This article illustrates the use of test sensitivity, specificity, and disease prevalence in calculating the predictive value of a test\'s results.

Many, if not most, of the decisions made in modern medicine are based on a continuously increasing battery of clinical tests.  Every day, physicians interpret various test results, answer questions about these results, and give treatment and management recommendations based on the findings these results are assumed to imply.  But what do these results truly imply?

The example discussed here is meant to illustrate what a positive (or negative) test result actually means, or, more precisely, how the inherent statistical properties of a test—namely its sensitivity and specificity, as well as the prevalence of a disease, affect the significance of a test's results.  What we  find is that when it comes to clinical tests, everything is in terms of probabilities, and absolute certainties hardly ever exist. 

This discussion assumes a basic level of familiarity with statistical concepts such as 2x2 contingency table analysis, true positives/negatives, specificity, sensitivity, etc.

Example:

A 60 year old man in good health, with no lifestyle cardiac risk factors, who eats healthy and exercises regularly, comes in wishing to have a cardiac stress test to rule out coronary artery disease (CAD).  Although he is in good health, his father died of a heart attack at the age of 63 and he would like to rule out any potential problems.  You know that the prevalence of CAD in his population (based on age group and risk factors) is about 2%, and that the sensitivity of the exercise stress EKG test is 33% and its specificity is 99%.  What would the findings of such a test mean in this patient, and would you even recommend the test?

Step 1:

To start, we first set up our 2x2 table given the information known.  What we know is that the disease prevalence is 2%, the test’s sensitivity is 33%, and its specificity is 99%. 

First, we set up our 2x2 table assuming a nice round number group of subjects, say 1000 to make it easy to work with.

            Disease

            +          -

+          TP        FP

Test

-           FN       TN

            20        980      =1,000 total

Out of 1000 people in this patient’s population:

2% of 1000 = 20 have CAD

1000-20 = 980 don’t have CAD

The 2% prevalence is also known as the pre-test probability, or the probability that the person has the disease before the test is administered.

Step 2:

Next, knowing the sensitivities and specificity of the test, and the definition of these terms, we fill in the rest of the table:

Sensitivity = TP / (TP + FN) = 33%

Therefore: TP = 33% X 20 = 7, FN = 20-7 = 13

Specificity = TN / (TN + FP) = 99%

Therefore: TN = 99% X 980 = 970, FP = 980-970=10

            Disease

            +          -

+          7          10

Test

-           13        970

            20        980      = 1,000 total

Step 3:

Lastly, we calculate what’s known as the positive predictive value (PPV) and negative predictive value (NPV) of this test based on the table above:

PPV = TP / (TP+FP) = 7/(7+10) = 0.41

NPV = TN / (FN+TN) = 970/(7+970) = 0.99

 

Interpretation:

So what does this mean?  Simply stated, the PPV indicates the probability of disease given a POSITIVE test result, while the NPV indicates the probability of NOT having the disease given a NEGATIVE result.  In this case, if the patient had a positive stress test result, he would still only have about a 41% chance of actually having CAD, while if the result was negative, he would have only about a 1% chance of having CAD (100-99).  Even though his chances of having CAD are still less than 50% even with a positive result, this is still a significant probability, and given the potential repercussions of a missed diagnosis, a test would probably be warranted here. 

The calculations described above are summarized by what’s known as Bayes Theorem.  Without going into the details of derivation, Bayes theorem allows us to directly calculate the post-test probabilities PPV and NPV using the following formulas:

PPV = (prev x sens)  /  [ prev x sens + (1- prev) x (1-spec) ]

NPV =  [spec x (1 - prev)]  /  [spec x (1-prev) + (1-sens) x prev]

Those who prefer the “plug-and-chug” approach might appreciate deriving the PPV and NPV from these formulas vs. going through the 2x2 table calculations described above.

Summary:

The main take-home point is that the significance of a test’s result depends not only on that test’s properties (sensitivity and specificity), but also on the prevalence of the disease being tested.  In general, it is said that a high sensitivity test rules in a disease on a positive test result, while a high specificity test rules out a disease on a negative test result (Mnemonic: SnNOut, SpPIn).  

Diseases with a very low prevalence require a test with a high sensitivity and specificity to be accurately detected.  Tests with high sensitivities are good at detecting a given disease, but they also tend to have high false positive rates, meaning the disease is often falsely detected.  For a test to have a low false positive rate, and therefore accurately detect the disease most of the time, it also needs to have a high specificity.  For this reason, high sensitivity tests are used as screening tests, while confirmatory tests have a high sensitivity and a high specificity. 

Notes:

The example used in this article is based on material from the Statistics chapter of Borm Bruckmeier’s upcoming title, Wards 101, by Jed Katzel MD.