Likelihood Ratio

A likelihood ratio expresses the probability of a given test result in a patient with the disease relative to the probability of that same result in a patient without the disease. LR+ (positive likelihood ratio) = Sensitivity / (1 - Specificity). LR- (negative likelihood ratio) = (1 - Sensitivity) / Specificity. An LR+ of 10 means a positive result is 10 times more likely in a diseased patient than a non-diseased one.

Unlike Positive Predictive Value (PPV) and Negative Predictive Value (NPV), likelihood ratios are theoretically independent of disease prevalence. This makes them portable across different clinical settings. A test's LR+ remains constant whether applied in a tertiary referral center with high prevalence or a community screening program with low prevalence, allowing clinicians to apply the metric directly to their specific patient population.

LRs are interpreted on a continuous scale of diagnostic impact:

• LR+ > 10 or LR- < 0.1: Large, often conclusive shifts in probability
• LR+ 5-10 or LR- 0.1-0.2: Moderate shifts
• LR+ 2-5 or LR- 0.2-0.5: Small but potentially important shifts
• LR+ 1-2 or LR- 0.5-1: Rarely clinically significant A test with an LR+ of exactly 1 provides no diagnostic information.

Likelihood ratios are applied clinically using Fagan's nomogram, a graphical tool that converts pre-test probability to post-test probability. The clinician draws a line from the patient's estimated pre-test probability through the calculated LR to read the post-test probability. This Bayesian approach integrates the test result with clinical judgment, moving beyond binary 'positive/negative' interpretations to a probabilistic diagnostic framework.

For tests producing continuous or ordinal results, multi-level likelihood ratios can be calculated for each result interval rather than a single cutoff. This preserves more diagnostic information than dichotomizing results. For example, a biomarker assay might report LRs for ranges: < 1.0 ng/mL (LR 0.05), 1.0-3.0 ng/mL (LR 0.8), 3.1-10.0 ng/mL (LR 4.5), and > 10.0 ng/mL (LR 25).

Like all statistical estimates, LRs should be reported with 95% confidence intervals to convey precision. Wide confidence intervals indicate uncertainty due to small sample sizes. An LR+ of 8.0 (95% CI: 2.1-30.5) suggests a potentially useful test but with substantial imprecision. Regulatory submissions for diagnostic AI often require demonstrating that the lower confidence bound of the LR exceeds a clinically meaningful threshold.

The proportion of true positive cases correctly identified by a diagnostic test. A highly sensitive test rarely misses disease, making it ideal for screening where the cost of a false negative is catastrophic. Sensitivity directly feeds into the positive likelihood ratio (LR+) calculation.

• Formula: TP / (TP + FN)
• A test with 95% sensitivity misses 5% of diseased patients
• Prioritized when ruling out dangerous conditions (SNNOUT)

The proportion of true negative cases correctly identified. A highly specific test rarely mislabels healthy patients as diseased, making it essential for confirmatory diagnosis where false positives cause unnecessary intervention. Specificity anchors the negative likelihood ratio (LR-) calculation.

• Formula: TN / (TN + FP)
• A test with 98% specificity generates false alarms in 2% of healthy patients
• Prioritized when ruling in conditions (SPPIN)

The probability that a patient truly has the disease given a positive test result. Unlike sensitivity and specificity, PPV is prevalence-dependent—the same test will have a higher PPV in a high-risk population than in general screening.

• Formula: TP / (TP + FP)
• A test with 99% sensitivity can still have low PPV if the disease is rare
• Directly answers the clinical question: 'Given this positive result, what are the odds my patient is sick?'

The probability that a patient truly does not have the disease given a negative test result. Like PPV, NPV shifts with disease prevalence—a negative result is more reassuring in a low-prevalence setting.

• Formula: TN / (TN + FN)
• High NPV is critical for rule-out tests in emergency medicine
• Works in tandem with LR- to quantify how much a negative result reduces disease probability

The Receiver Operating Characteristic curve plots sensitivity against 1-specificity across all possible decision thresholds. The Area Under the Curve (AUC) summarizes overall discriminative ability independent of any single cutoff.

• AUC of 1.0 = perfect discrimination; 0.5 = random guessing
• Unlike likelihood ratios, ROC-AUC does not directly inform post-test probability
• The DeLong test statistically compares AUCs between competing diagnostic models

A contingency table that lays out the raw counts of true positives, true negatives, false positives, and false negatives. Every diagnostic performance metric—including likelihood ratios—derives from these four fundamental values.

• Rows represent predicted class; columns represent actual class
• The foundation for calculating sensitivity, specificity, PPV, NPV, and LR+/LR-
• Essential for visualizing where a model makes errors before computing summary statistics