PPI++ Estimator — msd_ppi • mixedsubjects

Power-tuned Prediction-Powered Inference estimator for ATE. PPI++ Estimator

Computes the PPI++ (power-tuned prediction-powered inference) estimator for the average treatment effect (ATE). This estimator uses a single tuning parameter lambda that is estimated via cross-fitting to minimize variance.

Usage

msd_ppi(
  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: Estimated tuning parameter (single value)

Details

The PPI++ estimator uses a single tuning parameter lambda across both arms: $$\hat{\mu}_d^{PPI}(\lambda) = \bar{Y}_{\mathcal{O}_d} + \lambda(\bar{S}^{(d)}_{\mathcal{U}_d} - \bar{S}^{(d)}_{\mathcal{O}_d})$$

The tuning parameter is estimated via cross-fitting:

Split labeled data into K folds
For each fold k, estimate lambda on the opposite folds
Compute the fold-k estimate using the estimated lambda
Average across folds

Lambda is chosen to minimize the combined variance across arms.

Note

PPI++ uses a single lambda across arms, unlike D-T which uses arm-specific tuning parameters. For arm-specific tuning, use msd_dt.

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_ppi(msd)

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