Covariate balance, with treatment/covariate association tests
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balanceTest(fmla, data, strata = NULL, report = c("std.diffs", "z.scores", "adj.means", "adj.mean.diffs", "chisquare.test", "p.values", "all")[1:2], unit.weights, stratum.weights = harmonic_times_mean_weight, subset, include.NA.flags = TRUE, covariate.scaling = NULL, post.alignment.transform = NULL, p.adjust.method = "holm")
A formula containing an indicator of treatment assignment on the left hand side and covariates at right.
A data frame in which
A list of right-hand-side-only formulas containing
the factor(s) identifying the strata, with
Character vector listing measures to report for each
stratification; a subset of
Per-unit weight, or 0 if unit does not meet condition specified by subset argument. If there are clusters, the cluster weight is the sum of unit weights of elements within the cluster. Within each stratum, unit weights will be normalized to sum to the number of clusters in the stratum.
Function returning non-negative weight for each stratum; see details.
Optional: condition or vector specifying a subset of observations to be permitted to have positive unit weights.
Present item missingness comparisons as well as covariates themselves?
A scale factor to apply to covariates in
Optional transformation applied to covariates just after their stratum means are subtracted off.
Method of p-value adjustment.
Given a grouping variable (treatment assignment, exposure status, etc) and variables on which to compare the groups, compare averages across groups and test hypothesis of no selection into groups on the basis of that variable. The multivariate test is the method of combined differences discussed by Hansen and Bowers (2008, Statist. Sci.), a variant of Hotelling's T-squared test; the univariate tests are presented with multiplicity adjustments, the details of which can be controlled by the user. Clustering, weighting and/or stratification variables can be provided, and are addressed by the tests.
The function assembles various univariate descriptive statistics
for the groups to be compared: (weighted) means of treatment and
control groups; differences of these (adjusted differences); and
adjusted differences as multiples of a pooled S.D. of the variable
in the treatment and control groups (standard differences). This
is done separately for each provided stratifying factor and, by
default, for the unstratified comparison, in each case reflecting
a standardization appropriate to the designated (post-)
stratification of the sample. In the case without stratification
or clustering, the only weighting used to calculate treatment and
control group means is that provided by the user as
unit.weights; in the absence of such an argument, these
means are unweighted. When there are strata, within-stratum means
of treatment or of control observations are calculated using
unit.weights, if provided, and then these are combined
across strata according to a ‘effect of treatment on
treated’-type weighting scheme. (The function's
stratum.weights argument figures in the function's
inferential calculations but not these descriptive calculations.)
To figure a stratum's effect of treatment on treated weight, the
sum of all
unit.weights associated with treatment or
control group observations within the stratum is multiplied by the
fraction of clusters in that stratum that are associated with the
treatment rather than the control condition. (Unless this
fraction is 0 or 1, in which case the stratum is downweighted to
The function also calculates univariate and multivariate inferential
statistics, targeting the hypothesis that assignment was random within strata. These
calculations also pool
unit.weights-weighted, within-stratum group means across strata,
but the default weighting of strata differs from that of the descriptive calculations.
stratum.weights=harmonic_times_mean_weight (the default), each stratum
is weighted in proportion to the product of the stratum mean of
and 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 and Hansen and
Bowers 2008, Sections 3.2 and 5). The multivariate assessment is based on a Mahalanobis-type
distance that combines each of the univariate mean differences while accounting
for correlations among them. It's similar to the Hotelling's T-squared statistic,
except standarized using a permutation covariance. See Hansen and Bowers (2008).
In contrast to the earlier function
xBalance that it is intended to replace,
balanceTest accepts only binary assignment variables (for now).
stratum.weights must be a function of a single argument,
a data frame containing the variables in
returning a named numeric vector of non-negative weights identified by stratum.
(For an example, enter
If the stratifying factor has NAs, these cases are dropped. On the other hand, if NAs in a covariate are found then those observations are dropped for descriptive calculations and "imputed" to the stratum mean of the variable for inferential calculations. When covariate values are dropped due to missingness, proportions of observations not missing on that variable are recorded and returned. The printed output presents non-missing proportions alongside of the variables themselves, distinguishing the former by placing them at the bottom of the list and enclosing the variable's name in parentheses. If a variable shares a missingness pattern with other another variable, its missingness information may be labeled with the name of the other variable in the output.
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, 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 as the treatment assignments of clusters are freely permuted. For stratified comparisons, the reference distributions describes re-randomizations of this type performed separately in each stratum. 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 balanceTest(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n, data=nuclearplants) ##No strata, all output balanceTest(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n, data=nuclearplants, report=c("all")) ##Stratified, all output balanceTest(pr~.-cost-pt + strata(pt), data=nuclearplants, report=c("adj.means", "adj.mean.diffs", "chisquare.test", "std.diffs", "z.scores", "p.values")) ##Comparing unstratified to stratified, just adjusted means and #omnibus test balanceTest(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n + strata(pt), data=nuclearplants, report=c("adj.means", "chisquare.test")) ##Comparing unstratified to stratified, just adjusted means and #omnibus test balanceTest(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n + strata(pt), data=nuclearplants, report=c("adj.means", "chisquare.test")) ##Missing data handling. testdata<-nuclearplants testdata$date[testdata$date<68]<-NA ##Comparing unstratified to stratified, just one-by-one wilcoxon #rank sum tests and omnibus test of multivariate differences on #rank scale. balanceTest(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n + strata(pt), data=nuclearplants, report=c("adj.means", "chisquare.test"), post.alignment.transform=rank)
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