14.2 Diagnostic Testing: Sensitivity, Specificity, PPV & NPV

14.2 Diagnostic Testing: Sensitivity, Specificity, PPV & NPV

The 2x2 Contingency Matrix & Primary Test Metrics

Evaluating diagnostic tests requires placing clinical test results alongside a gold-standard reference method in a standard 2x2 contingency matrix. The true disease status (Disease Present vs. Disease Absent) is positioned along the columns, and test results (Test Positive vs. Test Negative) are positioned along the rows.

Sensitivity (True Positive Rate)

Sensitivity measures the probability that a diagnostic test yields a positive result in a patient who truly has the disease: $\text{Sensitivity} = \frac{\text{TP}}{\text{TP} + \text{FN}}$

• Clinical Mnemonic: SnNOut — A test with high Snsitivity, when Negative, rules Out the disease. High-sensitivity tests produce very few false negatives, making them ideal for initial screening of serious conditions (e.g., HIV screening with ELISA, D-dimer for pulmonary embolism).

Specificity (True Negative Rate)

Specificity measures the probability that a diagnostic test yields a negative result in a patient who is free of the disease: $\text{Specificity} = \frac{\text{TN}}{\text{TN} + \text{FP}}$

• Clinical Mnemonic: SpPIn — A test with high Specificity, when Positive, rules In the disease. High-specificity tests produce very few false positives, making them ideal for confirmatory testing after a positive screening test (e.g., Western blot/differentiation assay for HIV, lumbar puncture after non-contrast CT for SAH).

Complementary Rates

• False Negative Rate (FNR) $= 1 - \text{Sensitivity} = \frac{\text{FN}}{\text{TP} + \text{FN}}$
• False Positive Rate (FPR) $= 1 - \text{Specificity} = \frac{\text{FP}}{\text{TN} + \text{FP}}$

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Predictive Values & Disease Prevalence

While sensitivity and specificity are intrinsic properties of a diagnostic test (remaining fixed across populations), predictive values measure clinical usefulness in real-world practice and depend heavily on disease prevalence in the population tested.

Positive Predictive Value (PPV)

PPV is the proportion of patients with a positive test result who truly have the disease: $\text{PPV} = \frac{\text{TP}}{\text{TP} + \text{FP}}$

Negative Predictive Value (NPV)

NPV is the proportion of patients with a negative test result who are truly free of the disease: $\text{NPV} = \frac{\text{TN}}{\text{TN} + \text{FN}}$

Impact of Disease Prevalence

• Increasing Disease Prevalence: As prevalence increases, the absolute number of True Positives increases while False Positives decrease relative to true cases. Consequently, PPV increases while NPV decreases.
• Decreasing Disease Prevalence: In a low-prevalence screening population (e.g., screening asymptomatic young adults for rare disorders), PPV drops dramatically (most positive results are false positives), while NPV approaches 100%.

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Diagnostic Cutoff Adjustments & ROC Curves

When diagnostic tests rely on continuous laboratory variables (e.g., blood glucose, serum troponin, prostate-specific antigen), clinicians must choose a numeric cutoff value to classify results as positive or negative. Shifting this threshold alters the distribution of true positives, false positives, true negatives, and false negatives.

       Disease Absent (Healthy)          Disease Present (Sick)
         /------------------\            /------------------\
        /                    \          /                    \
       /                      \        /                      \
      +------------------------+------+------------------------+
                               ^      ^
                           Cutoff B  Cutoff A

Effects of Shifting Cutoffs

• Lowering the Cutoff Value (Moving Cutoff Left toward healthy distribution):

  • Captures more true cases $\rightarrow$ Increases Sensitivity and True Positives.
  • Decreases missed cases $\rightarrow$ Decreases False Negatives.
  • Mistakenly classifies more healthy people $\rightarrow$ Decreases Specificity and True Negatives.
  • Increases incorrect positives $\rightarrow$ Increases False Positives.
  • Clinical Indication: Preferred when missing a diagnosis is life-threatening (e.g., cardiac troponin for acute coronary syndrome).

• Raising the Cutoff Value (Moving Cutoff Right toward diseased distribution):

  • Requires higher threshold $\rightarrow$ Increases Specificity and True Negatives.
  • Decreases false alarms $\rightarrow$ Decreases False Positives.
  • Misses mild cases $\rightarrow$ Decreases Sensitivity and True Positives.
  • Increases missed cases $\rightarrow$ Decreases False Negatives.
  • Clinical Indication: Preferred when false positive diagnoses cause severe harm, unnecessary invasive surgery, or severe psychological distress.

Receiver Operating Characteristic (ROC) Curves

An ROC curve plots Sensitivity ($y$-axis) against $1 - \text{Specificity}$ (False Positive Rate, $x$-axis) for every possible cutoff threshold of a continuous test.

• Area Under the Curve (AUC): Quantifies overall diagnostic accuracy across all cutoffs.
  • $\text{AUC} = 1.0$: Perfect diagnostic test (100% sensitivity and 100% specificity).
  • $\text{AUC} = 0.5$: Useless test (diagonal line of chance, equivalent to a coin flip).
• The optimal cutoff point on an ROC curve is typically the point closest to the upper-left corner $(0, 1)$, maximizing both sensitivity and specificity.

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Likelihood Ratios & Fagan Nomogram

Unlike PPV and NPV, Likelihood Ratios (LRs) combine sensitivity and specificity into single metrics that are independent of disease prevalence. They quantify how much a test result modifies the pre-test probability of disease into a post-test probability.

Formulas

• Positive Likelihood Ratio ($LR+$): $\frac{\text{Sensitivity}}{1 - \text{Specificity}} = \frac{\text{True Positive Rate}}{\text{False Positive Rate}}$
  • $LR+ > 10$: Generates a large, conclusive increase in disease probability (rules IN disease).
  • $LR+ = 5 - 10$: Moderate increase in post-test probability.
• Negative Likelihood Ratio ($LR-$): $\frac{1 - \text{Sensitivity}}{\text{Specificity}} = \frac{\text{False Negative Rate}}{\text{True Negative Rate}}$
  • $LR- < 0.1$: Generates a large, conclusive decrease in disease probability (rules OUT disease).
  • $LR- = 0.1 - 0.2$: Moderate decrease in post-test probability.

Application via Odds

$\text{Pre-test Odds} = \frac{\text{Pre-test Probability}}{1 - \text{Pre-test Probability}}$ $\text{Post-test Odds} = \text{Pre-test Odds} \times \text{Likelihood Ratio}$ $\text{Post-test Probability} = \frac{\text{Post-test Odds}}{1 + \text{Post-test Odds}}$

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Diagnostic & Screening Algorithm

When selecting and interpreting diagnostic studies on Step 2 CK:

1. What is the goal of testing?

   • Initial Screening / Exclude Disease $\rightarrow$ Choose test with High Sensitivity (low false negatives).
   • Confirm Suspected Diagnosis $\rightarrow$ Choose test with High Specificity (low false positives).

2. How does moving from a community setting to a specialist clinic affect metrics?

   • Disease prevalence increases $\rightarrow$ PPV increases, NPV decreases, Sensitivity & Specificity remain constant.

3. What happens if the diagnostic threshold cutoff is reduced?

   • Cutoff lowered $\rightarrow$ Sensitivity $\uparrow$, False Positives $\uparrow$, Specificity $\downarrow$, False Negatives $\downarrow$.