuous treatment variable
• But MSM parameter estimates could be highly
  sensitive to a number of issues
• Given the lack of reported statistical work in
  this area, how good is our continuous MSM
  parameter estimate? in other words how well is
  the MSM model adjusting for time dependent
  confounding?

1. ETA assumption
2. Observed Counterfacturals is it a
   representative sample

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Study design Observed data

• 60,000 patients in the database
• 7/2000 6/2002 up to 2 years of data
• EPO dose of every administration, Hb on average
  every 2 weeks
• 6 months baseline, 12 months follow up
• Data aggregated to bi-weekly

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MSM Simulation Approach 1
Baseline Month i1
Age N(Omean , Ovar) truncated at 18 and 100 c1
b10 b11 Age t1 b10 b11 Age b12
c1 logit(Y1) b10 b11 Age b12 c1 a
t1 logit(cen1) b10 b11 Age b12 c1 b13 t1
Months i2 to 12
ci bi0 bi1 Age bi2 ci-1 bi3 ti-1 ti bi0
bi1 Age bi2 ci bi3 ti-1 logit(Yi) bi0
bi1 Age bi2 ci a ti logit(ceni) bi0 bi1
Age bi2 ci bi3 ti
a fixed at log(0.85)
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MSM Simulation 1 Results
Results log(0.85)
MSM Estimates N600 Mean0.919 Median0.908
CI0.6101.229 PHREG Estimates N600
Mean0.859 Median0.858 CI0.8460.871
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MSM Simulation Approach 2
Design
Build two simulated datasets. Dataset A EPOHGB
independent. Dataset B EPOHGB related as in
obs data. Model datasets A B with
PHREG/compare. Model dataset B with MSM/compare
with A.
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MSM Simulation 2
Design Dataset A

• Sample log EPO, Hb, censoring
• separately from observed data.
• Model Mortality Curr Log EPO Curr Hb
• using observed data ß0.73 for dose
• Fit sampled log EPO Hb to model
• obtain a predicted probability of mortality.

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MSM Simulation 2
Design Dataset B
Keep Mortality, Hb, censoring as in dataset
A. Model logEPOLag1 Hb using observed
data. Fit bootstrapped Lag1 Hb to model obtain
a predicted logEPO.
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MSM Simulation 2 Results
n520 runs
Design Dataset A
PHREG mean0.750 median0.750 CI0.7456-0.7541 MS
M mean0.737 median0.737 CI0.7314-0.7419
Design Dataset B
PHREG mean0.999 median0.999 CI0.9877-1.0112 MS
M mean0.883 median0.874 CI0.7220-1.0434
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Simulation Results Summary

• Simulation 1 hard to interpret as the truth is
  unknown
• Simulation 2 Dataset A
• MSM ? PHREG (as expected)
• Simulation 2 Dataset B
• Truth lt MSM lt PHREG
• Suggestion of adjustment for confounding
• No over-adjustment, ie, not biased in the other
  direction

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MSM Assumptions

• Consistency Assumption (CA)
• the observed outcome equals the treatment
  regimen counterfactural outcome
• a violation of the above would occur if the
  patient was not treatment compliant
• Sequential Randomization Assumption (SRA)
• at each time point, conditional on the
  observed past, treatment assignment
• at this time point is strongly ignorable and
  there are no unmeasured confounders
• Under SRA patients are conditionally
  exchangeable
• Coarsening at Random (CAR)
• at each time point, conditional on the
  observed past, the censoring mechanism
• at this time point is strongly ignorable
  (missing at random MAR)
• Experimental Treatment Assignment (ETA)
• the probability of observing a specific
  treatment regimen is gt 0 and lt 1 that is
• the treatment decision is not a deterministic
  function of the past
• MSM Assumptions not easy or even impossible to
  check

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Observed data analysis

• Analysis model
• Mortality is modeled with weighted,
  time-dependent, 2-month lagged EPO
• Treatment weight models and censoring models
• 2-month lagged EPO and censoring are modeled with
  2.5-, 3-, 3.5-, and 4-month lagged EPO and Hb,
  plus baseline covariates.

Dose Hb
Dose Hb
Dose Hb
Dose Hb
2 months
2 wk
2 wk
2 wk
2 wk
Mortality
EPO dose (for inference)
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Continuous MSM, non-HA, 5 Truncation
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Continuous MSM, HA, 5 Truncation
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Categorical MSM, Non-HA, 5 Truncation
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Marginal Structural Models (MSM)

• Censoring weights are created similarly
• This weighting creates a pseudo-population,
  without confounding between A and L
• Stabilized vs. non-stabilized weights
• History-adjusted MSM

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Categorical MSM, HA, 5 Truncation
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Categorical MSM, Non-HA, 2 Truncation
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Categorical MSM, HA, 2 Truncation
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Categorical MSM, Non-HA, 1 Truncation
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Categorical MSM, HA, 1 Truncation
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Continuous vs categorical Caution!
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Conclusions

• The value of repeated Epogen and Hemoglobin data
  over time, and the granularity of these data, are
  key to understanding their relationship with
  mortality.
• MSM IPTW estimation can be a challenging problem
  over time when there are many time points and a
  large number of treatment levels.
• MSM models are promising for adjusting for
  confounding by indication in a variety of
  treatment variable types
• Simulations seem to point to some adjustment for
  confounding by indication with continuous
  treatment but residual bias may still be
  unaccounted for.