D-T (Doubly-Tuned) Estimator — msd

D-T estimator with arm-specific tuning parameters for ATE. D-T (Doubly-Tuned) Estimator

Computes the Doubly-Tuned (D-T) estimator for the average treatment effect (ATE). This estimator uses arm-specific tuning parameters (lambda_1 and lambda_0) estimated via cross-fitting.

Usage

msd_dt(
  formula_or_data,
  data = NULL,
  observed = NULL,
  unobserved = NULL,
  n_folds = 2,
  conf_level = 0.95,
  seed = NULL
)

Arguments

formula_or_data: Either an msd_data object created by msd_data, or a formula of the form outcome ~ treatment | prediction.
data: If formula_or_data is a formula, this should be either: an msd_data object, a combined dataframe, or NULL (if using observed/unobserved).
observed: If using formula with separate dataframes, the observed data.
unobserved: If using formula with separate dataframes, the unobserved data.
n_folds: Number of folds for cross-fitting (default 2)
conf_level: Confidence level for the confidence interval (default 0.95)
seed: Random seed for fold splitting (optional)

Value

An msd_result object containing:

estimate: Point estimate of the ATE
variance: Estimated variance (delta-method)
se: Standard error
ci_lower, ci_upper: Confidence interval bounds
method: Name of the estimation method
lambda: Vector of arm-specific tuning parameters (lambda_1, lambda_0)

Details

The D-T estimator uses arm-specific tuning parameters: $$\hat{\mu}_d^{D-T}(\lambda_d) = \bar{Y}_{\mathcal{O}_d} + \lambda_d(\bar{S}^{(d)}_{\mathcal{U}_d} - \bar{S}^{(d)}_{\mathcal{O}_d})$$

Each lambda_d is chosen to minimize the variance in arm d: $$\lambda_d^* = \frac{Cov(Y(d), S^{(d)}) / n_d}{Var(S^{(d)})(1/m_d + 1/n_d)}$$

The tuning parameters are estimated via cross-fitting to avoid bias.

Note

D-T differs from PPI++ by using separate tuning parameters for each arm, which can improve efficiency when the prediction quality differs between treatment and control.

Examples

# Create sample data
set.seed(123)
n <- 100
obs_df <- data.frame(
  Y = rnorm(n),
  S0 = rnorm(n, 0, 0.5),
  S1 = rnorm(n, 0.2, 0.5),
  D = rep(c(1, 0), each = n/2)
)
obs_df$Y <- obs_df$Y + 0.3 * obs_df$D
obs_df$S1[obs_df$D == 1] <- obs_df$S1[obs_df$D == 1] + 0.5 * obs_df$Y[obs_df$D == 1]
obs_df$S0[obs_df$D == 0] <- obs_df$S0[obs_df$D == 0] + 0.5 * obs_df$Y[obs_df$D == 0]

unobs_df <- data.frame(
  S0 = rnorm(200, 0, 0.5),
  S1 = rnorm(200, 0.2, 0.5),
  D = rep(c(1, 0), each = 100)
)

msd <- msd_data(observed = obs_df, unobserved = unobs_df)
result <- msd_dt(msd)

# Using formula interface
result2 <- msd_dt(Y ~ D | S1 + S0, observed = obs_df, unobserved = unobs_df)