# Negative predictive value

Negative predictive value

In statistics and diagnostic testing, the negative predictive value (NPV) is a summary statistic used to describe the performance of a diagnostic testing procedure. It is defined as the proportion of subjects with a negative test result who are correctly diagnosed. A high NPV means that when the test yields a negative result, it is most likely correct in its assessment. In the familiar context of medical testing, a high NPV means that the test only rarely misclassifies a sick person as being healthy. Note that this says nothing about the tendency of the test to mistakenly classify a healthy person as being sick.

## Definition

The Negative Predictive Value is defined as:

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

where a "true negative" is the event that the test makes a negative prediction, and the subject has a negative result under the gold standard, and a "false negative" is the event that the test makes a negative prediction, and the subject has a positive result under the gold standard.

The following diagram illustrates how the positive predictive value, negative predictive value, sensitivity, and specificity are related.

 Condition (as determined by "Gold standard") Positive Negative Test outcome Positive True Positive False Positive (Type I error) → Positive predictive value = $\,\!\tfrac{\Sigma\text{ True Positive}}{\Sigma\text{ Test outcome Positive}}$ Negative False Negative (Type II error) True Negative → Negative predictive value = $\,\!\tfrac{\Sigma\text{ True Negative}}{\Sigma\text{ Test outcome Negative}}$ ↓ Sensitivity = $\,\!\tfrac{\Sigma\text{ True Positive}}{\Sigma\text{ Condition Positive}}$ ↓ Specificity = $\,\!\tfrac{\Sigma\text{ True Negative}}{\Sigma\text{ Condition Negative}}$

Note that the positive and negative predictive values can only be estimated using data from a cross-sectional study or other population-based study in which valid prevalence estimates may be obtained. In contrast, the sensitivity and specificity can be estimated from case-control studies.

If the prevalence, sensitivity, and specificity are known, the negative predictive value can be obtained from the following identity:

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

## Worked example

Suppose that a fecal occult blood (FOB) screen test is used in 2030 people to detect bowel cancer:

 Patients with bowel cancer (as confirmed on endoscopy) Positive Negative Fecal occult blood screen test outcome Positive True Positive (TP) = 20 False Positive (FP) = 180 → Positive predictive value = TP / (TP + FP) = 20 / (20 + 180) = 20 / 200 = 10% Negative False Negative (FN) = 10 True Negative (TN) = 1820 → Negative predictive value = TN / (FN + TN) = 1820 / (10 + 1820) = 1820 / 1830 ≈ 99.5% ↓ Sensitivity = TP / (TP + FN) = 20 / (20 + 10) = 20 / 30 ≈ 66.67% ↓ Specificity = TN / (FP + TN) = 1820 / (180 + 1820) = 1820 / 2000 = 91%

In this setting, with NPV = 99.5%, a negative test result may provide some reassurance that the subject is unlikely to have cancer. This high NPV value would be particularly notable if the cancer were relatively common. For example, if 5% of people in the population had bowel cancer, then a NPV of 99.5% would indicate that a person with a negative test result has much lower than the average population risk for bowel cancer. However if the prevalence of bowel cancer were 0.5%, a negative test result in this setting would be uninformative.

## Relation to negative post-test probability

Although sometimes used synonymously, a negative predictive value generally refers to what is established by control groups, while a negative post-test probability rather refers to a probability for an individual. Still, if the individual's pre-test probability of the target condition is the same as the prevalence in the control group used to establish the negative predictive value, then the two are numerically equal.

## References

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