This page explains the details of estimating weights from Bayesian additive regression trees (BART)-based propensity scores by setting method = "bart" in the call to weightit() or weightitMSM(). This method can be used with binary, multi-category, and continuous treatments.
In general, this method relies on estimating propensity scores using BART and then converting those propensity scores into weights using a formula that depends on the desired estimand. This method relies on dbartsbart2 from the dbarts package.
Binary Treatments
For binary treatments, this method estimates the propensity scores using dbartsbart2. The following estimands are allowed: ATE, ATT, ATC, ATO, ATM, and ATOS. Weights can also be computed using marginal mean weighting through stratification for the ATE, ATT, and ATC. See get_w_from_ps() for details.
Multi-Category Treatments
For multi-category treatments, the propensity scores are estimated using several calls to dbartsbart2, one for each treatment group; the treatment probabilities are not normalized to sum to 1. The following estimands are allowed: ATE, ATT, ATC, ATO, and ATM. The weights for each estimand are computed using the standard formulas or those mentioned above. Weights can also be computed using marginal mean weighting through stratification for the ATE, ATT, and ATC. See get_w_from_ps() for details.
Continuous Treatments
For continuous treatments, weights are estimated as \(w_i = f_A(a_i) / f_{A|X}(a_i)\), where \(f_A(a_i)\) (known as the stabilization factor) is the unconditional density of treatment evaluated the observed treatment value and \(f_{A|X}(a_i)\) (known as the generalized propensity score) is the conditional density of treatment given the covariates evaluated at the observed value of treatment. The shape of \(f_A(.)\) and \(f_{A|X}(.)\) is controlled by the density argument described below (normal distributions by default), and the predicted values used for the mean of the conditional density are estimated using BART as implemented in dbartsbart2. Kernel density estimation can be used instead of assuming a specific density for the numerator and denominator by setting density = "kernel". Other arguments to density() can be specified to refine the density estimation parameters.
Longitudinal Treatments
For longitudinal treatments, the weights are the product of the weights estimated at each time point.
Missing Data
In the presence of missing data, the following value(s) for missing are allowed:
"ind"(default)First, for each variable with missingness, a new missingness indicator variable is created which takes the value 1 if the original covariate is
NAand 0 otherwise. The missingness indicators are added to the model formula as main effects. The missing values in the covariates are then replaced with the covariate medians. The weight estimation then proceeds with this new formula and set of covariates. The covariates output in the resultingweightitobject will be the original covariates with theNAs.
Details
BART works by fitting a sum-of-trees model for the treatment or probability of treatment. The number of trees is determined by the n.trees argument. Bayesian priors are used for the hyperparameters, so the result is a posterior distribution of predicted values for each unit. The mean of these for each unit is taken for use in computing the (generalized) propensity score. Although the hyperparameters governing the priors can be modified by supplying arguments to weightit() that are passed to the BART fitting function, the default values tend to work well and require little modification (though the defaults differ for continuous and categorical treatments; see the dbartsbart2 documentation for details). Unlike many other machine learning methods, no loss function is optimized and the hyperparameters do not need to be tuned (e.g., using cross-validation), though performance can benefit from tuning. BART tends to balance sparseness with flexibility by using very weak learners as the trees, which makes it suitable for capturing complex functions without specifying a particular functional form and without overfitting.
Reproducibility
BART has a random component, so some work must be done to ensure reproducibility across runs. See the Reproducibility section at dbartsbart2 for more details. To ensure reproducibility, one can do one of two things: 1) supply an argument to seed, which is passed to dbarts::bart2() and sets the seed for single- and multi-threaded uses, or 2) call set.seed(), though this only ensures reproducibility when using single-threading, which can be requested by setting n.threads = 1. Note that to ensure reproducibility on any machine, regardless of the number of cores available, one should use single-threading and either supply seed or call set.seed().
Additional Arguments
All arguments to dbartsbart2 can be passed through weightit() or weightitMSM(), with the following exceptions:
test,weights,subset,offset.testare ignoredcombine.chainsis always set toTRUEsampleronlyis always set toFALSE
For continuous treatments only, the following arguments may be supplied:
densityA function corresponding to the conditional density of the treatment. The standardized residuals of the treatment model will be fed through this function to produce the numerator and denominator of the generalized propensity score weights. If blank,
dnorm()is used as recommended by Robins et al. (2000). This can also be supplied as a string containing the name of the function to be called. If the string contains underscores, the call will be split by the underscores and the latter splits will be supplied as arguments to the second argument and beyond. For example, ifdensity = "dt_2"is specified, the density used will be that of a t-distribution with 2 degrees of freedom. Using a t-distribution can be useful when extreme outcome values are observed (Naimi et al., 2014).Can also be
"kernel"to use kernel density estimation, which callsdensity()to estimate the numerator and denominator densities for the weights. (This used to be requested by settinguse.kernel = TRUE, which is now deprecated.)bw,adjust,kernel,nIf
density = "kernel", the arguments todensity(). The defaults are the same as those indensity()except thatnis 10 times the number of units in the sample.plotIf
density = "kernel", whether to plot the estimated densities.
Additional Outputs
objWhen
include.obj = TRUE, thebart2fit(s) used to generate the predicted values. With multi-category treatments, this will be a list of the fits; otherwise, it will be a single fit. The predicted probabilities used to compute the propensity scores can be extracted using 2dbartsbartfitted.
References
Hill, J., Weiss, C., & Zhai, F. (2011). Challenges With Propensity Score Strategies in a High-Dimensional Setting and a Potential Alternative. Multivariate Behavioral Research, 46(3), 477–513. doi:10.1080/00273171.2011.570161
Chipman, H. A., George, E. I., & McCulloch, R. E. (2010). BART: Bayesian additive regression trees. The Annals of Applied Statistics, 4(1), 266–298. doi:10.1214/09-AOAS285
Note that many references that deal with BART for causal inference focus on estimating potential outcomes with BART, not the propensity scores, and so are not directly relevant when using BART to estimate propensity scores for weights.
See method_glm for additional references on propensity score weighting more generally.
See also
weightit(), weightitMSM(), get_w_from_ps()
method_super for stacking predictions from several machine learning methods, including BART.
Examples
library("cobalt")
data("lalonde", package = "cobalt")
#Balancing covariates between treatment groups (binary)
(W1 <- weightit(treat ~ age + educ + married +
nodegree + re74, data = lalonde,
method = "bart", estimand = "ATT"))
#> A weightit object
#> - method: "bart" (propensity score weighting with BART)
#> - number of obs.: 614
#> - sampling weights: none
#> - treatment: 2-category
#> - estimand: ATT (focal: 1)
#> - covariates: age, educ, married, nodegree, re74
summary(W1)
#> Summary of weights
#>
#> - Weight ranges:
#>
#> Min Max
#> treated 1.000 || 1.0000
#> control 0.002 |---------------------------| 9.9715
#>
#> - Units with the 5 most extreme weights by group:
#>
#> 5 4 3 2 1
#> treated 1 1 1 1 1
#> 524 569 592 374 608
#> control 2.1032 2.8928 3.1713 3.5302 9.9715
#>
#> - Weight statistics:
#>
#> Coef of Var MAD Entropy # Zeros
#> treated 0.00 0.00 0.000 0
#> control 1.86 0.94 0.747 0
#>
#> - Effective Sample Sizes:
#>
#> Control Treated
#> Unweighted 429. 185
#> Weighted 96.35 185
bal.tab(W1)
#> Balance Measures
#> Type Diff.Adj
#> prop.score Distance 0.4928
#> age Contin. 0.0712
#> educ Contin. -0.0109
#> married Binary -0.0322
#> nodegree Binary 0.0291
#> re74 Contin. -0.0635
#>
#> Effective sample sizes
#> Control Treated
#> Unadjusted 429. 185
#> Adjusted 96.35 185
#Balancing covariates with respect to race (multi-category)
(W2 <- weightit(race ~ age + educ + married +
nodegree + re74, data = lalonde,
method = "bart", estimand = "ATE"))
#> A weightit object
#> - method: "bart" (propensity score weighting with BART)
#> - number of obs.: 614
#> - sampling weights: none
#> - treatment: 3-category (black, hispan, white)
#> - estimand: ATE
#> - covariates: age, educ, married, nodegree, re74
summary(W2)
#> Summary of weights
#>
#> - Weight ranges:
#>
#> Min Max
#> black 1.2422 |---------------| 8.5207
#> hispan 2.5673 |------------------------| 13.1425
#> white 1.0644 |--------------| 7.9128
#>
#> - Units with the 5 most extreme weights by group:
#>
#> 226 181 244 423 231
#> black 6.9311 7.3318 7.797 8.1914 8.5207
#> 346 426 512 570 564
#> hispan 11.7516 12.6563 12.7555 13.1103 13.1425
#> 68 23 60 76 140
#> white 4.4648 5.0231 5.4995 7.6074 7.9128
#>
#> - Weight statistics:
#>
#> Coef of Var MAD Entropy # Zeros
#> black 0.569 0.369 0.123 0
#> hispan 0.385 0.315 0.074 0
#> white 0.451 0.318 0.082 0
#>
#> - Effective Sample Sizes:
#>
#> black hispan white
#> Unweighted 243. 72. 299.
#> Weighted 183.81 62.83 248.54
bal.tab(W2)
#> Balance summary across all treatment pairs
#> Type Max.Diff.Adj
#> age Contin. 0.1957
#> educ Contin. 0.1559
#> married Binary 0.0536
#> nodegree Binary 0.0267
#> re74 Contin. 0.1125
#>
#> Effective sample sizes
#> black hispan white
#> Unadjusted 243. 72. 299.
#> Adjusted 183.81 62.83 248.54
#Balancing covariates with respect to re75 (continuous)
#assuming t(3) conditional density for treatment
(W3 <- weightit(re75 ~ age + educ + married +
nodegree + re74, data = lalonde,
method = "bart", density = "dt_3"))
#> A weightit object
#> - method: "bart" (propensity score weighting with BART)
#> - number of obs.: 614
#> - sampling weights: none
#> - treatment: continuous
#> - covariates: age, educ, married, nodegree, re74
summary(W3)
#> Summary of weights
#>
#> - Weight ranges:
#>
#> Min Max
#> all 0.0805 |---------------------------| 20.3566
#>
#> - Units with the 5 most extreme weights:
#>
#> 486 310 469 487 484
#> all 7.1071 7.6036 9.4356 19.3029 20.3566
#>
#> - Weight statistics:
#>
#> Coef of Var MAD Entropy # Zeros
#> all 1.143 0.474 0.276 0
#>
#> - Effective Sample Sizes:
#>
#> Total
#> Unweighted 614.
#> Weighted 266.55
bal.tab(W3)
#> Balance Measures
#> Type Corr.Adj
#> age Contin. 0.0282
#> educ Contin. 0.0504
#> married Binary 0.0794
#> nodegree Binary -0.0805
#> re74 Contin. 0.1147
#>
#> Effective sample sizes
#> Total
#> Unadjusted 614.
#> Adjusted 266.55