Estimating a causal effect using observational data
Aad van der VaartAfdeling Wiskunde, Vrije Universiteit Amsterdam
Joint with Jamie Robins, Judith Lok, Richard Gill
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
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 | A=1 ) – E( Y | A=0 )
However, if treatment is not randomly assigned this is NOT what we want to know
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
Counter factuals (3)
ASSUMPTION: there exists a measured covariate Z with
A (Y0, Y1 ) given Z
means “are statistically independent”
Under ASSUMPTION:
E Y1 – E Y0 = {E (Y | A=1, Z=z) - E (Y | A=1, Z=z) } 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”
Longitudinal Data
times:
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
Longitudinal Data (2)times:
treatments: a = (a0, a1, . . . , aK )
observed treatments: A = (A0, A1, . . . , AK )
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”
observed covariates: Z = (Z0, Z1, . . . , ZK )
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.
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.
Structural Nested Models
shift map corresponding to these distributions,
transforms into
IDEA: model by a parameter and estimate it
treatment until time k
outcome of this treatment
Structural Nested Models (2)
treatment until time k
outcome of this treatment
transforms into
positive effectno effectnegative effect
no effect
negative effect
timek-1 k
Estimation
• Make regression model for
• Make model for
• Add as explanatory variable
•Estimate by the value such that does NOT add explanatory value.
Under ASSUMPTION:
• is distributed as
•
Estimation (2)
Example: if treatment A is binary, then we might use a logistic regression model
We estimate ( by standard software for given The “true” is the one such that the estimated is zero.
We can also test whether treatment has an effect at all by testing H0: =0 in this model with Y instead of Y
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