Negative predictive value

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

Worked example

A worked example

A diagnostic test with sensitivity 67% and specificity 91% is applied to 2030 people to look for a disorder with a population prevalence of 1.48%

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

Related calculations

• False positive rate (α) = type I error = 1 − specificity = FP / (FP + TN) = 180 / (180 + 1820) = 9%
• False negative rate (β) = type II error = 1 − sensitivity = FN / (TP + FN) = 10 / (20 + 10) ≈ 33%
• Power = sensitivity = 1 − β
• Positive likelihood ratio = sensitivity / (1 − specificity) ≈ 0.67 / (1 − 0.91) ≈ 7.4
• Negative likelihood ratio = (1 − sensitivity) / specificity ≈ (1 − 0.67) / 0.91 ≈ 0.37
• Prevalence threshold = $PT={\frac {{\sqrt {TPR(-TNR+1)}}+TNR-1}{(TPR+TNR-1)}}$ ≈ 0.2686 ≈ 26.9%

This hypothetical screening test (fecal occult blood test) correctly identified two-thirds (66.7%) of patients with colorectal cancer. Unfortunately, factoring in prevalence rates reveals that this hypothetical test has a high false positive rate, and it does not reliably identify colorectal cancer in the overall population of asymptomatic people (PPV = 10%).

On the other hand, this hypothetical test demonstrates very accurate detection of cancer-free individuals (NPV ≈ 99.5%). Therefore, when used for routine colorectal cancer screening with asymptomatic adults, a negative result supplies important data for the patient and doctor, such as reassuring patients worried about developing colorectal cancer.

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,

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

See also

• Positive predictive value
• Binary classification
• OpenEpi software program

References

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