Spike and Slab Regression for Nonstationary Gaussian Linear Mixed Effects Modeling of Rapid Disease Progression

Abstract

Select measures of social and environmental determinants of health (referred to as “geomarkers”), predict rapid lung function decline in cystic fibrosis (CF), defined as a prolonged decline relative to patient and/or center-level norms. The extent to which hyper-localization, defined as increasing the spatiotemporal precision of geomarkers, aids in prediction of rapid lung decline remains unclear. Linear mixed effects (LME) models with specialized covariance functions have been used for predicting rapid lung function decline, but there are few options to properly incorporate spatial correlation into the covariance functions while inducing simultaneous variable selection. Our innovative Bayesian model uses a spike and slab prior for simultaneous variable selection and offers additional advantages when coupled with nonstationary Gaussian LME modeling. This model also incorporates spatial correlation through an additional random effect term that accounts for spatial correlation based on ZIP code distances. We validated the model with simulations and applied it to real CF data from a Midwestern CF Center. We demonstrate how a combination of demographic, clinical, and geomarker variables can be selected as optimal predictors using Bayesian false discovery rate controlling rule. Our results indicate that incorporating spatiotemporal effects and geomarkers into this novel Bayesian stochastic LME model enhances the dynamic prediction of rapid CF disease progression.

1 |. Introduction

Cystic fibrosis (CF), a chronic and progressive genetic disorder, primarily affects the respiratory and digestive systems, leading to complex clinical challenges that require routine clinical follow-up and monitoring (Morgan et al. 2013). The disease’s progression is often marked by a rapid decline in lung function, a critical determinant of both quality of life and survival in CF patients. Lung function is measured through percent predicted forced expiratory volume in 1 s (FEV1%) using standard reference equations (Wang et al. 1993; Hankinson, Odencrantz, and Fedan 1999). Historically, the prediction and monitoring of this decline have relied heavily on clinical and demographic characteristics. However, recent research indicates that a broader perspective, one that encompasses social and environmental determinants of health, could provide a more accurate predictive framework. The role of the built environment and community context as determinants of health has gained increasing recognition, particularly with evolving care needs in chronic diseases like CF. The use of “geomarkers” to capture these determinants has shown great potential for predicting rapid lung function decline in CF (Szczesniak et al. 2020; Gecili et al. 2023; Peterson et al. 2024). The extent to which hyper-localization – increasing the spatiotemporal precision of social and environmental exposures – can help to predict rapid lung function decline is still an open research area.

Traditional methods, for example, linear mixed effects (LME) models, have made significant contribution to predict rapid lung function decline in CF (Taylor et al. 2012; Szczesniak et al. 2013). Nonetheless, these forms of LME fall short in adequately incorporating spatial correlation and facilitating simultaneous variable selection, which are crucial for a nuanced understanding of disease progression. Various regression regularization procedures have been developed for estimation and variable selection. However, approaches have mostly focused on linear regression models that ignore correlation arising from long sequences of repeated measurements and spatial correlation in the outcome that arises based on location and varies according to distance between locations. In a previous work (Gecili et al. 2022), we implemented four different types of regularized regression models to identify proteomic biomarkers of rapid lung-function decline in CF patients by variable selection in an LME setting with temporal correlation but with no spatial correlation. Specifically, we considered the Bayesian lasso, adaptive lasso, ridge and elastic-net models with different priors and compared them using simulations and several real data examples. Note that these methods allow for penalized variable selection. Although these models ignore spatial variability, they have been used to identify predictive geomarkers for rapid lung function decline in CF (Peterson et al. 2024). The spike and slab prior may offer additional advantages when coupled with nonstationary Gaussian LME modeling while accounting for spatiotemporal variability in lung function. The spike and slab prior is a popular discrete shrinkage prior that is often considered as the “gold standard” for sparse Bayesian estimation. Emerging from a class of sparsity-enforcing priors, this prior is formed by a mixture of priors on regression coefficients with density functions that are peaked at zero and also have large probability mass for a wide range of nonzero values. Using such priors plays a similar role as the $L_1$ norm penalty in the Bayesian lasso and induces adaptive shrinkage on coefficients. Bayesian spike and slab priors have been used for predictive modeling and variable selection in large scale genomic studies (Zhou, Carbonetto, and Stephens 2013; Lu et al. 2015; Tang et al. 2017; Ročková and George 2014). Spike and slab priors and their variants are extensively discussed by others (Ročková and George 2018; Ray, Ročková, and George 2021).

While spike-and-slab priors, Gaussian LME models, and spatial correlation modeling have each been employed independently in various studies, the novelty of our approach lies in the integration of these components into a unified Bayesian framework specifically tailored for CF data. Our method combines a Bayesian spike-and-slab prior for simultaneous variable selection with a nonstationary Gaussian LME model that incorporates both temporal and spatial variability through an integrated Brownian motion process and spatial random effects based on ZIP code distances. This comprehensive model allows us to capture the complex interplay between clinical, demographic, and environmental factors influencing rapid lung function decline in CF patients. To our knowledge, this is the first application of such an integrated model in the context of CF, particularly utilizing geomarker data to enhance predictive accuracy.

Our modeling framework allows for capturing both spatial and temporal variability in lung function decline. This approach also allows for obtaining multiplicity adjusted posterior inclusion probabilities (PIPs) for each candidate predictors that are used to make variable selection. This approach not only addresses the shortcomings of traditional models but also aligns with the growing emphasis on precision medicine. The modeling framework that we propose is a Bayesian spatial stochastic LME model that leverages select predictive geomarkers to predict rapid lung function decline. We evaluate its performance through simulations under different scenarios as well as a real dataset from patients followed at a Midwestern CF Center. These analyses enable us to identify the optimal combination of demographic, clinical, and geomarker variables for predicting rapid lung function decline in CF.

The remainder of this manuscript is organized as follows: In Section 2, we describe the longitudinal model and notation, including the temporal and spatial components of the model. Section 3 presents the Bayesian spike and slab linear mixed-effects model, with details on the variable selection procedure and the priors used. In Section 4, we provide results from simulation studies that evaluate the performance of the proposed model. Section 5 applies the model to real data from a Midwestern CF cohort. Finally, in Section 6, we provide conclusions and discuss the key findings, limitations, and potential future directions of this research.

2 |. Longitudinal Model and Notation

We use a nonstationary Gaussian linear mixed effects model (Diggle, Sousa, and Asar 2015) for modeling longitudinal FEV1% trajectories and include the geomarkers, in addition to routinely collected demographic and clinical variables as potential predictors of rapid CF lung disease progression. This model choice is supported by previous CF applications (Szczesniak et al. 2020; Su et al. 2021; Gecili et al. 2023). The model specification is provided below.

$$Y_{ijk}={\mu }_i\left(t_{ij},s_{ijk}\right)+U_i+W_i\left(t_{ij}\right)+S_i\left(s_{ijk}\right)+Z_{ijk}$$ (1)

where $Y_{ijk}$ denotes the FEV1% response for subject $i(i=1,\dots ,N)$ at time $t_{ij}\left(j=1,\dots ,n_i\right)$ and location $s_{ijk}$ $(k=1,\dots ,m_i$; that is, the specific location at time $t_{ij}$). Above ${\mu }_i(t,s)$ is the mean response function at time $t$ and location $s$. The $U_i$ is independent $N\left(0,{\omega }^2\right)$ random variable that represents time- and location-constant differences between patients. $W_i\left(t\right)$ represents independent copies of a continuous time stochastic process which we specified as an integrated Brownian motion (IBM), representing stochastic change in an individual patient’s lung function over time. We chose IBM for modeling the temporal variability in lung function because it captures both short-term fluctuations and long-term cumulative effects. This choice is particularly suitable for CF data, where lung function, measured through FEV1%, exhibits both random and gradual changes over time (Szczesniak et al. 2020; Su et al. 2021; Gecili et al. 2023). IBM models this cumulative variability by integrating random changes over time, which aligns well with the longitudinal nature of our data, capturing the irregular progression of CF. In our previous research, we found that using IBM improved the accuracy of predictions for rapid lung function decline in CF patients (Su et al. 2021). Additionally, exploratory analyses, such as variograms, further supported that IBM is appropriate for the underlying temporal structure of our data, as it provides a more realistic model of how lung function evolves in a stochastic manner over time (Gecili et al. 2022). The $S_i\left(s\right)$ represents change (spatial variability) in a patient’s lung function based on location (e.g., ZCTA/ZIP codes). The $Z_{ijk}$ terms are mutually independent and identical $N\left(0,{\tau }^2\right)$ random variables representing measurement error in the determination of $Y_{ijk}$.

In our model, $W_i\left(t\right)$ is modeled as the integral of a continuous-time random walk,

$$W_i(t)={\int }_0^t{B}_i(v)dv$$ (2)

where $B_i(t)$, the rate of change at time $t$, is Brownian motion. Set $B_i(0)=0$ for all $i$. The conditional distribution of $B_i(t)$ given $B_i(h)$ for some $h<t$ is Normal, with mean $B_i(h)$ and variance ${\sigma }^2(t-h)$. The marginal distribution of $B_i(t)$ is $N\left(0,{\sigma }^{2}t\right)$ and $Cov\left(B_i(h),B_i(t)\right)={\sigma }^2\mathrm{m}\mathrm{i}\mathrm{n}(h,t)$. Hence, $W_i(t)~N\left(0,\frac{{\sigma }^2{t}^3}{3}\right)$ and $\mathrm{C}\mathrm{o}\mathrm{v}\left(W_i(h),W_i(t)\right)={\sigma }^2\frac{\mathrm{m}\mathrm{i}\mathrm{n}(h,t{)}^2}{2}\left(\mathrm{m}\mathrm{a}\mathrm{x}(h,t)-\frac{\mathrm{m}\mathrm{i}\mathrm{n}(h,t)}{3}\right)$. The bivariate process $\left(B_i(h),W_i(t)\right)$ is bivariate Gaussian with mean 0 and cross-covariance structure $Cov\left(B_i(h),W_i(t)\right)={\sigma }^2\frac{\mathrm{m}\mathrm{i}\mathrm{n}(h,t{)}^2}{2}$. Note that $B_i(t)$ and the bivariate process ($B_i(t),W_i(t)$) are Markov, whereas $W_i(t)$ is non-Markov.

We assume that $S_i(s)$ has a Normal distribution with mean 0 and covariance matrix ${\mathrm{\Sigma }}_i$ where ${\mathrm{\Sigma }}_i$ consists of exponential covariance function for all pairs of locations and $Cov\left(S_i\left(s_i\right),S_i\left(s_j\right)\right)={\eta }^{2}exp\left(-d\left(s_i,s_j\right)\right)$ where $d\left(s_i,s_j\right)$ is the distance between two ZIP codes (locations). Here, we assume an exponential covariance function for the spatial variability, that induces an exponentially decaying correlation as the distance between two measurements (location) increases. We chose an exponential decay function for modeling spatial correlation due to its simplicity and the practical advantage of providing a rapidly decreasing correlation as distance increases. This ensures that locations further apart have little influence on each other, which reflects real-world phenomena where spatial dependencies weaken quickly with distance. The exponential decay also produces a smooth surface for spatial variability, which aligns well with the geographic scale of our data and allows for efficient computation. This approach has been commonly used in spatial modeling due to its ability to capture fast-decaying correlations without overcomplicating the model (Matern 2013). While more flexible alternatives like the Matérn function (Matern 2013) could be considered, we opted for exponential decay to balance model complexity and computational efficiency.

The mean response function, ${\mu }_i\left(t_{ij}\right)$ can be decomposed into, ${\mathrm{\Psi }}_i\left(t_{ij}\right)\alpha +X_i\left(t_{ij}\right)\beta$, where ${\mathrm{\Psi }}_i\left(t_{ij}\right)$ denotes a set of covariates that we do not perform variable selection and $\alpha$ denotes the corresponding set of population-averaged parameters not to be shrunk. Similarly, $X_i\left(t_{ij}\right)$ denotes a set of covariates that we perform variable selection and $\beta$ denotes the corresponding set of population-averaged parameters to be shrunk.

The above model specifications induce the following multivariate Normal distribution on $Y_i$:

$$Y_i~MVN\left({\mathrm{\Psi }}_i\alpha +X_i\beta ,V_i(\varphi )\right)$$ (3)

where $Y_i={\left[Y_i\left(t_{i1}\right),\dots ,Y_i\left(t_{i{n}_i}\right)\right]}^T$, ${\mathrm{\Psi }}_i={\left(\begin{array}{rrrr}1& 1& \dots & 1\\ t_{i1}& t_{i2}& \dots & t_{i{n}_i}\end{array}\right)}^T$, $\alpha =\left(\genfrac{}{}{0pt}{}{{\alpha }_0}{{\alpha }_1}\right)$, $X_i={\left[X_i{\left(t_{i1}\right)}^T,\dots ,X_i{\left(t_{i{n}_i}\right)}^T\right]}^T$ with $X_i\left(t_{ij}\right)=\left[X_{i1}\left(t_{ij}\right),\dots ,X_{ip}\left(t_{i{n}_i}\right)\right]$, $\beta ={\left({\beta }_0,\dots ,{\beta }_p\right)}^T$. $V_i(\varphi )={\omega }^2{J}_i+{\sigma }^2{R}_i+{\eta }^2{T}_i+{\tau }^2{I}_i$ where $\varphi =\left({\omega }^2,{\sigma }^2,{\eta }^2,{\tau }^2\right),J_i$ is an $n_i\times n_i$ matrix of ones, $R_i$ is an $n_i\times n_i$ matrix with the ($j,k$)th element

$$\frac{\mathrm{m}\mathrm{i}\mathrm{n}{\left(t_{ij},t_{ik}\right)}^2}{2}\left[\mathrm{m}\mathrm{a}\mathrm{x}\left(t_{ij},t_{ik}\right)-\frac{\mathrm{m}\mathrm{i}\mathrm{n}\left(t_{ij},t_{ik}\right)}{3}\right]$$

$I_i$ is an $n_i\times n_i$ identity matrix and $T_i$ is an $n_i\times n_i$ matrix with $(j,k)$ th element $\mathit{exp}\left[-d\left(s_{ij},s_{ik}\right)\right]$ and $\mathit{diag}\left(T_i\right)=1$.

Specifically, the model we apply to the CF data has the following form:

$$Y_{ijk}={\alpha }_0+{\alpha }_{1}Ag{e}_i+\sum _{l=1}^p{\beta }_l{X}_{il}+U_i+W_i\left(t_{ij}\right)+S_i\left(s_{ijk}\right)+Z\left(t_{ijk}\right)$$ (4)

In model (4), our main interest is shrinking ${\beta }_l$’s to identify potential predictive predictors. The spike and slab prior peak at zero and also have large probability mass for a wide range of nonzero values. Hence, based on the data, regression coefficients close to zero will be assigned to the spike, resulting in shrinkage towards 0, while coefficients that deviate substantially from zero will be assigned to the slab, resulting in (almost) no shrinkage. It is not one of the primary interests to shrink the coefficient for the time variable especially in longitudinal models. If we have any additional variables for which we do not want to shrink their coefficients (e.g., gender, genotype), we can retain them in the model in a similar fashion. It can be of biological importance to simultaneously adjust the model for other variables while performing variable selection.

In this study, we model time and space as independent components for simplicity and computational efficiency. While this assumption may not fully capture complex spatiotemporal interactions, it allows us to focus on the primary temporal trends and spatial variability in lung function decline. It is worth noting that some interaction terms may indeed mimic latent space-time effects, even though such interactions are not explicitly included in the current model. Future work could explore models that incorporate spatiotemporal interactions to capture these latent effects more accurately.

In our model, nonstationarity primarily arises in the temporal domain, where we use IBM to capture the irregular, cumulative changes in lung function over time. IBM introduces nonstationarity by allowing the process to evolve over time without reverting to a fixed mean, reflecting the progressive nature of CF. This nonstationarity in the temporal component captures the disease’s gradual and unpredictable decline. On the other hand, we assume stationarity in the spatial domain by using an exponential covariance structure. The spatial random effects are modeled isotropically, meaning that the correlation between two locations depends only on the distance between them, not on their specific coordinates. This assumption simplifies the model and reflects the relatively stable nature of spatial variability in our data, where spatial effects are not expected to vary significantly across locations. The combination of temporal nonstationarity and spatial stationarity allows us to capture complex longitudinal patterns while maintaining a manageable level of model complexity. We recognize that nonstationarity could theoretically arise from other sources, such as spatial effects, but based on the structure of our data and previous exploratory analyses, the primary source of nonstationarity is temporal.

3 |. Bayesian Spike and Slab Linear Mixed Effects Model

Our novel Bayesian model utilizes a spike and slab prior for simultaneous variable selection and accounts for spatial variability, which is particularly beneficial when coupled with nonstationary Gaussian linear mixed effects modeling. While spike-and-slab priors are well-established for variable selection in high-dimensional settings (Tang et al. 2017; Ročková and George 2014, 2018; Ray, Ročková, and George 2021), our application within a nonstationary Gaussian LME model that includes both integrated Brownian motion and spatial random effects represents a novel methodological advancement. This novel model accounts for spatial variability by including an additional random effect term based on ZIP code distances in the model. We assume an exponential covariance function for the spatial variability, which induces an exponentially decaying correlation as the separation in distance between two measurements (location) increases. Hypothesis tests designed for Bayesian variable selection often require correction for multiplicity (Scott and Berger 2010; Bühlmann, Kalisch, and Meier 2014; Li et al. 2014). Our proposed model based on the spike and slab prior enables obtaining multiplicity adjusted PIPs for each candidate predictors, which can be used to make variable selection. Nonetheless, using these PIPs allows us to pursue a false discovery rate controlling approach by estimating Bayesian false discovery rate (BFDR) (Ventrucci, Scott, and Cocchi 2011; Heuven and Janss 2010; Li et al. 2014). A decision rule based on BFDR can be used to determine key predictors while controlling BFDR under a prespecified threshold, which enables multiplicity adjustment.

3.1 |. Model Specification

In the Bayesian framework, the spike and slab prior is commonly used as a probabilistic construct for high dimensional modeling to induce variable selection. Using the spike and slab prior is typically implemented by introducing latent binary indicator variables for each predictor. Inference is then carried out using posterior summaries of these latent variables. Prior choices for the parameters and posterior computation for the proposed model are provided below.

The variable selection procedure can be seen as deciding which of the coefficients, the ${\beta }_l$’s, are equal to zero. Each ${\beta }_l$ should therefore have a spike and slab prior, with a spike either exactly at or around zero, and a flat slab elsewhere. For this, we may use an auxiliary indicator variable ${\pi }_l$ where ${\pi }_l=1$ indicates presence, and ${\pi }_l=0$ absence of predictor $l$ in the model to denote whether the variable is in the slab or spike part of the prior. The spike and slab prior specified below plays a similar role as the $L_1$ penalty in LASSO, facilitating variable selection (Ray, Ročková, and George 2021; Habier et al. 2011).

For measurement error ${\tau }^2$ in model (4), the noninformative scale-invariant prior $1/{\tau }^2$ is used which causes no issue with propriety but any inverse-gamma prior would be appropriate. Conditioning the prior of $\beta$ on the measurement error ${\tau }^2$ guarantees the unimodality of posteriors which accelerate convergence of the MCMC sampler and makes point estimates more meaningful. The Bayesian expression of the model would be as follows:

$$\begin{array}{r}{Y}_i~MVN\left({\mathrm{\Psi }}_i\alpha +X_i\beta ,V_i(\varphi )\right)\\ V_i(\varphi )={\omega }^2{J}_i+{\sigma }^2{R}_i+{\eta }^2{T}_i+{\tau }^2{I}_i\\ \beta ~\left(1-{\pi }_l\right){\delta }_0+{\pi }_{l}N\left(0,s^2{\tau }^2\right)\\ s^2~Inv-Gamma\left(1/2,{\lambda }^2/2\right)\\ {\tau }^2~Inv-Gamma\left(b_1,b_2\right)\\ {\pi }_l~Bern(\theta )\\ \theta ~Beta\left(a_1,a_2\right)\end{array}$$ (5)

For the indicator ${\pi }_l$ (which is prior probability of ${\beta }_l$ is being signal/relevant predictor), a Bernoulli prior is assigned. In our analysis, a $Beta\left(a_1,a_2\right)$ prior is assigned to $\theta$ based on the previous research (Scott and Berger 2006, 2010; Gecili and Sivaganesan 2023). A default choice for the parameters of this Beta prior would be $a_1=a_2=1$, which implies setting prior mean of $\theta$ to 0.5, so that we let data to mainly determine ${\pi }_j$. However, one may assume smaller prior mean for $\theta$, by choosing appropriate values for $a_1$ and $a_2=1$, to reflect the prior knowledge on the true number of signals (predictors). With a well-chosen prior on $\theta$, the prior ${\pi }_l~Bern(\theta )$ also automatically favors parsimonious models in high dimensions, thus avoiding the curse of dimensionality (Ray, Ročková, and George 2021). As discussed before, in model (4), we only aim to penalize the ${\beta }_l$’s, but not ${\alpha }_0$ and ${\alpha }_1$. Hence, ${\alpha }_0$ and ${\alpha }_1$ are modelled separately from ${\beta }_l$’s and are assumed to have noninformative uniform priors such that ${\alpha }_0,{\alpha }_1~Uni(-\mathrm{\infty },\mathrm{\infty })$. Further, to simplify the computations we reparametrize the variance parameters as ${\omega }^2={\varphi }_1{\tau }^2,{\sigma }^2={\varphi }_2{\tau }^2$, and ${\eta }^2={\varphi }_3{\tau }^2$, so that $V_i(\varphi )={\tau }^2\left({\varphi }_1{J}_i+{\varphi }_2{R}_i+{\varphi }_3{T}_i+I_i\right)={\tau }^2{V}_i^0(\varphi )$ where ${\varphi }_1,{\varphi }_2,{\varphi }_3>0$. This reparametrisation allows us to have a closed form for the full conditional posterior distribution of ${\tau }^2$, which speeds up our MCMC algorithm. The choice of priors for ${\varphi }_1,{\varphi }_2$, and ${\varphi }_3$ is discussed below to complete model specification. By following a previous work (Gecili et al. 2022), we assume that $\frac{{\varphi }_1}{{\varphi }_1+1}~Uni(\mathrm{0,1})$. This induces that $\pi \left({\varphi }_1\right)=\frac{1}{{\left(1+{\varphi }_1\right)}^2}$. Similarly it can be done for ${\varphi }_2$ and ${\varphi }_3$ and we obtain that $\pi \left({\varphi }_2\right)=\frac{1}{{\left(1+{\varphi }_2\right)}^2}$ and $\pi \left({\varphi }_3\right)=\frac{1}{{\left(1+{\varphi }_3\right)}^2}$. Assigning these priors for ${\varphi }_1,{\varphi }_2$, and ${\varphi }_3$ implies that $\pi \left({\sigma }^2\mid {\tau }^2\right)=\frac{{\tau }^2}{{\left({\tau }^2+{\sigma }^2\right)}^2},\pi \left({\omega }^2\mid {\tau }^2\right)=\frac{{\tau }^2}{{\left({\tau }^2+{\omega }^2\right)}^2}$, and $\pi \left({\eta }^2\mid {\tau }^2\right)=\frac{{\tau }^2}{{\left({\tau }^2+{\eta }^2\right)}^2}$. These priors have been previously used for the multiple testing problem of Binomial proportions and Normally distributed data (Scott and Berger 2006, 2010; Gecili et al. 2023). In our computations, we set $a_1=a_2=1,b_1=b_2=0.01$, and $\lambda =0.5$.

3.2 |. Posterior Computation

Unknown parameters and hyper-parameters of the model are sampled from their conditional posterior distributions through the MCMC algorithm. Given the priors and the data likelihood, the posterior distributions of all unknown parameters can be obtained by using the Bayes’ theorem. Below, we omit derivation of all conditional posterior distributions for the unknowns but provide their final forms. The conditional posterior distributions of many of the unknown parameters are available in closed form due to conjugacy, which improves the convergence and reduces the computation time for our MCMC algorithm.

The full conditional posterior distribution of $\alpha$ is $MVN\left({\mu }_{\alpha },{\mathrm{\Sigma }}_{\alpha }\right)$, where

$${\mathrm{\Sigma }}_{\alpha }={\left(\sum _{i=1}{\mathrm{\Psi }}_i^T{V}_i(\varphi {)}^{-1}{\mathrm{\Psi }}_i\right)}^{-1}\text{and}$$

$${\mu }_{\alpha }={\mathrm{\Sigma }}_{\alpha }{\left(\sum _{i=1}{\left(Y_i-X_i\beta \right)}^T{V}_i(\varphi {)}^{-1}{\mathrm{\Psi }}_i\right)}^T$$

Similarly, the full conditional posterior distribution of $\beta$ is $MVN(\mu ,\mathrm{\Sigma })$, where

$$\mathrm{\Sigma }={\left({\left({\tau }^2{s}^{2}I\right)}^{-1}+\sum _{i=1}{X}_i^T{V}_i(\varphi {)}^{-1}{X}_i\right)}^{-1}\text{and}$$

$$\mu =\mathrm{\Sigma }{\left(\sum _{i=1}{\left(Y_i-{\mathrm{\Psi }}_i\alpha \right)}^T{V}_i(\varphi {)}^{-1}{X}_i\right)}^T$$

The posterior distribution of $\theta$ is a Beta distribution,

$$\pi (\theta \mid .)~Beta\left(a+\sum {\pi }_i,b+\sum \left(1-{\pi }_i\right)\right)$$

The posterior distribution of $s^2$ is inverse Gamma distribution,

$$\pi \left(s^2\mid .\right)~\mathit{Inv}-\mathit{Gamma}\left(\frac{1}{2}+\frac{\sum {\pi }_i}{2},\frac{{\lambda }^2}{2}+\frac{{\beta }^T\beta }{2{\tau }^2}\right)$$

The posterior distribution of ${\tau }^2$ is inverse Gamma distribution,

$$\pi \left({\tau }^2\mid .\right)~Inv-Gamma\left(a_1+\sum n_i+p,a_2+\frac{A}{2}\right)$$

where $A=\sum {\left(Y_i-{\mathrm{\Psi }}_i\alpha -X_i\beta \right)}^T{V}_i^0(\varphi {)}^{-1}\left(Y_i-{\mathrm{\Psi }}_i\alpha -X_i\beta \right)+{\beta }^T\beta /2$.

The conditional posterior of ${\pi }_l=0$ is a Bernoulli distribution with $1-{\eta }_l$ chance parameter:

$$1-{\eta }_l=\frac{(1-\theta )}{\begin{array}{c}{\left(s^2{\tau }^2\right)}^{-\frac{1}{2}}\mathit{exp}\left(\frac{{\left(\sum _{i=1}^n{x}_i{z}_i\right)}^2}{2{\tau }^2\left(\sum _{i=1}^n{x}_i^2+\frac{1}{s^2}\right)}\right){\left(\frac{{\tau }^2}{\left(\sum _{i=1}^n{x}_i^2+\frac{1}{s^2}\right)}\right)}^{1/2}\\ \theta +(1-\theta )\end{array}}$$

where $z_l=y_l-{\psi }_l\alpha -x_{-l}{\beta }_{-l}$ changes depending which ${\beta }_l$ is being sampled.

Finally, we derive the conditional posterior distribution for ${\varphi }_1,{\varphi }_2$, and ${\varphi }_3$ from the joint posterior distribution. Here, the conditional posterior distributions of ${\varphi }_1,{\varphi }_2$, and ${\varphi }_3$ have the same form and they are given below. And, the corresponding Metropolis-Hastings algorithm is developed to update ${\varphi }_r$ for $r=1,2,3$ based on the below expression.

$$\begin{gathered} \pi \left({\varphi }_r\mid .\right)\propto \prod _{i=1}^M{\left|V_i^0(\varphi )\right|}^{-1/2}exp\left(-\frac{1}{2{\tau }^2}{\left(Y_i-{\mathrm{\Psi }}_i\alpha -X_i\beta \right)}^T{V}_i^0(\varphi {)}^{-1}\right. \\ \left.\left(Y_i-{\mathrm{\Psi }}_i\alpha -X_i\beta \right)\right)\pi \left({\varphi }_r\right) \end{gathered}$$

Upon deriving all the necessary conditional posterior distributions for unknown parameters, we use MCMC to estimate the posterior distribution of each parameter by sampling posterior samples from the corresponding posterior distribution, given the current values of all other parameters and the observed data. In the following sections, the posterior distributions are summarized by using maximum a-posteriori (MAP) estimates along with median, mean, standard deviation (SD), and 95% credible intervals (CI) for the posterior mean estimates upon convergence. Convergence is assessed through inspecting trace plots and $\hat{R}$ (Vehtari et al. 2021). The described procedures and MCMC algorithms were implemented in R (R Core Team 2020), and the computer code is publicly available at https://github.com/emrahgecili/SSSlme.

3.3 |. Decision Making

After obtaining the posterior samples from the posterior distribution of indicators and regression coefficients, the posterior inclusion probability (PIP), can simply be calculated as the mean value of the indicator ${\pi }_l$’s for all posterior samples. Thus, the higher the posterior inclusion probability of the indicator variable, the higher is the empirical evidence that the corresponding predictor’s coefficient might be different from zero and therefore have a nonzero impact on the response variable.

As Efron et al. (2001). and Ventrucci, Scott, and Cocchi (2011). have shown, the posterior probability of the lth predictor being irrelevant (noise), $1-\hat{P}\left({\pi }_l=1\mid y\right)$, can be interpreted as an approximation of a local false discovery rate (lFDR). A BFDR-based decision rule can be derived to identify relevant predictors while controlling the BFDR below a predetermined threshold, which induces multiplicity adjustment. In particular, the lFDRs for each predictor are arranged in ascending order. The average value of the lFDRs for the first $q$ predictors $(q=1,\dots ,p)$ is defined as a BFDR for these predictors. We then find the highest possible value of BFDR (calculated as the average of the $q$ smallest lFDRs), which remains below a specific threshold, and the corresponding predictors are classified to be significant (Li et al. 2014). In the following sections, we mainly used the BFDR-based decision rule with commonly used prespecified thresholds for our simulations and the real data applications.

4 |. Simulation Studies

We used simulation studies to investigate variable selection performance of the proposed spike and slab regression model. In our simulations, we mimic our motivating research question and data by following Gecili et al. (2022). We generate data following the setting based on model (4) with different numbers of predictors: $p=20,100$, or $500$. We assume a total of $n=150$ patients, aged between 6 to 18 years, each with a number of repeated measurements ranging from 20 to 100. The predictors/geomarkers were generated from a p-dimensional multivariate Normal distribution with mean 0 and compound symmetric covariance matrix $A_{p\times p}$, where off-diagonal elements of $A$ correspond to $\rho =0.1,0.6$, and $0.9$, and the diagonal elements are 1. In our simulations we assume that 5 of the $p$ predictors have nonzero effects and the remaining $p-5$ predictors have zero effects. We assume the true coefficient vector $\beta =\{90,3,2,2,1,1,0,\dots ,0{\}}_{p+1}$ to generate the response variable FEV1%. When number of signals fixed at 5, increasing $p$ allows us to consider sparse scenarios, reflective of variable selection in sparse and high dimensional problems. Here, the mean effect was fixed at 90 and we fixed ${\sigma }^2=0.75,{\omega }^2=132,{\tau }^2=59$, and ${\eta }^2=10$ in $V_i$ based on our available data from the CF cohort to simulate more realistic values for the response variable. While varying these parameters could provide deeper insights into the model’s robustness, we focused on a fixed set of values to streamline the initial study and simplify the comparison across different scenarios.

In each simulation setting, we generated 100 data sets and implemented our MCMC algorithms to obtain posterior samples, as previously described. These samples were used to calculate BFDRs through obtaining multiplicity adjusted posterior inclusion probabilities for each predictors. We evaluated the variable selection performance of the proposed model by considering various thresholds (e.g., 0,01 or 0,05) for the BFDR-based rule to select relevant predictors. The simulation results, summarized over 100 replications, include the average numbers of true discoveries (TD; correctly identified nonzero effects) and false discoveries (FD; incorrectly identified nonzero effects), presented for different simulation settings in Table 1. Additionally, we report the rates of correct and incorrect inclusions of predictors in the model. In evaluating the performance of the proposed procedure, it is noteworthy that the nonzero coefficients that we used in our simulations are {3, 2, 2, 1, 1}, and these signals are not strong (not quite far from zero). Correctly classifying such observations as signals is difficult, as the signal-to-noise ratio is relatively low.

TABLE 1 |.

Variable selection performance based on simulations when BFDR-based decision rule is used. In this simulations, we assume that the true values of 5 nonzero coefficients are {3, 2, 2, 1, 1}, $\rho =\{0.1,0.6,0.9\}$ and variance terms in $V_i$ are fixed as ${\sigma }^2=0.75,{\omega }^2=132,{\tau }^2=59$, and ${\eta }^2=10$.

		ρ = 0.1				ρ = 0.6				ρ = 0.9
Cutoff	p	TD	FD	RRA	RRN	TD	FD	RRA	RRN	TD	FD	RRA	RRN
0.001	20	3.98	0	0.796	0	2.38	0	0.476	0	0.01	0	0.002	0
	100	4	0	0.800	0	0.03	0	0.006	0	0	0	0	0
	500	3.68	0	0.736	0	0	0	0	0	0	0	0	0
0.005	20	4.35	0	0.87	0	3.13	0	0.626	0	0.33	0	0.066	0
	100	4.22	0	0.844	0	0.53	0	0.106	0	0.02	0	0.008	0
	500	4.03	0	0.806	0	0.04	0	0.008	0	1.53	0.04	0.306
0.01	20	4.47	0	0.894	0	3.46	0	0.692	0	0.90	0	0.18	0
	100	4.33	0	0.866	0	1.71	0	0.342	0	0.40	0	0.080	0
	500	4.21	0	0.842	0	0.66	0	0.132	0	4.97	4.38	0.994	0.009
0.05	20	4.82	0.04	0.964	0.003	4.14	0	0.828	0	3.51	0	0.702	0
	100	4.68	0.02	0.936	0	4.02	0	0.804	0	5	14.43	1.000	0.152
	500	4.51	0.02	0.902	0	5	2.66	1.000	0.005	5	495	1.000	1.000
0.1	20	4.92	0.10	0.984	0.007	4.61	0.10	0.922	0.007	4.9	0.95	0.980	0.063
	100	4.85	0.06	0.970	0.001	4.78	0.35	0.956	0.004	5	92.62	1.000	0.975
	500	4.72	0.10	0.944	0	5	15.03	1.000	0.030	5	495	1.000	1.000

Note: RRA is the average rate of rejecting true alternative hypothesis (average rate of correctly including covariates with nonzero effects in the final model). Similarly, RRN is the average rate of rejecting the true null hypothesis (average rate of incorrectly including covariates with zero coefficients in the final model). TD and FD stand for the average numbers of true and false discoveries, respectively.

Based on our simulation experiments, the average rate of correctly-included covariates with nonzero effects in the final model, RRA, is high across most scenarios, suggesting that the proposed procedure effectively selects relevant predictors using the BFDR-based rule. However, it can decrease slightly with increased number of predictors or $\rho$. The average numbers of correctly identified predictors (TD) varies between 3.68 and 4.92 out of 5 as the threshold increases for different values of $p$ and $\rho =0.1$. The RRA values are notably high (approximately 0.80 to 0.98) for $\rho =0.1$ with $p=20$ or $100$, even when a conservative threshold is used for the BFDR-based rule. This rate slightly decreases to ~ 0.74 − 0.94 as $p$ increases to 500. The RRA ranges from 0.80 to 1, while the rate of correctly excluded irrelevant covariates, RRN, ranges from 0 to 0.03 when $\rho =0.6$, with thresholds set at 0.05 and 0.1, respectively. Higher values of $\rho$ tend to result in more FD, especially noticeable when $p$ is large (e.g., 500). This indicates that as the predictors become more correlated, it becomes more challenging for our model to distinguish between relevant and irrelevant predictors. For $\rho =0.9$ and $p=500$, the challenge in variable selection becomes apparent as FD increases significantly, showing that high correlation among a large number of predictors complicates the model’s ability to correctly identify relevant predictors. However, having this high correlation among predictors might be unrealistic for our motivating real data problem. Nonetheless, a threshold of 0.01 still yields the highest average number of true discoveries (TD) (4.97 out of 5) while maintaining the RRA below 0.01. In especially high correlation setting, selecting the threshold for BFDR-based rule requires caution to avoid either zero discovery or an inflated number of false discoveries. The algorithm becomes more conservative as the threshold decreases, which is expected.

According to additional simulation results provided in Tables A1 and A2 in Supporting Information, as $\rho$ increases the PIPs of the noises increase (so the estimated BFDRs decreases), which escalate possibility of making a false discovery. However, these PIPs are still lower than 0.5 for a $\rho \le 0.8$. In contrast, these PIPs were quite high (BFDRs are so low) when $\rho =0.9$ for our simulation setting when we have mixed 5 (later 10) signals with 95 (90) noises. The details of these simulations provided in the Supporting Information. The best threshold seems to depend on the values of $\rho$ and true numbers of irrelevant and relevant predictors in our simulations.

5 |. Midwestern Cystic Fibrosis Cohort Application

The proposed model was implemented to select prognostic predictors among a large set of clinical/ demographic characteristics and geomarkers for predicting rapid lung function decline in CF. The Cincinnati CF center data is acquired from patient’s information at the Cincinnati Children’s Hospital Cystic Fibrosis Center (CCHMC) in Cincinnati, Ohio, USA. This study cohort includes 147 individuals with CF aged 6–20 years old and received care at Cincinnati CF center between 2012 and 2017. The data consists of 2865 observations. The outcome variable is lung function which is measured as percent predicted forced expiratory volume in 1 second (FEV1%). The data includes routinely examined patients’ clinical/demographic variables and regional geomarkers derived from geocoded residential addresses and comprise factors. Geocoding of the candidate geomarkers have been performed by using DeGAUSS (Brokamp 2018). This process and more details about the geomarkers are previously described in another work (Gecili et al. 2023). The average (range) number of observations per individual patient was 19.5 (1–53), with an average (range) follow-up time of 3.85 (0–6.97) years.

The 53 candidate geomarkers were derived to measure characteristics of the built environment using geocoded exact residential addresses and comprise factors. These geomarkers are detailed in Table A4 in the Supporting Information. The list geomarkers include socioeconomic status (7 covariates), access to healthcare (1 covariate), roadway proximity (10 covariates), crime (10 covariates), land cover and impervious descriptors (12 covariates), weather and atmosphere (7 covariates), and air pollution (6 covariates). The census-tract-level socioeconomic status covariates were obtained from the 2015 American Community Survey (ACS) (American Communities Project 2018). The healthcare access markers were calculated by using drivetime isochrones from openrouteservice and “as the crow flies” distances from CCHMC. Roadway proximity covariates were generated by comparing address locations to U.S. Census Bureau shapefiles of primary and secondary roads. Crime covariates were obtained from an analysis of crime data from most law enforcement jurisdictions across the nation from 2011 to 2017 (Solutions 2018). Land cover and impervious descriptor covariates were derived from the National Land Cover Database (NLCD) developed by the Multi-Resolution Land Characteristic (MRLC) Consortium (Jin et al. 2019). Weather data was obtained from the National Centers for Environmental Prediction (NCEP) North American Regional Reanalysis (NARR) daily weather data (Mesinger et al. 2006). Air pollution covariates are generated by ambient air sampling in Cincinnati temporally adjusted using data from EPA monitoring stations (elemental carbon attributable to traffic or ECAT) or spatiotemporal hybrid satellite daily exposure predictions (atmospheric particulate matter) (Brokamp et al. 2017).

We implemented our novel spatial spike and slab linear mixed effects (SSSlme) model for this local data to select predictive variables among commonly examined clinical/demographic characteristics and 53 candidate regional geomarkers in addition to their interaction with encounter age (e.g., predictor × Age). The model utilized age at the clinical encounter as the time variable, alongside time-invariant predictors such as sex, genotype (categorized as F508del homozygous, heterozygous mutation, or neither/unknown), and time-variant variables including pancreatic insufficiency (indicated by the use of pancreatic enzymes), medicaid insurance use, Pseudomonas aeruginosa (PA) infection, Methicillin-resistant Staphylococcus aureus (MRSA) infection, cystic fibrosis-related diabetes mellitus (CFRD), rolling numbers of clinic visits in year to each clinical encounter (numVisityr). All the regressors were standardized. Initially, this model had 122 $\beta$’s for the main and interactions effects to shrink to induce simultaneous variable selection.

In our analysis, we obtained posterior distributions of all the model parameters by using previously stated posterior distributions, the data, and MCMC algorithm. These posterior distributions are summarized in Table A4 in the Supporting Information, providing MAP estimates alongside the median, mean and standard deviation (SD). Additionally, 95% CIs for the posterior mean estimates and estimated Bayesian false discovery rates (BFDR) for each regressors are provided in the same table. Figure 1 presents the posterior mean estimates of coefficients with their corresponding 95% CIs for the selected predictors with estimated BFDR < 0.005. Figure A1 in the Supporting Information visualizes estimated BFDR. The additional forest plots, Figures A2 and A3 for the main effects present the shift in overall FEV1% while Figures A4 and A5, the ones for the interactions provide rate of decline in FEV1%. The computation time of the proposed model was 4~ hours for the Cincinnati CF center data example with ~ 122 predictors for MCMC size 25 K with a regular Windows computer (Intel(R) Core(TM) i5–10310U CPU @ 1.7 GHz, 2.21 GHz, 8 GB RAM).

FIGURE 1 |.

Posterior mean estimates of coefficients and 95% credible intervals (CIs) from the SSSlme model that includes clinical/demographic predictors + 53 geomarkers + interaction of all 62 predictors with encounter age (e.g., predictor × Age). The points and lines represent posterior means for coefficient estimates and their 95% CIs. Coefficients with 95% CIs that do not cross zero are highlighted in brown. On the right, Bayesian false discovery rate (BFDR) values are provided in increasing order, with variables having BFDR < 0.005 displayed.

Using the BFDR-based decision rule that controls BFDR below 0.001 (would be a choice if sparsity is desired), we selected a total of 14 main and interaction terms which are provided in Figure 1 (see variables with green BFDR). The selected predictors includes MRSA infection, exposure to elemental carbon attributable to traffic sources (ECAT), and air temperature, along with the age interaction of ECAT, number of hospital visits in the previous year, PA infection, percent of greenspace, enzymes use, humidity, CFRD, sex, medicaid insurance use, ozone concentration, and length (meters) of S1200 roads (arterial roads and state highways) within 400 m. Additionally, 26 more main and interaction terms are selected when threshold of 0.005 is used for BFDR-based rule (see variables with blue BFDR in Figure 1). These additional predictors include CFRD, F508del homozygous, enzymes use, land cover and impervious descriptors (percent of non-road urban, percent of urban secondary road, percent of tree canopy, percent of impervious, percent of greenspace, percent of not urban secondary road, percent of non-impervious, percent of tertiary road urban, and percent of not urban primary road), socioeconomic status (fraction of high school education and fraction assisted income), assault crime, and length (meters) of S1200 roads (arterial roads and state highways) within 400 m, along with the age interaction of fraction of high school education, median income, deprivation index, air temperature, surface pressure, pm concentration, respiratory hazard index, diesel pm, percent of non-road not urban, percent of not urban secondary road, planetary boundary layer height, motor vehicle theft, and distance (meters) to S1200 roads. As presented in Table A4, some additional predictors and their interactions with encounter age are being selected when a more liberal threshold 0.01 is used. We observed that an increase in ECAT, ozone concentration, humidity, and length (meters) of S1200 roads (arterial roads and state highways) within 400 m are associated with more rapid lung function decline even when the model is adjusted for routinely used clinical and demographic characteristics and a very conservative BFDR-based rule is used for variable selection. When we use more liberal threshold for decision making, we additionally selected more geomarkers including deprivation index, air temperature, percent of non-road not urban, respiratory hazard index, motor vehicle theft, and distance to s1200 m, all of which are associated with increased rapid lung function decline. On the other hand, fraction of high school education, surface pressure, and percent of not urban secondary road are associated with decreased rapid lung function decline. Further details are provided in Figure 1 and Table A4.

Using BFDR-based rule allows controlling BFDR so it enables multiplicity adjustment. Above we reported results for threshold of 0.01 but one can use a more conservative (e.g., using 0.001 as threshold would results in selecting total of 14 main and interaction terms, the predictor with green colored BFDR in Figure 1) or liberal threshold (e.g., using 0.05 as threshold which would result in selecting all candidate predictors at increased risk of making more false discoveries) that is appropriate and guided by the study’s objectives, the nature of data, and the potential consequences of making false discoveries.

We performed a second analysis by including only the main effects of the aforementioned clinical/demographic covariates and geomarkers in our proposed model. In this analysis, the initial model included a total of 63 predictors. Similar to previous analyses, the posterior estimates of the model parameters are again summarized in Table A5. Figure 2 demonstrates the posterior mean estimates of coefficients with 95% CIs for the selected predictors, focusing on those with an estimated BFDR of less than 0.01.

FIGURE 2 |.

Posterior mean estimates of coefficients and 95% credible intervals (CIs) from the SSSlme model that includes clinical/demographic predictors + 53 geomarkers. Points and lines represent posterior means for coefficient estimates and their 95% CIs. Coefficients with 95% CIs that do not cross zero are highlighted in brown. On the right, Bayesian false discovery rate (BFDR) values are provided in increasing order, with variables having BFDR < 0.01 displayed.

In this application, we select sex, MRSA infection, PA infection, CFRD, enzymes use, medicaid insurance use, ECAT, air temperature, percents of not urban primary and secondary road, and length of S1200 roads (arterial roads and state highways) within 400 m when we used BFDR-based rule with threshold of 0.001. In addition to these variables, greenspace %, distance (meters) to S1100 roads, percent of impervious, percent of urban tertiary road, and fractions of high school education and assisted income are selected when threshold of 0.005 is used. The number of hospital visits in the previous year, genotype (F508del homozygous), percent of non-impervious, motor vehicle theft, and rape are also selected in addition to others when threshold of 0.01 is used for the BFDR-based rule. Figure A6 in the Supporting Information shows estimated BFDR for the predictors, while Figures A7 and A8 in the Supporting Information for the main effects illustrate the shift in overall FEV1%.

A third analysis was conducted with a limited set of geomarkers, including ECAT, percent of greenspace, and deprivation index, which were previously studied Gecili et al. (2023). We implemented our model for the aforementioned demographic and clinical covariates and these three geomarkers. Similar to previous examples, the parameter estimates and BFDR values of all predictors are provided in Table A6. Figure 3 provides a forest plot presenting the posterior mean estimates of coefficients with their corresponding 95% CIs for the predictors with BFDR < 0.01. In this application, age interactions of all three geomarkers are selected when BFDR is even controlled under 0.005, see Figure 3. Additionally, Figure A9 in the Supporting Information presents estimated BFDR values while Figure A10 provides the shift in overall FEV1% (left graph) and rate of decline in FEV1% (right graph).

FIGURE 3 |.

Posterior mean estimates of coefficients and 95% credible intervals (CIs) from the SSSlme model that includes commonly used clinical/demographic predictors + 3 geomarkers, which are previously studied elsewhere. The points and lines represent posterior means for coefficient estimates and their 95% CIs. Coefficients with 95% CIs that do not cross zero are highlighted in brown. On the right, Bayesian false discovery rate (BFDR) values are provided in increasing order, with variables having BFDR < 0.01 displayed.

Similar to the initial analyses with larger number of geomarkers, the age interaction of the ECAT and percent of greenspace are again selected with a quite conservative threshold of 0.001, in addition some clinical/demographic covariates. When we use 0.01 as threshold, we additionally select the age interaction of deprivation index. Based on the estimated associations, we can state that all these three geomarkers are associated with more rapid decline. The estimated associations of ECAT and Deprivation index with lung function decline are similar to what we reported in a previous work (Gecili et al. 2023). Specifically, an increase of 0.1 μ g/m³ of ECAT was associated with 0.125% predicted/yr more rapid decline. Similarly, an increase of 0.1-unit in deprivation index was similarly associated to additional decline of 0.096% predicted/yr. In contrast to our previous findings (Gecili et al. 2023), in this work, we found that percent of greenspace is associated with more rapid decline in all three described analyses.

For this third analysis model, we additionally compared our spatial model with two other Bayesian models that we implemented and several frequentist LME models with different spatial correlation structures. We evaluated model fits and prediction accuracies using differences in the Akaike information criterion (dAIC) and Bayesian information criterion (dBIC) and root mean square error (RMSE). The results and details of each model are provided in Table 2.

TABLE 2 |.

Comparison of different models for the third analysis in terms of model fit and prediction accuracy.

Models	Spatial corr	dAIC	dBIC	RMSE	Model details
LME	Ignores			8.54	Traditional random intercept LME model
LME_Exp	Exponential	211	198	8.82	Random intercept LME model + Exponential spatial corr
LME_Gaus	Gaussian	−4	−16	8.54	Random intercept LME model + Gaussian spatial corr
LME_Ratio	Rational quadratic	214	202	8.83	Random intercept LME model + Rational quadratic spatial corr
LME_zip	Random intercept for ZIP code	69	63	8.20	Random intercept LME model + Intercept varying among ZIP code and subject within Zip code
Imenssp	Ignores	519	513	8.39	Frequentist stochastic LME model
SpatBEN	Exponential	527	521	8.09	Bayesian spatial stochastic elastic-net model
SSlme	Ignores	523	517	8.20	Bayesian stochastic spike and slab model
SSSlme	Exponential	539	527	8.04	Bayesian Spatial stochastic spike and slab model

Note: Compared our spatial model with two other Bayesian models that we implemented and several frequentist LME models with different spatial correlation structures. We evaluated model fits and prediction accuracy using differences in the Akaike information criterion (dAIC) and Bayesian information criterion (dBIC) and root mean square error (RMSE). dAIC = AIC_LME — AIC_model, so higher dAIC and dBIC are better for model fit; the smaller RMSE is better for prediction accuracy. Spatial Corr: spatial correlation structure that we assumed. The proposed model, highlighted in blue, outperforms the other models in terms of model fit and prediction accuracy.

The proposed novel Bayesian spatial stochastic LME model (SSSlme) resulted in better model fit (dAIC = 539, dBIC = 527) and greater prediction accuracy (RMSE = 8.04) than other Bayesian and frequentist LME models with different spatial correlation assumptions. The SSSlme model is followed by our Bayesian spatial elastic-net (SpatBEN) model (dAIC = 527, dBIC = 521, RMSE = 8.09). The best-performing frequentist models were the stochastic LME model ignoring spatial correlation, followed by the LME model with the rational quadratic spatial correlation structure (Table 2).

6 |. Conclusions

We incorporated a spatial spike and slab linear mixed effects model that simultaneously induces variable selection and accounts for spatial variability for a Gaussian linear mixed effects model with nonstationary covariance structure. Hence, this model incorporates the complicated structure of long sequences of repeated, irregular measurements on the response variable. We demonstrated through simulation studies that our framework is effective for identifying predictors when the overarching candidate set is of high dimension and the correlation between predictors are not that strong (≤ 0.8). We also illustrated our approach using a Midwestern cystic fibrosis cohort data that consists of routinely used clinical/demographic covariates along with residential addresses of the individuals with CF. The residential addresses were geocoded to obtain a set of potential social and environmental predictors of rapid CF disease progression.

After assessing a variety of scenarios through simulation studies, we found the proposed procedure to be effective in selecting relevant predictors by estimating the PIPs and BFDRs for each predictor. We mainly used a BFDR-based rule for making variable selection through our work but we also reported some simulation results from a case where we directly used PIPs to make a decision. The average rate of correctly including covariates with nonzero effects in the final model, RRA, are quite high across most scenarios especially when $\rho$ is low or $\rho$ is high and $p=20$ or $100$. This suggests that the proposed procedure effectively selects relevant predictors using the BFDR-based rule when appropriate threshold is used. However, it can decrease slightly with increased number of predictors or $\rho$. The average numbers of correctly identified predictors (TD) varies between 3.68 and 4.92 out of 5 as the threshold increases for different values of $p$ and $\rho =0.1$. The RRA values are notably high (approximately 0.80 to 0.98) for $\rho =0.1$ with $p=20$ or $100$, even when a conservative threshold is used for the BFDR-based rule. This rate slightly decreases to ~ 0.74 − 0.94 as $p$ increases to 500. The RRA ranges from 0.80 to 1, while the rate of correctly excluded irrelevant covariates, RRN, ranges from 0 to 0.03 when $\rho =0.6$, with thresholds set at 0.05 and 0.1, respectively. Higher values of $\rho$ tend to result in more FD, especially noticeable when $p$ is large (e.g., 500). This indicates that as the predictors become more correlated, it becomes more challenging for our model to distinguish between relevant and irrelevant predictors. For $\rho =0.9$ and $p=500$, the challenge in variable selection becomes apparent as FD increases significantly, showing that high correlation among a large number of predictors complicates the model’s ability to correctly identify relevant predictors. Despite this, the model performance remains acceptable even when $\rho =0.9$, and improves as $p$ increases ($p<100$) with a fixed number of relevant predictors. Furthermore, having this high correlation among predictors might be unrealistic for our motivating real data problem. Nonetheless, a threshold of 0.01 still yields the highest average number of true discoveries (TD) (4.97 out of 5) while maintaining the RRA below 0.01. As aforementioned, in especially high correlation setting, selecting the threshold for BFDR-based rule requires caution to avoid either zero discovery or an inflated number of false discoveries. The algorithm becomes more conservative as the threshold decreases, which is expected.

The additional simulation results provided in Tables A1 and A2 showed that as $\rho$ increases the PIPs of the nonrelevant predictors increase (so the BFDRs decreases), which increases possibility of making a false discovery. But, the PIPS are still lower than 0.5 for a $\rho \le 0.8$. On the other hand, these PIPs were quite high (so the BFDRs are quite small) when $\rho =0.9$ for our simulation scenarios when we mix 5 (later 10) signals with 95 (90) noises. The best threshold seems to depend on the values of $\rho$ and true numbers of irrelevant and relevant predictors in our simulations. In real life problems, the choice of the thresholds for both PIPs and BFDR-based rule depend on factors such as model complexity restrictions, degree of sparsity required, the cost of excluding a predictor, and context.

In our Midwestern CF cohort application, we successfully identified a set of predictive geomarkers while controlling the BFDR below some certain threshold (achieving multiplicity adjustment) and models are adjusted for the routinely used clinical/demographic covariates. Initially with a comprehensive set of geomarkers, we found that increases in ECAT, ozone concentration, humidity, and length of S1200 roads (arterial roads and state highways) within 400 m were associated with more rapid lung function decline, even after adjusting for clinical and demographic characteristics using a very conservative threshold for BFDR-based rule for variable selection. Adapting a more liberal threshold for decision-making allowed us to identify additional geomarkers, including the deprivation index, air temperature, percent of non-road not urban, respiratory hazard index, motor vehicle theft, distance to S1200 m, all of which are associated with increased rapid lung function decline. Conversely, a higher fraction of high school education, surface pressure, and percent of nonurban secondary road were associated with decreased rapid lung function decline. In the following analysis with only main effects, we identified a similar set of geomarkers (such as ECAT, air temperature, percents of not urban primary and secondary road, and length (meters) of S1200 roads (arterial roads and state highways) within a 400 m with threshold of 0.001, see Figure 2 and Table A5).

In the third analysis on the same cohort with limited number of geomarkers, the age interaction of the ECAT and percent of greenspace are selected with threshold of 0.001, in addition some clinical/demographic covariates. When 0.005 is used as threshold, the age interaction of deprivation index is also selected. Based on the estimated associations, we can state that all three geomarkers are associated with more rapid decline (Figure 3 and Table A6). The estimated associations of ECAT and Deprivation index with lung function decline are similar to what we reported in a previous work (Gecili et al. 2023). We also observed that when we include more geomarkers in the model as we did in our initial analysis above, we observed similar associations between the geomarkers and lung function decline, however the magnitude of the associations diminish as we include more geomarkers in the model. This is clear when we compare the coefficients estimates in Tables A4 to A6 in the Supporting Information.

Another CF geomarker related work has built a social-environmental adversity index based on principal components of various geomarkers using the US Cystic Fibrosis Foundation Patient Registry data (Knapp et al. 2016), and shown that timing of peak decline of lung function varies by this adversity index (Palipana et al. 2023). This confirms the geomarkers that we select have important role in rapid lung function decline in CF.

In more exploratory research or when dealing with highly complex data where true signals are difficult to detect, a higher threshold for the BFDR-based rule might be acceptable. Conversely, in confirmatory research or when the implications of false discoveries are significant, a lower BFDR threshold would be preferable.

In conclusion, our findings suggest that incorporating spatiotemporal effects and selected geomarkers in our novel Bayesian stochastic LME model based on spike and slab prior resulted in an improved prediction tool for predicting rapid CF disease progression. By identifying patients who are likely to experience rapid lung function decline, clinicians can proactively adjust treatment plans and improve patient outcomes. Although motivated by the CF lung function data, the methods described in our work are applicable to any dataset suitable to model (4), including large-scale health studies in which various variables are collected frequently over time.

The use of IBM in our model allowed us to capture the random, yet cumulative changes in lung function over time, which are characteristic of CF progression. This is crucial for predicting rapid lung function decline, as CF patients’ lung function often show irregular patterns that simpler temporal models cannot capture. The continuous accumulation of variability modeled by IBM aligns with both the short-term fluctuations and long-term trends observed in CF data, making it an ideal fit for our longitudinal modeling framework.

The proposed model framework could be extended in several ways. Although we assumed integrated Brownian motion for the stochastic process term $W_i(t)$ in model (4), one can consider alternative stationary or nonstationary processes– whichever is appropriate for the specific context. Similarly, one may consider a different covariance structure instead of exponential covariance that we assumed for the random effects term to account spatial variability. Depending on the process chosen, the model structure may remain unchanged, or it may require assigning priors to additional parameters introduced by the selected process. Findings from our applications are mixed with regard to novel discoveries and corroborations of past work, but the proposed method could be adapted to incorporate grouping structures of the geomarkers. Within this large set of geomarkers, those from the similar sources (e.g., socioeconomic status (7 covariates), roadway proximity (10 covariates), and so on) may form natural groups; these grouping structures should not be ignored. For instance, a Bayesian group lasso for longitudinal genome-wide association studies data under a different modeling approach was considered by Li et al. (2015), where they show improvement in the variable selection and prediction performance. Although our context is not genomics in this work, incorporating this grouping information of the candidate predictors (e.g., geomarkers) could potentially improve variable selection and prediction performance of the proposed model while still allowing for a response variable with complex longitudinal correlation and measurement error and accounting for spatial variability. Hence, as a future development, one can expand these established Bayesian penalized regression models to incorporate the grouping information of geomarkers. Another future research area would be employing our model to nationally representative data from the U.S. Cystic Fibrosis Foundation Patient Registry, which has broad national coverage and would allow confirming some of the select geomarkers to see whether it is still a valid predictor of lung function decline on other parts of the US.

One limitation of our current model is the assumption of independence between time and space, which may not fully reflect the complex interactions that exist in real-world data. While this simplification facilitated the implementation of the model and allowed us to focus on the primary temporal and spatial effects, it is possible that some selected interaction terms could be capturing latent spatiotemporal interactions. Future studies could consider models that explicitly include space-time interactions to further refine predictions of lung function decline in cystic fibrosis patients, particularly in the presence of environmental or social factors that vary across both time and space.

In this study, nonstationarity is primarily introduced through the temporal component of the model via integrated Brownian motion. This allows us to capture the cumulative and irregular changes in lung function that are characteristic of cystic fibrosis progression. The increments of the process remain stationary, but the overall process evolves in a nonstationary manner over time, providing flexibility in modeling long-term trends. Although nonstationarity could arise from other sources, such as spatial effects, we chose to model spatial variability using a stationary isotropic covariance function for simplicity and computational efficiency. This choice reflects the relatively stable spatial patterns in our data. However, the decision to model the temporal component as nonstationary has a significant impact on our ability to accurately predict lung function decline, as it aligns with the unpredictable and progressive nature of the disease. Future work could explore more complex models that account for potential nonstationarity in both the temporal and spatial domains.

While our simulation studies provided valuable insights into the performance of the proposed model, there are limitations to the current setup. In particular, we fixed parameters such as ${\sigma }^2,{\omega }^2,{\tau }^2$, and ${\eta }^2$, as well as the number of nonzero predictors, to values derived from available data. This allowed us to streamline the initial investigation, but it limits the generalizability of our findings. A potential future work would address these limitations by varying these parameters across a range of plausible values to assess the model’s sensitivity and robustness in more diverse scenarios.

Incorporating spatiotemporal effects and geomarkers into this innovative Bayesian stochastic LME model appears to offer a significant advancement in predicting rapid CF disease progression. By enhancing the prediction of lung function decline, this model holds the potential to transform clinical decision-making. Timely identification of rapid decline phases would allow clinicians to adjust treatment plans proactively, potentially improving patient outcomes significantly. Our findings contribute to a growing body of research that underscores the importance of integrating environmental and social factors into medical predictions and treatment strategies for chronic diseases like CF. This integration is not only a stride towards more comprehensive patient care but also a step towards the future of computational medicine in chronic disease management.

Supplementary Material

Supporting Information

Additional supporting information can be found online in the Supporting Information section.

Data Availability Statement

The data that support the findings of this study are available from the Cystic Fibrosis Center of Cincinnati Children’s Hospital Medical Center, but restrictions apply to the availability of these data, which were used under an information use agreement for the current study, and so are not publicly available. Requests for these data can be directed to the Cystic Fibrosis Center of Cincinnati Children’s Hospital Medical Center.