Covariate Balance Tables and Plots: A Guide to the cobalt Package

Citing cobalt

When using cobalt, please cite your use of it along with the conditioning package used. The full APA reference for cobalt is the following:

Greifer, N. (2022). cobalt: Covariate Balance Tables and Plots. R package version 4.4.1.

For example, if you use Matching for propensity score estimation and matching and cobalt for balance assessment and/or reporting, a possible citation might go as follows:

Matching was performed using the Matching package (Sekhon, 2011), and covariate balance was assessed using cobalt (Greifer, 2022), both in R (R Core Team, 2022).

Visual clarity

cobalt presents one table in its balance output, and it contains all the information required to assess balance. twang and CBPS present two tables, MatchIt presents three tables, and Matching presents as many tables as there are covariates. Although each of these tables contains valuable information, the bal.tab() function in cobalt allows for a quick and easy search for the information desired, which is often a single column containing a balance statistic (such as the standardized mean difference) for the adjusted sample.

Useful summaries

Although a thorough balance assessment requires examining the balance of each covariate individually, cobalt’s bal.tab() function can also produce quick balance summaries that can aid in model selection when there are many covariates or higher order terms to examine. These summaries include the proportion of covariates that have met a user-specified threshold for balance and the covariate with the highest degree of imbalance, two values that have been shown to be effective in diagnosing imbalance and potential bias (Stuart, Lee, and Leacy 2013).

One tool to rule them all

Because there is no a priori way to know which conditioning method will work best for a given sample, users should try several methods, and these methods are spread across various packages; for example, full matching is available only in MatchIt and optmatch, generalized boosted modeling only in twang, covariate balancing propensity score weighting only in CBPS, genetic matching only in MatchIt and Matching, and entropy balancing only in ebal². If a user wants to compare these methods on their ability to generate balance in the sample, they cannot do so on the same metrics and with the same output. Each package computes balance statistics differently (if at all), and the relevant balance measures are in different places in each package. By using cobalt to assess balance across packages, users can be sure they are using a single, equivalent balance metric across methods, and the relevant balance statistics will be in the same place and computed the same way regardless of the conditioning package used.

Flexibility

cobalt gives users choice in which statistics are presented and how they are calculated, but intelligently uses defaults that are in line with the goals of unified balance assessment and with the data available. Rather than displaying all values calculated, bal.tab() only displays what the user wants; at a bare minimum, the standardized mean difference for each covariate is displayed, which is traditionally considered sufficient for model selection and justification in preprocessing analysis for binary treatments. Even if the user doesn’t want other values displayed, they are all still calculated, and thus available for use in programming (though this can be disabled for increased speed).

Pretty plots

The main conditioning packages produce plots that can be useful in assessing balance, summarizing balance, and understanding the intricacies of the conditioning method for which simple text would be insufficient. Many of these plots are unique to each package, and cobalt has not attempted to replace or replicate them. For other plots, though, cobalt uses ggplot2 to present clean, clear, customizable, and high-quality displays for balance assessment and presentation. The two included plotting functions are bal.plot(), which generates plots of the distributions of covariates and treatment levels so that more complete distributional balance can be assessed beyond numerical summaries, and love.plot(), which generates a plot summarizing covariate balance before and after conditioning, popularized by Dr. Thomas E. Love. Because these plots use ggplot2 as their base, users familiar with ggplot2 can customize various elements of the plots for use in publications or presentations.

Unique features

There are unique features in cobalt that do not exist in any other package. These include the handling of clustered and grouped data and the handling of data generated with multiple imputation. These more advanced uses of cobalt are described in detail in the accompanying Appendix 2. In addition, cobalt includes tools for handling data sets with continuous and multi-category treatments. Data sets with longitudinal treatments, where time-varying confounding may be an issue, can be handled as well; these uses are described in the accompanying Appendix 3.

Utilities

In addition to its main functions, cobalt contains several utility functions, which include f.build(), splitfactor() and unsplitfactor(), and get.w(). These are meant to reduce the typing and programming burden that often accompany the use of R with a diverse set of packages. To simplify this vignette, descriptions of these functions are in Appendix 1. To understand the code in this vignette, you should be aware of f.build(), which creates a formula from its inputs, and get.w() which extracts weights from its input.

bal.tab()

bal.tab() is the primary function of cobalt. It produces balance tables for the objects given as inputs. The balance tables can be customized with a variety of inputs, which affect both calculation and presentation of values. It performs similar functions to summary() in MatchIt; bal.table(), summary(), and dx.wts() in twang; MatchBalance() and summary() in Matching; balance() in CBPS; and summarize() in sbw. It can be seen as a replacement or a supplement to these functions.

For more help using bal.tab(), see ?bal.tab in R, which contains information on how certain values are calculated and links to the help files for the bal.tab() methods that integrate with the above packages.

For simplicity, the description of the use of bal.tab() will be most complete in its use without any other package. The demonstration will display bal.tab()’s many options, several of which differ based on with which package, if any, bal.tab() is used. The other demonstrations will be minimal, highlighting how to use bal.tab() effectively with MatchIt and WeightIt, but not detailing all its possible options with these packages, to avoid redundancy. The use of bal.tab() with other packages is described in the accompanying Appendix 1, “Using cobalt with Other Preprocessing Packages”.

data("lalonde", package = "cobalt") #If not yet loaded
covs <- subset(lalonde, select = -c(treat, re78, nodegree, married))

# Generating ATT weights as specified in Austin (2011)
lalonde$p.score <- glm(treat ~ age + educ + race + re74 + re75,
                       data = lalonde,
                       family = "binomial")$fitted.values
lalonde$att.weights <- with(lalonde, treat + (1-treat)*p.score/(1-p.score))

bal.tab(covs, treat = lalonde$treat, weights = lalonde$att.weights)

## Balance Measures
##                Type Diff.Adj
## age         Contin.   0.1112
## educ        Contin.  -0.0641
## race_black   Binary  -0.0044
## race_hispan  Binary   0.0016
## race_white   Binary   0.0028
## re74        Contin.  -0.0039
## re75        Contin.  -0.0428
## 
## Effective sample sizes
##            Control Treated
## Unadjusted   429.      185
## Adjusted     108.2     185

bal.tab(treat ~ covs, data = lalonde,
        weights = "att.weights",
        distance = "p.score")

## Balance Measures
##                 Type Diff.Adj
## p.score     Distance  -0.0397
## age          Contin.   0.1112
## educ         Contin.  -0.0641
## race_black    Binary  -0.0044
## race_hispan   Binary   0.0016
## race_white    Binary   0.0028
## re74         Contin.  -0.0039
## re75         Contin.  -0.0428
## 
## Effective sample sizes
##            Control Treated
## Unadjusted   429.      185
## Adjusted     108.2     185

bal.tab(treat ~ covs, data = lalonde,
        weights = "att.weights",
        binary = "std", continuous = "std")

## Balance Measures
##                Type Diff.Adj
## age         Contin.   0.1112
## educ        Contin.  -0.0641
## race_black   Binary  -0.0120
## race_hispan  Binary   0.0068
## race_white   Binary   0.0093
## re74        Contin.  -0.0039
## re75        Contin.  -0.0428
## 
## Effective sample sizes
##            Control Treated
## Unadjusted   429.      185
## Adjusted     108.2     185

# Balance on all covariates in data set, including interactions and squares
bal.tab(treat ~ covs, data = lalonde,
        weights = "att.weights",
        addl = ~ nodegree + married,
        int = TRUE, poly = 2)

## Balance Measures
##                             Type Diff.Adj
## age                      Contin.   0.1112
## educ                     Contin.  -0.0641
## race_black                Binary  -0.0044
## race_hispan               Binary   0.0016
## race_white                Binary   0.0028
## re74                     Contin.  -0.0039
## re75                     Contin.  -0.0428
## nodegree                  Binary   0.1151
## married                   Binary  -0.0938
## age²                     Contin.  -0.0194
## educ²                    Contin.  -0.1353
## re74²                    Contin.   0.0675
## re75²                    Contin.   0.0196
## age * educ               Contin.   0.0950
## age * race_black         Contin.   0.0741
## age * race_hispan        Contin.  -0.0096
## age * race_white         Contin.  -0.0013
## age * re74               Contin.  -0.0498
## age * re75               Contin.  -0.0144
## age * nodegree_0         Contin.  -0.1950
## age * nodegree_1         Contin.   0.2507
## age * married_0          Contin.   0.3612
## age * married_1          Contin.  -0.2843
## educ * race_black        Contin.  -0.0441
## educ * race_hispan       Contin.   0.0122
## educ * race_white        Contin.   0.0085
## educ * re74              Contin.  -0.0149
## educ * re75              Contin.  -0.0834
## educ * nodegree_0        Contin.  -0.2655
## educ * nodegree_1        Contin.   0.3051
## educ * married_0         Contin.   0.2016
## educ * married_1         Contin.  -0.2453
## race_black * re74        Contin.   0.0477
## race_black * re75        Contin.  -0.0297
## race_black * nodegree_0   Binary  -0.1110
## race_black * nodegree_1   Binary   0.1067
## race_black * married_0    Binary   0.0535
## race_black * married_1    Binary  -0.0579
## race_hispan * re74       Contin.  -0.0129
## race_hispan * re75       Contin.   0.0258
## race_hispan * nodegree_0  Binary  -0.0068
## race_hispan * nodegree_1  Binary   0.0084
## race_hispan * married_0   Binary   0.0076
## race_hispan * married_1   Binary  -0.0060
## race_white * re74        Contin.  -0.2488
## race_white * re75        Contin.  -0.0926
## race_white * nodegree_0   Binary   0.0027
## race_white * nodegree_1   Binary   0.0001
## race_white * married_0    Binary   0.0327
## race_white * married_1    Binary  -0.0300
## re74 * re75              Contin.   0.0636
## re74 * nodegree_0        Contin.  -0.0464
## re74 * nodegree_1        Contin.   0.0467
## re74 * married_0         Contin.   0.1108
## re74 * married_1         Contin.  -0.1122
## re75 * nodegree_0        Contin.  -0.3656
## re75 * nodegree_1        Contin.   0.1487
## re75 * married_0         Contin.   0.1093
## re75 * married_1         Contin.  -0.1177
## nodegree_0 * married_0    Binary  -0.0134
## nodegree_0 * married_1    Binary  -0.1017
## nodegree_1 * married_0    Binary   0.1072
## nodegree_1 * married_1    Binary   0.0079
## 
## Effective sample sizes
##            Control Treated
## Unadjusted   429.      185
## Adjusted     108.2     185

# Balance tables with mean differences, variance ratios, and 
#  statistics for the unadjusted sample
bal.tab(treat ~ covs, data = lalonde,
        weights = "att.weights",
        disp = c("means", "sds"), un = TRUE, 
        stats = c("mean.diffs", "variance.ratios"))

## Balance Measures
##                Type    M.0.Un   SD.0.Un    M.1.Un   SD.1.Un Diff.Un V.Ratio.Un
## age         Contin.   28.0303   10.7867   25.8162    7.1550 -0.3094     0.4400
## educ        Contin.   10.2354    2.8552   10.3459    2.0107  0.0550     0.4959
## race_black   Binary    0.2028         .    0.8432         .  0.6404          .
## race_hispan  Binary    0.1422         .    0.0595         . -0.0827          .
## race_white   Binary    0.6550         .    0.0973         . -0.5577          .
## re74        Contin. 5619.2365 6788.7508 2095.5737 4886.6204 -0.7211     0.5181
## re75        Contin. 2466.4844 3291.9962 1532.0553 3219.2509 -0.2903     0.9563
##               M.0.Adj  SD.0.Adj   M.1.Adj  SD.1.Adj Diff.Adj V.Ratio.Adj
## age           25.0205   10.0330   25.8162    7.1550   0.1112      0.5086
## educ          10.4748    2.5918   10.3459    2.0107  -0.0641      0.6018
## race_black     0.8476         .    0.8432         .  -0.0044           .
## race_hispan    0.0578         .    0.0595         .   0.0016           .
## race_white     0.0945         .    0.0973         .   0.0028           .
## re74        2114.5263 4013.9557 2095.5737 4886.6204  -0.0039      1.4821
## re75        1669.9618 2974.6678 1532.0553 3219.2509  -0.0428      1.1712
## 
## Effective sample sizes
##            Control Treated
## Unadjusted   429.      185
## Adjusted     108.2     185

# Balance tables with thresholds for mean differences and variance ratios
bal.tab(treat ~ covs, data = lalonde,
        weights = "att.weights",
        thresholds = c(m = .1, v = 2))

## Balance Measures
##                Type Diff.Adj        M.Threshold V.Ratio.Adj  V.Threshold
## age         Contin.   0.1112 Not Balanced, >0.1      0.5086 Balanced, <2
## educ        Contin.  -0.0641     Balanced, <0.1      0.6018 Balanced, <2
## race_black   Binary  -0.0044     Balanced, <0.1           .             
## race_hispan  Binary   0.0016     Balanced, <0.1           .             
## race_white   Binary   0.0028     Balanced, <0.1           .             
## re74        Contin.  -0.0039     Balanced, <0.1      1.4821 Balanced, <2
## re75        Contin.  -0.0428     Balanced, <0.1      1.1712 Balanced, <2
## 
## Balance tally for mean differences
##                    count
## Balanced, <0.1         6
## Not Balanced, >0.1     1
## 
## Variable with the greatest mean difference
##  Variable Diff.Adj        M.Threshold
##       age   0.1112 Not Balanced, >0.1
## 
## Balance tally for variance ratios
##                  count
## Balanced, <2         4
## Not Balanced, >2     0
## 
## Variable with the greatest variance ratio
##  Variable V.Ratio.Adj  V.Threshold
##       age      0.5086 Balanced, <2
## 
## Effective sample sizes
##            Control Treated
## Unadjusted   429.      185
## Adjusted     108.2     185

# Generating ATT weights with different covariates
lalonde$p.score2 <- glm(treat ~ age + I(age^2) + race + educ + re74, 
                        data = lalonde,
                        family = "binomial")$fitted.values
lalonde$att.weights2 <- with(lalonde, treat + (1-treat)*p.score2/(1-p.score2))

bal.tab(treat ~ covs, data = lalonde,
        weights = c("att.weights", "att.weights2"),
        estimand = "ATT")

## Balance Measures
##                Type Diff.att.weights Diff.att.weights2
## age         Contin.           0.1112           -0.0400
## educ        Contin.          -0.0641           -0.0227
## race_black   Binary          -0.0044           -0.0005
## race_hispan  Binary           0.0016           -0.0010
## race_white   Binary           0.0028            0.0015
## re74        Contin.          -0.0039            0.0037
## re75        Contin.          -0.0428           -0.0083
## 
## Effective sample sizes
##              Control Treated
## All           429.       185
## att.weights   108.2      185
## att.weights2   72.04     185

# Subclassification for ATT with 5 subclasses
lalonde$p.score <- glm(treat ~ age + educ + race + re74 + re75,
                       data = lalonde, 
                       family = "binomial")$fitted.values
nsub <- 5 #number of subclasses
lalonde$subclass <- with(lalonde,
                         findInterval(p.score, 
                                      quantile(p.score[treat == 1], 
                                               seq(0, 1, length.out = nsub + 1)), 
                                      all.inside = TRUE))
                         
bal.tab(treat ~ covs, data = lalonde,
        subclass = "subclass", 
        which.subclass = .all,
        subclass.summary = TRUE)

## Balance by subclass
##  - - - Subclass 1 - - - 
##                Type Diff.Adj
## age         Contin.  -0.4889
## educ        Contin.   0.1324
## race_black   Binary   0.1934
## race_hispan  Binary   0.1230
## race_white   Binary  -0.3164
## re74        Contin.  -0.2334
## re75        Contin.  -0.0435
## 
##  - - - Subclass 2 - - - 
##                Type Diff.Adj
## age         Contin.  -0.4423
## educ        Contin.   0.3102
## race_black   Binary   0.0000
## race_hispan  Binary   0.0000
## race_white   Binary   0.0000
## re74        Contin.   0.0584
## re75        Contin.   0.3445
## 
##  - - - Subclass 3 - - - 
##                Type Diff.Adj
## age         Contin.   0.4968
## educ        Contin.  -0.1031
## race_black   Binary   0.0000
## race_hispan  Binary   0.0000
## race_white   Binary   0.0000
## re74        Contin.  -0.1250
## re75        Contin.  -0.1912
## 
##  - - - Subclass 4 - - - 
##                Type Diff.Adj
## age         Contin.   0.2665
## educ        Contin.  -0.1616
## race_black   Binary   0.0000
## race_hispan  Binary   0.0000
## race_white   Binary   0.0000
## re74        Contin.  -0.1082
## re75        Contin.  -0.0307
## 
##  - - - Subclass 5 - - - 
##                Type Diff.Adj
## age         Contin.   0.2946
## educ        Contin.  -0.0757
## race_black   Binary   0.0000
## race_hispan  Binary   0.0000
## race_white   Binary   0.0000
## re74        Contin.  -0.0546
## re75        Contin.  -0.1770
## 
## Balance measures across subclasses
##                Type Diff.Adj
## age         Contin.   0.0216
## educ        Contin.   0.0195
## race_black   Binary   0.0387
## race_hispan  Binary   0.0246
## race_white   Binary  -0.0633
## re74        Contin.  -0.0923
## re75        Contin.  -0.0170
## 
## Sample sizes by subclass
##           1  2  3  4  5 All
## Control 350 25 21 14 19 429
## Treated  37 37 34 40 37 185
## Total   387 62 55 54 56 614

data("lalonde", package = "cobalt")

# Nearest neighbor 2:1 matching with replacement
m.out <- MatchIt::matchit(treat ~ age + educ + race + re74 + re75, 
                          data = lalonde, method = "nearest", 
                          ratio = 1,  replace = TRUE)

bal.tab(m.out)

## Balance Measures
##                 Type Diff.Adj
## distance    Distance  -0.0010
## age          Contin.   0.1964
## educ         Contin.  -0.0726
## race_black    Binary   0.0108
## race_hispan   Binary  -0.0054
## race_white    Binary  -0.0054
## re74         Contin.   0.0667
## re75         Contin.   0.0936
## 
## Sample sizes
##                      Control Treated
## All                   429.       185
## Matched (ESS)          36.45     185
## Matched (Unweighted)   77.       185
## Unmatched             352.         0

data("lalonde", package = "cobalt") #If not yet loaded

#Generating propensity score weights for the ATT
W.out <- WeightIt::weightit(treat ~ age + educ + race + re74 + re75,
                            data = lalonde,
                            method = "ps",
                            estimand = "ATT")

bal.tab(W.out)

## Balance Measures
##                 Type Diff.Adj
## prop.score  Distance  -0.0397
## age          Contin.   0.1112
## educ         Contin.  -0.0641
## race_black    Binary  -0.0044
## race_hispan   Binary   0.0016
## race_white    Binary   0.0028
## re74         Contin.  -0.0039
## re75         Contin.  -0.0428
## 
## Effective sample sizes
##            Control Treated
## Unadjusted   429.      185
## Adjusted     108.2     185

bal.plot()

The gold standard for covariate balance is multidimensional independence between treatment and covariates. Because this is hard to visualize and assess with the large numbers of covariates typical of causal effect analysis, univariate balance is typically assessed as a proxy. Most conditioning packages, as well as cobalt, will provide numerical summaries of balance, typically by comparing moments between the treated and control groups. But even univariate balance is more complicated than simple numerical summaries can address; examining distributional balance is a more thorough method to assess balance between groups. Although there are statistics such as the Kolmogorov-Smirnov statistics and the overlapping coefficient that attempt to summarize distributional balance beyond the first few moments (Austin and Stuart 2015; Ali et al. 2015), complimenting statistics with a visual examination of the distributional densities can be an effective way of assessing distributional similarity between the groups (Ho et al. 2007).

bal.plot() allows users to do so by displaying density plots, histograms, empirical CDF plots, bar graphs, and scatterplots so that users can visually assess independence between treatment and covariates before and after conditioning. Below is an example of the use of bal.plot() after using propensity score weighting for the ATT using the output from WeightIt generated above.:

bal.plot(W.out, var.name = "age")
bal.plot(W.out, var.name = "race")

The first argument (or set of arguments) is the sufficient set of arguments for a simple call to bal.tab(), defining the data object (e.g., the output of a conditioning function), the treatment indicators, and the weights or subclasses. See above for examples. The next argument is the name of the covariate for which distributional balance is to be assessed. If subclassification is used (i.e., if subclasses are present in the input data object or arguments), an additional argument which.sub can be specified, with a number corresponding to the subclass number for which balance is to be assessed on the specified covariate; if it is not specified, plots for all subclasses will be displayed.

The user can also specify whether distributional balance is to be shown before or after adjusting or both by using the argument to which. If which = "unadjusted", balance will be displayed for the unadjusted sample only. If which = "both", balance will be displayed for both the unadjusted sample and the adjusted sample. The default is to display balance for the adjusted sample.

The output of bal.plot() is a density plot, histogram, or empirical CDF plot for the two groups on the given covariate, depending on the argument to type. For categorical or binary variables, a bar graph is displayed instead. When multi-category categorical variables are given, bars will be created for each level, unlike in bal.tab(), which splits the variable into several binary variables. The degree to which the densities for the two groups overlap is a good measure of group balance on the given covariate; significant differences in shape can be indicative of poor balance, even when the mean differences and variance ratios are well within thresholds. Strong distributional similarity is especially important for variables strongly related to the outcome of interest.

Distributional balance can also be assessed on the distance measure, and this can form an alternative to other common support checks, like MatchIt’s plot(..., type = "hist") or twang’s plot(..., plots = "boxplot"). To examine the distributions of the distance measure, the input to var.name must be the name of the distance variable. If the data input object doesn’t already contain the distance measure (e.g., if not using one of the conditioning packages), the distance measure must be manually added as an input to bal.plot() through distance, in addition to being called through var.name.

Below is an example using bal.plot() to display the distributions of propensity scores before and after weighting adjustment:

#Before and after weighting; which = "both"
bal.plot(W.out, var.name = "prop.score",
         which = "both",
         type = "histogram",
         mirror = TRUE)

Setting type = "histogram" produces a histogram rather than a density plot, and setting mirror = TRUE creates a mirrored plot rather than overlapping histograms. Mirroring only works with binary treatments.

It is generally not a useful assessment of balance to examine the overlap of the distance measure distributions after adjustment, as most conditioning methods will yield good distributional overlap on the distance measure whether or not balance is achieved on the covariates (Stuart, Lee, and Leacy 2013). However, it may be useful to see the new range of the distance measure if calipers or common support pruning are used.

The output plot is made using ggplot2, which means that users familiar with ggplot2 can adjust the plot with ggplot2 commands.

When the treatment variable is continuous, users can use bal.plot() to examine and assess dependence between the covariate and treatment. The arguments given to bal.plot() are the same as in the binary treatment case, but the resulting plots are different. If the covariate is continuous, a scatterplot between the covariate and the treatment variable will be displayed, along with a linear fit line, a Loess curve, and a reference line indicating linear independence. Used together, these lines can help diagnose departures from independence beyond the simple correlation coefficient. Proximity of the fit lines to the reference line is suggestive of independence between the covariate and treatment variable. If the covariate is categorical (including binary), density plots of the treatment variable for each category will be displayed. Densities that overlap completely are indicative of independence between the covariate and treatment. See the section “Using cobalt with continuous treatments” for more details and an example.

love.plot()

The Love plot is a summary plot of covariate balance before and after conditioning popularized by Dr. Thomas E. Love. In a visually appealing and clear way, balance can be presented to demonstrate to readers that balance has been met within a threshold, and that balance has improved after conditioning [which is not always the case; cf. King and Nielsen (2019)]. love.plot() does just this, providing the user with several options to customize their plot for presentation. Below is an example of its use:

data("lalonde", package = "cobalt")

# Nearest neighbor 1:1 matching with replacement
m.out <- MatchIt::matchit(treat ~ age + educ + married + race +
                              nodegree + re74 + re75,
                          data = lalonde, 
                          method = "nearest",
                          replace = TRUE)

love.plot(m.out, binary = "std", thresholds = c(m = .1))

love.plot() takes as its arguments the same ones that would go into a call to bal.tab(). In addition, it can take as its first argument the output of a call to bal.tab(); this can be accomplished simply by inserting the bal.tab() call into the first argument or by saving the result of a call to bal.tab() to an object and inserting the object as the argument. There are several other arguments, all of which control display, that are described below.

The output is a plot with the balance statistic on the X-axis and the covariates output in bal.tab() on the Y-axis. Each point represents the balance statistic for that covariate, colored based on whether it is calculated before or after adjustment. The dotted lines represent the threshold set in the threshold argument; if most or all of the points after adjustment are within the threshold, that is good evidence that balance has been achieved.

The default is to present the absolute mean differences as they are calculated in the call to bal.tab(); by specifying stats = "variance.ratios" or stats = "ks.statistics" (abbreviations allowed), the user can request variance ratios or KS statistics instead or in addition. Because binary variables don’t have variance ratios calculated, there will not be rows for these variables, but these rows can be added (with empty entries) to be in alignment with mean differences by setting drop.missing = FALSE.

The thresholds argument works similarly to how it does in bal.tab(); specifying it is optional, but doing so will provide an additional point of reference on which to evaluate the displayed balance measures.

If mean difference are requested, love.plot() will use the mean differences as they are calculated by bal.tab() and presented in the mean differences columns of the balance table. See the section on using bal.tab() to see what the default calculations are for these values. If abs = TRUE in love.plot(), the plot will display absolute mean differences, which can aid in display clarity since the magnitude is generally the more important aspect of the statistic.

The order of the covariates displayed can be adjusted using the argument to var.order. If left empty or NULL, the covariates will be listed in the order of the original dataset. If "adjusted", covariates will be ordered by the requested balance statistic of the adjusted sample. If "unadjusted", covariates will be ordered by the requested balance statistic of the unadjusted sample, which tends to be more visually appealing. Abbreviations are allowed. The distance variable(s), if any, will always be displayed at the top. They can be omitted by setting drop.distance = TRUE.

The plot uses the original variable names as they are given in the data set, which may not be the names desired for display in publication. By using the argument to var.names, users can specify their own variable names to be used instead. To specify new variable names with var.names, the user must enter an object containing the new variable names and, optionally, the old variable names to replace. For options of how to do so, see the help file for love.plot() with ?love.plot. Below is an example, creating a publication-ready plot with a few other arguments to customize output:

v <- data.frame(old = c("age", "educ", "race_black", "race_hispan", 
                        "race_white", "married", "nodegree", "re74", "re75", "distance"),
                new = c("Age", "Years of Education", "Black", 
                        "Hispanic", "White", "Married", "No Degree Earned", 
                        "Earnings 1974", "Earnings 1975", "Propensity Score"))

love.plot(m.out, stats = c("mean.diffs", "ks.statistics"), 
          threshold = c(m = .1, ks = .05), 
          binary = "std",
          abs = TRUE,
          var.order = "unadjusted",
          var.names = v,
          limits = c(0, 1),
          grid = FALSE,
          wrap = 20,
          sample.names = c("Unmatched", "Matched"),
          position = "top",
          shapes = c("circle", "triangle"),
          colors = c("red", "blue"))

This plot shows that balance was improved on almost all variables after adjustment, bringing all but two below the threshold of .1 for absolute mean differences.

A helper function, var.names() can be used to more easily create new variable names when many variables are present. See ?var.names for details.

When the treatment variable is continuous, love.plot() will display Pearson correlations between each covariate and treatment. The same arguments apply except that stats is ignored and threshold corresponds to r.threshold, the threshold for correlations.

Like the output of bal.plot(), the output of love.plot() is a ggplot2 object, which means ggplot2 users can modify the plot to some extent for presentation or publication. Several aspects of the appearance of the plot can be customized using the love.plot() syntax, including the size, shape, and color of the points, the title of the plot, whether to display grid lines, and whether to display lines connecting the points. See ?love.plot for details. It may be challenging to make adjustments to these aspects using ggplot2 syntax, so these arguments allow for some simple adjustments. See Appendix 4: Using love.plot To Generate Love Plots for information on more of love.plot()’s features.

Using cobalt with continuous treatments

Although the most common use of propensity scores is in the context of binary treatments, it is also possible to use propensity scores with continuous treatment to estimate dose-response functions while controlling for background variables (Hirano and Imbens 2005). As in the binary case, the goal of propensity score adjustment in the continuous case is to arrive at a scenario in which, conditional on the propensity score, treatment is independent of background covariates. When this is true (and there are no unmeasured confounders), treatment is also independent of potential outcomes, thereby meeting the strong ignorability requirement for causal inference.

Bia and Mattei (2008) describe the use of the gpscore function in Stata, which appears to be effective for estimating and assessing dose-response functions for continuous treatments. In R, there are not many ways to estimate and condition on the propensity score in these contexts. It is possible, using the formulas described by Hirano and Imbens (2005), to generate the propensity scores manually and perform weighting, subclassification, or covariate adjustment on them. The WeightIt package supports continuous treatments with a variety of options, including the CBPS method implemented in the CBPS package and described by Fong, Hazlett, and Imai (2018), GBM as described by Zhu, Coffman, and Ghosh (2015), and entropy balancing as described by Vegetabile et al. (2021), among others.

In cobalt, users can assess and present balance for continuous treatments using bal.tab(), bal.plot(), and love.plot(), just as with binary treatments. The syntax is almost identical in both cases regardless of the type of treatment variable considered, but there are a few differences and specifics worth noting. The approach cobalt takes to assessing balance is to display correlations between each covariate and the treatment variable, which is the approach used in CBPS and described in Zhu, Coffman, and Ghosh (2015) and Austin (2019), but not that described in Hirano and Imbens (2005) or implemented in gpscore (Bia and Mattei 2008), which involves stratifying on both the treatment variable and the propensity score and calculating mean differences. Note that the weighted correlations use the unweighted standard deviations of the treatment variable and covariate in the denominator, so correlations above 1 may be observed in rare cases.

In addition to assessing the treatment-covariate correlations, it is important to assess the degree to which the adjusted sample is representative of the original target population. If the weighted sample differs greatly from the original sample, the estimated effect may be biased for the target population of interest, even if the covariates are independent from treatment. cobalt offers methods to compare the weighted sample to the unweighted sample in the context of continuous treatments, such as computing the standardized mean difference or KS statistic between the weighted and unweighted sample for each covariate.

Below is an example of the workflow for using propensity scores for continuous treatments in the WeightIt package. To demonstrate, we use the lalonde package included in cobalt, using an arbitrary continuous variable as the treatment, though substantively this analysis makes little sense.

data("lalonde", package = "cobalt")

#Generating weights with re75 as the continuous treatment
W.out.c <- WeightIt::weightit(re75 ~ age + educ + race + married +
                                  nodegree + re74 + I(re74^2), 
                              data = lalonde,
                              method = "ps")

First, we can assess balance numerically using bal.tab(). The main balance statistic used is the Pearson correlation between each covariate and the treatment variable. A threshold for balance on correlations can be specified using thresholds; Zhu, Coffman, and Ghosh (2015) recommend using .1 as indicating balance, but in general lower is better. Because the goal is complete independence between treatment and covariates, not simply the absence of a linear correlation between treatment and covariates, including interactions and polynomial terms through the use of arguments to int and poly is recommended (we just display the use of poly here for brevity). In addition to treatment-covariate correlations, we request standardized mean difference between the weighted and unweighted samples by include "m" (for "mean.diffs.target") in the argument to stats along with "c" (for "correlations").

#Assessing balance numerically
bal.tab(W.out.c, stats = c("c", "m"), un = TRUE, 
        thresholds = c(cor = .1), poly = 3)

## Balance Measures
##                Type Corr.Un Corr.Adj        R.Threshold Diff.Adj
## age         Contin.  0.1400   0.0165     Balanced, <0.1  -0.1552
## educ        Contin.  0.0183   0.0573     Balanced, <0.1  -0.0392
## race_black   Binary -0.1405   0.0499     Balanced, <0.1   0.1139
## race_hispan  Binary  0.0616   0.0637     Balanced, <0.1  -0.1379
## race_white   Binary  0.0978  -0.0899     Balanced, <0.1  -0.0227
## married      Binary  0.3541   0.0249     Balanced, <0.1  -0.2152
## nodegree     Binary -0.0705  -0.0409     Balanced, <0.1  -0.0159
## re74        Contin.  0.5520  -0.0904     Balanced, <0.1  -0.3843
## age²        Contin.  0.0998  -0.0160     Balanced, <0.1  -0.1426
## educ²       Contin.  0.0312   0.0849     Balanced, <0.1  -0.0328
## re74²       Contin.  0.4607  -0.1631 Not Balanced, >0.1  -0.3904
## I(re74^2)²  Contin.  0.3142  -0.1591 Not Balanced, >0.1  -0.2725
## age³        Contin.  0.0627  -0.0436     Balanced, <0.1  -0.1293
## educ³       Contin.  0.0361   0.0911     Balanced, <0.1  -0.0257
## re74³       Contin.  0.3790  -0.1755 Not Balanced, >0.1  -0.3418
## I(re74^2)³  Contin.  0.2230  -0.0989     Balanced, <0.1  -0.1385
## 
## Balance tally for treatment correlations
##                    count
## Balanced, <0.1        13
## Not Balanced, >0.1     3
## 
## Variable with the greatest treatment correlation
##  Variable Corr.Adj        R.Threshold
##     re74³  -0.1755 Not Balanced, >0.1
## 
## Effective sample sizes
##             Total
## Unadjusted 614.  
## Adjusted   181.28

We can also visually assess balance using bal.plot(). For continuous covariates, bal.plot() displays a scatterplot of treatment against the covariate, and includes a linear fit line (red), a smoothed fit curve (blue), and a horizontal reference line (black) at the unweighted mean of the treatment variable, and a vertical line at the unweighted mean of the covariate. These lines can be used to diagnose dependence. If either fit line is not close to flat and not lying on top of the reference line, there may be some remaining dependence between treatment and the covariate. The points in the weighted plot are shaded according to the size of their corresponding weight. If the linear fit line (red) does not cross through the intersection of the black reference lines, the target population of the weighted estimate differs from the original population. For categorical covariates, including binary, bal.plot() displays a density plot of the treatment variable in each category. If treatment and the covariate are independent, the densities for each category should overlap with each other. A distinct lack of overlap is indicative of remaining dependence between treatment and the covariate.

#Assessing balance graphically
bal.plot(W.out.c, "re74", which = "both")

bal.plot(W.out.c, "married", which = "both")

When balance has been achieved to a satisfactory level, users can present balance improvements in a Love plot using the love.plot() command, just as with binary treatments.

#Summarizing balance in a Love plot
love.plot(W.out.c, stats = c("c", "ks"),
          thresholds = c(cor = .1), 
          abs = TRUE, wrap = 20,
          var.order = "unadjusted", line = TRUE)

Using cobalt with multi-category treatments

When multiple categorical treatment groups are to be compared with each other, it is possible to create balance across the treatment groups using preprocessing methods. Lopez and Gutman (2017) compare methods used to create balance with multi-category treatments and briefly describe balance assessment for these scenarios. An important note is the choice of estimand to be examined. The ATE represents the causal effect of moving from one treatment group to another for all units in the population; the ATT represents the causal effect of moving from one treatment group to another “focal” treatment group for just the units that would have been in the focal treatment group. The way balance is assessed in these scenarios differs. For the ATE, all possible treatment pairs must be assessed for balance because all possible comparisons are potentially meaningful, but for the ATT, only treatment pairs that include the focal treatment group can be meaningfully compared, so balance needs only to be assessed in these pairs.

In cobalt, users can assess and present balance for multi-category treatments using bal.tab(), bal.plot(), and love.plot(), just as with binary treatments. The output is slightly different, though, and is similar to the output generated when using these functions with clusters. bal.tab() computes balance statistics for all pairwise comparisons between treatment groups and a table containing the worst balance for each covariate across pairwise comparisons. For mean differences, this is described in Lopez and Gutman (2017) as “Max2SB,” or the maximum pairwise standardized bias. In cobalt, this has been extended to variance ratios and KS statistics as well. If the worst imbalance is not too great, then imbalance for all pairwise comparisons will not be too great either. When the ATT is desired, a focal group must be specified (unless done so automatically for some methods), and only the treatment group comparisons that involve that focal group will be computed and displayed.

love.plot() allows for the display of each pairwise treatment or the range of balance across treatment pairs for each covariate. bal.plot() displays distributional balance for the requested covariate across all treatment groups. Currently, multi-category treatments are not supported with clustered or multiply imputed data.

Below is an example of using cobalt with multi-category treatments. For this example, race will be the “treatment”; this type of analysis is not meant to be causal, but rather represents a method to examine disparities among groups accounting for covariates that might otherwise explain differences among groups. We will use WeightIt to generate balanced groups by estimating propensity score weights with multinomial logistic regression.

data("lalonde", package = "cobalt")

#Using WeightIt to generate weights with multinomial
#logistic regression
W.out.mn <- WeightIt::weightit(race ~ age + educ + married +
                                   nodegree + re74 + re75,
                               data = lalonde,
                               method = "ps",
                               use.mlogit = FALSE)

First, we can examine balance numerically using bal.tab(). There are three possible pairwise comparisons, all of which can be requested with which.treat = .all. See ?bal.tab.multi for more details.

#Balance summary across treatment pairs
bal.tab(W.out.mn, un = TRUE)

## Balance summary across all treatment pairs
##             Type Max.Diff.Un Max.Diff.Adj
## age      Contin.      0.3065       0.0504
## educ     Contin.      0.5861       0.1046
## married   Binary      0.3430       0.0355
## nodegree  Binary      0.2187       0.0438
## re74     Contin.      0.6196       0.1445
## re75     Contin.      0.3442       0.1462
## 
## Effective sample sizes
##            black hispan  white
## Unadjusted 243.   72.   299.  
## Adjusted   140.5  54.32 259.28

#Assessing balance for each pair of treatments
bal.tab(W.out.mn, un = TRUE,
        disp.means = TRUE,
        which.treat = .all)

## Balance by treatment pair
## 
##  - - - black (0) vs. hispan (1) - - - 
## Balance Measures
##             Type    M.0.Un    M.1.Un Diff.Un   M.0.Adj   M.1.Adj Diff.Adj
## age      Contin.   26.0123   25.9167 -0.0101   26.6302   27.0937   0.0491
## educ     Contin.   10.2346    9.0139 -0.4515   10.4977   10.2149  -0.1046
## married   Binary    0.2222    0.4444  0.2222    0.4124    0.3781  -0.0343
## nodegree  Binary    0.6955    0.7639  0.0684    0.5998    0.6436   0.0438
## re74     Contin. 2499.4492 4431.6260  0.3183 5447.2370 4836.5681  -0.1006
## re75     Contin. 1613.7752 2741.3920  0.3442 2654.8185 2196.7393  -0.1398
## 
## Effective sample sizes
##            black hispan
## Unadjusted 243.   72.  
## Adjusted   140.5  54.32
## 
##  - - - black (0) vs. white (1) - - - 
## Balance Measures
##             Type    M.0.Un    M.1.Un Diff.Un   M.0.Adj   M.1.Adj Diff.Adj
## age      Contin.   26.0123   28.8094  0.2963   26.6302   27.1058   0.0504
## educ     Contin.   10.2346   10.5987  0.1347   10.4977   10.2982  -0.0738
## married   Binary    0.2222    0.5652  0.3430    0.4124    0.4136   0.0011
## nodegree  Binary    0.6955    0.5452 -0.1503    0.5998    0.6265   0.0267
## re74     Contin. 2499.4492 6260.5029  0.6196 5447.2370 4570.1106  -0.1445
## re75     Contin. 1613.7752 2515.1320  0.2752 2654.8185 2175.8306  -0.1462
## 
## Effective sample sizes
##            black  white
## Unadjusted 243.  299.  
## Adjusted   140.5 259.28
## 
##  - - - hispan (0) vs. white (1) - - - 
## Balance Measures
##             Type    M.0.Un    M.1.Un Diff.Un   M.0.Adj   M.1.Adj Diff.Adj
## age      Contin.   25.9167   28.8094  0.3065   27.0937   27.1058   0.0013
## educ     Contin.    9.0139   10.5987  0.5861   10.2149   10.2982   0.0308
## married   Binary    0.4444    0.5652  0.1208    0.3781    0.4136   0.0355
## nodegree  Binary    0.7639    0.5452 -0.2187    0.6436    0.6265  -0.0171
## re74     Contin. 4431.6260 6260.5029  0.3013 4836.5681 4570.1106  -0.0439
## re75     Contin. 2741.3920 2515.1320 -0.0691 2196.7393 2175.8306  -0.0064
## 
## Effective sample sizes
##            hispan  white
## Unadjusted  72.   299.  
## Adjusted    54.32 259.28
##  - - - - - - - - - - - - - - - - - - - -

We can also assess balance graphically. The same guidelines apply for multi-category treatments as do for binary treatments. Ideally, covariate distributions will look similar across all treatment groups.

#Assessing balance graphically
bal.plot(W.out.mn, "age", which = "both")

bal.plot(W.out.mn, "married", which = "both")

Finally, we can use love.plot() to display balance across treatments. By default, love.plot() displays the values in the summary across pairwise comparisons. To request individual treatment comparisons, use which.treat = .all in love.plot().

#Summarizing balance in a Love plot
love.plot(W.out.mn, thresholds = c(m = .1), binary = "std",
          which.treat = .all, abs = FALSE)

Comparing balancing methods

It is possible to display balance for multiple balancing methods at the same time in bal.tab(), bal.plot(), and love.plot(). To do so, weights generated from each balancing method need to be supplied together in each call. This can be done by supplying the weights or the output objects themselves. For example, we can compare matching and inverse probability weighting for the ATT using the following code and the output generated above.

bal.tab(treat ~ age + educ + married + race +
            nodegree + re74 + re75, data = lalonde, 
        weights = list(Matched = m.out,
                       IPW = W.out),
        disp.v.ratio = TRUE)

## Balance Measures
##                Type Diff.Matched V.Ratio.Matched Diff.IPW V.Ratio.IPW
## age         Contin.       0.2395          0.5557   0.1112      0.5086
## educ        Contin.      -0.0161          0.5765  -0.0641      0.6018
## married      Binary       0.0595               .  -0.0938           .
## race_black   Binary       0.0054               .  -0.0044           .
## race_hispan  Binary      -0.0054               .   0.0016           .
## race_white   Binary       0.0000               .   0.0028           .
## nodegree     Binary       0.0054               .   0.1151           .
## re74        Contin.      -0.0493          1.0349  -0.0039      1.4821
## re75        Contin.       0.0087          2.1264  -0.0428      1.1712
## 
## Effective sample sizes
##         Control Treated
## All      429.       185
## Matched   43.49     185
## IPW      108.2      185

To use bal.plot(), the same syntax can be used:

bal.plot(treat ~ age, data = lalonde, 
         weights = list(Matched = m.out,
                        IPW = W.out),
         var.name = "age", which = "both")

With love.plot(), var.order can be "unadjusted", "alphabetical", or one of the names of the weights to order the variables. Also, colors and shapes should have the same length as the number of weights or have length 1.

love.plot(treat ~ age + educ + married + race +
              nodegree + re74 + re75,
          data = lalonde, 
          weights = list(Matched = m.out,
                         IPW = W.out),
          var.order = "unadjusted", binary = "std",
          abs = TRUE, colors = c("red", "blue", "darkgreen"), 
          shapes = c("circle", "square", "triangle"),
          line = TRUE)

Another way to compare weights from multiple objects is to call bal.tab() with one object as usual and supply the other(s) to the weights argument; see below for an example:

bal.tab(m.out, weights = list(IPW = W.out))

Using the prognostic score for balance assessment

The prognostic score is the model-predicted outcome for an individual, excluding the treatment variable in the model (Hansen 2008). Stuart, Lee, and Leacy (2013) found that prognostic scores can be an extremely effective tool for assessing balance, greatly outperforming mean differences on covariates and significance tests. This is true even if the prognostic score model is slightly misspecified. Although the use of prognostic scores appears to violate the spirit of preprocessing in that users observe the outcome variable prior to treatment effect estimation, typically the prognostic score model is estimated in just the control group, so that the outcome of the treated group (which may contain treatment effect information) is excluded from analysis.

Assessing balance on the prognostic score is simple in cobalt, and highly recommended when available. The steps are:

1. Estimate the outcome model in the control group
2. Generate model-predicted outcome values for both the treated and control groups
3. Assess balance on prognostic scores by comparing standardized mean differences

To use prognostic scores in cobalt, simply add the prognostic score as a variable in the argument to distance. Below is an example of how to do so after a call to matchit():

ctrl.data <- lalonde[lalonde$treat == 0,]
ctrl.fit <- glm(re78 ~ age + educ + race + 
                    married + nodegree + re74 + re75,
                data = ctrl.data)
lalonde$prog.score <- predict(ctrl.fit, lalonde)

bal.tab(m.out, distance = lalonde["prog.score"])

## Balance Measures
##                 Type Diff.Adj
## prog.score  Distance  -0.0485
## distance    Distance   0.0044
## age          Contin.   0.2395
## educ         Contin.  -0.0161
## married       Binary   0.0595
## race_black    Binary   0.0054
## race_hispan   Binary  -0.0054
## race_white    Binary   0.0000
## nodegree      Binary   0.0054
## re74         Contin.  -0.0493
## re75         Contin.   0.0087
## 
## Sample sizes
##                      Control Treated
## All                   429.       185
## Matched (ESS)          43.49     185
## Matched (Unweighted)   80.       185
## Unmatched             349.         0

Although the prognostic score is sensitive to the outcome estimation model used, a defensible prognostic score model can yield valid prognostic scores, which can then be used in balance assessment. In the above example, balance on the estimated prognostic score was good, so we can have some confidence that the effect estimate will be relatively unbiased, even though the “age” variable remains imbalanced. The logic is that that age is not a highly prognostic variable, which could be demonstrated by examining the standardized regression output of prognostic score model, so even though imbalance remains, such imbalance is unlikely to affect the effect estimate. The variables “re74” and “re75”, though, which are highly prognostic of the outcome, are quite balanced, thereby supporting an unbiased treatment effect estimate.

Variance in Standardized Mean Differences and Correlations

When computing a standardized mean difference, the raw mean difference is divided by a standard deviation, yielding a d-type effect size statistic. In bal.tab(), the user can control whether the standard deviation is that of the treated group or control group or a pooled estimate, calculated as the square root of the average of the group variances. In most applications, the standard deviation corresponding to the default for the method is the most appropriate.

A key detail is that the standard deviation, no matter how it is computed, is always computed using the unadjusted sample⁴. This is line with how MatchIt computes standardized mean differences, and is recommended by Stuart [-Stuart (2008); -Stuart (2010)]⁵. One reason to favor the use of the standard deviation of the unadjusted sample is that it prevents the paradoxical situation that occurs when adjustment decreases both the mean difference and the spread of the sample, yielding a larger standardized mean difference than that prior to adjustment, even though the adjusted groups are now more similar. By using the same standard deviation before and after adjusting, the change in balance is isolated to the change in mean difference, rather than being conflated with an accompanying change in spread.

The same logic applies to computing the standard deviations that appear in the denominator of the treatment-covariate correlations that are used to assess balance with continuous treatments. Given that the covariance is the relevant quality to be assessed and the correlation is just a standardization used to simplify interpretation, the standardization factor should remain the same before and after adjustment. Thus, the standard deviations of the unadjusted sample are used in the denominator of the treatment-covariate correlation, even when the correlation in question is for the adjusted sample.

Note that when sampling weights are used, values for the unadjusted sample will be computed incorporating the sampling weights; in that sense, they are “adjusted” by the sampling weights.

Weighted Variance

When using weighting or matching, summary values after adjustment are calculated by using weights generated from the matching or weighting process. For example, group means are computed using the standard formula for a weighted mean, incorporating the weighting or matching weights into the calculation. To estimate a weighted sample variance for use in variance ratios, sample standard deviations, or other statistics in the presence of sampling weights, there are two formulas that have been proposed:

1. \(\frac{\sum_{i=1}^{n} w_{i}(x_{i} - \bar{x}_{w})^2}{(\sum_{i=1}^{n} w_{i}) - 1}\)

2. \(\frac{\sum_{i=1}^{n} w_{i}}{(\sum_{i=1}^{n} w_{i})^2 - \sum_{i=1}^{n} w^2_{i}} \sum_{i=1}^{n} w_{i}(x_{i} - \bar{x}_{w})^2\)

The weights used in the first formula are often called “frequency weights”, while the weights in the second formula are often called normalized or “reliability weights”. MatchIt, twang, and Matching all use the first formula when calculating any weighted variance (CBPS does not compute a weighted variance). However, Austin (2008b) and Austin and Stuart (2015) recommend the second formula when considering matching weights for k:1 matching or weights for propensity score weighting. In cobalt, as of version 2.0.0, the second formula is used to remain in line with recommended practice. For some applications (e.g., when all weights are either 0 or 1, as in 1:1 matching), the two formulas yield the same variance estimate. In other cases, the estimates are nearly the same. For binary variables, the weighted variance is computed as \(\bar{x}_{w}(1-\bar{x}_{w})\) where \(\bar{x}_{w}\) is the weighted proportion of 1s in the sample.

Effective Sample Size for Weighting

Knowledge of the sample size after adjustment is important not just for outcome analysis but also for assessing the adequacy of a conditioning specification. For example, pruning many units through common support cutoffs, caliper matching, or matching with replacement can yield very small sample sizes that hurt both the precision of the outcome estimate and the external validity of the conclusion. In both matching and weighting, the adjusted sample size is not so straightforward because the purpose of weighting is to down- and up-weight observations to create two similar samples. The “effective sample size” (ESS) is a measure of the sample size a non-weighted sample would have to have to achieve the same level of precision as the weighted sample (Ridgeway 2006). This measure is implemented in twang using the following formula: \[ESS = \frac{(\sum_{i=1}^{n} w_{i})^2}{\sum_{i=1}^{n} w_{i}^2}\]Shook-Sa and Hudgens (2020) derived the specific relationship between the ESS and the standard error of a propensity score-weighted mean.

Test Statistics and P-values

Some of the early literature on propensity score matching included measures for balance assessment that relied on hypothesis tests for the independence of treatment assignment and covariates after adjustment (e.g., Hansen 2004; Rosenbaum and Rubin 1985). In a review of propensity score applications in the social sciences, Thoemmes and Kim (2011) found that over 66% of studies used significance tests to assess balance. Likewise, Austin (2008a) found that over 70% of studies using propensity scores in the medical literature used significance tests to assess balance, a finding replicated by Ali et al. (2015). These hypothesis tests can come in many forms: t-tests for the difference in means between groups, chi-square tests for the difference in proportion between groups, Kolmogorov-Smirnov tests for the difference in cumulative density between groups, or F-tests for the difference in means between groups across subclasses.

The use of hypothesis tests appears natural here: if balance is achieved, we would not expect extreme values for the test statistics, and we can quantify the probability of observing imbalance as extreme as the one observed if no imbalance is present as a way to assess whether there is balance, as we can do with standard hypothesis testing. But this view is not shared by the methodological community: many contemporary propensity score methodologists recommend against using hypothesis tests for balance assessment (e.g., Ho et al. 2007; Imai, King, and Stuart 2008; Austin 2011, 2009; Stuart 2010; Thoemmes and Kim 2011; Ali et al. 2015). There are logical reasons for this preference against hypothesis tests, noted clearly in Ali et al. (2015) and Linden (2014): they are influenced by sample size, which fluctuates during adjustment, and the theory behind them is inappropriate because balance is a quality solely of the sample in question, not in relation to a population. The relevant information in a hypothesis test for group differences is the standardized magnitude of the group difference, and so such a measure is preferred.

Because hypothesis tests can be misleading and their use is discouraged by leading methodologists, they have been completely excluded in cobalt in favor of summary statistics. This stands in contrast to twang, Matching, and RItools, all of which report hypothesis test p-values in their balance output.

Q-Q Plots and Summaries

Q-Q plots have been recommended as tools to assess distributional balance of covariates between groups (Ho et al. 2007), and are implemented in MatchIt and Matching (twang implements them but for a different purpose). Statistics summarizing the degree of imbalance in Q-Q plots are also reported in both MatchIt and Matching. MatchIt’s summary() command reports “eQQ Mean” and “eQQ Max” for each covariate before and after adjustment. These are the mean and maximum distance between the treated and control group empirical Q-Q plots on the scale of the covariate. Values close to 0 indicate good balance between groups for the given covariate.

A weakness of empirical Q-Q plots is that they don’t reveal much about the differences in the shapes of the distributions between the groups, which is a key aspect of distributional balance. A density plot essentially contains the same information, but is clearer and more intuitive. Although the assessment of balance using an empirical Q-Q plot is straightforward (i.e., deviations from the 45-degree line indicate imbalances), density plots present the information in a way more in line with the actual goals of conditioning, in particular, that the distributions of treated and control units are similar (Ho et al. 2007). Empirical Q-Q plot summary statistics may be useful in quantifying imbalance, but currently there are few recommendations for their use.

Density Plots

As mentioned above, cobalt displays density plots and bar charts rather than empirical Q-Q plots for the assessment of distributional similarity. These charts are not standard in any of the conditioning packages, but can be an intuitive and helpful tool for deciding whether adjustment has yielded similar distributions between the groups for given covariates. Though there are no obvious heuristics for deciding how much dissimilarity is too much dissimilarity, density plots do avoid the sometimes confusing logic of empirical Q-Q plots in favor of simplicity and interpretability. Austin (2009) and Linden (2014) consider density plots as a compliment to empirical Q-Q plots to more finely examine balance after adjusting.

Variance Ratios

Although mean differences (including t-tests and chi-square tests) are the most reported balance statistic (Thoemmes and Kim 2011), variance ratios have been recommended in the literature as a means to further examine balance between groups (Austin 2009; Ho et al. 2007; Imai, King, and Stuart 2008). When group variances are similar, the variance ratio will be close to 1. Common thresholds for the variance ratio for balanced groups are .5 and 2 (Stuart 2010; Rubin 2001), though ratios closer to 1 are preferred. Although bal.tab() in cobalt does not display variance ratios by default, they can be easily requested and have thresholds set.

Distinguishing Continuous and Binary Covariates

Continuous and binary covariates are treated differently by default in bal.tab() and bal.plot(). For continuous covariates, the standard summaries apply: standardized mean differences, variance ratios, and density plots. For binary covariates, raw differences in proportion and bar charts are preferable, and variance ratios are useless.

The value of standardized mean differences for continuous variables is that they are on the same scale so that they can be compared across variables, and they allow for a simple interpretation even when the details of the variable’s original scale are unclear to the analyst. None of these advantages are passed to binary variables because binary variables are already on the same scale (i.e., a proportion), and the scale is easily interpretable. In addition, the details of standardizing the proportion difference of a binary variable involve dividing the proportion difference by a variance, but the variance of a binary variable is a function of its proportion (Austin 2009). Standardizing the proportion difference of a binary variable can yield the following counterintuitive result: if \(X_{T} = .2\) and \(X_{C} = .3\), the standardized difference in proportion would be different from that if \(X_{T} = .5\) and \(X_{C} = .6\), even though the expectation is that the balance statistic should be the same for both scenarios because both would yield the same degree of bias in the effect estimate. In addition, Ali et al. (2014) found that the raw difference in proportion was a better predictor of bias than the standardized mean difference for binary variables⁶.

MatchIt allows users to view either standardized mean differences for all covariates or raw differences for all covariates, and twang and Matching display standardized differences for all variables but calculate test statistics depending on whether the covariate is continuous or binary (CBPS does not calculate mean differences, but presents both standardized and unstandardized means for all covariates). cobalt allows the user to select how the differences are to be calculated separately for continuous and binary variables, and uses the intuitive default that mean differences for continuous variables should be standardized while proportion differences for binary variables should not be.

Because the variance of a binary variable is a function only of its proportion, the variance ratio of a binary variable in two groups is a function only of their proportions, thereby containing no more information than the simple difference in proportion. Therefore, for binary variables, cobalt does not compute variance ratios, as they can be misleading.

Interactions and Polynomials

Because the goal of a balancing procedure is to achieve independence between treatment and the joint distribution of covariates, evaluating univariate distributional similarity may not be sufficient for assessing balance completely. Some writers have recommended the evaluation of distributional similarity of interaction and polynomial terms to account for this. Rather than requiring the user to create interaction variables by hand, bal.tab() can produce balance statistics for interactions by specifying int = TRUE, similar to MatchIt’s summary(), and polynomials by specifying a numeric argument to poly (e.g., 2 for squared terms).

When including categorical variables in balance assessment, bal.tab() makes a few adjustments under the hood that deserve explanation. First, if a variable is binary and is entered as a factor variable, balance statistics will only be displayed for one level of the variable (since the other is redundant), but balance on interaction terms will be displayed for all values of the variables.

For example, consider a binary variable “Sex” with values “Male” and “Female”. Many functions, including bal.tab(), lm(), and matchit(), will split this variable into two numeric dummy variables, “Male” and “Female”, each of which take on the values 0 and 1. One of these new variables is completely redundant: all of the relevant information is stored in just “Female”, so “Male” can be eliminated. Consider now a variable “Age”: what is desired in balance assessment is the interaction between Age and Sex; distributional similarity on the interaction between the treated and untreated groups is evidence of multivariate balance. Computing the interaction between “Female” and “Age” yields a variable that is “Age” when the unit is female and 0 otherwise. The average value of this variable is the average age of females in the sample, weighted by the proportion of females. If the two treatment groups have similar average values of this variable, this is taken as evidence for balance on the interaction between sex and age, though it is entirely possible that the average age of men differs greatly between the two groups. Thus, an additional variable computed as the product of “Male” and “Age” would be necessary to fully assess balance on the interaction between sex and age. bal.tab() produces this interaction term, which would otherwise be unobserved by the analyst⁷. The interactions among levels of a single factor, which always be equal to 0, are excluded in bal.tab().

Interactions between the distance measure and other variables have been excluded in bal.tab(), noting that balance on the distance measure is neither necessary nor sufficient for covariate balance.

Because the number of computations increases both with sample size and number of variables, computing interactions can be slow. In addition to taking the product of variables two at a time, bal.tab() checks variables to ensure no variables were created that contain only a single value (e.g., interactions between mutually exclusive covariates) or are redundant with respect to other variables (e.g., manually created interactions or manually split factors). This results in cleaner, more useful output, but also requires more computing time. It is advisable to store the results of a call to bal.tab() to a variable to be accessed later rather than to call bal.tab() several times when using it with interactions.

Clusters

The use of preprocessing techniques in the context of multilevel data (e.g., students within schools, patients within hospitals) has been increasing. Currently no other package allows for any balance assessment with respect to clusters, except by manually examining balance on clusters specified manually. It can be useful to examine balance within each cluster, but, especially if there are many clusters, it may also be useful to examine a summary of balance across clusters. In all of its functions, cobalt provides options for displaying balance on clustered data sets. This can occur either within specified clusters or across all clusters. Details on using cobalt with clustered data can be found in the accompanying Appendix 2.

Missing Data

Missing data is frequent in all research involving human subjects, and especially in the large survey data sets that are often used to answer causal questions in the social sciences. cobalt functions can assess balance not only on the observed covariates but also on the proportion of missing values for each covariate. Additionally, cobalt has features designed especially for assessing balance on preprocessed data sets that have been multiply imputed to address covariate missingness. Although the guidelines on assessing balance on multiply imputed data sets are scarce, it is valuable to assess balance within and across imputed data sets to ensure the preprocessing solution is applicable to all imputations. Details on using cobalt with multiply imputed data can be found in the accompanying Appendix 2.