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Predicted VAS Pain / 10
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VAS Reduction (points)
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Total Causal Effect (SMD)
0
Active Interventions
Visual Analogue Scale (VAS) — Pain Trajectory
Baseline (black marker) vs. Predicted (blue marker) — do(X = selected interventions)
0 — No Pain246810 — Worst Pain
Select interventions to see prediction.
P(VAS | do(X)) = Baseline - SMD_total × σ_pain
SMD_total = Σk(βk × dosek) × [1 + avg_ρ × (n−1)/2]−1 // correlation penalty
Mediation: SMDdirect = 0.65 × SMD_total; SMDindirect = 0.35 × SMD_total
SMD_total = Σk(βk × dosek) × [1 + avg_ρ × (n−1)/2]−1 // correlation penalty
Mediation: SMDdirect = 0.65 × SMD_total; SMDindirect = 0.35 × SMD_total
If-Not-For: Marginal Causal Contribution per Intervention Pearl do-operator
Bars show the incremental pain-SMD lost if this intervention were counterfactually removed while all others remain active. Longer bars = more causally necessary.
Mediation Decomposition
Category Contribution to Total SMD
Forest Plot: Hazard Ratios (95% CI) Backdoor adjusted
SMD on Pain Score — All Interventions MCID = −0.40
Evidence Summary Table
Pearl Counterfactual Framework (Tian & Pearl 2000): Three probabilities of causation are computed for each intervention under monotonicity and exchangeability assumptions.
PN (necessity): would OA have occurred without the intervention, given it didn't occur with it?
PS (sufficiency): would the intervention have prevented OA, given it occurred without it?
PNS: joint probability of both.
PN, PS, PNS — All Interventions Prevalence = 25%
PN = P(Yx=0=1 | X=1, Y=0) = (P(Y|X=0) − P(Y|X=1)) / (1 − P(Y|X=1))
PS = P(Yx=1=0 | X=0, Y=1) = (P(Y|X=0) − P(Y|X=1)) / P(Y|X=0)
PNS = P(Y1=0, Y0=1) ∈ [max(0, p₀₋p₁), min(p₀, 1−p₁)] // Balke-Pearl sharp bounds
PS = P(Yx=1=0 | X=0, Y=1) = (P(Y|X=0) − P(Y|X=1)) / P(Y|X=0)
PNS = P(Y1=0, Y0=1) ∈ [max(0, p₀₋p₁), min(p₀, 1−p₁)] // Balke-Pearl sharp bounds
ARR & NNT by Intervention
E-value (VanderWeele & Ding 2017): The minimum strength of an unmeasured confounder (measured as its maximum risk ratio with both exposure and outcome) needed to fully explain away the causal effect. E-values ≥ 2.0 are considered robust. The Balke-Pearl bounds are non-parametric sharp identification limits that hold regardless of distributional assumptions.
E-value Sensitivity Analysis Higher = More robust
Full Sensitivity Summary Table
Balke-Pearl Sharp Bounds on ACE (Average Causal Effect)
Hill Equation Dose-Response: E(D) = Emax × Dn / (EC50n + Dn), n=2.5. Shaded area = 95% CI propagated from meta-analytic SMD uncertainty. Orange dashed line = MCID threshold (SMD = −0.40).
Dose-Response Curve
Causal Priority Score = E-value × |SMD|
Mediation Pathways by Intervention
Mechanism of Action Reference