Sensitivity, specificity and predictive value

Formula

Pretest probability

Screening: Screening for breast cancer, where disease is rare (let’s say 10 in 10,000)

Surveillance: Surveillance in women diagnosed with breast cancer in the past two years, where recurrent or contralateral disease is much more common (let’s say 1 in 10 or 10%).

Sensitivity = TP / (TP + FN) = a / (a+c)

Specificity = TN / (FP + TN) = d / (b+d)

Positive predictive value (PV+): PV+ = TP / (TP + FP) = a / (a + b) = Post-test probability of a positive test having disease

Negative predictive value (PV-): PV- = TN / (TN + FN) = d / (c + d) = Post-test probability of a negative test not having disease

Examples

Low pretest probability

Pre-Test Probability: 10/10,000 = 0.1%

1. Sensitivity: 9/10 = 90%

2. Specificity: 9490/9990 = 95%

3. Positive predict value: 9/509 = 1.8%

4. Negative predict value: 9490/9491 = 99.99%

High pretest probability

Pre-Test Probability: 1000/10,000 = 10%

1. Sensitivity: 900/1000 = 90%

2. Specificity: 8550/9000 = 95%

3. Positive predict value: 900/1350 = 67% (up from 1.8%)

4. Negative predict value: 8550/8650 = 98.8% (down from 99.99%)

Likelihood ratios

We’d like the measure to be a feature of the test so it is stable across different prevalences/pretest probabilities

Likelihood ratio (LR): the ratio of the Probability that an individual with disease has the test result to the probability that an individual without disease has the test result.

1. Positive likelihood ratio (LR+): LR+ = 4, when 80% with disease have a positive test divided by 20% without disease have a positive test. LR+ = Sensitivity / (100 – specificity). “How well does a positive test help us rule in disease?”.

2. Negative likelihood ratio (LR-): LR+ = 0.1, when 9% with disease have a negative test divided by 90% without disease have a negative test. LR- = (100 - sensitivity) / specificity. “How well does a negative test help us rule-out disease?”.

Sensitivity and specificity are characteristics of the test and don’t change with pretest probability of prevalence. Because LR is calculated using sensitivity and specificity, it also doesn’t change when prevalence changes

Key

Using multiple categories of normal or abnormal can provide richer, more useful information from a test

Likelihood ratios can be used to interpret these “polytomous tests” and correspond to how we think about ruling in and ruling out disease