Estimating a causal effect using observational data

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Estimating a causal effect using observational
data

• Aad van der Vaart
• Afdeling Wiskunde, Vrije Universiteit Amsterdam
• Joint with Jamie Robins, Judith Lok, Richard Gill

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CAUSALITY
Operational Definition If individuals are
randomly assigned to a treatment and control
group, and the groups differ significantly after
treatment, then the treatment causes the
difference
We want to apply this definition with
observational data
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Counter factuals
treatment indicator A ? 0,1 outcome Y
Given observations (A, Y) for a sample of
individuals, mean treatment effect might be
defined as E( Y A1 ) E( Y
A0 )
However, if treatment is not randomly assigned
this is NOT what we want to know
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Counter factuals (2)
treatment indicator A ? 0,1 outcome Y
outcome Y1 if individual had been
treated outcome Y0 if individual had not been
treated mean treatment effect E Y1 E Y0
Unfortunately, we observe only one of Y1 and Y0,
namely Y YA
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Counter factuals (3)
ASSUMPTION there exists a measured covariate Z
with A ?? (Y0, Y1 )
given Z
Under ASSUMPTION E Y1 E Y0 ? E (Y
A1, Zz) - E (Y A1, Zz) dPZ(z)
CONSEQUENCE under ASSUMPTION the mean treatment
effect is estimable from the observed data (Y,Z,A)
ASSUMPTION is more likely to hold if Z is
bigger
?? means are statistically independent
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Longitudinal Data
times t0 lt t1 lt . . . . lt
tK treatments a (a0, a1, . . . , aK
) observed treatments A (A0, A1, . . . , AK )
counterfactual outcomes Ya observed outcome
YA
We are interested in E Ya for certain a
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Longitudinal Data (2)
times t0 lt t1 lt . . . . lt
tK treatments a (a0, a1, . . . , aK
) observed treatments A (A0, A1, . . . , AK )
observed covariates Z (Z0, Z1, . . . , ZK )
ASSUMPTION Ya ?? Ak given ( Zk , Ak-1 ),
for all k
Under ASSUMPTION E Ya can be expressed in
the distribution of the observed data (Y, Z, A )
It is the task of an epidemiologist to collect
enough information so that ASSUMPTION is
satisfied
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Estimation and Testing

• Under ASSUMPTION it is possible, in principle
• to test whether treatment has effect
• to estimate the mean counterfactual treatment
  effects

A standard statistical approach would be to
model and estimate all unknowns. However there
are too many. We look for a semiparametric
approach instead.
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Shift function
The quantile-distribution shift function is the
(only monotone) function that transforms a
variable distributionally into another
variable. It is convenient to model a change in
distribution.
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Structural Nested Models
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Structural Nested Models (2)
positive effect no effect negative effect
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Estimation
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Estimation (2)
Example if treatment A is binary, then we might
use a logistic regression model
We estimate (a , b, d ) by standard software for
given g. The true g is the one such that the
estimated d is zero.
We can also test whether treatment has an effect
at all by testing H0 d0 in this model with Y
instead of Yg .
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End
Lok, Gill, van der Vaart, Robins,
2004, Estimating the causal effect of a
time-varying treatment on time-to-event using
structural nested failure time models Lok,
2001 Statistical modelling of causal effects in
time Proefschrift, Vrije Universiteit