Point and Interval Estimation of Population Prevalence Using a Fallible Test and a Non-Probabilistic Sample: Post-Stratification Correction

Abstract

Accurate prevalence estimation is crucial for public health planning, particularly for rare diseases or low-prevalence conditions. This study evaluated frequentist and Bayesian methods for estimating prevalence, addressing challenges such as imperfect diagnostic tests, partial disease status verification, and non-probabilistic samples. Post-stratification was applied as a novel method and was used to improve representativeness and correct biases. Three scenarios were analyzed: (1) complete verification using a gold standard, (2) estimation with a diagnostic test of known sensitivity and specificity, and (3) partial verification of disease status limited to test positives. In all scenarios, post-stratification adjustments increased prevalence estimates and interval lengths, highlighting the importance of accounting for population variability. Bayesian methods demonstrated advantages in integrating prior information and modeling uncertainty, particularly under high-variability and low-prevalence conditions. Key findings included the flexibility of Bayesian approaches to maintain estimates within plausible ranges and the effectiveness of post-stratification in correcting biases in non-probabilistic samples. Frequentist methods provided narrower intervals but were limited in addressing inherent uncertainties. This study underscores the need for methodological adjustments in epidemiological studies, offering robust solutions for real-world challenges. These results have significant implications for improving public health decision-making and the design of prevalence studies in resource-constrained or non-probabilistic contexts.

1. Introduction

Estimating prevalence in epidemiological studies is a fundamental component of public health planning, particularly for pathological conditions with low population prevalence. However, this process faces numerous methodological challenges, such as the need for large sample sizes and the implementation of complex designs for probabilistic sample selection [1,2]. Additionally, using diagnostic tests with inaccuracies in the absence of a gold standard or the limited application of such tests to only a fraction of the sample adds complexity to the analysis and may introduce bias. Furthermore, it is crucial to employ advanced interval estimation methods that overcome the inherent limitations of extremely low prevalence rates, particularly those below $0.1$ [3].

Various authors have addressed the methodological challenges associated with estimating prevalence in the context of low-prevalence conditions and imperfect diagnostic tests. For example, Rogan and Gladen [4] proposed a classical approach for inference and bias correction in estimated prevalence when using tests with sensitivities and specificity below $1.0$. This approach has been revisited and expanded more recently by Izbicki [5]. Similarly, Reiczigel [6] proposed an exact method to construct confidence intervals when employing tests with known sensitivity and specificity, a contribution later developed further by Lang et al. [7].

Another methodological challenge in estimating low prevalence is the combined use of a fallible diagnostic test within a two-phase cross-sectional design, where verification via a gold standard is performed only on positive cases. This issue was addressed by Thomas et al. [8], who developed frequentist and Bayesian estimators to correct the bias introduced by this verification scheme.

However, while probabilistic samples enable robust population-level inference, their implementation often involves high costs and significant limitations in resource-constrained settings. In such contexts, convenience and nonprobabilistic samples present a practical alternative. Various methodological approaches have been proposed for the estimation with nonprobabilistic samples [9,10]. A notable example is the use of post-stratification estimators, as proposed by Smith [11], which leverage auxiliary population-level covariate distributions to correct prevalence estimates. Designed initially to mitigate nonresponse bias in survey sampling, this technique is an effective tool to adjust bias in nonprobabilistic samples by estimating selection probabilities.

This study aims to present and evaluate some of the methods described previously to estimate the prevalence of rare or orphan diseases, incorporating adjustments proposed by various authors. These methods address contexts involving fallible diagnostic tests and partial verification schemes applied exclusively to individuals who test positive. Additionally, a novel post-stratification approach is integrated as a strategy to correct prevalence estimates derived from nonprobabilistic samples. The methods and adjustments are applied and analyzed in different scenarios using real-world data.

2. Materials and Methods

Three scenarios were established using different interval estimation methods within both frequentist and Bayesian frameworks. For each approach, estimates with and without post-stratification adjustment were compared. This sensitivity analysis permits the evaluation of the impact of post-stratification on probably bias reduction, showing that although complete bias removal is not guaranteed, the adjustment likely improves the accuracy of prevalence estimates in non-probabilistic samples.

2.1. Estimation of a Proportion in Non-Probabilistic Samples with Post-Stratification Adjustment

Post-stratification is a weighting adjustment technique used in nonprobabilistic samples. It was initially proposed to correct various biases in probabilistic samples, such as under-coverage, over-coverage, and nonresponse bias. This technique involves adjusting the weights of the sample units to reflect the known proportions of a target population divided into strata. However, these strata are constructed after the sample has been selected. In the ideal case, using a gold standard test, the prevalence estimate adjusted for post-stratification, as proposed by Holt et al. [12], in standard notation, assumes that the population comprises N units, which can be uniquely divided into H strata of sizes $N_1,N_2,\dots ,N_H$, such that ${\sum }_{h=1}^H{N}_h=N$.

Let $y_i$ be a variable taking values $y_{hi}$, where $h=1,\dots ,H$ and $i=1,\dots ,N_h$. The selected sample is of fixed size n, which, after selection, is distributed among the strata according to the vector $n=(n_1,n_2,\dots ,n_H)$, where ${\sum }_{h=1}^H{n}_h=n$. The components of n are unknown until the sample is drawn. The prevalence estimator for the population in each stratum h is defined as:

$${\widehat{p}}_h={\displaystyle \frac{{\sum }_{i=1}^{n_h}{y}_{ih}}{n_h}}$$

(1)

where $y_i$ is an indicator variable such that $y_i=1$ if the individual in the sample is positive and $y_i=0$ otherwise. Furthermore, $x_h={\sum }_{i=1}^{n_h}{y}_{ih}$ represents the total number of cases identified in stratum h.

Thus, the population prevalence estimator adjusted for post-stratification, ${\widehat{p}}_w$, and its variance, $\widehat{V}\left({\widehat{p}}_w\right)$, are expressed as follows:

$${\widehat{p}}_w=\sum _{h=1}^H{\displaystyle \frac{N_h}{N}}{\widehat{p}}_h$$

(2)

The variance of the weighted prevalence estimator ${\widehat{p}}_w$ is derived using the formula for the variance of a weighted sum of independent random variables, incorporating the finite population correction factor for each stratum. Specifically, the expression considers the variance of ${\widehat{p}}_h$ in each stratum and the proportion of the total population in that stratum:

$$\widehat{V}\left({\widehat{p}}_w\right)=\sum _{h=1}^H\left(1-{\displaystyle \frac{n_h}{N_h}}\right){\left({\displaystyle \frac{N_h}{N}}\right)}^2{\displaystyle \frac{{\widehat{p}}_h(1-{\widehat{p}}_h)}{n_h-1}}$$

(3)

Here, $\widehat{V}\left({\widehat{p}}_h\right)={\displaystyle \frac{{\widehat{p}}_h(1-{\widehat{p}}_h)}{n_h-1}}$ represents the estimated variance for each stratum with the finite population correction factor $(1-{\displaystyle \frac{n_h}{N_h}})$ applied.

2.2. Scenario 1: Prevalence Estimation with Full Verification Using a Gold Standard

2.2.1. Frequentist Approach

Let P denote the unknown population prevalence to be estimated from a probabilistic sample of size n. Following the interval proposed by Agresti and Coull [3], we employ an adjustment—the commonly known “add two successes and two failures” method—that modifies both the sample size and the observed number of successes. Specifically, we define the adjusted sample size as follows:

$$\tilde{n}\equiv n+z_{\alpha }^2$$

where $z_{\alpha /2}$ denotes the critical value of the standard normal distribution corresponding to a significance level of $\alpha$. Likewise, the adjusted prevalence estimator is given as follows:

$$\tilde{p}={\displaystyle \frac{x+\frac{z_{\alpha /2}^2}{2}}{\tilde{n}}}$$

(4)

where x represents the number of observed successes. Based on these definitions, an interval for the population prevalence P is expressed as follows:

$$P\approx \tilde{p}\pm z_{\alpha }\sqrt{{\displaystyle \frac{\tilde{p}}{\tilde{n}}}(1-\tilde{p})}$$

(5)

• Post-Stratification Adjustment
  Let $\tilde{P}$ denote the adjusted prevalence, where $w_h$ represents the weight of stratum h in the population, and ${\widehat{p}}_h$ is the estimated prevalence within stratum h. The adjustment for the sample size per stratum, ${\tilde{n}}_h$, is given as follows:

  $${\tilde{n}}_h=n_h+z_{\alpha /2}^2$$
  The estimator $\tilde{p}$ must also be adjusted using the weights and prevalences of each stratum and would be expressed as follows:

  $${\tilde{p}}_h={\displaystyle \frac{x_h+\frac{z_{\alpha }^2}{2}}{{\tilde{n}}_h}}$$
  For the weighted prevalence, the formula is as follows:

  $$\tilde{P}=\sum _{h=1}^H{w}_h{\tilde{p}}_h$$

  (6)
  The weighted variance is given as follows:

  $$\mathrm{Var}\left(\tilde{P}\right)=\sum _{h=1}^H{w}_h^2{\displaystyle \frac{{\tilde{p}}_h(1-{\tilde{p}}_h)}{{\tilde{n}}_h}}$$

  (7)
  The confidence interval is as follows:

  $$P\approx \tilde{P}\pm z_{\alpha }\sqrt{\mathrm{Var}\left(\tilde{P}\right)}$$

  (8)

2.2.2. Bayesian Approach

From a Bayesian perspective, an interval for the parameter P can be derived using a beta-binomial model, which is recognized for its favorable frequentist properties [13]. This approach also accommodates post-stratification adjustments to account for sampling biases.

For the prevalence in each stratum, we use a beta distribution as the prior:

$$P_h\sim \mathrm{Beta}({\alpha }_h,{\beta }_h)$$

In this specific case, we use a non-informative prior $P_h\sim \mathrm{Beta}(1,1)$. The likelihood of observing $x_h$ positives in a sample of size $n_h$ in stratum h follows a binomial distribution; that is, $x_h\sim \mathrm{Binomial}(n_h,P_h)$. Consequently,

$$\mathbf{P}\left(x_h\right|n_h,P_h)=\left({\displaystyle \genfrac{}{}{0pt}{}{n_h}{x_h}}\right)P_h^{x_h}{(1-P_h)}^{n_h-x_h}$$

By applying Bayes’ theorem, the posterior distribution for $P_h$ is obtained by combining the prior and the likelihood:

$$\mathbf{P}\left(P_h\right|x_h,n_h)\propto \mathbf{P}\left(x_h\right|n_h,P_h)·\mathbf{P}\left(P_h\right)$$

Substituting the binomial likelihood and the $\mathrm{Beta}({\alpha }_h,{\beta }_h)$ prior into this expression yields the following:

$$\mathbf{P}\left(P_h\right|x_h,n_h)\propto P_h^{x_h}{(1-P_h)}^{n_h-x_h}·P_h^{{\alpha }_h-1}{(1-P_h)}^{{\beta }_h-1}.$$

(9)

where ${\alpha }_h=1$ and ${\beta }_h=1$, the posterior simplifies to the following:

$$\mathbf{P}\left(P_h\right|x_h,n_h)\propto P_h^{x_h}{(1-P_h)}^{n_h-x_h}$$

(10)

The adjusted prevalence, $\left(\tilde{P}\right)$, is a weighted combination of subgroup prevalences, where the weights are the proportions of each stratum in the total population. During the Bayesian inference process, samples are drawn from the posterior distribution of $P_h$ for each stratum h, which we denote as $P_h^s$, where s indicates a specific sample from the posterior distribution.

$${\tilde{P}}^s=\sum _{h=1}^H{w}_h{P}_h^s$$

(11)

Based on the above, a $95\%$ credible interval for the adjusted prevalence $\tilde{P}$ can be defined as follows:

$${\mathrm{CIr}}_{95\%}\left(\tilde{P}\right)=\left[{\tilde{P}}_{2.5},{\tilde{P}}_{97.5}\right]$$

(12)

where ${\tilde{P}}_{2.5}$ and ${\tilde{P}}_{97.5}$ denote the 2.5th and 97.5th percentiles, respectively, of the posterior distribution of $\tilde{P}$.

2.3. Scenario 2: Estimation with a Single Diagnostic Test with Known Sensitivity and Specificity

In most practical situations, using a gold standard test is not feasible either because of the complexity of its application or because of the high associated costs. This limitation is further exacerbated when, due to sampling schemes, whether simple or complex, the gold standard is not available for all selected individuals. In addition, its use may be contraindicated in some cases due to specific indications.

2.3.1. Frequentist Approach

Reiczigel [6] proposes the estimation of prevalence using an exact method that corrects the bias introduced by an imperfect diagnostic test, assuming that its sensitivity and specificity are known. Later, Lang et al. [7] extended this proposal by combining the formula of Rogan and Gladen with an adjustment based on the method proposed by Agresti and Coull.

Let $Se$ and $Sp$ denote the sensitivity and specificity obtained from an independent study of an imperfect diagnostic test used to estimate the prevalence P of a condition or disease. The apparent prevalence ($AP$) is the proportion of positive results obtained from the test in the target population. According to Rogan and Gladen, the point estimator of P is calculated as follows:

$$\widehat{p}={\displaystyle \frac{\widehat{AP}+\widehat{Sp}-1}{\widehat{Se}+\widehat{Sp}-1}}$$

(13)

if the adjustment is applied to the interval proposed by Agresti and Coull on $\widehat{AP}$:

$$\tilde{n}=n+z_{\alpha }^2$$

$$\widehat{{AP}^{\prime }}={\displaystyle \frac{n·\widehat{AP}+\frac{z_{\alpha /2}^2}{2}}{\tilde{n}}}$$

Thus, both the corrected point estimate of P and its confidence interval are constructed as follows:

$$\tilde{p}={\displaystyle \frac{\widehat{A{P}^{\prime }}+Sp-1}{Se+Sp-1}}$$

(14)

$$\tilde{p}\pm z_{\alpha /2}{\displaystyle \frac{1}{(Se+Sp-1)}}{\left({\displaystyle \frac{\widehat{{AP}^{\prime }}(1-\widehat{{AP}^{\prime })}}{n^{\prime }}}\right)}^{1/2}$$

(15)

• Post-Stratification Adjustment
  Applying the adjustment proposed by Agresti and Coull for each stratum h:

  $$\widehat{A{P}_h^{\prime }}={\displaystyle \frac{x_h+\frac{z_{\alpha /2}^2}{2}}{n_h+z_{\alpha /2}^2}}$$

  (16)

  $${\tilde{p}}_h={\displaystyle \frac{\widehat{A{P}_h^{\prime }}+Sp-1}{Se+Sp-1}}$$

  (17)
  The estimator for the prevalence adjusted by post-stratification and the known sensitivity and specificity of the test, $\tilde{P}$, is as follows:

  $$\tilde{P}=\sum _{h=1}^H{w}_h{\tilde{p}}_h$$

  (18)
  The variance for each stratum is as follows:

  $$\mathrm{Var}\left({\tilde{p}}_h\right)={\displaystyle \frac{1}{{(Se+Sp-1)}^2}}·{\displaystyle \frac{\widehat{A{P}_h^{\prime }}(1-\widehat{A{P}_h^{\prime }})}{{\tilde{n}}_h}}$$
  The combined total variance is as follows:

  $$\mathrm{Var}\left(\tilde{P}\right)=\sum _{h=1}^H{w}_h^2·\mathrm{Var}\left({\tilde{p}}_h\right)$$

  (19)
  Finally, the confidence interval adjusted for post-stratification is as follows:

  $$\tilde{P}\pm z_{\alpha /2}·\sqrt{\mathrm{Var}\left(\tilde{P}\right)}$$

  (20)

2.3.2. Bayesian Approach

Flor et al. [14] proposed a comparative evaluation of point and interval estimation methods utilizing both frequentist and Bayesian approaches to estimate prevalence with an imperfect diagnostic test. In this context, the Bayesian approach offers the advantage of incorporating prior information about the test’s sensitivity and specificity, which can improve the precision of the estimates.

In a Bayesian framework, the true prevalence $\left(P\right)$ is modeled as a random parameter with an associated prior distribution. The sensitivity $\left(Se\right)$ and specificity $\left(Sp\right)$ of the diagnostic test, previously defined, are treated as known parameters obtained from independent studies. The probability of observing x positive cases in a sample of size n, assuming that the probability of success is given by the apparent prevalence, is modeled by a binomial distribution $x\sim \mathrm{Binomial}(n,AP)$.

The apparent prevalence is related to the true prevalence P, as well as the sensitivity and specificity of the diagnostic test, by the following equation:

$$AP=P·Se+(1-P)·(1-Sp)$$

The likelihood function for observing x positive cases is as follows:

$$\mathbf{P}\left(x\right|n,P,Se,Sp)=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{x}}\right)·A{P}^x·{(1-AP)}^{n-x}.$$

(21)

Based on Bayes’ theorem, the posterior distribution for P is as follows:

$$\mathbf{P}\left(P\right|x,n,Se,Sp)\propto P(x|n,P,Se,Sp)·\mathbf{P}\left(P\right)$$

(22)

Substituting the binomial likelihood and non-informative prior yields the following:

$$\begin{array}{cc}\hfill \mathbf{P}\left(P\right|x,n,Se,Sp)& \propto \left({\displaystyle \genfrac{}{}{0pt}{}{n}{x}}\right)·{\left(P·Se+(1-P)·(1-Sp)\right)}^x·\hfill \\ \hfill & {\left(1-\left(P·Se+(1-P)·(1-Sp)\right)\right)}^{n-x}·Beta(1,1)\hfill \end{array}$$

(23)

The posterior simplifies to the following:

$$\begin{array}{cc}\hfill \mathbf{P}\left(P\right|x,n,Se,Sp)& \propto {\left(P·Se+(1-P)·(1-Sp)\right)}^x·\hfill \\ \hfill & {\left(1-\left(P·Se+(1-P)·(1-Sp)\right)\right)}^{n-x}\hfill \end{array}$$

(24)

Numerical methods such as MCMC (Markov Chain Monte Carlo) can be used to obtain the posterior distribution of P. This approach allows for the generation of samples from the posterior distribution of P and the provision of inferences about the true prevalence.

• Post-Stratification Adjustment
  The previous Bayesian estimation can be complemented with a post-stratification adjustment. For H strata, each with a prevalence $P_h$, and each stratum having a proportion $w_h$ of the total population, the total prevalence adjusted for post-stratification can be calculated as follows:

  $$\begin{array}{cc}\hfill \mathbf{P}\left(P_h\right|x_h,n_h,Se,Sp)& \propto {\left(P_h·Se+(1-P_h)·(1-Sp)\right)}^{x_h}·\hfill \\ \hfill & {\left(1-\left(P_h·Se+(1-P_h)·(1-Sp)\right)\right)}^{n_h-x_h}\hfill \end{array}$$

  (25)

  $$\tilde{P}=\sum _{h=1}^H{w}_h·P_h$$

  (26)
  However, when estimating the prevalence $P_h$ for each stratum h under a Bayesian approach, we do not obtain a single point estimate but rather a posterior distribution for $P_h$. Thus, this post-stratification must be based on posterior distributions.
  Suppose we have obtained S samples from the posterior distribution of $P_h$ for H subgroups. We denote these samples for each subgroup h as follows:

  $$P_h^{\left(1\right)},P_h^{\left(2\right)},\dots ,P_h^{\left(S\right)}$$
  For each iteration s in the posterior, we weigh the prevalence $P_h^{\left(s\right)}$ by the proportion $w_h$ of the stratum h in the total population. In this way, the adjusted prevalence in iteration s is as follows:

  $${\tilde{P}}^{\left(s\right)}=\sum _{h=1}^H{w}_h·P_h^{\left(s\right)}$$
  This process is repeated for each sample $s=1,\dots ,S$, resulting in a combined posterior distribution for the total prevalence adjusted by post-stratification:

  $${\tilde{P}}^{\left(1\right)},{\tilde{P}}^{\left(2\right)},\dots ,{\tilde{P}}^{\left(S\right)}$$
  From the samples ${\tilde{P}}^{\left(s\right)}$, it is possible to calculate any credible interval. For example, the 2.5th and 97.5th percentiles of the posterior distribution of $\tilde{P}$ can be used to define a credible interval $95\%$. Similarly, the point estimate is obtained as the median of the posterior distribution of $\tilde{P}$.

2.4. Scenario 3: Estimation When the Status of the Disease Is Verified Only Among Test Positives

2.4.1. Frequentist Approach

In diagnostic processes or public health surveys, verifying the true disease status is common only in individuals who test positive on an initial screening test. For cost or ethical reasons, this verification is typically restricted to those with a high probability of disease based on screening results. In this context, Thomas et al. [8] derived maximum likelihood estimators for disease prevalence using both frequentist and Bayesian approaches, assuming that the sensitivity $\left(Se\right)$ and specificity $\left(Sp\right)$ of the screening test are known with certainty. The key steps are briefly described below (for additional algebraic details, we refer the reader to the supplementary materials of Thomas et al. [8], where a complete derivation is provided):

The likelihood function is given as follows:

$$\begin{array}{c}\hfill \mathcal{L}\left(P\right)={[(1-Se)P+Sp(1-P)]}^{n_{.0}}{(Se·P)}^{n_{11}}{\left((1-Sp)(1-P)\right)}^{n_{01}}\end{array}$$

(27)

where $n_{11}$ denotes the number of true positives, $n_{01}$ denotes the number of false positives among those tested positive, $n_{.0}$ denotes the total number of test negatives, and P denotes the prevalence and the sensitivity and specificity.

The likelihood function is constructed based on the observed values and the probability of test results given the true health condition of the individuals (see [8] for more details). The maximum likelihood estimator (MLE) of prevalence, $\widehat{p}$, arises from differentiating the log-likelihood with respect to P and setting it to zero. Upon expanding and simplifying the derivative of

$$log\mathcal{L}\left(P\right)=n_{.0}log[(1-Se)P+Sp(1-P)]+n_{11}log\left(SeP\right)+n_{01}log\left[(1-Sp)(1-P)\right]$$

a quadratic equation in P is obtained, and the explicit solution to this equation is given as follows:

$$\begin{array}{c}\hfill \widehat{p}={\displaystyle \frac{-B-\sqrt{B^2-4AC}}{2A}}\end{array}$$

(28)

where the terms A, B, and C are defined as follows:

$$A=-(1-Se-Sp)n$$

$$B=(1-Se-Sp)(n_{.0}+n_{11})-Sp(n_{11}+n_{01})$$

$$C=n_{11}Sp$$

Here, n represents the total sample size (the sum of individuals who tested positive and negative on the screening test).

The asymptotic variance of $\widehat{p}$ follows from the standard large-sample theory for maximum likelihood estimators, where it is given by the inverse of the Fisher information (i.e., the negative second derivative of the log-likelihood) evaluated at $\widehat{p}$. Substituting $\widehat{p}$ into the resulting expression yields the following:

$$\begin{array}{c}\hfill {\widehat{\sigma }}^2={\left[{\displaystyle \frac{Se}{\widehat{p}}}+{\displaystyle \frac{1-Sp}{1-\widehat{p}}}+{\displaystyle \frac{{(1-Se-Sp)}^2}{(1-Se)\widehat{p}+Sp(1-\widehat{p})}}\right]}^{-1}\end{array}$$

(29)

Using this variance, an asymptotic $100(1-\alpha )\%$ confidence interval can be constructed as follows:

$$\begin{array}{c}\hfill \widehat{p}\pm z_{1-\alpha /2}{\displaystyle \frac{\widehat{\sigma }}{\sqrt{n}}}\end{array}$$

(30)

However, in daily practice, screening programs are typically not based on a random sample from the population. Therefore, disease prevalence estimates require adjustments to reduce the bias associated with this practice. In this context, post-stratification is a suitable methodological option for such adjustments.

• Post-Stratification Adjustment
  The prevalence for each stratum is calculated using the method proposed by [8]. A ${\widehat{p}}_h$ value is defined for each of the H strata in the sample:

  $$\begin{array}{c}\hfill \widehat{p_h}={\displaystyle \frac{-B_h-\sqrt{B_h^2-4A_h{C}_h}}{2A_h}}\end{array}$$

  (31)

  where $A_h$, $B_h$, and $C_h$ are defined previously (28). The variance for each stratum is as follows:

  $$\begin{array}{c}\hfill {\widehat{\sigma }}_h^2={\left[{\displaystyle \frac{Se}{\widehat{p_h}}}+{\displaystyle \frac{1-Sp}{1-\widehat{p_h}}}+{\displaystyle \frac{{(1-Se-Sp)}^2}{(1-Se)\widehat{p_h}+Sp(1-\widehat{p_h})}}\right]}^{-1}\end{array}$$

  (32)
  The overall prevalence adjusted by post-stratification is as follows:

  $$\begin{array}{c}\hfill \tilde{p}=\sum _{h=1}^H{w}_h·{\widehat{p}}_h\end{array}$$

  (33)
  The weighted variance is given as follows:

  $$\begin{array}{c}\hfill Var\left(\tilde{p}\right)=\sum _{h=1}^H{\left({\displaystyle \frac{N_h}{N}}\right)}^2{\widehat{\sigma }}_h^2\end{array}$$

  (34)
  A confidence interval for $\tilde{p}$ is as follows:

  $$\begin{array}{c}\hfill CI\left(\tilde{p}\right)=\tilde{p}\pm z_{1-\alpha /2}\sqrt{Var\left(\tilde{p}\right)}\end{array}$$

  (35)

2.4.2. Bayesian Approach

In the same work published by Thomas et al. [8], the Bayesian version for prevalence estimation is presented for cases where verification using a gold standard is performed only on individuals who test positive in a screening test. Starting from the previously defined likelihood and assuming that $Se$ and $Sp$ are known and fixed for the screening test, the prevalence is estimated as the parameter of interest under a non-informative prior, defined as the following distribution: $P\sim Beta(1,1)$. Thus, the posterior distribution for P is as follows:

$$\begin{array}{c}\hfill \mathbf{P}\left(P\mid \mathbf{D}\right)\propto \mathbf{P}\left(\mathbf{D}\mid P\right)·\mathbf{P}\left(P\right)\end{array}$$

(36)

where $\mathbf{D}$ corresponds to true positives $\left(n_{11}\right)$, false positives among those who tested positive in the test $\left(n_{01}\right)$, total negatives according to the test $\left(n_{.0}\right)$, and sensitivity and specificity of the test $(Se,Sp)$.

$$\begin{array}{c}\hfill \mathbf{P}\left(P\mid \mathbf{D}\right)={[(1-Se)P+Sp(1-P)]}^{n_{.0}}{(Se·P)}^{n_{11}}{\left((1-Sp)(1-P)\right)}^{n_{01}}·Beta(1,1)\end{array}$$

(37)

Finally, to obtain the posterior distribution of P, MCMC (Markov Chain Monte Carlo) methods can be implemented, with the calculation of a credible interval for P derived from the estimated posterior distribution.

• Post-Stratification Adjustment
  The previous Bayesian estimation can also include a post-stratification adjustment. In this case, as previously defined, for the H strata formed, $P_h$ is estimated by stratifying the posterior of P:

  $$\begin{array}{c}\hfill \mathbf{P}\left(P_h\mid \mathbf{D}\right)={[(1-Se)P_h+Sp(1-P_h)]}^{n_{.0_h}}{(Se·P_h)}^{n_{{11}_h}}{\left((1-Sp)(1-P_h)\right)}^{n_{{01}_h}}·Beta(1,1)\end{array}$$

  (38)

  $$\begin{array}{c}\hfill \tilde{P}=\sum _{h=1}^H{w}_h·P_h\end{array}$$

  (39)
  As previously defined, when estimating the prevalence $P_h$ for each stratum h under a Bayesian approach, a single point estimate is not obtained; instead, a posterior distribution for $P_h$ is generated. In this context, post-stratification must be based on these posterior distributions, allowing for the incorporation of uncertainty into the stratum-adjusted estimates. We have obtained S samples from the posterior distribution of $P_h$ for H strata. These samples for each stratum h are denoted as follows:

  $$P_h^{\left(1\right)},P_h^{\left(2\right)},\dots ,P_h^{\left(S\right)}$$
  For each iteration s of the posterior distribution, we weight the prevalence $P_h^{\left(s\right)}$ by the proportion $w_h$ of the stratum h in the total population. In this way, the adjusted prevalence in iteration s is as follows:

  $$\begin{array}{c}\hfill {\tilde{P}}^{\left(s\right)}=\sum _{h=1}^H{w}_h·P_h^{\left(s\right)}\end{array}$$

  (40)
  This process is repeated for each sample $s=1,\dots ,S$, resulting in a combined posterior distribution for the total prevalence adjusted by post-stratification:

  $${\tilde{P}}^{\left(1\right)},{\tilde{P}}^{\left(2\right)},\dots ,{\tilde{P}}^{\left(S\right)}$$
  From the samples ${\tilde{P}}^{\left(s\right)}$, we can calculate any credible interval.

2.5. Application of Prevalence Estimation for Mutation in Familial Chylomicronemia with Post-Stratification Adjustment

Data from a two-phase descriptive cross-sectional study, in which one of the authors (JMEA) participated [15], were used to evaluate the applicability of the proposed post-stratification adjustment. The study began with an initial nonprobabilistic sample of n = 74,145 adult subjects from Pereira, Colombia. The progressive selection of patients was carried out in two phases. In the first phase, positive cases were identified by applying the FCS clinical score (Family Chylomicronemia Syndrome), considered a fallible diagnostic test. With a cutoff point $\ge 8$, this score has a reported sensitivity of $0.88$ and a specificity of $0.85$ for detecting mutations associated with FCS [16]. As a result of this stage, $n=18$ positive subjects were identified.

In the second phase, these 18 individuals underwent a gold standard test using molecular sequencing of the coding region of the genome (exome: 20,000 genes), with coverage greater than $98\%$ and a minimum depth of $20X$. This analysis included the APOA5, APOC2, GPIHBP1, LMF1, and LPL genes, covering all pathogenic variants in exonic regions, splicing sites, as well as small insertions and deletions. As a final result, $n=6$ subjects were confirmed to have mutations associated with familial chylomicronemia syndrome.

The strata used for the post-stratification adjustment were based on sex and single-year age (ranging from 18 to 85 years or older), obtained from the official population projections published by The National Administrative Department of Statistics (DANE) of Colombia.

Frequentist estimates were performed using the R software, version 4.2.3 while Bayesian estimates were conducted using Stan through the RStan library. In the MCMC simulation, the convergence towards a stationary distribution was verified by running four chains with 1000 iterations each, ensuring that $\widehat{R}<1.1$ is used as the convergence criterion. This approach yields the complete posterior distribution, thereby facilitating the selection of appropriate $95\%$ credible intervals [17].

3. Results

Table 1. Point and interval estimates of the population prevalence $\left(P\right)$ obtained using the frequentist and Bayesian methods, both unadjusted and adjusted by post-stratification.

Scenario	Method	Unadjusted (%)			Adjusted (%)
		P	Lower	Upper	P	Lower	Upper
Scenario 1	Agresti–Coull	0.0081	0.0032	0.0181	0.4621	0.4015	0.5226
	Bayesian (beta-binomial)	0.0088	0.0028	0.0156	0.2417	0.2014	0.2886
Scenario 2	Lang	0.0000	0.0000	0.0162	0.0000	0.0000	0.0836
	Bayesian–Flor	0.0016	0.0000	0.0058	0.2740	0.2280	0.3286
Scenario 3	Frequentist–Thomas	0.0094	0.0020	0.0169	0.0075	0.0014	0.0135
	Bayesian–Thomas	0.0104	0.0045	0.0201	0.2797	0.2328	0.3352

presents the results obtained for the estimation with and without post-stratification adjustment, corresponding to each scenario.

Table 2. Comparison of Interval Lengths by Estimation Method.

Method	Without Adjustment	With Adjustment
Agresti–Coull	0.0149	0.1212
Bayesian (beta-binomial)	0.0128	0.0872
Lang	0.0162	0.0836
Flor–Bayesian	0.0058	0.1005
Frequentist–Thomas	0.0149	0.0121
Bayesian–Thomas	0.0156	0.1025

shows the lengths of each interval with and without post-stratification adjustment. According to each scenario, the following was found:

3.1. Scenario 1: Estimation with Full Verification Using a Gold Standard

In the first scenario, the Agresti–Coull method showed a point estimate of prevalence of $0.0081\%$ without post-stratification adjustment, with a confidence interval of $[0.0032\%,0.0181\%]$. After applying the post-stratification adjustment, the estimate increased to P = 0.4621%. This result highlights a significant increase in the prevalence estimate, likely due to the correction of biases in the sample.

The confidence interval length without post-stratification was $0.0149$, while it increased to $0.1211$ after adjustment, reflecting a notable increase. This result aligns with the objective of post-stratification: to account for variability previously unconsidered due to using a nonprobabilistic sample.

Similarly, the Bayesian approach initially presented a prevalence of P = 0.0088%, with a $95\%$ credible interval ($CI{r}_{95\%}$) of $[0.0028\%,0.0156\%]$. With adjustment, the estimate increased to P = 0.2417%, with an adjusted interval of $[0.2014\%,0.2886\%]$. The interval length increased from $0.0128$ without post-stratification to $0.0872$ with adjustment, reflecting the same trend observed with the frequentist method.

3.2. Scenario 2: Estimation with a Diagnostic Test of Known Sensitivity and Specificity

In this scenario, the Lang method presented an initial prevalence of P = 0.000% without post-stratification adjustment, indicating severe underestimation due to the limited sensitivity of the method. After applying post-stratification, the estimator increased to $0.0836\%$, highlighting the importance of adjustment to improve bias in designs with imperfect diagnostic tests. The interval length without post-stratification was $0.0162$, while it increased to $0.0836$ with post-stratification.

The Bayesian method proposed by Flor showed an initial prevalence of P = 0.0016%, with a credible interval $95\%$ ($CI{r}_{95\%}$) of $[0.0000\%,0.0058\%]$. After adjustment, the estimator increased significantly, as shown in Table 1; this result emphasizes the flexibility of the Bayesian approach to address the limitations of fallible diagnostic tests and adjust prevalence estimates. In this case, the interval length increased from $0.0058$ to $0.1006$. This significant increase is due to the Bayesian method explicitly incorporating the additional uncertainty associated with the nonprobabilistic sample.

3.3. Scenario 3: Estimation When the Status of the Disease Is Verified Only Among Positives

In this final scenario, the frequentist method of Thomas estimated an initial prevalence of P = 0.0094%, with a confidence interval $95\%$ ($C{I}_{95\%}$) of $[0.0020\%,0.0169\%]$. After applying the correction by the inverse of inclusion in the sample, the estimator decreased, resulting in a narrower interval. The interval length was $0.0149$ without post-stratification and decreased slightly to $0.0121$ with adjustment. This result is unique to this scenario, as post-stratification appears to reduce variability by addressing the specific limitations of partial verification. This change reflects the adjustment’s ability to improve precision in partial verification designs, a situation more representative of daily practice.

The Bayesian approach by Thomas et al. initially estimated a prevalence of P = 0.0104%, with a credible interval 95% ($CI{r}_{95\%}$) of $[0.0045\%,0.0201\%]$. The adjusted estimator using the post-stratification method increased, demonstrating the ability of Bayesian estimates to integrate additional information and adjust prevalence estimates in partial verification scenarios. The length of the credible interval increased from $0.0156$ to $0.1024$ with adjustment. This considerable increase is consistent with the Bayesian approach’s ability to model the additional uncertainty introduced by post-stratification.

4. Discussion

Accurate prevalence estimation in epidemiological studies is essential for public health planning, particularly for rare diseases or conditions with low prevalence. The methods evaluated in this study address key challenges such as imperfect diagnostic tests, partial verification of disease status, and reliance on nonprobabilistic samples. These limitations can lead to biased estimates and inadequate public health decisions if not corrected. Thus, this study highlights the need to implement robust methodological adjustments to improve the accuracy of prevalence estimates. A comparison of frequentist and Bayesian approaches revealed significant differences in their ability to address inherent limitations in study designs and diagnostic tests. These observations align with McNamee et al. [18], who noted that despite their complexity, two-phase designs can optimize precision and reduce costs in prevalence studies.

4.1. First Scenario

In the first scenario, the results show that both methodological approaches indirectly mitigate the biases in the unadjusted estimates, as evidenced by the observed differences in the point estimates and the length of the intervals. However, post-stratification adjustment had a significant impact by incorporating additional information on the distribution of population variables. This result is consistent with Holt et al. [12], who reported the effectiveness of post-stratification in addressing the lack of representativeness in nonprobabilistic samples. Furthermore, the Bayesian approach adjusted with post-stratification generated wider confidence intervals than the Agresti–Coull method, as observed in the study by Flor et al. [14]. This behavior reflects the Bayesian approach’s ability to explicitly model uncertainty, making it an advantageous alternative in contexts of high variability.

4.2. Second Scenario

The second scenario, where a diagnostic test with known sensitivity and specificity was performed, highlights the importance of considering the quality of the initial tests in the study design. As noted by Shrout and Newman [19], the limited sensitivity of screening methods can significantly underestimate prevalence, consistent with the unadjusted results obtained in this investigation. However, post-stratification adjustments and Bayesian estimates allowed for correction of these limitations, aligning with previous studies that highlight the flexibility of the Bayesian approach in incorporating prior information and improving precision. Regarding interval lengths, unadjusted intervals were observed to be shorter, which could create false confidence in underrepresented estimates. In contrast, post-stratification adjustment, while increasing interval lengths, corrected the inherent bias in the data and improved population representativeness.

A particular situation, well documented in the literature, became evident in the interval proposed by Lang and was also observed in this study. When using the adjusted prevalence estimator proposed by Rogan and Gladen, the adjusted estimate may fall outside the allowed range $[0,1]$, especially for negative values when the true prevalence is very low. This situation, previously studied by authors such as Viana [20] and Hasselt [21], was reflected in the present study, where the reported prevalence (Table 1) was zero, consistently underestimating the true prevalence in low-prevalence scenarios.

In contrast, the Bayesian approach proved beneficial, as evidenced by the results of this study. Using a prior distribution P $\sim Beta(\alpha ,\beta )$, this approach ensures that the estimated prevalence remains within the allowed range of $[0,1]$, even in situations of extremely low prevalence. This advantage was also reflected in the Bayesian approach adjusted by post-stratification used in Scenario 2, where a more robust and representative estimate was achieved, effectively addressing the limitations observed with frequentist methods.

4.3. Third Scenario

In Scenario 3, where the disease status is verified only among positives, the results confirm the advantages of combining maximum likelihood methods and post-stratification adjustments. The frequentist approach, while traditional, showed a reduction in the interval length after adjustment, highlighting its ability to address specific biases in such designs. On the other hand, the Bayesian approach provided wider intervals but with a better representation of uncertainty, consistent with the observations of Flor et al. [14] in partial verification scenarios.

Regarding the interval lengths, the unadjusted intervals were shorter, potentially leading to overconfidence in underrepresented estimates. In contrast, post-stratification adjustment, while increasing interval lengths, corrected inherent bias in the data and improved population representativeness.

In this scenario, a different pattern was observed in the lengths of the confidence intervals. The frequentist approach showed a reduction in interval length after adjustment. In contrast, Bayesian approaches presented wider intervals post-adjustment, reflecting their ability to incorporate the inherent uncertainty of partial verification designs using a gold standard test. These results underscore the importance of selecting the most appropriate method based on the study context, as suggested in previous research [18].

Finally, Bayer et al. [22] proposed an innovative approach for estimating prevalence derived from complex sampling surveys based on a melding method that uses gamma and beta distributions. While their method is particularly robust in addressing bias introduced by the imperfect sensitivity and specificity of diagnostic tests, its context is limited to probabilistic samples, ensuring population representativeness from the sampling design but not directly addressing bias in nonprobabilistic samples or post-stratification adjustments.

In contrast, the approach in this study focuses on contexts where samples are nonprobabilistic, presenting an additional challenge. Integrating Bayesian methods with post-stratification adjustments allows for correcting biases related to population representativeness, a dimension not explicitly addressed in Bayer’s method. This methodological difference makes this work complementary to that of Bayer et al. It provides a flexible solution for situations where practical limitations—such as the inability to use probabilistic sampling designs or incomplete verification of diagnostic tests—necessitate methods that balance representativeness and adaptability.

In epidemiological studies, frequentist methods offer computational efficiency and straightforward interpretation in large samples but may underestimate uncertainty in low-prevalence settings or designs with partial verification. In contrast, Bayesian methods integrate prior information and provide more comprehensive uncertainty quantification through credible intervals. The integration of both approaches, along with post-stratification adjustments, effectively corrects biases related to nonprobabilistic sampling and imperfect diagnostics. This combined strategy offers a robust framework for prevalence estimation, enhancing representativeness and supporting more informed public health decision-making.

Despite favorable findings, this study has some limitations. Reliance on known metrics, such as the sensitivity and specificity of diagnostic tests, can restrict the applicability of methods in contexts where these parameters are not well established. Additionally, while robust, the Bayesian approach critically depends on the quality of prior information (i.e., without reliable data, using noninformative priors may lead to less precise estimates).

It was not feasible to generate or acquire a synthetic dataset that accurately represents the complexities associated with nonprobabilistic sampling processes, which are a crucial part of our study design. This limitation underscores the need for future research to rigorously evaluate the coverage properties of intervals adjusted through post-stratification and quantify the bias linked to point estimators under different conditions. These evaluations are essential for determining the robustness and validity of the proposed methodologies, especially when compared to alternative approaches.

The results have important implications for public health practice and epidemiological research. Adjusting prevalence estimates through post-stratification significantly improves representativeness in contexts with nonprobabilistic samples and imperfect diagnostic tests. Bayesian methods have proven valuable tools in high-uncertainty scenarios, particularly when prior information is incorporated.