Causal Interaction in Factorial Experiments: Application to Conjoint Analysis
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1 Causal Interaction in Factorial Experiments: Application to Conjoint Analysis Kosuke Imai Department of Government Department of Statistics Harvard University Talk at the Harvard Data Science Initiative Annual Conference October 18, 2018 Kosuke Imai (Harvard Gov/Stat) Causal Interaction Harvard DSI (Oct. 18, 2018) 1 / 12
2 Causal Heterogeneity and Interaction Effects 1 Moderation: How does the effect of a treatment vary across individuals? Interaction between the treatment variable and pre-treatment covariates 2 Causal interaction: What combination of treatments is efficacious? Interaction among multiple treatment variables Kosuke Imai (Harvard Gov/Stat) Causal Interaction Harvard DSI (Oct. 18, 2018) 2 / 12
3 Conjoint Analysis Survey experiments with a factorial design Respondents evaluate several pairs of randomly selected profiles defined by multiple factors Frequently used for consumer research Social scientists use it to analyze multidimensional preferences Example: Immigration preference (Hopkins and Hainmueller 2014) representative sample of 1,407 American adults each respondent evaluates 5 pairs of immigrant profiles gender 2, education 7, origin 10, experience 4, plan 4, language 4, profession 11, application reason 3, prior trips 5 What combinations of immigrant characteristics do Americans prefer? High dimension: over 1 million treatment combinations Methodological challenges: Many interaction effects false positives, difficulty of interpretation Very few applied researchers study interaction Kosuke Imai (Harvard Gov/Stat) Causal Interaction Harvard DSI (Oct. 18, 2018) 3 / 12
4 The Methodological Contributions 1 New causal estimand: Average Marginal Interaction Effect (AMIE) relative magnitude does not depend on baseline condition intuitive interpretation even for high dimension estimation using ANOVA with weighted zero-sum constraints regularization done directly on AMIEs 2 Comparison with the conventional interaction effect: lack of invariance to the choice of baseline condition difficulty of interpretation for higher-order interaction Kosuke Imai (Harvard Gov/Stat) Causal Interaction Harvard DSI (Oct. 18, 2018) 4 / 12
5 Factorial Experiments with Two Treatments Two factorial treatments (e.g., gender and race): A A = {a 0, a 1,..., a LA 1} B B = {b 0, b 1,..., b LB 1} Assumption: Full factorial design 1 Randomization of treatment assignment {Y (a l, b m )} al A,b m B {A, B} 2 Non-zero probability for all treatment combination Pr(A = a l, B = b m ) > 0 for all a l A and b m B Kosuke Imai (Harvard Gov/Stat) Causal Interaction Harvard DSI (Oct. 18, 2018) 5 / 12
6 Main Causal Estimands in Factorial Experiments 1 Average Combination Effect (ACE): Average effect of treatment combination (A, B) = (a l, b m ) relative to the baseline condition (A, B) = (a 0, b 0 ) τ AB (a l, b m ; a 0, b 0 ) = E{Y (a l, b m ) Y (a 0, b 0 )} Effect of being Asian male 2 Average Marginal Effect (AME; Hainmueller et al. 2014; Dasgupta et al. 2015): Average effect of treatment A = a l relative to the baseline condition A = a 0 averaging over the other treatment B ψ A (a l, a 0 ) = E{Y (a l, B) Y (a 0, B)}dF (B) Effect of being male averaging over race Kosuke Imai (Harvard Gov/Stat) Causal Interaction Harvard DSI (Oct. 18, 2018) 6 / 12
7 The New Causal Interaction Effect Average Marginal Interaction Effect (AMIE): π AB (a l, b m ; a 0, b 0 ) = τ AB (a l, b m ; a 0, b 0 ) ψ }{{} A (a l, a 0 ) ψ }{{} B (b m, b 0 ) }{{} ACE of (a l, b m) AME of a l AME of b m Interpretation: additional effect induced by A = a l and B = b m together beyond the separate effect of A = a l and that of B = b m Additional effect of being Asian male beyond the sum of separate effects for being male and being Asian Invariance: relative magnitude of AMIE does not depend on the choice of baseline condition Generalizable to higher-order interaction ANOVA with direct regularization on AMEs and AMIEs Kosuke Imai (Harvard Gov/Stat) Causal Interaction Harvard DSI (Oct. 18, 2018) 7 / 12
8 Conjoint Analysis of Ethnic Voting in Africa Ethnic voting and accountability (Carlson 2015, World Politics) Do voters prefer candidates of same ethnicity regardless of their prior performance? Do ethnicity and performance interact? Conjoint analysis in Uganda: 547 voters from 32 villages Each voter evaluates 3 pairs of hypothetical candidates 5 factors: Coethnicity 2, Prior record 2, Prior office 4, Platform 3, Education 8 Kosuke Imai (Harvard Gov/Stat) Causal Interaction Harvard DSI (Oct. 18, 2018) 8 / 12
9 Ranges of Estimated AMEs and AMIEs Selection Range prob. AME Record Coethnicity Platform Degree AMIE Coethnicity Record Record Platform Platform Coethnic Coethnicity Degree Platform Degree Record Degree Factor selection probability based on bootstrap Kosuke Imai (Harvard Gov/Stat) Causal Interaction Harvard DSI (Oct. 18, 2018) 9 / 12
10 Effect of Regularization on AMIEs Yes/Village Yes/District Yes/MP No/Village No/District No/MP No/Business Yes/Village Yes/District Yes/MP No/Village No/District No/MP No/Business Platform Coethnicity Yes/Village Yes/District Yes/MP No/Village No/District No/MP No/Business Record Yes/Village Yes/District Yes/MP No/Village No/District No/MP No/Business Record Without Regularization With Regularization Kosuke Imai (Harvard Gov/Stat) Causal Interaction Harvard DSI (Oct. 18, 2018) 10 / 12
11 Concluding Remarks Interaction effects play an essential role in causal heterogeneity 1 moderation 2 causal interaction Randomized experiments with a factorial design 1 useful for testing multiple treatments and their interactions 2 social science applications: audit studies, conjoint analysis 3 challenge: estimation and interpretation in high dimension Average Marginal Interaction Effect (AMIE) 1 invariant to baseline condition 2 straightforward interpretation even for high order interaction 3 enables regularization through ANOVA Open questions: 1 designing factorial experiments 2 combining moderation and causal interaction Kosuke Imai (Harvard Gov/Stat) Causal Interaction Harvard DSI (Oct. 18, 2018) 11 / 12
12 References 1 Egami, Naoki and Kosuke Imai. Causal Interaction in Factorial Experiments: Application to Conjoint Analysis. Journal of the American Statistical Association, Forthcoming 2 Egami, Naoki, Marc Ratkovic, and Kosuke Imai. FindIt: Finding Heterogeneous Treatment Effects. R package available at CRAN Send comments and suggestions to Imai@Harvard.Edu Kosuke Imai (Harvard Gov/Stat) Causal Interaction Harvard DSI (Oct. 18, 2018) 12 / 12
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