Concept Architecture
Concept
Theoretically, Prior Odds are the odds that a hypothesis or condition is true before new evidence is observed. They represent the initial degree of belief derived from prior information, such as disease prevalence, previous studies or expert judgement, and form the starting point for Bayesian inference. The concept is founded on Bayesian probability theory and exists to enable the systematic updating of beliefs when additional evidence becomes available.
Mathematically, Prior Odds are obtained by converting Prior Probability into odds. Unlike probabilities, which range from 0 to 1, odds express the ratio of the probability that an event occurs to the probability that it does not occur. Prior Odds are subsequently multiplied by the appropriate Likelihood Ratio to produce Posterior Odds according to Bayes' Theorem.
In practice, Prior Odds are estimated from disease prevalence, epidemiological studies, clinical prediction models or expert opinion before diagnostic testing. They are widely applied in diagnostic medicine, Bayesian evidence synthesis, health technology assessment and decision-analytic modelling to provide the baseline against which new evidence is evaluated.
Purpose
Used to quantify the baseline odds of a hypothesis before new evidence is observed, support Bayesian inference, estimate diagnostic probabilities and improve clinical and health economic decision-making.
Mathematical Formulae
Primary Formula
Prior Odds = Prior Probability / (1 ? Prior Probability)
Supporting Formulae
Posterior Odds = Prior Odds ? Likelihood Ratio
Posterior Probability = Posterior Odds / (1 + Posterior Odds)
Prior Probability = Prior Odds / (1 + Prior Odds)
where:
- Likelihood Ratio = positive or negative likelihood ratio depending on the observed evidence
Related Mathematical Methods
- Bayes' Theorem
- Prior Probability
- Posterior Odds
- Posterior Probability
- Likelihood Ratio
- Positive Likelihood Ratio
- Negative Likelihood Ratio
- Bayesian Inference
Example
A patient has an estimated pre-test probability of disease of 20%.
Prior Odds:
Prior Odds = 0.20 / (1 ? 0.20)
Prior Odds = 0.20 / 0.80 = 0.25
If a diagnostic test subsequently produces a Positive Likelihood Ratio of 8, the Prior Odds provide the baseline for calculating Posterior Odds and Post-Test Probability.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Division | =B2/(1-B2) | Converts Prior Probability into Prior Odds before Bayesian updating. |
| Multiplication | =C2*D2 | Calculates Posterior Odds using the selected Likelihood Ratio. |
| Division | =E2/(1+E2) | Converts Posterior Odds into Posterior Probability. |
| IF | =IF(B2>0.50,"High baseline odds","Low baseline odds") | Classifies patients according to baseline disease odds. |
VBA (Optional)
A VBA macro can automatically calculate Prior Odds from prevalence estimates and perform Bayesian updating across multiple diagnostic scenarios.
Sources
- Bayes T. An Essay towards Solving a Problem in the Doctrine of Chances. 1763.
- Gelman A, Carlin JB, Stern HS, et al. Bayesian Data Analysis. 4th ed.
- Spiegelhalter DJ, Abrams KR, Myles JP. Bayesian Approaches to Clinical Trials and Health-Care Evaluation.
- Deeks JJ, Altman DG. Diagnostic tests 4: likelihood ratios. BMJ. 2004;329:168?169.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What are prior odds?
The odds that a hypothesis is true before incorporating new evidence, the starting point for Bayesian updating alongside a likelihood ratio.
Source: Bayes 1763
What do prior odds represent at the start of Bayesian updating?
Prior odds represent the odds that a hypothesis is true before any new evidence is considered, the starting point from which Bayesian updating proceeds. In diagnosis they are the odds a patient has a disease based on background knowledge, such as its prevalence and the clinical picture, before a test is run. Multiplying them by a likelihood ratio produces the posterior odds after the evidence. Because they anchor the update, a mistaken prior distorts the conclusion. The belief before the evidence is what they capture. Sackett and colleagues (1991) describe this.
Source: Sackett et al. 1991
How are prior odds obtained?
Prior odds are obtained by converting the prior probability of the hypothesis into odds, dividing the probability that the hypothesis is true by the probability that it is false. In diagnosis, the prior, or pre-test, probability of disease, based on prevalence and clinical features, is converted to prior odds. So prior odds are obtained from the prior probability through the relationship between probability and odds, which expresses the initial belief in a form convenient for Bayesian updating, since working on the odds scale allows the evidence, summarised by a likelihood ratio, to be applied by simple multiplication to give the posterior odds.
Source: Bayes 1763
How do prior odds relate to posterior odds?
Prior odds relate to posterior odds as the starting point that evidence updates: multiplying the prior odds by the likelihood ratio of the evidence gives the posterior odds. The prior odds represent belief before the evidence, and the posterior odds belief afterward. So prior odds and posterior odds are the before and after of Bayesian updating on the odds scale, connected by the likelihood ratio, which determines how far the evidence shifts the odds, so that the stronger the evidence, the further the posterior odds move from the prior odds toward or away from the hypothesis being true.
Source: Bayes 1763
Why do prior odds matter?
Prior odds matter because the conclusion after evidence, the posterior odds, depends on them as well as on the evidence, so the same evidence leads to different posterior beliefs depending on the prior odds. Neglecting the prior odds, for example the low prior probability of a rare condition, leads to overestimating the hypothesis after positive evidence. So prior odds matter for correct Bayesian reasoning, since ignoring them produces errors such as base-rate neglect, and in diagnosis the pre-test odds must be considered when interpreting a test, because a positive result carries different meaning depending on how likely the condition was beforehand.
Source: Bayes 1763
How are prior odds used in diagnosis?
In diagnosis, prior odds are used as the pre-test odds of disease, derived from the pre-test probability, which are multiplied by the test's likelihood ratio to obtain the post-test, or posterior, odds. This is the mechanism by which a test result updates the probability of disease. So prior odds are used in diagnostic reasoning as the starting point for interpreting a test, since the post-test odds, and hence the post-test probability, depend on both the prior odds and the likelihood ratio, which is why estimating the pre-test probability, and thus the prior odds, is a necessary step in the correct interpretation of diagnostic tests.
Source: Sackett et al. 1991
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 10 Dec 2025
Content version: 1.0.0
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- Persistent URI
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- Term code
- HE-ES-RM-030
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