Estimating a causal effect using observational data Aad van der Vaart Afdeling 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 Y 1 if individual had been treated outcome Y 0 if individual had not been treated mean treatment effect E Y 1 – E Y 0 Unfortunately, we observe only one of Y 1 and Y 0, namely: Y= Y A

Counter factuals (3) ASSUMPTION: there exists a measured covariate Z with A  (Y 0, Y 1 ) given Z  means “are statistically independent” Under ASSUMPTION: E Y 1 – E Y 0 =  {E (Y | A=1, Z=z) - E (Y | A=1, Z=z) } dP Z (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 = (a 0, a 1,..., a K ) observed treatments: A = (A 0, A 1,..., A K ) counterfactual outcomes: Y a observed outcome: Y A We are interested in E Y a for certain a

Longitudinal Data (2) times:       treatments: a = (a 0, a 1,..., a K ) observed treatments: A = (A 0, A 1,..., A K ) ASSUMPTION: Y a  A k given ( Z k, A k-1 ), for all k Under ASSUMPTION E Y a 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 = (Z 0, Z 1,..., Z K )

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 effect no effect negative effect no effect negative effect time  k-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 H 0 :  =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