Given covariates, a treatment variable, and a stratifying factor, calculates standardized mean differences along each covariate, with and without the stratification and tests for conditional independence of the treatment variable and the covariates within strata.
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xBalance(fmla, strata = list(unstrat = NULL), data, report = c("std.diffs", "z.scores", "adj.means", "adj.mean.diffs", "adj.mean.diffs.null.sd", "chisquare.test", "p.values", "all")[1:2], stratum.weights = harmonic, na.rm = FALSE, covariate.scaling = NULL, normalize.weights = TRUE, impfn = median, post.alignment.transform = NULL)
A formula containing an indicator of treatment assignment on the left hand side and covariates at right.
A list of right-hand-side-only formulas containing
the factor(s) identifying the strata, with
A data frame in which
Character vector listing measures to report for each
stratification; a subset of
Weights to be applied when aggregating
across strata specified by
Whether to remove rows with NAs on any variables
mentioned on the RHS of
A scale factor to apply to covariates in
A function to impute missing values when
Optional transformation applied to covariates just after their stratum means are subtracted off.
In the unstratified case, the standardized difference of covariate
means is the mean in the treatment group minus the mean in the
control group, divided by the S.D. (standard deviation) in the
same variable estimated by pooling treatment and control group
S.D.s on the same variable. In the stratified case, the
denominator of the standardized difference remains the same but
the numerator is a weighted average of within-stratum differences
in means on the covariate. By default, each stratum is weighted
in proportion to the harmonic mean 1/[(1/a +
1/b)/2]=2*a*b/(a+b) of the number of treated units (a) and
control units (b) in the stratum; this weighting is optimal under
certain modeling assumptions (discussed in Kalton 1968, Hansen and
Bowers 2008). This weighting can be modified using the
stratum.weights argument; see below.
When the treatment variable, the variable specified by the
left-hand side of
fmla, is not binary,
calculates the covariates' regressions on the treatment variable,
in the stratified case pooling these regressions across strata
using weights that default to the stratum-wise sum of squared
deviations of the treatment variable from its stratum mean.
(Applied to binary treatment variables, this recipe gives the same
result as the one given above.) In the numerator of the
standardized difference, we get a “pooled S.D.” from separating
units into two groups, one in which the treatment variable is 0 or
less and another in which it is positive. If
includes "adj.means", covariate means for the former of these
groups are reported, along with the sums of these means and the
covariates' regressions on either the treatment variable, in the
unstratified (“pre”) case, or the treatment variable and the
strata, in the stratified (“post”) case.
stratum.weights can be either a function or a numeric
vector of weights. If it is a numeric vector, it should be
non-negative and it should have stratum names as its names. (i.e.,
its names should be equal to the levels of the factor specified by
strata.) If it is a function, it should accept one
argument, a data frame containing the variables in
stratum.code, and return a
vector of non-negative weights with stratum codes as names; for an
covariate.scaling is not
NULL, no scaling is
applied. This behavior is likely to change in future versions.
(If you want no scaling, set
covariate.scaling=1, as this
is likely to retain this meaning in the future.)
adj.mean.diffs.null.sd returns the standard deviation of
the Normal approximated randomization distribution of the
strata-adjusted difference of means under the strict null of no
An object of class
c("xbal", "list"). There are
xtable methods for class
Evidence pertaining to the hypothesis that a treatment variable is not associated with differences in covariate values is assessed by comparing the differences of means (or regression coefficients), without standardization, to their distributions under hypothetical shuffles of the treatment variable, a permutation or randomization distribution. For the unstratified comparison, this reference distribution consists of differences (more generally, regression coefficients) when the treatment variable is permuted without regard to strata. For the stratified comparison, the reference distribution is determined by randomly permuting the treatment variable within strata, then re-calculating the treatment-control differences (regressions of each covariate on the permuted treatment variable). Significance assessments are based on the large-sample Normal approximation to these reference distributions.
Ben Hansen and Jake Bowers and Mark Fredrickson
Hansen, B.B. and Bowers, J. (2008), “Covariate Balance in Simple, Stratified and Clustered Comparative Studies,” Statistical Science 23.
Kalton, G. (1968), “Standardization: A technique to control for extraneous variables,” Applied Statistics 17, 118–136.
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data(nuclearplants) ##No strata, default output xBalance(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n, data=nuclearplants) ##No strata, all output xBalance(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n, data=nuclearplants, report=c("all")) ##Stratified, all output xBalance(pr~.-cost-pt, strata=factor(nuclearplants$pt), data=nuclearplants, report=c("adj.means", "adj.mean.diffs", "adj.mean.diffs.null.sd", "chisquare.test", "std.diffs", "z.scores", "p.values")) ##Comparing unstratified to stratified, just adjusted means and #omnibus test xBalance(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n, strata=list(unstrat=NULL, pt=~pt), data=nuclearplants, report=c("adj.means", "chisquare.test")) ##Comparing unstratified to stratified, just adjusted means and #omnibus test xBalance(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n, strata=data.frame(unstrat=factor('none'), pt=factor(nuclearplants$pt)), data=nuclearplants, report=c("adj.means", "chisquare.test")) ##Missing data handling. testdata<-nuclearplants testdata$date[testdata$date<68]<-NA ##na.rm=FALSE by default xBalance(pr ~ date, data = testdata, report="all") xBalance(pr ~ date, data = testdata, na.rm = TRUE,report="all") ##To match versions of RItools 0.1-9 and older, impute means #rather than medians. ##Not run, impfn option is not implemented in the most recent version ## Not run: xBalance(pr ~ date, data = testdata, na.rm = FALSE, report="all", impfn=mean.default) ## End(Not run) ##Comparing unstratified to stratified, just one-by-one wilcoxon #rank sum tests and omnibus test of multivariate differences on #rank scale. xBalance(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n, strata=data.frame(unstrat=factor('none'), pt=factor(nuclearplants$pt)), data=nuclearplants, report=c("adj.means", "chisquare.test"), post.alignment.transform=rank)
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