Overlap Propensity Score Weighting to Balance Covariates

Transcription

1 Overlap Propensity Score Weighting to Balance Covariates Kari Lock Morgan Department of Statistics Penn State University JSM 2016 Chicago, IL Joint work with Fan Li (Duke) and Alan Zaslavsky (Harvard) Kari Lock Morgan (Penn State) JSM 2016 August 3rd, / 14

2 Outline 1 Introduce a general class of balancing weights 2 Propose overlap weights 3 Illustrate with an example Kari Lock Morgan (Penn State) JSM 2016 August 3rd, / 14

3 Balancing weights Notation: Population density of the covariates X is f (x) Density for group Z {0, 1} is f z (x) = P(X = x Z = z) Propensity score is e(x) Pr(Z i = 1 X i = x) We propose the following class of balancing weights: w 1 (x) h(x) e(x) w 0 (x) h(x) 1 e(x) which balance the weighted covariate distributions: f 1 (x)w 1 (x) f 0 (x)w 0 (x) f (x)h(x). Kari Lock Morgan (Penn State) JSM 2016 August 3rd, / 14

4 Examples of target population and balancing weights target population h(x) estimand ( weight (w 1 ), w 0 ) 1 combined 1 ATE e(x), 1 1 e(x) [HT] treated e(x) ATT ( ) e(x) 1, 1 e(x) control 1 e(x) ATC ( ) 1 e(x) e(x), 1 truncated 1(α < e(x) < 1 α) ATTrunc combined ( 1(α<e(x)<1 α) e(x), 1(α<e(x)<1 α) 1 e(x) ) overlap e(x)(1 e(x)) ATO (1 e(x), e(x)) Kari Lock Morgan (Penn State) JSM 2016 August 3rd, / 14

5 Overlap weights We propose the overlap weights with h(x) = e(x)(1 e(x)): w 1 (x) 1 e(x) w 0 (x) e(x). Target density f (x)e(x)(1 e(x)) defined by covariate overlap: Kari Lock Morgan (Penn State) JSM 2016 August 3rd, / 14

6 Minimizing Asymptotic Variance Weighted estimator: ˆτ w h = i:z i =1 Y iw 1 (x i ) i:z i =1 w 1(x i ) i:z i =0 Y iw 0 (x i ) i:z i =0 w 0(x i ) Theorem Assuming v 0 (x) v 1 (x) v, the overlap weights with h(x) = e(x)(1 e(x)) give the smallest asymptotic variance for ˆτ h w, among the class of balancing weights, and min{v[ˆτ h w ]} = v f (x)e(x)(1 e(x))µ(dx). N Kari Lock Morgan (Penn State) JSM 2016 August 3rd, / 14

7 Exact Balance Theorem When the propensity scores are estimated from a logistic regression model with main effects, logit{e(x i )} = β 0 + β x i, the overlap weights lead to exact balance in the means of any included covariate between treatment and control groups: i x i,kz i (1 ê i ) i i Z = x i,k(1 Z i )ê i i(1 ê i ) i (1 Z. i)ê i Kari Lock Morgan (Penn State) JSM 2016 August 3rd, / 14

8 Target Population of Scientific Interest Overlap population, f (x)e(x)(1 e(x)), gives more weight to units with e(x) = 1/2 units who, based on their covariates, could be in either treatment group marginal units who may get either treatment units in clinical equipoise" Kari Lock Morgan (Penn State) JSM 2016 August 3rd, / 14

9 Racial Disparity in Medical Expenditure Goal: estimate racial disparity in medical expenditures after balancing covariates (Le Cook et al., 2010) Data: 2009 Medical Expenditure Panel Survey 9830 non-hispanic Whites (Z = 1) 4020 Blacks 1446 Asians 5280 Hispanics Three comparisons: comparing Whites to each minority 31 covariates (5 continuous, 26 binary) Logistic regression to estimate propensity scores Kari Lock Morgan (Penn State) JSM 2016 August 3rd, / 14

10 Racial Disparity in Medical Expenditure Figure: Covariate balance (absolute standardized bias) with no weights, overlap weights, and HT weights. White Black White Asian White Hispanic Absolute Standardized Bias Unweighted Overlap HT Unweighted Overlap HT Unweighted Overlap HT Kari Lock Morgan (Penn State) JSM 2016 August 3rd, / 14

11 Racial Disparity in Medical Expenditure Figure: White and Asian BMI distributions unweighted and with overlap and HT weights Unweighted Overlap HT White Asian BMI BMI BMI Kari Lock Morgan (Penn State) JSM 2016 August 3rd, / 14

12 Racial Disparity in Medical Expenditure White and Asian comparison, inverse probability weighting: One Asian woman has over 30% of the weight! 78 year old Asian lady with a BMI of 55.4: e(x) = Common practice: Eliminate or truncate cases with e(x) close to 0 or 1 Can lead to ad hoc changes to target population Results can be very sensitive to truncation choice The overlap weights avoid these extreme weights and avoid an abrupt threshold for elimination or truncation. Kari Lock Morgan (Penn State) JSM 2016 August 3rd, / 14

13 Racial Disparity in Medical Expenditure Results: Table: Unweighted, overlap weights, and HT weighting estimates (SE) for difference in average yearly medical expenditure. Unweighted Overlap HT White - Black $786 (222) $824 (185) $856 (200) White - Asian $2764 (209) $1227 (205) $2167 (640) White - Hispanic $2599 (174) $1212 (171) $596 (323) Kari Lock Morgan (Penn State) JSM 2016 August 3rd, / 14

14 Advantages of the overlap weights Statistical advantages Minimizes asymptotic variance among balancing weights Perfect (exact small-sample) balance for means Weights are bounded, avoiding explosive weights or the need for arbitrary truncation Scientific advantages Upweights marginal" units who, based on their covariates, could be in either treatment group Rather than focusing on atypical individuals, focuses on the naturally comparable overlap" population Kari Lock Morgan (Penn State) JSM 2016 August 3rd, / 14