Kernel-Distance-Based Covariate Balancing

We use the symbol ‘$\stackrel{{\scriptstyle\triangle}}{{=}}$’ to distinguish definitions from equalities. Suppose the observed dataset consists of a random sample or finite population of $N$ units, each assigned to one of the groups, treatment or control, for which covariate-balanced comparisons are of interest. For each unit $i~{}(i=1,\cdots,N)$, we can observe $\{(Y_{i},T_{i},\bm{X}_{i}):~{}i=1,\cdots,N\}$ of $\{(Y,T,\bm{X})\}$, where $Y$ is an outcome variable, $T$ is a treatment indicator and $\bm{X}=(X_{1},\cdots,X_{D})$ with expectation $\bm{\mu}$ and covariance matrix $\bm{\Sigma}$ is a set of $D$ covariates. Every $T_{i}$ takes values 1 or 0, which correspondingly indicates the unit $i$ is assigned to the treatment or the control group. We define the sample size in the treatment and control group respectively as $n_{1}$ and $n_{0}$. The observed data in the treatment and control groups are denoted as $\{(Y_{1i},T_{1i},\bm{X}_{1i}):~{}i=1,\cdots,n_{1}\}$ and $\{(Y_{0j},T_{0j},\bm{X}_{0j}):~{}j=1,\cdots,n_{0}\}$, respectively, where

The probability of assignment to the treatment group given the covariates is the so-called propensity score (Rosenbaum and Rubin, 1983)

which plays a central role in causal inference.

Under the potential outcome framework (Neyman, 1923; Rubin, 1974), we define a pair of potential outcomes $\{Y_{i}(0),Y_{i}(1)\}$ for every unit $i$ under treatment 0 or 1 respectively. The observed oucome can be expressed by the pair of potential outcomes under the consistency assumption:

In this paper, we focus on two causal estimands: the average treatment effect (ATE), that is

where $\mu_{t}:=E[Y(T=t)]$, and the average treatment effect on the treated (ATT), that is

where $\mu_{t|1}\equiv E[Y(T=t)|T=1]$ for $t=0,1$. The expectations for the potential outcomes given the covariates are

For causal comparisons, we make the following assumptions (Rosenbaum and Rubin, 1983) throughout this article so that unbiased estimators of ATE and ATT are available in observational studies.

According to Ma and Wang (2010), the IPW assigns a weight

to $Y_{1i}$ (the outcome in the treatment group), $i=1,2,\cdots,n_{1}$, and a weight

to $Y_{0j}$ (the outcome in the control group), $j=1,2,\cdots,n_{0}$, to estimate ATE, thereby deriving a natural estimator of ATE, that is,

Besides, a natural estimator of ATT using inverse probability weighting is

Therefore, the weights assigned to $Y_{1i},~{}i=1,2,\cdots,n_{1}$ for estimating ATT is

and the weights assigned to $Y_{0j},~{}j=1,2,\cdots,n_{0}$ is

The entropy balancing (EB) scheme by Hainmueller (2012) considered:

The weights $\widehat{p}_{i}^{EB}=\frac{1}{n_{1}}~{}(1,\cdots,n_{1})$ and $\widehat{q}_{j}^{EB}=w_{j}~{}(j=1,\cdots,n_{0})$ of EB method can be directly calculated with the R package “ebal”.

Imai and Ratkovic (2014) proposed a relatively more stable method, namely the Covariate Balance Propensity Score (CBPS), for estimating propensity score so that covariate balance is optimized. CBPS can significantly improve the poor empirical performance of propensity score matching and weighting methods. The key idea is using propensity score weighting to characterize the covariate balance:

where $\tilde{\bm{X}}_{i}=f(\bm{X}_{i})$ is an D-dimensional vector-valued measurable function of $\bm{X}_{i}$ specified by the researcher. Equation (3.3) holds for any function of covariates. Selecting a specific $f(\cdot)$, for example, setting $\tilde{X}_{i}=(X_{i}^{T}X_{i}^{2T})^{T}$, Equation (3.3) helps to balance both the first and second moments of all covariates.

The CBPS method can be implemented through the open-source R package “CBPS”.

A general class of calibration (CAL) estimators of ATE is established with a wide class, such as exponential tilting, of calibration weights. The weights are constructed to achieve an exact moment balance of observed covariates among the treated, the control, and the combined group. Global semiparametric efficiency has been established for the CAL estimators.

For a fixed $g\in\mathbb{R}$, let $D(f,g)$ denote a continuously differentiable distance measure for $f\in\mathbb{R}$. Being nonnegative and strictly convex in $f$, $D(f,g)=0$ if and only if $f=g$. The general idea of calibration as in Deville and Särndal (1992) is to minimize the aggregate distance between the final weights $w=(w_{1},\cdots,w_{N})$ to a given vector of design weights $d=(d_{1},\cdots,d_{N})$ subject to moment constraints. Avoiding estimating the design weights, CAL first used a vector of misspecified uniform design weights $d^{*}=(1,\cdots,1)$, and constructed the final weights $w$ by solving the following constrained optimization problem:

This CAL method can be implemented through an open-source R package “ATE” for estimating both the ATE and ATT.

Let $X_{1},\cdots,X_{n}$ and $Y_{1},\cdots,Y_{m}$ be random samples from two unknown probability measures, $\mathbb{P}$ and $\mathbb{Q}$, then researchers are often interested in detecting the difference and estimating the discrepency between them. In this paper, we use a probability metric with $\zeta$-structure (Zolotarev, 1983)

where $\mathscr{F}$ indicates a class of functions, to measure the distance between the treatment and the control covariates.

As for the implementation of $\gamma(P,Q)$, we can define $||\cdot||_{\mathscr{K}}$ to be a norm of a Reproducing Kernel Hilbert Spaces (RKHS) $\mathscr{K}$ and set $\mathscr{F}=\{f:||f||_{\mathscr{K}}\neq 1\}$, by which we can get a “kernelized” version of the total variation distance. Restricting $k$ to be a strictly positive definite kernel function corresponding to RKHS $\mathscr{K}$, then we can get a closed-form solution for $\gamma(P,Q)$ (Sriperumbudur et al., 2012), that is

where $P_{n_{1}}$ and $Q_{n_{0}}$ respectively denote the empirical measure of $\bm{X}|T=1$ and $\bm{X}|T=0$, and $w_{i}$ is the sample weight for the unit $i\in\{1,\cdots,N\}$. Let us define $w_{i}=1/n_{1}$ if $T_{i}=1$ and $w_{i}=1/n_{0}$ if $T_{i}=0$, then we can get a simple as well as closed-form analytical solution for the empirical estimate of the probability metric under the assumption that the function class represents an RKHS. The computation of $\gamma_{k}(P_{n_{1}},Q_{n_{0}})$ is $O(N^{2})$ but it is also independent of the dimension of the confounders.

We can use $\gamma_{k}(P_{n_{1}},Q_{n_{0}})$ as a diagnostic for covariate balance, the smaller values of which denote better performance.

Given two probability measures, $\mathbb{P}$ and $\mathbb{Q}$ defined on a measurable space S, the integral probability metric (IPM) (Sriperumbudur et al., 2012), also called probability metrics with a $\zeta$-structure, between $\mathbb{P}$ and $\mathbb{Q}$ is defined as

where $\mathcal{F}$ is a class of real-valued bounded measurable functions on S. The choice of functions is the crucial distinction between different IPMs and various popular distance measures, for example, the Kantorovich metric and the total variation distance, can be obtained by choosing appropriate $\mathcal{F}$ in (4.1). When $\mathcal{F}=\{f:||f||_{\mathcal{H}}\leq 1\}$, the corresponding $\gamma_{\mathcal{F}}$ is called kernel distance or maximum mean discrepancy, where $\mathcal{H}$ represents a reproducing kernel Hilbert space (RKHS) with $k$ as its reproducing kernel (r.k.). In this situation, we can rewrite the function space $\mathcal{F}$ as $(\mathcal{H},k)$ and the IPM $\gamma_{\mathcal{F}}$ as $\gamma_{\mathcal{F}_{k}}$. The kernel distance has been widely used in statistical applications, including homogeneity testing, independence testing, conditional independence testing, and mixture density estimation.

Let $\left\{X_{P,i}:~{}i=1,2,\cdots,n\right\}$ and $\left\{X_{Q,j}:~{}j=1,2,\cdots,m\right\}$ be i.i.d. samples randomly drawn from $\mathbb{P}$ and $\mathbb{Q}$, respectively, the empirical estimator of $\gamma_{\mathcal{F}}(\mathbb{P},\mathbb{Q})$ is given by

where $\mathbb{P}_{n}(x):=\sum_{i=1}^{n}\frac{1}{n}I(X_{P,i}\leq x)$ and $\mathbb{Q}_{m}(x):=\sum_{j=1}^{m}\frac{1}{m}I(X_{Q,j}\leq x)$ represent the empirical distributions of $\mathbb{P}$ and $\mathbb{Q}$, respectively, $I(\cdot)$ is an indicator function, $N=n+m$, $w_{i}^{E}=\frac{1}{n}$ when $X_{i}=X_{P,i}$ for $i=1,\cdots,n$ and $w_{j}^{E}=-\frac{1}{m}$ when $X_{j}=X_{Q,j}$ for $j=1,\cdots,m$.

Correspondingly, we define the weighted estimator of of $\gamma_{\mathcal{F}}(\mathbb{P},\mathbb{Q})$ as

where $\mathbb{P}^{w}_{n}(x):=\sum_{i=1}^{n}p_{i}I(X_{P,i}\leq x)$ and $\mathbb{Q}_{m}(x):=\sum_{j=1}^{m}q_{j}I(X_{Q,j}\leq x)$ represent the wighted distributions of $\mathbb{P}$ and $\mathbb{Q}$, respectively, $w_{i}=p_{i}$ when $X_{i}=X_{P,i}$ for $i=1,\cdots,n$ and $w_{j}=-q_{j}$ when $X_{j}=X_{Q,j}$ for $j=1,\cdots,m$. $\{p_{i}:i=1,\cdots,n\}$ and $\{q_{j}:j=1,\cdots,m\}$ are the balancing weights derived frome some weighting methods.

Following the Cauchy-Schwartz inequality, which states that for all vectors $u$ and $v$ of an inner product space it is true that

where $\langle\cdot,\cdot\rangle$ is the inner product, we have

Therefore, we can set

Since $k$ is strictly positive definite, $\gamma_{k,N}(\mathbb{P}_{n},\mathbb{Q}_{m})=0$ if and only if $\mathbb{P}_{n}=\mathbb{Q}_{m}$, which therefore ensures that (4.4) is well-defined.

According to Rasmussen (2003), one of the choice of kernel $k$ for covariate balance is the Gaussian kernel, that is

where the parameter $\sigma$ determines the width of the Gaussian kernel and it can either be fixed or estimated from the data. In this article, we estimate $\sigma$ with the median of all possible pairwise squared Euclidean distances between all pairs of subjects.

The Information Matrix $\bm{K}_{G}$ corresponding to the Gaussian kernel $K(\bm{x},\bm{x^{\prime}};\sigma^{2})$ is

In this article, we propose a preprocessing procedure, the kernel-distance-based covariate balancing method, to create balanced samples for evaluating the treatment effects. In this procedure, we create two groups of scalar weights for the treatment and the control groups so that the reweighted treatment and control groups can match precisely at the specified moments.