ABSTRACT
Laboratories use group (pooled) testing with multiplex assays to reduce the time and cost associated with screening large populations for infectious diseases. Multiplex assays test for multiple diseases simultaneously, and combining their use with group testing can lead to highly efficient screening protocols. However, these benefits come at the expense of a more complex data structure which can hinder surveillance efforts. To overcome this challenge, we develop a general Bayesian framework to estimate a mixed multivariate probit model with data arising from any group testing protocol that uses multiplex assays. In the formulation of this model, we account for the correlation between true disease statuses and heterogeneity across population subgroups, and we provide for automated variable selection through the adoption of spike and slab priors. To perform model fitting, we develop an attractive posterior sampling algorithm which is straightforward to implement. We illustrate our methodology through numerical studies and analyze chlamydia and gonorrhea group testing data collected by the State Hygienic Laboratory at the University of Iowa.
Keywords: generalized linear mixed model, latent variable model, multiplex assay, multivariate probit model, pooled testing
1. INTRODUCTION
The World Health Organization recently identified multiple health challenges for the next decade. These include outbreaks of common and novel diseases, a lack of access to health care, and the emergence of drug-resistant pathogens. In many instances, these challenges could be lessened with robust screening and surveillance programs that detect infected individuals and identify risk factors of disease. The primary barrier to such programs is usually the cost of implementation. One potential way to alleviate cost constraints is to amplify the use of group (pooled) testing. Group testing confers savings by testing pools of individual specimens, such as blood, urine, or swabs. Individuals in a pool that tests negatively are classified as such at the expense of a single assay, while positive pools are resolved through further testing; see Kim et al. (2007) for a review. Because of its potential to reduce cost, group testing has been adopted in many areas, including infectious disease testing (Krajden et al., 2014), animal health surveillance (Dhand et al., 2010), entomology (Speybroeck et al., 2012), and environmental monitoring (Heffernan et al., 2014).
Motivated by infectious disease testing practices at the State Hygienic Laboratory (SHL) at the University of Iowa, new group testing protocols using multiplex assays have been proposed recently (Hou et al., 2017, 2020; Tebbs et al., 2013; Bilder et al., 2019). Multiplex assays, unlike their single-disease predecessors, test for multiple diseases at once. Examples include the Procleix Ultrio Assay which tests for HIV, hepatitis B, and hepatitis C, the CDC Flu SC2 Multiplex Assay which tests for influenza A/B and SARS-CoV-2, and the Aptima Combo 2 Assay (AC2A) which tests for chlamydia and gonorrhea. The benefit of multiplex assays is their high-throughput potential which offers a more comprehensive assessment and a shorter turnaround time. Combining multiplex assays with group testing offers the opportunity to screen populations even more efficiently. For example, the SHL tests thousands of Iowa residents each year for chlamydia and gonorrhea using group testing and the AC2A. Annual savings are approximately 
600,000, a practically significant figure for a state-run public health laboratory.
Group testing data can have a complex structure, especially when pools are potentially misclassified. Various authors have considered estimating a regression function from parametric (Vansteelandt et al., 2000; Xie, 2001), semiparametric (Wang et al., 2014), nonparametric (Delaigle and Meister, 2011; Delaigle and Hall, 2012), and Bayesian (McMahan et al., 2017; Joyner et al., 2020; Liu et al., 2021) perspectives. However, this existing work is equipped to analyze group testing data from single-disease assays. Extending group testing estimation methods to the multiplex setting is more challenging. Initial contributions by Hughes-Oliver and Rosenberger (2000), Tebbs et al. (2013), and Warasi et al. (2016) developed prevalence estimators for multiple diseases. In a regression setting, only Zhang et al. (2013) and Lin et al. (2019) have proposed approaches to model multivariate group testing data. The former considers responses from initial pools only (ie, no retesting results are used) and the latter was designed only for the protocol in Tebbs et al. (2013). Neither work accounts for heterogeneity across population subgroups.
In many large-scale screening programs, individual specimens are collected at different clinic sites throughout a geographic region and are transported to a central location for testing. Given the inherent differences among areas in a region (eg, rural, urban, suburban, etc.) and the types of clinics providing the specimens (eg, primary care, community health, sexual health, etc.), it is natural to expect that heterogeneity will exist across various subpopulations. Accounting for this heterogeneity in group testing can be difficult, especially when pools are formed with specimens collected at different clinic sites. In fact, most existing regression methods for group testing data do not account for this type of variability. Chen et al. (2009) and Joyner et al. (2020) have considered incorporating heterogeneity through random effects, but neither work is applicable in the multiplex assay setting.
In this paper, we develop a general methodology to estimate a mixed probit model (Chib and Greenberg, 1998) for multivariate group testing data. We use fixed effects to describe population-level characteristics and random effects to account for heterogeneity across population subgroups. There are several enticing features of this work. First, our methodology is completely general, allowing one to analyze data arising from any group testing protocol that uses multiplex assays. Second, our use of a multivariate model acknowledges dependence that may exist between (or among) different diseases. Third, we cast the problem within a Bayesian framework and adopt spike and slab priors to facilitate variable selection for both fixed and random effects. Finally, we develop a Markov chain Monte Carlo (MCMC) algorithm that consists entirely of Gibbs steps with all but one involving sampling from common distributions. Acting in unison, these features make possible the regression analysis of multiplex group testing data while accounting for its highly complex structure.
Subsequent sections are organized as follows. Section 2 provides information on the mixed multivariate probit model, modeling assumptions, the observed data likelihood, and prior model elicitation. Section 3 provides an overview of the posterior sampling algorithm and data augmentation steps. Section 4 reports the results of simulation studies to assess the performance of our approach. Section 5 presents an analysis of chlamydia and gonorrhea group testing data collected by the SHL. Section 6 concludes with a discussion.
2. METHODOLOGY
Suppose 
individuals are tested for 
diseases simultaneously through a group testing protocol. We assume the protocol makes use of multiplex assays and the specimens (eg, blood, urine, swabs, etc.) are collected from individuals at 
clinics. A few initial comments are in order. First, because different clinics serve different populations, a substantial amount of heterogeneity may exist across clinic sites. Second, a group testing protocol could be performed “in-house” (ie, at a clinic site) or at a regional laboratory like the SHL. The former would involve pooling individuals within each site, while the latter would allow for pooling individuals across sites. Third, given the nature of most diseases tested by a multiplex assay, it is expected that true disease statuses within each individual are correlated. Our methodology accounts for all of these features among others.
Let 
if the 
th individual is truly positive for the 
th disease, 
otherwise, for 
and 
. We aggregate the true disease statuses for the 
th individual into the vector 
and define 
. Denote by 
and 
the 
and 
vectors of covariates corresponding to fixed and random effects, respectively, such that 
is a subvector of 
. We relate the individuals’ true disease statuses to their covariates through a mixed multivariate probit model (Chib and Greenberg, 1998). Under this model, the distribution of 
given the covariates and model parameters is
 |
(1) |
where 
, 
is a vector of regression coefficients for the 
th disease, 
, 
is a vector of random effects for 
th individual associated with the 
th disease, 
is the density of a 
-variate normal random vector with mean 
and correlation matrix 
, 
is the linear predictor, and
 |
, are regions of integration. Note that 
must be restricted to be a correlation matrix to ensure identifiability (Chib and Greenberg, 1998). To account for heterogeneity across clinic sites, we adopt the convention that 
if the 
th individual presents at the 
th clinic. We assume the 
’s are mutually independent 
random vectors.
The model specification in (1) leads to several challenges, for example, how to identify a subset of important predictors corresponding to the random effects and specifying the covariance structure. To overcome these difficulties, we reparameterize (1) using the proposal of Chen and Dunson (2003). Using a modified Cholesky decomposition, we write the covariance matrices of the random effects as 
, for 
, where 
is a 
diagonal matrix with nonnegative elements 
and 
is a 
lower triangular matrix with unit diagonal elements and free elements 
. Aggregating 
and 
, the reparameterized model is
 |
(2) |
where 
and 
, where 
is a standardized random effect for the 
th individual associated with the 
th disease. We specify 
if the 
th individual presents at the 
th clinic and assume 
.
The reparameterized model in (2) has several advantages. First, it is no longer necessary to posit prior models for the covariance matrices 
. Instead, 
is estimated through the elements of 
and 
. Second, by specifying spike and slab priors for the elements in 
, we develop an automated model selection strategy that identifies predictors with associated random effects. Note that by setting a diagonal element of 
equal to 0 results in the corresponding diagonal element of 
being set to 0, which drops the corresponding random effect from the model. Under (2), posterior inference would be available from existing work if the individual disease statuses 
were observed. However, because specimens are pooled in group testing and because all specimens (pooled and individual) are potentially misclassified, the 
’s are best regarded as latent.
The observed data from a group testing protocol consist of test results on a collection of pools, some of which may be of size one (ie, individual testing). Several protocols using multiplex assays have been proposed, including those referenced in Section 1. To develop a methodology that accommodates all protocols, we track pool membership via the index set 
, for 
, where 
if and only if the 
th individual is tested in the 
th pool. Therefore, the true status of the 
th pool for the 
th disease is 
; that is, the 
th pool is positive for the 
th disease if at least one of its members is positive for the 
th disease, and these statuses are aggregated into 
. The observed test result from assaying the 
th pool is 
, where 
if the 
th pool tests positively for the 
th disease, 
otherwise. We let 
and 
denote the sensitivity and specificity, respectively, of the multiplex assay used to test the 
th pool for the 
th disease.
Defining 
and 
to be pool-dependent allows for changes in these probabilities which may be attributed to different multiplex assays or other factors which could impact assay performance; for example, the size of the 
th pool, the specimen type, etc. We assume these probabilities do not vary within the strata created by cross-classifying these factors. For example, if the 
th and the 
th pool are of the same size, contain the same type of specimens, and are tested using the same assay, we assume 
and 
for 
. This notion is captured mathematically by defining index sets 
so that 
and 
for all 
, for 
. We regard 
and 
as unknown which are estimated alongside the other model parameters.
The conditional distribution of the observed testing outcomes 
given the covariates and the model parameters can be expressed as
 |
(3) |
where 
and 
aggregates all model parameters. Equation (3) is derived by making mild assumptions. First, testing outcomes for each disease are conditionally independent given the true pool statuses; that is, 
is independent of 
for 
, where 
, and the conditional distribution 
does not depend on the covariates. Second, individual disease statuses 
are conditionally independent given the covariates and the random effects. The first assumption is common in the group testing literature (Hou et al., 2017; McMahan et al., 2017), while the second is ubiquitous in the literature for mixed models (see, eg, Demidenko, 2013).
Our description of the model is completed by eliciting prior distributions. To facilitate variable selection, both in the fixed and random effects components, we use spike and slab priors for 
and 
, for 
. For the 
th disease, prior specifications for the fixed effects are
 |
whereas for the random effects,
 |
In the priors above, 
is the Dirac delta function, 
denotes the truncated normal distribution that restricts a normal distribution with mean 
and variance 
to the interval 
, and 
, 
, 
, 
, 
, and 
are hyperparameters. The remaining model parameters for the 
th disease are the free elements 
in the Cholesky decomposition matrix 
and the 
assay accuracy probabilities. Prior models for these are 
, 
, and 
, for 
, where 
, 
, 
, 
, 
, and 
are hyperparameters.
We use a Dirac delta function for the spike components, and slab distributions are chosen to be normal and truncated normal for fixed and random effects, respectively. Variance components of the slab distributions (
and 
) should be large to provide a diffuse proposal; see Wagner and Duller (2012). However, specifying 
and 
should be done informatively. Not doing so induces a strong a priori specification for the correlation between any 2 random effects for the 
th disease (Chen and Dunson, 2003). Finally, uninformative priors for the mixing probability hyperparameters and the assay accuracy probabilities can be specified by setting 
and 
, respectively. If historical information from assay validation studies is available, informative priors for the sensitivity and specificity parameters can be used; see Section 5.
The final parameter is the correlation matrix 
. We follow Zhang et al. (2006) and specify a joint prior for 
and an extra variance parameter matrix 
; that is,
 |
(4) |
where 
, 
is a scale matrix, and 
denotes the operator 
. It is straightforward to show 
follows a Wishart distribution with 
degrees of freedom and scale matrix 
; that is , 
.
3. DATA AUGMENTATION AND POSTERIOR SAMPLING
3.1. Data augmentation
Our goal is to estimate the multivariate probit model in (2) with the observed group testing responses in 
. However, working with the observed data model 
in (3) is prohibitive as it involves 
terms. We propose a 2-stage data augmentation strategy which leads to a convenient posterior sampling algorithm. The first stage introduces the individual disease statuses 
as latent random variables, producing the joint distribution
 |
The second stage introduces a latent random vector 
for each individual and defines 
, if 
, and 
otherwise, for 
. We regard 
to be mutually independent 
random vectors. This stage decomposes the multivariate probit model and leads to the joint conditional distribution
 |
(5) |
where 
and 
, where 
is the indicator function. Given the form of (5) and the priors elicited in Section 2, it is possible to derive closed-form full conditional distributions of the latent variables 
and 
and all model parameters except the correlation matrix 
.
3.2. Posterior sampling
Our sampling algorithm consists entirely of Gibbs steps with all but one involving sampling from common distributions. Web Appendix A in the Supplementary Material provides derivations of the following full conditional distributions and gives expressions for the parameters in these distributions. For the latent variables in Section 3.1,
 |
where 
is the vector of all disease statuses for the 
th individual excluding the 
th one and TMN denotes the truncated multivariate normal distribution. The full conditional for 
above reminds the reader why our methodology can be used for any group testing protocol. Different protocols will produce different sets of observed group testing responses in 
, but it suffices to keep track of the index sets 
defined in Section 2 and the Bernoulli mean 
does this; see Web Appendix A.
For the fixed effects, the full conditional distribution of 
is degenerate at 0 if 
, while the nonzero elements of 
, say 
, has the full conditional distribution 
, where 
, 
, 
, and 
. In addition, 
and 
, where 
denotes the vector 
with the 
entry removed. For the random effects,
 |
where 
is defined in Web Appendix A. The remaining conditionals are 
, 
, and 
.
To sample 
, we implement the parameter-extended Metropolis–Hastings (PX-MH) algorithm proposed by Zhang et al. (2006). This avoids having to acknowledge the inherent constraints placed on the form of 
by sampling it jointly with 
. Moreover, the algorithm leverages the fact that 
is a covariance matrix to design a proposal distribution from which it is easy to sample. The PX-MH algorithm is described below.
PX-MH ALGORITHM
Based on the current pair
, compute 
.Sample
from a Wishart
distribution.Compute
based on 
.- Generate
according to
The acceptance probability in Step 4 is
 |
where 
is the proposal density based on 
and 
is the joint posterior density of 
, which is proportional to 
. The density 
is the product of the Jacobian 
, where 
is the 
th diagonal element of 
, and the Wishart
density. The acceptance probability 
is controlled by selecting 
appropriately; larger values of 
increase this probability.
4. SIMULATION EVIDENCE
We performed various simulation experiments to examine the performance of our estimation and model selection methods. All experiments were designed to emulate the data application in Section 5. Our primary experiment considers 
individuals tested for 
diseases across 
distinct clinic sites (200 individuals per site). For each individual, we generated the covariate 
, where 
, 
, 
, and 
. We then set 
, where 
denotes the vector 
after being standardized, and generated the true individual disease status 
according to
 |
where 
, 
, 
, 
, 
, 
, where the elements of 
, 
, are shown in Table 1, and 
is a 
correlation matrix with off diagonal elements set to 0.6. These configurations provide an overall prevalence of about 13% and 6% for diseases 1 and 2, respectively. We repeated this process independently 500 times.
TABLE 1.
Simulation study. Average bias (Bias) of the posterior mean estimates, sample standard deviation (SSD) of the estimates, and average estimated posterior probability of inclusion (PI) for the associated fixed and random effects.
| Disease 1 | Disease 2 | ||||||
|---|---|---|---|---|---|---|---|
| Parameter | Bias | SSD | PI | Parameter | Bias | SSD | PI |
|
0.01 |
0.16 | 1.00 |
|
0.01 | 0.17 | 1.00 |
|
0.01 | 0.15 | 0.99 |
|
0.00 | 0.02 | 0.02 |
|
0.00 | 0.05 | 1.00 |
|
0.00 |
0.01 |
0.01 |
|
0.00 |
0.01 |
0.01 |
|
0.00 | 0.03 | 1.00 |
|
0.00 |
0.01 |
0.01 |
|
0.00 | 0.03 | 1.00 |
|
0.04 | 0.13 | 1.00 |
|
0.05 | 0.14 | 1.00 |
|
0.02 | 0.09 | 1.00 |
|
0.02 | 0.09 | 1.00 |
|
0.00 | 0.05 | 0.99 |
|
0.00 | 0.05 | 0.99 |
|
0.00 |
0.01 |
0.01 |
|
0.00 |
0.01 |
0.01 |
|
0.00 |
0.01 |
0.01 |
|
0.00 |
0.01 |
0.01 |
|
0.01 |
0.17 |
|
|
0.03 |
0.19 |
|
|
0.00 | 0.24 |
|
|
0.00 | 0.25 |
|
|
0.00 | 0.22 |
|
|
0.01 |
0.24 |
|
|
0.10 |
0.02 |
|
|
0.10 |
0.03 |
|
|
0.00 | 0.02 |
|
|
0.00 | 0.02 |
|
|
0.20 |
0.02 |
|
|
0.20 |
0.03 |
|
|
0.10 |
0.02 |
|
|
0.10 |
0.02 |
|
|
0.50 |
0.03 |
|
|
0.50 |
0.03 |
|
|
0.20 |
0.02 |
|
|
0.20 |
0.03 |
|
|
0.50 |
0.02 |
|
|
0.50 |
0.02 |
|
|
0.00 | 0.01 |
|
|
0.00 | 0.01 |
|
|
0.00 | 0.01 |
|
|
0.00 |
0.01 |
|
|
0.01 |
0.01 |
|
|
0.00 | 0.01 |
|
|
0.00 |
0.01 |
|
|
0.00 |
0.01 |
|
|
0.19 |
0.04 | |||||
Averaged posterior mean estimates of the elements of 
, 
the assay accuracy probabilities, and the correlation matrix element 
are also shown.
For each data set, we simulated the execution of the 2-stage Dorfman protocol used by the SHL and described in Tebbs et al. (2013). Under this protocol, each individual is randomly assigned to an initial pool of size 4. This method of assignment allows for pools to consist of individuals from different clinics. Each pool is tested for both diseases using a multiplex assay. If a pool tests positively for either disease (or both), then each individual is retested for both diseases using the same assay. Individuals in pools that test negatively for both diseases are diagnosed as negative. The testing result for the 
th pool is simulated as 
, where 
is the true status of the 
th pool. We consider 2 strata for the assay accuracy probabilities. The first stratum 
applies to initial pools, and the second stratum 
applies to individuals who are retested from the first stage. Based on the multiplex assay used at the SHL, we set 
, 
, 
, and 
, for 
. However, when we estimated the model for each of the 500 group testing data sets, these quantities were treated as unknown and were assigned uniform priors.
In the spike and slab distributions, we set 
in the slab components to provide diffuse prior information, and we used uniform priors for all mixing weights; that is, 
. The latter specification mandates that no prior information is used in model selection for the fixed and random effects. Following Chen and Dunson (2003), we set 
, 
, 
, to avoid specifying a strong prior correlation between any 2 random effects, and we set 
and 
, where 
is a 
identity matrix, to provide a diffuse prior in (4). We used our sampling algorithm from Section 3.2 to draw 100 000 MCMC iterates and retained every 10th iterate after discarding the first 50 000. We set the degrees of freedom in the PX-MH algorithm to be 
, which led to acceptance rates between 20% and 40%. Standard MCMC diagnostics were used to ensure convergence and point estimates were recorded as sample means of the posterior draws.
Table 1 summarizes the results. Of primary interest are the fixed and random effects parameters 
and 
. For these, the average bias across the 500 group testing data sets is close to 0, and sample standard deviations are small relative to the true values. The results also show our methods reliably identify fixed and random effects. This can be seen from the posterior probabilities of inclusion, which are unity for nearly all nonzero effects and are close to 0 when the effects are vacuous. For the remaining parameters, assay accuracy probabilities 
and 
are estimated nearly perfectly despite the fact that uniform priors were used, and the nuisance parameters 
, 
, that is, those associated with nonzero random effects, are estimated with little or no bias. Inflated bias in the 
parameters, for 
, is expected because these are associated with null random effects; that is, 
. As shown in Web Appendix A in the Supplementary Material, if 
, then 
is sampled from its zero-mean prior distribution for nearly all iterations. The correlation 
, perhaps also best regarded as a nuisance parameter, is negatively biased.
We performed 4 additional simulation studies that complement the findings of our primary experiment. These studies and their conclusions are summarized below with complete details given in Web Appendix B in the Supplementary Material. First, to demonstrate our methods can be applied with other group testing protocols, we examined estimation and model selection when a non-adaptive, single-stage protocol was used. We observed nearly identical results to those in Table 1. Second, we compared our proposed modeling methods to the marginal (single-disease) modeling approach in Joyner et al. (2020). As expected, our multivariate approach outperforms single-disease methods in terms of estimation efficiency. Third, we examined the possible benefit of constructing pools homogeneously in terms of their covariates and site effects (instead of pooling individuals at random). On the basis of bias, precision, and model selection, we observed no noticeable benefit of doing so when positive pools were resolved using Dorfman’s 2-stage protocol. Fourth, we performed 2 robustness studies to examine the impact of model misspecification in the linear predictor and the link function. When model violations are present, not surprisingly, our approach can provide estimates which are biased. However, even under severe misspecification, our approach continues to reliably identify nonzero fixed and random effects.
5. IOWA DATA ANALYSIS
Even at the height of the COVID-19 pandemic, the United States Centers for Disease Control and Prevention reported approximately 2.3 million new cases of chlamydia and gonorrhea in 2020 (Centers for Disease Control and Prevention, 2020), making these 2 of the most common sexually transmitted diseases (STDs). Coinfection can be common, and both diseases are associated with the same symptoms, including painful urination and chronic pelvic pain (Creighton et al., 2003; Workowski, 2013). At the same time, a large percentage of infected individuals are asymptomatic which makes screening critical (Low, 2007). Both diseases can be cured with antibiotics; however, treatment is becoming challenging as some antibiotics are now failing as a result of overuse. Given the high prevalence of both diseases, their possible long-term complications, and the looming threat of antibiotic resistance, chlamydia and gonorrhea continue to pose a serious threat to public health.
In the United States, many state-run public health laboratories have enacted screening programs which regularly test for chlamydia and gonorrhea. In Iowa, the SHL has tested thousands of residents each year dating back to the creation of the Infertility Prevention Project in 1988. Urine and swab specimens are sent to the laboratory daily from different locations throughout the state and from different types of clinics (eg, family planning clinics, STD clinics, etc.). Due to their higher prevalence, male specimens are tested individually, whereas most female specimens are tested by using the 2-stage Dorfman group testing protocol described in Section 4. The SHL uses the AC2A, which is manufactured by Hologic, Inc., to test pooled and individual specimens for both diseases simultaneously. In our analysis, we seek to identify risk factors associated with chlamydia and gonorrhea for female subjects tested at the SHL.
The data provided by our collaborators consist of testing results from female subjects in 2014. There are 4316 individual urine specimens, 416 individual cervical swab specimens, and 2286 cervical swab pool specimens (1 pool of size 2, 12 pools of size 3, and 2273 pools of size 4), as well as the additional individual test results required to resolve swab pools which test positively. These specimens represent a total of 
individuals from 
clinics. In addition to the test results, several individual-level covariates were recorded, including age (in years, denoted by 
), a race indicator (
if Caucasian, 
otherwise), an indicator denoting whether the subject reported a new sexual partner in the last 90 days (
if yes), an indicator of whether the subject reported having multiple sexual partners in the last 90 days (
if yes), an indicator of whether the subject reported sexual contact with an STD-infected partner in the previous year (
if yes), and an indicator of whether the subject presented at a clinic with symptoms (
if yes). We relate the individual disease statuses to these covariates through the mixed probit model
 |
where 
and 
. In the linear predictor, we set 
, where 
denotes the vector of covariates 
after being standardized. Standardization was used so the spike and slab distributions would have the same impact on the regression coefficients across all covariates. For each of the 64 clinics, a random effect vector 
is conceptualized for each disease, with the convention that 
if the 
th individual was seen at the 
th clinic site.
In our analysis, we used the same prior models as in Section 4 except for the assay accuracy probabilities, which we model informatively. We conceptualize 3 strata for each disease: 
and 
for swab specimens tested individually, 
and 
for urine specimens tested individually, and 
and 
for swab specimens tested in pools. To set informative priors for these 12 parameters, we used results from AC2A validation studies, which were published in the Hologic product literature and reported in Gaydos et al. (2003). Web Appendix C in the Supplementary Material reproduces these results and describes prior model construction. To estimate the model above, we used our posterior sampling algorithm to draw 100 000 MCMC iterates, retaining every 10th iterate after discarding the first half. We again used 
as the proposal degrees of freedom in the PX-MH algorithm.
Tables 2 and 3 summarize the results of our analysis for chlamydia and gonorrhea, respectively, which include posterior means and standard deviations and posterior probabilities of inclusion for the fixed and random effects. The direction of the estimates of the fixed effects are consistent with known epidemiological patterns of both diseases (US Preventive Services Task Force, 2021). In particular, the risk of chlamydia tends to decrease overall with age and Caucasian females are associated with a lower risk. Having contact with a sexual partner recently diagnosed with a STD is clearly associated with increased risk for both diseases. Our analysis also identifies a random intercept parameter for both diseases and a random effect for new sexual partner associated with chlamydia, indicating evidence of heterogeneity across clinics sites. Finally, the posterior mean and standard deviation of the correlation 
is 0.46 and 0.04, respectively. This strongly supports using a joint model for these data.
TABLE 2.
Iowa data application. Fixed and random effects results for chlamydia.
| Parameter | Description | Estimate | ESD | PI |
|---|---|---|---|---|
|
Intercept |
1.46 |
0.03 | 1.00 |
|
Age |
0.23 |
0.02 | 1.00 |
|
Race |
0.04 |
0.03 | 0.66 |
|
New partner | 0.02 | 0.03 | 0.29 |
|
Multiple partners | 0.03 | 0.03 | 0.44 |
|
Contact with STD | 0.15 | 0.01 | 1.00 |
|
Symptoms | 0.00 | 0.02 | 0.09 |
|
Intercept | 0.16 | 0.03 | 1.00 |
|
Age | 0.00 | 0.01 | 0.01 |
|
Race | 0.00 |
0.01 |
0.01 |
|
New partner | 0.06 | 0.05 | 0.70 |
|
Multiple partners | 0.00 | 0.01 | 0.07 |
|
Contact with STD | 0.00 |
0.01 |
0.01 |
|
Symptoms | 0.00 |
0.01 |
0.01 |
|
Swab individual | 0.98 |
0.01 |
|
|
Urine individual | 0.99 |
0.01 |
|
|
Swab pool | 0.99 |
0.01 |
|
|
Swab individual | 0.98 |
0.01 |
|
|
Urine individual | 0.99 |
0.01 |
|
|
Swab pool | 0.99 |
0.01 |
|
The posterior mean estimate, the estimated posterior standard deviation (ESD), and the posterior probability of inclusion (PI) are shown.
TABLE 3.
Iowa data application. Fixed and random effects results for gonorrhea.
| Parameter | Description | Estimate | ESD | PI |
|---|---|---|---|---|
|
Intercept |
2.55 |
0.08 | 1.00 |
|
Age | 0.00 |
0.01 |
0.01 |
|
Race |
0.06 |
0.06 | 0.54 |
|
New partner | 0.00 | 0.01 | 0.01 |
|
Multiple partners | 0.00 | 0.01 | 0.02 |
|
Contact with STD | 0.18 | 0.02 | 1.00 |
|
Symptoms | 0.00 | 0.01 | 0.01 |
|
Intercept | 0.35 | 0.07 | 1.00 |
|
Age | 0.01 | 0.02 | 0.07 |
|
Race | 0.04 | 0.07 | 0.25 |
|
New partner | 0.00 |
0.01 |
0.01 |
|
Multiple partners | 0.00 | 0.02 | 0.03 |
|
Contact with STD | 0.00 | 0.01 | 0.01 |
|
Symptoms | 0.00 |
0.01 |
0.01 |
|
Swab individual | 1.00 |
0.01 |
|
|
Urine individual | 1.00 |
0.01 |
|
|
Swab pool | 1.00 |
0.01 |
|
|
Swab individual | 1.00 |
0.01 |
|
|
Urine individual | 1.00 |
0.01 |
|
|
Swab pool | 1.00 |
0.01 |
|
The posterior mean estimate, the estimated posterior standard deviation (ESD), and the posterior probability of inclusion (PI) are shown.
We performed a sensitivity analysis to assess the impact of assigning informative priors for the assay accuracy probabilities (all other model parameters were assigned flat or diffuse priors). Fixed and random effects estimation, as well as model selection, were largely unchanged for both diseases when we used uniform (0,1) priors for these probabilities, but we did observe minor changes in the estimates for 
and 
, the AC2A sensitivity associated with urine specimens for the 2 diseases; see Web Appendix C. We also formulated a simulation-based procedure to detect overall lack of fit when estimating the probit model with group testing data having the same structure as the Iowa data. This procedure, also described in Web Appendix C, does not indicate evidence of lack of fit in our analysis.
6. DISCUSSION
We have formulated a Bayesian approach to model the relationship between multiple disease statuses and covariates with group testing data from multiplex assays. We estimate population-level characteristics and incorporate heterogeneity across population subgroups while identifying significant effects of each. Although we have focused on the multivariate probit model, other parametric approaches may be adaptable to group testing outcomes, including logistic regression (Glonek and McCullagh, 1995). At the same time, probit models enjoy advantages such as marginal interpretation and are amendable to augmentation strategies that lead to efficient posterior sampling. Motivated by ecological applications, Chakraborty et al. (2024) recently investigated the multivariate probit model for high-dimensional binary responses. Future work could generalize their methods for group testing responses from multiplex assays. For example, Koehler et al. (2018) report the development of a multiplex assay that tests for 164 different viruses, bacteria, and parasites simultaneously. Given technological advances in modern assay development, it may soon become commonplace to test specimens for a very large number of diseases at once.
Although our paper offers a comprehensive approach to analyze multiplex data from any group testing protocol, our methodology is parametric in nature, and, as with any Bayesian analysis, one should be concerned about assessing model fit. This is a nonstandard problem with group testing data because the true individual disease statuses are not observed. Thus, the common practice of comparing posterior model-based predictions to these statuses is not available when individuals are pooled and all specimen diagnoses are potentially error-laden. Our strategy in Web Appendix C in the Supplementary Material, referenced in Section 5, was formulated to assess global goodness-of-fit, namely, whether there is an overall departure in the diagnosed statuses and those which would be expected from the probit model fit. On the other hand, anonymous reviewers have suggested it would be of interest to diagnose specific model departures, such as a violation of linearity in the fixed and/or random effects, the choice of link function, or normality assumptions for random effects. All 3 of these assumptions are potentially testable by nesting our parametric model within various nonparametric extensions and testing point null hypotheses via the Savage–Dickey ratio (Verdinelli and Wasserman, 1995), an approach espoused by various authors using Polya trees (Hanson, 2006; Jara et al., 2009) and transformed Bernstein polynomials (Zhou and Hanson, 2018). Adapting this approach for group testing data would require new methodological development
both for single and multiple diseases.
Supplementary Material
Web Appendices and code referenced in Sections 3–6 are available with this paper at the Biometics website on Oxford Academic. R programs for data analysis are available at https://github.com/mcmaha2/probit_gt.
ACKNOWLEDGMENTS
We are grateful to the Editor, Associate Editor, and two referees for their comments on earlier versions of this article. We thank Jeffrey Benfer and Kristofer Eveland at the State Hygienic Laboratory for their continued collaboration.
Contributor Information
Christopher S McMahan, School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC 29634, United States.
Chase N Joyner, School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC 29634, United States.
Joshua M Tebbs, Department of Statistics, University of South Carolina, Columbia, SC 29208, United States.
Christopher R Bilder, Department of Statistics, University of Nebraska-Lincoln, Lincoln, NE 68583, United States.
FUNDING
This research was funded by the National Institutes of Health grant R01 AI121351.
CONFLICT OF INTEREST
None declared.
DATA AVAILABILITY
The Iowa data used in this paper are not available for public use. Our GitHub website contains a simulated data set with a similar structure which can be analyzed using the R programs on this site. These R programs and the simulated data are available at https://github.com/mcmaha2/probit_gt.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Supplementary Materials
Web Appendices and code referenced in Sections 3–6 are available with this paper at the Biometics website on Oxford Academic. R programs for data analysis are available at https://github.com/mcmaha2/probit_gt.
Data Availability Statement
The Iowa data used in this paper are not available for public use. Our GitHub website contains a simulated data set with a similar structure which can be analyzed using the R programs on this site. These R programs and the simulated data are available at https://github.com/mcmaha2/probit_gt.