Epidemiological modelling and its use to manage COVID-19

#7 – If my test is positive, what does that mean?

Let's step outside of modeling for a moment to discuss an important question in epidemiology: what exactly does the result of a screening test mean?

A test is characterized by its sensitivity (Se) and specificity (Sp). The sensitivity of a test is its ability to give a positive result when an infected individual is tested (although one must know for sure that he or she is infected). The specificity of a test is its ability to give a negative result when an uninfected individual is tested (again, one must be sure that the individual is not infected). To determine the Se and Sp values of a test, experimental work is needed to compare the results given by this test with those obtained with another test with well-known characteristics (usually a gold standard).

However, when you go for a test, you do not know in advance what your health status is since you are trying to determine it. Once the test done, what is the probability of actually being infected when the test is positive? What is the probability of being truly uninfected when the test is negative? To find out, you have to calculate the positive (PPV) and negative (NPV) predictive values of the test. We must therefore know not only the characteristics of the test (Se and Sp), but also a third key information: the prevalence of infection p in the population which the tested indidivual belongs, i.e. the proportion of people who are actually infected in this population. This gives the following table:

The predictive values can be calculated as follows:

VPP = VP / (VP+FP) = Se x p / (Se x p + (1-Sp) x (1-p))

VPN = VN / (VN+FN) = Sp x (1-p) / (Sp x (1-p) + (1-Se) x p)

Let's take as an example a large population like France (70 million inhabitants), where 10% of the people are tested (7 million). Let us imagine that the prevalence of infection is p=1%, and that the test has a sensitivity Se=90% and a specificity Sp=98%. The table gives:

Applying the above formulas to calculate the predictive values, we find that a person with a positive test result has a 31% chance (PPV) of actually being infected (and a 69% chance of not being infected). A person with a negative result has a 99.9% chance (NPV) of being truly uninfected. Other examples are shown in the figure below. The negative predictive value is very good when the prevalence is low, while the positive predictive value is less good, but allows the right precautions to be taken.

Positive (VPP) and negative (VPN) predictive values of the test according to its sensitivity (in blue: Se=0.9; in black: Se=0.5), specificity (in blue: Sp=0.99; in black: Sp=0.98) and the prevalence of infection.

To sum up: tests do of course play a key role in controlling an epidemic, as they help to identify infected and uninfected individuals, but their use is not that simple. The uncertainty resulting from their sensitivity, specificity and prevalence of infection (which, remember, varies over time) must be taken into account in order to make a relevant use of their results.