A Generalized Bootstrap Procedure of the Standard Error and Confidence Interval Estimation for Inverse Probability of Treatment Weighting

Abstract

The inverse probability of treatment weighting (IPTW) approach is commonly used in propensity score analysis to infer causal effects in regression models. Due to oversized IPTW weights and errors associated with propensity score estimation, the IPTW approach can underestimate the standard error of causal effect. To remediate this, bootstrap standard errors have been recommended to replace the IPTW standard error, but the ordinary bootstrap (OB) procedure might still result in underestimation of the standard error because of its inefficient resampling scheme and untreated oversized weights. In this paper, we develop a generalized bootstrap (GB) procedure for estimating the standard error and confidence intervals of the IPTW approach. Compared with the OB procedure and other three procedures in comparison, the GB procedure has the highest precision and yields conservative standard error estimates. As a result, the GB procedure produces short confidence intervals with highest coverage rates. We demonstrate the effectiveness of the GB procedure via two simulation studies and a dataset from the National Educational Longitudinal Study-1988 (NELS-88).

Keywords:

• Propensity score analysis
• inverse probability treatment weighting
• bootstrap
• causal inference
• unequal probability sampling

Notes

1 We used 1000 bootstrap iterations to ensure the stability of bootstrap standard error estimates, and running 1000 iterations was computationally affordable (typically less than 10 seconds in a laptop).

2 The ordinal scale is coded as follows based on annual family income in 1988: 1-No Income; 2-Less than $1000; 3-[$1000,$2999]; 4-[$3000,$4999]; 5-[$5000,$7499]; 6-[$7500,$9999]; 7-[$10000,$14999]; 8-[$15000,$19999]; 9-[$20000,$24999]; 10-[$25000,$34999]; 11-[$35000,$49999]; 12-[$50000,$74999]; 13-[$75000,$99999]; 14-[$100000,$199999]; 15-More than $200000.