# Implement a Caliper Using a Penalty Function in Optimal Matching A caliper forces a close match on a continuous covariate, often the propensity score (Rosenbaum and Rubin 1985). Alternatively or additionally, a caliper may be imposed on a prognostic score estimated from some other independent data set (Hansen 2008). Rosenbaum and Rubin (1985) suggested minimizing the Mahalanobis distance within calipers defined by the propensity score. If the caliper is implemented using a penalty function, the caliper cannot create infeasibility. Implementation of a caliper using a penalty function is discussed in Chapter 9 of "Design of Observational Studies", second edition. ```r addcaliper(dmat, z, p, caliper = 0.1, penalty = 1000) ``` ## 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'. - `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. - `p`: A vector covariate to which the caliper will be applied. Often, p is the propensity score. length(p) must equal length(z). - `caliper`: A positive number. A penalty will be added to row i and column j of the distance matrix dmat if treated unit i and control unit j have values of p that differ in absolute value by more than caliper*sd(p), the default being a tenth of the standard deviation of p. This is different from the code in the first edition of "Design of Observational Studies", where caliper defaulted to 0.2, rather than 0.1. - `penalty`: A positive number. penalty determines the magnitude of the penalty that is added to the distance matrix when the caliper is violated. Let pt and pc be the values of p for a treated individual and a control. If |pt-pc| <= caliper*sd(p), then nothing is added to dmat for these individuals. Otherwise, define gap = |pt-pc|-caliper*sd(p); then, the distance between these individuals in dmat is increased by gap*penalty. So there is no penalty inside the caliper, but dmat is increased rapidly as violation of the caliper increases in magnitude. A reasonable value of penalty will depend upon the units in which p is measured, and the default values are reasonable starting points for the propensity score expressed as a probability. ## Details By calling 'addcaliper' several times, calipers may be imposed on several variables, say a propensity score and age. If you use several penalties, say from addcaliper, addalmostexact, and fine, then you need to pay some attention to their relative magnitudes to ensure that they express your view of the relative importance of the conditions they impose. ## Returns Returns the penalized distance matrix. ## Note The use of a penalty function to implement a caliper is discussed in chapter 9 of the second edition of "Design of Observational Studies". It is also discussed in chapter 8 of the first edition. The matching functions in the 'DOS2' package are aids to instruction or self-instruction while reading "Design of Observational Studies". As in the book, these functions break the task of matching into small steps so they are easy to understand in depth. In practice, matching entails a fair amount of book-keeping best done by a package that automates these tasks. Consider R packages 'optmatch', 'rcbalance', 'DiPs', 'designmatch' or 'bigmatch'. Section 14.10 of "Design of Observational Studies", second edition, discusses and compares several R packages for optimal matching. ## References Hansen, B. B. and Klopfer, S. O. (2006) "Optimal full matching and related designs via network flows". Journal of Computational and Graphical Statistics, 15(3), 609-627. ('optmatch' package) Hansen, B. B. (2007) "Flexible, optimal matching for observational studies". R News, 7, 18-24. ('optmatch' package) Hansen, B. B. (2008) "The prognostic analogue of the propensity score". Biometrika, 95(2), 481-488. Rosenbaum, P. R. and Rubin, D. B. (1985) "Constructing a control group using multivariate matched sampling methods that incorporate the propensity score". The American Statistician, 39, 33-38. ## Author(s) Paul R. Rosenbaum ## Examples ```r 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 9.5 in Design of Observational # Studies, 2nd edition. # 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 9.5 in "Design of Observational # Studies", 2nd edition. ## Not run: # Find the minimum distance match within propensity score calipers. optmatch::pairmatch(dmat,data=costa) ## 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 9.5 in "Design of Observational # Studies", 2nd edition. ```

addcaliper function