Abstract

Quantile treatment effects can be important causal estimands in evaluation of biomedical treatments or interventions for health outcomes such as medical cost and utilisation. We consider their estimation in observational studies with many possible covariates under the assumption that treatment and potential outcomes are independent conditional on all covariates. To obtain valid and efficient treatment effect estimators, we replace the set of all covariates by lower dimensional sets for estimation of the quantiles of potential outcomes. These lower dimensional sets are obtained using sufficient dimension reduction tools and are outcome specific. We justify our choice from efficiency point of view. We prove the asymptotic normality of our estimators and our theory is complemented by some simulation results and an application to data from the University of Wisconsin Health Accountable Care Organization.

References

1. Abadie, A., Angrist, J., & Imbens, G. W. (2002). Instrumental variables estimates of the effect of subsidized training on the quantiles of trainee earnings. Econometrica, 70, 91–117. doi: 10.1111/1468-0262.00270 [Crossref], [Web of Science ®], [Google Scholar]
2. Bickel, P. J., Klaassen, C. J., Ritov, Y., & Wellner, J. (1993). Efficient and adaptive inference in semiparametric models. Baltimore: Johns Hopkins University Press. [Google Scholar]
3. Brookhart, M., Schneeweiss, S., Rothman, K., Glynn, R., Avorn, J., & Sturmer, T. (2006). Variable selection for propensity score models. American Journal of Epidemiology, 163, 1149–1156. doi: 10.1093/aje/kwj149 [Crossref], [Web of Science ®], [Google Scholar]
4. Cattaneo, M. D. (2010). Efficient semiparametric estimation of multi-valued treatment effects under ignorability. Journal of Econometrics, 155, 138–154. doi: 10.1016/j.jeconom.2009.09.023 [Crossref], [Web of Science ®], [Google Scholar]
5. Cattaneo, M. D., Drukker, D. M., & Holland, A. D. (2013). Estimation of multivalued treatment effects under conditional independence. The Stata Journal, 13, 407–450. doi: 10.1177/1536867X1301300301 [Crossref], [Web of Science ®], [Google Scholar]
6. Chen, X., Wan, A. T. K., & Zhou, Y. (2015). Efficient quantile regression analysis with missing observations. Journal of the American Statistical Association, 110, 723–741. doi: 10.1080/01621459.2014.928219 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
7. Cheng, P. E., & Chu, C. (1996). Kernel estiamation of distribution functions and quantiles with missing data. Statistica Sinica, 6, 63–78. [Web of Science ®], [Google Scholar]
8. Chernozhukov, V., & Hansen, C. (2005). An IV model of quantile treatment effects. Econometrica, 73, 245–261. doi: 10.1111/j.1468-0262.2005.00570.x [Crossref], [Web of Science ®], [Google Scholar]
9. Cook, R. D., & Weisberg, S. (1991). Discussion of ‘Sliced inverse regression for dimension reduction’. Journal of the American Statistical Association, 86, 328–332. [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
10. De Luna, X., Waernbaum, I., & Richardson, T. S. (2011). Covariate selection for the nonparametric estimation of an average treatment effect. Biometrika, 98, 861–875. doi: 10.1093/biomet/asr041 [Crossref], [Web of Science ®], [Google Scholar]
11. Doksum, K. (1974). Empirical probability plots and statistical inference for nonlinear models in the two-sample case. The Annals of Statistics, 2, 267–277. doi: 10.1214/aos/1176342662 [Crossref], [Web of Science ®], [Google Scholar]
12. Firpo, S. (2007). Efficient semiparametric estimation of quantile treatment effects. Econometrica, 75, 259–276. doi: 10.1111/j.1468-0262.2007.00738.x [Crossref], [Web of Science ®], [Google Scholar]
13. Frölich, M., & Melly, B. (2010). Estimation of quantile treatment effects with Stata. The Stata Journal, 10, 423–457. doi: 10.1177/1536867X1001000309 [Crossref], [Web of Science ®], [Google Scholar]
14. Frölich, M., & Melly, B. (2013). Unconditional quantile treatment effects under endogeneity. Journal of Business and Economic Statistics, 31, 346–357. doi: 10.1080/07350015.2013.803869 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
15. Hahn, J. (1998). On the role of the propensity score in efficient semiparametric estimation of average treatment effects. Econometrica, 66, 315–331. doi: 10.2307/2998560 [Crossref], [Web of Science ®], [Google Scholar]
16. Hahn, J. (2004). Functional restriction and efficiency in causal inference. Review of Economics and Statistics, 86, 73–76. doi: 10.1162/003465304323023688 [Crossref], [Web of Science ®], [Google Scholar]
17. Hardle, W., Muller, M., Sperlich, S., & Werwatz, A. (2004). Nonparametric and semiparametric models. Heidelberg: Springer-Verlag. [Crossref], [Google Scholar]
18. Hsing, T., & Carroll, R. J. (1992). An asymptotic theory for sliced inverse regression. The Annals of Statistics, 20, 1040–1061. doi: 10.1214/aos/1176348669 [Crossref], [Web of Science ®], [Google Scholar]
19. Hu, Z., Follmann, D. A., & Wang, N. (2014). Estimation of mean response via the effective balancing score. Biometrika, 101, 613–624. doi: 10.1093/biomet/asu022 [Crossref], [Web of Science ®], [Google Scholar]
20. Koenker, R., & Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33–50. doi: 10.2307/1913643 [Crossref], [Web of Science ®], [Google Scholar]
21. Lehman, E. L. (1975). Nonparametrics: Statistical methods based on ranks. San Francisco: Holden-Day. [Google Scholar]
22. Li, K. C. (1991). Sliced inverse regression for dimension reduction. Journal of the American Statistical Association, 86, 316–327. doi: 10.1080/01621459.1991.10475035 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
23. Rosenbaum, P. R., & Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika, 70, 41–55. doi: 10.1093/biomet/70.1.41 [Crossref], [Web of Science ®], [Google Scholar]
24. Shortreed, S. M., & Ertefaie, A. (2017). Outcome-adaptive Lasso: Variable selection for causal inference. Biometrics, 73, 1111–1122. doi: 10.1111/biom.12679 [Crossref], [Web of Science ®], [Google Scholar]
25. Silverman, B. W. (1986). Density estimation for statistics and data analysis. London: Chapman and Hall. [Crossref], [Google Scholar]
26. Xia, Y., Tong, H., Li, W. K., & Zhu, L. X. (2002). An adaptive estimation of dimension reduction space. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64, 363–410. doi: 10.1111/1467-9868.03411 [Crossref], [Web of Science ®], [Google Scholar]
27. Zhou, Y., Wan, A. T. K., & Wang, X. (2008). Estimating equations inference with missing data. Journal of the American Statistical Association, 103, 1187–1199. doi: 10.1198/016214508000000535 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
28. Zhu, L. X., & Ng, K. W. (1995). Asymptotics of sliced inverse regression. Statistica Sinica, 5, 727–736. [Web of Science ®], [Google Scholar]
29. Zhu, L. P., Zhu, L. X., & Feng, Z. H. (2010). Dimension reduction in regressions through cumulative slicing estimation. Journal of the American Statistical Association, 105, 1455–1466. doi: 10.1198/jasa.2010.tm09666 [Taylor & Francis Online], [Web of Science ®], [Google Scholar]