ROC Curves and Diagnostic Accuracy

Sensitivity and Specificity

Sensitivity and specificity are the fundamental measures of a diagnostic test's performance at a given cutoff threshold.

• ●Sensitivity (True Positive Rate) = TP / (TP + FN): The proportion of true cases correctly identified. High sensitivity → few false negatives. A sensitive test is good for screening (you don't want to miss any cases).
• ●Specificity (True Negative Rate) = TN / (TN + FP): The proportion of true non-cases correctly identified. High specificity → few false positives. A specific test is good for confirmation (you need high confidence when positive).
• ●Tradeoff: Lowering the threshold increases sensitivity but decreases specificity, and vice versa.

The ROC Curve

The Receiver Operating Characteristic (ROC) curve plots sensitivity (y-axis) against 1 - specificity (false positive rate, x-axis) across all possible cutoff values. It visualizes the tradeoff between sensitivity and specificity across the full range of thresholds.

• ●A perfect test: The curve goes straight up to (0, 1) then across → AUC = 1.0
• ●A useless test: The curve follows the diagonal → AUC = 0.5 (no better than random)
• ●A good test: The curve bulges toward the top-left corner
• ●The optimal cutoff: Often chosen using the Youden index = Sensitivity + Specificity - 1 (maximizes correct classification)

Area Under the Curve (AUC / C-statistic)

AUC (Area Under the ROC Curve) is the single-number summary of a test's discriminative ability. It represents the probability that a randomly selected positive case will have a higher test value than a randomly selected negative case.

• ●AUC = 0.50: No discrimination (useless)
• ●AUC 0.50–0.70: Poor
• ●AUC 0.70–0.80: Acceptable
• ●AUC 0.80–0.90: Excellent
• ●AUC > 0.90: Outstanding

Compare two tests or models using DeLong's test for the difference between AUC values. A larger AUC is not always clinically better — consider the relative costs of false positives vs. false negatives in your clinical context.

Positive and Negative Predictive Values

Sensitivity and specificity are fixed properties of a test. Predictive values depend on disease prevalence in your target population.

• ●PPV (Positive Predictive Value): Given a positive test, what is the probability the patient truly has the disease? PPV decreases as prevalence decreases.
• ●NPV (Negative Predictive Value): Given a negative test, what is the probability the patient is truly disease-free?
• ●A test with excellent sensitivity/specificity can still have poor PPV in a low-prevalence population (high false-positive rate in screening programs).