Select the smallest acceptable weak-component weight — egidi_select_weak

Implements the Egidi, Pauli and Torelli selection rule $$\hat\psi = \inf\{\psi \in [0, 1] : P_\psi(t_{obs}) \ge \alpha_{PC}\},$$ separately for every replicate and using only the observed target statistic.

Usage

egidi_select_weak_weight(
  t_obs,
  s_target,
  mu_p,
  tau_p,
  mu_q,
  tau_q,
  alpha_pc = 0.05,
  weight_grid_step = 0.001,
  weight_scan_step = 0.02,
  weight_tolerance = 1e-12
)

Arguments

t_obs: Observed target treatment effect estimate, one per replicate.
s_target: Target standard error, one per replicate.
mu_p, tau_p: Informative component prior mean and standard deviation.
mu_q, tau_q: Weak component prior mean and standard deviation.
alpha_pc: Conflict threshold, 0.05 in the primary analysis.
weight_grid_step: Resolution of the weight scan.
weight_scan_step: Resolution of the coarse stage of the scan.
weight_tolerance: Width of the weight bracket at which the crossing is considered found.

Value

A data frame with one row per replicate and the columns psi_weak, pvalue_informative, pvalue_weak, pvalue_selected, initial_conflict and conflict_unresolved.

Details

Both ends of the interval are exact and are tried first, which decides most replicates without any search. $P_0$ is the conflict p-value under the informative component alone: when it already reaches the threshold there is no conflict to resolve and the selected weight is zero. $P_1$ is the conflict p-value under the weak component alone: when even that falls short, no weight removes the conflict, and the rule returns one while flagging the conflict as unresolved rather than treating it as resolved.

With common centres the mixture predictive is symmetric, $P_\psi$ is exactly $(1 - \psi)P_p + \psi P_q$, and the crossing is solved in closed form. Otherwise the weight is found by scanning upwards for the first crossing and refining it, since monotonicity in $\psi$ is not guaranteed.

The scan is run in two stages, a coarse one to locate the crossing and a fine one at weight_grid_step inside it, and the crossing is then refined by vectorised_bracketed_root(). The two-stage scan agrees with a single scan at weight_grid_step unless a crossing both starts and ends inside one coarse step; weight_scan_step = weight_grid_step disables the coarse stage.