# Sensitivity and specificity

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## Biostatistics and epidemiology

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A novel blood test is currently under study for the evaluation of deep vein thrombosis. After obtaining initial data, the sensitivity of the test is found to be 0.77 and the specificity is 0.89. Which of the following best describes the positive likelihood ratio of this test?

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#### Content Reviewers:

Rishi Desai, MD, MPH

#### Contributors:

Let’s say a new screening test is developed to figure out if people have diabetes before they start showing symptoms. Before using the test, we have to make sure that the test works - in other words, can the test correctly identify if a person has diabetes or not? This is the test’s validity, and it has two components - sensitivity and specificity.

A test with high sensitivity will correctly identify most people who have the condition, and a test with high specificity will correctly identify most people who don’t have the disease.

So let’s say that we recruit a 1000 people - 100 people who have diabetes and 900 people who don’t to put our diabetes test to the test!

We can organize the results using a 2 by 2 table, where the true disease status, positive or negative, of the individual is on the top of the box and the results of the screening test, positive or negative, are on the side, and each of the cells is labeled a, b, c, or d. In this situation, a positive test indicates that a person has diabetes.

So let’s look at this table closer, a person who gets a positive test result and has positive disease status, so has diabetes, is called a true positive.

A person who gets a negative test result and a negative disease status, so doesn’t have diabetes, would be a true negative.

A person who gets a positive test result even though they don’t have diabetes, would be a false positive.

And lastly a person who gets a negative test result even though they have diabetes, would be a false negative.

To calculate sensitivity, we divide the number of true positives by the total number of people who have diabetes - so cell a divided by the sum of cell a and cell c.

A test with perfect sensitivity would have 100 true positives in cell a, because the test would correctly identify everyone who has diabetes, and zero false negatives in cell c.

To calculate specificity, we divide the number of true negatives by the total number of people who do not have diabetes - so cell d divided by the sum of cell d and cell b.

A test with perfect specificity would have 900 true negatives in cell d, because the test would correctly identify everyone who doesn’t have diabetes, and zero false positives, in cell b.

But no test is 100% perfect, so let’s say that cell a contains 80 true positives, cell b contains 100 false positives, cell c contains 20 false negatives, and cell d contains 800 true negatives.

In this situation, the sensitivity would be 80%, because there are 100 people who truly have diabetes - cell a plus cell c, and 80 of them - in cell a - are true positives. In other words, this test will correctly identify 80% of people who have diabetes.

The specificity would be 89%, because 900 people don’t have diabetes, cell d plus cell b, and 800 of them - in cell d - are true negatives. In other words, this test will correctly identify 89% of people who don’t have diabetes.

Sometimes deciding if a person has a condition is a matter of yes or no - like having a cavity or not having a cavity - but other conditions are on the continuous scale, like blood glucose level. For these conditions, there has to be a cut- off point that makes the test result either positive or negative.

For example, let’s say there are 20 individuals with diabetes and 20 individuals without diabetes who get their blood sugar levels tested.

Even though the blood sugar levels of people with diabetes tends to be higher, there’s no clear cut-off point because there’s a lot of overlap in the blood glucose levels of people with and without diabetes.

Let’s say that we pick a high cut- off value. Then, bringing back our 2 by 2 table, we can see that we might end up with only 2 false positives, but 15 false negatives, so the test has high specificity and low sensitivity, and a large number of people that have diabetes won’t be diagnosed.

That can be a serious problem for conditions that need to be diagnosed early, either because they develop quickly or can only be cured in the early stages.

On the other hand, let’s say that we pick a low cut- off value. Then we might end up with only 3 false negatives, but 14 false positives, so the test has high sensitivity and low specificity, and a large number of people that don’t have diabetes will be misdiagnosed as having it!

That can lead to extra anxiety and a lot of extra expense related to unnecessary medical workups for something that they don’t even have.

So there’s always a trade- off between sensitivity and specificity, so choosing an appropriate cut- off point should depend on the severity of the disease being screened for, the effectiveness of available treatments, and whether the effectiveness is greater if the treatment is administered earlier rather than later.

Now, to calculate sensitivity and specificity, we would have to know the number of individuals who truly have the condition or don’t.

But this usually isn’t the case, since the whole purpose of testing is to figure out if a person has a certain condition.

So, a newly developed test is often compared to an existing gold standard test - the best known test that already exists.

For example, the gold standard test for diabetes is the oral glucose tolerance test, which involves measuring blood glucose levels before and after drinking a certain amount of glucose sugar.

But the oral glucose tolerance test takes multiple hours to complete and it can’t really be used to screen a lot of people at a time, so a new test called the fasting plasma glucose test was developed because it’s simpler and faster.

The fasting plasma glucose test only requires one blood sample and is usually taken in the morning, after a person has fasted overnight.

But before it was put into practice, the fasting plasma glucose test was compared to the oral glucose tolerance test, to make sure it had a relatively high sensitivity and specificity.

Oftentimes, an individual will be tested using two tests, either sequentially or simultaneously.

Summary

Sensitivity and specificity are two important statistical measures used to evaluate the performance of medical tests, such as diagnostic tests for diseases.

Sensitivity measures the ability of a test to correctly identify those who have the disease. It is the proportion of people with a disease who test positive. This means that a test with high sensitivity can correctly identify most individuals who have the disease, while a test with low sensitivity will miss many cases of the disease. Sensitivity is calculated by dividing the number of true positives, by the total number of all people who have the condition - the true positives and false negatives.

Specificity, on the other hand, measures the ability of a test to correctly identify those who do not have the disease. It is the proportion of people without a disease who correctly test negative. This means that a test with high specificity can correctly identify most individuals who do not have the disease, while a test with low specificity will result in many false positive results. Specificity is calculated by dividing the number of true negatives by the total number of all people who don't have the condition - the true negatives and false positives.