A likelihood ratio describes how a finding changes the balance of evidence. It does not provide a diagnosis by itself, and it does not tell you to multiply the starting probability directly. The update has a particular sequence: probability becomes odds, the finding changes the odds, and the result becomes a probability again.
This article extends QuietStride’s diagnostic-table walkthrough. The task here is updating a probability and deciding whether the evidence has already been counted. Every condition, test and numerical estimate below is invented for teaching; none is a recommendation to use a clinical test.
Read what the ratio compares
For a particular result, the likelihood ratio compares the probability of that result among people with the target condition with its probability among people without it. It is not the probability of the condition among people with that result. Deeks and Altman explain this direction and the odds form of Bayes’ rule. Diagnostic tests: likelihood ratios.
For a binary test, the familiar formulas are:
LR+ = sensitivity / (1 − specificity)
LR− = (1 − sensitivity) / specificity
The plus and minus refer to the result. They do not mean that the ratio must be positive or negative in the mathematical sense: both are nonnegative. A result with LR = 1 leaves the odds unchanged. A result with LR above 1 increases them; one below 1 decreases them.
A fictional update, line by line
A teaching case supplies a 20% pretest probability of condition Z. The specified test has sensitivity 80% and specificity 80% in a relevant fictional validation population. For this exercise, treat those values as known and applicable.
The positive likelihood ratio is 0.80/0.20 = 4. The negative likelihood ratio is 0.20/0.80 = 0.25.
For a positive result:
- Pretest odds = 0.20 / 0.80 = 0.25, or one to four.
- Post-test odds = 0.25 × 4 = 1, or one to one.
- Post-test probability = 1 / (1 + 1) = 0.50, or 50%.
The finding moves the supplied probability from 20% to 50%. It does not move it to 80%, which is what multiplying 20% by four would suggest. That shortcut confuses odds with probability; at higher starting values it can even produce an impossible probability above 100%.
For a negative result, start again from the original pretest odds. The updated odds are 0.25 × 0.25 = 0.0625. Converting back gives 0.0625/1.0625, approximately 5.9%.
These are alternative results from one test. Do not apply both LR+ and LR− to the same observation.
The starting probability remains part of the answer
Hold the fictional LR+ at four and change only the starting probability:
| Supplied pretest probability | Pretest odds | Post-test probability after a positive result |
|---|---|---|
| 5% | 0.05/0.95 | 17.4% |
| 20% | 0.20/0.80 | 50.0% |
| 50% | 0.50/0.50 | 80.0% |
A positive result is evidence in the same direction in each row, yet the resulting probabilities differ substantially. “This result rules it in” would hide the starting point and whatever decision threshold the speaker had in mind.
In a real appraisal, ask where the pretest estimate came from and whether it fits the referral setting and presenting population. A study’s condition frequency is not automatically the appropriate starting probability for another setting. Also check how the test was performed, its threshold and the reference standard. The FDA’s diagnostic-study guidance addresses such design and interpretation issues; it is used here for statistical principles, not Canadian professional regulation. FDA diagnostic-study guidance.
Do not turn two labels into two independent findings
Now suppose the positive result is copied into a second report under a different heading. The second entry is an exact duplicate, not a new measurement.
If you multiply the updated odds of one by four again, you obtain odds of four and a probability of 80%. But no new evidence arrived. The justified probability in this constructed case remains 50%.
This extreme counterexample exposes a subtler problem with related examination findings. Two different labels can share information. To use two ordinary likelihood ratios successively, you need appropriate conditional independence given condition status and given its absence, or evidence supporting the combined rule. Otherwise, the second update needs a likelihood ratio conditional on what is already known.
For the exact duplicate, once the first positive is known, the duplicate positive is certain in either condition group. Its conditional likelihood ratio is therefore 1/1 = 1. It changes nothing. That is why a list of apparently supportive findings cannot be converted automatically into repeated multiplication.
The same caution applies if the information used to choose the pretest probability already includes the very finding you are about to apply as a test result.
Put uncertainty back into the calculation
The earlier arithmetic deliberately treated the inputs as fixed. Suppose a sensitivity check uses LR+ values from two to six while retaining the 20% starting probability. Updated probabilities range from 33.3% to 60.0%.
That range is an illustration of dependence on an input, not automatically a 95% interval for the final probability. A full uncertainty analysis would need to address uncertainty in the starting estimate, the likelihood ratio and their relationship. Reporting many decimal places cannot make uncertain inputs precise.
Nor does reaching 50% determine a clinical action. Consequences, alternative explanations and the relevant assessment context remain outside this arithmetic exercise.
Transfer exercise: explain the next update
A new fictional case supplies a pretest probability of 10% and an applicable LR+ of six. A positive result appears, followed by a second entry that merely repeats the first result. What probability is supported by the supplied model?
The pretest odds are 0.10/0.90 = 1/9. Multiplying by six gives 2/3. The probability is (2/3) / (1 + 2/3) = 0.40, or 40%. The duplicate provides no additional information, so it does not justify another update.
If the second entry were an independently validated new finding, you would first need its appropriate likelihood ratio and evidence about dependence. “It is a different test” would not, by itself, supply either.
For your next diagnostic paper, write a four-line note: starting probability and its basis; result-specific likelihood ratio and population; odds calculation; and information already counted. That record makes the reasoning inspectable without pretending that an arithmetic result is a complete diagnosis.
Sources and further reading
- Deeks and Altman — Diagnostic tests 4: likelihood ratios — Likelihood-ratio definitions and odds-based updating; publisher-indexed text consulted because direct full-text access was unavailable.
- FDA — Statistical guidance on diagnostic-test studies — Diagnostic-study population, reference-standard and interpretation concerns.