Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions Alessandra Mattei Department of Statistics, Informatics, Applications University of Florence mattei@ds.unifi.it Joint work with Fan Li (Duke) & Fabrizia Mealli (Florence) Symposium: Causal Mediation Analysis Center for Statistics - Ghent University (Belgium) January 28 − 29, 2013
Motivation Shed light on crucial issues about defining, identifying and estimating causal mechanisms We use potential outcomes to discuss ( a ) research questions, which motivate focus on understanding causal mechanisms; ( b ) alternative definitions of causal estimands; ( c ) identifying assumptions We clarify the role of the alternative (structural and distributional) assumptions, separating and critically discussing those allowing one to carry out extrapolation to recover never observable quantities and those on potentially observable sub-populations. Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions
The Potential Outcome Approach to Causal Inference Units: i = 1 ,..., n Treatment variable: Z i = z z = 0 ⇐ ⇒ Control/standard treatment z = 1 ⇐ ⇒ Active/new treatment The Stable Unit Treatment Value Assumption (SUTVA; Rubin, 1980) is assumed Potential outcomes − Intermediate variable: ( S i ( 0 ) , S i ( 1 )) − Primary outcome: ( Y i ( 0 ) , Y i ( 1 )) A causal effect of the treatment Z on the outcome Y is defined as a comparison of the potential outcomes Y ( 1 ) and Y ( 0 ) on a common set of units Pre-treatment variables: X i Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions
Research Question Understanding the causal pathways by which a treatment affects an outcome Intermediate variables may mediate the effect of the treatment on the outcome, in some way channelling a part of the treatment effect Objective: “Disentangling” direct and indirect effects Examples � Assessing the efficacy of drug treatment having side-effects (Pearl, 2001) � Assessing the effect of physical activity on circulation diseases, not channelled through body mass index (Sj¨ olander et al., 2009) � Understanding to what extent the effects of a training program on participants’ employment and earnings is mediated by the achievement of a secondary educational degree (Flores and Flores-Lagunes, 2010) Untying the direct and mediated effects may help understanding and answers policy-related questions of practical significance Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions
Direct & Indirect Effects A direct effect should measure the effect of Z on Y not mediated through S : part of the effect of Z on Y that is not due to a change in S caused by the treatment. � Effect of physical activity on circulation diseases that is not due to a change in body mass index caused by physical activity An indirect effect should measure the extent to which an intervention, Z , affects the outcome, Y , through the mediator, S : the effect of a change in S , which is due to Z , on the outcome Y � Effect of a change in body mass index due to physical activity on circulation diseases Which causal estimands answer these research questions? Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions
Potential Outcomes of the Form Y i ( z , S i ( z ′ )) Y i ( z , S i ( z ′ )) = Potential outcomes of Y if treatment Z were set to the value z and the mediator S were set to the value it would have taken if Z had been set to z ′ Y i ( z , S i ( z ′ )) are a priori counterfactuals for units with S i ( z ) ≠ S i ( z ′ ) , because in one specific experiment, they can be never observed for such type of units For units with S i ( z ) ≠ S i ( z ′ ) , Y i ( z , S i ( z ′ )) is not in the data, and in a specific experiment or study, it cannot be observed, not even on units of the same type assigned the opposite treatment Y i ( z , S i ( z ′ )) are ill-defined quantities: The assumption of “no hidden versions of treatment” implies that given a fixed treatment level, say z , no matter the mediator S is forced to change its value for unit i from S i ( z ) to another value, S i ( z ′ ) , z ′ ≠ z the outcome Y i ( z , S i ( z ′ )) would remain the same (Rubin 2013) Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions
Compound Assignment Mechanism for Z and S When an hypothetical intervention on the intermediate variable is conceivable and S can be regarded as an additional treatment, there are no ‘a priori counterfactuals’ Potential outcomes have to be defined as a function of a multivariate treatment variable, ( Z , S ) , and a compound assignment mechanism should be specified All values Y i ( z , s ) are potentially observable, although only one will ultimately be realized and therefore possibly observed: the potential outcome corresponding to the treatment actually assigned Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions
Natural Direct and Indirect Effects (Robins and Greenland 1992; Pearl 2001) Average Natural Direct Effect: Effect of Z on Y intervening to fix the mediator to the value it would have taken if Z had been set to z NDE ( z ) = E [ Y i ( 1 , S i ( z ))− Y i ( 0 , S i ( z ))] z = 0 , 1 Average Natural Indirect Effect: Effect on the outcome Y of intervening to set the mediator to what it would have been if Z were z = 1 in contrast to what it would have been if Z were z = 0 NIE ( z ) = E [ Y i ( z , S i ( 1 ))− Y i ( z , S i ( 0 ))] z = 0 , 1 Average total Causal Effect = NDE + NIE ACE = NDE ( 0 )+ NIE ( 1 ) ACE = NDE ( 1 )+ NIE ( 0 ) Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions
Natural Direct and Indirect Effects: Descriptive Tools NDE and NIE are descriptive tools for attributing part of the effect of an intervention to an intermediate variable (Pearl 2001) Asymmetric roles of S i ( 0 ) and S i ( 1 ) S i ( 0 ) and S i ( 1 ) describe how an individual reacts to a treatment � ⇒ both values are natural The joint value of S i ( 0 ) and S i ( 1 ) is essentially a characteristic of a subject, so that conceiving a manipulation of one of the two values is like considering changing the value of a pre-treatment characteristic Consistency assumptions (e.g., Imai et al. 2013) are rarely credible: The action taken to modify the value of an intrinsic characteristic of a subject has no consequence on the value of the outcome There may be subsets of subjects for whom a level of S equal to S i ( z ′ ) under treatment z can never be reached. Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions
Principal Stratification & Principal Causal Effects (Frangakis & Rubin, 2002) The basic principal stratification with respect to a posttreatment variable S is the partition of subjects into sets such that all subjects in the same set have the same vector ( S i ( 0 ) ; S i ( 1 )) If S is a binary variable, then ( S i ( 0 ) ; S i ( 1 )) ∈ {( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 )} A principal stratification with respect to posttreatment variable S is a partition of the units whose sets are unions of sets in the basic principal stratification: S is unaffected by Z ∶ { i ∶ S i ( 0 ) = S i ( 1 )} = { i ∶ S i ( 0 ) = S i ( 1 ) = 0 }∪{ i ∶ S i ( 0 ) = S i ( 1 ) = 1 } S is affected by Z ∶ { i ∶ S i ( 0 ) ≠ S i ( 1 )} = { i ∶ S i ( 0 ) = 1 , S i ( 1 ) = 0 }∪{ i ∶ S i ( 0 ) = 0 , S i ( 1 ) = 1 } Principal causal effects: PCE ( s 0 , s 1 ) = E [ Y i ( 1 )− Y i ( 0 ) ∣ S i ( 0 ) = s 0 , S i ( 1 ) = s 1 ] Principal strata are not affected by treatment assignment � ⇒ Principal effects are always well-defined causal effects Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions
Associative & Dissociative Principal Causal Effects Associative Principal Causal Effects: Causal effects within principal strata where the posttreatment variable is affected by treatment in this study PCE ( s 0 , s 1 ) = E [ Y i ( 1 )− Y i ( 0 ) ∣ S i ( 0 ) = s 0 , S i ( 1 ) = s 1 ] s 0 ≠ s 1 Dissociative Principal Causal Effects: Causal effects within principal strata where the posttreatment variable is unaffected by treatment in this study PCE ( s ) ≡ PCE ( s , s ) = E [ Y i ( 1 )− Y i ( 0 ) ∣ S i ( 0 ) = S i ( 1 ) = s ] Principal stratification makes it clear that only in strata where the intermediate variable is unaffected by the treatment can we hope to learn something about the direct effect of the treatment A dissociative PCE measures an effect on the outcome that is dissociative with an effect on the intermediate variable Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions
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