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Controlling for Time Dependent Confounding Using Marginal Structural Models in the Case of a Continuous Treatment O Wang 1, T McMullan 2 1 Amgen, Thousand Oaks, CA 2 inVentiv Clinical, Collegeville,PA

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2 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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3 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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4 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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5 MSM Illustrated Severely Sick PatientsMildly Sick Patients IPTW High dose Low dose High dose Low dose High dose Low dose

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6 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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7 MSM weights For patient i at time point K

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8 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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9 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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10 MSM Simulation Approach 1 Baseline Month i=1 Age ~ N(O mean, O var ) truncated at 18 and 100 c 1 ~ 10 + 11 Age t 1 ~ 10 + 11 Age + 12 c 1 logit(Y 1 ) ~ 10 + 11 Age + 12 c 1 + t 1 logit(cen 1 ) ~ 10 + 11 Age + 12 c 1 + 13 t 1 Months i=2 to 12 c i ~ i0 + i1 Age + i2 c i-1 + i3 t i-1 t i ~ i0 + i1 Age + i2 c i + i3 t i-1 logit(Y i ) ~ i0 + i1 Age + i2 c i + t i logit(cen i ) ~ i0 + i1 Age + i2 c i + i3 t i fixed at log(0.85)

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11 MSM Simulation 1 Results Results: log(0.85) MSM Estimates: N=600 Mean=0.919 Median=0.908 CI=0.610:1.229 PHREG Estimates: N=600 Mean=0.859 Median=0.858 CI=0.846:0.871

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12 MSM Simulation Approach 2 Design: Build two simulated datasets. Dataset A: EPO~HGB independent. Dataset B: EPO~HGB related as in obs data. Model datasets A & B with PHREG/compare. Model dataset B with MSM/compare with A.

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13 MSM Simulation 2 Design: Dataset A 1.Sample log EPO, Hb, & censoring separately from observed data. 2.Model Mortality ~ Curr Log EPO + Curr Hb using observed data – β=0.73 for dose 3.Fit sampled log EPO & Hb to model & obtain a predicted probability of mortality.

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14 MSM Simulation 2 Design: Dataset B Keep Mortality, Hb, & censoring as in dataset A. Model logEPO~Lag1 Hb using observed data. Fit bootstrapped Lag1 Hb to model & obtain a predicted logEPO.

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15 MSM Simulation 2 : Results Design: Dataset A Design: Dataset B n=520 runs PHREG: mean=0.750 median=0.750 CI=0.7456-0.7541 MSM: mean=0.737 median=0.737 CI=0.7314-0.7419 PHREG: mean=0.999 median=0.999 CI=0.9877-1.0112 MSM: mean=0.883 median=0.874 CI=0.7220-1.0434

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16 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 < MSM < PHREG Suggestion of adjustment for confounding No over-adjustment, ie, not biased in the other direction

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17 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 > 0 and < 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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18 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. MortalityEPO dose (for inference) Dose Hb Dose Hb Dose Hb Dose Hb 2 wk 2 months

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19 Continuous MSM, non-HA, 5% Truncation

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20 Continuous MSM, HA, 5% Truncation

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21 Categorical MSM, Non-HA, 5% Truncation

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22 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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23 Categorical MSM, HA, 5% Truncation

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24 Categorical MSM, Non-HA, 2% Truncation

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25 Categorical MSM, HA, 2% Truncation

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26 Categorical MSM, Non-HA, 1% Truncation

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27 Categorical MSM, HA, 1% Truncation

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28 Continuous vs categorical: Caution!

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29 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.

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