The Essential Role of Pair Matching in Cluster-Randomized Experiments, with Application to the Mexican Universal Health Insurance Evaluation Kosuke Imai Princeton University Joint work with Gary King (Harvard) & Clayton Nall (Stanford) September 6, 2012 Published in Statistical Science (2009) with discussions Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 1 / 16
Cluster-Randomized Experiments (CREs) Problem of many field experiments: unit of randomization = clusters of individuals unit of interest = individuals Public health and medicine: CREs have “risen exponentially since 1997” (Campbell, 2004) Political science: About 2/3 of field experiments are CREs Education: Randomization of classrooms and schools Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 2 / 16
Advantages of CRE Feasibility Cluster-level treatment Interference between units Standard potential outcomes framework: Y i ( T i = 1 ) and Y i ( T i = 0 ) Potential outcomes of one unit may depend on treatment status of other units: many potential outcomes for each unit Examples: peer effects, contagion, spill-over effects Causal inference with such interference is notoriously difficult Cluster randomization limits the number of potential outcomes: all units in the same cluster receives the treatment vs. no unit does Avoids the interference problem rather than “solving” it Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 3 / 16
Main Disadvantage of CREs and Possible Solution Problem: Loss of efficiency CRE variance = usual variance × { 1 + ( n − 1 ) ρ } where n is the cluster size and ρ is the intracluster correlation coefficient Number of clusters is often small Matched-Pair Designs (MPDs) to improve efficiency: Pair clusters based on background characteristics 1 Within each pair, randomly assign one cluster to the treatment 2 group and the other to the control group Idea: Eliminate as much difference between treated and control groups as possible before randomization of treatment assignment Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 4 / 16
Common Arguments Against MPDs “Analytical limitations” of MPDs (Klar and Donner, 1997): inability to test for homogeneity of causal effects across clusters 1 difficulties in estimating the intracluster correlation coefficient 2 Concerns about losing both clusters in a pair in event of 3 randomization failure (Donner and Klar, 2000) In 10 or fewer pairs, MPDs can lose power (Martin et al. 1993) Our paper shows that these concerns are unfounded Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 5 / 16
Contributions of Our Paper Conclusion: pair-matching should be used whenever feasible MPDs improve bias, efficiency, and power Not pairing = throwing away data! Existing estimator is based on a highly restrictive model Propose new simple design-based estimators and s.e.’s Demonstrate advantages using data from the Mexico study Present quantities of interest for CREs Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 6 / 16
Design-based Analysis of CREs under MPDs Existing Model-based approach: assume DGP for observed data The standard estimator assumes homogeneity across clusters = ⇒ no point of matching to begin with! Our Design-based approach avoids modeling assumptions (Neyman, 1923) Randomness comes from: randomization of treatment assignment 1 random sampling of clusters and units within clusters 2 Recommendation: match on cluster sizes and prognostic covariates Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 7 / 16
Motivating Study: Seguro Popular de Salud (SPS) Article 4 of the Mexican constitution: all persons have a right to the protection of their health SPS provides medical services, preventive care, pharmaceuticals, and financial health protection Voluntary and available for everyone but free to the poor Beneficiaries: intended to cover (by 2012) all 50M Mexicans who otherwise have no access to the healthcare system A key goal: reduce out-of-pocket health expenditures Randomized evaluation commissioned by the Fox administration One of the largest policy experiments to date Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 8 / 16
Detailed Design Summary Define 12,284 “health clusters” that tile Mexico’s 31 states; each 1 includes a health clinic and catchment area Persuaded 13 of 31 states to participate (7,078 clusters) 2 Match clusters in pairs on background characteristics. 3 Select 74 pairs (based on necessary political criteria, closeness of 4 the match, likelihood of compliance) Randomly assign one in each pair to receive encouragement to 5 affiliate, better health facilities, drugs, and doctors Conduct baseline survey of each cluster’s health facility 6 Survey ≈ 32,000 random households in 50 of the 74 treated and 7 control unit pairs (chosen based on likelihood of compliance with encouragement and similarity of the clusters within pair) Repeat surveys in 10 months and subsequently to see effects 8 Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 9 / 16
Clusters are Representative On Measured Variables 0.4 25 4 20 0.3 3 15 Density Density Density 0.2 2 10 0.1 1 5 0.0 0 0 0.0 0.4 0.8 0 2 4 6 8 10 0.00 0.10 0.20 0.30 Prop earning <2 min wages Mean Years Education Prop aged 0−4 years old 10 8 4 8 6 3 6 Density Density Density 4 2 4 2 1 2 0 0 0 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 Prop Employed Prop Female−headed HH Prop w/o Soc Sec Rights Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 10 / 16
Quantities of Interest Depend on Sampling Units within Quantities Clusters Clusters Inferential Target SATE Observed Observed Observed sample CATE Observed Sampled Population within observed clusters UATE Sampled Observed Observable units within pop. of clusters PATE Sampled Sampled Population Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 11 / 16
Main Finding: Effect of SPS on % of Households with Catastrophic Expenditures All Study Participants Experimental Compliers Average ITT SE Average CACE SE (Control) (Control) All 8 . 4 1 . 9 ∗ (0.9) 9 . 5 5 . 2 ∗ (2.3) Low Asset 9 . 9 3 . 0 ∗ (1.3) 11 . 0 6 . 5 ∗ (2.5) High Asset 7 . 1 0 . 9 (0.8) 7 . 9 3 . 0 (2.7) Female-Headed 8 . 5 1 . 4 (1.1) 10 . 6 3 . 8 (3.0) “Catastrophic expenditures”: out-of-pocket health expenses > 30% of post-subsistence income Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 12 / 16
Efficiency Gains: MPD vs. Complete Randomization Unit ATE: MPDs 1 . 1 to 2 . 9 times more efficient Population ATE: MPDs 1 . 8 to 38 . 3 times more efficient! ● ● 30 ● Relative Efficiency, PATE 20 ● ● ● ● 10 ● ● ● ● ● ● ● 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Relative Efficiency, UATE Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 13 / 16
Bias and Inefficiency of Existing Approach Simulations Based on Mexico Data 1.00 Coverage Probability of 90% CIs 1.0 0.95 0.8 ● 0.6 Density 0.90 ● ●●●●●●●●● ● ● ● ● ● ● ● ● ● 0.4 0.85 0.2 Existing CIs ● New CIs 0.80 0.0 0 50 100 150 200 0.5 1.0 1.5 2.0 2.5 Number of pairs Ratio of Existing (Biased) SE to New (Unbiased) SE 1.00 80 Coverage Probability of 90% CIs ● ● ● ● ● Minus Arithmetic Estimator Squared Bias ● 60 Harmonic Estimator 0.95 ● ● ● ● ● ● MSE ● ● ● Existing CIs 40 ● ● ● New CIs ● 0.90 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 20 ●●●●●● ● ● ● ● ● ● 0.85 ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● Variance 0.80 −20 0 50 100 150 200 1.0 1.5 SPS 2 2.5 Number of pairs Average Cluster−Size Ratio Across Pairs Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 14 / 16
Other Findings of SPS Evaluation Positive effects detected: Catastrophic expenditures slashed In-patient out-of-pocket expenditures drastically reduced Out-patient out-of-pocket expenditures drastically reduced Citizen satisfaction is high Positive effects not yet seen: Expenditures on medicines Utilization (preventative and procedures) Risk factors Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 15 / 16
Concluding Remarks Field experiments often require cluster randomization Problem: Loss of statistical efficiency Our recommendation: MPDs for CREs Select quantities of interest 1 Identify pre-treatment covariates for matching 2 Pair clusters based on the covariates and cluster sizes 3 Randomize treatment within each pair 4 Use design-based methods to analyze the data 5 Our design-based estimators avoid modeling assumptions MPDs are preferred from perspectives of bias, efficiency, & power May affect CONSORT, Cochrane Collaboration, Council guidelines, etc. Kosuke Imai (Princeton) Matched-Pair Cluster-Randomized Design ACF/OPRE M EETING 16 / 16
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