2.1 Leishmaniasis

Leishmaniasis is a vector-borne disease caused by infection with protozoan parasites of the genus Leishmania. Dogs serve as the primary reservoir for human infection with L. infantum and the immunopathology of CanL is very similar to human disease [5]. Both human VL and CanL are chronic, progressive diseases that lead to a state of immunosuppression and symptomatic cases that are fatal without treatment [5]. Available treatments are not ideal and no vaccine exists for humans. There is an ongoing critical need for novel therapeutics for VL/CanL, however, the complicated nature of the immune response to Leishmania parasites has impeded progress.

Dogs infected with L. infantum parasites can maintain a state with no clinical signs for an extended period of time before progressing to a state with clinical manifestations of CanL; some may never experience clinical disease. There has been much research done to identify immune factors involved in maintaining subclinical infection with interesting results. There is a need to incorporate these findings with statistical modeling to establish a deeper and more comprehensive understanding of the relationship between specific immune response components and pathogen control. A Type 1 immune response with the secretion of IFN-γ, produced predominately by CD4+ T cells, is required for clinical protection to Leishmania parasites [6]. Solano-Gallego et. al. [7] assayed cytokine production by cells from dogs infected with L. infantum and found a significant negative relationship between IFN-γ production and parasitemia. IFN-γ is a highly inflammatory cytokine that suppresses parasite replication by activating the microbicidal activity of infected host cells to maintain a low pathogen load. Regulatory immune factors that antagonize the effects of IFN-γ facilitate parasite replication. The immunoregulatory cytokine IL-10 is a potent antagonist of IFN-γ [8–10]. Additionally, although also a marker of activation, regulatory surface receptors on T cells, such as programmed cell death 1 (PD-1), send inhibitory signals upon ligation to curb T cell effector functions such as proliferation, cytokine production, and cytotoxicity [11]. Esch et al. [1] investigated T cell PD-1 expression in dogs with progressive CanL and showed a role for this receptor in limiting the ability of T cells to induce parasite control within infected host cells. While CD8+ T cells are thought to play a relatively minor protective role in the immunopathogenesis of VL and CanL compared to their CD4+ counterparts, these cells express IFN-γ and IL-10, to a lesser extent, and are well characterized in their expression of PD-1 [12, 13]. Finally, B cells produce antibodies during CanL, however, these antibodies are non-productive, and mainly correlate with an enhanced disease rather than limiting parasite burden [14]. These complex inflammatory and regulatory arms of the immune response exist in a delicate balance to eliminate pathogens while also limiting immune-mediated tissue pathology associated with sustained exposure to inflammation. Understanding how different immune response components interact to impact the overall disease state would help identify candidates with immunotherapeutic potential.

Further complicating the situation, dogs with CanL experience a high rate of co-infections [15]. Pathogen composition can help model co-infection with different organisms or different strains of the same pathogen [16]. It is known that multiple pathways of the adaptive immune system coordinate to enact responses linked to different infectious agents [17]. By including both CD4+ and CD8+ T cell variables such as proliferation and expression of IFN-γ, IL-10, or PD-1 collected from a prospective cohort of dogs with CanL, we are able to more closely model real-world disease progression. In addition to these immune responses, the Leishmania-specific antibody level, pathogen burden, and disease state classification were also measured and considered within the model specification for identifying the risk of death due to progressive disease. Models of CanL that include aspects of the immune response will more accurately approximate natural disease evolution, which may offer an important tool for researchers targeting this deadly disease.

3.1 cohort selection and diagnostics

A cohort of client-owned dogs naturally exposed to L. infantum in the Midwestern United States was selected based on a positive Leishmania diagnostic test or diagnostic positive dam or full sibling. Diagnostic tests included Real-Time quantitative Polymerase Chain Reaction (RT-qPCR) of Leishmania DNA from peripheral blood or Canine Visceral Leishmaniasis ® Dual Path Platform (DPP) serological test [18, 19]. Inclusion criteria included negative 4Dx Plus SNAP test (IDEXX Reference Labs) result indicating no recent exposure to tick-borne bacteria Borrelia burgdorferi, Ehrlichia spp., Anaplasma spp., or heartworm, and 2 or fewer outward clinical signs of CanL. Fifty dogs were enrolled in the study and assessed at three-month intervals over the course of 18 months. The cohort was block-randomized into two age and sex-matched groups where one group received a tick-preventative drug and the second group received a placebo. All dog caretakers gave signed informed consent, following a protocol approved by the University of Iowa Institutional Animal Care and Use Committee (IACUC).

3.2 Measurements of state components

At each study collection visit, pathogen load was quantified as parasite equivalents per mL of peripheral blood using RT-qPCR against a standard curve of blood derived from an unexposed dog spiked with known quantities of L. infantum promastigotes. To keep things simple, we focus here on a single infectious agent (L. infantum) and incorporate tick-borne bacterial co-infection as an indicator variable. Antibody levels were measured in dog serum via indirect enzyme-linked immunosorbent assay (ELISA). ELISA plates were coated with soluble Leishmania antigen (SLA) prepared from freeze-thawed stationary phase L. infantum promastigotes. The optical density (OD) ratio was calculated as the OD of test wells divided by a cutoff of the average OD plus 3 standard deviations of wells containing serum from unexposed negative control dogs. An OD ratio greater than 1 indicates a positive serological result. Disease status was determined using the LeishVet staging guidelines using complete blood count, serum chemistry panel, and physical examination information assessed by board-certified veterinarians [14].

Inflammatory and regulatory immune response variables were quantified from peripheral blood mononuclear cells (PBMCs) isolated by Ficoll-Paque PLUS density gradient. PBMCs were stimulated with 10 ug/mL total Leishmania antigen for seven days and then processed for surface and intracellular staining with fluorophore-conjugated antibodies. Cell events were read on an LSR II flow cytometer and analyzed using FlowJo^™ software [20]. Immune response readouts are presented as the percentage of cells positive for IFN-γ, IL-10, or PD-1 expression among CD4+ or CD8+ lymphocytes expressing high levels of CD49d, a marker of antigen-experienced T cells [21, 22]. Proliferation was measured using carboxyfluorescein succinimidyl ester (CFSE) dilution assay and presented as percent CFSE low cells among CD4+ or CD8+ CD49dhi lymphocytes.

3.3 Model specification: Longitudinal submodel

In Fig 1, we show an abbreviated illustration of the lag-1 temporal dependence structure of the Bayesian model, relating a given time index t to the following time across the state components. We consider N = 50 dogs during the T = 7 time points. Mechanically, the state components are organized into a subject-specific row-vector M_i,t of length 24, where i = 1, 2, …, N and t = 1, 2, …, T. For the corresponding indexes i and t, the row-vector M_i,t includes terms representing the interplay of disease state with the rest of the model components: pathogen load, antibody level, three inflammatory responses, and three regulatory responses. This specification allows us to assess the effects of the eight key drivers based on the dog’s disease state. For clarity, we organize the non-time-varying predictors into separate subject-specific row-vectors X_i of length 6, which include an intercept, indicators for age groups, 4Dx SNAP test result for tick-borne co-infection status, DPP result, and an indicator for tick prevention treatment. These two subject-specific row vectors drive the expected values of model components for each time point conditional on the previous time. This dependency occurs as a result of updated functions denoted by f(⋅) as presented in Fig 1. In this graphical display, we separate the univariate components of the model such as disease state, antibody level, and pathogen load, from those given a multivariate structure. Since there are known relationships between CD4 and CD8 T cells, all of the measurements coming from them could be potentially correlated. Thus, the inflammatory and regulatory responses are being modeled using a multivariate structure. Other specifications could consider a global multivariate vector autoregressive structure, but the chosen approach allows us to adapt the form of the model to known processes and relationships even in this relatively small sample setting.

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Fig 1. Dynamic process.

Graphical display of how the disease state (D), antibody level (A), pathogen load (P), and inflammatory (I) and regulatory (R) responses are related to predicting the future value of each state for each dog i at time t + 1. Non-time-varying predictors are organized into a subject-specific row-vector X_i. Expected future values of model components are determined based on corresponding functional forms f(⋅). For the case of inflammatory and regulatory responses, the term IR corresponds to a vector of length 6 including the indicated immunologic responses.

https://doi.org/10.1371/journal.pone.0297175.g001

To assess the CanL disease progression of subjects over time, we took into account LeishVet score, pathogen load, and level of anti-Leishmania antibodies as well as inflammatory and regulatory immune response elements, incorporated jointly in a longitudinal Bayesian submodel. To determine the clinical status of each dog based on professional veterinarian assessment and laboratory tests, the LeishVet scoring system [14] was used to classify the dogs into disease scores ranging from 0 to 4, with 0 indicating no signs of disease and 4 indicating severe disease. We then define less granular categories [3], based on LeishVet scores, consisting of four categories: healthy, mild, severe, and removed, as previously described by Pabon-Rodriguez et al. [3]. In this work, we go further and rewrite the disease state as a three-categorization outcome, where the removed state is now transformed into a whole new time-to-death submodel which is described in a subsequent section. The disease state variable is denoted by D_i,t for the ith dog at time t, indicating the category into which the dog’s state is classified. This categorization was then encoded through separate indicator variables for each score over time for each dog. Once an individual is infected with the parasite L. infantum, replication may occur. Concurrently, the host immune mounts a response against the parasite. To measure and quantify the pathogen status component of the overall dynamic process, we define P_i,t to denote the pathogen load for the ith dog at time t. Further, let A_i,t denote the anti-Leishmania antibody level for the same indexes. We denote the inflammatory state of an immune response by I_i,t for i-th dog at time t. Since we consider several inflammatory signaling or responses, we then divided I_i,t to define I1_i,t to represent the proportion of IFN-γ positive cells among CD4+ T cells. In a similar way, I2_i,t and I3_i,t are defined to capture the proportion of proliferating CD4+ and CD8+ T cells, respectively. The regulatory response R_i,t components focus on CD4+ cells: IL-10 expression, and cell-surface PD-1 expression. As before, an example implementation of the regulatory state of an immune response might be separated into three components. First, R1_i,t denotes the proportion of IL-10+ expressing CD4+ T cells, while R2_i,t and R3_i,t represent proportion of PD-1+ expressing CD4+ and CD8+ T cells, respectively.

Next, we present the Longitudinal Autoregressive Moving Average (ARMA) Bayesian model used for this work. Here, the AR component is used to model a linear combination of past time series values (lag-p values) and the MA piece for the linear combination of the past error (residuals) terms, in order to further improve predictions. Since each time point is equally spaced with 3 elapsed months, we decided to use an ARMA(p, q) component of orders p = 1 and q = 1 for the continuous responses, which means that only the information and errors from the immediate previous time point are being used in the model. For the case of disease state, only an AR component of order p = 1 is considered. In addition, Fig 2 shows a graphical display of how the longitudinal outcomes are structurally associated with the risk of death. A lagged structure is used to look at the association between the longitudinal outcome at the current time point or the last observed time with the risk of death at the following time point. The risk of death due to progressive Leishmania infection is modeled through a survival submodel via a proportional hazard specification with a Weibull baseline hazard function, with the goal of examining the association between longitudinal and survival processes.

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Fig 2. Joint dynamic process.

Graphical display of how the longitudinal outcomes are structurally associated with the risk of death. A lagged structure is used to look at the association between the longitudinal outcome at the current time point (t) or the last observed time (t*) into the risk of death (hazard) at the following time point (t + 1). The diagram shows an example of a possible univariate outcome and the hazard. The diagram serves as a visual representation, and the trajectories are not the actual data values.

https://doi.org/10.1371/journal.pone.0297175.g002

3.4 Model specification: Survival submodel

To measure the impact of longitudinal outcomes on the risk of death due to progressive Leishmaniosis, a survival submodel is defined to examine the association between longitudinal outcomes, predictors, and the risk of death due to progressive disease. Here, the survival time for individual i is modeled from a proportional hazard specification with a Weibull baseline hazard function h₀ as shown in Eq 7. (7)

This submodel requires the definition of an association structure between the longitudinal and survival data, an important component in joint modeling [26]. In this case, the association structures, denoted by m_i,j(t*), use a lagged effect property. They are defined as the expected value of the longitudinal outcome j at the last observed time t* of subject i. The association parameters ω’s quantify the associations between each longitudinal outcome and the risk of death due to severe disease, and γ’s examine the effect of the non-time-varying covariates. For instance, since pathogen load is the first component we explained in Subsubsection 3.3.1, let us assume j = 1, then m_i,1(t*) = E[logit(P_i,t*)], and w₁ measures the association of pathogen load on the risk of death. While the longitudinal submodel uses information on time points from TP1 to TP7, the survival submodel on the other hand, uses information from TP1 to TP7 as well as two post-follow-up time points after the study ended (TP8 and TP9) where more deaths due to severe cases of diseases and unrelated to it were identified.

3.5 Computation and model diagnostics

To fit the presented Bayesian model, we used the NIMBLE [27] package in R [28, 29] along with the parallel [28] package to obtain posterior samples of model parameters and latent quantities using Markov Chain Monte Carlo (MCMC) techniques. A general description of MCMC techniques is provided by Liu [30] and Carlin [31]. In this case, due to the selection of priors and definition of the model components, default samplers from NIMBLE seemed appropriate and hence were used via the configuration tool provided in the package.

In Bayesian analysis, the prior distribution represents the knowledge or belief about the parameters before the data is observed. The choice of the prior distribution and its parameters can have a significant impact on the posterior distribution, which is the updated knowledge or belief about the parameters after the data is observed. Independent normal prior distributions with zero mean, , were used for the conditional and unconditional effects (β’s) in order to shrink effects towards zero, where gamma prior distributions Γ(1, 1) was used for the variance of the effects (), which is a hyperparameter. All of the parameters associated with the non-time-varying predictors (α’s), the moving-average parameters associated with pathogen load, and antibody levels (θ_p, θ_a) use independent standard normal prior distributions . This choice of prior indicates that there is no strong prior information or belief about the magnitude or direction of the effect of the non-time-varying predictors or moving-average parameters on the outcomes, and allows the data to drive the posterior inference.

Elements of the matrix T_ir uses independent normal prior distributions . A gamma prior distribution Γ(1, 1) was also used for the variance terms of the model components (, ). The prior for the covariance matrix was chosen to be an Inverse Wishart, , where N is the number of subjects, while p and S are the dimensions and the sample covariance of the Inflammatory-Regulatory responses, respectively. The white noise (w’s) or residual terms of the continuous responses are derived parameters within the model. In the case of the survival submodel, the association parameters (ω’s) assumed standard normal prior distributions , while the covariate effects (γ’s) use independent normal prior distributions with zero means and each with different variance term with a gamma prior distribution Γ(1, 1).

The normal distribution is a commonly used prior distribution for model parameters because it is mathematically convenient and flexible. While the normal distribution is a popular choice for prior distributions, it is not always appropriate or necessary. Some reasons that led us to use this distribution are:

1. (i). Conjugacy: The normal distribution is a conjugate prior for the parameters of many commonly used likelihood functions. This means that the posterior distribution will also be a normal distribution, making computation of the posterior distribution relatively straightforward.
2. (ii). Flexibility: The normal distribution is a flexible distribution that can be parameterized to allow for varying degrees of prior information or uncertainty. Specifically, the mean and variance parameters can be used to represent the prior mean and precision (or equivalently, the prior standard deviation) of the parameter values.
3. (iii). Familiarity: The normal distribution is a well-known and widely used distribution, making it familiar to many researchers and practitioners. This can make it easier to communicate results and compare findings across studies.

Among the hyperparameters within the presented Bayesian model, the variance parameter of the prior distribution for the conditional and unconditional effects as well as for the association parameters controls the amount of information or uncertainty that is incorporated into the prior distribution. A smaller variance indicates a more informative prior, where the prior distribution is concentrated around a particular value, indicating a stronger prior belief in the parameter value. A larger variance indicates a less informative prior, where the prior distribution is more spread out, indicating weaker prior belief in the parameter value. By varying the variance parameter of the prior distribution, you can explore the impact of different levels of prior information or uncertainty on the posterior distribution. A sensitivity analysis can help us determine how sensitive the posterior distribution is to changes in the variance parameter and can inform decisions about the appropriateness of the prior distribution and the level of prior information or uncertainty to use in the analysis.

In the current work, we performed a sensitivity analysis (see supplementary file S2 File), targeting the prior information on the variance of the model parameters, particularly on the driver effects and association parameters between the longitudinal and survival submodels. For variance parameters (or standard deviations), we used gamma-distributed priors as previously discussed. Thus, we varied the shape and scale parameters of the gamma distribution to create different levels of prior information or uncertainty. Specifically, we considered the following scenarios with different levels of prior variance:

1. (i). Gamma distribution: The first scenario uses the Γ(1, 1) distribution mentioned above. This results in a prior distribution that is weakly informative and places a lot of mass near zero but allows for a wide range of possible values. This prior can be useful when you have very little information about the variance and want to avoid overfitting by not allowing the variance to become too large.
2. (ii). Uniform distribution: This scenario considers a uniform distribution to model prior uncertainty about the coefficient variance across the entire range of possible values. We used a U(0, 100) distribution, a relatively uninformative prior, which assumes equal probability for any value of the variance between 0 and 100. We assumed no prior knowledge or strong beliefs about the possible range of values for the variance.
3. (iii). Inverse Gamma distribution: For the third scenario, we specified a Γ⁻¹(2, 0.5), which can be considered an informative prior for the variance parameters. This would put more prior weight on larger values of the variance compared to the previous scenarios but still allow for a wide range of possible values. The shape parameter of 2 indicates some prior belief that the variance is not very small, while the scale parameter of 0.5 indicates some prior belief that the variance is not very large.

For the case of unobserved or missing longitudinal outcomes (latent quantities), we used Bayesian imputation within the analysis to obtain the posterior distribution of these quantities of interest. In Bayesian approaches for longitudinal data, we assume that the missing data are missing at random (MAR), which means that the probability of missingness depends only on observed data, and not on the missing values themselves after conditioning on the observed data. On the other hand, in the survival submodel, missing data can occur when an individual is lost to follow-up or drops out of the study before the end of the follow-up period. The probability of being lost to follow-up or dropping out may depend on observable characteristics, such as age or sex, or unobservable characteristics, such as health status or treatment response. When missing data occur due to unobservable characteristics, the data are said to be missing not at random (MNAR), since the missingness mechanism is related to the unobserved outcomes. While MNAR and MAR are mutually exclusive assumptions in general and cannot occur at the same time, in this particular case of the joint modeling of longitudinal and survival data, it is possible to have both. This is because the missingness mechanism can be different for the longitudinal and survival data components. For example, the missingness in the longitudinal data may depend on the observed longitudinal data and some additional unobserved covariates, while the missingness in the survival data may depend on the observed survival data and some other unobserved covariates. In this case, the longitudinal data would be MAR with respect to the observed data and the unobserved covariates, while the survival data would be MNAR with respect to the unobserved covariates.

Posterior results are based on a set of three (3) MCMC chains of 20, 000 iterations on each chain, after the burn-in period. We ran the Bayesian model on a laptop computer with an AMD Ryzen 9 processor and 24 GB of memory in order to evaluate the computing performance. We achieved a running time of around 13 hours for the number of iterations per chain ran in parallel, which is shorter than the running time obtained in Pabon-Rodriguez et al. [3]. While the running time was reduced in the current work for a more complex model specification, we still believe there are numerous opportunities for optimization.

To assess the convergence of the parameters, a Gelman-Rubin diagnostic was computed using the coda package [32] in R. As a rule to assess convergence, any potential scale reduction factor below 1.2 indicates that the corresponding parameter has reached a level of convergence. In this case, all parameters achieved an estimated factor below the chosen threshold of 1.2. We also computed the Monte Carlo Standard Error (MCSE), which is a measure of the precision of the posterior distribution obtained from an MCMC algorithm. The purpose of MCSE is to provide a measure of the accuracy of the estimate of the posterior distribution, which is the distribution of the parameters of interest after taking into account the data and prior information [33]. For the parameters in the model, we obtained MCSE values between 0.00033 and 0.06583, with a mean of 0.00568 and a median of 0.00328. The full MCMC summary and diagnostics can be found in the supplementary file S1 File.

The code implementing the sampler using the NIMBLE system, and the posterior summary, diagnostics, sensitivity analysis, and code for the forecasting are provided in the supplemental material in the GitHub repository.

3.6 Posterior predictive distribution and p-values

The posterior sampling process (simulation) was used to obtain posterior predictive trajectories. The idea is to use the posterior distribution of the model parameters and the model specification provided throughout previous subsections. The posterior predictive trajectories are obtained by following the steps:

1. (i). Sample a value from the posterior distributions of each model parameter,
2. (ii). Obtain the estimated model components (predicted outcomes) from the Bayesian model specification from Eqs 1 to 6,
3. (iii). Repeat steps (i) and (ii) for S = 1000 times (s = 1, 2, 3…, S), where S represents the number of simulations performed, and
4. (iv). Plot the resulting posterior predictive trajectories.

Posterior predictive p-values are a Bayesian statistical method used to assess the fit of a model to observed data [31]. They measure the probability of observing data as extreme or more extreme than the observed data, given the model and the prior distribution. In the Bayesian framework, a posterior predictive p-value is an important tool for model checking and selection [33]. It allows us to evaluate whether the model adequately explains the observed data and whether the model is consistent with prior knowledge and assumptions. This approach can help identify potential flaws in the model and guide the choice of alternative models. To compute the posterior predictive p-value, a test statistic is required to summarize the observed data [33]. This test statistic is then compared to a distribution of the same test statistic generated from samples from the posterior predictive distribution. In the current work, we used the mean, standard deviation, and mean squared error (MSE) as test statistics.

To obtain the posterior predictive p-values, we followed these steps:

1. (i). Fit a Bayesian model to the data and obtain samples from the posterior distribution using MCMC methods
2. (ii). Generate new data from the posterior predictive distribution using the sampled parameters from the posterior distribution as described at the beginning of this subsection
3. (iii). Compute the test statistic on the observed data and on each simulated dataset from the posterior predictive distribution
4. (iv). Calculate the p-value as the proportion of simulated test statistics that are as extreme or more extreme than the observed test statistic

4.1 Posterior predictive trajectories

We examined the extent to which the presented Bayesian model could forecast trajectories of CanL progression. We considered four dogs within the cohort showing outcome patterns of decreasing, stable, or growing clinical status. One disadvantage of longitudinal forecasts for dynamic processes generally is that predictions further forward in time increase model uncertainty. Accordingly, we assumed only data from the first two time points were available and used a six month forecast window, which is a clinically relevant time frame for this disease. All of the posterior predicted trajectories are presented in the supplementary figures from S1 to S9 Figs. In addition, we only provide the 95% credible bounds of the posterior predictive trajectories. Here, we noticed that the posterior predictive intervals are wider when looking at time point 4 (six month forecast window) compared to the narrower intervals obtained when predicting responses for time point 3 (three month forecast window).

As discussed in Subsubsection 3.3.1, pathogen load was defined based on a censoring approach, therefore the model should be able to detect a trajectory of censored values. S1 Fig shows the posterior predictive trajectories of the pathogen load of the four example subjects. Subjects 2, 3, and 4 had censored values, and the model generated a median predicted trajectory similar to the observed trajectory at time point 3, but a slight increase is shown in pathogen load from time point 3 to 4, which was accompanied by a more notable change in antibody production as shown in S2 Fig. This caused a change in the probabilities of disease state classification as shown in S3 Fig. At the second time point, subject 2 is in LeishVet score 1 with probability 1, but the probability of remaining score 1 decreased gradually, while the probability of progressing to LeishVet score 2 or 3 increased. The increase in pathogen load and antibody levels in this subject caused deterioration in the probability of improving clinical status, as seen in the graph, when the probability of LeishVet score 1 decreases from 1 at time point 2 to 0.70 by time point 4, while the marginal probability of LeishVet score 2 for this subject is at approximately 0.20 by time point 4. In the case of subject 3, which is in LeishVet score 2 with probability 1 at time point 2, its marginal probability of LeishVet score 2 is reduced to around 0.45, while the marginal probability of clinical improvement to LeishVet score 1 disease is around 0.41 at time point 4. Subsequent graphs in the supplementary figures S4 to S9 Figs show posterior predictive trajectories of immune components, where S4–S6 Figs correspond to responses associated with the inflammatory immune response, while S7–S9 Figs are associated with the regulatory immune response. Among these, the model performed well in predicting the proportion of IFN-γ+ and IL-10+ CD4+ T cells, which are of great importance for immunopathogenesis during CanL.

The posterior predictive p-values for pathogen load and antibody levels readouts are 0.242 and 0.486, respectively. For the case of the inflammatory and regulatory responses of the immune system, we obtained the posterior p-values: 0.348, 0.868, 0.563, 0.801, 0.494, and 0.847. These posterior p-values correspond, respectively, to the readouts as presented in Fig 4. While there is no universal rule of thumb for posterior p-values, on many occasions, a common approach considers a posterior p-value close to 0.50 to indicate that the observed data is perfectly consistent with the model and prior assumptions, but other research works have also used the rule of a p-value between 0.15 and 0.85 to be acceptable. It is worth noting that the interpretation of a posterior p-value can also depend on the sample size, as larger samples may lead to smaller posterior p-values even if the data are relatively consistent with the model and prior assumptions. In this work, we handled a small sample setting, which could lead to larger posterior p-values. Based on the estimated posterior p-values, we can say that our observed data are relatively consistent with the model and prior assumptions.