1 Definition and Basic Concepts
1.1 Odds
Odds are a numerical expression of the likelihood of an event, defined as the ratio of the probability that the event occurs to the probability that it does not occur. If an event has probability \(p\), the odds are \(p/(1-p)\). Odds can range from 0 to infinity. An odds of 1 indicates equal probability of occurrence and non-occurrence. Odds are commonly used in gambling, epidemiology, and Bayesian statistics because they facilitate multiplicative updating with likelihood ratios.
1.1.1 Probability to Odds Conversion
Given a probability \(p\) (where \(0 \le p \le 1\)), the odds \(O\) are computed as:
\[ O = \frac{p}{1-p}. \]
For example, if \(p = 0.75\), the odds are \(0.75 / 0.25 = 3\), often written as “3 to 1” in favor of the event. Probabilities near 0 yield odds close to 0; probabilities near 1 yield very large odds.
1.1.2 Odds to Probability Conversion
Conversely, given odds \(O\), the corresponding probability \(p\) is recovered by:
\[ p = \frac{O}{O+1}. \]
For instance, odds of 4 (or 4:1) correspond to a probability of \(4/5 = 0.8\). This transformation is essential when interpreting odds in contexts where probabilities are more intuitive, such as risk communication.
1.2 Likelihood Ratios
A likelihood ratio (LR) quantifies how much evidence—such as a test result or observation—changes the odds of a hypothesis. It is the ratio of the probability of observing the evidence given one hypothesis to the probability of that same evidence under an alternative hypothesis.
1.2.1 Definition and Formula
For a hypothesis \(H\) and its complement \(\neg H\), and evidence \(E\), the likelihood ratio is:
\[ LR = \frac{P(E \mid H)}{P(E \mid \neg H)}. \]
If \(LR > 1\), the evidence supports \(H\) over \(\neg H\); if \(LR < 1\), it supports \(\neg H\); if \(LR = 1\), the evidence is neutral.
1.2.2 Positive and Negative Likelihood Ratios
In diagnostic testing, two specific likelihood ratios are used:
- Positive Likelihood Ratio (LR+): the ratio of the probability of a positive test result in diseased individuals to the probability of a positive result in non-diseased individuals. It measures the ability of a positive test to confirm disease.
- Negative Likelihood Ratio (LR-): the ratio of the probability of a negative test result in diseased individuals to the probability of a negative result in non-diseased individuals. It measures the ability of a negative test to rule out disease.
Mathematically, LR+ = sensitivity / (1 – specificity) and LR- = (1 – sensitivity) / specificity.
1.3 Relationship Between Odds and Likelihood Ratios
Bayes’ theorem links prior odds and posterior odds through the likelihood ratio:
\[ \text{Posterior odds} = \text{Prior odds} \times \text{Likelihood ratio}. \]
This multiplicative relationship allows evidence to update beliefs directly on the odds scale. For example, if prior odds of a disease are 1:9 (probability 0.1) and the test has LR+ = 10, the posterior odds become 10:9, corresponding to a probability of about 0.526. This simple multiplication is a cornerstone of Bayesian inference.
2 Calculation and Interpretation
2.1 Computing Likelihood Ratios from Contingency Tables
Likelihood ratios are derived from the performance characteristics of a test, summarized in a 2×2 contingency table comparing test results with true disease status.
2.1.1 Sensitivity and Specificity
- Sensitivity (true positive rate): the proportion of diseased individuals who test positive.
- Specificity (true negative rate): the proportion of non-diseased individuals who test negative.
These are computed as:
\[ \text{Sensitivity} = \frac{TP}{TP+FN}, \quad \text{Specificity} = \frac{TN}{TN+FP}. \]
2.1.2 Positive Likelihood Ratio (LR+)
\[ LR+ = \frac{\text{Sensitivity}}{1 - \text{Specificity}}. \]
A high LR+ (e.g., >10) indicates a very strong positive confirmation.
2.1.3 Negative Likelihood Ratio (LR-)
\[ LR- = \frac{1 - \text{Sensitivity}}{\text{Specificity}}. \]
A low LR- (e.g., <0.1) indicates that a negative result effectively rules out the condition.
2.2 Interpreting Likelihood Ratios
2.2.1 Scale of Evidence (e.g., Small, Moderate, Large)
Likelihood ratios are often interpreted using qualitative scales:
| LR+ value | Interpretation | LR- value | Interpretation | |
|---|---|---|---|---|
| >10 | Large increase in disease probability | <0.1 | Large decrease | |
| 5–10 | Moderate increase | 0.1–0.2 | Moderate decrease | |
| 2–5 | Small increase | 0.2–0.5 | Small decrease | |
| 1–2 | Negligible change | 0.5–1 | Negligible change |
These thresholds are guidelines; precise interpretation depends on context.
2.2.2 Bayes’ Theorem in Odds Form
The odds form of Bayes’ theorem is:
\[ \text{Posterior odds} = \text{Prior odds} \times LR. \]
This form avoids converting to probabilities until the final step and highlights the multiplicative nature of evidence. For example, if prior odds are 1:4 and LR = 8, posterior odds = 8:4 = 2:1, corresponding to a posterior probability of 2/3.
2.3 Example Calculations
2.3.1 Diagnostic Test Example
A test for a disease has sensitivity 90% and specificity 80%. The contingency table (for 1000 people, 100 diseased) yields:
- LR+ = 0.90 / (1 – 0.80) = 0.90 / 0.20 = 4.5.
- LR- = (1 – 0.90) / 0.80 = 0.10 / 0.80 = 0.125.
Thus, a positive test increases the odds of disease by a factor of 4.5; a negative test reduces them by a factor of 0.125.
2.3.2 Bayesian Updating Example
Prior probability of disease = 0.2 (odds = 0.2/0.8 = 0.25). After a positive test with LR+ = 4.5, posterior odds = 0.25 × 4.5 = 1.125, giving posterior probability = 1.125/(1+1.125) ≈ 0.529. After a negative test with LR- = 0.125, posterior odds = 0.25 × 0.125 = 0.03125, probability ≈ 0.0303.
3 Applications
3.1 Medical Diagnostics
Likelihood ratios are central to evidence-based medicine, allowing clinicians to update the probability of a disease given test results.
3.1.1 Pre-Test and Post-Test Probabilities
Pre-test probability is the estimated disease prevalence or clinical suspicion before testing. Post-test probability is derived by converting pre-test odds → multiply by LR → convert back. This process quantifies the diagnostic value of a test.
3.1.2 Fagan’s Nomogram
Fagan’s nomogram is a graphical tool that relates pre-test probability, likelihood ratio, and post-test probability. A straight line drawn from the pre-test probability (left axis) through the LR (center axis) intersects the right axis at the post-test probability. It avoids calculations and is widely used in clinical teaching.
3.2 Forensic Science
In forensic science, likelihood ratios evaluate the strength of evidence under competing hypotheses (e.g., prosecution vs. defense).
3.2.1 DNA Evidence Evaluation
DNA profile matching uses likelihood ratios to compare the probability of observing a match given that the suspect is the source versus that the suspect is not the source (and a random person is). A typical LR for a full 13-locus STR profile can be billions, representing extremely strong evidence for the prosecution hypothesis.
3.2.2 Fingerprint Analysis
Likelihood ratios are applied to fingerprint comparisons by modeling the frequency of minutiae patterns. The LR indicates how much more likely the observed features are under the hypothesis of a common source versus different sources, aiding in transparent reporting of evidential weight.
3.3 Machine Learning and Classification
Likelihood ratios are used in classification tasks, particularly in Bayesian classifiers and decision theory.
3.3.1 Likelihood Ratio as a Decision Criterion
For binary classification, the likelihood ratio of the observed features under the two classes can be compared to a threshold (derived from costs and prior probabilities) to decide the class. This is equivalent to the Bayes decision rule.
3.3.2 Log-Likelihood Ratio
In practice, log-likelihood ratios (LLR) are often used to avoid numerical underflow and to make additive rather than multiplicative updates. The LLR is the logarithm of the LR: \(LLR = \log(LR)\). A positive LLR favors class 1; negative favors class 0. LLR is also a fundamental component of sequential probability ratio tests (SPRT).
4 Relation to Other Measures
4.1 Relative Risk and Odds Ratio
Relative risk (RR) is the ratio of probabilities of an outcome between two groups. The odds ratio (OR) is the ratio of odds of the outcome between groups. While the OR approximates the RR for rare events, it is distinct from a likelihood ratio. The LR compares probabilities of evidence given hypotheses, whereas OR compares odds of outcome given exposure. Both are cross-sectional measures of association.
4.2 Bayes Factor
The Bayes factor is the Bayesian analogue of the likelihood ratio when comparing two models or hypotheses. It is the ratio of marginal likelihoods. Unlike LR, the Bayes factor integrates over prior distributions for parameters, making it a tool for model comparison. In simple hypothesis testing with no nuisance parameters, the Bayes factor equals the likelihood ratio.
4.3 Information-Theoretic Measures
The log-likelihood ratio is related to Kullback–Leibler divergence, which measures the information lost when using one distribution to approximate another. The expected log-likelihood ratio under one hypothesis is the KL divergence separating the two distributions. Thus, likelihood ratios connect to fundamental information-theoretic quantities.
5 Limitations and Common Misconceptions
5.1 Misinterpretation of Odds
Odds are often confused with probabilities. For example, odds of 3:1 correspond to a probability of 0.75, not 0.33. In medical contexts, reporting odds instead of probabilities can exaggerate perceived risk. Also, odds are not directly comparable across different baselines, which can lead to misinterpretation in case-control studies.
5.2 Assumptions for Likelihood Ratios
Likelihood ratios assume that the evidence is independent of other factors conditional on the hypothesis—the so-called conditional independence assumption. In diagnostic tests, this means that the test’s sensitivity and specificity do not depend on the patient’s other characteristics. When violated, LRs can mislead. Additionally, LRs from one population may not generalize to another with different disease spectrum.
5.3 Pitfalls in Practice
- Base rate neglect: Even with a high LR+, a low pre-test probability can yield a modest post-test probability. Clinicians often overlook this.
- Non-dichotomous tests: For continuous or multi-level tests, using a single LR+ and LR- discards information; likelihood ratio functions are more appropriate.
- Improper use of Bayes factor: Using the LR from a simple test as a Bayes factor without accounting for prior model probabilities can inflate evidence in complex settings.
- Interpretation on probability scale: Multiplying probabilities directly by an LR is incorrect; odds must be used first. This common error yields inaccurate posterior estimates.