Covariate balancing propensity score

bw_cbps() specifies the covariate balancing propensity score for balance(). A propensity model is fit so that its parameters satisfy covariate balancing moment conditions rather than the maximum-likelihood score alone. In the just-identified form the moment conditions equal the parameter count, so balance on the requested moments is exact by construction and the weights solve smooth estimating equations. The covariate balancing propensity score supports binary, categorical, and continuous exposures, and is the only method that supports the overlap estimand "ato" for a binary exposure.

Usage

bw_cbps(
  ...,
  over_identified = FALSE,
  two_step = TRUE,
  link = c("logit", "probit", "cloglog"),
  convergence_tolerance = 1e-10,
  max_iterations = NULL
)

Arguments

...

Reserved for future extensions; must be empty. Tuning parameters must be passed by name.

over_identified

Whether to add the response-residual moments and minimize the generalized-method-of-moments criterion. FALSE fits the just-identified form, whose balance is exact. Binary exposures only; ignored, with a warning, for a categorical or continuous exposure.

two_step

Whether to use the two-step weighting matrix for the over-identified criterion, rather than the continuously updating criterion. Ignored, with a warning, whenever the fit is not over-identified.

link

The propensity link, one of "logit", "probit", or "cloglog". Binary and categorical exposures both fit a propensity model and consume it. A continuous exposure fits none, so the setting is ignored, with a warning, there.

convergence_tolerance

The solver convergence tolerance.

max_iterations

The maximum solver iterations, or NULL for the core default.

Value

A bw_cbps specification, a balance_method.

Details

For a binary exposure the just-identified fit weights a unit by a function of its modeled propensity determined by the estimand: the inverse propensity for the average treatment effect, the inverse odds for the average treatment effect on the treated, and the overlap factor for the overlap estimand. With mean balance and the logit link the just-identified average-treatment- effect-on-the-treated fit solves the same treated-target moment conditions as entropy balancing and inverse probability tilting, so the three methods produce the same weights.

Setting over_identified = TRUE stacks the response-residual moments sum(s * (t - p) * x) onto the balancing conditions and minimizes a generalized-method-of-moments criterion. Those moments are the propensity model's own score only under the canonical logit link. Under a probit or complementary log-log link they stay a valid moment condition, since the response residual has mean zero at the true parameters whatever the link, but they are not that model's score and the fit is not a likelihood-augmented one. Balance is then approximate, the fit records the criterion value on its objective, and it supplies no estimating equations. The recorded criterion carries no units from the sampling weights: the mean moment and the moment covariance the weighting matrix inverts are each divided by the average sampling weight, so the same design expressed in survey-expansion units reports the same criterion and meets convergence_tolerance at the same fit. A converged verdict on that criterion does not always mean the gradient reached convergence_tolerance: the solver also certifies convergence when a full Newton step's predicted decrease falls at or below the objective's own floating-point resolution, which is the numerical minimum whatever the gradient reads. A squared criterion reaches that floor with a gradient near the square root of the arithmetic's precision, so this is the ordinary outcome rather than an exception. That criterion is defined for a binary exposure alone: a categorical or continuous exposure has no over-identified form, so the request is warned and ignored and the fit balances its moment conditions exactly. two_step selects the two-step weighting matrix for that criterion; it has no effect on a fit that is not over-identified and is warned and ignored there. Every just-identified discrete fit, the overlap estimand included, supplies estimating equations; only the binary over-identified form and a continuous exposure do not.

For a continuous exposure the covariate balancing conditions require the weighted exposure mean to match the sample mean and the weighted covariance between the exposure and every covariate to vanish. The weights that meet these conditions with the least departure from uniformity are the minimum-divergence exponential tilt, the same reweighting the continuous form of entropy balancing uses. This is the nonparametric reading of covariate balancing for a continuous exposure and departs from the parametric generalized propensity score of the Fong, Hazlett, and Imai reference, which derives the weights from a Gaussian density ratio and can be unstable; the two share the balancing conditions but not the weight family.

References

Imai, K. and Ratkovic, M. (2014). Covariate balancing propensity score. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 76(1), 243-263.

Fong, C., Hazlett, C., and Imai, K. (2018). Covariate balancing propensity score for a continuous treatment: Application to the efficacy of political advertisements. The Annals of Applied Statistics, 12(1), 156-177.

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_cbps())
#> ℹ Treating `.exposure` as binary
fit
#> 
#> ── Covariate balancing propensity score ────────────────────────────────────────
#> Exposure: "exposure" (binary)
#> Estimand: "ate"
#> Observations: 200
#> Solver: converged in 3 iterations
#> Constraints: 2 terms (tolerance 0)
#> Largest imbalance: 0.0000 (standardized mean difference)