Stable Balancing Weights

Estimates stable balancing weights for the supplied treatments and covariates. The degree of balance for each covariate is specified by tols and the target population can be specified with targets or estimand. See Zubizarreta (2015) and Wang & Zubizarreta (2020) for details of the properties of the weights and the methods used to fit them.

Usage

optweight(
  formula,
  data = NULL,
  tols = 0,
  estimand = "ATE",
  targets = NULL,
  s.weights = NULL,
  b.weights = NULL,
  focal = NULL,
  norm = "l2",
  min.w = 1e-08,
  verbose = FALSE,
  ...
)

optweight.fit(
  covs,
  treat,
  tols = 0,
  estimand = "ATE",
  targets = NULL,
  s.weights = NULL,
  b.weights = NULL,
  focal = NULL,
  norm = "l2",
  std.binary = FALSE,
  std.cont = TRUE,
  min.w = 1e-08,
  verbose = FALSE,
  solver = NULL,
  ...
)

Arguments

formula: a formula with a treatment variable on the left hand side and the covariates to be balanced on the right hand side, or a list thereof. Interactions and functions of covariates are allowed.
data: an optional data set in the form of a data frame that contains the variables in formula.
tols: a vector of balance tolerance values for each covariate. The resulting weighted balance statistics will be at least as small as these values. If only one value is supplied, it will be applied to all covariates. Can also be the output of a call to process_tols(). See Details. Default is 0 for all covariates.
estimand: a string containing the desired estimand, which determines the target population. For binary treatments, can be "ATE", "ATT", "ATC", or NULL. For multi-category treatments, can be "ATE", "ATT", or NULL. For continuous treatments, can be "ATE" or NULL. The default for both is "ATE". estimand is ignored when targets is non-NULL. If both estimand and targets are NULL, no targeting will take place. See Details.
targets: an optional vector of target population mean values for each covariate. The resulting weights will yield sample means within tols/2 units of the target values for each covariate. If NULL or all NA, estimand will be used to determine targets. Otherwise, estimand is ignored. If any target values are NA, the corresponding variable will not be targeted and its weighted mean will be wherever the weights yield the smallest variance; this is only allowed for binary and multi-category treatments. Can also be the output of a call to process_targets(). See Details.
s.weights: a vector of sampling weights. For optweight(), can also be the name of a variable in data that contains sampling weights.
b.weights: a vector of base weights. If supplied, the desired norm of the distance between the estimated weights and the base weights is minimized. For optweight(), can also the name of a variable in data that contains base weights.
focal: when multi-category treatments are used and estimand = "ATT", which group to consider the "treated" or focal group. This group will not be weighted, and the other groups will be weighted to be more like the focal group. If specified, estimand will automatically be set to "ATT".
norm: character; a string containing the name of the norm corresponding to the objective function to minimize. Allowable options include "l1" for the L1 norm, "l2" for the L2 norm (the default), "linf" for the L\(\infty\) norm, "entropy" for the negative entropy, and "log" for the sum of the logs. See Details.
min.w: numeric; a single value less than 1 for the smallest allowable weight. Some analyses require nonzero weights for all units, so a small, nonzero minimum may be desirable. The default is 1e-8 (\(10^{-8}\)), which does not materially change the properties of the weights from a minimum of 0 but prevents warnings in some packages that use weights in model fitting. When norm is "entropy" or "log" and min.w <= 0, min.w will be set to the smallest nonzero value.
verbose: logical; whether information on the optimization problem solution should be printed. Default is FALSE.
...: for optweight(), additional arguments passed to optweight.fit(), including options that are passed to the settings function corresponding to solver.
covs: a numeric matrix of covariates to be balanced.
treat: a vector of treatment statuses. Non-numeric (i.e., factor or character) vectors are allowed.
std.binary, std.cont: logical; whether the tolerances are in standardized mean units (TRUE) or raw units (FALSE) for binary variables and continuous variables, respectively. The default is FALSE for std.binary because raw proportion differences make more sense than standardized mean difference for binary variables. These arguments are analogous to the binary and continuous arguments in bal.tab() in cobalt.
solver: string; the name of the optimization solver to use. Allowable options depend on norm. Default is to use whichever eligible solver is installed, if any, or the default solver for the corresponding norm. See Details for information.

Value

For optweight(), an optweight object with the following elements:

weights: The estimated weights, one for each unit.
treat: The values of the treatment variable.
covs: The covariates used in the fitting. Only includes the raw covariates, which may have been altered in the fitting process.
s.weights: The provided sampling weights.
b.weights: The provided base weights.
estimand: The estimand requested.
focal: The focal variable if the ATT was requested with a multi-category treatment.
call: The function call.
tols: The tolerance values for each covariate.
duals: A data.frame containing the dual variables for each covariate. See Details for interpretation of these values.
info: A list containing information about the performance of the optimization at termination.
norm: The norm used.
solver: The solver used.

For optweight.fit(), an optweight.fit object with the following elements:

w: The estimated weights, one for each unit.
duals: A data.frame containing the dual variables for each covariate.
info: A list containing information about the performance of the optimization at termination.
norm: The norm used.
solver: The solver used.

Details

optweight() is the primary user-facing function for estimating stable balancing weights. The optimization is performed by the lower-level function optweight.fit(), which transforms the inputs into the required inputs for the optimization functions and then supplies the outputs (the weights, dual variables, and convergence information) back to optweight(). Little processing of inputs is performed by optweight.fit(), as this is normally handled by optweight().

For binary and multi-category treatments, weights are estimated so that the weighted mean differences of the covariates are within the given tolerance thresholds (unless std.binary or std.cont are TRUE, in which case standardized mean differences are considered for binary and continuous variables, respectively). For a covariate \(x\) with specified tolerance \(\delta\), the weighted means of each each group will be within \(\delta\) of each other. Additionally, when the ATE is specified as the estimand or a target population is specified, the weighted means of each group will each be within \(\delta/2\) of the target means; this ensures generalizability to the same population from which the original sample was drawn.

If standardized tolerance values are requested, the standardization factor corresponds to the estimand requested: when the ATE is requested or a target population specified, the standardization factor is the square root of the average variance for that covariate across treatment groups, and when the ATT or ATC are requested, the standardization factor is the standard deviation of the covariate in the focal group. The standardization factor is computed accounting for s.weights.

For continuous treatments, weights are estimated so that the weighted correlation between the treatment and each covariate is within the specified tolerance threshold. The means of the weighted covariates and treatment are restricted to be exactly equal to those of the target population to ensure generalizability to the desired target population, regardless of tols. The weighted correlation is computed as the weighted covariance divided by the product of the unweighted standard deviations. The means used to center the variables in computing the covariance are those specified in the target population.

Target and balance constraints are applied to the product of the estimated weights and the sampling weights. In addition, the sum of the product of the estimated weights and the sampling weights is constrained to be equal to the sum of the product of the base weights and sampling weights. For binary and multi-category treatments, these constraints apply within each treatment group.

`norm`

The objective function for the optimization problem is \(f\left(w_i, b_i, s_i\right)\), where \(w_i\) is the estimated weight for unit \(i\), \(s_i\) is the sampling weight for unit \(i\) (supplied by s.weights) and \(b_i\) is the base weight for unit \(i\) (supplied by b.weights). The norm argument determines \(f(.,.,.)\), as detailed below:

when norm = "l2", \(f\left(w_i, b_i, s_i\right) = \frac{1}{n} \sum_i {s_i(w_i - b_i)^2}\)
when norm = "l1", \(f\left(w_i, b_i, s_i\right) = \frac{1}{n} \sum_i {s_i \vert w_i - b_i \vert}\)
when norm = "linf", \(f\left(w_i, b_i, s_i\right) = \max_i {\vert w_i - b_i \vert}\)
when norm = "entropy", \(f\left(w_i, b_i, s_i\right) = \frac{1}{n} \sum_i {s_i w_i \log \frac{w_i}{b_i}}\)
when norm = "log", \(f\left(w_i, b_i, s_i\right) = \frac{1}{n} \sum_i {-s_i \log \frac{w_i}{b_i}}\)

By default, s.weights and b.weights are set to 1 for all units unless supplied. b.weights must be positive when norm is "entropy" or "log", and norm = "linf" cannot be used when s.weights are supplied.

When norm = "l2" and both s.weights and b.weights are NULL, weights are estimated to maximize the effective sample size. When norm = "entropy", the estimated weights are equivalent to entropy balancing weights (Källberg & Waernbaum, 2023). When norm = "log", b.weights are ignored in the optimization, as they do not affect the estimated weights.

Dual Variables

Two types of constraints may be associated with each covariate: target constraints and balance constraints. Target constraints require the mean of the covariate to be at (or near) a specific target value in each treatment group (or for the whole group when treatment is continuous). Balance constraints require the means of the covariate in pairs of treatments to be near each other. For binary and multi-category treatments, balance constraints are redundant if target constraints are provided for a variable. For continuous variables, balance constraints refer to the correlation between treatment and the covariate and are not redundant with target constraints. In the duals component of the output, each covariate has a dual variable for each nonredundant constraint placed on it.

The dual variable for each constraint is the instantaneous rate of change of the objective function at the optimum corresponding to a change in the constraint. Because this relationship is not linear, large changes in the constraint will not exactly map onto corresponding changes in the objective function at the optimum, but will be close for small changes in the constraint. For example, for a covariate with a balance constraint of .01 and a corresponding dual variable of 40, increasing (i.e., relaxing) the constraint to .025 will decrease the value of the objective function at the optimum by approximately \((.025 - .01) * 40 = .6\).

For factor variables, optweight() takes the sum of the absolute dual variables for the constraints for all levels and reports it as the the single dual variable for the variable itself. This summed dual variable works the same way as dual variables for continuous variables do.

`solver`

The solver argument controls which optimization solver is used. Different solvers are compatible with each norm. See the table below for allowable options, which package they require, which function does the solving, and which function controls the settings.

`solver`	`norm`	Package	Solver function	Settings function
`"osqp"`	`"l2"`, `"l1"`, `"linf"`	osqp	`osqp::solve_osqp()`	`osqp::osqpSettings()`
`"highs"`	`"l2"`, `"l1"`, `"linf"`	highs	`highs::highs_solve()`	`highs::highs_control()` / `highs::highs_available_solver_options()`
`"lpsolve"`	`"l1"`, `"linf"`	lpSolve	`lpSolve::lp()`	.
`"scs"`	`"entropy"`, `"log"`	scs	`scs::scs()`	`scs::scs_control()`
`"clarabel"`	`"entropy"`, `"log"`	clarabel	`clarabel::clarabel()`	`clarabel::clarabel_control()`

Note that "lpsolve" can only be used when min.w is nonnegative.

The default solver for each norm is as follows:

`norm`	Default `solver`
`"l2"`	`"osqp"`
`"l1"`	`"highs"`
`"linf"`	`"highs"`
`"entropy"`	`"scs"`
`"log"`	`"scs"`

If the package corresponding to a default solver is not installed but the package for a different eligible solver is, that will be used. Otherwise, you will be asked to install the required package. osqp is required for optweight, and so will be the default for the "l1" and "linf" norms if highs is not installed. The default package is the one has shown good performance for the given norm; generally, all eligible solvers perform about equally well in terms of accuracy but differ in time taken.

Solving Convergence Failure

Sometimes the optimization will fail to converge at a solution. There are a variety of reasons why this might happen, which include that the constraints are nearly impossible to satisfy or that the optimization surface is relatively flat. It can be hard to know the exact cause or how to solve it, but this section offers some solutions one might try. Typically, solutions can be found most easily when using the "l2" norm; other norms, especially "linf" and "l1", are more likely to see problems.

Rarely is the problem too few iterations, though this is possible. Most problems can be solved in the default 200,000 iterations, but sometimes it can help to increase this number with the max_iter argument. Usually, though, this just ends up taking more time without a solution found.

If the problem is that the constraints are too tight, it can be helpful to loosen the constraints. Sometimes examining the dual variables of a solution that has failed to converge can reveal which constraints are causing the problem.

Sometimes a suboptimal solution is possible; such a solution does not satisfy the constraints exactly but will come pretty close. To allow these solutions, the argument eps can be increased to larger values. This is more likely to occur when s.weights are supplied.

Sometimes using a different solver can improve performance. Using the default solver for each norm, as described above, can reduce the probability of convergence failures.

References

Chattopadhyay, A., Cohn, E. R., & Zubizarreta, J. R. (2024). One-Step Weighting to Generalize and Transport Treatment Effect Estimates to a Target Population. The American Statistician, 78(3), 280–289. doi:10.1080/00031305.2023.2267598

de los Angeles Resa, M., & Zubizarreta, J. R. (2020). Direct and Stable Weight Adjustment in Non-Experimental Studies With Multivalued Treatments: Analysis of the Effect of an Earthquake on Post-Traumatic Stress. Journal of the Royal Statistical Society Series A: Statistics in Society, 183(4), 1387–1410. doi:10.1111/rssa.12561

Källberg, D., & Waernbaum, I. (2023). Large Sample Properties of Entropy Balancing Estimators of Average Causal Effects. Econometrics and Statistics. doi:10.1016/j.ecosta.2023.11.004

Wang, Y., & Zubizarreta, J. R. (2020). Minimal dispersion approximately balancing weights: Asymptotic properties and practical considerations. Biometrika, 107(1), 93–105. doi:10.1093/biomet/asz050

Zubizarreta, J. R. (2015). Stable Weights that Balance Covariates for Estimation With Incomplete Outcome Data. Journal of the American Statistical Association, 110(511), 910–922. doi:10.1080/01621459.2015.1023805

Examples

library("cobalt")
#>  cobalt (Version 4.6.1, Build Date: 2025-08-20)
data("lalonde", package = "cobalt")

# Balancing covariates between treatment groups (binary)
(ow1 <- optweight(treat ~ age + educ + married +
                    nodegree + re74, data = lalonde,
                  tols = c(.01, .02, .03, .04, .05),
                  estimand = "ATE"))
#> An optweight object
#>  - number of obs.: 614
#>  - norm minimized: "l2"
#>  - sampling weights: present
#>  - base weights: present
#>  - treatment: 2-category
#>  - estimand: ATE
#>  - covariates: age, educ, married, nodegree, re74
bal.tab(ow1)
#> Balance Measures
#>             Type Diff.Adj
#> age      Contin.    -0.00
#> educ     Contin.     0.02
#> married   Binary    -0.03
#> nodegree  Binary     0.04
#> re74     Contin.    -0.05
#> 
#> Effective sample sizes
#>            Control Treated
#> Unadjusted   429.    185. 
#> Adjusted     415.3   125.3

# Exactly alancing covariates with respect to race (multi-category)
(ow2 <- optweight(race ~ age + educ + married +
                    nodegree + re74, data = lalonde,
                  tols = 0, estimand = "ATT",
                  focal = "black"))
#> An optweight object
#>  - number of obs.: 614
#>  - norm minimized: "l2"
#>  - sampling weights: present
#>  - base weights: present
#>  - treatment: 3-category (black, hispan, white)
#>  - estimand: ATT (focal: black)
#>  - covariates: age, educ, married, nodegree, re74
bal.tab(ow2)
#> Balance summary across all treatment pairs
#>             Type Max.Diff.Adj
#> age      Contin.            0
#> educ     Contin.            0
#> married   Binary            0
#> nodegree  Binary            0
#> re74     Contin.            0
#> 
#> Effective sample sizes
#>            hispan  white black
#> Unadjusted  72.   299.     243
#> Adjusted    45.96 181.39   243

# Balancing covariates between treatment groups (binary)
# and requesting a specified target population
targets <- process_targets(~ age + educ + married +
                             nodegree + re74,
                           data = lalonde,
                           targets = c(26, 12, .4, .5,
                                       1000))

(ow3a <- optweight(treat ~ age + educ + married +
                     nodegree + re74, data = lalonde,
                   targets = targets,
                   estimand = NULL))
#> An optweight object
#>  - number of obs.: 614
#>  - norm minimized: "l2"
#>  - sampling weights: present
#>  - base weights: present
#>  - treatment: 2-category
#>  - estimand: targets
#>  - covariates: age, educ, married, nodegree, re74

bal.tab(ow3a, disp.means = TRUE)
#> Note: `s.d.denom` not specified; assuming "pooled".
#> Balance Measures
#>             Type M.0.Adj M.1.Adj Diff.Adj
#> age      Contin.    26.0    26.0       -0
#> educ     Contin.    12.0    12.0       -0
#> married   Binary     0.4     0.4       -0
#> nodegree  Binary     0.5     0.5       -0
#> re74     Contin.  1000.0  1000.0       -0
#> 
#> Effective sample sizes
#>            Control Treated
#> Unadjusted  429.    185.  
#> Adjusted    158.04   64.09

# Balancing covariates between treatment groups (binary)
# and not requesting a target population
(ow3b <- optweight(treat ~ age + educ + married +
                     nodegree + re74, data = lalonde,
                   targets = NULL,
                   estimand = NULL))
#> An optweight object
#>  - number of obs.: 614
#>  - norm minimized: "l2"
#>  - sampling weights: present
#>  - base weights: present
#>  - treatment: 2-category
#>  - estimand: targets
#>  - covariates: age, educ, married, nodegree, re74

bal.tab(ow3b, disp.means = TRUE)
#> Note: `s.d.denom` not specified; assuming "pooled".
#> Balance Measures
#>             Type   M.0.Adj   M.1.Adj Diff.Adj
#> age      Contin.   26.4160   26.4160       -0
#> educ     Contin.   10.3547   10.3547       -0
#> married   Binary    0.3615    0.3615       -0
#> nodegree  Binary    0.6305    0.6305       -0
#> re74     Contin. 3908.9059 3908.9059       -0
#> 
#> Effective sample sizes
#>            Control Treated
#> Unadjusted  429.    185.  
#> Adjusted    382.74  139.23

# Using a different norm
(ow1b <- optweight(treat ~ age + educ + married +
                    nodegree + re74, data = lalonde,
                  tols = c(.01, .02, .03, .04, .05),
                  estimand = "ATE",
                  norm = "l1"))
#> An optweight object
#>  - number of obs.: 614
#>  - norm minimized: "l1"
#>  - sampling weights: present
#>  - base weights: present
#>  - treatment: 2-category
#>  - estimand: ATE
#>  - covariates: age, educ, married, nodegree, re74

summary(ow1b, weight.range = FALSE)
#>                   Summary of weights
#> 
#> 
#> - Weight statistics:
#> 
#>           L2    L1     L∞ Rel Ent # Zeros
#> treated 2.16 0.422 26.361   0.664       0
#> control 1.04 0.165 14.038   0.228       0
#> 
#> - Effective Sample Sizes:
#> 
#>            Control Treated
#> Unweighted  429.    185.  
#> Weighted    206.18   32.65
summary(ow1, weight.range = FALSE)
#>                   Summary of weights
#> 
#> 
#> - Weight statistics:
#> 
#>            L2    L1    L∞ Rel Ent # Zeros
#> treated 0.69  0.536 3.419   0.198       0
#> control 0.182 0.165 0.409   0.017       0
#> 
#> - Effective Sample Sizes:
#> 
#>            Control Treated
#> Unweighted   429.    185. 
#> Weighted     415.3   125.3

# Allowing for negative weights
ow4 <- optweight(treat ~ age + educ + married + race +
                   nodegree + re74 + re75,
                 data = lalonde,
                 estimand = "ATE",
                 min.w = -Inf)

summary(ow4)
#>                   Summary of weights
#> 
#> - Weight ranges:
#> 
#>            Min                                 Max
#> treated -0.987 |---------------------------| 7.254
#> control  0.407     |-----|                   2.17 
#> 
#> - Units with the 5 most extreme weights by group:
#>                                       
#>            137   124    68    23    10
#>  treated 5.193 5.206 6.116 6.205 7.254
#>            388   375   226   196   118
#>  control 2.109  2.11 2.111 2.133  2.17
#> 
#> 
#> - Weight statistics:
#> 
#>            L2    L1    L∞ # Zeros
#> treated 1.608 1.216 6.254       0
#> control 0.499 0.39  1.17        0
#> 
#> - Effective Sample Sizes:
#> 
#>            Control Treated
#> Unweighted  429.    185.  
#> Weighted    343.49   51.57

# Using `optweight.fit()`
treat <- lalonde$treat
covs <- splitfactor(lalonde[2:8], drop.first = "if2")

ow.fit <- optweight.fit(covs,
                        treat = treat,
                        tols = .02,
                        estimand = "ATE",
                        norm = "l2")