Joint likelihood for correlated observations O-17, O-31, O-29, O-18, O-43, O-19, O-15 (shared basis: D-7, D-9, D-25). Step 8 writes the single ## Likelihood block here; the member edges point back via group.
Likelihood
# CG-8 (HC-3) — ONE joint estimate over E-55/E-56/E-57/E-60/E-61/E-62/E-63: all seven observations rest on the
# overlapping Harvard / Swedish / pooled-meta cohort corpus (D-7, D-9, D-25), so they are judged as ONE pattern
# (rule 1), not seven independent witnesses. The whole pattern: a US-positive / Europe-null / Asia-inverse
# gradient (O-17, O-31; p-interaction 0.01, US I2=0%), a functionally-null register-based European cohort
# (O-29), the association concentrating at LOW diet quality (O-18), eggs co-occurring with the unhealthy
# Western basket (O-43), benefit only when eggs are swapped for PRUDENT foods and none vs other animal proteins
# (O-19), and a fragile, sex-varying US signal null in men (O-15). Members in HC-3.hypotheses order
# [H-8 region-dependent-causal, H-9 uniform-null-via-confounding, H-51 residual/uniform-effect].
# Anchored on H-9 = 1 (best fit); others priced as ratios (rule 7).
lik_grad_H9 = 1.0 # anchor: the confounding account predicts the WHOLE pattern — the gradient arises because
# confounding strength varies by region (eggs mark the diabetogenic Western pattern in the
# US, prudent diets in Asia); O-18 (stronger where diet quality is low) is where residual
# confounding is strongest; O-43 is the confounder-correlation H-9 requires; O-19's
# marker-substitution signature (benefit only vs prudent foods, neutral vs the animal-food
# class eggs belong to) and O-15's sex-fragility both fit a confounded, non-causal signal.
# A-5/A-10/A-16 (approved) and A-6 (corrected) all read these features as confounding.
lik_grad_H8 = 0.45 # ~0.45x the anchor: a genuinely region-dependent CAUSAL effect predicts the bare gradient
# (O-17/O-31/O-29) about as well as H-9 does — E-55/E-56/E-57 note the gradient does not
# separate the two — but it gives no strong reason for the WITHIN-population diet-quality
# modification (O-18), the marker-substitution pattern (O-19: a real egg effect should lower
# risk when eggs are replaced by ANY neutral food, yet swaps vs other animal proteins are
# null), or the sex-fragility (O-15, null in HPFS men). Those four features tilt the joint
# pattern toward confounding, so H-8 sits below the anchor.
lik_grad_H51 = 0.25 # ~0.25x: a UNIFORM modest effect (positive or inverse) across all regions predicts
# homogeneous estimates — directly contradicted by the region heterogeneity (p-interaction
# 0.01; US 1.18 I2=0% vs non-US 0.97), the register-based Swedish null (O-29) and the
# null-in-men (O-15). Below both listed members, but kept well above zero: "some other
# structure" is unconstrained (rule 3) and a uniform inverse could weakly accommodate the
# Asia end, so the heterogeneity does not forbid it outright.
t_gradient = 0.75 # cap = min trust over {S-4 0.78, S-8 0.75, S-19 0.82} = 0.75 (S-8). Held at the cap: the
# region gradient is a straightforward description of published, independently reproduced
# meta-analytic estimates (S-4 and S-8 report it separately); the overlapping-cohort
# correlation is handled by this being ONE joint call, not by t, and the FFQ/observational
# limitations are already priced into the step-2 trust scores.
evidence("HC-3", ["O-17", "O-31", "O-29", "O-18", "O-43", "O-19", "O-15"], [lik_grad_H8, lik_grad_H9, lik_grad_H51], t=t_gradient)