How Do You Calculate False Negative Rate

The false negative rate (FNR) is a key metric in statistical testing and diagnostic accuracy. It measures the proportion of actual positive cases that are incorrectly identified as negative. This guide explains how to calculate FNR, its importance, and practical applications.

What is False Negative Rate?

The false negative rate (FNR) represents the probability that a test result will be negative when the condition being tested for is actually present. In medical testing, this means a patient who has a disease tests negative. In quality control, it might mean a defective product passes inspection.

FNR is calculated by dividing the number of false negatives by the total number of actual positives. A lower FNR indicates a more accurate test, while a higher FNR suggests more false negatives are occurring.

Formula

False Negative Rate (FNR) = (False Negatives) / (True Positives + False Negatives)

Where:

False Negatives = Number of actual positives incorrectly classified as negative
True Positives = Number of actual positives correctly classified as positive

The result is typically expressed as a decimal or percentage. For example, an FNR of 0.10 means 10% of positive cases are incorrectly identified as negative.

How to Calculate

Identify the number of false negatives in your test results
Identify the number of true positives in your test results
Add the false negatives and true positives together
Divide the number of false negatives by this sum
Multiply by 100 to get a percentage if desired

For continuous data, you may need to establish thresholds to determine what counts as a false negative.

Example Calculation

Suppose a medical test is evaluated with the following results:

Actual Condition	Test Result	Count
Positive (Disease Present)	Positive	80
Positive (Disease Present)	Negative	20
Negative (Disease Absent)	Positive	15
Negative (Disease Absent)	Negative	85

To calculate FNR:

False Negatives = 20 (actual positives incorrectly classified as negative)
True Positives = 80
Total positives = 20 (false negatives) + 80 (true positives) = 100
FNR = 20 / 100 = 0.20 or 20%

This means 20% of people with the disease tested negative, which would be considered a high false negative rate for a medical test.

Interpreting Results

A high false negative rate indicates that the test is missing many actual positive cases. This is particularly concerning in medical testing where it could mean patients with a disease are not being properly identified. In quality control, it might indicate that defective products are slipping through inspection.

To improve FNR:

Improve test sensitivity
Use more accurate diagnostic methods
Implement stricter quality control measures
Consider additional testing for high-risk individuals

Note: False negative rate should be considered alongside false positive rate and overall accuracy when evaluating test performance.

FAQ

What is the difference between false negative rate and false positive rate?: The false negative rate measures missed positive cases, while the false positive rate measures incorrect positive identifications. Both are important for understanding test accuracy.
How does false negative rate affect medical testing?: A high false negative rate means patients with a disease might not be properly identified, potentially delaying treatment and worsening outcomes.
Can false negative rate be zero?: In theory, a perfect test would have a zero false negative rate, meaning no actual positives are missed. In practice, no test is perfect.
What is an acceptable false negative rate?: Acceptable rates vary by application. In medical testing, rates below 5% are generally considered good, while in quality control, rates below 1% might be expected.
How can I reduce false negative rate?: Improving test sensitivity, using more accurate methods, and implementing better quality control measures can help reduce false negative rates.