Page 1

ORIGINAL REPORT

Reweighted Mahalanobis distance matching for cluster-randomized

trials with missing data

Robert A. Greevy Jr.1,2*, Carlos G. Grijalva4, Christianne L. Roumie1,3, Cole Beck1,2, Adriana M. Hung1,3,

Harvey J. Murff1,3, Xulei Liu1,2and Marie R. Griffin1,3,4

1VA Tennessee Valley Geriatric Research Education Clinical Center (GRECC), Nashville, TN, USA

2Department of Biostatistics, Vanderbilt University, Nashville, TN, USA

3Department of Medicine, Vanderbilt University, Nashville, TN, USA

4Department of Preventive Medicine, Vanderbilt University, Nashville, TN, USA

ABSTRACT

Purpose

clinical and other knowledge regarding the relative importance of variables used in matching and allows for multiple types of missing data.

The method is illustrated in the context of a cluster-randomized trial. A Web application and an R package are introduced to implement the

method and incorporate recent advances in the area.

Methods

Reweighted Mahalanobis distance (RMD) matching incorporates user-specified weights and imputed values for missing data.

Weight may be assigned to missingness indicators to match on missingness patterns. Three examples are presented, using real data from

a cohort of 90 Veterans Health Administration sites that had at least 100 incident metformin users in 2007. Matching is utilized to balance

seven factors aggregated at the site level. Covariate balance is assessed for 10000 randomizations under each strategy: simple randomization,

matched randomization using the Mahalanobis distance, and matched randomization using the RMD.

Results

The RMD matching achieved better balance than simple randomization or MD randomization. In the first example, simple and

MD randomization resulted in a 10% chance of seeing an absolute mean difference of greater than 26% in the percent of nonwhite patients

per site; the RMD dramatically reduced that to 6%. The RMD achieved significant improvement over simple randomization even with as

much as 20% of the data missing.

Conclusions

Reweighted Mahalanobis distance matching provides an easy-to-use tool that incorporates user knowledge and missing data.

Copyright © 2012 John Wiley & Sons, Ltd.

This paper introduces an improved tool for designing matched-pairs randomized trials. The tool allows the incorporation of

Received 1 September 2011; Revised 20 February 2012; Accepted 21 February 2012

INTRODUCTION

In trials designed to reflect routine care, the cluster-

randomized trial is appealing. By randomly assigning

interventions to physicians or hospitals instead of

directly to patients, routine care settings may be stud-

ied under experimental interventions while problems

such as treatment contamination can be prevented.

However, the number of clusters being randomized is

often relatively small. A study including thousands

of patients may have randomized only a dozen

hospitals. In this situation, assigning treatments with

a simple randomization, for example, drawing the

names of half of the hospitals out of a hat, is unaccept-

ably risky. When the number of units being random-

ized is small, there is substantial risk that severe

imbalanceinimportantcovariateswilloccurbychance.1

Restricted randomization methods are commonly

used to reduce this risk. Cluster-randomized trials

frequently have the advantage of covariate information

being available on all units prior to randomization;

those units are randomized all at once or in a few

batches. Stratified randomization is a commonly used

restricted randomization method that creates strata

based on a few important covariates and then randomly

assigns half of the units in each stratum to one treatment

and half to the other. While providing some benefit,

this approach is limited to including only a few of

the important covariates and the categorization of

continuous covariates into a few bins. In spite of its

*Correspondence to: Robert A. Greevy, Jr., PhD, Vanderbilt University School

of Medicine, Department of Biostatistics, Nashville, TN 37232-2158, U.S.A.

E-mail: robert.greevy@vanderbilt.edu

Copyright © 2012 John Wiley & Sons, Ltd.

pharmacoepidemiology and drug safety 2012; 21(S2): 148–154

Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/pds.3260

Page 2

limitations, the general concept is sound. The ideal

stratification would contain exactly two similar units

within each stratum. Matching prior to randomization

achieves this without requiring categorization of con-

tinuous covariates, without severely limiting the num-

ber of covariates being balanced and without requiring

units that match perfectly to achieve balance between

the study arms.

The benefits of any restricted randomization method

depend on its ability to balance important covariates,

the strength of the association between the covariates

and the outcome, and the study’s sample size. In a

non-clustered study of 132 patients who were randomly

assigned treatment, Greevy et al. demonstrated that

optimal nonbipartite matching on the Mahalanobis

distance (MD) derived from 14 covariates resulted in

anaverageincreaseinpowerequivalenttoa7%increase

in sample size.1Moreover, this approach eliminated the

rare but severe imbalances that may occur with simple

randomization. Despite the method’s superior perfor-

mance, itpresentlyremainsless widelyusedthan simple

or stratified randomization. In cluster-randomized trials,

wideradoption hasbeenhinderedby misunderstandings

about matching and an absence of user-friendly tools

to implement the method.

In their 2009 paper, Imai et al. dispelled the major

misconceptions surrounding matched-pair cluster-

randomization (MPCR).2For example, they examine

the assumptions leading Martin et al. to recommend

against MPCR in small samples.3When the assump-

tion of equal cluster sizes is relaxed, as is appropriate

for most practical scenarios, the MPCR that matches

on cluster size and pre-treatment covariates will im-

prove the study’s efficiency and power over unmatched

cluster-randomization, even with as few as six clusters.

In a discussion of Imai et al., Zhang and Small show

the utility of optimal nonbipartite matching for achiev-

ing pre-treatment covariate balance in MPCR and for

optimally selecting a set of units for study when the

number of units available is greater than the number

needed.4For observational studies utilizing matching,

Rosenbaum presents a method of augmenting the

distance matrix to optimally choose the number of units

to study for a specified level of quality of match.5When

the quality of matches is of greater concern than the

exact number of units included, this approach can be

very useful in the MPCR setting. Both approaches are

incorporated into the methods presented here.

To fully realize the benefits of MPCR with several

pre-treatmentcovariates,including continuousmeasures

and cluster size, a multivariate distance measure is

needed. To balance the cluster-specific covariate distri-

butions, appropriate summary measures are chosen.

Categorical variables may be summarized with pro-

portions, for example, the percentage of patients taking

statins. Likewise, when the shape of the distributions

is not highly variable, a single summary measure may

suffice for continuous covariate distributions, for exam-

ple, mean low-density lipoprotein (LDL). Otherwise,

multiple measures may be used, for example, the mean

and standard deviation of LDL or the 10th, 50th, and

90th percentiles. Once a continuous multivariate dis-

tance measure is developed, the optimal set of matches

is the set that minimizes the average distance between

pairs. Lu et al. recently released an R package and a

Web application that takes a user-created matrix of

distances between units and solves for the optimal

matches.6However, the creation of the distance matri-

ces may create an obstacle for some researchers, and

improving the utility of distance measures is an open

area of research.

This paper addresses the need for a customizable

distance measure that incorporates clinical and other

knowledge regarding the importance of the covariates

while also allowing the inclusion of covariates with

missing values. The method we propose may incorpo-

rate two ways to exclude units when more units are

available than can be included in the study. We in-

troduce two user-friendly tools to implement the meth-

odology in the form of a Web application and an R

package. To aid the development of the distance mea-

sure, the Web application includes tools for assessing

the quality of the matches prior to randomization and

comparing them to benchmark values to assist the user

in choosing covariate weights. Once the choice of

weights has been finalized, the application allows the

user to perform the official randomization with a

user-specified random seed to allow reproducibility

and, if needed, the randomization of additional

study units to be added after the first set of treatment

assignments has been made. The Web application

and instructions on downloading the R package

nbpMatching are available at http://biostat.mc.vander-

bilt.edu/MatchedRandomization. Examples using real

data from Veterans Health Administration (VHA) sites

are presented to illustrate the method.

METHODS

Mahalanobis distance and reweighted Mahalanobis

distance

The MD is a multivariate distance measure akin to

the familiar Euclidean Distance; however, it has two

additional benefits. First, it is scale invariant, for

example, including a site’s pre-treatment mean LDL

in mg/dL will yield the same results as LDL in

reweighted mahalanobis distance matching149

Copyright © 2012 John Wiley & Sons, Ltd.

Pharmacoepidemiology and Drug Safety, 2012; 21(S2): 148–154

DOI: 10.1002/pds

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mmol/L. Second, it incorporates the correlations be-

tween the covariates. The effect may be thought of as

down-weighting a difference in one covariate that is

expected based on the differences observed in the

other covariates. The MD may be written as

MD xi; xj

??¼

xi? xj

??TS?1xi? xj

??

hi1

2 =

where xiis the ith row of the (n?p) covariate matrix

X, with n subjects in the rows and p covariates in the

columns, and S is the (p?p) covariance matrix of X.

A limitation of the MD is that is the influence that

the covariates have on the distance is driven purely

by their covariance structure, not their clinical impor-

tance. The reweighted Mahalanobis distance (RMD)

incorporates user-specified weights, imputed values

for missing covariate data, and indicators of covariate

missingness. We refer to the distance as “reweighted”

to distinguish it from similarly named measures used

in different settings.7–10The RMD may be written as

RMD xi; xj;W

??¼

x ~i? x ~j

??TWS~?1W x ~i? x ~j

??

hi1

2 =

where X~is the (n?p+q) covariate matrix consisting

of X with the addition of indicator variables for the q

covariates with missingness and missing values

replaced with imputed values, x ~iis the ith row of X~,

S~ is the (p+qxp+q) covariance matrix of X~, and

W is a (p+q?p+q) diagonal matrix of user-specified

weights. Various methods may be used to impute

the missing values, provided that the imputation is

estimating an expected value without random noise

added. The application presented here currently uses

the R package transcan, which transforms the covari-

ates to have their maximum correlation with the best

linear combination of the other covariates and returns

expected values on the original scale that may be

interpreted as an expected median or mode for contin-

uous or categorical variables, respectively.11

The usefulness of the imputed values will depend

partially on the validity of the missing-at-random

(MAR) assumption on which they are based.12If the

MAR assumption is in question, researchers may wish

to match on missingness patterns more than on the

imputed missing values. This may be achieved through

adjusting the weights for the missingness indicators.

The Web application currently uses the same weight

for all missingness indicators and a small default value

of 0.1, giving preference to the MAR assumption. A

weight of 0 can be used to completely eliminate the

impact of the indicators.

Optimality and limitation on the number of clusters

With the use of the RMD, an (n?n) matrix of dis-

tances between units is created. Unlike a retrospective

cohort study where units from one group are matched

to units in another group, any of the n units may be

matched to any of the other units. The optimal set of

matches is the set that yields the smallest average

RMD distance between the matched pairs. Thanks to

advanced methods for solving this so-called optimal

nonbipartite matching problem, the computational

complexity no longer appreciably limits its use.13

The Web application presented here has successfully

handled up to 5000 units, well beyond the typical

number of clusters in a cluster-randomized trial.

Optimally selecting a subset of clusters

In cluster-randomized studies, the cost of each hospital

or experimental unit included in the study often limits

the number of units that may be used because more

units may be available for participation than can be

included. Typically, the units that are excluded are

chosen via ad hoc procedures. Some reasons to

exclude a unit may be obvious, such as logistical

difficulties unique to that unit. When there is no clear

choice, the matching method can optimally select

which units to drop by removing those that would

create the greatest imbalance between the groups.4

The user specifies a number of units to exclude, say

k units. The application adds k units to the cohort that

have the special property that they match every other

unit perfectly. These units are usually referred to as

“sinks” and are labeled “phantoms” by the applica-

tions presented here. Units that are matched to the

phantoms are excluded from the study. In an alternate

approach, the user specifies a threshold for acceptable

distances, d, that matches must meet to be included in

the study. The distance matrix used for matching is

augmented to select the optimal set of units satisfying

the threshold.5This approach is equivalent to adding

N units to the study that are distance d from all other

units. Any unit that is matched to one of these “near-

matchers,” or “chameleons,” as they are called in our

applications, is excluded from the study.

Evaluating performance

If examining the average difference between groups

over all possible randomizations, almost any random-

ized method appears to balance the variables well

because the expected mean difference for all of these

methods is zero. However, a particular randomization

may be quite poor, showing a large imbalance in the

r. a. greevy et al.150

Copyright © 2012 John Wiley & Sons, Ltd.

Pharmacoepidemiology and Drug Safety, 2012; 21(S2): 148–154

DOI: 10.1002/pds

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mean difference for a variable. Thus, balance is

assessed through the 90th percentile of the absolute

mean differences (AMD_90). The AMD_90 is empir-

ically estimated via 10000 randomizations for each

strategy, and all standard errors are ≤0.1 unless

specified otherwise. The simple randomization used

here balances only the number of sites in each arm,

drawing from a set of size

n

n=2

??

possible randomi-

zations for n sites. The MPCR balances the covariates

of interest by restricting to a smaller set of size 2n/2

possible randomizations.

EXAMPLE STUDY

As our motivating example, we consider designing a

trial to study the effects of early intensification with

insulin on LDL at 12months post-intensification. In

2006,theAmericanDiabetesAssociationrecommended

that metformin be used as the first-line agent unless

contraindicated.14No standard protocol currently exists

for patients failing their oral anti-diabetic monother-

apy shortly after starting it, that is, those who show

glycosylated hemoglobin (A1c) levels of 7–9% within

3–12months after initiation. Those failing metformin

monotherapy could remain on their current treatment,

intensify with insulin, or intensify to a metformin–

sulfonylureadualtherapy.Weconsiderrandomlyassign-

ing VHA sites to early intensification with insulin or

early intensification with dual therapy. The method pro-

posedwill beused to balance (between thetwo treatment

arms)thedistributionsofcovariatesknowntoaffectLDL

in the VHA patient population.15

RESULTS

To illustrate the method, a hypothetical cohort of poten-

tial study sites is drawn from the National VHA data-

bases, which include pharmacy, inpatient, outpatient,

and laboratory records. For the VHA fiscal year 2007,

90 VHA sites serving 100–500 incident metformin

users were identified. Covariate information for each

patient was derived from the 365days preceding their

starting treatment; see Roumie et al. for covariate

definitions.15Matching is utilized to balance seven fac-

tors aggregated at the site level: percentage of nonwhite

patients, percent female, percent on statins, mean

systolic blood pressure (mmHg), body mass index

(BMI), A1c (%), and the number of incident metformin

users (N). Three examples are presented. The first

compares the performance of the MD and RMD to

simple matching of 12 preselected sites. The second

example illustrates the performance when selecting 12

sites out of the 90 potential sites via the use of phan-

toms or chameleons. The third example illustrates the

performance when missingness is induced completely

at random.

Example 1

The correlations and standard deviations for the 12

sites are shown in Table 1. The comparatively large

standard deviation and low correlations for race sug-

gest that it will be more difficult to balance than the

other variables. Table 2 shows the AMD_90 for four

methods: simple randomization; MD matching (MD);

RMD matching with weight=1 for race and 0 for all

Table 1. Variable correlations and standard deviations (12 preselected sites)

Nonwhite (%) Female (%) On statins (%) Systolic BP (mmHg)BMIA1c (%)

N patients

Nonwhite

Female

On statins

Systolic BP

BMI

A1c

N patients

35.2

0.2

0

0.2

?0.4

0.1

0.6

0.2

2.3

0.2

0.3

0.2

0.3

0.3

0.0

0.2

6.0

0.2

0.3

?0.4

0.2

0.2

0.4

1.4

0.6

?0.6

0.1

0.3

0.6

0.3

?0.2

2.6

0.4

0.7

0.1

?0.1

0.7

0.6

0.2

?0.4

?0.1

0.1

?0.6

?0.4

133.4

?0.2

0.2

?0.1

?0.1

Standard deviations shown in italics. N sites=12.

Table 2.

four randomization methods (12 preselected sites)

90th percentile of the absolute mean difference (AMD_90) for

VariableSimpleMDRMD_race RMD_race+

Nonwhite (%)

Female (%)

On statins (%)

Systolic BP (mmHg)

BMI

A1c (%)

N patients

26.8

1.6

6.4

1.5

1.3

0.1

100.5

26.1

1.8

6.2

1.4

0.7

0.1

110.2

4.8

1.6

3.7

2.3

1.4

0.3

6.4

2.3

6.3

2.1

0.8

0.2

33.2120.5

Methods: Simplerandomization,Mahalanobisdistancematching(MD),RMD

matching with weight=1 for race and weights=0 for the other variables

(RMD_race), and RMD with weights=10 for race and BMI, 5 for statin use,

and 1 otherwise (RMD_race+). 10000 simulations run for each method.

Standard error for N patients ≤0.3, and ≤0.1 for all other variables.

reweighted mahalanobis distance matching 151

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Pharmacoepidemiology and Drug Safety, 2012; 21(S2): 148–154

DOI: 10.1002/pds

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other variables (RMD_race); and RMD with weights=

10 for race and BMI, 5 for statin use, and 1 otherwise

(RMD_race+). The AMD_90 for simple randomiza-

tion was 26.8%. In other words, simple randomization

yielded a 10% chance of a study having a mean

difference of at least 26.8% in the percent of nonwhite

patients between study arms. MD showed a small

improvement with 26.1%. RMD_race dramatically re-

duced the AMD_90 to 4.8% and RMD_race+ reduced

it to 6.4%. The benefit of MD was primarily seen in

BMI, reducing the AMD_90 to 0.7 from simple rando-

mization’s 1.3 and 1.4 for RMD_race. RMD_race+

had comparable balance on BMI at 0.8. Compared

with the MD, RMD_race+ achieved dramatically

better balance on race and cluster size at a small cost

of slightly less balance on the other variables.

Example 2

Where example 1 preselected 12 from the 90 sites, MD

and RMD can select an optimal subset via the use of

phantoms or chameleons. The balance for sites selected

via four different methods is presented in Table 3. For

comparison,500,000 setsof12 wereselectedviasimple

random samples. On average, simple randomization

performed similarly to how it did in example 1. By

utilizing 78 phantoms to optimally select 12 sites, MD

yielded slightly better balance on the preselected sites

than RMD_race+. In this setting RMD_race+ did not

balancethenumberofpatientspersite;thus,thevariable

“N patients” was also given a weight of 5, yielding

RMD_race++. As in example 1, RMD_race++ outper-

formed MD in terms of balancing the difficult variables

ofraceandnumberofpatientspersite,whileperforming

almost as well as on the other variables. When selecting

anoptimalsubsetusingchameleons,setwithathreshold

equal to the 0.2 percentile of the distance matrix, the

method selected 16 sites with performance similar to

the set selected via phantoms.

Example 3

The performance of simple randomization will improve

as the number of sites increases. When randomizing all

90 sites, simple randomization has an AMD_90 of 7.2

for race and 60.5 for patients per site. With as many as

20% of the covariate values missing, RMD_race++

outperformed simple randomization on all variables,

especially race and patients per site.

DISCUSSION

The RMD provides a user-friendly method for

researchers to incorporate into the matching process

their clinical knowledge and the relative difficulty of

balancing important covariates. Greevy et al. have

shown that matching prior to randomization outper-

forms unmatched randomization in non-clustered

randomized controlled trials (RCTs),1and Imai et al.

have shown its benefits in clustered RCTs.2Zhang

and Small have shown that optimal nonbipartite

matching using an MD may outperform other match-

ing methods,4and the current paper shows that

the RMD may yield results superior to the MD when

the perceived quality of the matching depends on the

relative clinical importance of the variables. Moreover,

the RMD may account for missing data in a sophisti-

cated, yet highly automated, process. Table 4 shows

benefits over simple randomization with up to 20% of

the data missing.

Analysis for MPCR designs is a growing area of

research. Recently, Imai et al. introduced a har-

monic mean estimator for which the study infer-

ences can be justified by the study design alone.2

Zhang, Traskin, and Small have developed a robust

test statistic for MPCR trials that outperforms linear

mixed models for heavy-tailed distributions and

performs nearly as well in the special case where

the mixed model assumptions are true.16The statis-

tic may optionally include covariate adjustment

while still relying on the study design to justify

inferencesviatheapproachdevelopedbyRosenbaum.17

Many studies will include the covariates used in the

matching as variables in the analysis model. To avoid

potential bias, we discourage including the indicators

for missingness that are created by RMD.18In addition,

Table 3.

12 sites selected from 90 via four methods

90th percentile of the absolute mean difference (AMD_90) for

Variable

Simple

random

sample

MD with

phantoms

RMD_race

++ with

phantoms

RMD_race

++ with

chameleons

Nonwhite (%)

Female (%)

On statins (%)

Systolic

BP (mmHg)

BMI

A1c (%)

N patients

23.1

2.1

5.4

2.1

3.9

0.9

0.4

0.5

1.7

0.9

0.8

0.7

2.0

1.1

0.7

0.8

0.7

0.2

0.1

0.1

28.8

0.1

0.1

17.3

0.1

0.1

17.0 177.0

A total of 12 sites were selected via simple random samples for the simple

randomization, 12 sites optimally selected using phantoms for MD and

RMD_race++, and 16 sites selected using chameleons with threshold equal

to the 0.2 percentile of the distance matrix. Note that a threshold equal to the

0.17 percentile yields the same 12 sites as using phantoms. RMD_race++ has

weights=10forraceandBMI,5forstatinuseandNpatients,and1otherwise.

A total of 500000 simulations were performed for simple randomization and

10000 for each of the other methods. Standard error for N patients ≤0.3, and

≤0.1 for all other variables.

r. a. greevy et al.152

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DOI: 10.1002/pds

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the form of the covariates used in the model may also

vary from the form used in the matching; for example,

the model may benefit from the transformation of a

covariate to account for a nonlinear association with

the outcome.

In situations where the potential for imbalance was

low, for example, low variability in the covariates, the

benefit of up-weighting variables purely for their clini-

cal importance was small. The choice of weights was

best made with a combination of clinical knowledge

andexaminationofthepre-randomizationcovariatedata.

Users may influence the impact of individual variables

on the MD by dropping variables entirely or applying

nonlinear transformations to them, for example, rank

or log. When matching observational data with highly

non-normal distributions, Rosenbaum recommends

using a rank-based MD (MD_rank).19The MD_rank is

equivalent to an RMD that uses a rank transformation

of each variable (i.e., replacing a variable with the rank

ordering of the variable and using average ranks for ties)

and uses weights equal to the standard deviation of the

rank-transformed variables divided by what the standard

deviationoftherankswouldbeiftherewerenoties.This

serves to down-weight variables with ties. Ranking is

particularly useful for covariates that have outliers, and

it is often desirable to down-weight a variable with

numerous ties, for example, a binary variable such as

“teaching hospital Y/N.” The clinical importance of a

particular variable may discourage researchers from

down-weighting, or even to up-weight, that variable.

Because the aggregated measures used in the examples

have sufficient precision to prevent any ties, the

MD_rank is equivalent to the MD on rank-transformed

data.Theeffectsoftheranktransformationarepresented

in online Supplementary Tables 1 and 2.

For missing data, a user may wish to use an impu-

tation method that is more sophisticated than the

highly automated procedure used here. The Web

application and the R package allow more advanced

users to use a covariate matrix that incorporates their

customized changes or use their own customized dis-

tance matrix. The methods applied here provide a

straightforward, easily implemented method for

creating optimally matched clusters for randomiza-

tion in an MPCR study.

CONFLICTS OF INTEREST

The authors of this research are responsible for its

content. Statements here should not be construed as

endorsement by the Agency for Healthcare Research

and Quality or the US Department of Health and

Human Services. There were no conflicts of interest

with this research.

ACKNOWLEDGEMENTS

ThisprojectwasfundedinpartbytheAgencyforHealth-

careResearchandQuality,USDepartmentofHealthand

Human Services, Contract No. HHSA2902010000161,

as part of the Developing Evidence to Inform Decisions

about Effectiveness (DEcIDE 2) program. The work of

R.G. was supported in part by a grant from the National

InstitutesofHealth,P60AR056116.TheworkofA.M.H.

was supported in full by the Career Development

Program from the Department of Veterans Affairs CDA

(2-031-09S) from CSR&D.

Supporting Information

Additional supporting information may be found in the

online version of this article:

Supplementary Table 1: 90th percentile of the

absolute mean difference (AMD_90) comparing three

rank-based methods (12 preselected sites)

Supplementary Table 2: 90th percentile of the

absolutemeandifference(AMD_90)for12sitesselected

from 90 comparing three rank-based methods

Please note: Wiley-Blackwellare not responsiblefor the

content or functionality of any supporting materials

suppliedbytheauthors.Anyqueries(otherthanmissing

material) should be directed to the corresponding author

for the article.

Table 4.

RMD_race++ with three levels of induced missingness

90th percentile of the absolute mean difference (AMD_90) for

Percent

missing:5% 10%20%

VariableSimple

RMD_race

++

RMD_race

++

RMD_race

++

Nonwhite (%)

Female (%)

On statins (%)

Systolic BP

(mmHg)

BMI

A1c (%)

N patients

7.2

0.7

1.8

0.8

3.2

0.6

1.2

0.8

3.8

0.6

1.3

0.8

4.9

0.7

1.5

0.8

0.3

0.1

60.5

0.1

0.1

33.8

0.1

0.1

37.4

0.2

0.1

44.5

A total of 90 sites matched. Balance is for the true covariate values without

missingness. Missingness was induced completely at random (MCAR)

over all seven covariates. RMD_race++ has weights=10 for race and

BMI, 5 for statin use and N patients, and 1 otherwise. A total of 10000

simulations run were performed for each method. Standard errors for N

patients are ≤0.7, and ≤0.1 for all other variables.

reweighted mahalanobis distance matching153

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Pharmacoepidemiology and Drug Safety, 2012; 21(S2): 148–154

DOI: 10.1002/pds

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