Computes the joint density for the two-split detection criterion where both split-halves must exceed individual thresholds and their average must exceed a consistency threshold.
density_threshold_both(x, theta, prop_cens = 0.3, n_sg, k_avg, k_ind)Numeric. Joint density value at x, or 0 if thresholds not met.
Numeric vector of length 2. Log hazard ratio estimates from the two split-halves.
Numeric. True hazard ratio in the subgroup.
Numeric. Proportion censored (0-1). Default: 0.3
Integer. Subgroup sample size.
Numeric. Threshold for average log(HR) across splits. Typically log(hr.threshold).
Numeric. Threshold for individual split log(HR). Typically log(hr.consistency).
The detection criterion requires:
Average of two splits: (x1 + x2)/2 >= k_avg
Individual splits: x1 >= k_ind AND x2 >= k_ind
Under the asymptotic approximation, each split-half log(HR) estimator follows N(log(theta), 8/d) where d = n_sg * (1 - prop_cens) / 2 is the expected number of events per split.