Bayesian hierarchical profile regression for binary covariates

Abstract

Dysphagia, a common result of other medical conditions, is caused by malfunctions in swallowing physiology resulting in difficulty eating and drinking. The Modified Barium Swallow Study (MBSS), the most commonly used diagnostic tool for evaluating dysphagia, can be assessed using the Modified Barium Swallow Impairment Profile (MBSImP^™). The MBSImP assessment tool consists of a hierarchical grouped data structure with multiple domains, a set of components within each domain which characterize specific swallowing physiologies, and a set of tasks scored on a discrete scale within each component. We lack sophisticated approaches to extract patterns of physiologic swallowing impairment from the MBSImP task scores within a component while still recognizing the nested structure of components within a domain. We propose a Bayesian hierarchical profile regression model, which uses a Bayesian profile regression model in conjunction with a hierarchical Dirichlet process mixture model to (1) cluster subjects into impairment profile patterns while respecting the hierarchical grouped data structure of the MBSImP, and (2) simultaneously determine associations between latent profile cluster membership for all components and the outcome of dysphagia severity. We apply our approach to a cohort of patients referred for an MBSS and assessed using the MBSImP. Our research results can be used to inform appropriate intervention strategies, and provide tools for clinicians to make better multidimensional management and treatment decisions for patients with dysphagia.

1 |. INTRODUCTION

Dysphagia is a condition where subjects face difficulty when eating and drinking due to physiologic malfunction when swallowing. While not a disease itself, dysphagia is often a comorbidity resulting from multiple medical conditions and diseases such as stroke or cancer.¹ Clinically, dysphagia can be assessed using the Modified Barium Swallow Study (MBSS), the preferred approach for characterizing swallowing impairment in patients with dysphagia.^1–3 The MBSS is a videofluoroscopic examination which utilizes standardized, customized, and commercially prepared barium consistencies to allow clinicians to visually assess any abnormalities associated with the swallowing processes. The Modified Barium Swallow Impairment Profile (MBSImP^™) is a validated tool which was created to standardize the examination protocol, visual observations and interpretation of swallowing impairment when using the MBSS.^1,3,4

The MBSImP consists of a naturally formed, hierarchical multiple group data structure. It includes three domains: oral, pharyngeal, and esophageal. Within each domain, there exist a set of components which characterize specific swallowing physiologies (eg, Lip Closure, Tongue Control During Bolus Hold, Bolus Preparation/Mastication, etc.). Within each component there are a set of tasks evaluated, which are a variety of standardized bolus viscosities and volumes known to influence swallowing physiology.^3,4 The tasks that can be evaluated within a component include: thin 5 mL (T5), thin cup sip (TCS), thin sequential swallow (TSS), nectar 5 mL (N5), nectar cup sip (NCS), nectar sequential swallow (NSS), thin-honey (H), pudding (P), and a cookie coated in barium (C). The number of tasks administered is specific to each component and task scores can be characterized as either unimpaired or impaired. The MBSImP allows for an objective and standardized characterization of the underlying mechanistic causes of dysphagia.^4,5 This quantification of the swallowing physiology ultimately facilitates better communication between clinicians when discussing patient dysphagia attributes and allows for research into the physiologic malfunctions which result in the manifesting dysphagia.^1,5,6

Our research goals regarding the motivating MBSImP data are: (1) to identify clusters which represent latent patterns of physiologic swallowing impairment, (2) allow for the identified latent impairment patterns to be shared across components, and (3) characterize the associations between latent impairment pattern cluster membership and the outcome of clinical dysphagia severity. In recent work by Beall et al.,⁶ the authors performed a latent class analysis (LCA) to develop a classification system for dysphagia severity by assuming MBSImP scores are manifestations of a latent ordinal disease severity class structure. While this work revealed latent clusters in a sample of patients with dysphagia, there were two main limitations of this work with respect to addressing our research goals: the lack of acknowledgment of the hierarchical multiple group data structure of the MBSImP and the inability to jointly estimate the latent impairment patterns and their association with clinical dysphagia severity.

First, the LCA did not consider the MBSImP’s hierarchical grouped data structure, since a standard LCA cannot accommodate this type of data. As an alternative to a finite mixture model, a Dirichlet process (DP) mixture model (DPMM) can be used when the number of distinct clusters in the data is unknown, as the DP allows for the number of non-empty clusters to be inferred.^7,8 However, a DPMM, similar to LCA, cannot accommodate data with a hierarchical multiple group structure and it lacks the capability to link the identified clusters with the outcome of interest. Similar to the DPMM is the hierarchical Dirichlet process mixture model (HDPMM) detailed by Teh et al.,⁹ which allows for the number of clusters to be inferred by the model, can accommodate a hierarchical multiple group data structure, and allows for information to be shared among the covariate groups. Applications of HDPMM as well as other methods such as Bayesian consensus clustering (BCC) and multivariate latent class linear mixed effects models (MLCLMEM) demonstrate the utility of being able to estimate latent profiles among data which have a hierarchical multiple group structure.^10–12 For example, Lu and Lou¹¹ present a novel adaptation of BCC for multivariate longitudinal data, specifically longitudinal biomarkers for primary biliary cirrhosis. While these works demonstrate the ability to cluster hierarchically grouped data, assessments for associations between cluster memberships and any outcomes is often completed in an independent step instead of jointly estimated with the cluster memberships.^10,11

Second, standard LCA and other methods discussed are not designed to jointly estimate latent class membership and the associations between latent class membership and an outcome of interest. Standard regression based approaches attempting to quantify the associations between patterns in the task level data within a component and dysphagia severity are challenged by the dimensionality of the MBSImP data being considered. Following an explanation by Molitor et al.,¹³ regression based approaches face difficulty drawing clinically relevant inference when a large number of covariates, often confounded, are included in the primary analysis model. In the context of our motivating data, if only four components of the MBSImP are considered, there are a maximum of 36 tasks which would be evaluated. If, within component, only two way interactions were considered in a linear model, then there would be 181 predictors in the model (1 intercept, 36 main effects, 144 two way interaction terms). As such, attempting to characterize the association between task level data and a measure of dysphagia severity using standard regression based approaches is prevented due to the dimensionality of the data. Molitor et al.¹³ introduced Bayesian profile regression (BPR) to address this dimensionality issue, which is an approach used to identify latent clusters within a set of covariates and jointly assesses the associations between cluster membership and a clinically relevant outcome. Howlett et al.¹⁰ combined BPR with a MLCLMEM in a two-stage model. The MLCLMEM is utilized in the first stage to discover latent class phenotypes among a set of longitudinally collected outcomes and estimate subject class membership. The second stage of the model uses the estimated class memberships in a BPR model to assess for complex associations between the subject class memberships and several outcomes of interest. Dunson et al.¹⁴ developed a similar method to BPR which allows for the estimation of changes in a response density with respect to a set of estimated predictor cluster memberships, where cluster memberships are estimated using a DP. While both of these methods allow for the estimation of latent clusters and the subsequent associations between cluster membership and some outcome of interest, neither method is capable of identifying latent clusters within data which have a hierarchical multiple group structure. That is, in relation to our motivating data, these methods can be only applied to a the individual MBSImP components.^13–16

Given the observed limitations of existing methods to address our research goal, we propose Bayesian hierarchical profile regression (BHPR) for binary data which uses the approach of BPR in conjunction with the HDPMM. The utilization of the HDPMM allows for the estimation of cluster memberships for data which have a multiple hierarchical grouped data structure and the approach of BPR allows for the estimation of the associations of cluster memberships with a clinical outcome of interest. Our proposed model has several attractive features. First, it can identify latent patterns of impairment given MBSImP task scores within each component. Second, the proposed method is able to identify latent patterns for multiple components and allows latent patterns to be shared across components. Lastly, it characterizes the associations between the latent impairment profile clusters for each component and clinical dysphagia severity.

The development of this statistical tool will facilitate the ability of clinicians to discover clusters of patients characterized by patterns in swallowing impairment and simultaneously assess for associations between cluster membership and clinical dysphagia severity, while taking into account the hierarchical multiple group data structure of the MBSImP. Such a tool may assist clinicians in deciding treatment plans for patients with dysphagia in different swallow impairment patterns and at different levels of dysphagia severity.

2 |. METHODS

In this section, we describe the details of the BHPR. The proposed BHPR model estimates a cluster membership for each subject for each covariate group according to their impairment profile within that covariate group and estimates the association of cluster memberships with the outcome. The primary assumption for our model is conditional independence (CI) between the manifest variables within task group, which means that all manifest variables with covariate group are independent given the latent impairment profile cluster memberships.

BHPR, similar to BPR detailed by Reference 13, consists of two “sub-models”, the covariate group cluster assignment sub-model and disease sub-model. The covariate group cluster assignment sub-model estimates subject membership in the latent impairment profile pattern clusters, henceforth referenced as clusters, for each subject for each covariate group. The estimated cluster memberships are then used as covariates in the disease sub-model, where the effect of cluster membership on the outcome is assessed. Using the covariate group cluster assignment sub-model we are able to group patients into clusters at the covariate group level. These clusters are characterized by their impairment profile for that covariate group. We adapted a HDPMM to facilitate cluster discovery, as it allows for clusters to share information across the covariate groups by constructing two levels of mixtures: global and covariate group specific.⁹ The global mixture acts as a reserve of the estimated impairment patterns which can then be allocated to the individual covariate group level mixtures according to covariate group specific mixture weights.

2.1 |. Covariate group cluster assignment sub-model

In the covariate group cluster assignment sub-model, henceforth referred to as the assignment sub-model, for each subject i where $i=1,\dots ,N$, we denote the collection of all observed covariate data by ${\mathbf{x}}_i=\left({\mathbf{x}}_{i1},\dots ,{\mathbf{x}}_{iJ}\right)$, where this collection represents the variables that are naturally grouped into J groups. For each of the J covariate groups, let ${\mathbf{x}}_{ij}=\left(x_{ij1},\dots ,x_{ijK}\right)$ represent a set of K observed manifest variables within the covariate group j. Let $x_{ijk}=1$ represent an indicator of impairment for subject i on manifest variable k in covariate group j. Additionally, let $z_{ij}$ represent the latent cluster membership for subject i on covariate group j, ${\pi }_{cj}=P\left(z_{ij}=c\right)$ for $c=1,\dots C$ represent the cluster prevalences within covariate group j, and ${\varphi }_{cj}^k=P\left(x_{ijk}=1\mid z_{ij}=c\right)$ represent the latent covariate group specific item response probabilities for subject i on manifest variable k in covariate group j given membership in cluster c. The probability of observing a covariate group specific score profile is shown in Equation (1).

$$P\left(x_{ij}\right)=\sum _{c=1}^C{\pi }_{cj}\prod _{k=1}^K{\varphi }_{cj}^{k_{ijk}}{\left(1-{\varphi }_{cj}^k\right)}^{1-x_{ijk}}$$ (1)

The probability of observing a specific score profile across all covariate groups is shown in Equation (2).

$$P\left(x_i\right)=\prod _{j=1}^J\left(\sum _{c=1}^C{\pi }_{cj}\left(\prod _{k=1}^K{\varphi }_{cj}^{k_{ijk}}{\left(1-{\varphi }_{cj}^k\right)}^{1-x_{ijk}}\right)\right)$$ (2)

We estimate the latent covariate group specific item response probabilities by assuming there exist a set of latent covariate group cluster specific cutpoints, ${\mathbf{d}}_{cj}=\left(d_{cj}^1,\dots ,d_{cj}^K\right)$, such that ${\varphi }_{cj}^k=1-\mathrm{\Phi }\left(d_{cj}^k\right)$, where $\mathrm{\Phi }$ is the standard normal cumulative distribution function. These cutpoints are estimated using a HDPMM.⁹ For the assignment sub-model, we assume both the global and covariate group specific dispersion parameters, γ and α_j respectively, follow a uniform distribution, Unif(.1, 10), for all j. This choice of prior is similar to the dispersion parameter specified for BPR.¹³

We begin with a base distribution P, which is assumed to be a multivariate normal distribution, MVN(0, Σ) where Σ is a K × K identity matrix. Using P, we define two measures at the global level and the covariate group level, respectively. The measure at the global level is $P_0$, where $P_0\mid \gamma$, $P\sim DP(\gamma ,P)$, which is called the global probability measure and γ being referenced as the global dispersion parameter. The second measure at the covariate group level, P_j , is the covariate group specific probability measures and defined as $P_j\mid \alpha$, $P_0\sim DP\left({\alpha }_j,P_0\right)$, and α_j is the covariate group specific dispersion parameter for covariate group j. All dispersion parameters play important roles in the cluster weights. Specifically, smaller values of the parameters lead to greater weights assigned to the initial clusters, thus leading to a smaller number of overall clusters formed in the modeling process.¹⁷

Both measures, the global and the covariate group specific, can be constructed using a modified “stick-breaking” construction.⁸ The details of “stick-breaking” construction for HDPMM are presented in the paper by Reference 9. Similar to the DP, we impose a maximum for the number of clusters allowed to represent the infinite mixture as a finite approximation. To make sure that we allow for enough clusters, we use 10 for the maximal allowable number of clusters per covariate group.

2.2 |. Disease sub-model

The disease sub-model links the estimated latent cluster assignments with an outcome of interest for each subject by using the estimated latent cluster memberships as categorical predictors in a logistic regression model. For individual i we let y_i represent a binary outcome of interest and define the corresponding probability ${\lambda }_i=P\left(y_i=1\right)$. Let ${\theta }_{cj}$, where $c=z_{ij}$, be an assigned parameter that measures the influence of the cluster membership for subject i of component j on the outcome. The disease sub-model is shown in Equation (3).

$$\text{log}\left(\frac{{\lambda }_i}{1-{\lambda }_i}\right)=\sum _{j=1}^J\sum _{c=1}^C{\theta }_{cj}I\left(z_{ij}=c\right)$$ (3)

The ${\theta }_{cj}$ represent cluster level random effects and their sum across all components can be interpreted as baseline log odds of clinical dysphagia severity for individual i. Each subject assigned to the same cluster has the same log odds. Although the inclusion of additional covariates would be simple, only the cluster level random effects are included in this paper. For the disease sub-model, the component cluster specific effects, ${\theta }_{cj}$, are assigned priors with normal distribution such that ${\theta }_{cj}\sim \mathrm{N}\left(0,{\tau }_{cj}\right)$ with ${\tau }_{cj}$ being assigned a hyperprior of gamma distribution such that ${\tau }_{cj}\sim \mathrm{G}\mathrm{a}\mathrm{m}\left(\mathrm{2,0.5}\right)$. This prior was selected through a simulation study (results are not presented). Compared to other priors in the simulation study, this prior resulted in equivalent performance in the assignment sub-model and smaller confidence intervals, reduced mean square error, and adequate coverage probability in the disease sub-model.

2.3 |. Evaluating model output

In order to interpret the model clustering results, we adopt the post-processing of the MCMC results used in BPR (Molitor et al., 2010). We first utilize the Partitioning Around Mediods (PAM) hard clustering procedure to determine the typical latent impairment profile cluster membership for each subject in each component.¹⁸ Then, we implement the model averaging approach used in Reference 13 to characterize the typical partition and associated parameters in both sub-models while incorporating uncertainty from the cluster assignments.

2.4 |. Correction for nuisance clusters

References 13 and 19 indicate in their respective works that inference can be inhibited when there are small clusters or clusters which are ill defined by their features are present. Additionally, the number of estimated clusters often exceeds that of the true number of clusters in the data when using DPMMs and LCA.^20,21 The existence of sparsely populated clusters is possible, especially in the case of small sample sizes. As such, it is important to distinguish sparse clusters from nuisance clusters and correct for the presence of nuisance clusters. We define nuisance clusters as those clusters which do not exist in the data, but are an artifact of the model’s observed ability to over estimate the number of populated clusters. Sparse clusters are those clusters which have a small prevalence in the dataset; however, the clusters reflect a true sub-group in the data.

We propose an approach to correct for nuisance clusters to improve the ability of the model to correctly identify the number of clusters across components. For our approach, we first determine the typical number of clusters for the data using the PAM procedure. PAM calculates the silhouette width for a set of assumed clustering structures, with the clustering structure having the largest silhouette width being labeled as the typical number of clusters for that data.¹⁸ Once the initial clustering structure is defined, we determine if there exist any clusters for which the number of subjects in that cluster is below a threshold value T, which is the minimum allowed cluster size. If the threshold is not met, that is, there is at least one cluster whose size is less than the threshold T, then we observe the next best clustering structure. This next best clustering structure has the next largest silhouette width and we can evaluate the clusters associated with this next best clustering structure to determine if the minimum cluster size threshold is maintained. If this next best clustering structure meets the minimum cluster size criteria, then the algorithm ends. Otherwise, the algorithm continues until the minimum cluster size requirement is maintained. This approach makes no assumptions regarding the change in cluster structure required to meet the cluster size threshold, as the correction could increase or decrease the number of populated clusters. Rather, this correction simply places a minimum on the number of subjects allowed in a populated cluster, as we assume that spurious classes will reflect a small percentage of the overall sample size.

3 |. SIMULATION MODEL

To evaluate the proposed model, we conducted a series of simulation studies. Furthermore, we select the suggested threshold T by evaluating the performance of the proposed nuisance cluster correction algorithm using all simulated datasets. For all simulations, 100 simulated datasets were generated. Models were given a burn-in of 75,000 iterations, a total chain length of 100,000 iterations, and samples were thinned such that every fifth sample was retained. This resulted in an analysis chain length of 5,000 iterations. Convergence was checked using a combination of visual assessments and the Geweke statistic.²² For the estimation of parameters in both the assignment sub-model and the disease sub-model, we used Markov Chain Monte Carlo (MCMC) methods as implemented in the R library nimble.^23,24

3.1 |. Simulation conditions

Simulations were conducted across two sample sizes: 150 and 500 subjects. For the assignment sub-model, data were generated for 3, 6, or9 covariate groups, with each covariate group having 3, 6, or9 tasks and consisting of 2, 3, 4, or 5 latent impairment profile clusters. Each cluster was characterized by its’ covariate group specific item response probabilities, ${\varphi }_{cj}^k$, and prevalence, ${\psi }_{cj}$. Each subject was assumed to be a member of a cluster for each covariate group. For the disease sub-model, each cluster had specific cluster level random effect, ${\theta }_{cj}$. Details of the simulation models and all parameter values in the models are presented in Web Appendix A and Web Tables 3–6.

We tested model performance under 15 distinct cases, one ideal case to determine baseline model performance and 14 cases to test robustness of the model performance using imperfect datasets. The dataset for the ideal case was one where cluster prevalences are sufficiently large, clusters are unique, and all model assumptions are maintained. The additional seven cases, which we collectively call the “stress-tests”, have data which have: (1) small cluster prevalences for covariate group 1 cluster 3 and covariate group 2 cluster 1, (2) similar impairment patterns for all clusters within covariate group 3 and all other clusters in covariate group 1 and 2 are well defined as in the ideal case, (3) similar impairment patterns for all clusters in covariate group 3 with an increase in the number of shared patterns between covariate group 3 and covariate groups 1 and 2, (4) and (5) a violation of the CI assumption such that task scores for the first three tasks in cluster 1 in covariate groups 1 and 2 are conditional dependent given subjects’ cluster membership where the correlation coefficient is 0.1, and 0.3, respectively, (6) and (7) correlated cluster membership among all three covariate groups with correlation coefficient set to 0.3 and 0.5, respectively. (8) and (9) 6 and 9 covariate groups, respectively. Each covariate group consists of 9 tasks and 3 latent impairment profiles. (10) and (11) 6 and 3 tasks within each covariate group. There are three covariate groups and three latent impairment profiles in each covariate group. (12) and (14) 5, 4, and 2 latent impairment profiles within each covariate group, respectively. There are three covariate groups and each covariate group consists of nine tasks.

3.2 |. Simulation results

In this section, we first provide a general rule on how to choose the threshold for correcting model results for nuisance clusters described in Section 3.2.1, and then present model performances for all simulated datasets. Only simulations which correctly identified the clustering structure for all three covariate groups were included to summarize the results.

3.2.1 |. Nuisance cluster correction

Figure 1 shows the performance of the proposed algorithm to correct for nuisance clusters. where the y-axis is the probability that a simulation recaptured the true clustering structure for all three covariate groups among 100 simulations and the x-axis is a range of thresholds, T, which range from 0% to 10% of the total sample size.

FIGURE 1.

Probability of recovering true clustering structure for all three covariate groups across varying thresholds for nuisance cluster correction using simulated data.

We see that when no correction for nuisance clusters is applied, equivalent to T = 0, the probability that the model can recover the true clustering structure across a set of covariate groups is around 60% when the model is supplied with ideal data, henceforth Stress Test 0, for both sample sizes. However, cluster identification performance significantly improves when even slight corrections are applied, such as T = 4% or T = 6% of the total sample size. In these cases, the probability of recovering the true cluster structure is close to 100% for both sample sizes.

When the model is supplied with data which violate the CI assumption, as in Stress Tests 4 and 5, Figure 1 shows that the severity of the violation affects the probability of recapturing the true clustering structure. That is, in Stress Test 4 where the violation of the CI assumption is weak, the probability of recapturing the true clustering structure is nearly identical to Stress Test 0. When there is a moderate violation of this assumption, as in Stress Test 5, we see that the probability of recapturing the clustering structure decreases severely. However, the most notable improvements in the ability of the model to correctly identify the number of clusters across all simulated covariate groups by increasing T can be seen in Stress Test 5. For this simulation, we see dramatic increase in the probability of recapturing the correct clustering structure as the correction threshold increases. Stress Test 4, which has a lesser violation of the CI assumption, also sees a benefit from the nuisance cluster correction. However, the performance of Stress Test 4 is most similar to Stress Test 0 with respect to the recapture of the true clustering structure under varying thresholds for the nuisance cluster correction.

Contrary to the results for Stress Test 5 is Stress Test 1, where results show that increasing the threshold can be detrimental to model performance if the threshold becomes too large. For Stress Test 1, the model recaptures the correct clustering structure across all three covariate groups at a rate similar to that of the results for Stress Test 0 when using a larger sample size and T does not exceed the cluster size of the cluster with the smallest prevalence. However, once T exceeds this value, we see a decline in model performance relative to Stress Test 0.

When patterns are not well defined, as in Stress Tests 2 or 3, no threshold was shown to improve model performance. While performance was globally poor in both Stress Test 2 and 3, the lackluster results were driven by and underestimation of the number of clusters in covariate group 3 due to the similarity of the latent impairment profiles for the clusters in covariate group 3. However, for covariate groups 1 and 2 in Stress Tests 2 and 3, where the latent impairment profiles were well defined for all clusters, the model correctly identified the number of clusters at a rate similar to that of Stress Test 0 (results not presented). For Stress Tests 6 and 7, where the cluster memberships across covariate groups are correlated, recapture of the true number of clusters is similar to the Stress Test 0 results at all threshold values.

When varying the attributes of the data, namely the number of covariate groups, tasks, and latent impairment profiles, as seen in Stress Tests 8 through 14, cluster identification performance shows similarities to that of Stress Test 0. That is, slight corrections result in meaningful increases in the ability to recapture the true clustering structure. A notable exception to this is Stress Test 11, where there are only 3 tasks within each covariate group. This simulation demonstrates very poor performance, a likely demonstration that the model does not perform well when provided with few tasks.

Given the above results, we recommend that the threshold be carefully chosen with clinical guidance and consideration of the features of the data and any discovered clusters which could either benefit or be harmed by the correction for nuisance clusters. Based on the empirical results in Figure 1, we choose the threshold as 6% of the total sample size for all simulations, that is, T = 6%×N. This threshold has shown to not degrade model performance relative to no correction when clusters with small prevalences are present and provides a noticeable improvement for almost all other simulations.

3.2.2 |. Recapture of the true number of clusters and prevalences

Web Table 14 presents the proportion of simulations which recaptured the true number of clusters across all covariate groups when applying the algorithm of correction for nuisance cluster at our chosen threshold. Simulation results demonstrated that BHPR is 70% or more likely to accurately recapture the true number of clusters across all covariate group with the exception of Stress Tests 2 and 3.

Web Figures 1 through 4 show the true and estimated prevalence for each simulated case. These figures reveal that, when the true number of clusters can be recaptured for each covariate group, the estimated cluster prevalences are close to the truth. Interestingly, although estimation of the cluster prevalences was poor in covariate group 3 for Stress Tests 2 and 3, for covariate groups 1 and 2, which had well defined latent impairment profiles, estimation of the cluster prevalences is similar to that of Stress Test 0.

FIGURE 4.

Identified latent patterns for selected components in the oral domain of the modified Barium swallow impairment profile (MBSImP). LC, Lip Closure; BT/LM, Bolus Transport/Lingual Motion; OR, Oral Residue; IPS, Initiation of Pharyngeal Swallow. The shaded areas in the left panel represent 95% credible intervals. The intervals in the right panel represent 95% credible intervals for parameter estimates.

3.2.3 |. Recapture of the true latent impairment profile pattern

Figure 2 (N = 150) and Web Figure 3 (N = 500) illustrate the capability of the proposed approach to recapture the true latent impairment patterns in Stress Test 0, with the left panel representing the true and estimated impairment profiles for the assignment sub-model and the right panel showing the true and estimated parameter values involved in the disease sub-model. Similar figures can be seen in Web Figures 7–34 for Stress Tests 1–14 with sample size of 150 and 500.

FIGURE 2.

Latent impairment patterns and parameter estimation in Stress Test 0 forN = 150. For the left panel, the true values are represented by x’s, the median across medians for all 100 simulations are represented by dots, the lower bound of the shaded region is the median 2.5% among all medians, and the upper bound of the shaded region is the median 97.5% among all medians. For the right panel, the true values are represented by x’s, the median across simulation medians are represented by dots, the lower bound of the interval is the median 2.5% among all medians, and the upper bound of the interval is the median 97.5% among all medians.

For Stress Test 0, when the model correctly identifies the number of clusters in all three covariate groups, the model performs well at recapturing the true impairment patterns. This result also held true for nearly all Stress Tests (Web Figures 7–34).

3.2.4 |. Accuracy of parameter estimation

Table 1 shows the results of parameter estimation for the disease sub-model for Stress Test 0. The mean square error (MSE) is calculated as the average squared difference between the true parameter and the simulation medians, coverage probability (CP) is the proportion of simulations for which the 95% credible interval for each associated parameter contained the true value for that parameter, and credible interval width (Width) is the median credible interval width across simulations for the associated parameters. We use cluster 1 and 3 in covariate group 1 and 2 as examples to illustrate how to interpret the results. When comparing the estimation of the cluster level random effects for clusters 1 and 3 in covariate group 1, when there is a reduction in cluster prevalence there is an increase in MSE, a decrease in CP, and an increase in the Width. However, when comparing the estimation for the cluster level random effects in clusters 1 and 3 in covariate group 2, where cluster prevalences are closer in magnitude, we see that estimation of the clusters effects are roughly equivalent. These results are consistent across the sample sizes tested.

TABLE 1.

Parameter estimation for disease sub-model in stress test 0.

N	Covariate group/cluster	θcj	$\widehat{{\mathit{\theta }}_{\mathit{c}\mathit{j}}}$	MSE	CP	Width
150	1/1	−2	−1.51	0.37	0.96	2.43
	1/2	0	0.04	0.02	1	1.57
	1/3	2	1.51	0.75	0.92	2.38
	2/1	−1	−0.58	0.36	0.97	1.80
	2/2	0	0.06	0.02	1	1.55
	2/3	1	0.54	0.26	0.96	1.77
	3/1	0	0.02	0.04	1	1.44
	3/2	0	0.00	0.04	1	1.44
	3/3	0	0.01	0.04	1	1.44
500	1/1	−2	−1.78	0.16	0.97	1.91
	1/2	0	0.01	0.02	1	1.56
	1/3	2	1.76	0.18	1	1.89
	2/1	−1	−0.82	0.09	0.99	1.60
	2/2	0	0.04	0.02	1	1.46
	2/3	1	0.78	0.09	1	1.60
	3/1	0	−0.01	0.02	1	1.21
	3/2	0	0.02	0.02	1	1.21
	3/3	0	0.02	0.02	1	1.21

Abbreviations: CP, coverage probability; MSE, mean square error; Width, credible interval width.

While many of the stress Tests are similar in the estimation of their cluster level random effects, Stress Tests 2 and 3 have notable exceptions. The results (not shown) in these two stress Tests demonstrate that in covariate groups where the clusters do not have well defined latent impairment profiles the estimation of model parameters is poor, as indicated by the right panel in Web Figures 9–12. Further, in these cases, increases in sample size do not aid in the estimation of these parameters. However, in covariate groups where the patterns are well defined, the estimation of model parameters is similar to that of Stress Test 0.

In summary, the simulation results demonstrate that the proposed approach does well with respect to recapturing the true clustering structure in the following situations: (1) when cluster prevalences are small (Stress Test 1), (2) when there is a mild violation of the CI assumption (Stress Tests 4), and (3) when cluster memberships are correlated (Stress Test 6 and 7). However, if patterns within or across given covariate groups are similar (Stress Test 2 and 3) or there is a moderate violation of the CI assumption (Stress Test 5), then recapture of the true clustering structure declines. Further, poor estimation due to imperfect data, such as a violation of the CI assumption or small class prevalences, appears to only impact the affected covariate groups and does not hinder cluster identification and parameter estimation in other covariate groups.

3.2.5 |. Comparison to BPR

We now compare BHPR to BPR. For this comparison, we will observe the number of latent pattern class combinations across MBSImP components for BHPR and the total number of clusters discovered for BPR. For BPR, as previously mentioned, the model is unable to delineate between covariate groups; therefore, we are unable to assess then number of clusters discovered within each covariate group and can only assess the total number of clusters discovered across covariate groups. The analogous comparison for BHPR would be to observe the total number of combinations of latent pattern clusters across components. This is calculated as the number of unique combinations of estimated latent pattern cluster memberships across the covariate groups. For all simulation conditions, given that there were three latent impairment clusters generated for each of the three covariate groups, we would expect to observe at most 27 combinations of latent impairment cluster memberships. Ideally, we would then expect the number of clusters discovered by BPR to be near to the true number of combinations of latent impairment clusters that were generated for that data. Similarly, we would expect the estimated number of discovered combinations of latent impairment clusters across all covariate groups to be near to the true number of combinations. For these simulation results, we will include results were BHPR did not correctly identify the correct number of clusters across all three simulated covariate groups.

Figure 3 presents a boxplot of the number of true cluster combinations generated by the simulation model, the number of clusters discovered by BPR, and the number of cluster combinations discovered by BHPR. We can see that, while the median number of cluster combinations discovered by BHPR is slightly overestimated when compared to that of the true number of cluster combinations generated by the simulation model, the estimated number of clusters from BPR is greatly reduced relative to the true value for Stress Test 0. For Stress Test 1, we see that BPR underestimates the number of cluster combinations in data, with the number of cluster combinations discovered by BHPR remaining consistent with the number of clusters that were generated in the simulated datasets. For Stress Tests 2 and 3, we see that both BPR and BHPR underestimate the number of clusters in the data; however, we see that BHPR is nearer to the truth. The poor performance of BHPR is due to the model underestimating the number of clusters in the covariate groups where the clusters were ill defined, resulting in the poor recapture of the true number of cluster combinations across the covariate groups. For Stress Test 5, where there is a moderate violation of the conditional independence assumption, we see that there is an increase in the estimated number of clusters for BHPR relative to the true number of clusters generated by the simulation model. However, when there is a mild violation of this assumption, as in Stress Test 4, we see that the number of cluster combinations discovered by BHPR is consistent with the number generated by the simulation model. Further, for both Stress Test 4 and Stress Test 5, we again see that BPR under estimates the true number of clusters. For Stress Tests 6 and 7, we see that the results are similar to that of the results for Stress Test 0 for both BPR and BHPR.

FIGURE 3.

Estimated number of cluster/cluster combinations from BPR and BHPR. For each simulation condition, the dark gray box represents the estimated number of cluster combinations presenting in the real data across the simulated datasets, the light gray box represents the number of clusters discovered by BPR across the simulated datasets, and the white box represents the number of cluster combinations discovered by BHPR across the simulated datasets.

For Stress Tests 8 and 9, where there are six and nine covariate groups, respectively, there are similar findings to that of Stress Test 0, where there are three covariate groups. That is, BHPR estimates the number of cluster combinations near to the number of cluster combinations found in the data. For these results, given the increased number of covariate groups, it is noted that the number of cluster combinations greatly increases. In these simulations, the poorer performance of BPR is displayed relatively to BHPR.

For Stress Tests 10 and 11, where there are six and three tasks within each covariate group, respectively, both simulations demonstrate that BHPR and BPR underestimate the combinations of cluster assignments. However, as in previous simulations, BHPR is closer to the true number of combinations. These results demonstrate that as the number of task data within a covariate group decreases, the model is less able to correctly estimate the number of clusters within a covariate group.

Lastly, for Stress Test 12, 13 and 14, where the number of latent profiles within each covariate group is varied, we note that, as before, both BHPR and BPR tend to underestimate the number of cluster combinations with BHPR having better performance when compared to BPR. An exception in Stress Test 14, where there are two latent profiles per covariate group, when the estimation for the number of clusters per covariate group is nearly identical to the generated data.

4 |. APPLICATION TO MOTIVATING DATA

In this section, we applied the proposed model to our motivating dataset. Our motivating dataset is derived from a MBSImP database of 345 adults who were referred for a routine MBSS between the years 2005 and 2008 at either St. Joseph’s Hospital in Atlanta, Georgia or the Medical University of South Carolina in Charleston, South Carolina. For our work, multiple swallow studies from the same patient were excluded, with only the first swallow study being retained for analysis. The exclusion of multiple swallow studies from the same patient maintains independence between swallow studies. Additionally, subjects without a valid study identification number were excluded. With these exclusions, our final analysis dataset consists of 319 subjects. A subset of components from the oral domain of the MBSImP were included in the analysis, with the MBSImP components representing the covariate groups. Specifically, we included: lip closure (LC), bolus transport/lingual motion (BT/LM), oral residue (OR), and initiation of pharyngeal swallow (IPS). The outcome of interest is a measure of clinical dysphagia severity. This measure of clinical dysphagia severity differs from a physiologic dysphagia severity measure as it is derived from patient dietary restrictions and feeding tube status instead of patient physiology. A full description of how this variable is constructed can be seen in Web Table 1. Clinical dysphagia severity was dichotomized such that patients were classified as having either functional/mild or moderate/severe clinical dysphagia. Therefore, our final analysis dataset consisted of 319 subjects with 36 tasks scored across four components. Web Table 2 shows the dichotomization scheme for the task level data within each component, where scores are dichotomized in order to represent impaired swallowing physiology and unimpaired swallowing physiology.

4.1 |. Missing data

Following the discussion by Beall et al.,⁶ 43% of subjects were missing at least one task score for the entire oral domain of the MBSImP. Further, it was suggested that the exclusion of subjects who did not have complete task data could produce a cohort which was not reflective of the general population of patients who were referred for a MBSS, with the complete case analysis reflecting a milder overall population. Hence, we constructed 25 multiple imputed datasets to mitigate the potential bias of conducting a complete case analysis. Full details of the multiple imputation can be found in the paper by Beall et al.⁶

4.2 |. Model fitting

The BHPR model was fit to each of our imputed datasets using on the discussed MCMC post-processing approach. All priors remain unchanged from the simulation model and the threshold for correction of nuisance clusters is set to 6% of the total sample size. Regarding the maximum number of allowed clusters in the assignment sub-model, we increase this value to 15 to ensure that we allowed for enough clusters. We ran the MCMC algorithm using a burn-in chain length of 40,000 iterations and a total chain length of 50,000 iterations, with the remaining samples thinned such that every fifth iteration was retained. Thus, our analysis chain length for each imputed dataset was 2,000 iterations. The resulting posterior chains from the 25 individual model runs were pooled, resulting in an analysis chain length of 50,000 iterations.²⁵ Posterior estimates are summarized using medians and 95% credible intervals, with convergence assessed using Geweke’s statistic and visual assessments of appropriate trace plots.²² All analyses were completed using R Version 3.5.1, with the models constructed using the NIMBLE library.^23,24

4.3 |. Results

Table 2 presents the clustering results. For the LC and OR components, the model identified three clusters. The majority of subjects are in cluster 2 for the LC component. For the OR component, clusters are roughly equivalent with respect to their prevalences. For the BT/LM and IPS component, the model identifies four clusters with the majority of subjects in clusters 3 and 1.

TABLE 2.

Number of subjects (percent of total sample size) in each component-specific clusters for the oral domain of the modified Barium swallow impairment profile (MBSImP).

MBSImP Comp.	Cluster 1	Cluster 2	Cluster 3	Cluster 4
LC	61 (19.1%)	235 (73.7%)	20 (6.3%)	—
BT/LM	107 (33.5%)	55 (17.2%)	118 (37.0%)	36 (11.3%)
OR	102 (32.0%)	120 (37.6%)	94 (29.5%)	—
IPS	185 (58.0%)	54 (16.9%)	39 (12.2%)	38 (11.9%)

Figure 4 presents the identified latent patterns for each cluster in each component (left panel) and the estimates of the cluster level random effects from the disease sub-model (right panel). The x-axis for the left panel represents the evaluated tasks for these components and the y-axis represents the probability of impairment for each task. The shaded areas in the left panel and the intervals in the right panel represent 95% credible intervals.

For the LC component, cluster 2 is characterized by a low probability of impairment across all nine tasks and a decreased probability of moderate/severe clinical dysphagia (parameter estimate (log odds) in the right panel is less than 0). Clusters 1 and 3 are characterized by an increased probability of impairment across all tasks when compared to cluster 2 and an increased probability of moderate/severe dysphagia.

For the BT/LM component, Figure 4 shows that cluster 2’s patterns are similar to cluster 3’s for the thin and nectar thickened liquid tasks (T5, TCS, TSS, N5, NCS and NSS), with the probability of impairment being low. However, for solid and semi-solid task (H, P, C), where subjects in cluster 3 maintains a low probability of impairment, subjects in cluster 2 demonstrate an increasing probability of an impaired task score. The impairment pattern for cluster 4 is similar to cluster 1 with a relatively high probability of an impaired task score for most tasks. However, where subjects in cluster 1 have an elevated impairment probability for all tasks, subjects in cluster 4 demonstrate an increasing probability of impairment across tasks. Similar to the LC component, there are differences in the effect of cluster membership on the probability of clinical dysphagia severity. Notably, membership in cluster 3 appears to reduce the probability of clinical dsyphagia severity relative to other discovered clusters in BT/LM.

For the OR component, cluster 2 has a similar pattern as cluster 1 with a relatively high probability of an impaired task score for most tasks. However, where cluster 1 demonstrates an increasing probability of impairment with each task, cluster 2 demonstrates a consistently elevated probability of impairment. Cluster 3 is characterized by a low probability of impairment for the thin and nectar thickened liquid tasks (T5, TCS, TSS, N5, NCS and NSS), with the probability of impairment increasing for the remainder of the tasks. Unlike the LC and BT/LM components, there appear to be no significant effects of cluster membership for any of the discovered clusters in this component since the 95% credible intervals include 0 for all parameter estimates.

For the IPS component, cluster 1 is defined by a high probability of impairment across all tasks. Clusters 2 and 4 share similar patterns in their impairment profiles, with there being a repeating pattern of increasing probability of impairment seen in each triplet of tasks. Similar to the OR component, no clusters demonstrate a significant effect of cluster membership on the outcome of clinical dysphagia severity.

Furthermore, in addition to observing the discovered clusters within each component, we can evaluate the combinations of possible impairment patterns across all four components for all subjects. Figure 5 shows the top five popular combinations and the subsequent estimated probability of moderate/severe clinical dysphagia and the corresponding 95% credible intervals for that combination. For example, the first row illustrates the combination of patterns expressed by 19 subjects in the dataset, with this combination of patterns having an estimated probability of moderate/severe probability being below 25%. Similarly, row 4 shows a combination of patterns expressed by 18 subjects, with these subjects demonstrating an increase in the probability of moderate/severe dysphagia. While Figure 4 represents the possible impairment patterns revealed by subjects in each component, Figure 5 demonstrates the most common manifestations of impairment patterns across all components in our sample. Figure 5 allows for the characterization of both the patterns expressed within and across components as well as the associated probability of dysphagia severity. Results in Figure 5 extract the structure of physiologic swallowing impairment severity from videofluoroscopic images obtained during the MBSS, which fills in the clinical practice gap and provides a useful tool for clinicians to better manage and treatment patients with dysphagia.

FIGURE 5.

Top five most popular combinations of latent impairment patterns in the oral domain of Modified Barium Swallow Impairment Profile (MBSImP). LC, Lip Closure; BT/LM, Bolus Transport/Lingual Motion; OR, Oral Residue; IPS, Initiation of Pharyngeal Swallow; M/S, Moderate/Severe. The shaded areas in the left panel represent 95% credible intervals. The intervals in the right panel represent 95% credible intervals for parameter estimates.

5 |. DISCUSSION

In this work, we proposed the BHPR model which uses BPR in conjunction with a HDPMM to discover latent impairment profiles using hierarchical grouped data. The proposed BHPR model not only reveals patterns of swallowing impairment through MBSImP scores, but also provides the link between the clinical dysphagia severity and the identified MBSImP impairment patterns. The simulation studies and the application of the proposed model on our motivating dataset demonstrate the adequate ability of the model to recapture underlying component specific latent profile patterns and subsequent associations with disease severity. We also assessed the robustness of the proposed model given several imperfect scenarios. Further, we proposed a method to correct model results for the presence of nuisance clusters.

There are a number potential extensions of this work which would remove some limitations of the current implementation. One of the most appropriate extensions of the model is to allow ordinal task data in addition to binary task scores. This would more align the model with the motivating data, as tasks scored using MBSImP protocols are ordinal. Two additional limitations of our model are the CI assumption and the requirement that each component have an equal number of tasks. A relaxation of the CI assumption, as this is one of the main assumptions for the model, and allowing for a different number of tasks within each component and additional covariates in the disease sub-model would further align our model to the motivating data and increase the applicability of our model to other disease areas.

Additionally, while the MBSImP is characterized as having a naturally formed, hierarchical, multiple group structure, other data sources can be similarly characterized. A key potential domain of application for BHPR is multivariate longitudinal data. For example, data collected on longitudinal environmetal exposure levels or repeated biomarker data are representative of data which resemble the data structure of the MBSImP. Additionally, these data sources are found in the literature where clustering methods are applied.^10–12

Most importantly, the proposed model provides a sophisticated statistical approach allowing clinicians to discover profiles of physiologic swallowing impairment from videofluoroscopic images scored using the MBSImP, which fills in the current clinical gap. BHPR and its’ results can be used to inform appropriate intervention strategies, provides a tool which can allow for clinicians to make better multidimensional management and aid in treatment decisions for patients with dysphagia.

Supplementary Material

Additional supporting information can be found online in the Supporting Information section at the end of this article.

DATA AVAILABILITY STATEMENT

Data sharing is not applicable to this article as no new data were created or analyzed in this study.