Everything Ends on Wednesday

The Brazilian Carnival just ended this week, but for some people it is time to starting worry about crazy things that may have happened over the days of the flesh festival.

Watching the news, the spokesperson of the Test and Prevention Center (CTA) in Brasilia estimated that the number of people seeking counseling and test kits increases on average 40% the day after the carnival (Wednesday). He also disclosed that 2 out of 124 performed tests (most likely staff finger stick tests) turned out positive.

After the news I started thinking of HIV incidence among Brazilians and how likely it is to be tested positive when incidence levels also change; every year these numbers go up and dow accordingly to the government policies.

After a little research, I found several sources with estimated quantities of the incidence of HIV/AIDS. The one I chose, says for instance, in 2011 the incidence rate of AIDS (stage when the disease manifests itself in the patient) in Brazil was 17.9 to 100,000 inhabitants. This number varies significantly across the regions, so it’s higher in South-East and lower in the Midwest of the country, with others regions falling in between. The incidence rate also varies between males and females, and among age groups. For the sake of simplicity, I will not consider those differences here. It would get too complicated for a blog post post-carnival.

Consider that the enzyme-linked immunosorbent assay (ELISA screening test) for testing a blood sample for the HIV antibodies being present in human blood has the following properties:



Sensitivity is the percentage of individuals with HIV infection (based on ELISA reading) whom correctly identified as having infection (aka the true positive rate). Specificity is the percentage of individuals without HIV infection (based on ELISA reading) whom correctly identified as being free of infection (aka true negative rate). No test is perfect, as a consequence, few individuals will receive false negatives and others false positives.

Suppose the incidence of HIV in the population being tested is denoted by p . Replacing: p = 18/100000 or 0.00018. Using the Law of Total Probability we can show this relation as:

Here is the output and R code computing this for various values of p with a 25 interval.


cases <- seq(0, 10000, 25)

p <- c(cases/100000)
prob <- p*.977 / (p*.977 + (1-p)*.074)
> head(round(cbind(p,prob),3), 10)
          p  prob
 [1,] 0.000 0.000
 [2,] 0.000 0.003
 [3,] 0.000 0.007
 [4,] 0.001 0.010
 [5,] 0.001 0.013
 [6,] 0.001 0.016
 [7,] 0.002 0.019
 [8,] 0.002 0.023
 [9,] 0.002 0.026
[10,] 0.002 0.029
 library(SciencesPo) # for using theme_538
ggplot() + 
geom_line(aes(prob, p), size=1) +


I’m not in the field of epidemiology or biostatistics, but this simple experiment teaches two things. First, it is really important to keep the incidence rate low. Assuming we randomly selected you for the test, when p is really small, it’s much more likely that you got a false positive (you are positive when you are in fact negative). But as p gets larger, it becomes more likely that you do have the infection and the test result is accurate. Second, the accuracy of the test “improves” as a by-product of the incidence/prevalence among elements within the population.

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