Negative predictive value

In statistics, 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 uncommon that the result should have been positive. In the familiar context of medical testing, a high NPV means that the test only rarely mistakenly classifies 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.

		True condition
	Total population	Condition positive	Condition negative	Prevalence = ⁠Σ Condition positive/Σ Total population⁠		Accuracy (ACC) = ⁠Σ True positive + Σ True negative/Σ Total population⁠
Predicted condition	Predicted condition positive	True positive	False positive, Type I error	Positive predictive value (PPV), Precision = ⁠Σ True positive/Σ Predicted condition positive⁠		False discovery rate (FDR) = ⁠Σ False positive/Σ Predicted condition positive⁠
	Predicted condition negative	False negative, Type II error	True negative	False omission rate (FOR) = ⁠Σ False negative/Σ Predicted condition negative⁠		Negative predictive value (NPV) = ⁠Σ True negative/Σ Predicted condition negative⁠
		True positive rate (TPR), Recall, Sensitivity (SEN), probability of detection, Power = ⁠Σ True positive/Σ Condition positive⁠	False positive rate (FPR), Fall-out, probability of false alarm = ⁠Σ False positive/Σ Condition negative⁠	Positive likelihood ratio (LR+) = ⁠TPR/FPR⁠	Diagnostic odds ratio (DOR) = ⁠LR+/LR−⁠	Matthews correlation coefficient (MCC) = √TPR·TNR·PPV·NPV − √FNR·FPR·FOR·FDR	F1 score = ⁠2 · PPV · TPR/PPV + TPR⁠ = ⁠2 · Precision · Recall/Precision + Recall⁠
		False negative rate (FNR), Miss rate = ⁠Σ False negative/Σ Condition positive⁠	Specificity (SPC), Selectivity, True negative rate (TNR) = ⁠Σ True negative/Σ Condition negative⁠	Negative likelihood ratio (LR−) = ⁠FNR/TNR⁠

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

The fecal occult blood (FOB) screen test was used in 203 people to look for bowel cancer:

		Patients with bowel cancer (as confirmed on endoscopy)
	Total population (pop.) = 2030	Condition positive	Condition negative	Prevalence = (TP + FN) / pop. = (20 + 10) / 2030 ≈ 1.48%	Accuracy (ACC) = (TP + TN) / pop. = (20 + 1820) / 2030 ≈ 90.64%
Fecal occult blood screen test outcome	Test outcome positive	True positive (TP) = 20 (2030 × 1.48% × 67%)	False positive (FP) = 180 (2030 × (100% − 1.48%) × (100% − 91%))	Positive predictive value (PPV), precision = TP / (TP + FP) = 20 / (20 + 180) = 10%	False discovery rate (FDR) = FP / (TP + FP) = 180 / (20 + 180) = 90.0%
	Test outcome negative	False negative (FN) = 10 (2030 × 1.48% × (100% − 67%))	True negative (TN) = 1820 (2030 × (100% − 1.48%) × 91%)	False omission rate (FOR) = FN / (FN + TN) = 10 / (10 + 1820) ≈ 0.55%	Negative predictive value (NPV) = TN / (FN + TN) = 1820 / (10 + 1820) ≈ 99.45%
		True positive rate (TPR), recall, sensitivity = TP / (TP + FN) = 20 / (20 + 10) ≈ 66.7%	False positive rate (FPR), fall-out, probability of false alarm = FP / (FP + TN) = 180 / (180 + 1820) = 9.0%	Positive likelihood ratio (LR+) = ⁠TPR/FPR⁠ = (20 / 30) / (180 / 2000) ≈ 7.41	Diagnostic odds ratio (DOR) = ⁠LR+/LR−⁠ ≈ 20.2	F1 score = 2 × ⁠precision × recall/precision + recall⁠ ≈ 0.174
		False negative rate (FNR), miss rate = FN / (TP + FN) = 10 / (20 + 10) ≈ 33.3%	Specificity, selectivity, true negative rate (TNR) = TN / (FP + TN) = 1820 / (180 + 1820) = 91%	Negative likelihood ratio (LR−) = ⁠FNR/TNR⁠ = (10 / 30) / (1820 / 2000) ≈ 0.366

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, however, pick up 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).

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.

See also

• Positive predictive value
• Sensitivity and specificity
• Binary classification

References

• Altman DG, Bland JM (1994). "Diagnostic tests 2: Predictive values". BMJ. 309 (6947): 102. PMC 2540558. PMID 8038641. {{cite journal}}: Unknown parameter |month= ignored (help)