Assessing Balance

Introduction

Covariate balance is the degree to which the distribution of covariates is similar across levels of the treatment. It has three main roles in causal effect estimation using matching: 1) as a target to optimize with matching, 2) as a method of assessing the quality of the resulting matches, and 3) as evidence to an audience that the estimated effect is close to the true effect. When covariate balance is achieved, the resulting effect estimate is less sensitive to model misspecification and ideally close to true treatment effect. The benefit of randomization is that covariate balance is achieved automatically (in expectation), which is why unadjusted effects estimated from randomized trial data (in the absence of drop-out) can be validly interpreted as causal effects. When using matching to recover causal effect estimates form observational data, balance is not guaranteed and must be assessed.

This document provides instructions for assessing and reporting covariate balance as part of a matching analysis. The tools available in MatchIt for balance assessment should be used during the process of selecting a good matching scheme and ensuring that the chosen scheme is adequate. These tools implement the recommendations of Ho et al. (2007) and others for assessing balance.

In addition to the tools available in MatchIt, the cobalt package has a suite of functions designed to assess and display balance and is directly compatible with MatchIt objects. cobalt has extensive documentation, but we describe some of its functionality here as a complement to the tools in MatchIt.

The structure of this document is as follows: first, we describe some of the recommendations for balance checking and their rationale; next, we describe the tools for assessing balance present in MatchIt and display their use in evaluating several matching schemes; finally; we briefly describe some of the functionality in cobalt to extend that in MatchIt.

Recommendations for Balance Assessment

Assessing balance involves assessing whether the distributions of covariates are similar between the treated and control groups. Balance is typically assessed by examining univariate balance summary statistics for each covariate, though more complicated methods exist for assessing joint distributional balance as well. Visual depictions of distributional balance can be a helpful complement to numerical summaries, especially for hard to balance and prognostically important covariates.

Many recommendations for balance assessment have been described in the methodological literature. Unfortunately, there is no single best way to assess balance or to weigh balance summary statistics because the degree and form of balance that will yield the least bias in an effect estimate depends on unknown qualities of the outcome data-generating model. Nonetheless, there are a number of valuable recommendations that can be implemented to ensure matching is successful at eliminating or reducing bias. We review some of these here.

Common recommendations for assessing balance include the following:

Standardized mean differences. The standardized mean difference (SMD) is the difference in the means of each covariate between treatment groups standardized by a standardization factor so that it is on the same scale for all covariates. The standardization factor is typically the standard deviation of the covariate in the treated group when targeting the ATT or the pooled standard deviation across both groups when targeting the ATE. The standardization factor should be the same before and after matching to ensure changes in the mean difference are not confounded by changes in the standard deviation of the covariate. SMDs close to zero indicate good balance. Several recommended thresholds have been published in the literature; we recommend .1 and .05 for prognostically important covariates. Higher values may be acceptable when using covariate adjustment in the matched sample. In addition to computing SMDs on the covariates themselves, it is important to compute them on squares, cubes, and higher exponents as well as interactions between covariates. Several empirical studies have examined the appropriateness for using SMDs in balance assessment, including Belitser et al. (2011), Ali et al. (2014), and Stuart, Lee, and Leacy (2013); in general, there is often a high correlation between the mean or maximum absolute SMD and the degree of bias in the treatment effect.
Variance Ratios. The variance ratio is the ratio of the variance of a covariate in one group to that in the other. Variance ratios close to 1 indicate good balance because they imply the variances of the samples are similar (Austin 2009).
Empirical CDF Statistics. Statistics related to the difference in the empirical cumulative distribution functions (eCDFs) of each covariate between groups allow assessment of imbalance across the entire covariate distribution of that covariate rather than just its mean or variance. The maximum eCDF difference, also known as the Kolmogorov-Smirnov statistic, is sometimes recommended as a useful supplement to SMDs for assessing balance (Austin and Stuart 2015) and is often used as a criterion to use in propensity score methods that attempt to optimize balance (e.g., McCaffrey, Ridgeway, and Morral 2004; Diamond and Sekhon 2013). Although the mean eCDF difference has not been as well studied, it provides a summary of imbalance that may be missed by relying solely on the maximum difference.
Visual Diagnostics. Visual diagnostics such as eCDF plots, empirical quantile-quantile (eQQ) plots, and kernel density plots can be used to see exactly how the covariate distributions differ from each other, i.e., where in the distribution the greatest imbalances are (Ho et al. 2007; Austin 2009). This can help to figure out how to tailor a matching method to target imbalance in a specific region of the covariate distribution.
Prognostic scores. The prognostic score is an estimate of the potential outcome under control for each unit (Hansen 2008). Balance on the prognostic score has been shown to be highly correlated with bias in the effect estimate, making it a useful tool in balance assessment (Stuart, Lee, and Leacy 2013). Estimating the prognostic score requires having access to the outcome data, and using it may be seen as violating the principle of separating the design and analysis stages of a matching analysis (Rubin 2001). However, because only the outcome values from the control group are required to use the prognostic score, some separation is maintained.

Several multivariate statistics exist that summarize balance across the entire joint covariate distribution. These can be functions of the above measures, like the mean or maximum absolute SMD or the generalized weighted distance [GWD; Franklin et al. (2014)], which is the sum of SMDs for the covariates and their squares and interactions, or separate statistics that measure quantities that abstract away from the distribution of individual covariates, like the L1 distance (Iacus, King, and Porro 2011), cross-match test (Heller, Rosenbaum, and Small 2010), or energy distance (Huling and Mak 2020).

Balance on the propensity score has often been considered a useful measure of balance, but we do not necessarily recommend it except as a supplement to balance on the covariates. Propensity score balance will generally be good with any matching method regardless of the covariate balancing potential of the propensity score, so a balanced propensity score does not imply balanced covariates (Austin 2009). Similarly, it may happen that covariates may be well balanced even if the propensity score is not balanced, such as when covariates are prioritized above the propensity score in the matching specification (e.g., with genetic matching). Given these observations, the propensity score should not be relied upon for assessing covariate balance. Simulation studies by Stuart, Lee, and Leacy (2013) provide evidence for this recommendation against relying on propensity score balance.

There has been some debate about the use of hypothesis tests, such as t-tests or Kolmogorov-Smirnov tests, for assessing covariate balance. The idea is that balance tests test the null hypothesis that the matched sample has equivalent balance to a randomized experiment. There are several problems with balance tests, described by Ho et al. (2007) and Imai, King, and Stuart (2008): 1) balance is a property of the sample, not a of a population from which the sample was drawn; 2) the power of balance tests depends on the sample size, which changes during matching even if balance does not change; and 3) the use of hypothesis tests implies a uniform decision criterion for rejecting the null hypothesis (e.g., p-value less than .05, potentially with corrections for multiple comparisons), when balance should be improved without limit. MatchIt does not report any balance tests or p-values, instead relying on the descriptive statistics described above.

Recommendations for Balance Reporting

A variety of methods should be used when assessing balance to try to find an optimal matched set that will ideally yield a low-error estimate of the desired effect. However, reporting every balance statistic or plot in a research report or publication can be burdensome and unnecessary. That said, it is critical to report balance to demonstrate to readers that the resulting estimate is approximately unbiased and relies little on extrapolation or correct outcome model specification. We recommend the following in reporting balance in a matching analysis:

Report SMDs before and after matching for each covariate, any prognostically important interactions between covariates, and the prognostic score; this can be reported in a table or in a Love plot.
Report summaries of balance for other statistics, e.g., the largest mean and maximum eCDF difference among the covariates and the largest SMD among squares, cubes, and interactions of the covariates.

MatchIt provides tools for calculating each of these statistics so they can be reported with ease in a manuscript or report.

Assessing Balance with `MatchIt`

MatchIt contains several tools to assess balance numerically and graphically. The primary balance assessment function is summary.matchit(), which is called when using summary() on a MatchIt object and produces several tables of balance statistics before and after matching. plot.summary.matchit() generates a Love plot using R’s base graphics system containing the standardized mean differences resulting from a call to summary.matchit() and provides a nice way to display balance visually for inclusion in an article or report. plot.matchit() generates several plots that display different elements of covariate balance, including propensity score overlap and distribution plots of the covariates. These functions together form a suite that can be used to assess and report balance in a variety of ways.

To demonstrate MatchIt’s balance assessment capabilities, we will use the Lalonde data included in MatchIt and used in vignette("MatchIt"). We will perform 1:1 nearest neighbor matching with replacement on the propensity score, though the functionality is identical across all matching methods except propensity score subclassification, which we illustrate at the end.

library("MatchIt")
data("lalonde", package = "MatchIt")

#1:1 NN matching w/ replacement on a logistic regression PS
m.out <- matchit(treat ~ age + educ + race + married + 
                   nodegree + re74 + re75, data = lalonde,
                 replace = TRUE)
m.out

## A `matchit` object
##  - method: 1:1 nearest neighbor matching with replacement
##  - distance: Propensity score
##              - estimated with logistic regression
##  - number of obs.: 614 (original), 267 (matched)
##  - target estimand: ATT
##  - covariates: age, educ, race, married, nodegree, re74, re75

`summary.matchit()`

When summary() is called on a matchit object, several tables of information are displayed. These include balance statistics for each covariate before matching, balance statistics for each covariate after matching, the percent reduction in imbalance after matching, and the sample sizes before and after matching. summary.matchit() has four additional arguments that control how balance is computed:

interactions controls whether balance statistics for all squares and pairwise interactions of covariates are to be displayed in addition to the covariates. The default is FALSE, and setting to TRUE can make the output massive when many covariates are present, but it is important to ensure no important interactions remain imbalanced.
addlvariables allows for balance to be assessed on variables other than those inside the matchit object. For example, if the distance between units only relied on a subset of covariates but balance needed to be achieved on all covariates, addlvariables could be used to supply these additional covariates. In addition to adding other variables, addlvariables can be used to request balance on specific functions of the covariates already in the matchit object, such as polynomial terms or interactions. The input to addlvariables can be a one-sided formula with the covariates and any desired transformations thereof on the right hand side, just like a model formula (e.g., addlvariables = ~ X1 + X2 + I(X1^2) would request balance on X1, X2, and the square of X1). Additional variables supplied to addlvariables but not present in the matchit object can be supplied as a data frame using the data argument.
standardize controls whether standardized or unstandardized statistics are to displayed. Standardized statistics include the standardized mean difference and eCDF statistics; unstandardized statistics include the raw difference in means and eQQ plot statistics. (Regardless, the variance ratio will always be displayed.). The default is TRUE for standardized statistics, which are more common to report because they are all on the same scale regardless of the scale of the covariates¹.
pair.dist controls whether within-pair distances should be computed and displayed. These reflect the average distance between units within the same pair, standardized or unstandardized according to the argument to standardize. The default is TRUE. With full matching, exact matching, coarsened exact matching, and propensity score subclassification, computing pair distances can take a long time, and so it may be beneficial to set to FALSE in these cases.

In addition, the arguments un (default: TRUE) and improvement (default: FALSE) control whether balance prior to matching should be displayed and whether the percent balance improvement after matching should be displayed. These can be set to FALSE to reduce the output.

Below, we call summary.matchit() with addlvariables to display balance on covariates and a few functions of them in the matched sample. In particular, we request balance on the square of age, the variables representing whether re74 and re75 were equal to 0, and the interaction between educ and race.

summary(m.out, addlvariables = ~ I(age^2) + I(re74==0) + 
          I(re75==0) + educ:race)

## 
## Call:
## matchit(formula = treat ~ age + educ + race + married + nodegree + 
##     re74 + re75, data = lalonde, replace = TRUE)
## 
## Summary of Balance for All Data:
##                  Means Treated Means Control Std. Mean Diff. Var. Ratio eCDF Mean eCDF Max
## distance                 0.577         0.182           1.794      0.921     0.377    0.644
## age                     25.816        28.030          -0.309      0.440     0.081    0.158
## educ                    10.346        10.235           0.055      0.496     0.035    0.111
## raceblack                0.843         0.203           1.762          .     0.640    0.640
## racehispan               0.059         0.142          -0.350          .     0.083    0.083
## racewhite                0.097         0.655          -1.882          .     0.558    0.558
## married                  0.189         0.513          -0.826          .     0.324    0.324
## nodegree                 0.708         0.597           0.245          .     0.111    0.111
## re74                  2095.574      5619.237          -0.721      0.518     0.225    0.447
## re75                  1532.055      2466.484          -0.290      0.956     0.134    0.288
## I(age^2)               717.395       901.779          -0.428      0.363     0.081    0.158
## I(re74 == 0)TRUE         0.708         0.261           0.983          .     0.447    0.447
## I(re75 == 0)TRUE         0.600         0.312           0.587          .     0.288    0.288
## educ:raceblack           8.697         2.047           1.580      0.980     0.354    0.645
## educ:racehispan          0.578         1.263          -0.294      0.487     0.046    0.078
## educ:racewhite           1.070         6.925          -1.767      0.365     0.279    0.555
## 
## Summary of Balance for Matched Data:
##                  Means Treated Means Control Std. Mean Diff. Var. Ratio eCDF Mean eCDF Max Std. Pair Dist.
## distance                 0.577         0.576           0.004      0.992     0.003    0.049           0.013
## age                     25.816        24.103           0.239      0.557     0.077    0.341           1.262
## educ                    10.346        10.378          -0.016      0.577     0.022    0.059           1.086
## raceblack                0.843         0.838           0.015          .     0.005    0.005           0.045
## racehispan               0.059         0.065          -0.023          .     0.005    0.005           0.297
## racewhite                0.097         0.097           0.000          .     0.000    0.000           0.054
## married                  0.189         0.130           0.152          .     0.059    0.059           0.511
## nodegree                 0.708         0.703           0.012          .     0.005    0.005           0.868
## re74                  2095.574      2336.463          -0.049      1.036     0.041    0.216           0.609
## re75                  1532.055      1503.929           0.009      2.129     0.068    0.238           0.650
## I(age^2)               717.395       670.946           0.108      0.510     0.077    0.341           1.196
## I(re74 == 0)TRUE         0.708         0.492           0.476          .     0.216    0.216           0.975
## I(re75 == 0)TRUE         0.600         0.362           0.485          .     0.238    0.238           1.037
## educ:raceblack           8.697         8.589           0.026      0.869     0.024    0.054           0.468
## educ:racehispan          0.578         0.638          -0.026      0.827     0.007    0.022           0.336
## educ:racewhite           1.070         1.151          -0.024      0.846     0.005    0.022           0.220
## 
## Sample Sizes:
##               Control Treated
## All            429.       185
## Matched (ESS)   46.31     185
## Matched         82.       185
## Unmatched      347.         0
## Discarded        0.         0

Let’s examine the output in detail. The first table (Summary of Balance for All Data) provides balance in the sample prior to matching. The included statistics are the mean of the covariates in the treated group (Means Treated), the mean of the covariate in the control group (Means Control), the SMDs (Std. Mean Diff.), the variance ratio (Var. Ratio), the average distance between the eCDFs of the covariate across the groups (eCDF Mean), and the largest distance between the eCDFs (eCDF Max). Setting un = FALSE would have suppressed the creation of this table.

The second table (Summary of Balance for Matched Data) contains all the same statistics in the matched sample. Because we implicitly request pair distance, an additional column for standardized pair distances (Std. Pair Dist.) is displayed.

The final table (Sample Sizes) contains the sizes of the samples before (All) and after (Matched) matching, as well as the number of units left unmatched (Unmatched) and the number of units dropped due to a common support restriction (Discarded).

The SMDs are computed as the mean difference divided by a standardization factor computed in the unmatched sample. An absolute SMD close to 0 indicates good balance; although a number of recommendations for acceptable values have appeared in the literature, we recommend absolute values less than .1 and less than .05 for potentially prognostically important variables.

The variance ratios are computed as the ratio of the variance of the treated group to that of the control group for each covariate. Variance ratios are not computed for binary covariates because they are a function of the prevalence in each group, which is captured in the mean difference and eCDF statistics. A variance ratio close to 1 indicates good balance; a commonly used recommendation is for variance ratios to be between .5 and 2.

The eCDF statistics correspond to the difference in the overall distributions of the covariates between the treatment groups. The values of both statistics range from 0 to 1, with values closer to zero indicating better balance. There are no specific recommendations for the values these statistics should take, though notably high values may indicate imbalance on higher moments of the covariates. The eQQ statistics produced when standardize = FALSE are interpreted similarly but are on the scale of the covariate.

All these statistics should be considered together. Imbalance as measured by any of them may indicate a potential failure of the matching scheme to achieve distributional balance.

`plot.summary.matchit()`

A Love plot is a clean way to visually summarize balance. Using plot on the output of a call to summary() on a matchit object produces a Love plot of the standardized mean differences. plot.summary.matchit() has several additional arguments that can be used to customize the plot.

abs controls whether standardized mean difference should be displayed in absolute value or not. Default is TRUE.
var.order controls how the variables are ordered on the y-axis. The options are "data" (the default), which orders the variables as they appear the in the summary.matchit() output; "unmatched", which orders the variables based on their standardized mean differences before matching; "matched", which orders the variables based on their standardized mean differences after matching; and "alphabetical", which orders the variables alphabetically. Using "unmatched" tends to result in attractive plots and ensures the legend doesn’t overlap with points in its default position.
threshold controls where vertical lines indicating chosen thresholds should appear on the x-axis. Should be a numeric vector. The default is c(.1, .05), which display vertical lines at .1 and .05 standardized mean difference units.
position controls the position of the legend. The default is "bottomright", which puts the legend in the bottom right corner of the plot, and any keyword value available to supplied to x in legend() is allowed.

Below we create a Love plot of the covariates.

m.sum <- summary(m.out, addlvariables = ~ I(age^2) + I(re74==0) + 
                   I(re75==0) + educ:race)
plot(m.sum, var.order = "unmatched")

A love plot with most matched dots below the threshold lines, indicaitng good balance after matching, in contrast to the unmatched dots far from the treshold lines, indicating poor balance before matching.

From this plot it is clear to see that balance was quite poor prior to matching, but full matching improved balance on all covariates, and most within a threshold of .1. To make the variable names cleaner, the original variables should be renamed prior to matching. cobalt provides many additional options to generate and customize Love plots using the love.plot() function and should be used if a plot beyond what is available with plot.summary.matchit() is desired.

`plot.matchit()`

In addition to numeric summaries of balance, MatchIt offers graphical summaries as well using plot.matchit() (i.e., using plot() on a matchit object). We can create eQQ plots, eCDF plots, or density plots of the covariates and histograms or jitter plots of the propensity score. The covariate plots can provide a summary of the balance of the full marginal distribution of a covariate beyond just the mean and variance.

plot.matchit() has a few arguments to customize the output:

type corresponds to the type of plot desired. Options include "qq" for eQQ plots (the default), "ecdf" for eCDF plots, "density" for density plots (or bar plots for categorical variables), "jitter" for jitter plots, and "histogram" for histograms.
interactive controls whether the plot is interactive or not. For eQQ, eCDF, and density plots, this allows us to control when the next page of covariates is to be displayed since only three can appear at a time. For jitter plots, this can allow us to select individual units with extreme values for further inspection. The default is TRUE.
which.xs is used to specify for which covariates to display balance in eQQ, eCDF, and density plots. The default is to display balance on all, but we can request balance just on a specific subset. If three or fewer are requested, interactive is ignored. The argument can be supplied as a one-sided formula with the variables of interest on the right or a character vector containing the names of the desired variables. If any variables are not in the matchit object, a data argument can be supplied with a data set containing the named variables.

Below, we demonstrate the eQQ plot:

#eQQ plot
plot(m.out, type = "qq", which.xs = ~age + nodegree + re74)

eQQ plots of age, nodegree, and re74 in the unmatched and matched samples.

The y-axis displays the each value of the covariate for the treated units, and the x-axis displays

Assessing Balance

Noah Greifer

2025-05-29

Introduction

Recommendations for Balance Assessment

Recommendations for Balance Reporting

Assessing Balance with `MatchIt`

`summary.matchit()`

`plot.summary.matchit()`

`plot.matchit()`

Assessing Balance

Noah Greifer

2025-05-29

Introduction

Recommendations for Balance Assessment

Recommendations for Balance Reporting

Assessing Balance with MatchIt

summary.matchit()

plot.summary.matchit()

plot.matchit()

Assessing Balance with `MatchIt`

`summary.matchit()`

`plot.summary.matchit()`

`plot.matchit()`