Osteoporosis and low bone mass affect ~54 million United States adults aged 50+, and a hip fracture carries a ~20–30% one-year mortality. This oracle estimates the causal reduction in fracture / mortality achievable by combining bone-directed therapies, using hazard ratios (HRs) from named published trials — flagging where the endpoint is a surrogate or the evidence is weak.
Tick the interventions to combine. Each shows its trial hazard ratio, 95% confidence interval (CI), E-value, mechanism, and citation. ★ = in the current Pareto effective set but not yet ticked.
Headline is the front-door estimate: shared bone-density-mediator overlap removed via dose-response saturation; residual direct-effect overlap removed via the eigenvalue model at ρ̄.
Under monotonicity + exogeneity (E-value bounds the exogeneity assumption).
Faithful causal directed acyclic graph (DAG). Each intervention is a distinct node; effects flow through named mediators — the front-door path via bone mineral density (BMD), the BMD-independent path via bone quality/turnover and the muscle→falls chain — into fracture, which itself mediates mortality. Estrogen carries a competing-harm path (breast cancer / venous thromboembolism / stroke → mortality). Named confounders open back-door paths (adjusted; robustness bounded by the E-value). This graph is illustrative of structure, not yet the identification engine.
Eigenvalue diagnostics for the selected interventions under an equicorrelation matrix (off-diagonal ρ̄). A large λmax relative to k signals redundancy; n_eff is the effective number of independent interventions actually contributing.
On mechanistic grounds ρ̄ ≈ 0.30 is defensible for this domain: incretin drugs, surgery, diet, and activity overwhelmingly share the weight-loss / insulin-sensitivity pathway, so ~30% of their nominal effects overlap. A mediation analysis of semaglutide found ~80% of its MACE benefit is not mediated by weight loss, which is why ρ̄ is user-adjustable rather than fixed.
Minimum-effective-set analysis. Set a target combined risk reduction; the model finds the smallest set of interventions — accounting for front-door mediator overlap — that reaches it, and highlights them. If the target exceeds what all interventions together can achieve, the full set is shown (never an empty one). "Apply" ticks exactly that set.
Monte Carlo propagation. Each selected intervention's log-hazard-ratio is sampled from a normal distribution implied by its 95% CI; samples are combined with the same eigenvalue overlap discount. 5,000 draws.
Intervening on the selected set S with Pearl's do-operator (setting the interventions, not merely observing them). Contrast against do(∅) = no intervention.
For each intervention: "if not for this one, the combined front-door hazard ratio would be…". Isolates each intervention's marginal causal contribution after mediator-overlap removal, so shared-pathway agents are not double-credited.
| If not for… | HR without it | HR with full set | marginal RRR lost |
|---|
One-at-a-time sensitivity. Each intervention's hazard ratio is swung across its 95% confidence interval (others held at point estimate); the bar is the resulting swing in the combined front-door HR. A long bar means the combined estimate leans heavily on that single trial's precision.
Front-door (mediation) decomposition. Bone therapies act partly through one shared mediator — bone mineral density (BMD) gain. Each log-hazard-ratio is split into a BMD-mediated (indirect) and a BMD-independent (direct) part (bone quality/turnover, and the fall pathway). Indirect parts are pooled through the mediator with dose-response saturation (BMD gain cannot stack without limit), removing the mediator cross-correlation; direct parts keep the residual eigenvalue correlation at ρ̄. This is why exercise/fall-prevention (which acts on falls, not density) is NOT discounted for overlap with bone drugs.
| Intervention | HR | %BMD | BMD-med | indirect logHR | direct logHR |
|---|
Which % of cross-correlation is appropriate? Not one number. The mediator overlap is fixed empirically by the BMD dose-response saturation (currently removing — of the summed BMD-mediated effect when bone drugs are stacked); the direct overlap is the residual ρ̄. BMD-mediated fractions are cited from published mediation analyses (risedronate ~11–18%; denosumab ~35–51%) — a striking finding that most anti-fracture benefit is NOT explained by density.
Front-door caveat (antithesis): BMD is a weak mediator here — it explains a minority of the anti-fracture effect, so a BMD-only model badly understates efficacy. The BMD→fracture path is also confounded (bone quality co-varies with density). This is a natural direct/indirect-effect decomposition in the front-door spirit, not a clean front-door-structured mediation; indirect estimates are bounds.
Select interventions to generate a plain-language summary.