13 Research methods, statistics, critical review and evidence-based practice13.5 Diagnostic studies
13.5.5 Applying likelihood ratios and nomograms
3 min read2 figures
Written by Resident Doctor Committee · Published
Likelihood ratios
A likelihood ratio (LR) is a multiplier, not a percentage. Each one is calculated within the diseased and non-diseased groups, so prevalence is not needed.
Ratio | Formula | Meaning |
|---|---|---|
Positive likelihood ratio (LR+) | sensitivity / (1 - specificity) | True positive rate over false positive rate: how many times more likely a positive result is in someone with the disorder than in someone without it |
Negative likelihood ratio (LR-) | (1 - sensitivity) / specificity | The factor by which a negative result multiplies the pre-test odds; a positive number below 1 for a useful test |
A memory clinic screen with sensitivity 0.90 and specificity 0.85 has LR+ = 0.90 / 0.15 = 6 and LR- = 0.10 / 0.85, about 0.12.
The threshold with the highest LR+ is found by calculating sensitivity / (1 - specificity) at each threshold. It may differ from the threshold with the highest sensitivity, specificity or Youden's index.
The condition with the smallest LR- is the one a negative screen rules out most effectively.
An LR+ above 10 or an LR- below 0.1 shifts the odds a lot, while an LR of 0.5 to 2 shifts them little.
Why likelihood ratios transfer
Likelihood ratios carry a test's accuracy to a population whose prevalence differs from the study sample, which predictive values cannot do.
Spectrum and setting can still alter them: an LR validated on severe cases and healthy controls may poorly represent mild presentations.
From pre-test to post-test probability
Pre-test probability is the prevalence in the population where the test will be used, not in the study sample.
Likelihood ratios act on odds, not probabilities, so the update takes three steps:
Pre-test odds = p / (1 - p).
Post-test odds = pre-test odds x LR.
Post-test probability = odds / (1 + odds).
Multiplying 40% straight by 6 would give an impossible 240%. After a positive result, the post-test probability equals the positive predictive value from the 2x2 table.
Worked example: one screen in two settings
The screen has LR+ 6 and LR- 0.12.
Setting | Pre-test probability | Pre-test odds | After a positive screen | After a negative screen |
|---|---|---|---|---|
Memory clinic | 40% | 0.40 / 0.60 = 0.67 | Odds 0.67 x 6 = 4.0, so 4.0 / 5.0 = 80% | Odds 0.67 x 0.12 = 0.08, so 0.08 / 1.08 = 7.4% |
General practice | 5% | 0.05 / 0.95 = 0.053 | Odds 0.32, so 24% | Odds 0.0063, so 0.6% |
The results differ only because the starting probabilities differ.
A negative screen still leaves about 7% in the memory clinic, so persistent clinical concern is not dismissed on a screening score.
Fagan's nomogram
Fagan's (Bayes') nomogram does the same update graphically, without converting to odds by hand.
It has three parallel scales: pre-test probability on the left, likelihood ratio in the centre (0.001 to 1000, centred on 1) and post-test probability on the right.
A straight line through any two values gives the third.
A line from the pre-test probability through the LR+ gives the post-test probability after a positive result.
A line through the LR- gives a lower post-test probability after a negative result.
Chaining two tests
After the positive screen (odds 4.0), an informant test with LR+ 3 gives odds 12 and a probability of 12 / 13, about 92%.
Multiplying is legitimate only under conditional independence: within cases and within non-cases, one result must not change the other's probability. Tests sharing items can overstate the evidence.
Forest plots
A study on a forest plot is significant at the 5% level only when its 95% confidence interval excludes the line of no effect, which is 1 for an odds ratio.
The size of each box shows the study's weight, not its sample size.