# How to make sense of both high FNR and NPV?

So, even though this is one of the most basic and already explained things about statistics I always seem to find way not to get it.

Having this table which shows high Negative Predictive Value (352/369 = 95%), which I understand as once a person having a negative test result he will most probably be free of disease, how can I make sense of a relatively high False Negative Rate (17/28 = 60 %) which i understand as an ill person testing negative for the test?.

Also, having low sensitivity but high NPV makes for a good screening test?

Thank you in advance!

• The estimate of FNR is based on a small sample (N=28) & thus may be unreliable. On the other hand NPV should be pretty accurate since it is based on a relatively large sample size (369). Since there are 2 types of errors for your screening test you may be able to trade off an increase in one error for a decrease in the other. This depends on how serious the disease is. For a disease like cancer FNR is more critical to keep low. Nov 18 '19 at 21:13
• So I should take it as a problem with the sample more than a problem on the test itself? In this case the disease is cancer so even if it had a higher sample for people with the disease but still a relatively high FNR (also maintaining a high NPV), the test shouldn't be recommended as a screening method right? Thank you very much! It helped me a lo to understand more whats going on! Nov 18 '19 at 21:26
• I think you are getting the idea. You won't find a perfect test but you should aim for one that has both error rates very low. Sometimes the test is based on using a threshold for measurements. That is a situation where changing the threshold can be used to trade off the 2 errors. Nov 18 '19 at 22:24

The high FNR means that testing negative does not provide much evidence that you are disease-free.

The NPV is relatively high because the disease is rare. The NPV of tossing a coin would be the population frequencym, 379/407=93% and this test, at 95%, doesn't do much better.

Whether this is a good screening test depends on a lot of additional factors about the test and disease, but the low specificity (high FNR) argues against it.

Your table set-up uses a template that is well-documented. TRUTH IS AT THE TOP.

https://en.wikipedia.org/wiki/Template:SensSpecPPVNPV

The calculation of the negative predictive value (NPV) (0.95 or 95%) and the false negative rate (FNR) (0.60 or 60%) are correct.

The table provided in the question permits an assessment of how “good” this (hypothetical) test might be considering not just the specificity of the test but also the sensitivity and the prevalence of disease in this (hypothetical) population. NOTE: Assume that there is no sample size issue.

The sensitivity of the test that is examined—the number of positive tests among the people who were screened divided by the number of people who truly have the disease—is low.

Sensitivity=TP/(TP + FN)=True Positives/(True Positive + False Negative)=11/28=0.40 or 40%


This means that the test identifies only 40% of the people truly have the disease. The remaining 60% of people who truly have the disease have a negative test—the test is a false negative.

The positive predictive value—the number of people with a positive test who truly have the disease--is also low in the population with a prevalence of disease that is 0.07 (7%).

 PPV=TP/(TP + FP)=True Positive/(True Positive + False Positive)=11/38=0.29 or 29%


This means the only 29% of the people with a positive test truly have the disease. The remaining 71% of people who have a positive test do not truly have the disease—the test is a false positive.

The classic description of the principles of screening for diseases is by Wilson and Jungner.

Wilson JMG, Jungner G. Principles and practices of screening for disease. Geneva, Switzerland: World Health Organization; 1968. Report No.: Public Health Papers No. 34. Available from: http://whqlibdoc.who.int/php/WHO_PHP_34.pdf.

In it, the authors state:

Ideally ……a test should be highly sensitive and should miss very few persons with the disease, though a relatively high proportion of false positives can be accepted…page 22

Writing in 2011, several current and former members of the United States Preventive Services Task Force state this another way:

Test must be easy and quick, may be less sensitive and specific than a diagnostic test. In a screening test, one may accept a higher false-positive rate, but a high false-negative rate would not be acceptable. Table 1.

Harris R, Sawaya GF, Moyer VA, Calonge N. Reconsidering the criteria for evaluating proposed screening programs: reflections from 4 current and former members of the US Preventive Services Task Force. Epidemiologic Reviews. 2011;33:20-35. https://doi.org/10.1093/epirev/mxr005

Your instinct is correct. This is NOT a good screening test in a population with this disease prevalence. It has both a high false negative rate (60%) and a high false positive rate (71%).