A primer on Bayesian estimation of prevalence of COVID-19 patient outcomes.

A common research task in COVID-19 studies often involves the prevalence estimation of certain medical outcomes. Although point estimates with confidence intervals are typically obtained, a better approach is to estimate the entire posterior probability distribution of the prevalence, which can be easily accomplished with a standard Bayesian approach using binomial likelihood and its conjugate beta prior distribution. Using two recently published COVID-19 data sets, we performed Bayesian analysis to estimate the prevalence of infection fatality in Iceland and asymptomatic children in the United States.

LAY SUMMARY

We illustrate a Bayesian approach for prevalence estimation using two recently published COVID-19 data sets.

INTRODUCTION

Many COVID-19 studies are interested in estimating the prevalence of certain medical outcomes of interest. Typically, the prevalence was reported as a point estimate accompanied by a 95% confidence interval (95% CI). For example, in a study recently published by Gudbjartsson et al, the authors estimated the prevalence of COVID-19 deaths in Iceland, obtaining the infection fatality risks of 0.1% (95% CI 0.0–0.3%), 2.4% (95% CI 0.6–6.2%), and 11.2% (95% CI 3.6–24.0%) for those 70 years old or younger, those between 70 and 80 years of age, and those older than 80, respectively. In another recent study published by Sola et al, the authors estimated the prevalence of infected children without any COVID-19 symptoms for multiple regions in the United States, showing a pooled asymptomatic prevalence of 0.65% (95% CI 0.47–0.83%).

There are three main limitations with the traditional biostatistical methods used to obtain the above estimations. First, the above studies only obtained point estimates for the prevalence inferred from the available data. Although point estimates may be the most likely values of the unknown prevalence, values other than the point estimates may also have a non-negligible high probability. Since there always exists uncertainty associated with any inferred values for prevalence, the uncertainty should be ideally measured by a probability distribution that assigns a precise probability to every possible value of the unknown prevalence (ie, values with higher likelihood get higher probability). Second, even though 95% CIs were reported, it is important to note that 95% confidence intervals do not represent a range of values with a 95% probability in containing the point estimates. Instead, 95% CIs are a range produced by a statistical procedure that, in repeated sampling, has a 95% probability of containing the true value of the unknown parameter. In other words, confidence intervals evaluate the reliability of the statistical procedures rather than the parameters. In addition, confidence intervals do not provide a probabilistic measurement of the uncertainty associated with the possible values for prevalence. Since no probability was assigned to any value within the range of the confidence intervals, it is not possible to evaluate which value is more likely than others. Third, the above estimations cannot incorporate prior existing knowledge of prevalence into the analysis, which may be critical for obtaining accurate estimations when the true prevalence is low and the available sample size is relatively small. Therefore, we would like to advocate the use of Bayesian methods for researchers who work in this important field for COVID-19 research, as it enables them to overcome the above limitations by deriving a probability for every possible value of the unknown parameter of interest.

BAYESIAN MODELING

Two essential elements are required in any Bayesian model: (1) likelihood functions for describing the mathematical relationship between observed data and unknown parameters and (2) prior probability distributions for unknown parameters. As mentioned above, a common parameter of interest in COVID-19 studies is the unknown prevalence of certain medical outcomes, for example, the prevalence of death or asymptomatic status in people who were infected by the SARS-CoV-2 virus. Let θ, y, and N denote the unknown prevalence, the observed number of medical outcomes of interest (eg, the number of death or asymptomatic infection), and the total sample size, respectively. The mathematical relationship among θ, y, and N can be described with the following binomial likelihood function:

In Eq. (1), only θ is the unknown parameter, whose possible values are typically modeled using a beta probability distribution:

The beta distribution in Eq. (2) has two shape parameters, a and b, whose values represent different degrees of prior knowledge or belief on the likely values of θ. In COVID-19 studies, researchers are typically faced with no prior data to derive informative prior probability distributions. In that case, both a and b can be set to 1 as a flat noninformative prior distribution for θ, which essentially means that θ has an equal chance to be any value between 0 and 100%.

Based on the likelihood function and prior probability distribution, a probability distribution for the unknown parameters (called posterior probability distribution in Bayesian terminology) is derived either analytically or sampled through Markov chain Monte Carlo (MCMC) techniques. In reality, many Bayesian models do not have an analytical solution and thus require specialized software for MCMC sampling (eg, WinBUGS, OpenBUGS, JAGS, Stan). However, for the prevalence estimation in many COVID-19 studies, the posterior probability distribution can be easily derived analytically. Specifically, beta prior distributions have a special mathematical relationship with binomial likelihoods (beta distributions are called conjugate priors for binomial likelihoods), so that the posterior distribution for θ is also a beta distribution with the two shape parameter values updated as (a + y) and (b + N − y), respectively.

APPLICATION TO COVID-19 DATA

We have applied the above binomial and beta model to perform Bayesian analysis on two recently published COVID-19 data sets (Table 1). Since we did not have any prior knowledge on the infection fatality rate or the asymptomatic prevalence, we used a noninformative beta prior (ie, both its shape parameters, a and b, were set to the value of 1). We then plugged in the necessary numbers to calculate the posterior distributions by updating the parameters of the beta distributions (Table 1). For example, for the age group 0–70 years old in Iceland, there were three deaths (y) out of a total of 3012 infections (N), so the posterior probability distribution of the infection fatality risk for this age group is beta(3 + 1, 1 + 3012 − 3). Similarly, out of a total of 15 311 infected children (N) in the West region of United States, 120 were asymptomatic (y), so the posterior distribution for the prevalence of asymptomatic children in the West region of U.S. is beta(1 + 120, 1 + 15 311 − 120).

Table 1.

Bayesian analysis of two published COVID-19 data sets

Study	Age groups (years old)	Death (y)	Infection (N)	Prior Beta(a, b)	Posterior Beta(a + y, b + N − y)	Posterior median (95% credible interval) (%)
Infection fatality rates in Iceland1	0–70	3	3012	Beta(1)	Beta(4, 3010)	0.12 (0.04–0.29)
	70–80	3	128	Beta(1)	Beta(4, 126)	2.84 (0.85–6.65)
	>80	4	38	Beta(1)	Beta(5, 35)	11.87 (4.30–24.22)
Study	Regions	ASX (y)	Infection (N)	Prior Beta(a, b)	Posterior Beta(a+y, b+N − y)	Posterior median (95% credible interval), %
Asymptomatic (ASX) children in U.S.2	West	120	15311	Beta(1)	Beta(121, 15192)	0.79 (0.66 - 0.94)
	Midwest	40	5217	Beta(1)	Beta(41, 5178)	0.78 (0.56 - 1.04)
	South	49	8354	Beta(1)	Beta(50, 8306)	0.59 (0.44 - 0.78)
	Northeast	41	4159	Beta(1)	Beta(42, 4119)	1.00 (0.73 - 1.33)

After obtaining the posterior distributions (ie, the beta distributions with updated parameters), we can visualize the distributions by randomly sampling from them and plotting the samples. Figures 1 and 2 depict the posterior distributions for infection fatality rates in Iceland and the prevalence of asymptomatic children in the United States, respectively, which provide a complete probabilistic landscape for those parameters. Besides plotting, the posterior distributions are also often characterized by summary statistics, for example, medians and 95% credible intervals (Table 1). It is important to note that contrary to confidence intervals, credible intervals represent the likely ranges of the true values of the unknown parameter. We provided an example R programming script (Supplementary File S1) for plotting the posterior distributions and calculating the summary statistics. Although our current estimations were based on noninformative prior probability distributions for prevalence, informative priors can be used if relevant information is available. In fact, our current estimates can become informative priors for future updates using the same Bayesian framework.

Figure 1.

The posterior probability densities of infection fatality rate for different age groups in Iceland: (A) 0–70, (B) 70–80, and (C) >80.

Figure 2.

The posterior probability density of the prevalence of asymptomatic children in four different US regions: (A) West, (B) Midwest, (C) South, and (D) Northeast.

DISCUSSION

Bayesian analyses are often perceived as complicated. It is true that applying Bayesian analyses may require highly customized modeling procedures. For example, we have recently published COVID-19 related studies using Bayesian approaches,, which required (1) developing customized likelihood functions and (2) the estimation of the posterior distributions by MCMC. However, as illustrated above via the reanalysis of the two published COVID-19 data sets, estimating prevalence can be easily achieved using a simple Bayesian model based on binomial likelihood and its beta conjugate prior, which is mathematically straightforward and well applicable for prevalence estimation in real-world data analysis. As researchers around the world are gathering more and more COVID-19 data for estimating the prevalence of various medical outcomes, we hope that Bayesian approaches will be widely utilized. In our own experience, the presented Bayesian model is a stepping stone for beginners to appreciate the power of Bayesian approaches before learning more complicated models (eg, Bayesian hierarchical modeling) and computational techniques (eg, MCMC).

SUPPLEMENTARY MATERIAL

Supplementary material is available at Journal of the American Medical Informatics Association online.

AUTHORS’ CONTRIBUTIONS

X.G. performed data analysis. Q.D. drafted the manuscript. Both conceived the project.

Conflict of interest statement

None declared.

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