Diagnostic Test Statistics: Sensitivity, Specificity, PPV/NPV, Likelihood Ratios & ROC

• Sensitivity = TP / (TP + FN) - of all who HAVE the disease, the fraction the test catches (down the "disease present" column).
• Specificity = TN / (TN + FP) - of all who do NOT have the disease, the fraction the test clears (down the "disease absent" column).
• Positive predictive value (PPV) = TP / (TP + FP) - of all who test POSITIVE, the fraction truly diseased (across the "test positive" row).
• Negative predictive value (NPV) = TN / (TN + FN) - of all who test NEGATIVE, the fraction truly well (across the "test negative" row).

Sensitivity and specificity are properties of the test; predictive values are properties of the test applied to a particular population. As prevalence (pre-test probability) falls, the pool of truly diseased people shrinks relative to the well, so even a small false-positive rate generates many false positives - and PPV falls (while NPV rises). The identical test therefore has a high PPV in a high-prevalence (specialist clinic) setting and a low PPV in a low-prevalence (population screening) setting. This is the single most important caveat when applying published test data to your own patients.

Likelihood ratios (LRs) combine sensitivity and specificity into a single number that tells you how much a given result changes the odds of disease - and, unlike predictive values, they are independent of prevalence:

• LR+ = sensitivity / (1 - specificity) - how much MORE likely a positive result is in disease than in health.
• LR- = (1 - sensitivity) / specificity - how much LESS likely a negative result makes disease.

You apply them as: pre-test odds x LR = post-test odds (convert probability to odds, multiply, convert back; a Fagan nomogram does this graphically).

Using the Lachman data (sensitivity 85%, specificity 94%): LR+ = 0.85 / (1 - 0.94) = 0.85 / 0.06 = ~14 (a positive Lachman strongly raises the probability of ACL rupture), and LR- = (1 - 0.85) / 0.94 = 0.15 / 0.94 = ~0.16 (a negative Lachman meaningfully lowers it). The high LR+ explains why a positive Lachman is so persuasive.

For a test measured on a continuous (or ordinal) scale, the ROC curve plots sensitivity (true positive rate) on the y-axis against 1 - specificity (false positive rate) on the x-axis as the threshold is varied across its full range. A test with no discriminating ability follows the diagonal (the line of chance); a good test bows toward the top-left corner. The area under the curve (AUC) summarises overall discrimination in a single number:

The ROC curve makes the threshold-independent performance of a test visible and lets you compare two tests (the one with the larger AUC discriminates better overall) and choose an operating point that balances the costs of false negatives and false positives for your clinical question.