What is the likelihood ratio in biostatistics?

Understanding Likelihood Ratios in Biostatistics

Likelihood ratios are diagnostic test performance measures that quantify how many times more (or less) likely patients with a disease are to have a particular test result compared to patients without the disease, providing a direct method to revise probability estimates from pre-test to post-test values. 1

Core Concept and Definition

The positive likelihood ratio (LR+) indicates how much more likely a person with the disease is to have a positive test result than a person without the disease. 1 For example, an LR+ of 46 means that a person with the disease is 46 times more likely to test positive than someone without the disease. 1

The negative likelihood ratio (LR−) indicates how much less likely a person with the disease is to have a negative test result than a person without the disease. 1 An LR− of 0.05 means that a person with disease is 20 times less likely (1/0.05) to test negative than a person without the disease. 1

Clinical Interpretation Thresholds

A widely accepted rule of thumb for "high" clinical validity is an LR+ ≥10 and an LR− ≤0.1. 1 These thresholds indicate:

• LR+ ≥10: High likelihood that disease is present when the test is positive 1
• LR− ≤0.1: High likelihood that disease is absent when the test is negative 1

More granular interpretation shows that LR+ values of 2,5, and 10 correspond to approximately 15%, 30%, and 45% changes in pretest probability, respectively. 1 Similarly, LR− values of 0.5,0.2, and 0.1 indicate 15%, 30%, and 45% changes in pretest probability for negative tests to "rule-out" a diagnosis. 1

Mathematical Relationship to Other Test Characteristics

Likelihood ratios are calculated from sensitivity and specificity but provide more clinically useful information than these measures alone. 2, 3 The formulas are:

• LR+ = Sensitivity / (1 - Specificity) 2, 4
• LR− = (1 - Sensitivity) / Specificity 2, 4

The diagnostic odds ratio (DOR) equals LR+ divided by LR−, representing the ratio of odds of disease with a positive test relative to odds of disease with a negative test. 1 The DOR is independent of disease prevalence and useful for comparing diagnostic accuracy between tests in meta-analyses. 1

Clinical Application Using Bayes Theorem

Likelihood ratios enable conversion of pretest probability to post-test probability through Bayesian reasoning. 2, 3 This process:

1. Start with pretest probability (clinical estimate before testing) 3
2. Convert to pretest odds 3
3. Multiply by the likelihood ratio 3
4. Convert back to post-test probability 3

The Fagan nomogram provides a graphical tool for this calculation in routine clinical practice, allowing direct visualization of how test results shift probability estimates. 3

Advantages Over Sensitivity and Specificity

Likelihood ratios have three main advantages: they are intuitive, they simplify predictive value calculations, and they facilitate evaluation of sequential testing. 5 Unlike sensitivity and specificity, which cannot estimate probability of disease in individual patients, likelihood ratios provide immediate and direct description of a test's ability to revise initial probability of disease upward or downward. 4

Likelihood ratios can be calculated for multiple levels of test results, not just dichotomous (positive/negative) outcomes. 2, 4 This allows more nuanced interpretation of tests with continuous or multi-level results, such as creatine kinase levels or imaging findings. 2

Important Caveats and Limitations

Confidence intervals around likelihood ratios are essential but frequently omitted in medical literature. 5 Broad confidence intervals indicate less certainty in point estimates, particularly when based on smaller numbers of studies. 1

Likelihood ratios near unity (LR+ ≈1, LR− ≈1) have minimal effect on clinical decision-making and indicate poor test performance. 2 Such values suggest the test provides little diagnostic information.

When interpreting likelihood ratios across different testing strategies, avoid assuming "more is better" without considering confidence intervals. 1 Statistical differences may not translate to clinically meaningful differences if confidence intervals overlap substantially. 1

Likelihood ratios require recalculation of probabilities in odds format, which can be less intuitive than direct probability estimates. 5 Additionally, the non-linearity of the relationship between odds and probability can complicate interpretation. 5