Likelihood Ratio for Non-Detection¶
Method — instantiates Null Finding Warrant Calibration
Quantifies how much less likely the null finding is under target presence than target absence.
The archetype's whole complaint is that people forget one term: not "we found nothing," but "how surprising is finding nothing if the target were really there?" The Likelihood Ratio for Non-Detection is the method that computes exactly that missing term. It forms the ratio of two probabilities — the chance of observing nothing given the target is present, over the chance of observing nothing given the target is absent — and reports it as a single number that says how far the null finding should move belief. Its defining idea is that it prices surprise: a ratio near one means the null is uninformative (you'd probably have seen nothing either way), while a ratio far below one means the silence genuinely argues against presence. Unlike the minimum-detectable-presence table, which states the smallest level you could have seen, the likelihood ratio states how much the null should shift a prior toward absence.
Example¶
A field team surveys a river reach for a rare fish believed possibly to persist there, using environmental-DNA sampling and baited cameras, and detects nothing. The bare result is "not observed." The likelihood ratio makes it usable. From the detection function they estimate the probability of no detection if the fish is present at a plausible low density — say the gear and effort would have caught it seventy percent of the time, leaving a thirty percent chance of a false null. Against a near-certain "no detection if truly absent," the ratio is about 0.3 to 1: the survey makes presence at that density roughly threefold less likely than before. Conditioned on a higher assumed density the ratio plunges — strong evidence of absence; conditioned on a trace population it climbs toward one — near silence. Reporting the ratio as a curve across densities, rather than one number, is what turns "not observed" into a defensible statement about how much was learned.
How it works¶
- Model the numerator. Estimate P(null finding | target present) from the detection function, folding in the ways a present target leaves no trace — imperfect detection, evasion, decay, sampling that missed it.
- Set the denominator. P(null finding | target absent) is usually near one, discounted only by the false-positive rate that would have fired anyway.
- Take the ratio. The quotient is a Bayes factor for the non-detection — the multiplicative weight the null contributes to the odds of absence.
- Combine and map. Multiply prior odds by the ratio to get posterior odds, and map the ratio's magnitude to the warrant ladder — near one is silence, far below one is strong evidence of absence or exclusion.
Tuning parameters¶
- Assumed presence scenario — the density, effect size, or prevalence the numerator is conditioned on. This is the highest-leverage dial: the ratio is only meaningful relative to an assumed "if present, how much."
- False-negative modeling — how generously the numerator admits present-but-unseen mechanisms. Optimism here shrinks the numerator and overstates the evidence.
- Single value vs. curve — reporting one ratio at one scenario, or a curve across the range of plausible presence levels.
- Prior odds source — where the base rate comes from, since the posterior depends on it as much as on the ratio.
When it helps, and when it misleads¶
Its strength is that it makes the archetype's omitted term explicit and quantitative: it forces the question "how expected is this silence under presence?" into a number a decision can hang on, and it exposes when a null is genuinely strong evidence versus mere absence of looking.
Its failure mode is a flattering numerator — an over-optimistic detection model makes P(null | present) too small, so the ratio overstates how much the null argues against presence; and pinning the ratio to a single assumed presence level hides that the answer swings with that assumption. The method is the general-evidence cousin of the diagnostic negative likelihood ratio, LR−,[n1] and inherits its discipline: a likelihood ratio is only as honest as the sensitivity estimate feeding it. The guarding discipline is to report the ratio as a curve over presence scenarios and to source the numerator from measured detection performance, not from hope.
How it implements the components¶
The method realizes the quantify-the-evidence side of the archetype:
detection_power_model— the numerator P(null | present) is the detection power expressed as a likelihood; the method's core computation.non_detection_alternative_set— the present-but-unseen mechanisms are exactly what lift the numerator above zero, and the method requires them enumerated to be honest.absence_warrant_grade— the ratio's magnitude maps directly onto the warrant ladder, converting a number into a graded conclusion.
It weighs how surprising the null is but does not state the detection floor: minimum_detectable_presence_threshold and upper_bound_statement are the output of its nearest twin, the Minimum Detectable Presence Table — the ratio says how much the null moves belief, the table says what level the null could have caught.
Related¶
- Instantiates: Null Finding Warrant Calibration — the method supplies the archetype's omitted likelihood term as an explicit number.
- Sibling mechanisms: Null Finding Warrant Memo · Search Sensitivity Matrix · Detection Power Checklist · Negative Test Interpretation Protocol · Minimum Detectable Presence Table · Coverage Map and Blind-Spot Review · Silent Monitor Assurance Review
Editorial Notes¶
Form Classification¶
Form family: Analysis, Modeling & Optimization
Rationale: Likelihood Ratio for Non-Detection operates as a computation, comparison, model, or analytic representation used to infer, estimate, or choose because it quantifies how much less likely the null finding is under target presence than target absence.
Independent corroboration: The frozen evidence defines Likelihood Ratio for Non-Detection as 'Quantifies how much less likely the null finding is under target presence than target absence', so its operative form is Analysis, Modeling & Optimization.
Review outcome: Independent reviewer agreement; high confidence.
Origin Attribution¶
Primary origin: Statistics & Experimental Design
Origin pattern: Cross-disciplinary synthesis
Present-day reach: Multi-domain
Rationale: Likelihood ratios for negative or absent findings are rooted in statistical hypothesis comparison and diagnostic-test theory.
Related originating lineages:
- Criminology & Forensic Studies — Forensic evaluation uses non-detection likelihoods to weigh absence of a trace under competing propositions.
- Medicine & Healthcare — The negative likelihood ratio from diagnostic testing materially established the canonical non-detection form.
Review resolution: Both independent reviews assign primary provenance to statistics_experimental_design. The queued secondary differences (alternate_origin_disagreement, origin_mode_disagreement) are reconciled by retaining criminology_forensic, medicine_healthcare only as formative or independently established lineage(s), not merely as application domains. origin_mode=cross_disciplinary_synthesis records the provenance relationship, while domain_reach=multi_domain separately records applicability breadth. confidence=high preserves the more cautious assessment, and encyclopedia_synthesis=false records whether either reviewer identified a corpus-specific synthesis.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
[n1] The negative likelihood ratio, LR− = (1 − sensitivity) / specificity, is the diagnostic-testing special case of this method: how much more likely a negative result is in the diseased than the disease-free. Generalized, the same ratio quantifies any non-detection — an empty net, a silent sensor, a blank archive. ↩