Entropy balancing: a maximum-entropy reweighting scheme to adjust for coverage error

Abstract

This paper presents a newly available technique to adjust for bias in non-probabilistically selected samples. To date, applications of this innovative technique—termed entropy balancing—have been restricted to evaluation settings, where the goal is to reduce model dependence prior to the estimation of treatment effects. In a novel application, we demonstrate the technique’s utility in cases where the goal is to correct for sample bias originating in coverage error. The appeal of entropy balancing in this latter setting lies in its capacity to optimise the twin goals of improved balance in covariate distribution and maximum retention of information. Entropy balancing combines the opportunity to incorporate a large set of moment conditions in the calculation of weights, with the ability to directly implement exact balance. The technique thus builds upon the theoretical appeal of the more widely known and applied propensity score adjustment method, while addressing that method’s practical limitations. We demonstrate the utility of the entropy balancing technique empirically, through an example using the Young Lives Project survey data for rural Andhra Pradesh, South India. We conclude by summarising the potential of this procedure to contribute to robust survey-based research more widely.

Appendix : A condensed version of the theoretical framework for entropy balancing

Under ebalance, weights are selected to minimize the entropy distance metric:

$$ {\mathop {\hbox{min} \,H(w)}\limits_{{W_{1} }}} = \sum\limits_{{\left\{ {i\left| {D = 0} \right.} \right\}}} {w_{i} \log (w_{i} /q_{i} )} $$

(1)

where w _i is the weight selected for each non-random sample units.

D_i $\in \left\{ {1,0} \right\}$ is a binary indicator coded 1 unit i is drawn from the reference sample or 0 if it is drawn from the non-random sample. q _i  = 1/n₀ and is a base weight.

The selection of weights is subjected to the balance constraints defined in Eq. 2.1, the normalising constraints defined in Eq. 2.2, and the non-negativity constraints defined in Eq. 2.3:

$$ \sum\limits_{{\left\{ {i\left| {D = 0} \right.} \right\}}} {w_{i} c_{ri} (X_{i} )\,\, = \,\,m_{i} } \;\rm with\;r \in 1, \ldots ,{\text{R}} $$

(2.1)

$$ \sum\limits_{{\left\{ {i\left| {D = 0} \right.} \right\}}} {w_{i} = 1} $$

(2.2)

$$ w_{i} \ge 0 \quad {\text{for}}\;{\text{all}}\;i\;{\text{such}}\;{\text{that}}\;D = 0 $$

(2.3)

X is a matrix that contains the data of J exogenous pre-treatment covariates with X _ij denoting the values of the j-th covariate characteristic for unit i.

$C_{ri} (X_{i} ) = m_{r}$ describes a set of R balance constraints imposed on the covariate moments of the reweighted non-random sample group.

The ebalance approach accommodates high dimensionality to assign one weight to each control unit. The weights that solve the entropy balancing scheme are computed from a dual problem that is unconstrained and reduced to a system of non-linear equations in R Langrange multipliers. The dual problem is given by:

$$ \mathop {\hbox{min}\; L^{d} }\limits_{Z} = \log (Q^{\prime}\exp - (C^{\prime}Z)) + M^{\prime}Z $$

(3)

where ${\text{Z = }}\left\{ {\lambda_{1} , \ldots ,\lambda_{\text{R}} } \right\}^{\prime }$ is a vector (Z*) of Langrange multiplier for the balance constraints, rewritten in matrix form as CW=M with the $({\text{R}}\; \times \;{\text{n}}_{0} )$ constraint matrix, C=[c₁(X_i),…,c_R(X_i)]′, and the moment vector, $M\, = \,[m_{1} , \ldots ,m_{R} ]^{\prime}$. Thevector Z* that solves the dual problem also the primal problem. The solution weights are recover using:

$$ W^{*} = \frac{{Q.\exp ( - C^{\prime}Z^{*} )}}{{Q^{\prime}.\exp ( - C^{\prime}Z).}} $$

(4)

An iterative Levenberg–Marquardt scheme exploits second order information to solve the dual problem:

$$ {\text{Z}}^{\text{new}} = \;\;{\text{Z}}^{\text{old}} - {\text{I}}\nabla^{2} {}_{\text{Z}}({\text{L}}^{\text{d}} )^{ - 1} \nabla {}_{\text{Z}}({\text{L}}^{\text{d}} ) $$

(5)

Here,I is a scalar denoting the step length. The optimal step length (either the full newton step or I) is selected for each iteration (Hainmueller and Xu 2013).