fine {DOS}

Expand a Distance Matrix for Matching with Fine Balance.

Description

In optimal pair matching with fine balance, expand a distance matrix to become a square matrix to enforce fine balance. The method is discussed in Chapter 10 of Design of Observational Studies (2010), and it is conceptually the simplest way to implement fine balance; therefore, it remains very useful for teaching and for self-study. See details. For practical work, consider also the rcbalance package, the designmatch package and the bigmatch package; see the references.

Usage

fine(dmat, z, f, mult = 100)

Arguments

dmat

A distance matrix with one row for each treated individual and one column for each control. Often, this is either the Mahalanobis distance based on covariates, mahal(), or else a robust variant produced by smahal(). The distance matrix dmat may have been penalized by addalmostexact() or addcaliper(). An error will result unless dmat has more columns than rows.

z

z is a vector that is 1 for a treated individual and 0 for a control. The number of treated subjects, sum(z), must equal the number of rows of dmat, and the number of potential controls, sum(1-z), must equal the number of columns of dmat.

f

A factor or vector to be finely balanced. Must have length(f)=length(z).

mult

A positive number, mult>0. Determines the penalty used to enforce fine balance as max(dmat)*mult. The distance matrix dmat may have been penalized by addalmostexact() or addcaliper(), and in this case it makes sense to set mult=1 or mult=2, rather than the default, mult=100. If dmat is already penalized, taking mult>1 creates a larger penalty for deviations from fine balance than the exisiting penalties.

Details

The method is discussed in Chapter 10 of Design of Observational Studies (2010), and it is conceptually the simplest way to implement fine balance. However, the expanded distance matrix can become quite large, so this simplest method is not the most efficient method in its use of computer storage. A more compact implementation uses minimum cost flow in a network (Rosenbaum 1989). Additionally, there are several extensions of fine balance, including near-fine balance (Yang et al. 2012), fine balance for several covariates via integer programming (Zubizarreta 2012, designmatch R-package), and refined balance (Pimentel et al. 2015, rcbalance R-package). Ruoqi Yu's bigmatch R-package implements fine balance and near-fine balance in very large matching problems.

Value

An expanded, square distance matrix with "extra" treated units for use in optimal pair matching. Any control paired with an "extra" treated unit is discarded, as are the "extra" treated units.

Author(s)

Paul R. Rosenbaum

References

Hansen, B. B. (2007). Flexible, optimal matching for observational studies. R News, 7, 18-24. (optmatch package)

Pimentel, S. D., Kelz, R. R., Silber, J. H. and Rosenbaum, P. R. (2015). Large, sparse optimal matching with refined covariate balance in an observational study of the health outcomes produced by new surgeons. Journal of the American Statistical Association, 110, 515-527. Introduces an extension of fine balance called refined balance.

Pimentel, S. D. (2016) Large, Sparse Optimal Matching with R Package rcbalance. Observational Studies, 2, 4-23. An introduction to the rcbalance package.

Pimentel, S. D. (2016). R Package rcbalance. A recommended R package implementing both fine balance and an extension, refined balance.

Rosenbaum, P. R. (1989). Optimal matching for observational studies. Journal of the American Statistical Association, 84(408), 1024-1032. Discusses and illustrates fine balance using minimum cost flow in a network.

Rosenbaum, P. R., Ross, R. N. and Silber, J. H. (2007). Minimum distance matched sampling with fine balance in an observational study of treatment for ovarian cancer. Journal of the American Statistical Association, 102, 75-83. Discusses and illustrates fine balance using optimal assignment.

Rosenbaum, P. R. (2010). Design of Observational Studies. New York: Springer. The method and example are discussed in Chapter 10.

Yang, D., Small, D. S., Silber, J. H. and Rosenbaum, P. R. (2012). Optimal matching with minimal deviation from fine balance in a study of obesity and surgical outcomes. Biometrics, 68, 628-636. Extension of fine balance useful when fine balance is infeasible. Comes as close as possible to fine balance. Implemented in Pimentel's rcbalance package.

Yu, Ruoqi (2018). R package bigmatch.

Zubizarreta, J. R., Reinke, C. E., Kelz, R. R., Silber, J. H. and Rosenbaum, P. R. (2011). Matching for several sparse nominal variables in a case-control study of readmission following surgery. The American Statistician, 65(4), 229-238. Combines near-exact matching with fine balance for the same covariate.

Zubizarreta, J. R. (2012). Using mixed integer programming for matching in an observational study of kidney failure after surgery. Journal of the American Statistical Association, 107, 1360-1371. Extends the concept of fine balance using integer programming. Implemented in R in the designmatch package.

Zubizarreta, J. R. designmatch. A recommended R package for fine balance using integer programming.

Examples

data(costa)
z<-1*(costa$welder=="Y")
aa<-1*(costa$race=="A")
smoker=1*(costa$smoker=="Y")
age<-costa$age
x<-cbind(age,aa,smoker)
dmat<-mahal(z,x)
# Mahalanobis distances
round(dmat[,1:6],2) # Compare with Table 8.5 in Design of Observational Studies (2010)
# Impose propensity score calipers
prop<-glm(z~age+aa+smoker,family=binomial)$fitted.values # propensity score
# Mahalanobis distanced penalized for violations of a propensity score caliper.
# This version is used for numerical work.
dmat<-addcaliper(dmat,z,prop,caliper=.5)
round(dmat[,1:6],2) # Compare with Table 8.5 in Design of Observational Studies (2010)
# Because dmat already contains large penalties, we set mult=1.
dmat<-fine(dmat,z,aa,mult=1)
dmat[,1:6] # Compare with Table 10.1 in Design of Observational Studies (2010)
dim(dmat) # dmat has been expanded to be square by adding 5 extras, here numbered 48:52
# Any control matched to an extra is discarded.
## Not run: 
# Find the minimum distance match within propensity score calipers.
optmatch::pairmatch(dmat)
# Any control matched to an extra is discarded.  For instance, the optimal match paired
# extra row 48 with the real control in column 7 to form matched set 1.22, so that control
# is not part of the matched sample.  The harmless warning message from pairmatch
# reflects the divergence between the costa data.frame and expanded distance matrix.

## End(Not run)
# Conceptual versions with infinite distances for violations of propensity caliper.
dmat[dmat>20]<-Inf
round(dmat[,1:6],2) # Compare with Table 10.1 in Design of Observational Studies (2010)