D-T DiP (Doubly-Tuned Difference-in-Predictions) Estimator

D-T DiP estimator with arm-specific tuning for ATE. D-T DiP (Doubly-Tuned Difference-in-Predictions) Estimator

Computes the D-T DiP estimator for the average treatment effect (ATE). This estimator uses paired predictions S^(1) and S^(0) for each unlabeled unit, with arm-specific tuning parameters (lambda_1 and lambda_0) estimated via cross-fitting.

Usage

msd_dt_dip(
  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 | pred_treated + pred_control.
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 DiP estimator is: $$\hat{\tau}^{D-T DiP} = \frac{1}{|\mathcal{U}|} \sum_{i \in \mathcal{U}} (\lambda_1 S_i^{(1)} - \lambda_0 S_i^{(0)}) + \frac{1}{n_1}\sum_{i \in \mathcal{O}_1}(Y_i - \lambda_1 S_i^{(1)}) - \frac{1}{n_0}\sum_{i \in \mathcal{O}_0}(Y_i - \lambda_0 S_i^{(0)})$$

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

The tuning parameters are estimated via cross-fitting:

Split labeled data into K folds
For each fold k, estimate (lambda_1, lambda_0) on opposite folds
Compute the fold-k estimate using estimated lambdas
Average across folds with equal weights

Note

D-T DiP requires BOTH S0 and S1 predictions for ALL units. This corresponds to 2 predictions per unlabeled unit.

D-T DiP combines the benefits of:

DiP: exploiting positive correlation between S^(1) and S^(0)
D-T: arm-specific tuning for heterogeneous prediction quality

Examples

# Create sample data with both predictions
set.seed(123)
n <- 100
obs_df <- data.frame(
  Y = rnorm(n),
  D = rep(c(1, 0), each = n/2)
)
obs_df$Y <- obs_df$Y + 0.3 * obs_df$D
obs_df$S1 <- 0.6 * obs_df$Y + rnorm(n, 0, 0.4)
obs_df$S0 <- 0.4 * obs_df$Y + rnorm(n, 0, 0.5)

unobs_df <- data.frame(
  S0 = rnorm(300, 0, 0.5),
  S1 = rnorm(300, 0.2, 0.4),
  D = rep(c(1, 0), 150)
)
# Add correlation between S0 and S1
unobs_df$S1 <- unobs_df$S1 + 0.5 * unobs_df$S0

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

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