Causal Analysis in Population Studies

Transkript

1 Causal Analysis in Population Studies Prof. Dr. Henriette Engelhardt Prof. Dr. Alexia Prskawetz

2 Randomized experiments and observational studies for causal inference inference Historical dichotomy between randomized experiments and (nonrandomized) observational studies for causal effects Causal inference based on randomized experiments dates dates back to Fisher (1935), Kempthorne (1952), Cochran & Cox (1950), Cox (1958) Causal inference based on observational data described, e.g., Blalock (1964), Kenny (1979), Campbell & Stanley (1963), Cook & Campbell (1979) Change in 1970 s with Rubin s causal model: concept of potential outcomes was used to define causal effects in randomized experiments and observational studies (Rubin 1974)

3 The appeal of randomized experiments Randomized experiments are the golden standard in natural and social sciences for drawing inferences about causal effects Randomized experiments often infeasable due to ethical and practical reasons Features of randomized experiments : Random assignment to treatment and control condition Probabilities (or: propensity scores) are known from the design of the experiment Randomized experiments achieve balance on all pre-treatmentassignment variables (i.e., covariates), both measured and unmeasured the distribution of covariates differ only randomly between the treatment and control units

4 Observational studies as approximations of randomized experiments Rubin causal model (Rubin 1975, 1975, 1976, 1977, 1978, 1979, 1980): Defines causal effects as comparisons of potential outcomes or counterfactuals Considers explicitely the assignment mechanism A counterfactual statement is an assertion of the form If X had been the case, then Y would have happened, made when it is known to be false that X is the case. Famous historical counterfactual, opined by Pascal (1669): Le nez de Cléopâtre: s il eût été plus court, toute la face de la terre aurait changé. (Dawid 2000: 408)

5 Rubins framework for causal inference Units Treatment Outcome Covariates D Y 1 Y 0 X 1 1? 2 0? 3 0? 4 1?... N 1? Notes: = data observed;? = data are missing

6 Counterfactual account to causal inference Individual causal effect: δ(i) = Y 1 (i) Y 0 (i) the Fundamental Problem of Causal Inference is that causal inference is impossible (Holland 1986: 947). Statistical solution is to compute the average causal effect: δ = E(Y 1 Y 0 ) = E(Y 1 ) E(Y 0 )

7 Notation Let Y 1 and Y 0 denote the outcome with and without the treatment D, where Y ( X, U ) if D Y ( X, U ) if D The choice of treatment is assumed to be a latent variable, where D = 1 when > 0 and D = 0 otherwise: ( Z, U ) D 1 ( 0) X and Z denote factors observed by the analyst U 0, U 1 and U denote factors unobserved by the analyst

8 Notation (cont d) In a regression context (with an additively separable specification): Y ( X ) U, E( U ) 0, ( X ) X ' Y ( X ) U, E( U ) 0, ( X ) X ' ( Z) U, E( U ) 0, ( Z) Z ' X and Z denote factors observed by the analyst U 0, U 1 and U denote factors unobserved by the analyst 1

9 Parameters of interest Three different treatment effects are typically of interest 1. The average treatment effect ATE : E( Y Y X ) The average treatment effect of the treated ATT : E( Y Y X, D 1) The marginal treatment effect MTE : E ( Y,, ) 1 Y0 X Z U u

10 The selection problem in a regression context The fundamental problem is that each individual is only observed in one state of the world; i.e., we only observe Y DY (1 D) Y 1 0 D[ ( X ) U ] (1 D)[ ( X ) U ] ( X ) D[ ( X ) ( X )] ATE( X ) Where DU (1 D) U 1 0 Unfortunately, unless the treatment assignment is randomized, E( X, D) 0

11 The biases From the samples, we can compute Integrating out Z yields E( Y X, Z, D 1) E( Y X, Z, D 1) E( Y X, Z, D 0) E( Y X, Z, D 0) E( Y X, D 1) and E( Y X, D 0) The resulting bias from comparing (D = 1) and (D = 0) means Bias ATE [ E( Y X, D 1) E( Y X, D 0)] E( Y Y X ) = E( Y X, D 1) E( Y X ) E( Y X, D 0) E( Y X )

12 The biases (cont d) For ATT and MTE Bias ATT [ E( Y X, D 1) E( Y X, D 0)] E( Y Y X, D 1) = E( Y X, D 1) E( Y X, D 0) 0 0 Bias MTE [ E( Y X, Z, D 1) E( Y X, Z, D 0)] 1 0 E( Y Y X, Z, U u ) 1 0 = E( Y X, Z, D 1) E( Y X, Z, U u ) 1 1 E( Y X, Z, D 0) E( Y X, Z, U u ) 0 0

13 Estimation of causal effects based on a counterfactual approach with selection on observables: Regression methods Propensity score matching with selection on unobservables: Instrumental variable approach (I) Difference in difference estimators (DiD) Selection models

14 Estimation of causal effects based on a counterfactual approach with cross-sectional data: Regression discontinuity design Propensity score matching Instrumental variable approach (I) with panel data: Before after estimators Difference in difference estimators (DiD) Propensity score matching + DiD