Stable balancing weights — bw

bw_sbw() specifies stable balancing weights for balance(). Among all weightings that hold each reweighted exposure group's covariate means inside a tolerance band, stable balancing weights select the one of least dispersion, following Zubizarreta. The default "l2" norm minimizes the sum of squared weights, so for a fixed per-group total it minimizes the weight variance. Stable balancing weights support binary, categorical, and continuous exposures.

Usage

bw_sbw(
  ...,
  norm = c("l2", "l1", "linf"),
  min_weight = 1e-08,
  convergence_tolerance = NULL,
  max_iterations = NULL
)

Arguments

...: Reserved for future extensions; must be empty. Tuning parameters must be passed by name.
norm: The weight-dispersion norm to minimize, one of "l2" (the sum of squared weights, minimum variance), "l1" (the sum of absolute deviations from one), or "linf" (the largest absolute deviation from one).
min_weight: The smallest permitted weight.
convergence_tolerance: The quadratic-program solver tolerance, or NULL for the core default.
max_iterations: The maximum solver iterations, or NULL for the core default.

Value

An bw_sbw specification, a balance_method.

Details

The balance tolerance is the method's central tuning parameter. It is set through constraints = balance_terms(tolerance = ...) in balance(), not on the method spec, and it must be positive: with an exact (zero) tolerance the problem reduces to exact moment balance, which abandons the minimum-variance rationale and is prone to infeasibility, so a fit without a positive tolerance is refused. Given a positive tolerance the weights hold each group's weighted covariate means within the band while minimizing the weight dispersion. For a discrete exposure no feasible reweighting has smaller dispersion than the fit returns.

A single scalar tolerance applies to every covariate; a named vector sets a tolerance per covariate. A covariate left at zero in a named vector demands exact balance on that covariate while the others are relaxed, the infeasibility-prone case, so a mixed specification is an explicit choice rather than a convenience.

For a discrete exposure the constraints default to first-moment balance; pass balance_terms() to balance higher moments, interactions, or quantiles, each inside its tolerance band. For a focal estimand the focal group keeps unit weight and the other groups are pulled to the focal group's covariate means.

For a continuous exposure the weighted exposure-covariate correlations are held within the tolerance. The quadratic program bounds a linearized correlation whose scales are fixed at the sample, so the fit tightens that internal bound over a few passes until the reported weighted correlation sits inside the requested band. The returned weights therefore minimize dispersion over the tightened internal band rather than over every weighting that meets the reported band, so the strict minimum-dispersion guarantee is stated for discrete exposures only.

Stable balancing weights belong to the quadratic-program family, which has no estimating equations, so a fit produces no estimating-equations container.

The norm argument selects how the weight dispersion is measured, always against the uniform baseline of one within each reweighted group. "l2" minimizes the sum of squared weights, so for a fixed per-group total it minimizes the weight variance. "l1" minimizes the sum of absolute deviations from one, which tends to leave many weights untouched and concentrate the reweighting on a few units. "linf" minimizes the single largest absolute deviation from one, which spreads the reweighting as evenly as the balance constraints allow. The "l1" and "linf" problems are linear programs solved through the same quadratic-program backends as "l2"; their solutions can be non-unique, so a fit reports the achieved dispersion rather than promising a unique weighting.

References

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.

Examples

n <- 200
x1 <- rnorm(n)
x2 <- rnorm(n)
df <- data.frame(
  exposure = rbinom(n, 1, plogis(0.5 * x1 - 0.5 * x2)),
  x1 = x1,
  x2 = x2
)
fit <- balance(
  df,
  exposure,
  c(x1, x2),
  method = bw_sbw(),
  constraints = balance_terms(tolerance = 0.05)
)
#> ℹ Treating `.exposure` as binary
fit
#> 
#> ── Stable balancing weights ────────────────────────────────────────────────────
#> Exposure: "exposure" (binary)
#> Estimand: "ate"
#> Observations: 200
#> Solver: converged in 75 iterations
#> Constraints: 2 terms (tolerance 0.05)
#> Largest imbalance: 0.1000 (standardized mean difference)