Two Positive Tests Are Not Always Twice the Evidence
October 8, 2026 · 6 min read · Aqrab editorial team
A study reports a positive likelihood ratio (LR+) of 5. A patient tests positive, then tests positive again. Can you multiply their starting odds by 5 twice? Only if the second result has the same evidential value after the first result is known. Repeating a measurement does not establish that.
What the Second Result Must Add
An LR compares how often a result occurs among people with the target condition with how often it occurs among people without it. For a sequence, the second update needs a conditional LR: the probability of a second positive result among people with the condition who already tested positive, divided by that probability among people without the condition who already tested positive.
Multiplying two stand-alone LRs assumes the tests are conditionally independent given true condition status. In other words, once you know whether the condition is present, the first test result must tell you nothing about the second. Shared sample quality, biology, instrument behavior, or a repeated reading can defeat that assumption. Correlation among everyone pooled together is not the test of this assumption; dependence is assessed separately within those with and without the condition.
One Starting Risk, Two Very Different Updates
Take a deliberately hypothetical 10% starting probability and a first positive result with LR+ 5. The starting odds are 1:9. After that result, the odds are 5:9, or a probability of about 35.7%.
| Assumption about second positive | Second conditional LR | Probability after both |
|---|---|---|
| Independent within condition groups; same LR | 5 | about 73.5% |
| Exact duplicate of the first result | 1 | about 35.7% |
Under the first assumption, odds become 25:9 and probability is 25/34, or about 73.5%. Under the second, the repeat gives no new information: the odds stay 5:9. Real sequences are not automatically between these two examples; their conditional LR must be estimated from data collected in the relevant testing pathway.
The practical takeaway
When a paper combines sequential results, ask for the accuracy of the whole sequence or the second test's performance conditional on the first result. Two separate sensitivity/specificity tables do not supply that answer.
Four Questions for a Testing Pathway
- Who reaches test two? Is it offered to everyone or only after a particular first result? State the pathway and its denominators.
- What is repeated? Is it the same specimen, device, target, reader, or underlying measurement process?
- What joint data exist? Among people with and without the condition, how often does each pair of results occur?
- What was verified? Was condition status established for the full relevant sequence, including discordant and untested participants?
A test evaluated alone in one population may perform differently after patients are selected by an earlier result. If joint results or reference-standard follow-up are missing, the incremental evidence from test two is uncertain. Do not replace that gap with a second multiplication of the marginal LR.
Where Aqrab Fits
Use Aqrab to pressure-test whether a diagnostic or clinical-AI study evaluated the pathway it proposes, reported joint result counts, and justified any independence assumption. A methods critique can surface missing evidence; it cannot infer the conditional LR from two stand-alone accuracy estimates or prescribe a patient's next test.
Source Anchors
Deeks and Altman (BMJ, 2004) give the LR-to-odds update. Fanshawe and colleagues (2024) review analysis of sequential diagnostic tests, including conditional dependence and testing only after a first result. Lau (1991) models dependent repeated screening tests. None supplies the invented probabilities above.
Keep reading
Don't stop at one method.
Good methods judgment comes from contrast. Read the neighboring guides, see where the assumptions diverge, and avoid treating every observational problem like it needs the same hammer.
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Positive Predictive Value: Why a Good Test Can Still Produce Mostly False Alarms
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