The apparent prevalence, the true prevalence

Serologic tests are important for conducting seroepidemiologic and prevalence studies. However, the tests used are typically imperfect and produce false-positive and false-negative results. This is why the seropositive rate (apparent prevalence) does not typically reflect the true prevalence of the disease or condition of interest. Herein, we discuss the way the true prevalence could be derived from the apparent prevalence and test sensitivity and specificity. A computer simulation based on the Monte-Carlo algorithm was also used to further examine a situation where the measured test sensitivity and specificity are also uncertain. We then complete our review with a real example. The apparent prevalence observed in many prevalence studies published in medical literature is a biased estimation and cannot be interpreted correctly unless we correct the value.


Introduction
Serologic tests are commonly used in seroepidemiologic and prevalence studies (1). The design is typically conducted to understand the current situation of a condition of interest, say a disease. For example, over the past two years, soon after the announcement of the coronavirus disease pandemic, many serologic tests have been developed for diagnosis of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2); numerous seroepidemiologic studies have been conducted to determine the prevalence of the disease in various parts of the world. For example, a populationbased seroprevalence study revealed a SARS-CoV-2 seroprevalence of 9.7% in the Principality of Andorra (2). The results obtained from seroepidemiologic studies are generally used by health care researchers to understand where we do stand by estimating the health burden and the economic impact of a disease, and policy-makers to better identify the priorities and planning (3). But, are the values obtained from these studies valid?
At the heart of the design is the method by means of which we identify the condition of interest. We usually use a diagnostic test to detect the condition (e.g., a disease). However, a diagnostic test is usually not perfect; it may give false-positive and false-negative results; not all people with positive tests are diseased, and not all with negative tests are disease-free (4). This is why the prevalence derived from these studies, the so-called "apparent prevalence" (pr), is not necessarily an unbiased estimation of the true prevalence (π), the true proportion of diseased people in the population or the study sample. Herein, we are going to discuss how we can derive an unbiased estimation of π from the obtained pr and the test sensitivity (Se) and specificity (Sp). We also used a computer simulation program to better investigate the situation.
Electronic supplementary material available online for this article.
The apparent prevalence, the true prevalence

Prevalence
The pr (the apparent prevalence) is defined as the portion of tested people with a positive test (T + ) (5). Therefore: where TPR and FPR are true-positive and false-positive rates, respectively. Substituting the TPR and FPR, we have (4): Solving the above equation for π (the true prevalence), yields: This shows that the true prevalence (π) and the apparent prevalence (pr) are linearly related ( Figure 1).
If we take into account the uncertainty existing in the measured estimates of pr, Se, and Sp, Eq. 3 becomes: where x (any variable with a hat, e.g., π or pr) represents an estimation for x (e.g., π or pr). Assuming that pr and the test Se and Sp are independent, employing basic calculus and using a first-order Taylor series expansion, we have (6,7): (Eq. 5) x represents the variance of x. Based on the results, we can calculate the 95% confidence interval (CI) of the true prevalence (8)(9)(10). To portray the effect of variations in estimates of pr, Se, and Sp on the π, we conducted a Monte-Carlo simulation program.

Computer simulation
We assumed that the Se and Sp of a diagnostic test were measured in a hypothetical validity study on 225 individuals: 75 in whom the disease was confirmed and 150 without the disease (     To further investigate the situation, we used a Monte-Carlo simulation ( Table 2, Supplementary material). We assumed an arbitrarily chosen population size of 1,000,000 people and assumed that 200,000 of whom had a disease -i.e., a population true prevalence of 0.20. We randomly selected a sample of 300 individuals from the population. Each person in the study sample was then tested with a diagnostic test with Se and Sp values randomly selected from the above-mentioned distributions (supposed to be Gaussian with a mean of 93% and variance of 8.3 x10 -4 for the Se, and a mean of 90% and variance of 6.0 x10 -4 for the Sp) ( Table 2). The π s , the proportion of individuals in the sample with the disease (true prevalence of the disease in the sample); the pr s , the proportion of people in the sample with a positive test (apparent prevalence of the disease in the sample); and the calculated true prevalence (in the sample), π c , derived from Eq. 4 and 5, were then estimated for each sample. The above steps were repeated for an arbitrarily chosen 200,000 samples. The frequency distributions of π s , pr s , and π c were then plotted and compared. Linear regression analysis (no intercept model) was used to determine the relationship between the π s and π c .

Simulation results and discussion
The mean true prevalence (π s ) was 0.20 (95% CI 0.16 to 0.25) -as expected, equal to the population true prevalence (π) of 0.20. The mean apparent prevalence (pr s ) was 0.27 (0.20 to 0.33), a biased estimate of the true prevalence (π s ) ( Figure 1). The mean calculated true prevalence (π c ), 0.20 (0.14 to 0.26), however, was an unbiased estimation for the true prevalence (π s ) ( Figure 2). The slope of the regression line was almost 1; the model could explain almost all of the variance observed in the π c (Figure 3). The observed variance of the π s distribution was less than that of the pr s ( Figure 2). The former was attributed to the sampling variation; the second, to the sampling variation and variability in the test Se and Sp distribution used for each sample. The variance of the π c distribution (similar to that of the pr s ) was also due to the variations in estimating the pr s and the test Se and Sp (Eq. 4). It is important to note that the term "test" in this context should be construed in a general way as any means for classifying individuals, either a laboratory test for checking a biomark- The apparent prevalence, the true prevalence er, an imaging procedure examination, or a physical examination to check presence or absence of a sign (11,12). To elaborate on the topic presented, let us examine the following example.

Example
In the first round of a population-based seroprevalence study on SARS-CoV-2 serological screening, conducted in the Principality of Andorra, the researchers found that 6816 of 70,389 tested people were seropositive, translating into a seroprevalence, pr s , of 9.7% (95% CI 9.5% to 9.9%) (2

Conclusion
Depending on the Se and Sp of the diagnostic test used in a given prevalence study, the results ob-  tained are generally biased estimates of the true prevalence of the condition of interest (e.g., a disease). The derived apparent prevalence values should therefore be corrected. Based on the variances of the seroprevalence, and the test Se and Sp, it is possible to calculate an unbiased estimation of the true prevalence.

Data availability statement
The R codes are available from the journal website as Supplementary material. Running the codes results in the data file.