GREG Estimator — msd_greg • mixedsubjects

Generalized Regression (GREG) calibration estimator for ATE. GREG Estimator

Computes the Generalized Regression (GREG) calibration estimator for the average treatment effect (ATE). This is the untuned calibration estimator with tuning parameter $\lambda = 1$. GREG, the original Prediction-Powered Inference (PPI) estimator, and Design-based Supervised Learning (DSL) are all the same untuned ($\lambda = 1$) estimator. It is distinct from the power-tuned PPI++ estimator in msd_ppi, which estimates $\lambda$ via cross-fitting to minimize variance.

Usage

msd_greg(
  formula_or_data,
  data = NULL,
  observed = NULL,
  unobserved = NULL,
  conf_level = 0.95
)

Arguments

formula_or_data: Either an msd_data object created by msd_data, or a formula of the form outcome ~ treatment | prediction. For GREG, the formula specifies which prediction column(s) to use.
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.
conf_level: Confidence level for the confidence interval (default 0.95)

Value

An msd_result object containing:

estimate: Point estimate of the ATE
variance: Estimated variance
se: Standard error
ci_lower, ci_upper: Confidence interval bounds
method: Name of the estimation method
lambda: Tuning parameter (always 1 for GREG)

Details

The GREG estimator for arm $d$ is: $$\hat{\mu}_d^{GREG} = \bar{S}^{(d)}_{\mathcal{U}_d} + (\bar{Y}_{\mathcal{O}_d} - \bar{S}^{(d)}_{\mathcal{O}_d})$$

The ATE estimate is: $$\hat{\tau}^{GREG} = \hat{\mu}_1^{GREG} - \hat{\mu}_0^{GREG}$$

The variance is: $$\widehat{Var}(\hat{\tau}^{GREG}) = \sum_{d \in \{0,1\}} \left[\frac{s^2_{S^{(d)}}}{m_d} + \frac{Var(Y(d) - S^{(d)})}{n_d}\right]$$

Note

GREG requires predictions for each unit's assigned arm:

Treatment arm units need S1
Control arm units need S0

Examples

# Using msd_data object
obs_df <- data.frame(
  Y = c(1.2, 1.4, 0.8, 0.6),
  S0 = c(1.0, 1.2, 0.7, 0.5),
  S1 = c(1.1, 1.3, 0.9, 0.7),
  D = c(1, 1, 0, 0)
)
unobs_df <- data.frame(
  S0 = c(1.1, 0.9, 1.0, 0.8),
  S1 = c(1.2, 1.0, 1.1, 0.9),
  D = c(1, 1, 0, 0)
)
msd <- msd_data(observed = obs_df, unobserved = unobs_df)
result <- msd_greg(msd)

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