---
title: "Covariate Balance Tables and Plots: A Guide to the `cobalt` Package"
author: "Noah Greifer"
date: "`r Sys.Date()`"
output: 
    html_vignette:
        df_print: kable
        toc: true
vignette: >
  %\VignetteEncoding{UTF-8}
  %\VignetteIndexEntry{Covariate Balance Tables and Plots: A Guide to the `cobalt` Package}
  %\VignetteEngine{knitr::rmarkdown_notangle}
editor_options: 
  chunk_output_type: console
bibliography: references.bib
link-citations: true
---

```{r, include = FALSE}
knitr::opts_chunk$set(message = FALSE)
if (any(!sapply(c("WeightIt", "MatchIt"), requireNamespace, quietly = TRUE))) knitr::opts_chunk$set(eval = FALSE)
```

This is an introductory guide for the use of `cobalt` in most common scenarios. Several appendices are available for its use with more complicated data scenarios and packages not demonstrated here.[^1]

[^1]: If none of the examples are working, this may be because `MatchIt`, `twang`, or `WeightIt` are missing. Such issues should be fixed soon.

## Introduction

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 [@hoMatchingNonparametricPreprocessing2007]. 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 @stuartMatchingMethodsCausal2010 and @austinIntroductionPropensityScore2011, 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 @kellerPropensityScoreAnalysis2016. These include `MatchIt` [@JSSv042i08], `twang` [@ridgewayToolkitWeightingAnalysis2016], `Matching` [@sekhonMultivariatePropensityScore2011], `optmatch` [@hansenOptimalFullMatching2006], `CBPS` [@fongCBPSCovariateBalancing2019], `ebal` [@hainmuellerEbalEntropyReweighting2014], `sbw` [@zubizarretaSbwStableBalancing2021], `designmatch` [@zubizarretaDesignmatchMatchedSamples2018], `WeightIt` [@greiferWeightItWeightingCovariate2021], `MatchThem` [@pishgarMatchThemMatchingWeighting2021], and `cem` [@iacusCemSoftwareCoarsened2009]; 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 [@stuartMatchingMethodsCausal2010]:

1.  Decide on covariates for which balance must be achieved
2.  Estimate the distance measure (e.g., propensity score)
3.  Condition on the distance measure (e.g., using matching, weighting, or subclassification)
4.  Assess balance on the covariates of interest; if poor, repeat steps 2-4
5.  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 [@thoemmesSystematicReviewPropensity2011]. 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 [@hoMatchingNonparametricPreprocessing2007].

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 [@thoemmesSystematicReviewPropensity2011]. 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.

#### Citing `cobalt`

When using `cobalt`, please cite your use of it along with the conditioning package used. 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, `r format(Sys.Date(), "%Y")`), both in R (R Core Team, `r R.version$year`).

 Use `citation("cobalt")` to generate the correct citation for `cobalt`.

## Why `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:

#### 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 [@stuartPrognosticScorebasedBalance2013].

#### 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 `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.

[^2]: `WeightIt` is a wrapper for `twang`, `CBPS`, `ebal`, and other packages for point treatments, so it can perform their functions as well.

#### 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 `vignette("segmented-data")`. 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 `vignette("longitudinal-treat")`.

## How To Use cobalt

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:

```{r, include = F}
library("cobalt")
```

```{r, eval = F}
install.packages("cobalt")
library("cobalt")
```

### Utilities

In addition to its main functions, `cobalt` contains several utility functions, which include `f.build()`, `splitfactor()` and `unsplitfactor()`, `get.w()`, and `bal.init()` and `bal.compute()`. 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 `vignette("other-packages")`.

### `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 `vignette("other-packages")`.

#### Using `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:

```{r}
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)

```

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. See `?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:

```{r}
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`.

```{r}
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.

```{r}
# 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 @austin2009, 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].

[^3]: Older versions used the arguments `disp.means` and `disp.sds`, and these are still allowed.

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.coefficients"` for the complement of the overlapping coefficient (abbreviations are allowed). See `help("balance-statistics")` 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 [@austin2009]. 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. @austinMovingBestPractice2015 recommend its use, and it or a variant appears as a default balance statistic in `MatchIt`, `twang`, and `Matching`. On the other hand, @belitserMeasuringBalanceModel2011, @stuartPrognosticScorebasedBalance2013, and @aliPropensityScoreBalance2014 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 @franklinMetricsCovariateBalance2014, the complement is used so that 0 indicates perfectly overlapping distributions and 1 indicates perfectly non-overlapping distributions. The overlapping coefficient complement functions similarly to the KS statistic in that it summarizes imbalance across the whole distribution of the covariate, not just its mean or variance, but avoids some of the weaknesses of 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 de-clutter the output to maintain the spotlight on the group balance after adjustment.

```{r}
# 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 @stuartPrognosticScorebasedBalance2013 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`).

```{r}
# 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".

```{r}
# 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")
```

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.

```{r}
# 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)
```

When using `bal.tab()` with continuous treatments, the default balance statistic presented is the (weighted) Pearson correlation between each covariate and treatment. @zhuBoostingAlgorithmEstimating2015 recommend that absolute correlations should be no greater than 0.1, but correlations should ideally be as close to zero as possible. Spearman correlations and distance 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.

#### Using `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:

```{r}
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)
```

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.

#### Using `bal.tab()` with `WeightIt`

The `WeightIt` package provides functions and utilities for estimating balancing weights 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`:

```{r}
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 = "glm",
                            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 statistic and the overlapping coefficient complement that attempt to summarize distributional balance beyond the first few moments [@austinMovingBestPractice2015; @aliReportingCovariateSelection2015], complimenting statistics with a visual examination of the distributional densities can be an effective way of assessing distributional similarity between the groups [@hoMatchingNonparametricPreprocessing2007].

`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.:

```{r, fig.show = "hold", fig.width = 3.5, fig.height=2.75}
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:

```{r, fig.width = 5}
#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 [@stuartPrognosticScorebasedBalance2013]. 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. @kingWhyPropensityScores2019]. `love.plot()` does just this, providing the user with several options to customize their plot for presentation. Below is an example of its use:

```{r, fig.width = 5}
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:

```{r, fig.width = 5}
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 `vignette("love.plot")` for information on more of `love.plot()`'s features.

## Additional 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 [@hiranoPropensityScoreContinuous2005]. 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.

@biaStataPackageEstimation2008 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 @hiranoPropensityScoreContinuous2005, 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 @fongCovariateBalancingPropensity2018, GBM as described by @zhuBoostingAlgorithmEstimating2015, and entropy balancing as described by @vegetabileNonparametricEstimationPopulation2020, 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 @zhuBoostingAlgorithmEstimating2015 and @austinAssessingCovariateBalance2019, but not that described in @hiranoPropensityScoreContinuous2005 or implemented in `gpscore` [@biaStataPackageEstimation2008], 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. Correlations can be computed as Pearson correlations, Spearman correlations (Pearson correlations on the ranks), or distance correlations.

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.

```{r}
data("lalonde", package = "cobalt")

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

First, we can assess balance numerically using `bal.tab()`. The default 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`; @zhuBoostingAlgorithmEstimating2015 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). Requesting the distance correlation can also be useful for assessing independence because they are 0 only when the treatment and covariate are completely independent, not just uncorrelated. In addition to treatment-covariate correlations, we request KS statistics between the weighted and unweighted samples by include `"ks"` (for `"ks.statistics.target"`) in the argument to `stats` along with `"c"` (for `"correlations"`).

```{r}
#Assessing balance numerically
bal.tab(W.out.c, stats = c("c", "k"), 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.

```{r, fig.width = 5}
#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.

```{r, fig.width = 6, fig.height = 3}
#Summarizing balance in a Love plot
love.plot(W.out.c, stats = c("c", "ks"),
          thresholds = c(cor = .1), 
          abs = TRUE, wrap = 20,
          limits = list(ks = c(0, .5)),
          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. @lopezEstimationCausalEffects2017 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 @lopezEstimationCausalEffects2017 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.

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 generalized propensity score weights using multinomial logistic regression.

```{r}
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 = "glm")
```

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.

```{r}
#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",
        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.

```{r, fig.width = 5, fig.height=2.75}
#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()`.

```{r, fig.width = 7}
#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.

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

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

```{r, fig.width=7}
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.

```{r, fig.width=5}
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:

```{r, eval = FALSE}
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 [@hansenPrognosticAnaloguePropensity2008]. @stuartPrognosticScorebasedBalance2013 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()`:

```{r}
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"])
```

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.

## Details on Calculations

There are calculations in `cobalt` that may be opaque to users; this section explains them. For even more details, see `vignette("faq")`.

### 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* sample[^4]. This is line with how `MatchIt` computes standardized mean differences[^5], and is recommended by Stuart [-@stuartDevelopingPracticalRecommendations2008; -@stuartMatchingMethodsCausal2010]. One reason to favor the use of the standard deviation of the unadjusted sample is that it prevents the [paradoxical situation](https://stats.stackexchange.com/a/565705/116195) 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.

[^4]: Unless common support is used in `Matching`, in which case the standard deviation is computed using the remaining unadjusted sample, or `s.d.enom` is set to `"weighted"`, which uses the standard deviation computed in the weighted (or matched) sample.

[^5]: 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.

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, @austinAssessingBalanceMeasured2008 and @austinMovingBestPractice2015 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 [@ridgewayAssessingEffectRace2006]. 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-saPowerSampleSize2020b derived the specific relationship between the ESS and the standard error of a propensity score-weighted mean.

## What's Missing in `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.

### 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., @hansenFullMatchingObservational2004; @rosenbaumConstructingControlGroup1985]. In a review of propensity score applications in the social sciences, @thoemmesSystematicReviewPropensity2011 found that over 66% of studies used significance tests to assess balance. Likewise, @austinCriticalAppraisalPropensityscore2008 found that over 70% of studies using propensity scores in the medical literature used significance tests to assess balance, a finding replicated by @aliReportingCovariateSelection2015. 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., @hoMatchingNonparametricPreprocessing2007; @imaiMisunderstandingsExperimentalistsObservationalists2008; @austinIntroductionPropensityScore2011; @austin2009; @stuartMatchingMethodsCausal2010; @thoemmesSystematicReviewPropensity2011; @aliReportingCovariateSelection2015]. There are logical reasons for this preference against hypothesis tests, noted clearly in @aliReportingCovariateSelection2015 and @linden2014: 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 [@hoMatchingNonparametricPreprocessing2007], 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 [@hoMatchingNonparametricPreprocessing2007]. Empirical Q-Q plot summary statistics may be useful in quantifying imbalance, but currently there are few recommendations for their use.

## What's Added in `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.

### 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. @austin2009 and @linden2014 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 [@thoemmesSystematicReviewPropensity2011], variance ratios have been recommended in the literature as a means to further examine balance between groups [@austin2009; @hoMatchingNonparametricPreprocessing2007; @imaiMisunderstandingsExperimentalistsObservationalists2008]. 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 [@stuartMatchingMethodsCausal2010; @rubinUsingPropensityScores2001], 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 [@austin2009]. 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, @aliPropensityScoreBalance2014 found that the raw difference in proportion was a better predictor of bias than the standardized mean difference for binary variables[^6].

[^6]: 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.

`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 analyst[^7]. The interactions among levels of a single factor, which always be equal to 0, are excluded in `bal.tab()`.

[^7]: 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.

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 `vignette("segmented-data")`.

### 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 `vignette("segmented-data")`.

## For Programmers: Integrating `cobalt` with Your Package

If 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 overlapping statistics [@franklinMetricsCovariateBalance2014], 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). See `help("balance-summary")` for details.

The functions `bal.init()` and `bal.compute()` can be used to compute scalar balance statistics efficiently, which can be useful in tuning a parameter to achieve balance or quickly summarizing balance across many specifications. See `vignette("optimizing-balance")` for more information on how and when to use these functions.

## Acknowledgments

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.

## References
