`cobalt`

PackageThis is an introductory guide for the use of `cobalt`

in
most common scenarios. Three appendices are available for its use with
more complicated data scenarios and packages not demonstrated here.^{1}

Preprocessing data through matching, weighting, or subclassification can be an effective way to reduce model dependence and improve efficiency when estimating the causal effect of a treatment (Ho et al. 2007). Propensity scores and other related methods (e.g., coarsened exact matching, Mahalanobis distance matching, genetic matching) have become popular in the social and health sciences as tools for this purpose. Two excellent introductions to propensity scores and other preprocessing methods are Stuart (2010) and Austin (2011), which describe them simply and clearly and point to other sources of knowledge. The logic and theory behind preprocessing will not be discussed here, and reader’s knowledge of the causal assumption of strong ignorability is assumed.

Several packages in R exist to perform preprocessing and causal
effect estimation, and some were reviewed by Keller and Tipton (2016). These include
`MatchIt`

(Ho et al. 2011), `twang`

(Ridgeway et al. 2016),
`Matching`

(Sekhon 2011), `optmatch`

(Hansen and Klopfer 2006),
`CBPS`

(Fong et
al. 2019), `ebal`

(Hainmueller 2014), `sbw`

(Zubizarreta, Li, and Kim 2021),
`designmatch`

(Zubizarreta, Kilcioglu, and Vielma
2018), `WeightIt`

(Greifer 2021), `MatchThem`

(Pishgar et al. 2021), and
`cem`

(Iacus,
King, and Porro 2009); these together provide a near complete
set of preprocessing tools in R to date.

The following are the basic steps in performing a causal analysis using data preprocessing (Stuart 2010):

- Decide on covariates for which balance must be achieved
- Estimate the distance measure (e.g., propensity score)
- Condition on the distance measure (e.g., using matching, weighting, or subclassification)
- Assess balance on the covariates of interest; if poor, repeat steps 2-4
- Estimate the treatment effect in the conditioned sample

Steps 2, 3, and 4 are accomplished by all of the packages mentioned above. However, Step 4, assessing balance, is often overlooked in propensity score applications, with researchers failing to report the degree of covariate balance achieved by their conditioning (Thoemmes and Kim 2011). Achieving balance is the very purpose of preprocessing because covariate balance is what justifies ignorability on the observed covariates, allowing for the potential for a valid causal inference after effect estimation (Ho et al. 2007).

In addition to simply achieving balance, researchers must also report balance to convince readers that their analysis was performed adequately and that their causal conclusions are valid (Thoemmes and Kim 2011). Covariate balance is typically assessed and reported by using statistical measures, including standardized mean differences, variance ratios, and t-test or Kolmogorov-Smirnov-test p-values. Balance can be reported in an article by means of balance tables or plots displaying the balance measures before and after conditioning. If a defensible measure of balance is used and presented, readers are empowered to judge for themselves whether the causal claim made is valid or not based on the methods used and covariates chosen.

`cobalt`

is meant to supplement or replace the balance
assessment tools in the above packages and allow researchers to assess
and report balance on covariates simply, clearly, and flexibly before
and after conditioning. It integrates seamlessly with the above packages
so that users can employ both the conditioning package of their choice
and `cobalt`

in conjunction to assess and report balance. It
is important to note that `cobalt`

does not replace the
highly sophisticated *conditioning* tools of these packages, as
it does no conditioning or estimation of its own.

The rest of this guide explains how to use `cobalt`

with
some of the above packages and others, as well as the choices instituted
by the functions and customizable by the user. Other vignettes describe
the use of `cobalt`

with packages not mentioned here, with
multiply imputed and clustered data, and with longitudinal
treatments.

`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. (2023). cobalt: Covariate Balance Tables and Plots. R package version 4.5.0.

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, 2023), both in R (R Core Team, 2022).

`cobalt`

?If most of the major conditioning packages contain functions to
assess balance, why use `cobalt`

at all? `cobalt`

arose out of several desiderata when using these packages: to have
standardized measures that were consistent across all conditioning
packages, to allow for flexibility in the calculation and display of
balance measures, and to incorporate recent methodological
recommendations in the assessment of balance. However, some users of
these packages may be completely satisfied with their capabilities and
comfortable with their output; for them, `cobalt`

still has
value in its unique plotting capabilities that make use of
`ggplot2`

in R.

The following are some reasons why `cobalt`

may be
attractive to users of `MatchIt`

, `twang`

,
`Matching`

, `optmatch`

, `CBPS`

,
`ebal`

, `sbw`

, `designmatch`

,
`WeightIt`

, and other conditioning packages:

`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.

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).

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 `ebal`

^{2}. 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.

`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).

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.

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.

There are three main functions for use in `cobalt`

:
`bal.tab()`

, `bal.plot()`

, and
`love.plot()`

. There are also several utility functions which
can be used to ease the use of `cobalt`

and other packages.
The next sections describe how to use each, complete with example code
and output. To start, install and load `cobalt`

with the
following code:

```
install.packages("cobalt")
library("cobalt")
```

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”.

`bal.tab()`

on its own`bal.tab()`

can take in any data set and set of weights,
subclasses, or matching strata and evaluate balance on them. This can be
useful if propensity score weights, subclasses, or matching strata were
generated outside of the supported packages, if balance assessment is
desired prior to adjustment, or if package output is adjusted in such a
way as to make it unusable with one of `bal.tab()`

’s other
methods (e.g., if cases were manually removed or weights manually
changed). In `twang`

, the function `dx.wts()`

performs a similar action by allowing for the balance assessment of
groups weighted not using `twang`

functions, though it is
more limited in the types of data or conditioning strategies allowed.
Below is an example of the use of `bal.tab()`

with ATT
weights generating using logistic regression for a weighting-by-the-odds
analysis:

```
data("lalonde", package = "cobalt") #If not yet loaded
<- subset(lalonde, select = -c(treat, re78, nodegree, married))
covs
# Generating ATT weights as specified in Austin (2011)
$p.score <- glm(treat ~ age + educ + race + re74 + re75,
lalondedata = lalonde,
family = "binomial")$fitted.values
$att.weights <- with(lalonde, treat + (1-treat)*p.score/(1-p.score))
lalonde
bal.tab(covs, treat = lalonde$treat, weights = lalonde$att.weights)
```

Displayed first is the balance table, and last is a summary of sample
size information. Because weighting was specified as the method used,
effective sample sizes are given, as is done in `twang`

. See
the `twang`

documentation, `?bal.tab`

, or “Details
on Calculations” below for details on this calculation.

There are several ways to specify input to `bal.tab()`

when using data outside a conditioning package. The first, as shown
above, is to use a data frame of covariates and vectors for treatment
status and weights or subclasses. The user can additionally specify a
vector of distance measures (e.g., propensity scores) if balance is to
be assessed on those as well. If `weights`

is left empty,
balance on the unadjusted sample will be reported. The user can also
optionally specify a data set to the `data`

argument; this
makes it so that the arguments to `treat`

,
`weights`

, `distance`

, `subclass`

, and
others can be specified either with a vector or with the name of a
variable in the argument to `data`

that contains the
respective values.

Another way to specify input to `bal.tab()`

is to use the
formula interface. Below is an example of its use:

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

To use the formula interface, the user must specify a formula
relating treatment to the covariates for which balance is to be
assessed. If any of these variables exist in a data set, it must be
supplied to `data`

. Here, the `covs`

data frame
was used for simplicity, but using `f.build()`

or the
traditional formula input of `treat ~ v1 + v2 + v3 + ...`

is
also acceptable. As above, the arguments to `weights`

,
`distance`

, `subclass`

, and others can be
specified either as vectors or data frames containing the values or as
names of the variables in the argument to `data`

containing
the values. In the above example, an argument to `distance`

was specified, and balance measures for the propensity score now appear
in the balance table.

By default, `bal.tab()`

outputs standardized mean
differences for continuous variables and raw differences in proportion
for binary variables. For more details on how these values are computed
and determined, see `?bal.tab`

or “Details on Calculations”
below. To see raw or standardized mean differences for binary or
continuous variables, you can manually set `binary`

and/or
`continuous`

to `"raw"`

or `"std"`

.
These can also be set as global options by using, for example,
`set.cobalt.options(binary = "std")`

, which allows the user
not to type a non-default option every time they call
`bal.tab`

.

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

Users can specify additional variables for which to display balance
using the argument to `addl`

, which can be supplied as a
data.frame, a formula containing variables, or a string of names of
variables. Users can also add all two-way interactions between
covariates, including those in `addl`

, by specifying
`int = TRUE`

, and can add polynomials (e.g., squares) of
covariates by specifying a numeric argument to `poly`

.
Interactions will not be computed for the distance measure (i.e., the
propensity score), and squared terms will not be computed for binary
variables. For more details on interactions, see “Details on
Calculations”, below. To only request a few desired interaction terms,
these can be entered into `addl`

using a formula, as in
`addl = ~ V1 * V2`

. Below, balance is requested on the
variables stored in `covs`

, the additional variables
`nodegree`

and `married`

, and their interactions
and squares.

```
# 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)
```

Standardized mean differences can be computed several ways, and the
user can decide how `bal.tab()`

does so using the argument to
`s.d.denom`

, which controls whether the measure of spread in
the denominator is the standard deviation of the treated group
(`"treated"`

), most appropriate when computing the ATT; the
standard deviation of the control group (`"control"`

), most
appropriate when computing the ATC; the pooled standard deviation
(`"pooled"`

), computed as in Austin
(2009), most
appropriate when computing the ATE; or another value (see
`?col_w_smd`

for more options). `bal.tab()`

can
generally determine if the ATT or ATC are being estimated and will
supply `s.d.denom`

accordingly. Otherwise, the default is
`"pooled"`

.

The next options only affect display, not the calculation of any
statistics. First is `disp`

, which controls whether sample
statistics for each covariate in each group are displayed. Options
include `"means"`

and `"sds"`

, which will request
group means and standard deviations, respectively^{3}.

Next is `stats`

, which controls which balance statistics
are displayed. For binary and multi-category treatments, options include
`"mean.diffs"`

for (standardized) mean differences,
`"variance.ratios"`

for variance ratios,
`"ks.statistics"`

for Kolmogorov-Smirnov (KS) statistics, and
`"ovl.coefficient"`

for the complement of the overlapping
coefficient (abbreviations are allowed). See `?balance.stats`

for details. By default, standardized mean differences are displayed.
Variance ratios are another important tool for assessing balance beyond
mean differences because they pertain to the shape of the covariate
distributions beyond their centers. Variance ratios close to 1 (i.e.,
equal variances in both groups) are indicative of group balance (Austin
2009). KS statistics measure the greatest distance between
the empirical cumulative distribution functions (eCDFs) for each
variable between two groups. The statistic is bounded at 0 and 1, with 0
indicting perfectly identical distributions and 1 indicating perfect
separation between the distributions (i.e., no overlap at all); values
close to 0 are thus indicative of balance. The use of the KS statistic
to formally assess balance is debated. Austin and
Stuart (2015) recommend its use, and it or a
variant appears as a default balance statistic in `MatchIt`

,
`twang`

, and `Matching`

. On the other hand, Belitser et al. (2011), Stuart,
Lee, and Leacy (2013), and Ali et
al. (2014) all found that global balance
assessments using the KS statistic performed uniformly worse than
standardized mean differences, especially at sample sizes less than 1000
in their simulations. The overlapping coefficient measures the amount of
overlap in the covariate distributions between two groups. As in Franklin et al. (2014), the complement is used so that 0
indicates perfectly overlapping distributions and 1 indicates perfectly
non-overlapping distributions. It functions similarly to the KS
statistic.

Next is `un`

, which controls whether the statistics to be
displayed should be displayed for the unadjusted group as well. This can
be useful the first time balance is assessed to see the initial group
imbalance. Setting `un = FALSE`

, which is the default, can
declutter the output to maintain the spotlight on the group balance
after adjustment.

```
# 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"))
```

See `?display_options`

for the full list of display
options. They can also be set as global options by using
`set.cobalt.options()`

.

Finally, the user can specify a threshold for balance statistics
using the `threshold`

argument. Thresholds can be useful in
determining whether satisfactory balance has been achieved. For
standardized mean differences, thresholds of .1 and .25 have been
proposed, but Stuart, Lee, and Leacy (2013) found that a threshold of .1 was
more effective at assessing imbalance that would lead to biased effect
estimation. In general, standardized mean differences should be as close
to 0 as possible, but a conservative upper limit such as .1 can be a
valuable heuristic in selecting models and defending the conditioning
choice. The What Works Clearinghouse Standards Handbook recommends
standardized mean differences of less than .05 (What Works
Clearinghouse, 2020).

When thresholds are requested, a few components are added to the
balance output: an extra column in the balance table stating whether
each covariate is or is not balanced according to the threshold, an
extra table below the balance table with a tally of how many covariates
are or are not balanced according to the threshold, and a notice of
which covariate has the greatest imbalance after conditioning and
whether it exceeded the threshold. Below, thresholds are requested for
mean differences (`m`

) and variance ratios
(`v`

).

```
# 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))
```

To simplify output when many covariates are included or when
`int = TRUE`

is specified, `imbalanced.only`

can
be set to `TRUE`

, which will only reveal imbalanced
covariates in the output. These are covariates that have failed to meet
any of the balance thresholds set. In addition,
`disp.bal.tab`

can be set `FALSE`

, which will hide
the balance table (revealing only the balance summaries accompanying the
threshold).

If sampling weights are used and are to be applied to both the
adjusted and unadjusted groups, they can be specified with an argument
to `s.weights`

, which can be specified either by providing a
vector of sampling weights for each unit or by providing the name of a
variable in `data`

containing the sampling weights. The
adjusted and unadjusted samples will each be weighted by the sampling
weights by multiplying the adjustment weights (if any) by the sampling
weights.

It is possible to view balance for more than one set of weights at a
time. The input to `weights`

should be either the names of
variables in `data`

containing the desired weights or a named
data frame containing each set of weights. The arguments to
`s.d.denom`

or `estimand`

must have the same
length as the number of sets of weights, or else be of length 1,
applying the sole input to all sets of weights. Below is an example
comparing the weights estimated above to a new set of weights. Another
example can be found in the section “Comparing Balancing Methods”.

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

When subclassification is used in conditioning, an argument to
`subclass`

must be specified; this can be a vector of
subclass membership or the name of a variable in `data`

containing subclass membership. `bal.tab()`

produces a
different type of output from when matching or weighting are used,
though it has all of the same features. The default output is a balance
table displaying balance aggregated across subclasses; this can be
controlled with the `subclass.summary`

options. Each cell
contains the average statistic across the subclasses. Using the
arguments discussed above will change the output as it does when only
matching or weighting is used.

To examine balance within each subclass, the user can specify
`which.subclass = .all`

, which will produce output for the
subclasses in aggregate. Within subclasses, all the information above,
including other requested statistics, will be presented, except for
statistics for the unadjusted groups (since the adjustment occurs by
creating the subclasses), as specified by the user. See
`?bal.tab.subclass`

for more details.

```
# Subclassification for ATT with 5 subclasses
$p.score <- glm(treat ~ age + educ + race + re74 + re75,
lalondedata = lalonde,
family = "binomial")$fitted.values
<- 5 #number of subclasses
nsub $subclass <- with(lalonde,
lalondefindInterval(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)
```

When using `bal.tab()`

with continuous treatments, the
default balance statistic presented is the (weighted) Pearson
correlation between each covariate and treatment. Zhu, Coffman, and Ghosh (2015) recommend that absolute
correlations should be no greater than 0.1, but correlations should
ideally be as close to zero as possible. Spearman correlations can also
be requested. See the section “Using `cobalt`

with continuous
treatments” and `?balance.stats`

for more details.

The next two sections describe the use of `bal.tab()`

with
the `MatchIt`

and `WeightIt`

. As stated above, the
arguments controlling calculations and display are largely the same
across inputs types, so they will not be described again except when
their use differs from that described in the present section.

`bal.tab()`

with `MatchIt`

When using `bal.tab()`

with `MatchIt`

, fewer
arguments need to be specified because information is stored in the
`matchit`

object, the output of a call to
`matchit()`

. `bal.tab()`

is used very similarly to
`summary()`

in `MatchIt`

: it takes in a
`matchit`

object as its input, and prints a balance table
with the requested information. Below is a simple example of its
use:

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

The output looks very similar to `MatchIt`

’s
`summary()`

function: first is the balance table, and second
is a summary of the sample size before and after adjustment.

Setting `binary = "std"`

in `bal.tab()`

will
produce identical calculations to those in `MatchIt`

’s
`summary(m.out, standardize = TRUE)`

, which produces
standardized differences for binary variables as well as continuous
variables. The other arguments to `bal.tab()`

when using it
with `MatchIt`

have the same form and function as those given
when using it without a conditioning package. The output when using
`MatchIt`

for subclassification is the same as that displayed
previously.

`bal.tab()`

with `WeightIt`

The `WeightIt`

package is a convenient wrapper for the
weighting functions of several packages, including `twang`

,
`CBPS`

, `ebal`

, `sbw`

, and
`ATE`

, and allows for the estimation of weights for binary,
multi-category, and categorical treatments and both point and
longitudinal treatments. It was designed to work seamlessly with
`cobalt`

, so using it with `cobalt`

is very
straightforward. Below is a simple example of using
`bal.tab()`

with `WeightIt`

:

```
data("lalonde", package = "cobalt") #If not yet loaded
#Generating propensity score weights for the ATT
<- WeightIt::weightit(treat ~ age + educ + race + re74 + re75,
W.out data = lalonde,
method = "ps",
estimand = "ATT")
bal.tab(W.out)
```

`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
<- MatchIt::matchit(treat ~ age + educ + married + race +
m.out + re74 + re75,
nodegree 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:

```
<- data.frame(old = c("age", "educ", "race_black", "race_hispan",
v "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.

`cobalt`

with continuous treatmentsAlthough 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
<- WeightIt::weightit(re75 ~ age + educ + race + married +
W.out.c + re74 + I(re74^2),
nodegree 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)
```

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)
```

`cobalt`

with multi-category treatmentsWhen 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
<- WeightIt::weightit(race ~ age + educ + married +
W.out.mn + re74 + re75,
nodegree 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)
#Assessing balance for each pair of treatments
bal.tab(W.out.mn, un = TRUE,
disp.means = TRUE,
which.treat = .all)
```

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)
```

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 +
+ re74 + re75, data = lalonde,
nodegree weights = list(Matched = m.out,
IPW = W.out),
disp.v.ratio = TRUE)
```

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 +
+ re74 + re75,
nodegree 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))`

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:

- Estimate the outcome model in the control group
- Generate model-predicted outcome values for both the treated and control groups
- 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()`

:

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

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.

There are calculations in `cobalt`

that may be opaque to
users; this section explains them.

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* sample^{4}. 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.

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:

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

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

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.

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.

`cobalt`

A fair amount is missing in `cobalt`

that is present in
other packages. Though there is value in many of the aspects that
`cobalt`

lacks, many were purposefully excluded based on
methodological recommendations that conflict with the current use of
some other packages. Below are aspects that are intentionally missing
from `cobalt`

that users may be used to from other packages.
Their reasons for exclusion are included, with the hope that users of
`cobalt`

will be satisfied with what is available and be
confident they are using the most methodologically sound tools for
balance assessment.

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 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.

`cobalt`

There are several features in `cobalt`

that are present in
few, if any, balance assessment tools in the major packages. These come
from methodological recommendations and requests by members of the
methodological community.

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.

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.

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 variables^{6}.

`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.

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 analyst^{7}. 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.

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 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.

`cobalt`

with Your
PackageIf you are designing a new R package for preprocessing that performs
similar functions to `MatchIt`

, `twang`

,
`Matching`

, `optmatch`

, `CBPS`

,
`ebal`

, `designmatch`

, `cem`

, or
`WeightIt`

, you might consider integrating
`cobalt`

into your package to avoid programming your own
balance assessment tool. The simplest way to do so is to have the output
of your preprocessing function contain sufficient elements for use with
the default method for `bal.tab()`

and
`bal.plot()`

. See `?bal.tab.default`

for more
information on how this might work.

As `cobalt`

is updated to remain in line with
methodological recommendations, the balance assessment capabilities for
your function’s output will also improve if `cobalt`

is a
balance assessment tool for your package. In this way, users of your
package can use the most up-to-date balance assessment tools programmed
in `cobalt`

without you having to update your package.

If you develop a new balance assessment tool, this may also be able
to be integrated into `cobalt`

, especially if it would be
applicable to balance assessment generally in matching, weighting, or
subclassification. Incorporating a new tool into `cobalt`

may
be a good way to broaden its use.

`cobalt`

includes a small suite of functions that compute
balance statistics on matrices of covariates with and without weights.
These include `col_w_smd`

for computing (standardized) mean
differences, `col_w_vr`

for computing variance ratios,
`col_w_ks`

for computing KS statistics,
`col_w_ovl`

for computing the complement of the overlap
statistics (Franklin et al. 2014), and
`col_w_corr`

for computing treatment-covariate correlations.
These all run fairly quickly but are still quite flexible, so they can
be used to quickly and simply compute balance statistics in other
packages. They are used internally in `bal.tab`

, and they
might be able to be used internally in other functions (e.g., to choose
a tuning parameter based on a measure of balance).

I thank Kirsten Kainz and Elizabeth Stuart for their support and
advice on `cobalt`

’s creation. I thank Zachary Fisher and
Brian Barkley for their advice on developing an R package.

Ali, M. Sanni, Rolf H. H. Groenwold, Svetlana V. Belitser, Wiebe R.
Pestman, Arno W. Hoes, Kit C. B. Roes, Anthonius de Boer, and Olaf H.
Klungel. 2015. “Reporting of Covariate Selection and Balance
Assessment in Propensity Score Analysis Is Suboptimal: A Systematic
Review.” *Journal of Clinical Epidemiology* 68 (2):
122–31. https://doi.org/10.1016/j.jclinepi.2014.08.011.

Ali, M. Sanni, Rolf H. H. Groenwold, Wiebe R. Pestman, Svetlana V.
Belitser, Kit C. B. Roes, Arno W. Hoes, Anthonius de Boer, and Olaf H.
Klungel. 2014. “Propensity Score Balance Measures in
Pharmacoepidemiology: A Simulation Study.”
*Pharmacoepidemiology and Drug Safety* 23 (8): 802–11. https://doi.org/10.1002/pds.3574.

Austin, Peter C. 2008a. “A Critical Appraisal of Propensity-Score
Matching in the Medical Literature Between 1996 and 2003.”
*Statistics in Medicine* 27 (12): 2037–49. https://doi.org/10.1002/sim.3150.

———. 2008b. “Assessing Balance in Measured Baseline Covariates
When Using Many-to-One Matching on the Propensity-Score.”
*Pharmacoepidemiology and Drug Safety* 17 (12): 1218–25. https://doi.org/10.1002/pds.1674.

———. 2009. “Balance Diagnostics for Comparing the Distribution of
Baseline Covariates Between Treatment Groups in Propensity-Score Matched
Samples.” *Statistics in Medicine* 28 (25): 3083–3107. https://doi.org/10.1002/sim.3697.

———. 2011. “An Introduction to Propensity Score Methods for
Reducing the Effects of Confounding in Observational Studies.”
*Multivariate Behavioral Research* 46 (3): 399–424. https://doi.org/10.1080/00273171.2011.568786.

———. 2019. “Assessing Covariate Balance When Using the Generalized
Propensity Score with Quantitative or Continuous Exposures.”
*Statistical Methods in Medical Research* 28 (5): 1365–77. https://doi.org/10.1177/0962280218756159.

Austin, Peter C., and Elizabeth A. Stuart. 2015. “Moving Towards
Best Practice When Using Inverse Probability of Treatment Weighting
(IPTW) Using the Propensity Score to Estimate Causal
Treatment Effects in Observational Studies.” *Statistics in
Medicine* 34 (28): 3661–79. https://doi.org/10.1002/sim.6607.

Belitser, Svetlana V., Edwin P. Martens, Wiebe R. Pestman, Rolf H. H.
Groenwold, Anthonius de Boer, and Olaf H. Klungel. 2011.
“Measuring Balance and Model Selection in Propensity Score
Methods.” *Pharmacoepidemiology and Drug Safety* 20 (11):
1115–29. https://doi.org/10.1002/pds.2188.

Bia, Michela, and Alessandra Mattei. 2008. “A Stata
Package for the Estimation of the
Dose-Response Function Through
Adjustment for the Generalized Propensity
Score.” *The Stata Journal: Promoting Communications on
Statistics and Stata* 8 (3): 354–73. https://doi.org/10.1177/1536867X0800800303.

Fong, Christian, Chad Hazlett, and Kosuke Imai. 2018. “Covariate
Balancing Propensity Score for a Continuous Treatment: Application to
the Efficacy of Political Advertisements.” *The Annals of
Applied Statistics* 12 (1): 156–77. https://doi.org/10.1214/17-AOAS1101.

Fong, Christian, Marc Ratkovic, Chad Hazlett, Xiaolin Yang, and Kosuke
Imai. 2019. *CBPS: Covariate Balancing Propensity
Score*.

Franklin, Jessica M., Jeremy A. Rassen, Diana Ackermann, Dorothee B.
Bartels, and Sebastian Schneeweiss. 2014. “Metrics for Covariate
Balance in Cohort Studies of Causal Effects.” *Statistics in
Medicine* 33 (10): 1685–99. https://doi.org/10.1002/sim.6058.

Greifer, Noah. 2021. “WeightIt: Weighting for
Covariate Balance in Observational Studies.”

Hainmueller, Jens. 2014. “Ebal: Entropy Reweighting to Create
Balanced Samples.”

Hansen, Ben B. 2004. “Full Matching in an
Observational Study of Coaching for the
SAT.” *Journal of the American Statistical
Association* 99 (467): 609–18. https://doi.org/10.1198/016214504000000647.

———. 2008. “The Prognostic Analogue of the Propensity
Score.” *Biometrika* 95 (2): 481–88. https://doi.org/10.1093/biomet/asn004.

Hansen, Ben B, and Stephanie Olsen Klopfer. 2006. “Optimal
Full Matching and Related Designs via
Network Flows.” *Journal of Computational and
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Hirano, Keisuke, and Guido W. Imbens. 2005. “The Propensity
Score with Continuous Treatments.” In
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Ho, Daniel E., Kosuke Imai, Gary King, and Elizabeth A. Stuart. 2007.
“Matching as Nonparametric Preprocessing for
Reducing Model Dependence in Parametric Causal
Inference.” *Political Analysis* 15 (3): 199–236.
https://doi.org/10.1093/pan/mpl013.

———. 2011. “MatchIt: Nonparametric Preprocessing for
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Iacus, Stefano, Gary King, and Giuseppe Porro. 2009. “Cem:
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Imai, Kosuke, Gary King, and Elizabeth A. Stuart. 2008.
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Observationalists about Causal
Inference.” *Journal of the Royal Statistical Society.
Series A (Statistics in Society)* 171 (2): 481–502. https://doi.org/10.1111/j.1467-985X.2007.00527.x.

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King, Gary, and Richard Nielsen. 2019. “Why Propensity
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Linden, Ariel. 2014. “Combining Propensity Score-Based
Stratification and Weighting to Improve Causal Inference in the
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Weighting After Multiple Imputation.”
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Post-Traffic Stop Outcomes Using Propensity Scores.” *Journal
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———. 2010. “Matching Methods for Causal
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Stuart, Elizabeth A., Brian K. Lee, and Finbarr P. Leacy. 2013.
“Prognostic Score-Based Balance Measures Can Be a Useful
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69–110. https://doi.org/10.1007/s10742-020-00236-2.

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Zubizarreta, Jose R., Cinar Kilcioglu, and Juan P. Vielma. 2018.
“Designmatch: Matched Samples That Are Balanced and Representative
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Balancing Weights for Causal Inference and Missing Data*. Manual.

If none of the examples are working, this may be because

`MatchIt`

,`twang`

, or`WeightIt`

are missing. Such issues should be fixed soon.↩︎`WeightIt`

is a wrapper for`twang`

,`CBPS`

,`ebal`

, and other packages for point treatments, so it can perform their functions as well.↩︎Older versions used the arguments

`disp.means`

and`disp.sds`

, and these are still allowed.↩︎Unless common support is used in

`Matching`

, in which case the standard deviation is computed using the remaining unadjusted sample.↩︎It is important to note that both

`twang`

and`Matching`

calculate standardized mean differences using the standard deviation of the sample in question, not the unadjusted sample, though`CBPS`

uses the standard deviation of the unadjusted sample.↩︎Ali et al. (2014) compared the KS statistic to standardized mean differences, but the KS statistic is equivalent to the raw difference in proportion for binary variables.↩︎

This value is a linear function of the other observed values, so it would in principle be able to be computed by the analyst, but it seems more valuable to explicitly include this “redundant calculation” for the sake of ease and completeness.↩︎