University of Arkansas for Medical Sciences

BIOM 5173 Quantitative Epidemiology I

Spring 1997

 Calculation of Sensitivity, Specificity, and Predictive Value Using the 2X2 Table


Problem 7 inChapter 4 of the text requires the careful development of a 2X2 table. The 2X2 table is a tool that epidemiologists use frequently and serves as a basis for many of our calculations, so I am going to explain it in depth on this page.

This is what a typical 2X2 table looks like:

The way we develop a table like this in a real-world situation is by giving each person two tests: a diagnostic test that is almost always accurate and a screening test. We want to find screening tests that are cheap and easy to administer because most diagnostic are not either of these. So we get a large group of people, conduct the screening tests and put them into one of two categories: test positive or test negative. In terms of the letters above, test positive people would be a+b, and test negatives would be c+d.

Then the second step would be to give them all the diagnostic test and determine whether they really have the disease in question. Those who tested positive *and* were truly diseased would be put into the "a" cell and the others would be put in the "b" cell. From this simple table, then, we can calculate sensitivity, specificity, and predictive value.

Sensitivity is a/(a+c).

Specificity is d/(b+d).

Positive Predictive value (PPV) is a/(a+b).

Negative predictive value is d/(c+d).

Question 7 in your book gave you the prevalence, the sensitivity, and the specificity, and then asked you to calculate the PPV. You weren't given the numbers to put in cells a-d so how could you work this problem?

First you can enlarge on the 2X2 table in way that is very common: working on the margins, also known as the marginals. Basically this just means adding the cells by both columns and rows and putting the results outside the table, like so:


Since you are given that the prevalence is 12/1,000, you know that there are actually 12 sick people as determined by diagnostic testing, so your marginal (a+b) is =12, if you assume that you have a population of 1,000, which you should do since this is the most convenient way to work. Thus, if Total=1,000, then marginal (b+d)= 1,000-12=988. So now your table looks like this:

You were also told that the test is 70% sensitive, which means that it correctly finds 70% of the cases. If there are 12 cases and your screening test only finds 70% of them positive, then you have to calculate 70% of 12=8.4. Then you have 3.6 persons who were incorrectly labeled as negatives, and your table now looks like this:

Of course, there's no such thing as 8.4 people but these numbers are calculated from a prevalence rate which was probably rounded off anyway.

Now we need to calculate cell b. This involves the specificity value which was given as 75%. Which number do we multiply by 75%? If you said, "b+d," you're right. Basically we want to know of those people who are truly not sick, how many people will be labeled "negative" by the screening test? A specificity of 75% tells us that this test will correctly identify 75% of the people who do not have the disease. So 988*75%=741, and subtracting this from 988 yields b=247. Now our table looks like this:

The positive predictive value is the proportion of people who tested positive on the screening test who are actually sick. So of all the people testing positive, 8.4+247=255.4, what proportion were truly sick? This would be 8.4/255.4=3.3%.

In conclusion, this test is not good at all for finding people who have the disease. If you test a patient with this screening test, and 96.7% of those who test positive are not going to be sick, what would you tell the patient? "Well, your test came out positive, but I really doubt the results because it's wrong about 97% of the time."?

The patient would wonder why you even bothered with the test. However, you may have noticed the large number in cell "d" and if you calculate the negative predictive value, you'll find a 99.5% NPV, which means if the patient shows up negative, you can be almost completely sure that he/she is truly free of disease. In other words, a negative finding is a significant outcome of this test, but a positive outcome is very dubious.

Please note in Chap. 4 when 2X2 tables are used, and in later chapters. Also be sure to notice the categories on the top and the left. Some authors use them differently, i.e. some tables have disease/no disease on the left and the test results on the top. Just beware.


The End