## The Annals of Statistics

- Ann. Statist.
- Volume 47, Number 2 (2019), 965-993.

### Covariate balancing propensity score by tailored loss functions

#### Abstract

In observational studies, propensity scores are commonly estimated by maximum likelihood but may fail to balance high-dimensional pretreatment covariates even after specification search. We introduce a general framework that unifies and generalizes several recent proposals to improve covariate balance when designing an observational study. Instead of the likelihood function, we propose to optimize special loss functions—covariate balancing scoring rules (CBSR)—to estimate the propensity score. A CBSR is uniquely determined by the link function in the GLM and the estimand (a weighted average treatment effect). We show CBSR does not lose asymptotic efficiency in estimating the weighted average treatment effect compared to the Bernoulli likelihood, but CBSR is much more robust in finite samples. Borrowing tools developed in statistical learning, we propose practical strategies to balance covariate functions in rich function classes. This is useful to estimate the maximum bias of the inverse probability weighting (IPW) estimators and construct honest confidence intervals in finite samples. Lastly, we provide several numerical examples to demonstrate the tradeoff of bias and variance in the IPW-type estimators and the tradeoff in balancing different function classes of the covariates.

#### Article information

**Source**

Ann. Statist., Volume 47, Number 2 (2019), 965-993.

**Dates**

Received: March 2017

Revised: November 2017

First available in Project Euclid: 11 January 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1547197245

**Digital Object Identifier**

doi:10.1214/18-AOS1698

**Mathematical Reviews number (MathSciNet)**

MR3909957

**Zentralblatt MATH identifier**

07033158

**Subjects**

Primary: 62P10: Applications to biology and medical sciences

Secondary: 62C99: None of the above, but in this section

**Keywords**

Convex optimization kernel method inverse probability weighting proper scoring rule regularized regression statistical decision theory

#### Citation

Zhao, Qingyuan. Covariate balancing propensity score by tailored loss functions. Ann. Statist. 47 (2019), no. 2, 965--993. doi:10.1214/18-AOS1698. https://projecteuclid.org/euclid.aos/1547197245

#### Supplemental materials

- Supplement to “Covariate balancing propensity score by tailored loss functions”. In this supplement we provide the detailed proof for the theoretical results and some graphical illustration of the Beta-family of scoring rules.Digital Object Identifier: doi:10.1214/18-AOS1698SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.