independence_weights: Construction of distance covariance optimal weights weights

Description

Constructs independence-inducing weights (distance covariance optimal weights) for estimation of causal quantities for continuous-valued treatments

Usage

independence_weights(
  A,
  X,
  lambda = 0,
  decorrelate_moments = FALSE,
  preserve_means = FALSE,
  dimension_adj = TRUE
)

Arguments

vector indicating the value of the treatment or exposure variable. Should be a numeric vector.

matrix of covariates with number of rows equal to the length of A and each column is a pre-treatment covariate to be balanced between treatment groups.

lambda

tuning parameter for the penalty on the sum of squares of the weights

decorrelate_moments

logical scalar. Whether or not to add constraints that result in exact decorrelation of weighted first order moments of X and A. Defaults to FALSE.

preserve_means

logical scalar. Whether or not to add constraints that result in exact preservation of weighted first order moments of X and A. Defaults to FALSE.

dimension_adj

logical scalar. Whether or not to add adjustment to energy distance terms that account for the dimensionality of X. Defaults to TRUE.

Value

An object of class "independence_weights" with elements:

weights

A vector of length nrow(X) containing the estimated sample weights

Treatment vector

opt

The optimization object returned by osqp::solve_osqp()

objective

The value of the objective function at its optimal value. This is the weighted dependence statistic plus any ridge penalty on the weights.

D_unweighted

The value of the weighted dependence distance using all weights = 1 (i.e. unweighted)

D_w

The value of the weighted dependence distance of Huling, et al. (2021) using the optimal estimated weights. This is the weighted dependence statistic without the ridge penalty on the weights.

distcov_unweighted

The unweighted distance covariance term. This is the standard distance covariance of Szekely et al (2007). This term is always equal to D_unweighted.

distcov_weighted

The weighted distance covariance term. This term itself does not directly measure weighted dependence but is a critical component of it.

energy_A

The weighted energy distance between A and its weighted version

energy_X

The weighted energy distance between X and its weighted version

ess

The estimated effective sample size of the weights using Kish's effective sample size formula.

An object of class "independence_weights".

weights

the estimated weights, the distance covariance optimal weights (DCOWs)

the treatment vector

opt

the object returned by whatever optimization routine was used

objective

the value of the optimized objective function

distcov_unweighted

the unweighted distance covariance between treatment and covariates

distcov_weighted

the weighted distance covariance between treatment and covariates

energy_A

the (energy) distance between the treatment distribution and the weighted treatment distribution. Smaller values mean the marginal distribution of the treatment is preserved after weighting

energy_x

the (energy) distance between the covariate distribution and the weighted covariate distribution. Smaller values mean the marginal distribution of the covariates is preserved after weighting

ess

the expected sample size after weighting. Kish's approximation is used

References

Szekely, G. J., Rizzo, M. L., & Bakirov, N. K. (2007). Measuring and testing dependence by correlation of distances. Annals of Statistics 35(6) 2769-2794 10.1214/009053607000000505

Huling, J. D., Greifer, N., & Chen, G. (2021). Independence weights for causal inference with continuous exposures. arXiv preprint arXiv:2107.07086. https://arxiv.org/abs/2107.07086

Examples

Run this code

# NOT RUN {
simdat <- simulate_confounded_data(seed = 999, nobs = 500)

y <- simdat$data$Y
A <- simdat$data$A
X <- as.matrix(simdat$data[c("Z1", "Z2", "Z3", "Z4", "Z5")])

dcows <- independence_weights(A, X)

print(dcows)

# distribution of response:
quantile(y)

## create grid
trt_vec <- seq(min(simdat$data$A), 50, length.out=500)

## estimate ADRF
adrf_hat <- weighted_kernel_est(A, y, dcows$weights, trt_vec)$est

## estimate naively without weights
adrf_hat_unwtd <- weighted_kernel_est(A, y, rep(1, length(y)), trt_vec)$est

ylims <- range(c(simdat$data$Y, simdat$true_adrf(trt_vec)))
plot(x = simdat$data$A, y = simdat$data$Y, ylim = ylims, xlim = c(0,50))
## true ADRF
lines(x = trt_vec, y = simdat$true_adrf(trt_vec), col = "blue", lwd=2)
## estimated ADRF
lines(x = trt_vec, y = adrf_hat, col = "red", lwd=2)
## naive estimate
lines(x = trt_vec, y = adrf_hat_unwtd, col = "green", lwd=2)

# }