## The Annals of Statistics

- Ann. Statist.
- Volume 34, Number 4 (2006), 1814-1826.

### Affinely invariant matching methods with discriminant mixtures of proportional ellipsoidally symmetric distributions

Donald B. Rubin and Elizabeth A. Stuart

#### Abstract

In observational studies designed to estimate the effects of interventions or exposures, such as cigarette smoking, it is desirable to try to control background differences between the treated group (e.g., current smokers) and the control group (e.g., never smokers) on covariates *X* (e.g., age, education). Matched sampling attempts to effect this control by selecting subsets of the treated and control groups with similar distributions of such covariates. This paper examines the consequences of matching using affinely invariant methods when the covariate distributions are “discriminant mixtures of proportional ellipsoidally symmetric” (DMPES) distributions, a class herein defined, which generalizes the ellipsoidal symmetry class of Rubin and Thomas [*Ann. Statist.* **20** (1992) 1079–1093]. The resulting generalized results help indicate why earlier results hold quite well even when the simple assumption of ellipsoidal symmetry is not met [e.g., *Biometrics* **52** (1996) 249–264]. Extensions to conditionally affinely invariant matching with conditionally DMPES distributions are also discussed.

#### Article information

**Source**

Ann. Statist. Volume 34, Number 4 (2006), 1814-1826.

**Dates**

First available in Project Euclid: 3 November 2006

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1162567634

**Digital Object Identifier**

doi:10.1214/009053606000000407

**Mathematical Reviews number (MathSciNet)**

MR2283718

**Zentralblatt MATH identifier**

06075572

**Subjects**

Primary: 62D05: Sampling theory, sample surveys 62H05: Characterization and structure theory

Secondary: 60E05: Distributions: general theory 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20] 62K99: None of the above, but in this section

**Keywords**

Causal inference equal percent bias reducing (EPBR) matched sampling propensity scores

#### Citation

Rubin, Donald B.; Stuart, Elizabeth A. Affinely invariant matching methods with discriminant mixtures of proportional ellipsoidally symmetric distributions. Ann. Statist. 34 (2006), no. 4, 1814--1826. doi:10.1214/009053606000000407. http://projecteuclid.org/euclid.aos/1162567634.