Sensitivity, Specificity, Positive Predictive Value, Likelihood Ratio explained with an example

Sensitivity-SnOut: Sensitive test to rule-out a disease. Meaning it will try to not miss any case. Formula for Sensitivity is

Specificity-SpIn: Specific test to rule-in a disease. Meaning it will try not to misdiagnose a normal case as diseased. Formula for specificity is

Positive Predictive Value (PPV): Positive predictive value is also known as precision rate is the proportion of subjects with positive test results who are correctly diagnosed. It is a measure of the performance of a diagnostic method, as it reflects the probability that a positive test reflects the underlying condition being tested for. Its value does however depend on the prevalence of the outcome of interest. Hence the formula for PPV is as follows:

PPV can be calculated if the specificity, sensitivity and prevalence of the disease under question is known by the following formula.

Negative predictive value is the opposite of PPV hence the formula for NPV will be

Based on specificity, sensitivity and prevalence the formula is

Likelihood Ratio: The likelihood that a given test result would be expected in a patient with a disease compared to the likelihood that the same result would be expected in a patient without that disease.

Likelihood Ratio Positive (LR+): The odds that a positive test result would be found in a patient with, versus without, a disease. Formula for LR+ is as follows:
Likelihood Ratio Positive (LR+) = Sensitivity / (1 - Specificity).

Likelihood Ratio Negative (LR-): The odds that a negative test result would be found in a patient without, versus with, a disease.
Likelihood Ratio Negative (LR-) = (1- Sensitivity) / Specificity.

Now let us solve a question using all the above formulae


Imagine that a group of 203 patients had a Chest Xray to look for cancer. Of these 203 patients 20 patients had abnormal chest xray (positive test). 183 had a negative Chest XRay(negative test). 2 of the 20 with positive chest Xray actually had cancer while one of the normal chest xray patient had cancer.

True positives are those patients who had abnormal xray and had cancer so TP=2

False positives are those patients who had abnormal xray but did not have cancer so FP=18 (20 abnormal Xrays but only 2 cancers)

True negatives are those patients who had normal xray and did not have cancer so TN=182 (183 normal Xrays but 1 had cancer)

False negatives are those patients who had normal xray but had cancer so FN=1

Hence based on these numbers the sensitivity is:
TP/(TP+FN)
that is 2(2+1)=0.667 or 66.7%

Specificity is:
TN/(TN+FP)
that is 182/(182+18)=0.91 or 91%

PPV is:
TP/(TP+FP)
that is 2/(2+18)=0.1 or 10%

NPV is:
TN / (FN + TN)
that is 182/(1+182)= 0.995 or 99.5%