Having said that, I think you did notwe need to fully appreciate what "with replacement" entails. For starters, let'sLet's consider what will happen in the case of two sample that do not have substantial common support: When matching with replacement a lot of instances from the treatment group can be matched to the same instance from the control group. That is because the same control instance can be the nearest neighbour for many treatment instances. Now, recalling that
As the NSW sample and the PSID sample have a small overlap regarding their pre-treatment characteristics means that we arethis translates into not guaranteehaving a lot of uniquelyunique control samples picked. We would actually expect a relatively small sample
To that extent, yes, if we are using 1-to-1 matching with replacement we cannot ever have more control samples picked than the number of treatment samples.
One the other hand, with caliper matching we will use allall of the control units within a pre-defined propensity score radius (the caliper). In that sense, a larger sample size is not unexpected. Note that caliper matching is far from a silver-bullet. Some treatment units may not receive any matches because there are no neighbours within the given caliper. As Morgan & Winship, propose in Chapt. 5 you might be better to "use a hybrid approach, where in a second step all treatment cases without any caliper-based matches are then matched to a single nearest neighbor outside of the caliper."
In out1 we use 1-NN matching with replacement and then out2 1-NN matching without replacement. The matched datadataset created by out2 is actually smaller than out1 exactly because a number of control subjects are reused. The(The data lalonde is a subsample from the NSW study.)
Finally, you are right to question the actual numbernumbers reported in the paper. While very plausibleI think the wording leaves something to have a smallerbe desired. By 56 the authors most probably denote the number of matched sample when using 1-NN matchingcontrol units, rather than when using caliper matching (as explained above)"No. of observations" which itself is unambiguiduous if it refers to the whole matched sample or just the treatment sample of it.
Unfortunately I cannot replicate exactly because I do not have no idea how D&W end up with such a small numberthe full dataset; the unemployment figures are provided by the authors. If anything I would expect at least as many comparison units ascan replicate the analysis without these figures in which case they are 66 (not 56) matched control units against 185 treatment units. Clearly somethingSo 56, while quite low is omitted either internally from psmatch2 or the authorsnot totally improbable. I attach the R code used below:
rm(list=ls())
NSW = read.table('http://www.nber.org/~rdehejia/data/nswre74_treated.txt')
PSID = read.table('http://www.nber.org/~rdehejia/data/psid_controls.txt')
names(NSW) = c('treatment', 'age','education', 'black', 'hispanic',
'married', 'nodegree', 're74', 're75', 're78')
names(PSID) = names(NSW)
allData = rbind(NSW, PSID)
library(MatchIt)
Q = matchit(treatment ~ age + education + I(age^2) + I(education^2) +
black + hispanic + married + nodegree +
re74 + re75 + I(re74^2) + I(re75^2),
data = allData, method = "nearest",
distance = "logit", replace = TRUE )
Q
