A Decision-Theoretic Deployment Threshold for Multi-Cancer Early Detection Screening: Coupling Specificity, Prevalence, and Overdiagnosis

1. Introduction

Multi-cancer early detection (MCED) tests detect shared tumour signals (e.g. cell-free DNA methylation) and aim to screen for many cancers at once. A recurring reasoning error in screening advocacy is to treat detection as if it were benefit: a test that finds more cancers is assumed to help more patients. Decades of screening experience show this inference is unsafe, because of false positives, the gap between earlier detection and reduced mortality, and overdiagnosis [@welch2010overdiagnosis; @etzioni2003early]. This paper asks a precise question that does not require us to run a study: given honestly stated test characteristics and harms, when is population MCED screening expected to produce positive net utility, and where is the boundary? We answer with Bayes' rule and expected-utility theory. We perform no experiment, enrol no patients, and report no new measurements; every numeric input is a named parameter to be supplied by the primary literature or by a prospective trial.

2. The Screening Decision, Formally

Consider one screening decision in a population with prevalence \(\pi\) of a currently-present, detectable cancer. Let the test have aggregate sensitivity \(Se = \Pr(\text{positive} \mid \text{cancer})\) and specificity \(Sp = \Pr(\text{negative} \mid \text{no cancer})\). By Bayes' rule the positive predictive value is

PPV = (Se pi) / ( Se pi + (1 - Sp) * (1 - pi) ).

Two facts follow immediately and are often under-appreciated. First, at the low prevalence typical of unselected screening, even excellent specificity yields modest PPV: with Sp = 0.99 and pi = 0.005, the false-positive term (1-Sp)(1-pi) is about 0.00995, comparable to the true-positive term Se*pi, so most positives can be false. Second, PPV is governed by the ratio of true-positive to false-positive mass, so specificity and prevalence trade off multiplicatively — a point we now turn into a decision rule.

3. A Deployment Threshold

Detection is not the endpoint; averted cancer death (net of harms) is. We therefore attach utilities to outcomes rather than to test results. Per person screened, relative to not screening, define:

• \(B\): net benefit accrued when a true positive is one whose mortality outcome is genuinely improved by earlier detection;
• \(m \in [0,1]\): the actionable fraction — the share of screen-detected cancers for which earlier detection improves the outcome. \((1-m)\) captures overdiagnosis/overtreatment (cancers that would never have caused harm, or that are not curable earlier);
• \(H_{od}\): harm per overdiagnosed case (overtreatment morbidity);
• \(H_{fp}\): expected harm and cost of a false-positive work-up (imaging, biopsy, anxiety) ⁶;
• \(c\): cost/harm of the test itself.

The expected change in utility from screening one person is

E[dU] = pi Se ( m B - (1 - m) H_od ) - (1 - pi) (1 - Sp) H_fp - c.

The first term is the expected value of a true positive, corrected for overdiagnosis; the second is the expected false-positive harm; the third is the test burden. Screening is net-beneficial exactly when E[dU] > 0, i.e.

pi Se ( m B - (1 - m) H_od ) > (1 - pi) (1 - Sp) H_fp + c.

Ignoring the small per-test cost \(c\) and rearranging into odds form gives the deployment threshold:

pi / (1 - pi) > [ (1 - Sp) H_fp ] / [ Se ( m B - (1 - m) H_od ) ].

Interpretation. The prior odds of an actionable cancer must exceed the ratio of per-person false-positive harm to net per-true-case benefit. Three consequences are worth stating. (i) If \(m\,B \le (1-m)\,H_{od}\) — i.e. overdiagnosis harm cancels the benefit of the actionable cases — the right-hand side is non-positive-denominator and no prevalence makes screening worthwhile; detection performance is then irrelevant. (ii) Raising specificity shrinks the numerator but cannot, by itself, satisfy the inequality at sufficiently low prevalence. (iii) Sensitivity enters only through the denominator and with far less leverage than \(m\): a more sensitive test that mostly adds non-actionable detections (small \(m\)) can lower net utility.

4. Overdiagnosis as a First-Class Harm

Most reported MCED operating points emphasise specificity and stage-shift, not \(m\). Yet the algebra above shows \(m\) is load-bearing: it multiplies the entire benefit term and sets whether a beneficial regime exists at all. \(m\) is also the hardest quantity to estimate, because it is counterfactual — it asks what would have happened without detection — and is confounded by lead-time and length-time bias ⁵. This mirrors a pattern seen across decision-theoretic analyses of costly information-gathering: the binding uncertainty is not the easily-measured accuracy of the signal but the harder-to-forecast value of acting on it.

5. An Illustrative Calculation (parameters, not findings)

To show the threshold's behaviour we substitute illustrative values within ranges discussed in the MCED literature ³; these are inputs for exposition, not measurements we made. Take Sp = 0.995, Se = 0.5 (aggregate, stage-weighted), pi = 0.006. Then PPV = (0.5*0.006)/(0.5*0.006 + 0.005*0.994) = 0.003/0.00797 ≈ 0.38. With normalised utilities B = 1, H_od = 0.3, H_fp = 0.05 and an actionable fraction m = 0.6, the threshold RHS is (0.005*0.05)/(0.5*(0.6*1 - 0.4*0.3)) = 0.00025/(0.5*0.48) = 0.00104, while the prior odds pi/(1-pi) ≈ 0.00604 — so this regime clears the bar. But if m falls to 0.25, the benefit term 0.25 - 0.225 = 0.025 collapses, the RHS rises to 0.00025/(0.5*0.025) = 0.02, and the same test is now net-harmful. The qualitative lesson is robust to the exact numbers: feasibility is decided by \(m\), not by headline accuracy.

6. What the Model Does Not Establish

This is an analytic framework, not evidence of clinical benefit. It assumes single-round screening with a simple false-positive fallback; it folds a heterogeneous mixture of cancers into aggregate \(Se, Sp, m\), whereas these differ sharply by tumour type and stage; it treats utilities as known and commensurable; and it says nothing about repeated-screening dynamics, where cumulative false-positive probability grows ⁶. Most importantly, it cannot generate \(m\), \(B\), or \(H_{od}\): it only shows how sensitive the deploy/no-deploy decision is to them.

7. Validation Needed

The framework is falsifiable and prescribes its own test. A confirmatory randomised trial would need a mortality (or mortality-surrogate) endpoint, not a detection endpoint; per-tumour-type estimates of \(Se\) and \(Sp\) at the deployed threshold; an estimate of \(m\) via long-term follow-up of screen-detected versus clinically-detected cases to bound overdiagnosis; and measured \(H_{fp}\) from the realised work-up pathway. The pre-registered prediction is sharp: in any arm where the estimated prior odds fall below the computed RHS, screening should show no mortality benefit and net harm. Phase-structured biomarker evaluation ² supplies the staging for such estimates.

8. Relation to Decision-Theoretic Resource Use

The deployment rule has the same shape as cost-sensitive value-of-information boundaries used elsewhere (act only when the success-weighted gain exceeds the utility-scaled cost). The screening-specific contribution is the explicit overdiagnosis correction \((1-m)H_{od}\) inside the benefit term, which has no analogue in standard test-or-act rules and which can flip the sign of the decision even when raw detection accuracy is high ⁴.

9. Conclusion

Whether MCED screening helps a population is not settled by how many cancers it finds. Under an honest expected-utility model, deployment is justified only when the prior odds of an actionable cancer exceed the ratio of false-positive work-up harm to overdiagnosis-corrected per-case benefit. Specificity and sensitivity matter, but the actionable fraction \(m\) is decisive and least known. We offer the boundary as a falsifiable target for prospective evaluation, not as a clinical recommendation.