Fit the Egidi, Pauli and Torelli empirical mixture prior

Selects the mixture weight from the observed target summary and then updates the resulting prior with the same summary. The two prior components are the ones the robust mixture prior uses, so the only difference between the two methods is where the weight comes from.

Usage

fit_egidi_mixture(
  theta_target_hat,
  se_target,
  informative_component,
  weak_component,
  alpha_pc = 0.05,
  pvalue_method = c("exact", "mc"),
  weight_grid_step = 0.001,
  mc_draws = 1000L,
  seed = NULL,
  theta_0 = 0,
  null_space = "left",
  confidence_level = 0.95
)

Arguments

theta_target_hat

Observed target treatment effect estimate.

se_target

Target standard error.

informative_component

A list with mean and sd, the source-based prior \(p(\theta_T)\).

weak_component

A list with mean and sd, the weak or unit-information prior \(q(\theta_T)\).

alpha_pc

Prior-predictive conflict threshold. 0.05 in the primary analysis; 0.01 and 0.10 are the sensitivity values.

pvalue_method

"exact" for the deterministic calculation, or "mc" for the prior-predictive simulation fallback.

weight_grid_step

Resolution of the weight scan.

mc_draws

Number of hypothetical replications when pvalue_method is "mc".

seed

Seed for the simulation fallback. The same seed is used at every candidate weight, which is what makes the comparison use common random numbers.

theta_0

Null value of the treatment effect.

null_space

Either "left" or "right".

confidence_level

Credible interval level.

Value

A list with the selected and posterior weights, the three conflict p-values, the conflict flags, the posterior mixture and its summaries, the probability of success, and numerical diagnostics.

Details

The procedure is deliberately adaptive: the observed target data are used first to choose the weight on the weak component and then again to update the mixture. psi_weak is therefore not a prior probability chosen before the target data were seen, and should not be reported as one.

The selected weight and the posterior component weight are different quantities and are both returned. The first is the weight the prior is given; the second is the posterior probability that the treatment effect came from the informative component, obtained from the component marginal likelihoods on the log scale.

The robust mixture prior parameterises its mixture by the weight on the informative component, so w_informative_prior is 1 - psi_weak.