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# Negative predictive value

The negative predictive value is the proportion of patients with negative test results who are correctly diagnosed.

## Worked example

Relationships among terms
 Condition(as determined by "Gold standard") True False Testoutcome Positive True Positive False Positive(Type I error, P-value) → Positive predictive value Negative False Negative(Type II error) True Negative → Negative predictive value ↓Sensitivity ↓Specificity
A worked example
the Fecal occult blood (FOB) screen test is used in 203 people to look for bowel cancer:
 Patients with bowel cancer(as confirmed on endoscopy) True False ? FOBtest Positive TP = 2 FP = 18 = TP / (TP + FP)= 2 / (2 + 18)= 2 / 20 ≡ 10% Negative FN = 1 TN = 182 = TN / (TN + FN)182 / (1 + 182)= 182 / 183 ≡ 99.5% ↓= TP / (TP + FN)= 2 / (2 + 1)= 2 / 3 ≡ 66.67% ↓= TN / (FP + TN)= 182 / (18 + 182)= 182 / 200 ≡ 91%

Related calculations

• False positive rate (α) = FP / (FP + TN) = 18 / (18 + 182) = 9% = 1 - specificity
• False negative rate (β) = FN / (TP + FN) = 1 / (2 + 1) = 33% = 1 - sensitivity
• Power = 1 − β

Hence with large numbers of false positives and few false negatives, a positive FOB screen test is in itself poor at confirming cancer (PPV=10%) and further investigations must be undertaken, it will though pickup 66.7% of all cancers (the sensitivity). However as a screening test, a negative result is very good at reassuring that a patient does not have cancer (NPV=99.5%) and at this initial screen correctly identifies 91% of those who do not have cancer (the specificity).

## Definition

The Negative Predictive Value can be defined as:

$NPV = \frac{\rm number\ of\ True\ Negatives}{{\rm number\ of\ True\ Negatives}+{\rm number\ of\ False\ Negatives}}$

or, alternatively,

$NPV = \frac{({\rm specficity}) ({\rm 1 - prevalence})}{({\rm specificity}) ({\rm 1 - prevalence}) + (1 - {\rm sensitivity}) ({\rm prevalence})}$