Laboratory tests and probabilities.

Why the doctors don't have to ask for multiple laboratory test?

In the following lines, I will try to explain some probabilistic concepts with immediate application in the health system.

Suppose that the probability of a false positive result in any test performed by “Pepito” laboratory is 5%. What is the probability of having at least one false positive result if the doctor submits the patient to 12 simultaneous tests?

To answer this question we can develop the following reasoning:

The probability that a patient have “at least” one false positive result is X (he could have 1, 2, 3, and so on, up to 12 false positive results).
The probability that a patient have zero (0) false positive results is Y.

Now, if we add both probabilities (X + Y), we have the whole range of possible cases, so the sum of both probabilities must be 1 (all the events in the universe if we think in math sets).

Therefore:

X + Y = 1, and if we play with the algebra... X = 1 – Y.

Then, to answer the question in the beginning, we must calculate Y and subtract it from X. But, how can calculate Y?

Quite simple, if we think that each tests performed by the laboratory is an independent event (in probabilistic terms), the probability of occurrence for n independent events is the multiplication of the individual probabilities for each event, which, in this case, is the same for all the tests, ie, 95% (you have to remember that we are evaluating the presence of 0 false positive tests results, then 100% - (FP) 5% = (VP) 95%).

Now, we can determine X in the following way:

X = 1 – Y = 1 – (0.95) ^ 12 = 0.46 => 46%.

And we can extent the formula for any p (probability of true positives result) and for any n (number of tests):

X = 1 – Y = 1 – p ^ n, is a function X(p,n), it grows when p decrease and/or n increase.

Back into the question that arose this post, the probability of “at least” one false positive result in a submission of 12 tests is around 46%, a high percentage for any critical mind, especially taking into account all the implications derived by this fact: new tests, the patient and doctor concern, the submission to some other tests (maybe more invasive), secondary materials costs, etc.

What can we do against this problem?

The biochemists, who are in charge to “tune up” the techniques, have to improve the tests performance trough the increase of the probability of true positives results (p), and the consequently, the decrease in the probability of false positives results (the biochemists have two statistics, related to these probabilities, that measure the performance of the methodology: sensibility and specificity).

In the other way, the doctors must request the lower number of tests (n) capable to characterize the clinical picture.
The unnecessary and complementary tests just increase the n, and so, they increase the probability to get false positive results, and therefore, they must be avoided in the daily clinical practice.

Ok, it is all for today, I see you next time.

Reactivating this blog...

A lot of time has been passed since the last publication.
There were several causes that pushed me in this direction, but now, it is time to enter into the "rodeo" again.

We will see us soon...

Damián.