Game 3 · Diagnostic tests
How much should a test result change your diagnosis?
Likelihood ratios, pre-test and post-test probability, and Bayes' rule.
Read with: Bayes' rule and combining tests, Diagnostic tests10 DNB questions
Play
Guess the probability after the test
Each round gives you a patient, a pre-test probability and a test result with its likelihood ratio. Set your guess for the post-test probability, then see the working in odds and on a Fagan nomogram. Some rounds chain two tests, and some ask you to work out the likelihood ratios from sensitivity and specificity. You have three lives.
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The idea
A test result moves a probability you already had
Before any test, you already have an estimate of how likely the disease is. This is the pre-test probability. It comes from the prevalence of the disease among patients like this one in your setting, raised or lowered by the history and examination. The test result then moves it to the post-test probability.
The likelihood ratio (LR) says how far the result moves it. It tells you how many times more likely this result is in a person with the disease than in a person without it. An LR above 1 raises the probability, an LR below 1 lowers it, and an LR of 1 leaves it where it was. This is Bayes' rule written for a test result.
Formulas
Likelihood ratios work on odds
LR+ = sensitivity ÷ (1 − specificity)
LR− = (1 − sensitivity) ÷ specificity
post-test odds = pre-test odds × LR
- odds
- probability ÷ (1 − probability)
- probability
- odds ÷ (1 + odds)
- LR+
- the LR of a positive result
- LR−
- the LR of a negative result
Worked example from the diagnostic tests chapter: sensitivity 90%, specificity 80%. LR+ = 0.90 ÷ 0.20 = 4.5 and LR− = 0.10 ÷ 0.80 = 0.125. With a pre-test probability of 10%, the pre-test odds are 0.10 ÷ 0.90 = 0.111. After a positive result, 0.111 × 4.5 = 0.50 odds, which is 0.50 ÷ 1.50 = 33.3%. After a negative result, 0.111 × 0.125 = 0.0139 odds, which is 1.4%.
Fagan's nomogram (1975) does the same conversion with a ruler. The two outer scales are probabilities spaced by their log-odds, and the middle scale is the LR. A straight line from the pre-test probability through the LR meets the post-test probability.
Because likelihood ratios come from sensitivity and specificity, they do not change with prevalence the way PPV does. You can apply a published LR to your own patient's pre-test probability.
How big is the change
LR near 1 does little, LR above 10 or below 0.1 does a lot
McGee (2002) gave a bedside approximation that avoids odds. It is accurate to within 10 percentage points for pre-test probabilities between 10% and 90%.
| Likelihood ratio | Approximate change in probability |
|---|---|
| 10 | +45 percentage points |
| 5 | +30 |
| 2 | +15 |
| 1 | no change |
| 0.5 | −15 |
| 0.2 | −30 |
| 0.1 | −45 |
The gain from a test is largest when the pre-test probability is in the middle range. When you are already nearly sure, in either direction, even a strong result changes the probability only a little.
Myth: A positive result on a test with LR+ 10 means the patient almost certainly has the disease.
It depends on where you started. From a pre-test probability of 1%, LR+ 10 gives about 9%.
Myth: You can multiply the pre-test probability by the LR.
Multiply the odds, not the probability. Multiplying a probability can give values above 100%.
Two tests
Chaining two results
post-test odds = pre-test odds × LRA × LRB
The post-test probability after test A becomes the pre-test probability for test B. With odds, you multiply by each LR in turn. This is valid only if the tests are conditionally independent: among patients with the disease, a positive A does not make a positive B more likely, and the same holds among patients without it. Tests that detect the same abnormality tend to make the same mistakes, and then multiplying overstates the post-test probability. Use an LR measured for the combination where one exists, or report the chained result as an upper estimate and say that independence was assumed. The Bayes' rule chapter works through this.
Writing the answer
How to write this in a 10-mark DNB answer
- Define the likelihood ratio in one line: how many times more likely a result is in a person with the disease than in one without it.
- Draw the 2 × 2 table and give LR+ = sensitivity ÷ (1 − specificity) and LR− = (1 − sensitivity) ÷ specificity.
- Define pre-test and post-test probability, and write Bayes' rule in odds form: post-test odds = pre-test odds × LR.
- Work one example with round numbers, as above, for both a positive and a negative result.
- Draw a Fagan nomogram with one straight line, and give the interpretation: LR 1 no change, LR above 10 or below 0.1 a large change.
- End with the advantages (independent of prevalence, usable at the bedside, can be chained) and the caution that chaining assumes independent tests.
The one thing to remember
A test result does not give a diagnosis on its own. It multiplies the odds you started with by its likelihood ratio.
Past papers
DNB questions this game helps you answer
Questions from NBE theory papers for DNB, DrNB and Diploma courses, most often asked first. The tag after each question is the specialty and the year it was set.
- ×1Define sensitivity, specificity, likelihood ratios and ROC curve in research.
- ×1How does Bayes’ rule help in diagnostic probability?
- ×1Explain likelihood ratio.
- ×1Write briefly about Bayes’ Theorem and Fagan Nomogram with some examples in Hematology and how these impact testing in Hematology practice.
- ×1Discuss likelihood ratio with an example.
- ×1Likelihood Ratio.
- ×1Discuss about likelihood ratio giving an example.
- ×1Likelihood ratio.
- ×1Write short notes on Bayes Theorem.
- ×1What is Bayes' theorem? Discuss sensitivity and specificity of exercise electrocardiography in coronary artery disease.
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