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