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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: Zi = 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: (Si(0),Si(1)) − Primary outcome: (Yi(0),Yi(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: Xi

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¨

• lander 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

• f 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 Yi(z,Si(z′))

Yi(z,Si(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′ Yi(z,Si(z′)) are a priori counterfactuals for units with Si(z) ≠ Si(z′), because in

• ne specific experiment, they can be never observed for such type of units

For units with Si(z) ≠ Si(z′), Yi(z,Si(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 Yi(z,Si(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 Si(z) to another value, Si(z′), z′ ≠ z the outcome Yi(z,Si(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 Yi(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[Yi(1,Si(z))−Yi(0,Si(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[Yi(z,Si(1))−Yi(z,Si(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 Si(0) and Si(1) Si(0) and Si(1) describe how an individual reacts to a treatment ⇒ both values are natural The joint value of Si(0) and Si(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 Si(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 (Si(0);Si(1)) If S is a binary variable, then (Si(0);Si(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 ∶ Si(0) = Si(1)} = {i ∶ Si(0) = Si(1) = 0}∪{i ∶ Si(0) = Si(1) = 1} S is affected by Z ∶ {i ∶ Si(0) ≠ Si(1)} = {i ∶ Si(0) = 1,Si(1) = 0}∪{i ∶ Si(0) = 0,Si(1) = 1} Principal causal effects: PCE(s0,s1) = E[Yi(1)−Yi(0) ∣ Si(0) = s0,Si(1) = s1] 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(s0,s1) = E[Yi(1)−Yi(0) ∣ Si(0) = s0,Si(1) = s1] s0 ≠ s1 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[Yi(1)−Yi(0) ∣ Si(0) = Si(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

Dissociative Principal Causal Effects as Local Direct Effects

Dissociative PCEs naturally provide information on the existence of a direct causal effect

• f the treatment on the primary outcome

If PCE(s) = 0, for each s, then there is no evidence on the direct effect of the treatment after controlling for the mediator PCE(s) = 0 for each s ⇏ NDE(z) = 0 (VanderWeele, 2008) NDE(z) = E[Yi(1,Si(z))−Yi(0,Si(z))] ∑

s0=s1=s

PCE(s)πs,s + ∑

s0≠s1

E[Yi(1,Si(z))−Yi(0,Si(z)) ∣ Si(0) = s0,Si(1) = s1]πs0,s1 Associative PCEs generally combine direct and indirect effects ⇒ Associative effects do not correspond to indirect effects Average total Causal Effect = Weighted average of PCEs ACE = E[Yi(1)−Yi(0)] = ∑

s0,s1

PCE(s0,s1)πs0,s1 = ∑

s0=s1=s

PCE(s)πs,s + ∑

s0≠s1

PCE(s0,s1)πs0,s1 πs0,s1 = Proportion of units belonging to {i ∶ Si(0) = s0,Si(1) = s1}

Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions

PCEs: Identifying Assumptions

Assumptions of a different nature are required depending on if the focus is on principal causal effects, which are local treatment effects or natural direct and indirect effects, which are population causal effects In principal stratification analysis, challenges in identifying principal strata effects stem primarily from the fact that we cannot, in general, observe the principal stratum to which a subject belongs, because we cannot directly observe both Si(0) and Si(1) Principal Stratification Stratum Si(0) Si(1) 00 01 1 10 1 11 1 1 Observed Data Zi Sobs

i

Stratum 00∪01 1 10∪11 1 10∪00 1 1 01∪11 Assumptions that allow us to untie mixtures of PS are required to identify PCEs Identifying assumptions generally do not involve comparisons of units belonging to different strata: Partial identification versus Point identification (Distributional Assumptions: Bayesian Analysis)

Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions

Natural Effects: Identifying Assumptions

Natural direct and indirect effects involve causal effects for units for which the data contains no or little information. Assumptions allowing one to extrapolate information on potential outcomes of the form Yi(z,Si(z′)), z ≠ z′, from the observed data are required to identify NDEs and NIEs Assignment mechanism for the mediating variable, S: Assumptions implying that S could be, at least in principle, regarded as an additional treatment Sequential ignorability assumptions: For all s Yi(0,s),Yi(1,s),Si(0),Si(1) ⊥ Zi ∣ Xi Yi(0,s),Yi(1,s) ⊥ Si(z) ∣ Zi = z,Xi z = 0,1 Sequential ignorability implies that we can compare treated and untreated units with the same value of the covariates, also if they belong to different principal strata ⇒ Extrapolation across principal strata The nature of these extrapolations may be less credible than the inferences for a particular subpopulation

Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions

Principal Stratification: A principled Approach

If one is seeking natural direct and indirect effects, we suggest to preliminary conduct a principal stratification analysis, looking at the distribution of covariates and outcomes within each principal stratum The observed data, Xi, Zi Sobs

i

and Yobs

i

, contain information on the potential

• utcome Yi(z,Si(z′)), z ≠ z′ only for the subpopulation of units for which the

intermediate variable is unaffected by the treatment Si(0) = Si(1) ⇒ Yi(0,Si(1)) = Yi(0,Si(0)) and Yi(1,Si(0)) = Yi(1,Si(1)) and E[Yi(1,Si(z))−Yi(0,Si(z)) ∣ Si(0) = Si(1)] = ∑

s0=s1=s

PCE(s)πs,s/ ∑

s0=s1=s

πs,s The observed data are uninformative regarding the natural direct effects for other subpopulations of units, for which the treatment affects the intermediate variable, as associative effects combine direct and indirect effects

Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions

Principal Strata Assumptions versus Sequential Ignorability

NDE(0) = ∑

s0,s1

E[Yi(1,Si(0))−Yi(0) ∣ Si(0) = s0,Si(1) = s1]πs0,s1 = ∑

s0,s1

NDEs0s1πs0,s1 = ∑

s0=s1=s

PCE(s)πs,s + ∑

s0≠s1

E[Yi(1,Si(0))−Yi(0) ∣ Si(0) = s0,Si(1) = s1]πs0,s1 If E[Yi(0) ∣ Si(0) = s,Si(1) = 1−s] = E[Yi(0) ∣ Si(0) = s,Si(1) = s] s = 0,1 Then we can reasonably assume that E[Yi(0,Si(0)) ∣ Si(0) = s,Si(1) = 1−s] = E[Yi(0,Si(0)) ∣ Si(0) = s,Si(1) = s] and NDE(0) = NDE00(0)(π00 +π01)+NDE11(0)(π11 +π10) = PCE(0)(π00 +π01)+PCE(1)(π11 +π10)

Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions

Principal Strata Assumptions versus Sequential Ignorability

Under the sequential ignorability assumption (within cells defined by covariates) NDE(0) = E[Yi(1,Si(0))−Yi(0)] = [E[Yi ∣ Zi = 1,Si = 0]Pr(Si = 0∣Zi = 0)+E[Yi ∣ Zi = 1,Si = 1]Pr(Si = 1∣Zi = 0)]−E[Yi∣Zi = 0] But E[Yi(1,Si(0))] = [E[Yi ∣ Zi = 1,Si = 0]Pr(Si = 0∣Zi = 0)+E[Yi ∣ Zi = 1,Si = 1]Pr(Si = 1∣Zi = 0)] = (E[Yi(1) ∣ Si(0) = 0,Si(1) = 0] π00 π00 +π10 +E[Yi(1) ∣ Si(0) = 1,Si(1) = 0] π10 π00 +π10 )(π00 +π01)+ (E[Yi(1) ∣ Si(0) = 1,Si(1) = 1] π11 π01 +π11 +E[Yi(1) ∣ Si(0) = 0,Si(1) = 1] π01 π01 +π11 )(π10 +π11)

Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions

Numerical Example (1)

Z = Physical activity S = Body Mass Index Y = Survival (0 = Low level;1 = High level) (0 = Not obese;1 = Obese) (in years) πs0,s1 S(0) S(1) E[Y(0)] E[Y(1)] PCE 0.3 6 10 4 0.4 1 6 8 2 0.1 1 4 7 3 0.2 1 1 4 6 2 Associative PCE = 2⋅0.4+3⋅0.1 0.4+0.1 = 2.2 Dissociative PCE = 4⋅0.3+2⋅0.2 0.3+0.2 = 3.2 ACE = E[Y(1)]−E[Y(0)] = 8.1−5.4 = 2.7 NDE(0) = PCE(0)(π00 +π01)+PCE(1)(π11 +π10) = 4⋅(0.3+0.1)+2⋅(0.2+0.1) = 3.4 NIE(1) = ACE−NDE(0) = 2.7−3.4 = −0.7 Under the sequential ignorability assumption NDE(0) = (10 0.3 0.3+0.1 +7 0.1 0.3+0.1)(0.3+0.4)+(6 0.2 0.2+0.4 +8 0.4 0.2+0.4)(0.2+0.1) = 3.275 NIE(1) = ACE−NDE(0) = 2.7−3.275 = −0.575

Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions

Numerical Example (2)

Z = Physical activity S = Body Mass Index Y = Survival (0 = Low level;1 = High level) (0 = Not obese;1 = Obese) (in years) πs0,s1 S(0) S(1) E[Y(0)] E[Y(1)] PCE 0.25 6 10 4 0.25 1 6 8 2 0.25 1 4 7 3 0.25 1 1 4 4 Associative PCE = 2⋅0.25+3⋅0.25 0.25+0.25 = 2.5 Dissociative PCE = 4⋅0.25+0⋅0.25 0.25+0.25 = 2 ACE = E[Y(1)]−E[Y(0)] = 7.25−5.0 = 2.25 NDE(0) = PCE(0)(π00 +π01)+PCE(1)(π11 +π10) = 4⋅(0.25+0.25)+0⋅(0.25+0.25) = 2.0 NIE(1) = ACE−NDE(0) = 2.25−2.0 = 0.25 Under the sequential ignorability assumption NDE(0) = (100.25 0.5 +70.25 0.5 )(0.5)+(40.25 0.5 +80.25 0.5 )(0.5) = 2.25 NIE(1) = ACE−NDE(0) = 2.25−2.25 = 0

Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions

Concluding Remarks

Principal stratification does not always answer the causal question of primary interest, but it often provides useful insights A principal stratification analysis should involve all the principal strata, studying the characteristics of each principal stratum, and evaluating the distributions of potential outcomes in each principal stratum A stratification analysis that does not neglect any strata may provide substantial information on the causal problem at hand, by discovering heterogeneities in the treatment effects across principal strata, providing insights even on indirect effects

• f the treatment

Principal stratification is a principal framework for addressing causal inference problems in the presence of post-treatment variables

Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions

References

Flores, C. A. and Flores-Lagunes, A. (2010). Nonparametric partial identification of causal net and mechanism average treatment effects. Working Paper, Dept. Food and Resource Economics, Univ. of Florida. Frangakis, C. E. and Rubin, D. B. (2002). Principal stratification in causal inference. Biometrics 58, 21–29. Imai, K., Tingley, D. and Yamamoto, T. (2013). Experimental designs for identifying causal mechanisms (with discussion). J. R. Statist. Soc. A 176, 5–51. Pearl, J. (2001). Direct and indirect effects. In Proc. 17th Conf. Uncertainty in Artificial Intelligence (eds. J. S. Breese & D. Koller), 411–420. Morgan Kaufman, S. Francisco, CA. Rubin, D. B. (2013). Comment on ‘Experimental designs for identifying causal mechanisms’ by K. Imai, D., Tingley, and T. Yamamoto J. R. Statist. Soc. A 176 45. Rubin, D. B. (1980). Comment on ‘Randomization analysis of experimental Data: The Fisher randomization test’ by D. Basu. J. Amer. Statist. Assoc. 75, 591–593. Sj¨

• lander, A. Humphreys, K., Vansteelandt, S., Bellocco, R. and Palmgren, J. (2009). Sensitivity

Analysis for Principal Stratum Direct Effects,with an Application to a Study of Physical Activity and Coronary Heart Disease. Biometrics 65, 514–520. VanderWeele, T. L. (2008). Simple relations between principal stratification and direct and indirect

• effects. Statist. Probab. Lett. 78(17), 2957–2962.

Understanding Causal Mechanisms through Principal Stratification: Definitions and Assumptions