# Optimizing Tuning Parameters for Balance

Part of using balancing methods, including matching, weighting, and subclassification, involves specifying a conditioning method, assessing balance after applying that method, and repeating until satisfactory balance is achieved. For example, in propensity score score subclassification, one needs to decide on the number of subclasses to use, and one way to do so is to try a number of of subclasses, assess balance after subclassification, try another number of subclasses, assess balance, and so on. As another example, in estimating the propensity score model itself, one might decided which covariates to include in the model (after deciding on a fixed set of covariates to balance), which covariates should have squares or interactions, and which link function (e.g., probit, logit) to use. Or choosing the number of matches each unit should receive in k:1 matching, or which value of the propensity score should be used to trim samples, etc.

Essentially, these problems all involve selecting a specification by varying a number of parameters, which are sometimes called “tuning parameters”, in order to optimize balance. Many popular methods adjust tuning parameters to optimize balance as inherent parts of the method, like genetic matching (Diamond and Sekhon 2013), which tunes variance importance in the distance matrix, or generalized boosted modeling (McCaffrey, Ridgeway, and Morral 2004), which tunes the number of trees in the prediction model for propensity scores. This strategy tends to yield methods that perform better than methods that don’t tune at all or tune to optimize a criterion other than balance (e.g., prediction accuracy) Pirracchio and Carone (2018).

As of version 4.5.0, `cobalt` provides the functions `bal.compute()` and `bal.init()` to aid in selecting these tuning parameters in an efficient way without needing to manually program the computation of the balance statistic used as the criterion to optimize. This vignette explains how to use these functions, describes the balance statistics that are available, and provides examples of using these functions to implement new and existing balancing methods yourself. These functions are primarily for use inside other packages but may be useful to users experimenting with new methods. For a complete way to assess balance for a single specification, users should use `bal.tab()` and `bal.plot()` instead.

## `bal.compute()` and `bal.init()`

Broadly, these functions work by taking in the treatment, covariates for which balance is to be computed, and a set of balancing weights and return a scalar balance statistic that summarizes balance for the sample. `bal.compute()` does the work of computing the balance statistic, and `bal.init()` processes the inputs so they don’t need to be processed every time `bal.compute()` is called with a new set of weights.

For `bal.init()`, we need to supply the covariates, the treatment, the name of the balance statistic we wish to compute, sampling weights (if any), and any other inputs required, which depend on the specific balance statistic requested. `bal.init()` returns a `bal.init` object, which is then passed to `bal.compute()` along with a set of balancing weights (which may result from weighting, matching, or subclassification).

Below, we provide an example using the `lalonde` dataset. Our balance statistic will be the largest absolute standardized mean difference among the included covariates, which is specified as `"smd.max"`. We will first supply the required inputs to `bal.init()` and pass its output to `bal.compute()` to compute the balance statistic for the sample prior to weighting.

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

    covs <- subset(lalonde, select = -c(treat, race, re78))

    # Initialize the object with the balance statistic,
    # treatment, and covariates
    smd.init <- bal.init(covs,
                         treat = lalonde$treat,
                         stat = "smd.max",
                         estimand = "ATT")

    # Compute balance with no weights
    bal.compute(smd.init)
    #> [1] 0.8263093

    # Can also compute the statistic directly using bal.compute():
    bal.compute(covs,
                treat = lalonde$treat,
                stat = "smd.max",
                estimand = "ATT")
    #> [1] 0.8263093

The largest absolute standardized mean difference with no weights is 0.8263, which we can verify and investigate further using `bal.tab()`:

    bal.tab(covs,
            treat = lalonde$treat,
            binary = "std",
            estimand = "ATT",
            thresholds = .05)
    #> Balance Measures
    #>             Type Diff.Un      M.Threshold.Un
    #> age      Contin. -0.3094 Not Balanced, >0.05
    #> educ     Contin.  0.0550 Not Balanced, >0.05
    #> married   Binary -0.8263 Not Balanced, >0.05
    #> nodegree  Binary  0.2450 Not Balanced, >0.05
    #> re74     Contin. -0.7211 Not Balanced, >0.05
    #> re75     Contin. -0.2903 Not Balanced, >0.05
    #>
    #> Balance tally for mean differences
    #>                     count
    #> Balanced, <0.05         0
    #> Not Balanced, >0.05     6
    #>
    #> Variable with the greatest mean difference
    #>  Variable Diff.Un      M.Threshold.Un
    #>   married -0.8263 Not Balanced, >0.05
    #>
    #> Sample sizes
    #>     Control Treated
    #> All     429     185

We can see that the largest value corresponds to the covariate `married`.

Now, lets estimate weights using probit regression propensity scores in `WeightIt` and see whether this balance statistic decreases after applying the weights:

    library("WeightIt")
    w.out <- weightit(treat ~ age + educ + married + nodegree +
                          re74 + re75, data = lalonde,
                      method = "glm", estimand = "ATT",
                      link = "probit")

    # Compute the balance statistic on the estimated weights
    bal.compute(smd.init, get.w(w.out))
    #> [1] 0.06936946

After weighting, our balance statistic is 0.0694, indicating a significant improvement. Let’s try again with logistic regression:

    w.out <- weightit(treat ~ age + educ + married + nodegree +
                          re74 + re75, data = lalonde,
                      method = "glm", estimand = "ATT",
                      link = "logit")

    # Compute the balance statistic on the estimated weights
    bal.compute(smd.init, get.w(w.out))
    #> [1] 0.04791925

This is better, but we can do even better with bias-reduced logistic regression (Kosmidis and Firth 2020):

    w.out <- weightit(treat ~ age + educ + married + nodegree +
                          re74 + re75, data = lalonde,
                      method = "glm", estimand = "ATT",
                      link = "br.logit")

    # Compute the balance statistic on the estimated weights
    bal.compute(smd.init, get.w(w.out))
    #> [1] 0.04392724

Instead of writing each complete call one at a time, we can do a little programming to make this happen automatically:

    # Initialize object to compute the largest SMD
    smd.init <- bal.init(covs,
                         treat = lalonde$treat,
                         stat = "smd.max",
                         estimand = "ATT")

    # Create vector of tuning parameters
    links <- c("probit", "logit", "cloglog",
               "br.probit", "br.logit", "br.cloglog")

    # Apply each link to estimate weights
    # Can replace sapply() with purrr::map()
    weights.list <- sapply(links, function(link) {
        w.out <- weightit(treat ~ age + educ + married + nodegree +
                          re74 + re75, data = lalonde,
                      method = "glm", estimand = "ATT",
                      link = link)
        get.w(w.out)
    }, simplify = FALSE)

    # Use each set of weights to compute balance
    # Can replace sapply() with purrr:map_vec()
    stats <- sapply(weights.list, bal.compute,
                    x = smd.init)

    # See which set of weights is the best
    stats
    #>     probit      logit    cloglog  br.probit   br.logit br.cloglog
    #> 0.06936946 0.04791925 0.02726102 0.06444577 0.04392724 0.02457557
    stats[which.min(stats)]
    #> br.cloglog
    #> 0.02457557

Interestingly, bias-reduced complimentary log-log regression produced weights with the smallest maximum absolute standardized mean difference. We can use `bal.tab()` to more finely examine balance on the chosen weights:

    bal.tab(covs,
            treat = lalonde$treat,
            binary = "std",
            weights = weights.list[["br.cloglog"]])
    #> Balance Measures
    #>             Type Diff.Adj
    #> age      Contin.  -0.0089
    #> educ     Contin.  -0.0246
    #> married   Binary  -0.0012
    #> nodegree  Binary   0.0145
    #> re74     Contin.  -0.0209
    #> re75     Contin.  -0.0213
    #>
    #> Effective sample sizes
    #>            Control Treated
    #> Unadjusted  429.       185
    #> Adjusted    240.83     185

If balance is acceptable, you would move forward with these weights in estimating the treatment effect. Otherwise, you might try other values of the tuning parameters, other specifications of the model, or other weighting methods to try to achieve excellent balance.

## Balance statistics

Several balance statistics can be computed by `bal.compute()` and `bal.init()`, and the ones available depend on whether the treatment is binary, multi-category, or continuous. These are explained below and on the help page `?bal.compute`. A complete list for a given treatment type can be requested using `available.stats()`. Some balance statistics are appended with `".mean"`, `".max"`, or `".rms"`, which correspond to the mean (or L1-norm), maximum (or L-infinity norm), and root mean square (or L2-norm) of the absolute univariate balance statistic computed for each covariate.

#### `smd.mean`, `smd.max`, `smd.rms`

The mean, maximum, and root mean square of the absolute standardized mean differences computed for the covariates using `col_w_smd()`. The other allowable arguments include `estimand` (ATE, ATC, or ATT) to select the estimand, `focal` to identify the focal treatment group when the ATT is the estimand and the treatment has more than two categories, and `pairwise` to select whether mean differences should be computed between each pair of treatment groups or between each treatment group and the target group identified by `estimand` (default `TRUE`). Can be used with binary and multi-category treatments.

#### `ks.mean`, `ks.max`, `ks.rms`

The mean, maximum, or root-mean-squared Kolmogorov-Smirnov statistic, computed using `col_w_ks()`. The other allowable arguments include `estimand` (ATE, ATC, or ATT) to select the estimand, `focal` to identify the focal treatment group when the ATT is the estimand and the treatment has more than two categories, and `pairwise` to select whether statistics should be computed between each pair of treatment groups or between each treatment group and the target group identified by `estimand` (default `TRUE`). Can be used with binary and multi-category treatments.

#### `ovl.mean`, `ovl.max`, `ovl.rms`

The mean, maximum, or root-mean-squared overlapping coefficient compliment, computed using `col_w_ovl()`. The other allowable arguments include `estimand` (ATE, ATC, or ATT) to select the estimand, `focal` to identify the focal treatment group when the ATT is the estimand and the treatment has more than two categories, and `pairwise` to select whether statistics should be computed between each pair of treatment groups or between each treatment group and the target group identified by `estimand` (default `TRUE`). Can be used with binary and multi-category treatments.

#### `mahalanobis`

The Mahalanobis distance between the treatment group means, which is computed as \[ \sqrt{(\mathbf{\bar{x}}_1 - \mathbf{\bar{x}}_0) \Sigma^{-1} (\mathbf{\bar{x}}_1 - \mathbf{\bar{x}}_0)} \] where \(\mathbf{\bar{x}}_1\) and \(\mathbf{\bar{x}}_0\) are the vectors of covariate means in the two treatment groups, \(\Sigma^-1\) is the (generalized) inverse of the covariance matrix of the covariates (Franklin et al. 2014). This is similar to `"smd.rms"` but the covariates are standardized to remove correlations between and de-emphasize redundant covariates. The other allowable arguments include `estimand` (ATE, ATC, or ATT) to select the estimand, which determines how the covariance matrix is calculated, and `focal` to identify the focal treatment group when the ATT is the estimand. Can only be used with binary treatments.

#### `energy.dist`

The total energy distance between each treatment group and the target sample, which is a scalar measure of the similarity between two multivariate distributions. See Huling and Mak (2022) for details. The other allowable arguments include `estimand` (ATE, ATC, or ATT) to select the estimand, `focal` to identify the focal treatment group when the ATT is the estimand and the treatment has more than two categories, and `improved` to select whether the “improved” energy distance should be used, which emphasizes difference between treatment groups in addition to difference between each treatment group and the target sample (default `TRUE`). Can be used with binary and multi-category treatments.

#### `kernel.dist`

The kernel distance between treatment groups, which is a scalar measure of the similarity between two multivariate distributions. See Zhu, Savage, and Ghosh (2018) for details. Can only be used with binary treatments.

#### `l1.med`

The median L1 statistic computed across a random selection of possible coarsening of the data. See Iacus, King, and Porro (2011) for details. The other allowable arguments include `l1.min.bin` (default 2) and `l1.max.bin` default (12) to select the minimum and maximum number of bins with which to bin continuous variables and `l1.n` (default 101) to select the number of binnings used to select the binning at the median. `covs` should be supplied without splitting factors into dummies to ensure the binning works correctly. Can be used with binary and multi-category treatments.

#### `r2`, `r2.2`, `r2.3`

The post-weighting \(R^2\) of a model for the treatment given the covariates. Franklin et al. (2014) describe a similar but less generalizable metric, the “post-matching c-statistic”. The other allowable arguments include `poly` to add polynomial terms of the supplied order to the model and `int` (default `FALSE`) to add two-way interactions between covariates into the model. Using `"r2.2"` is a shortcut to requesting squares, and using `"r2.3"` is a shortcut to requesting cubes. Can be used with binary and continuous treatments. For binary treatments, the McKelvey and Zavoina \(R^2\) from a logistic regression is used; for continuous treatments, the \(R^2\) from a linear regression is used.

#### `p.mean`, `p.max`, `p.rms`

The mean, maximum, or root-mean-squared absolute Pearson correlation between the treatment and covariates, computed using `col_w_corr()`. Can only be used with continuous treatments.

#### `s.mean`, `s.max`, `s.rms`

The mean, maximum, or root-mean-squared absolute Spearman correlation between the treatment and covariates, computed using `col_w_corr()`. Can only be used with continuous treatments.

#### `distance.cov`

The distance covariance between the scaled covariates and treatment, which is a scalar measure of the independence of two possibly multivariate distributions. See Huling, Greifer, and Chen (2023) for details. Can only be used with continuous treatments.

### Choosing a balance statistic

Given all these options, how should one choose? There has been some research into which yields the best results (Franklin et al. 2014; Griffin et al. 2017; Stuart, Lee, and Leacy 2013; Belitser et al. 2011; Parast et al. 2017), but the actual performance of each depends on the unique features of the data and system under study. For example, in the unlikely case that the true outcome model is linear in the covariates, using the `"smd"` or `"mahalanobis"` statistics will work well for binary and multi-category treatments. In more realistic cases, though, every measure has its advantages and disadvantages.

For binary and multi-category treatments, only `"energy.dist"`, `"kernel.dist"`, and `"L1.med"` reflect balance on all features of the joint covariate distribution, whereas the others summarize across balance statistics computed for each covariate ignoring the others. Similarly, for continuous treatments, only `"distance.cov"` reflects balance on all features of the joint covariate distribution. Given these advantages, `"energy.dist"` and `"distance.cov"` are my preferences. That said, other measures are better studied, possibly more intuitive, and more familiar to a broad audience.

## Example

In this section, I will provide an example that demonstrates how these functions could be used to replicate the functionality of existing packages or develop new methods for optimizing balance. We will use these functions to replicate the functionality of `WeightIt` and `twang` for estimating propensity score weights for a binary treatment using generalized boosted modeling (GBM). See `help("bal.compute")` for another example that optimizes balance to find the number of subclasses in propensity score subclassification.

### Tuning GBM for balance

GBM has many tuning parameters that can be optimized, but the key parameter is the number of trees to use to calculate the predictions. `WeightIt` and `twang` both implement the methods described in McCaffrey, Ridgeway, and Morral (2004) for selecting the number of trees using a balance criterion. Here, we will do so manually both to understand the internals of these functions and illustrate the uses of `bal.compute()` and `bal.init()`.

    data("lalonde")

    # Initialize balance
    covs <- subset(lalonde, select = -c(treat, re78))
    ks.init <- bal.init(covs,
                        treat = lalonde$treat,
                        stat = "ks.max",
                        estimand = "ATT")

    # Fit a GBM model using `WeightIt` and `twang` defaults
    fit <- gbm::gbm(treat ~ age + educ + married + race +
                        nodegree + re74 + re75,
                    data = lalonde,
                    distribution = "bernoulli",
                    n.trees = 4000, interaction.depth = 3,
                    shrinkage = .01, bag.fraction = 1)

    trees_to_test <- seq(1, 4000)

    p.mat <- predict(fit, type = "response",
                     n.trees = trees_to_test)

    stats <- apply(p.mat, 2, function(p) {
        # Compute ATT weights
        w <- ifelse(lalonde$treat == 1, 1, p/(1-p))

        bal.compute(ks.init, weights = w)
    })

    stats[which.min(stats)]
    #>       1408
    #> 0.09563649

From these results, we see that using 1408 trees gives us the lowest maximum KS statistic of 0.0956. Out of interest, we can plot the relationship between the number of trees and our balance statistic to see what it looks like:

    library("ggplot2")
    ggplot() +
        geom_line(aes(x = trees_to_test, y = stats)) +
        theme_bw() +
        labs(y = "ks.max", x = "n.trees")

Let’s compare this to the output of `WeightIt`:

    library("WeightIt")
    w.out <- weightit(treat ~ age + educ + married + race +
                        nodegree + re74 + re75,
                    data = lalonde, estimand = "ATT",
                    method = "gbm", n.trees = 4000,
                    stop.method = "ks.max")

    # Display the best tree:
    w.out$info$best.tree
    #> [1] 1408

    # ks.max from weightit()
    bal.compute(ks.init, weights = get.w(w.out))
    #> [1] 0.09563649

We can see that `weightit()` also selects 1408 trees as the optimum and the resulting maximum KS statistic computed using the returned weights is equal to the one we computed manually. Using `twang` also produces the same results.

## References

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.
Diamond, Alexis, and Jasjeet S. Sekhon. 2013. “Genetic Matching for Estimating Causal Effects: A General Multivariate Matching Method for Achieving Balance in Observational Studies.” Review of Economics and Statistics 95 (3): 932945. https://doi.org/10.1162/REST_a_00318.
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.
Griffin, Beth Ann, Daniel F. McCaffrey, Daniel Almirall, Lane F. Burgette, and Claude Messan Setodji. 2017. “Chasing Balance and Other Recommendations for Improving Nonparametric Propensity Score Models.” Journal of Causal Inference 5 (2). https://doi.org/10.1515/jci-2015-0026.
Huling, Jared D., Noah Greifer, and Guanhua Chen. 2023. “Independence Weights for Causal Inference with Continuous Treatments.” Journal of the American Statistical Association 0 (0): 1–14. https://doi.org/10.1080/01621459.2023.2213485.
Huling, Jared D., and Simon Mak. 2022. “Energy Balancing of Covariate Distributions.” https://doi.org/10.48550/arXiv.2004.13962.
Iacus, Stefano M., Gary King, and Giuseppe Porro. 2011. “Multivariate Matching Methods That Are Monotonic Imbalance Bounding.” Journal of the American Statistical Association 106 (493): 345–61. https://doi.org/10.1198/jasa.2011.tm09599.
Kosmidis, Ioannis, and David Firth. 2020. “Jeffreys-Prior Penalty, Finiteness and Shrinkage in Binomial-Response Generalized Linear Models.” Biometrika 108 (1): 71–82. https://doi.org/10.1093/biomet/asaa052.
McCaffrey, Daniel F., Greg Ridgeway, and Andrew R. Morral. 2004. “Propensity Score Estimation With Boosted Regression for Evaluating Causal Effects in Observational Studies.” Psychological Methods 9 (4): 403–25. https://doi.org/10.1037/1082-989X.9.4.403.
Parast, Layla, Daniel F. McCaffrey, Lane F. Burgette, Fernando Hoces de la Guardia, Daniela Golinelli, Jeremy N. V. Miles, and Beth Ann Griffin. 2017. “Optimizing Variance-Bias Trade-Off in the TWANG Package for Estimation of Propensity Scores.” Health Services and Outcomes Research Methodology 17 (3): 175–97. https://doi.org/10.1007/s10742-016-0168-2.
Pirracchio, Romain, and Marco Carone. 2018. “The Balance Super Learner: A Robust Adaptation of the Super Learner to Improve Estimation of the Average Treatment Effect in the Treated Based on Propensity Score Matching.” Statistical Methods in Medical Research 27 (8): 2504–18. https://doi.org/10.1177/0962280216682055.
Stuart, Elizabeth A., Brian K. Lee, and Finbarr P. Leacy. 2013. “Prognostic Score-Based Balance Measures Can Be a Useful Diagnostic for Propensity Score Methods in Comparative Effectiveness Research.” Journal of Clinical Epidemiology 66 (8): S84. https://doi.org/10.1016/j.jclinepi.2013.01.013.
Zhu, Yeying, Jennifer S. Savage, and Debashis Ghosh. 2018. “A Kernel-Based Metric for Balance Assessment.” Journal of Causal Inference 6 (2). https://doi.org/10.1515/jci-2016-0029.