Compute Naive and Split-Derived 95% CIs for Pareto Frontier Members

For each candidate subgroup on fs$grp.consistency$out_sg$pareto_frontier, computes three 95\

Usage

compute_frontier_cis(
  fs,
  n_splits = 1000L,
  ci_level = 0.95,
  seed = NULL,
  verbose = FALSE
)

Arguments

fs: A forestsearch result object.
n_splits: Integer. Number of 50/50 random splits per frontier member for the Split CI. Default 1000L.
ci_level: Numeric in (0, 1). Default 0.95.
seed: Integer or NULL. Seed for reproducibility of the per-split halving. Default NULL (no explicit seed).
verbose: Logical. Default FALSE. When TRUE, prints per-frontier-member progress.

Value

A data.table keyed by frontier row index (m) with columns:

m: Candidate id (matches pareto_frontier$m).
estimate: Refit effect estimate on the natural scale.
naive_lcl, naive_ucl: Naive Wald CI bounds.
split_sd: Half-jackknife SE estimate: empirical SD of the $2S$ per-half effect estimates on the SE scale (log for ratio measures, natural for differences).
split_lcl, split_ucl: Split-derived CI bounds.
fsbc_estimate: Bias-corrected effect estimate on the natural scale ($2\hat\beta - \overline{\hat\beta^{(h)}}$ on the SE scale, then back-transformed).
fsbc_lcl, fsbc_ucl: FSBC-mimic CI bounds.
n_valid_splits: Number of splits that produced valid per-half estimates.

Rows where the refit fails (e.g., zero events in one arm) have NA in all CI columns.

Details

Naive CI: full-sample Wald CI from a Cox or GLM model refit on the subgroup's data. This is the textbook CI that ignores the multiple-testing / subgroup-search context. Anti-conservative by construction.
Split CI: a 95\ estimate, with SE estimated by the empirical standard deviation of the $2S$ individual half-sample effect estimates produced by n_splits random 50/50 splits. This is a half-jackknife SE (Shao1996). Naming: the $\sim$ in column labels marks this as a resampling-derived approximation, not a model-based CI.
FSBC-mimic CI: a bias-corrected interval inspired by the bootstrap bias-correction algorithm of Leon2024fs (eq 7), with the selection-on-bootstrap-data term $\eta_b^*(\hat H_b^*)$ zeroed out (the selected subgroup is treated as fixed across half-jackknife replicates). The bias-corrected estimate is $\hat\beta^{\mathrm{FSBC}} = 2\hat\beta - \overline{\hat\beta^{(h)}}$ where $\overline{\hat\beta^{(h)}}$ is the mean of the $2S$ half-sample estimates; the SE is the same half-jackknife SE as for the Split CI. See “Details — FSBC-mimic interpretation” below.

All three CIs are computed post-hoc on the returned forestsearch object; this function does not modify any internal state.

Details — Half-jackknife SE

For each of n_splits random 50/50 splits we fit the effect model separately on each half, obtaining $2S$ per-half estimates $\{\hat\beta^{(s,h)} : s=1,\ldots,S; h=1,2\}$. The half-jackknife SE estimator is $$\widehat{\mathrm{SE}} = \mathrm{sd}\bigl\{\hat\beta^{(s,h)}\bigr\},$$ i.e., the plain empirical SD of the $2S$ estimates. No $\sqrt{2}$ correction is applied: simulation studies confirm that this estimator's CI achieves nominal coverage in a no-selection setting (see the package's pressure-test script). Earlier versions of this function used the average $(\hat\beta^{(s,1)} + \hat\beta^{(s,2)})/2$ per split; that approach is invalid because the two halves are complementary and their average is nearly constant.

Details — FSBC-mimic interpretation

The FSBC-mimic CI is a quick approximation to the bootstrap bias-corrected estimator of Leon2024fs. Substituting the $2S$ per-half estimates for bootstrap replicates, and treating the selected subgroup $\hat H$ as fixed (i.e., omitting the $\eta_b^*(\hat H_b^*) = \hat\beta_b^*(\hat H_b^*) - \hat\beta(\hat H_b^*)$ term that captures variability in FS selection on the bootstrap data), the bias-corrected estimator from paper eq (7) collapses to $$\hat\beta^{\mathrm{FSBC}}(\hat H) = \hat\beta(\hat H) - \frac{1}{2S}\sum_{s,h}\bigl[\hat\beta^{(s,h)} - \hat\beta(\hat H)\bigr] = 2\hat\beta(\hat H) - \overline{\hat\beta^{(h)}}.$$ The SE is the same half-jackknife SE as for the Split CI.

The FSBC-mimic CI is not a full FSBC CI: it omits the selection-on-bootstrap-data variance. Use this column as a quick bias-corrected diagnostic, not for inference. For full FSBC CIs use forestsearch_bootstrap_dofuture.