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Part V: Further topics
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Compliance measurement error is real
• Compliance measurement error
• errors in compliance measurement are unavoidable
• Time-varying exposures
• common errors not only reflect random noise but also carry
a systematic component
• Dynamic treatment regimes
e.g. pill counts
e.g. chemical measures in body fluids capture only the
recently accumulated dose
e.g. white coat compliance
• Marginal models
• despite the introduction of electronic monitoring systems, a
margin of error remains
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Concerns
• Intent-to-treat analyses ignore compliance measurements
and thus remain valid with mismeasured doses
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G-estimation is robust to unbiased error
• Goetghebeur and Vansteelandt (2005) show that inference
with linear structural mean models is robust to unbiased
measurement error
compliance-adjusted analyses may not…
• They show that inference with generalized linear structural
mean models is highly, but not fully robust to unbiased
measurement error
• even in simple linear regression of outcome on observed
exposure, unbiased errors on exposure lead to less precise
and attenuated effect estimates
and how to correct for unbiased error with known variance
in generalized linear structural mean models (unpublished
work)
• Dunn (1999) studies this problem in a fully parametric
setting under the assumption that observed compliance to
assigned exposure is an attribute of the patient,
exchangeable between treatment arms
• Ongoing research on correcting for biased error
He finds that measurement error induces bias
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Potential impact in hypertension data
Simulation results
• 1000 datasets of 150 independent samples
•
A
B
C
normally distributed measurement error with
mean 0, standard deviation 0.1 ¥D
mean 0, standard deviation 0.2 ¥D
mean 0.5, standard deviation 0.1 ¥D
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Time-varying exposures
• Suppose that we measure the following observations
• When exposures are recorded in function of time, more
detailed causal conclusions can be obtained
L0,R
1. Sustained effects over time?
2. Does dose effect at time t depend on patient-characteristics
measured at that time?
D2
D1
L2
D2
Y
…
E(Y(d1,…,dt,0)- Y(d1,…,dt-1,0) | D1=d1, …,Dt=dt,L0,…,Lt)
U
D1
L1
• A structural nested mean model is a model for the effect of
a final blip dt in exposure at time t,
among people with a given covariate history (L1,…,Lt) who
received that exposure
e.g. is the effect of zidovudine intake at time t on mortality
modified by CD4-count at that time?
If so, is there an optimal dose to assign for a given CD4count?
optimal dynamic treatment regime
R
Structural nested mean models
• Example: fixed linear effects over time
E(Y(d1,…,dt,0)- Y(d1,…,dt-1,0) | D1=d1, …, Dt=dt) = ȥ dt
…
Y
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Examples
• Example: sustained linear effects in total dose taken
Examples
• Example: effect modification by time
E(Y(d1,…,dt,0)- Y(d1,…,dt-1,0) | D1=d1, …, Dt=dt) = ȥ1 dt + ȥ2 cum(dt-1)
E(Y(d1,…,dt,0)- Y(d1,…,dt-1,0) | D1=d1, …, Dt=dt) = (ȥ1 + ȥ2t)dt
where cum(dt-1) is the total exposure up to time t-1
dose effects may weaken over time
• Example: sustained linear effects in average dose taken
• Example: effect modification by current patient
characteristics
E(Y(d1,…,dt,0)- Y(d1,…,dt-1,0) | D1=d1, …, Dt=dt) = ȥ1 dt + ȥ2 cum(dt-1)
t-1
E(Y(d1,…,dt,0)- Y(d1,…,dt-1,0) | D1=d1, …, Dt=dt,L0,…,Lt) = (ȥ1 + ȥ2Lt)dt
dose dt may be beneficial for some and harmful for others
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Estimation principle
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Dynamic treatment regimes (Murphy, 2003)
• What is a dynamic treatment regime?
• Step 1: predict treatment free outcomes
A decision rule of what treatment to give at any point in time in
function of the past treatment and covariate history
e.g. under
E(Y(d1,…,dt,0)- Y(d1,…,dt-1,0) | D1=d1, …, Dt=dt) = ȥ dt
we can predict treatment free outcome as
Y - ȥ cum(dt)
e.g. under
E(Y(d1,…,dt,0)- Y(d1,…,dt-1,0)|D1=d1, …, Dt=dt,L0,…,Lt) = (ȥ1 + ȥ2Lt)dt
we can predict treatment free outcome as
Y – ȥ1 cum(dt) – ȥ2 cum(Ltdt)
• Step 2: find ȥ such that (predicted) treatment free outcomes
have equal means in both arms, conditional on baseline
covariates
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• The presence of effect modification raises the question of how
to determine optimal dynamic treatment regimes (Murphy,
2003; Robins, 2005; van der Laan and Pedersen, 2005)
• Previous structural models for compliance adjustment are not
directly useful to determine optimal dynamic regimes because
they merely compare received treatment with no treatment
• Assessing optimal dynamic regimes currently involves stronger
assumptions
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No unmeasured confounders
• One typically assumes that there are no unmeasured
confounders at each time t
No unmeasured confounders
• The validity of this assumption is not guaranteed by design
since there are likely many health-related factors influencing
patients’ decision to comply
i.e. that all prognostic factors used to decide on the next dose
level Dt were recorded in Ht=(L0,L1,D1,…,Lt-1,Dt-1,Lt)
i.e. Y(d) is conditionally independent of Dt, given Ht, for all
treatment regimes d and all times t
• Under this assumption, causal effects and optimal dynamic
treatment regimes can be inferred using structural nested
models
with the assumption of no unmeasured confounders in place of
the randomization assumption
• This assumption implies that average treatment effects
• Alternatively, causal effects can be inferred by constructing a
meta-population in which confounding by measured timevarying covariates has disappeared
E(Y(d1,…,dt,0)- Y(d1,…,dt-1,0)|D1=d1, …, Dt=dt,L0,…,Lt)
are the same for patients with different current doses Dt
and then analyzing the weighted data using a standard
parametric or semi-parametric model (Robins, Hernan and
Brumback, 2000)
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A lot is moving in this area
Exciting times
Acknowledgments
Colleagues and former students
• We have randomization based inference tools yielding
estimators of causal parameters which protect the null
hypothesis
• Work on optimal dynamic treatment regimes
• Dr. Krista Fischer-Lapp, Tartu University, Estonia
• Dr. Tom Loeys, MSD
It involves (even) more work than usual
• Phrasing the question and analysis a priori
• Postulating one (or more) causal models
• Issues of design: measure the necessary variables
as accurately as possible
• `Verifying’ assumptions
• Measurement of compliance
• Computation and analysis (gsmm in R and weights in SAS)
• Communication of results
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People who trusted us with their data
Abbott, ECOG, Johnson and Johnson, MSD, Pfizer, WHO
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Thank you!
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