From mixed effects modeling to spike and slab variable selection: A Bayesian regression model for group testing data

Summary:

Due to reductions in both time and cost, group testing is a popular alternative to individual-level testing for disease screening. These reductions are obtained by testing pooled biospecimens (e.g., blood, urine, swabs, etc.) for the presence of an infectious agent. However, these reductions come at the expense of data complexity, making the task of conducting disease surveillance more tenuous when compared to using individual-level data. This is because an individual’s disease status may be obscured by a group testing protocol and the effect of imperfect testing. Furthermore, unlike individual-level testing, a given participant could be involved in multiple testing outcomes and/or may never be tested individually. To circumvent these complexities and to incorporate all available information, we propose a Bayesian generalized linear mixed model that accommodates data arising from any group testing protocol, estimates unknown assay accuracy probabilities, and accounts for potential heterogeneity in the covariate effects across population subgroups (e.g., clinic sites, etc.); this latter feature is of key interest to practitioners tasked with conducting disease surveillance. To achieve model selection, our proposal uses spike and slab priors for both fixed and random effects. The methodology is illustrated through numerical studies and is applied to chlamydia surveillance data collected in Iowa.

1. Introduction

Group testing involves taking specimens (e.g., blood, urine, swabs, etc.) from different individuals and forming a pooled specimen that is then tested for disease. In most group testing protocols, if a pooled specimen tests negatively, then all individuals are declared to be disease free at the expense of a single diagnostic test. In contrast, if a pooled specimen tests positively, the pool is resolved algorithmically to determine which individuals are positive. Dorfman (1943) is credited with conceptualizing the group testing idea during World War II to screen military recruits for syphilis. Since then, group testing, or “pooling,” has become a mainstream approach to screen large populations for multiple diseases. The primary reason for pooling is to save money. For example, the State Hygienic Laboratory (SHL) at the University of Iowa has reported savings of approximately $3.1 million during a recent 5-year period after adopting a variant of Dorfman’s protocol to screen Iowa residents for chlamydia and gonorrhea; see Tebbs et al. (2013) and McMahan et al. (2017). Pooling biospecimens through group testing arises in other applications, including testing for HIV and HCV (Sarov et al., 2007; Krajden et al., 2014), environmental testing (Heffernan et al., 2014), and drug discovery (Hughes-Oliver, 2006).

While testing pools can be far more cost effective than performing individual tests, it also leads to a more complicated data structure. This is true because specimens are pooled and hence individual-level responses may never be observed. Recent statistical research has focused on developing regression methods to model the probability of disease for individuals based on pooled outcomes; e.g., see Vansteelandt et al. (2000), Bilder and Tebbs (2009), Huang (2009), Delaigle and Meister (2011), and Delaigle et al. (2014). Most of these methods are designed to analyze test results arising from assaying only the initially formed (master) pools; i.e., those formed by assigning each individual to exactly one initial pool for testing. As a consequence, these methods cannot incorporate retesting information that becomes available when positive pools are resolved (Kim et al., 2007) or when quality control steps are implemented (Gastwirth and Johnson, 1994; Johnson and Gastwirth, 2000). To incorporate retesting information, Xie (2001) developed an expectation-maximization algorithm to estimate the individual-level probability of disease for general regression models. Wang et al. (2014) developed a semiparametric framework to estimate single-index models. Most recently, McMahan et al. (2017) proposed a Bayesian approach to estimate generalized linear models while incorporating historical information on disease prevalence and uncertainty in assay performance.

At most public health laboratories like the SHL, individual specimens arrive at the lab from different locations throughout a particular geographic region. For example, in Iowa, specimens are collected at different types of clinics (e.g., STD clinics, family planning clinics, etc.) in multiple locations from all over the state and are then shipped to the SHL for testing. Given the vast differences among clinic types and the additional differences between rural and metropolitan areas, it is natural to suspect that heterogeneity may exist from location to location. However, when individual specimens are pooled together, it becomes a significant challenge to account for this source of variability while also estimating covariate effects like age, gender, race, and sexual history. In fact, most previous regression methods for group testing data, such as those referenced above, are not able to incorporate the effects due to observing data from different locations-especially when individual specimens from different locations are pooled together.

In this article, we develop a Bayesian generalized linear mixed model approach for group testing data which uses fixed effects to describe the population-level mean structure and random effects to account for differential variability among population subgroups. Our work generalizes the random effects modeling techniques for group testing data proposed by Chen et al. (2009) and simultaneously offers a far more flexible approach for data analysis. First, by taking a Bayesian point of view, we can incorporate historical information about disease prevalence, and our approach allows assay accuracy probabilities to be estimated from the observed data. Second, a limitation of Chen et al. (2009) is that it can incorporate only master pool responses; i.e., it does not allow one to include additional retests that will be performed for disease classification purposes. On the other hand, our estimation framework is flexible and, as in McMahan et al. (2017), it can accommodate data from any group testing protocol as well as back-end quality control procedures. Third, and perhaps most limiting, the methods in Chen et al. (2009) require pools to consist of individuals from within the same location. In practice, this can be markedly prohibitive because individual specimens are often pooled sequentially based on their arrival date for testing. Furthermore, for those locations performing a small number of tests, it may be impractical to wait and to pool within location. Our approach removes this limitation and includes “pooling within location” as a special case. Finally, given the complexity of the considered mixed effects model, we use spike and slab priors to perform variable selection-both within the fixed and random effects components. Three of the most common spike and slab priors are considered and details of implementation are provided under each. No existing group testing regression procedure has considered such an automated variable selection technique; i.e., for both fixed and random effects. We develop a computationally efficient Markov chain Monte Carlo (MCMC) sampling algorithm which can estimate the proposed model.

Subsequent sections of this article are organized as follows. Section 2 provides preliminary information regarding the proposed mixed effects model, the modeling assumptions, and the derivation of the observed data likelihood. Section 3 presents the specifics of the approach, including prior model specifications and data augmentation steps used to construct an efficient posterior sampling algorithm. Section 4 outlines the development of the full conditional distributions. Section 5 reports the results of a simulation study conducted to assess the performance of the proposed approach. Section 6 presents an analysis of chlamydia testing data collected by the SHL in Iowa. Section 7 concludes with a summary discussion. Additional details and simulation results are provided in the Supporting Information.

2. Notation and Preliminaries

Consider a setting in which N individuals are screened for an infectious agent by a group testing protocol. As a part of this process, each of the N individuals visit one of K distinct clinics where a specimen (e.g., blood, urine, swab, etc.) is collected. Testing is then performed either at the clinic site or at a regional laboratory; e.g., the SHL in Iowa. Note the former scenario would mandate pooling of individuals within clinic sites while the latter allows for pooling across sites, with our methodology being applicable in either case. Let ${\tilde{Y}}_i$ denote the true infection status of the ith individual, for i = 1, …, N, with ${\tilde{Y}}_i=1$ indicating the individual is truly positive and ${\tilde{Y}}_i=0$ otherwise. Furthermore, let $x_i={\left(1,x_{i1},\dots ,x_{i,q_1-1}\right)}^{\prime }$ and $t_i={\left(1,t_{i1},\dots ,t_{i,q_2-1}\right)}^{\prime }$ denote vectors of covariate values taken on the ith individual which correspond to fixed and random effects, respectively, where t_i is assumed to be a subvector of x_i. We assume throughout that individuals’ infection statuses are conditionally independent given the covariate information and the random effects. The individuals’ true infection statuses (i.e., the ${\tilde{Y}}_i$’s) are not observed due to the effect of imperfect testing, while the covariates for each individual are observed. For ease of exposition, we aggregate the individuals’ infection statuses as $\tilde{Y}={\left({\tilde{Y}}_1,\dots ,{\tilde{Y}}_N\right)}^{\prime }$ and denote X = (x₁ ⋯ x_N)′ and T = (t₁ ⋯ t_N)′ as the design matrices.

The goal of this work is to relate the individuals’ latent infection statuses to their covariate values through the following generalized linear mixed model

$$g^{-1}\left\{P\left({\tilde{Y}}_i=1\mid \beta ,{\gamma }_{k(i)}\right)\right\}=x_i^{\prime }\beta +t_i^{\prime }{\gamma }_{k(i)},$$ (1)

where g⁻¹(·) is a known link function, β is a q₁-dimensional vector of fixed effects, γ_k(i) ≔ γ_k if the ith individual presented at the kth clinic, and γ_k is a q₂-dimensional vector of clinic-specific random effects, for k = 1, …, K. It is assumed that the γ_k’s are independent and identically distributed and follow a mean zero multivariate Gaussian distribution with covariance matrix D; i.e., ${\gamma }_k\stackrel{\mathit{\text{iid}}}{~}N(0,D)$.

A typical challenge that arises in mixed modeling involves the selection of both the fixed and random effects components, which is tantamount to selecting the proper subsets of the available covariates to be retained in the final model. To accomplish this task, we adopt spike and slab priors (George and McCulloch, 1993, 1997; Kuo and Mallick, 1998). These specifications proceed as usual for the fixed effects and follow the proposal of Chen and Dunson (2003) for the random effects, which requires a reparameterization of the proposed model. The reparameterized model is

$$g^{-1}\left\{P\left({\tilde{Y}}_i=1\mid \beta ,\lambda ,a,b_{k(i)}\right)\right\}=x_i^{\prime }\beta +t_i^{\prime }\Lambda A{b}_{k(i)},$$ (2)

where b_k(i) ≔ b_k if the ith individual presented at the kth clinic, $b_k\stackrel{iid}{~}N(0,I)$, I is the identity matrix, Λ is a q₂ × q₂ diagonal matrix with non-negative diagonal entries, and A is a q₂ × q₂ lower triangular matrix with unit main diagonal elements and free elements given by a_ml, for l = 1, …, q₂ − 1; m = l + 1, …, q₂. For ease of exposition, we introduce $\lambda ={\left({\lambda }_1,\dots ,{\lambda }_{q_2}\right)}^{\prime }$ such that Λ = diag(λ) and a which denotes the vector of free elements of the matrix A; i.e., a = (a_ml : l = 1, …, q₂ − 1; m = l + 1, …, q₂)′. Note that the matrices Λ and A are obtained via a modified Cholesky decomposition and satisfy D = ΛAA′Λ. Under this reparameterization, if λ_l (the lth diagonal element of Λ) is zero, then so is the lth diagonal element of D. That is, if λ_l = 0, then the variance of the lth random effect is zero, which is equivalent to dropping the lth random effect from the model. Thus, to perform variable selection for the random effects, the proposed methodology places spike and slab priors on each λ_l. Our approach also models the a_ml values, which allows for the estimation of D without imposing any prior form or structure.

The observed data that arise from implementing a group testing protocol can be quite complex. First of all, there are many protocols available for use (e.g., see Dorfman, 1943; Phatarfod and Sudbury, 1994; Kim et al., 2007; Kim and Hudgens, 2009). Second, a given protocol often requires that individuals be tested in multiple (possibly overlapping) pools and may even mandate confirmatory testing for quality control purposes (Gastwirth and Johnson, 1994; Johnson and Gastwirth, 2000). Thus, to provide a general framework which can incorporate data from any group testing protocol, we define the index set ${\mathcal{P}}_j\subset \left\{1,\dots ,N\right\}$ which identifies the individuals contributing to the jth pool, for j = 1, …, J. Let ${\tilde{Z}}_j$ denote the true status of the jth pool, under the convention the pool is positive $\left({\tilde{Z}}_j=1\right)$ if it contains at least one infected individual and negative otherwise $\left({\tilde{Z}}_j=0\right)$; i.e., ${\tilde{Z}}_j=I\left({\sum }_{i\in {\mathcal{P}}_j}{\tilde{Y}}_i>0\right)$. Like the individuals’ true statuses, the ${\tilde{Z}}_j$’s are also unobserved due to the effect of imperfect testing. Instead, we observe the testing response Z_j which can be viewed as an error-contaminated version of ${\tilde{Z}}_j$, with Z_j = 1 indicating the jth pool tested positively and Z_j = 0 otherwise. To quantify the effect of imperfect testing, let $S_{ej}=P\left(Z_j=1\mid {\tilde{Z}}_j=1\right)$ and $S_{pj}=P\left(Z_j=0\mid {\tilde{Z}}_j=0\right)$ denote the sensitivity and specificity, respectively, of the assay used to test the jth pool. We allow S_ej and S_pj to be pool specific, thus allowing for the potential use of different types of assays and/or the potential effect that pool size (i.e., the cardinality of ${\mathcal{P}}_j$) may have on an assay’s performance.

To relate the individual-level model in (2) to the observed testing responses Z = (Z₁, …, Z_J)′, it is assumed that the responses in Z are conditionally independent given the true statuses $\tilde{Z}={\left({\tilde{Z}}_1,\dots ,{\tilde{Z}}_J\right)}^{\prime }$ and that the conditional distribution $Z\mid \tilde{Z}$ does not depend on the covariates. Under these assumptions, the conditional distribution of Z can be written as

$$\pi (Z\mid \beta ,\lambda ,a,b)=\sum _{\tilde{Y}\in {\{0,1\}}^N}\left[\prod _{j=1}^J{\left\{S_{ej}^{Z_j}{\left(1-S_{ej}\right)}^{1-Z_j}\right\}}^{{\tilde{Z}}_j}{\left\{{\left(1-S_{pj}\right)}^{Z_j}{S}_{pj}^{1-Z_j}\right\}}^{1-{\tilde{Z}}_j}\times \prod _{i=1}^{N}g{\left({\eta }_i\right)}^{{\tilde{Y}}_i}{\left\{1-g\left({\eta }_i\right)\right\}}^{1-{\tilde{Y}}_i}\right],$$ (3)

where ${\eta }_i={\text{x}}_i^{\prime }\beta +t_i^{\prime }\Lambda A{b}_{k(i)}$ and b = (b₁, …, b_K)′. Note that in (3) we are marginalizing the joint conditional distribution of the observed testing responses and the latent statuses of the individuals, denoted by $\pi (Z,\tilde{Y}\mid \beta ,\lambda ,a,b)$, over $\tilde{Y}$, that is, $\pi (Z\mid \beta ,\lambda ,a,b)={\sum }_{\tilde{Y}\in {\{0,1\}}^N}\pi (Z,\tilde{Y}\mid \beta ,\lambda ,a,b)$. As such, (3) involves a very high dimensional sum, which effectively renders direct numerical evaluation to be infeasible. To circumvent this issue, a two-stage data augmentation procedure in Section 3.2 is proposed which leads to an efficient posterior sampling algorithm.

3. Data Augmentation and Prior Specification

The full hierarchy of the proposed model is

$${\tilde{Y}}_i\mid {\eta }_i~\text{Bernoulli}\left\{g\left({\eta }_i\right)\right\},{\eta }_i=x_i^{\prime }\beta +t_i^{\prime }\Lambda A{b}_{k(i)}$$

$${\beta }_q\mid v_q~\left(1-v_q\right){\pi }_{\text{spike}}\left({\beta }_q\right)+v_q{\pi }_{\text{slab}}\left({\beta }_q\right),q=1,\dots ,q_1$$

$${\lambda }_l\mid w_l~\left(1-w_l\right){\pi }_{\text{spike}}\left({\lambda }_l\right)+w_l{\pi }_{\text{slab}}\left({\lambda }_l\right),l=1,\dots ,q_2$$

$$a~N\left(m_0,C_0\right),$$

$$b_k~N(0,I),k=1,\dots ,K$$

$$v_q\mid {\tau }_{v_q}~\text{Bernoulli}\left({\tau }_{v_q}\right),q=1,\dots ,q_1$$

$$w_l\mid {\tau }_{w_l}~\text{Bernoulli}\left({\tau }_{w_l}\right),l=1,\dots ,q_2$$

$${\tau }_{v_q}~\text{Beta}\left(a_v,b_v\right),q=1,\dots ,q_1$$

$${\tau }_{w_l}~\text{Beta}\left(a_w,b_w\right),l=1,\dots ,q_2,$$

where π_spike(·) and π_slab(·) denote the “spike” and “slab” components, respectively, of our prior distributions (for further details, see Section 3.1) and m₀, C₀, a_v, a_w, b_v, and b_w are hyperparameters. In specifying these hyperparameters, the prior on a should be made to be informative (e.g., specified with m₀ = 0 and C₀ = 0.5I) to avoid imposing a strong a priori correlation between any two random effects; see Chen and Dunson (2003). We also assume the a priori independence of the β_q’s and the λ_l’s. Doing so greatly simplifies the calculations necessary for posterior sampling and is common in the literature; e.g., see George and McCulloch (1993, 1997), Kuo and Mallick (1998), and Chen and Dunson (2003).

3.1. Spike and Slab Priors

The model hierarchy above provides a general representation of the spike and slab prior. To ground the description of our approach and to illustrate our methodology, we discuss three commonly used spike and slab priors: the stochastic search variable selection (SSVS), the normal mixture inverse gamma (NMIG), and the Dirac spike; see George and McCulloch (1993), George and McCulloch (1997), and Kuo and Mallick (1998), respectively.

The SSVS approach used herein makes use of spike and slab priors of the following form:

$${\beta }_q\mid v_q~N\left(0,r\left(v_q\right){\varphi }_q^2\right)$$ (4)

$${\lambda }_l\mid w_l~TN\left(0,r\left(w_l\right){\psi }_l^2,(0,\infty )\right),$$ (5)

where r(·) is a function serving as a binary switch (i.e., r(0) = r and r(1) = 1) that transitions the prior between the spike and the slab, ${\varphi }_q^2$ and ${\psi }_l^2$ are specified variance components, and TN(μ, ψ², (a, b)) denotes the truncated normal distribution which arises from restricting the support of a N(μ, ψ²) distribution to the interval (a, b). In (4) and (5), one should specify large values of ${\varphi }_q^2$ and ${\psi }_l^2$ and a small value for r. In particular, these specifications should be made such that r⁻¹ is sufficiently larger than the variance components, that is, $r^{-1}\gg {\varphi }_q^2$ and $r^{-1}\gg {\psi }_l^2$; for further discussion, see Wagner and Duller (2012). Proceeding in this fashion leads to a flat slab and a spike that is concentrated around zero. It is important to note that specifying appropriate values of the variance components can be challenging and moreover has the potential to greatly influence the analysis.

To avoid specifying the variance components, one could instead use the NMIG prior outlined in George and McCulloch (1997) and Ishwaran and Rao (2003). This approach proceeds identically to that of SSVS with the exception that the variance components are viewed as unknown and an inverse gamma prior is specified for them. That is, ${\varphi }_q^2~\text{Inv-Gamma}\left(a_{\varphi },b_{\varphi }\right)$ and ${\psi }_l^2~\text{Inv-Gamma}\left(a_{\psi },b_{\psi }\right)$. This addition to the hierarchy removes the need to specify these parameters and allows one to estimate them through data driven means. One still must specify the value of r; i.e., the proportional difference between the variance components of the spike and slab densities. Experience suggests the selection of r tends to impact the spike distribution far more than the slab, with the model selection process being too liberal when r is chosen too large and vice versa.

To avoid specifying r, a Dirac delta function could be used for the spike; see Kuo and Mallick (1998) and Wagner and Duller (2012). This can be viewed as a limiting case of SSVS where the variance of the continuous spike distribution is driven to zero. In this situation, v_q = 0 if and only if β_q = 0 and similarly for w_l and λ_l. Although this seems favorable, it also introduces an absorbing state in the Markov chain. To handle this issue, rather than sampling the binary variables from their full conditional distributions, β is integrated out when updating $v={\left(v_1,\dots ,v_{q_1}\right)}^{\prime }$ and λ is integrated out when updating $w={\left(w_1,\dots ,w_{q_2}\right)}^{\prime }$. Thus, to develop a computationally efficient posterior sampling algorithm, one must be able to analytically marginalize the posterior distribution over both β and λ. The ability to do so is inherently tied to the link function being used. Fortunately, this can be accomplished under both the probit and logistic link functions after a series of data augmentation steps; this process is outlined in Section 3.2. To complete the specification and to closely mimic the slab priors in SSVS and NMIG, we take a priori the β_q’s to be independent with slab component $N\left(0,{\varphi }_q^2\right)$ and the λ_l’s to be independent with slab component $TN\left(0,{\psi }_l^2,(0,\infty )\right)$, where ${\varphi }_q^2$ and ${\psi }_l^2$ are again specified to be large.

3.2. Data Augmentation

A two-stage data augmentation procedure is proposed which focuses on implementation under both the probit and logistic link functions. In the first stage, we introduce the individuals’ true statuses $\tilde{Y}$ as latent random variables and consider the joint conditional distribution of the observed testing responses and the latent statuses of the individuals, which is

$$\pi (Z,\tilde{Y}\mid \beta ,\lambda ,a,b)=\prod _{j=1}^J{\left\{S_{ej}^{Z_j}{\left(1-S_{ej}\right)}^{1-Z_j}\right\}}^{{\tilde{Z}}_j}{\left\{{\left(1-S_{pj}\right)}^{Z_j}{S}_{pj}^{1-Z_j}\right\}}^{1-{\tilde{Z}}_j}\times \prod _{i=1}^{N}g{\left({\eta }_i\right)}^{{\tilde{Y}}_i}{\left\{1-g\left({\eta }_i\right)\right\}}^{1-{\tilde{Y}}_i}.$$

In the second stage, a carefully constructed latent random variable, ω_i, is introduced for each of the individuals. Under the probit and logistic link functions, these random variables obey specifically structured normal and Pólya-Gamma distributions, respectively; for further details, see Albert and Chib (1993) and Polson et al. (2013). In either case, this stage yields the following joint conditional distribution

$$\pi (Z,\tilde{Y},\mathit{\omega }\mid \beta ,\mathit{\lambda },a,b)\propto \prod _{j=1}^J{\left\{S_{ej}^{Z_j}{\left(1-S_{ej}\right)}^{1-Z_j}\right\}}^{{\tilde{z}}_j}{\left\{{\left(1-S_{pj}\right)}^{Z_j}{S}_{pj}^{1-Z_j}\right\}}^{1-{\tilde{Z}}_j}\times \text{exp}\left\{-\frac{1}{2}(h-\eta {)}^{\prime }\Omega (h-\eta )\right\}\prod _{i=1}^N\xi \left({\omega }_i\right),$$ (6)

where ω = (ω₁, …, ω_N)′ and η = (η₁, …, η_N)′. Under the probit link, h = (ω₁, …, ω_N)′, Ω = I, and $\xi \left({\omega }_i\right)=I\left({\omega }_i⩾0,{\tilde{Y}}_i=1\right)+I\left({\omega }_i<0,{\tilde{Y}}_i=0\right)$ acts to control the support of ω_i, that is, given ${\tilde{Y}}_i=1$ or 0 results in ω_i being constrained to [0, ∞) or (−∞, 0), respectively. Under the logistic link, h = (κ₁/ω₁, …, κ_N/ω_N)′, ${\kappa }_i={\tilde{Y}}_i-1/2$, Ω = diag(ω), and $\xi \left({\omega }_i\right)=f\left({\omega }_i\mid 1,0\right)\text{exp}\left\{{\kappa }_i^2/\left(2{\omega }_i\right)\right\}$, where f(ω_i | a, b) denotes the Pólya-Gamma density with parameters (a, b); see Polson et al. (2013).

4. Posterior Computation and Inference

To facilitate estimation and inference, a posterior sampling algorithm consisting solely of Gibbs steps is constructed. In what follows, the necessary full conditional distributions used in this algorithm are provided. A symbolic representation of the entire posterior sampling algorithm is given in Web Appendix A in the Supporting Information.

Attention is first turned to the latent random variables introduced through the data augmentation procedure. The full conditional distribution of the individuals’ latent statuses is given by ${\tilde{Y}}_i\mid {\tilde{Y}}_{-i}$, Z, β, λ, a, $b_{k(i)}~\text{Bernoulli}\left\{p_{i1}^{\star }/\left(p_{i0}^{\star }+p_{i1}^{\star }\right)\right\}$, where

$$p_{i1}^{\star }=g\left({\eta }_i\right)\prod _{j\in {\mathcal{I}}_i}{S}_{ej}^{Z_j}{\left(1-S_{ej}\right)}^{1-Z_j}$$

$$p_{i0}^{\star }=\left\{1-g\left({\eta }_i\right)\right\}\prod _{j\in {\mathcal{I}}_i}{\left\{S_{ej}^{Z_j}{\left(1-S_{ej}\right)}^{1-Z_j}\right\}}^{I\left(s_{ij}>0\right)}{\left\{{\left(1-S_{pj}\right)}^{Z_j}{S}_{pj}^{1-Z_j}\right\}}^{I\left(s_{ij}=0\right)},$$

$s_{ij}={\sum }_{i^{\prime }\in {\mathcal{P}}_j:i^{\prime }\ne i}{\tilde{Y}}_{i^{\prime }}$, and the index set ${\mathcal{I}}_i=\left\{j:i\in {\mathcal{P}}_j\right\}$ keeps track of the indices of those pools to which the ith individual contributed. We also adopt the convention that V_−i represents the vector V after removing the ith component. The full conditional distribution of ω_i is link function dependent and is given by

$${\omega }_i\mid {\tilde{Y}}_i,\beta ,\lambda ,a,b_{k(i)}~\{\begin{array}{ll}TN\left\{{\eta }_i,1,[0,\infty )\right\},\hfill & \text{if}{\tilde{Y}}_i=1\hfill \\ TN\left\{{\eta }_i,1,(-\infty ,0)\right\},\hfill & \text{if}{\tilde{Y}}_i=0\hfill \end{array}$$

or

$${\omega }_i\mid \beta ,\lambda ,a,b_{k(i)}~\text{PG}\left(1,{\eta }_i\right)$$

under the probit and logistic link, respectively, where PG(a, b) denotes the Pólya-Gamma distribution with parameters (a, b); see Polson et al. (2013).

We now describe how to sample the fixed and random effects. Focusing on the quadratic form in the exponential in (6), we have that

$$\begin{gathered} {\left(h-\eta \right)}^{\prime }\mathbf{\Omega }(h-\eta )=\sum _{i=1}^N{\left(h_i-{\eta }_i\right)}^2{\Omega }_{ii} \\ =\sum _{i=1}^N{\left(h_i-{\mathbf{\text{x}}}_i^{\prime }\beta -t_i^{\prime }\Lambda A{b}_{k(i)}\right)}^2{\mathbf{\Omega }}_{ii} \\ =\sum _{i=1}^N{\left(h_{\beta i}-x_i^{\prime }\beta \right)}^2{\Omega }_{ii}={\left(h_{\beta }-X\mathit{\beta }\right)}^{\prime }\Omega \left(h_{\beta }-X\mathit{\beta }\right), \end{gathered}$$

where Ω_ii is the ith diagonal element of Ω, $h_{\beta i}=h_i-t_i^{\prime }\Lambda A{b}_{k(i)}$, h_i is the ith entry in h, and h_β = (h_β1, …, h_βN)′. Thus, under the SSVS and NMIG spike and slab priors, the full conditional distribution of β is given by

$$\beta \mid \tilde{Y},\omega ,\lambda ,a,b,v~N\left\{{\left(X^{\prime }\Omega X+{\Phi }^{-1}\right)}^{-1}{X}^{\prime }\Omega h_{\beta },{\left(X^{\prime }\Omega X+{\Phi }^{-1}\right)}^{-1}\right\},$$

where $\Phi =\text{diag}\left(r\left(v_1\right){\varphi }_1^2,\dots ,r\left(v_{q_1}\right){\varphi }_{q_1}^2\right)$. Under the Dirac spike, the full conditional distribution of β_q is degenerate at 0 if v_q = 0, while the non-zero elements of β, say β_v, have the following normal full conditional

$${\beta }_v\mid \tilde{Y},\omega ,\lambda ,a,b,v~N\left\{{\left(X_v^{\prime }\Omega X_v+{\Phi }_v^{-1}\right)}^{-1}{X}_v^{\prime }\Omega h_{\beta },{\left(X_v^{\prime }\Omega X_v+{\Phi }_v^{-1}\right)}^{-1}\right\},$$

where X_v is the design matrix consisting of those columns of X corresponding to non-zero elements of v and Φ_v is the diagonal matrix formed by retaining the diagonal elements of $\Phi =\text{diag}\left({\varphi }_1^2,\dots ,{\varphi }_{q_1}^2\right)$ corresponding to the non-zero elements of v. Due to the data augmentation steps described above, one can also obtain the following full conditionals

$${\lambda }_l\mid \tilde{Y},\omega ,\beta ,{\lambda }_{-l},a,b,w_l~TN\left\{{\mu }_{{\lambda }_l}\left(w_l\right),{\sigma }_{{\lambda }_l}^2\left(w_l\right),(0,\infty )\right\}$$

$$\text{a}\mid \tilde{Y},\omega ,\beta ,\lambda ,b~N\left({\mu }_a,{\Sigma }_a\right)$$

$$b_k\mid \tilde{Y},\omega ,\beta ,\lambda ,a~N\left({\mu }_{b_k},{\Sigma }_{b_k}\right),$$

where the specific forms of these distributions are provided in Web Appendix A. Sampling these parameters is equivalent to sampling the random effects as well as the covariance matrix of the distribution of the random effects.

The full conditional distributions of v_q and w_l are Bernoulli, where the success probabilities depend on which spike and slab prior is specified; see Web Appendix A. The full conditional distributions for the mixing weights ${\tau }_{v_q}$ and ${\tau }_{w_l}$ are ${\tau }_{v_q}\mid v_q~\text{Beta}\left(a_v+v_q,1-v_q+b_v\right)$ and ${\tau }_{w_l}\mid w_l~\text{Beta}\left(a_w+w_l,1-w_l+b_w\right)$, respectively. Under the NMIG prior, the full conditional distributions of the variance parameters are ${\varphi }_q^2\mid {\beta }_q$, $v_q~\text{Inv-Gamma}\left(a_{\varphi }+1/2,b_{\varphi }+{\beta }_q^2/2r\left(v_q\right)\right)$ and ${\psi }_l^2\mid {\lambda }_l$, $w_l~\text{Inv-Gamma}\left(a_{\psi }+1/2,b_{\psi }+{\lambda }_l^2/2r\left(w_l\right)\right)$.

Up until this point, the assay accuracy probabilities S_ej and S_pj have been assumed to be known. When these are unknown, we can estimate them along with the rest of the model parameters following the approach outlined in McMahan et al. (2017). This approach allows for different assays to be used throughout the testing process (e.g., screening and confirmatory testing) and/or can account for the effect of pool size on the accuracy of the assay; i.e., sensitivity and specificity might change with the pool size. Define the index set ${\mathcal{M}}_m$ which identifies the indices of the pools which were tested by the mth assay, for m = 1, …, M. Further, let S_e(m) and S_p(m) denote the sensitivity and specificity of the mth assay such that S_ej = S_e(m) and S_pj = S_p(m) for all $j\in {\mathcal{M}}_m$. Under these conventions, (6) can be written as

$$\pi \left(Z,\tilde{Y},\omega \mid \beta ,\lambda ,a,b,S_e,S_p\right)\propto \prod _{m=1}^M\prod _{j\in {\mathcal{M}}_m}{\left\{S_{e(m)}^{Z_j}{\left(1-S_{e(m)}\right)}^{1-Z_j}\right\}}^{{\tilde{Z}}_j}{\left\{{\left(1-S_{p(m)}\right)}^{Z_j}{S}_{p(m)}^{1-Z_j}\right\}}^{1-{\tilde{Z}}_j}\times \text{exp}\left\{-\frac{1}{2}{(h-\eta )}^{\prime }\Omega (h-\eta )\right\}\prod _{i=1}^N\xi \left({\omega }_i\right),$$

where S_e = (S_e(1), …, S_e(M))′ and S_p = (S_p(1), …, S_p(M))′. Given the form of the conditional distribution above, independent beta priors are a natural choice; i.e., S_e(m) ~ Beta(a_e(m), b_e(m)) and S_p(m) ~ Beta(a_p(m), b_p(m)). These specifications lead to the following full conditionals

$$S_{e(m)}\mid Z,\tilde{Y}~\text{Beta}\left(a_{e(m)}^{\star },b_{e(m)}^{\star }\right)$$

$$S_{p(m)}\mid Z,\tilde{Y}~\text{Beta}\left(a_{p(m)}^{\star },b_{p(m)}^{\star }\right),$$

where $a_{e(m)}^{\star }=a_{e(m)}+\sum _{j\in {\mathcal{M}}_m}{Z}_j{\tilde{Z}}_j$, $b_{e(m)}^{\star }=b_{e(m)}+\sum _{j\in {\mathcal{M}}_m}\left(1-Z_j\right){\tilde{Z}}_j$, $a_{p(m)}^{\star }=a_{p(m)}+\sum _{j\in {\mathcal{M}}_m}\left(1-Z_j\right)\left(1-{\tilde{Z}}_j\right)$, and $b_{p(m)}^{\star }=b_{p(m)}+\sum _{j\in {\mathcal{M}}_m}{Z}_j\left(1-{\tilde{Z}}_j\right)$. The other posterior distributions are left unchanged up to acknowledging dependence on the assay accuracy probabilities and accounting for the slight change in notation.

5. Simulation Study

To investigate the performance of our regression and variable selection methods, we designed a comprehensive simulation study which emulates the primary features of our Iowa chlamydia data application in Section 6. To this end, K = 50 clinic sites were conceptualized and the infection statuses for 100 individuals within each of these sites were generated; i.e., N = 5000. This sample size is roughly one-third of the sample size available in our data application. The individuals’ true statuses ${\tilde{Y}}_i$ were generated according to the following model

$$g^{-1}\left\{P\left({\tilde{Y}}_i=1\mid \beta ,\lambda ,a,b_{k(i)}\right)\right\}=x_i^{\prime }\beta +t_i^{\prime }\Lambda A{b}_{k(i)},\text{for}i=1,\dots ,N,$$

where g⁻¹(·) denotes the probit link, β = (−3, −1.5, 0.5, 0.25, 0, 0)′, λ = (1, 0.75, 0.25, 0, 0, 0)′, a = (1, 0.5, 0.7, 0, …, 0)′, b_k(i) = b_k if the ith individual presented at the kth clinic, and $b_k\stackrel{iid}{~}N(0,I)$. The covariate vectors x_i and t_i are taken to be equal and are standardized versions of $x_i^{*}={\left(1,x_{i1}^{*},x_{i2}^{*},x_{i3}^{*},x_{i4}^{*},x_{i5}^{*}\right)}^{\prime }$, where $x_{i1}^{*}$, $x_{i5}^{*}~N(0,1)$ and $x_{i2}^{*}$, $x_{i3}^{*}$, $x_{i4}^{*}~\text{Bernoulli}(0.5)$. Under these specifications, the generating model consists of four non-zero fixed effects (one intercept and three slopes) as well as three non-zero random effects (one intercept and two slopes). The parameter configurations above provide for an overall prevalence of approximately 9%, which is in keeping with the motivating data application. This model was used to generate 1000 independent data sets.

To generate the testing outcomes Z_j, we consider three group testing protocols; namely, master pool testing (MPT), Dorfman testing (DT), and array testing (AT). Under MPT, each individual is assigned to exactly one master pool which is tested and no further testing is performed regardless of the outcome. DT completes the decoding process initiated by MPT by retesting all individuals in positive master pools. Similarly, AT completes decoding in two stages, but it starts by assigning individuals to an array. In the first stage, AT tests pools formed by combining individuals who share a common row or column. The second stage retests individuals identified to be likely positives; e.g., individuals residing at the intersection of positive rows and columns. For the specific retesting protocol adopted for AT, see Kim et al. (2007). Following the pooling practices used in the motivating example, we implement MPT and DT using master pools of size 4 and AT using 4 × 4 arrays. For comparative purposes, individual testing (IT) was also implemented.

For each of the 1000 individual-level data sets, we simulate IT, MPT, DT and AT. To implement the group testing protocols, individuals were randomly assigned to pools (arrays), so that individuals would be pooled across sites rather than within sites. This poses the most difficult estimation configuration; that is, individuals within the same pool have different random effects. However, this configuration also mirrors how pooling is typically implemented in large-scale testing situations such as those at the SHL in Iowa. Under all protocols, the testing response for the jth pool was simulated as $Z_j\mid {\tilde{Z}}_j~\text{Bernoulli}\left\{S_{ej}{\tilde{Z}}_j+\left(1-S_{pj}\right)\left(1-{\tilde{Z}}_j\right)\right\}$, where ${\tilde{Z}}_j=I\left({\sum }_{i\in {\mathcal{P}}_j}{\tilde{Y}}_i>0\right)$. Two different simulation settings are considered for the assay accuracy probabilities. In the first, sensitivity and specificity are assumed to be known and are set to be S_ej = 0.95 and S_pj = 0.98 for all j = 1, …, J. The second setting allows for two assays, where the first (m = 1) is used to test pools and the second (m = 2) is used for individual-level testing with S_e = (S_e(1), S_e(2))′ = (0.95, 0.98)′ and S_p = (S_p(1), S_p(2))′ = (0.98, 0.99)′. Under this setting, we assume these accuracy probabilities are unknown and have to be estimated along with the other model parameters. We only implement DT and AT in the second setting because only these protocols mandate both pool and individual-level testing.

We assess the performance under the three spike and slab priors described in Section 3.1; we set m₀ = 0, C₀ = 0.5I, and used flat priors for all mixing weights and assay accuracy probabilities; i.e., Beta(1, 1). As noted earlier, we specify a slightly informative prior on a to avoid a strongly informative prior distribution on the correlation between any two random effects (Chen and Dunson, 2003). We chose r = 0.00025 for both SSVS and NMIG and a_ϕ = a_ψ = 5 and b_ϕ = b_ψ = 50 when using NMIG, closely resembling the values chosen in Scheipl (2011). To provide a fair comparison, the prior mean for the variance component under NMIG was used as the variance component in SSVS and the Dirac spike; i.e., ${\varphi }_q^2={\psi }_l^2=50/4$. To perform posterior estimation and inference, our MCMC algorithm was used to draw 100000 iterates, with every 50th being retained after a burn-in of 50000. Point estimates of the model parameters were obtained as the empirical means of the posterior distributions.

To examine the performance of the variable selection techniques, estimates of the posterior inclusion probabilities were calculated. These estimates were taken to be the sample mean of the posterior draws of v_q and w_l. To assess out-of-sample classification accuracy, we conducted the following receiver operating characteristic (ROC) curve analysis. For each model fit, we simulated 10000 new individuals and used our model fits to predict their infection probabilities. This provides 1000 ROC curve estimates which are summarized by using the average area under the curve (AUC). For purposes of comparison, we also fit the generalized linear model (GLM) described in McMahan et al. (2017). This is aimed at demonstrating the gains in classification accuracy that are possible by including site-specific random effects and using spike and slab priors to guide model selection.

Table 1 summarizes estimation performance under the Dirac spike in the first simulation setting, that is, when S_ej and S_pj are known. The analogous simulation results in this setting under SSVS and NMIG are shown in Web Appendix B. Overall, the results in Table 1 illustrate our approach provides reliable inference for the fixed and random effects; i.e., the empirical bias and the variability in the estimates are small relative to the true value of the corresponding parameter. These results also indicate the proposed methodology is adept at identifying non-zero fixed and random effects. That is, covariates with strong (no) effects almost always have posterior inclusion probabilities near 1 (0) in all data sets. Among the three priors, the Dirac spike tends to outperform both SSVS and NMIG in terms of variable selection. Finally, Web Table B.3 summarizes the ROC analysis described in the last paragraph. This table shows the average AUC for our model is notably larger than that for the simpler GLM in McMahan et al. (2017) across all configurations.

Table 1:

Simulation results with known assay accuracy probabilities (S_ej = 0.95 and S_pj = 0.98) under the Dirac spike. The average bias of the posterior mean estimates (Bias), the sample standard deviation of the estimates (SSD), and the posterior probability of inclusion (PI) are provided. The parameter d_ij denotes the ijth element of D. The total number of individuals is N = 5000 with a common pool size of 4 (for MPT, DT, and AT). The results for individual testing (IT) are also shown.

	IT			MPT			DT			AT
Parameter	Bias	SSD	PI	Bias	SSD	PI	Bias	SSD	PI	Bias	SSD	PI
β1 = −3	−0.06	0.27	1.00	0.10	0.40	1.00	−0.03	0.23	1.00	−0.02	0.22	1.00
β2 = −1.5	−0.01	0.20	1.00	0.08	0.27	1.00	−0.01	0.19	1.00	−0.01	0.19	1.00
β3 = 0.5	0.04	0.12	0.99	0.00	0.16	0.99	0.02	0.09	0.99	0.01	0.08	0.99
β4 = 0.25	0.00	0.06	0.99	−0.05	0.11	0.77	0.00	0.05	0.99	0.00	0.05	0.99
β5 = 0	0.00	0.01	0.03	0.00	0.02	0.04	0.00	0.01	0.03	0.00	0.01	0.03
β6 = 0	0.00	0.01	0.03	0.00	0.01	0.04	0.00	0.01	0.03	0.00	0.01	0.03
λ1 = 1	−0.01	0.32	0.93	−0.10	0.40	0.88	0.03	0.21	0.98	0.03	0.18	0.99
λ2 = 0.75	0.01	0.15	0.98	−0.13	0.32	0.80	0.04	0.11	0.99	0.02	0.10	1.00
λ3 = 0.25	−0.07	0.17	0.56	−0.16	0.18	0.21	−0.06	0.14	0.66	−0.06	0.13	0.69
λ4 = 0	0.00	0.00	0.01	0.00	0.00	0.01	0.00	0.00	0.01	0.00	0.00	0.01
λ5 = 0	0.00	0.00	0.01	0.00	0.00	0.01	0.00	0.00	0.01	0.00	0.00	0.01
λ6 = 0	0.00	0.00	0.01	0.00	0.00	0.01	0.00	0.00	0.01	0.00	0.00	0.01
d11 = 1	0.13	0.52	–	0.03	0.67	–	0.14	0.41	–	0.13	0.38	–
d22 = 1.125	0.07	0.40	–	−0.13	0.63	–	0.10	0.34	–	0.06	0.31	–
d33 = 0.109	0.03	0.18	–	−0.01	0.21	–	0.01	0.11	–	0.00	0.08	–
d21 = 0.75	0.01	0.37	–	−0.12	0.52	–	0.04	0.30	–	0.03	0.27	–
d31 = 0.125	−0.07	0.08	–	−0.11	0.06	–	−0.05	0.08	–	−0.05	0.08	–
d32 = 0.225	−0.07	0.16	–	−0.14	0.17	–	−0.06	0.13	–	−0.06	0.12	–

Among the group testing protocols presented in Table 1, MPT generally performs the worst in terms of estimation (see also Web Tables B.1 and B.2); i.e., estimates obtained from analyzing MPT data exhibit the most bias and variability. This is expected because MPT does not complete the classification process like the other protocols and therefore results in less information about the individuals’ latent statuses. In contrast, the estimation performance under the two classification protocols (DT and AT) is as good if not better than the performance under IT; furthermore, DT/AT estimates are obtained at about 60% of the testing cost on average when compared to IT. Specifically, 5000 tests are used to complete IT, while DT and AT require 2747 and 3258 tests on average, respectively. These results illustrate the “get more for less” phenomenon that has previously been reported with group testing regression (Zhang et al., 2013; McMahan et al., 2017).

Table 2 summarizes estimation performance under the Dirac spike when the assay accuracy probabilities are unknown; the corresponding SSVS and NMIG results are shown in Web Appendix B. In this second setting, the proposed methodology is tasked with estimating four additional parameters (S_e(1), S_p(1), S_e(2), and S_p(2)). The results in Table 2 indicate we can accurately estimate these parameters and the variability in the estimates is small relative to the true values. Moreover, there are no appreciable differences between the estimates in Tables 1 and 2 for DT and AT; i.e., inference for the fixed and random effects is not impacted by having to estimate these additional parameters. This is noteworthy because by selecting flat priors for S_e(1), S_p(1), S_e(2), and S_p(2), we have chosen to consider the most challenging scenario for estimation, that is, when no prior information about assay performance is available. In other settings, informative prior distributions could be used. For example, if one believes an assay’s sensitivity is around 0.95, then an informative prior could be specified as Beta(19c, c), where large (small) values of c would reflect strong (weak) prior belief. Informative priors can also be designated on the basis of assay validation trials as we describe in Section 6.

Table 2:

Simulation results with unknown assay accuracy probabilities under the Dirac spike. The average bias of the posterior mean estimates (Bias), the sample standard deviation of the estimates (SSD), and the posterior probability of inclusion (PI) are provided. The parameter d_ij denotes the ijth element of D. The total number of individuals is N = 5000 with a common pool size of 4 for DT and AT.

	DT			AT
Parameter	Bias	SSD	PI	Bias	SSD	PI
β1 = −3	−0.05	0.23	1.00	−0.04	0.21	1.00
β2 = −1.5	−0.04	0.19	1.00	−0.03	0.18	1.00
β3 = 0.5	0.03	0.09	0.99	0.01	0.09	0.99
β4 = 0.25	0.01	0.05	0.99	0.00	0.04	0.99
β5 = 0	0.00	0.01	0.03	0.00	0.01	0.03
β6 = 0	0.00	0.01	0.03	0.00	0.01	0.03
λ1 = 1	0.04	0.23	0.98	0.03	0.19	0.99
λ2 = 0.75	0.04	0.13	0.99	0.03	0.11	0.99
λ3 = 0.25	−0.05	0.14	0.68	−0.05	0.13	0.71
λ4 = 0	0.00	0.00	0.01	0.00	0.00	0.01
λ5 = 0	0.00	0.00	0.01	0.00	0.00	0.01
λ6 = 0	0.00	0.00	0.01	0.00	0.00	0.01
d11 = 1	0.17	0.44	–	0.14	0.38	–
d22 = 1.125	0.13	0.37	–	0.07	0.33	–
d33 = 0.109	0.02	0.13	–	0.01	0.09	–
d21 = 0.75	0.05	0.33	–	0.03	0.28	–
d31 = 0.125	−0.05	0.08	–	−0.06	0.08	–
d32 = 0.225	−0.05	0.14	–	−0.06	0.12	–
Se(1) = 0.95	−0.02	0.03	–	0.00	0.01	–
Se(2) = 0.98	−0.01	0.01	–	0.00	0.01	–
Sp(1) = 0.98	0.00	0.01	–	0.00	0.00	–
Sp(2) = 0.99	0.00	0.01	–	0.00	0.01	–

Finally, we have performed an additional simulation study to assess the robustness of our methodology to violations of the conditional independence assumption stated in Section 2; i.e., that the testing responses in Z are independent given the true statuses in $\tilde{Z}$. Web Appendix C in the Supporting Information provides details on how this study was conducted along with a summary discussion of the results. Overall, the study reveals our proposed methodology is not unduly affected even under a severe violation of this assumption.

6. Iowa Chlamydia Data Analysis

As Iowa’s public health and environmental laboratory, the SHL serves all of the state’s counties for infectious disease detection and surveillance. This includes annually screening thousands of residents for two of the most common sexually transmitted diseases (STDs): chlamydia and gonorrhea. This process begins with individual specimens (e.g., urine, swab, etc.) being collected from residents at different clinics (e.g., STD clinics, family planning clinics, etc.) throughout the state. These specimens are then transported to the SHL for testing. Current SHL screening practices mandate that all male specimens and female urine specimens be tested individually while a variant of Dorfman testing (DT) is used to classify female swab specimens; for further discussion, see Tebbs et al. (2013). The SHL uses the Aptima Combo 2 Assay (AC2A) to test both pooled and individual specimens. Swab master pools for females are formed chronologically based on the arrival date for testing.

Our analysis focuses on the chlamydia data collected on female patients during the 2014 calendar year. During this time period, K = 64 different clinics submitted specimens to the SHL for testing. The available data consist of test results on 4316 individual urine specimens, 416 individual swab specimens, and 2286 swab master pools (1 of size 2, 12 of size 3, and 2273 of size 4), as well as the additional test results required to resolve the positive master pools. In addition, several covariates were collected on each individual: age (in years, denoted by $x_1^{*}$), a race indicator ($x_2^{*}=1$ if Caucasian and $x_2^{*}=0$ otherwise), an indicator denoting whether the patient reported a new sexual partner in the last 90 days ($x_3^{*}=1$ if affirmative and $x_3^{*}=0$ otherwise), an indicator denoting whether the patient reported having multiple sexual partners in the last 90 days ($x_4^{*}=1$ if affirmative and $x_4^{*}=0$ otherwise), an indicator denoting whether the patient reported sexual contact with an STD-positive partner in the previous year ($x_5^{*}=1$ if affirmative and $x_5^{*}=0$ otherwise), and an indicator denoting whether the patient presented with symptoms ($x_6^{*}=1$ if affirmative and $x_6^{*}=0$ otherwise). To relate the individuals’ true chlamydia disease statuses to the covariate information, we assume

$$g^{-1}\left\{P\left({\tilde{Y}}_i=1\mid \beta ,\lambda ,a,b_{k(i)}\right)\right\}=x_i^{\prime }\beta +t_i^{\prime }\Lambda A{b}_{k(i)},$$

where g⁻¹(·) is the probit link. In this analysis, the covariate vectors x_i and t_i are taken to be equal and are standardized versions of $x_i^{*}={\left(1,x_{i1}^{*},x_{i2}^{*},x_{i3}^{*},x_{i4}^{*},x_{i5}^{*},x_{i6}^{*}\right)}^{\prime }$. Standardization was used so the spike and slab distributions have the same impact on the regression coefficients across all covariates. A random effect vector b_k is specified for each of the K = 64 clinics, with the convention that b_k(i) = b_k if the ith individual presented at the kth clinic.

Based on the simulation results in Section 5, we implement our methods for this analysis under the Dirac spike only. Other prior specifications were made in the exact same fashion as in Section 5. The only difference is that three sets of assay accuracy probabilities were conceptualized to account for the SHL’s testing protocol: S_e(1) and S_p(1) for swab specimens tested individually, S_e(2) and S_p(2) for urine specimens tested individually, and S_e(3) and S_p(3) for swab specimens tested in pools. Flat priors were initially specified for these parameters; i.e., S_e(m), S_p(m) ~ Beta(1, 1), for m = 1, 2, 3.

Table 3 displays estimates of the posterior mean and standard deviation for all model parameters and estimates of the posterior probabilities of inclusion for the fixed and random effects. Posterior mean estimates have been “unstandardized” to aid in their interpretation. We also display in Web Appendix D the six models with the highest posterior probabilities; these models were ranked in a manner similar to that in Kuo and Mallick (1998). The direction of the estimates of the fixed effects in Table 3 are expected in light of known epidemiological patterns of chlamydia infection. That is, the risk of infection tends to decrease with age and Caucasian females are associated with a lower risk when compared to females of other races. In contrast, having a new sexual partner, multiple partners, and contact with an STD are all associated with an increased risk. Our analysis also identifies the random intercept parameter to be strongly significant indicating clear evidence of heterogeneity across the clinics throughout the state. An anonymous referee has suggested these random intercepts may act as effective proxies for clinic-level unmeasured confounders such as socioeconomic status or possibly other predictor variables which may affect disease prevalence.

Table 3:

Iowa chlamydia data. Estimates of the posterior mean (Estimate), the posterior standard deviation (ESD), and the posterior probability of inclusion (PI). We use star notation on the regression parameters to emphasize the estimates are not standardized.

Parameter	Description	Estimate	ESD	PI
${\beta }_1^{\star }$	Intercept	−0.508	0.109	1.00
${\beta }_2^{\star }$	Age	−0.037	0.004	1.00
${\beta }_3^{\star }$	Race	−0.164	0.061	0.93
${\beta }_4^{\star }$	New partner	0.145	0.050	0.94
${\beta }_5^{\star }$	Multiple partners	0.137	0.093	0.74
${\beta }_6^{\star }$	Contact with STD	0.732	0.067	1.00
${\beta }_7^{\star }$	Symptoms	0.029	0.055	0.24
λ1	Intercept	0.174	0.036	1.00
λ2	Age	0.000	0.001	0.01
λ3	Race	0.000	0.001	0.00
λ4	New partner	0.003	0.017	0.03
λ5	Multiple partners	0.000	0.002	0.01
λ6	Contact with STD	0.000	<0.001	0.00
λ7	Symptoms	0.000	<0.001	0.01
Se(1)	Swab individual	0.998	0.002	–
Se(2)	Urine individual	0.792	0.090	–
Se(3)	Swab pool	0.909	0.062	–
Sp(1)	Swab individual	0.979	0.007	–
Sp(2)	Urine individual	0.987	0.007	–
Sp(3)	Swab pool	0.999	0.001	–

Shifting attention to the assay accuracy probabilities in Table 3, one notices lower estimates of S_e(2) and possibly S_e(3), suggesting potential underestimation. We also analyzed the Iowa data using highly informative (beta) prior distributions selected on the basis of validation trials reported in the product literature for the AC2A assay. This analysis is summarized in Web Appendix D. When comparing this new analysis to that in Table 3, we find there are no large differences in the regression parameter estimates although the estimated posterior standard deviations are slightly lower when modeling the accuracy probabilities informatively; see Web Appendix D. Even in the absence of a priori information, we find evidence that all assay accuracy probabilities are identifiable under the Iowa testing protocol, with S_e(2) and S_e(3) being “weaker learners” than S_e(1). This is perhaps not surprising; for example, testing all female urine specimens individually provides little to no confirmatory and/or counterfactual information that can be exploited to estimate S_e(2).

A final feature we highlight is our ability to estimate the posterior probability of infection for each individual given all of the observed data. Using our posterior sampling strategy, this can be accomplished by averaging over the sampled latent statuses for the ith individual; specifically, by computing $G^{-1}{\sum }_{g=1}^G{\tilde{Y}}_i^{(g)}$, where ${\tilde{Y}}_i^{(g)}$ is the gth posterior draw of ${\tilde{Y}}_i$ for g = 1, …, G. Web Figure D.1 displays these probabilities for the 13,862 individuals in the Iowa data set, cross classified by diagnosed status and whether informative priors were used for the assay accuracy probabilities. In addition to being used as a measure of diagnostic certainty, we conjecture these probabilities could be integrated into a back-end confirmatory testing strategy using the informative group testing ideas in McMahan et al. (2012).

7. Discussion

We have proposed a Bayesian approach to estimate generalized linear mixed models with data arising from any group testing protocol. When compared to existing regression techniques for group testing data, the appeal of our methodology is twofold. First, including random effects allows one to account for heterogeneity that may exist across subgroups of the population. Second, our approach employs automatic variable selection for both the fixed and random effects by using spike and slab priors. Through a series of data augmentation steps, we illustrate how our regression methods can be used with the probit and logistic link functions. R programs for data analysis are available in the Supporting Information. A description of the main functions is given in Web Appendix E.

Several alternative modeling approaches or extensions could be of interest. An anonymous referee has suggested one could generalize our framework to estimate additive models with group testing data and use variable selection strategies to differentiate between linear and nonlinear functions in these models. Another timely extension of this work would be the development of regression methods to analyze data from multiplex assays, where specimens (individuals and pools) are tested for multiple diseases simultaneously.