Biostatistics II: Sensitivity, Specificity, Correlation, and p-Value Interpretation

Core Review

The 2×2 Contingency Table

All diagnostic test evaluation begins with the 2×2 table, which cross-tabulates test results against the gold standard (truth):

Gold standard (reference standard): The definitive method for determining whether disease is truly present (e.g., histopathology for oral cancer, full-mouth periodontal charting with radiographic bone loss for periodontitis, clinical/radiographic exam for caries).

Sensitivity

Definition: The proportion of people WITH the disease who test positive. Sensitivity answers the question: "If the patient HAS the disease, how likely is the test to catch it?"

Formula: Sensitivity = TP / (TP + FN)

• A test with high sensitivity has FEW false negatives. It "catches" the disease reliably.
• SnNout: A highly Sensitive test, when Negative, rules out the disease. If sensitivity is very high (say, 99%), a negative result means it's very unlikely the patient has the disease — because the test almost never misses a true case.

Clinical example: A caries detection device with sensitivity of 95% means that 95% of truly carious surfaces will be identified. Only 5% of carious surfaces are missed (false negatives).

Specificity

Definition: The proportion of people WITHOUT the disease who test negative. Specificity answers: "If the patient does NOT have the disease, how likely is the test to correctly say so?"

Formula: Specificity = TN / (TN + FP)

• A test with high specificity has FEW false positives. It doesn't "cry wolf."
• SpPin: A highly Specific test, when Positive, rules in the disease. If specificity is very high (say, 98%), a positive result strongly suggests the patient truly has the disease — because the test almost never falsely flags healthy people.

Clinical example: A diagnostic test with specificity of 90% means that 90% of healthy (disease-free) surfaces test negative. 10% of healthy surfaces produce false alarms (false positives).

Worked 2×2 Example

A new oral cancer screening device is evaluated against biopsy (gold standard) in 1,000 patients:

Calculations:

• Sensitivity = 80 / (80 + 20) = 80/100 = 0.80 = 80% (the test catches 80% of cancers)
• Specificity = 840 / (840 + 60) = 840/900 = 0.933 = 93.3% (93.3% of non-cancer patients correctly test negative)
• Positive Predictive Value (PPV) = TP / (TP + FP) = 80 / 140 = 0.571 = 57.1% (If the test is positive, 57.1% probability the patient truly has cancer)
• Negative Predictive Value (NPV) = TN / (TN + FN) = 840 / 860 = 0.977 = 97.7% (If the test is negative, 97.7% probability the patient truly does NOT have cancer)

Critical distinction: Sensitivity and specificity are INHERENT PROPERTIES of the test and do not change with disease prevalence. PPV and NPV DO change with prevalence — as prevalence increases, PPV increases and NPV decreases. The same test performs differently in a high-risk specialty clinic (high prevalence → high PPV) versus a low-risk community screening (low prevalence → low PPV, more false positives).

SnNout and SpPin

This is tested by asking: "A test with 99% sensitivity is negative. What can you conclude?" Answer: Disease is very unlikely to be present (SnNout).

Correlation

Pearson correlation coefficient (r): Measures the strength and direction of a LINEAR relationship between two continuous variables.

• Range: −1.0 to +1.0
• r = +1.0: Perfect positive linear correlation (as X increases, Y increases proportionally)
• r = −1.0: Perfect negative linear correlation (as X increases, Y decreases proportionally)
• r = 0: No linear correlation
• r > 0: Positive correlation (both variables move in the same direction)
• r < 0: Negative correlation (variables move in opposite directions)

Interpretation guidelines (approximate):

• |r| < 0.3: Weak correlation
• 0.3 ≤ |r| < 0.7: Moderate correlation
• |r| ≥ 0.7: Strong correlation

Spearman rank correlation: Used when data are ordinal or when the relationship is monotonic but not necessarily linear. Based on ranks rather than raw values.

Correlation ≠ Causation

This is arguably the most important single concept in biostatistics — and one of the most frequently tested on the NBDHE.

A statistically significant correlation between two variables does NOT mean that one caused the other. Alternative explanations for a correlation include:

1. Confounding: A third variable causes both. Toothbrushing frequency may correlate with lower BMI, but the real confounder is likely health-consciousness/socioeconomic status.
2. Reverse causation: The direction of causality might be opposite to what is assumed. Does poor oral health cause diabetes, or does diabetes cause poor oral health? (Evidence supports bidirectionality.)
3. Coincidence: In large datasets with many variables tested, some correlations will reach statistical significance purely by chance (multiple comparison problem).

To establish causation, you need: temporality (cause precedes effect), biological plausibility, dose-response relationship, consistency across studies, and ideally experimental evidence (RCT).

The p-Value

The p-value is the most widely misunderstood statistical concept. Here is what the p-value IS and IS NOT:

What the p-value IS: The probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true.

In simpler terms: "If there were truly no effect (null hypothesis), how likely would it be to see data like these (or more extreme) purely due to random chance?"

What the p-value IS NOT (common NBDHE traps):

• ❌ The probability that the null hypothesis is true
• ❌ The probability that the results are due to chance
• ❌ The probability that the alternative hypothesis is false
• ❌ 1 − (probability that the alternative hypothesis is true)
• ❌ A measure of the size or importance of an effect

Conventional thresholds:

• p < 0.05: "Statistically significant" (results unlikely under the null; by convention, reject the null)
• p ≥ 0.05: "Not statistically significant" (fail to reject the null — note: you do NOT "accept" the null)

The p = 0.05 trap: If you run 20 independent tests where the null hypothesis is true, on average, one will produce p < 0.05 purely by chance. This is why exploratory analyses (fishing expeditions) without correction for multiple comparisons are problematic.

Statistical Significance ≠ Clinical Significance

A common board question presents a study with p < 0.001 (highly statistically significant) but an effect size that is trivially small. The correct interpretation: the result is statistically significant but may not be CLINICALLY significant.

• Statistical significance: Addresses whether the observed effect is likely to be real (not due to chance), given the sample size. With large enough sample sizes, even trivially small effects become statistically significant.

• Clinical significance: Addresses whether the effect is large enough to matter in practice. A new toothpaste that reduces plaque index by 0.05 on a 0–3 scale might be statistically significant with a sample of 10,000 subjects (p < 0.001), but is this reduction clinically meaningful? Probably not.

Always consider effect size (Cohen's d, absolute risk reduction, number needed to treat) alongside p-values.

Clinical/Board Application

Board-style question: "A new salivary diagnostic test for periodontitis has 95% sensitivity and 90% specificity. A patient tests positive. What does the positive result indicate, and which mnemonic applies?"

Answer: SpPin — a highly Specific test, when Positive, rules IN the disease. With 90% specificity, false positives are relatively uncommon, so a positive result is reasonably strong evidence for periodontitis (though PPV also depends on the pretest probability/prevalence).

Common Traps

• Trap: "The p-value is 0.03, so there is a 3% chance the null hypothesis is true." WRONG. The p-value is P(data | null), not P(null | data). You cannot flip the conditional probability.
• Trap: "The p-value is 0.08, so the null hypothesis is true / the intervention has no effect." WRONG. You fail to reject the null; you do not accept it. The study may simply have been underpowered.
• Trap: "A study found r = 0.45 between flossing frequency and lower CRP levels, therefore flossing reduces systemic inflammation." WRONG. This is correlation, not causation. A confounder (e.g., overall health-conscious behavior) could explain both.
• Trap: Confusing sensitivity and PPV. Sensitivity = "if diseased, test positive." PPV = "if test positive, is diseased." They answer opposite conditional questions.