UCSF Department Of Medicine | Evidence Based Management

Learn about Bayesian reasoning and likelihood ratios in the tutorial below
Search for test characteristics and likelihood ratios for various diseases by system using the navigation toolbar to the left
Calculate posttest probabilities using the Bayesian calculators below

General description of Bayesian analysis

Bayesian reasoning is a formal way of doing explicitly what most clinicians do implicitly, which is to interpret the results of a diagnostic test in light of the clinical situation.
Basic paradigm: What was thought before the test was done combined with the test result -> what is thought after the test result
To be more quantitative, this simple paradigm can be converted as follows:
- What was thought before the test was done = pretest probability of disease
- The test result = likelihood ratio (LR)
- What is thought after = posttest probability

Pretest and posttest probabilities

The sum of all pretest probabilities (for all diseases being considered in the differential diagnosis for a chief complaint) should be 100%. There must be an explanation for the patient’s problem, and the correct disease should hopefully be in the differential diagnosis and assigned a set pretest probability.
Pretest probability can also be thought of as the prevalence of a disease: the proportion of people with the target disorder in the population at risk at a specific time. There are various ways to determine the pretest probability of a disease prior to testing. For many conditions, a clinicians' gestalt about the pretest probability has been shown to be reasonably accurate. Thus, a formal algorithm (such as Wells criteria) is not always necessary to calculate pretest probability.
Posttest probabilities range from zero to 100%. When a posttest probability is obtained after testing, the clinician must decide if that result places a disease in question below a certain testing threshold (thus ceasing further testing for that diagnosis) or above a certain treating threshold (thus leading to treatment of that disease)
Quantitative Bayesian analysis actually uses pretest and posttest odds, as opposed to probabilities (which are more intuitive). However, probabilities can be calculated from odds and vice versa using the following equations, or more rapidly using the calculators (and nomograms) available below.
- Odds = probability/(1-probability)
- Probability = odds/(odds + 1)

Likelihood ratios

LR = (likelihood of test result in patients with disease)/(likelihood of test result in patients without disease)
Positive likelihood ratios (LR+) are greater than 1 and increase the probability of disease.
Negative likelihood ratios (LR-) are less than 1 and decrease the probability of disease.
Sensitivity and specificity are alternate ways to describe the characteristics of a test. Sensitivity and specificity are less useful than LRs in that they are not distinct terms; you cannot interpret one without knowing the other.
If the sensitivity and specificity of a test are known, likelihood ratios can be calculated:
- LR+ = sensitivity/(1-specificity)
- LR- = (1-sensitivity)/specificity
Likelihood ratios can be used in sequence, so long as the test results are independent. Thus, one can keep modifying the posttest probability based on a series of results.
In interpreting test characteristics, always be aware of what was used as the “gold standard” for diagnosis.

Example

Based on a recent meta-analysis, the pretest probability of septic arthritis in the general population presenting to an emergency department with an acutely swollen and tender joint is approximately 18%.
A patient presents with an acutely swollen and tender knee, and arthrocentecis is performed. The result is a synovial fluid WBC count of 55,000/µL. The LR+ for synovial fluid WBC count >50,000/µL is 7.7.
Using the Bayesian calculator below, the posttest probability of septic arthritis is determined at 62.8%, and the physician must decide whether to treat for septic arthritis or continue testing until a diagnosis is obtained. Most clinicians would at least start empiric antibiotic therapy (and consider orthopedic consultation) with a posttest probability of 62.8% for septic arthritis.
Alternatively, the synovial fluid WBC count in this patient is found to be <25,000/µL, which has an LR- of 0.32. Thus, the posttest probability in this case would be 6.6%. A posttest probability of 6.6% would force the clinician to consider alternate etiologies more highly.

Rule of thumb

For LR+ of 2, the pretest probability is increased by about 15%.
For LR+ of 5, the pretest probability is increased by about 30%.
For LR+ of 10, the pretest probability is increased by about 45%.
For LR- of 0.5, the pretest probability is decreased by about 15%.
For LR- of 0.2, the pretest probability is decreased by about 30%.
For LR- of 0.1, the pretest probability is decreased by about 45%.

Bayesian calculators

Use the following calculators to determine posttest probability after a test is performed

1)This calculator is useful in all cases, whether you know:

the pretest probability, sensitivity, and specificity
the pretest probability and the likelihood ratios
the raw data of the 2X2 table

http://www.healthcare.ubc.ca/calc/bayes.html

2) This calculator is useful if you know the pretest probability (or prevalence) and the likelihood ratio:

http://www.dokterrutten.nl/collega/LRcalcul.html

3)This calculator is useful if you know the pretest probability (or prevalence) and the sensitivity and specificity of the test. In addition, this link demonstrates graphically the Fagan Nomogram:

http://araw.mede.uic.edu/cgi-bin/testcalc.pl

References

Browner, WS. “Bayesian reasoning and diagnostic testing.” Hospital Medicine. Ed. Wachter RM, Goldman L, and Hollander H. Lippincott Williams & Wilkins: Philadelphia, 2000.
Margaretten ME, et al. Does this adult patient have septic arthritis? JAMA. 2007;297:1478-1488.
McGee S. Simplifying likelihood ratios. J Gen Intern Med. 2002;17(8):646-9.

UCSF Department of Medicine