Untreated caries of permanent teeth is the single most prevalent condition on Earth (~2.24 billion people), and oral conditions together affect ~3.7 billion. This oracle estimates the causal reduction in caries increment across topical-fluoride, public-health, barrier, arrest and dietary options, distinguishing agents that shift the mineral balance from those that remove the sugar substrate or physically seal the tooth. Prevented fractions are from Cochrane reviews. For education, not individual dental advice.
Tick the interventions to combine. Each shows its trial effect estimate, 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 demin 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). Fluoride agents act through the shared mediator de-/re-mineralisation balance, which governs lesion formation and thence caries increment (Y). Sealants (physical barrier) and sugar reduction (substrate removal) act independently of the mineral balance. Named confounders — sugar frequency, salivary flow, access — open back-door paths (adjusted). Mediator cascade: interventions attach to the node they act on (cariogenic → remineralisation → physical barrier → lesion arrest), which converge on the disease state and thence the endpoint — drawing the intermediate mediators explicitly is what exposes d-separation and per-channel saturation.
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: the topical-fluoride agents (toothpaste, rinse, varnish, water) all act through remineralisation, so combining them is partly redundant — their prevented fractions do NOT simply add. ρ̄ is user-adjustable because a physical sealant and a dietary sugar cut share no mechanism with fluoride. Most of the overlap is now handled structurally by the mediator nodes (same-node substitutes saturate); ρ̄ governs only the residual correlation among direct effects.
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-effect 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 effect would be…". Isolates each intervention's marginal causal contribution after mediator-overlap removal, so shared-pathway agents are not double-credited.
| If not for… | RR without it | RR with full set | marginal RRR lost |
|---|
One-at-a-time sensitivity. Each intervention's effect is swung across its 95% confidence interval (others held at point estimate); the bar is the resulting swing in the combined front-door effect. A long bar means the combined estimate leans heavily on that single trial's precision.
Front-door (mediation) decomposition. The fluoride agents act through one shared mediator — the mineral balance. Each log-effect is split into a remineralisation-mediated (indirect) and a direct part. Indirect parts are pooled through the mediator with dose-response saturation, removing the mediator cross-correlation; direct parts keep the residual eigenvalue correlation at ρ̄. Sealants (barrier) and sugar reduction (substrate), which act off the mineral balance, are NOT discounted against the fluoride agents. Here mediated effects are pooled WITHIN each cascade node (dose-response saturation of substitutes) and composed in SERIES across nodes (d-separated channels), with the per-node reductions reported so the channel structure is visible.
| Intervention | RR | %remin | med-frac | indirect log | direct log |
|---|
Which % of cross-correlation is appropriate? Not one number. The mediator overlap is fixed empirically by the demin saturation (currently removing — of the summed mediated effect when interventions are stacked). Note a domain caveat: the fluoride agents are mutually redundant (shared remineralisation), whereas sealants and sugar reduction add distinct mechanisms. Remineralisation-mediated fractions are transparent, adjustable priors from Cochrane prevented fractions.
Front-door caveat (antithesis): fluoride treats the mineral balance, not the sugar cause; SDF’s strong number is Cochrane-uncertain and stains teeth; the global burden is largely an access problem; and fluoride is non-monotonic (fluorosis at excess). Prevented fractions are not strictly additive.
Select interventions to generate a plain-language summary.