https://orcid.org/0000-0001-7747-5045
Figures
Abstract
The transition from milk to solid food during the weaning period exposes young animals to significant dietary and environmental stressors, which can profoundly affect mucosal immune responses and overall growth efficiency. This paper introduces a novel Bayesian hierarchical model to comprehensively assess the complex interactions between diet, environmental factors, intestinal microbiota, and immune markers in young animals’ small intestines. The model integrates data at both individual and group levels, providing a robust framework to understand how these stressors influence immune responses and growth outcomes. This hierarchical Bayesian approach captures individual variability and group-level effects by employing sophisticated interaction terms and data-dependent empirical priors, offering high-resolution uncertainty quantification. The model’s novelty lies in its ability to synthesize multiple sources of variability, offering insights that are not achievable through traditional statistical models.
Citation: Chatterjee D, Ghosh P (2025) Bayesian hierarchical modeling of mucosal immune responses and growth efficiency in young animals: Demonstrating the superiority of data-dependent empirical priors. PLoS One 20(6): e0326273. https://doi.org/10.1371/journal.pone.0326273
Editor: Fazul Nabi, Sindh Agriculture University, PAKISTAN
Received: January 12, 2025; Accepted: May 26, 2025; Published: June 25, 2025
Copyright: © 2025 Chatterjee, Ghosh. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data for this study are publicly available from the GitHub repository (https://github.com/debashisdotchatterjee/Bayesian-Hierarchical-Modeling-of-Mucosal-Immune-Responses-and-Growth-Efficiency-in-Young-Animals-).
Funding: The author(s) received no specific funding for this work.
Competing interests: No authors have competing interests.
1 Introduction
The early stages of an animal’s life, particularly the transition from milk to solid food during weaning, are critical periods that can significantly influence long-term health outcomes [29,19,33]. During this time, young animals face a variety of dietary and environmental stressors that can profoundly affect their mucosal immune responses and growth efficiency. Understanding the complex interactions between these factors is essential for developing nutritional and management strategies that optimize health and productivity.
Recent research has highlighted the importance of maternal nutrition and environmental conditions during gestation and early postnatal life in shaping the health of offspring. Studies have shown that maternal dietary intake, stress levels, and nutrient partitioning during pregnancy and lactation can affect the offspring’s growth and immune function. For instance, the work by [19,33] underscored the significant impact of maternal stress during late pregnancy on neonatal gut health and immune responses, revealing how these early-life events can predispose young animals to challenges like enteropathogenic infections. Additionally, research on the role of gut microbiota and mucosal immunity in young animals has demonstrated that these factors are crucial in determining the efficiency of nutrient absorption and overall growth.
The study of maternal nutrition, environmental stressors, and their impacts on offspring development has been a significant focus within animal sciences. Several key studies have explored these interactions, providing a foundation for understanding the complex relationships between maternal factors and young animals’ health and growth efficiency.
[1] discussed intrauterine growth retardation and its implications for animal sciences, providing insights into how maternal undernutrition can impair fetal development. [17] examined the epigenetic mechanisms by which maternal dietary protein and amino acids influence the growth and development of offspring, connecting dietary factors to developmental outcomes through epigenetic modifications. [18] focused on the effects of heat stress during late gestation on dry cows and their calves, showing how environmental stressors can negatively impact maternal health and offspring development.
In the context of gut health, [20,21] discussed the role of biologically active peptides and microRNAs in milk, respectively, and their potential impact on neonatal development. [22] explored the stability of long noncoding RNAs in bovine milk exosomes during digestion, contributing to understanding how maternal milk components influence gut health.
In the context of our study, which investigates the impact of maternal nutrition, environmental factors, and immune responses on the growth and development of young animals, several key studies provide important insights. [23] developed a noninvasive method for monitoring gastrointestinal development in infants using gene expression profiles from exfoliated epithelial cells, highlighting the importance of early gut development, which is crucial for understanding how maternal influences can shape offspring health. [24] explored the role of CD44 in neutrophil migration across the intestinal epithelium, providing a cellular perspective on immune response mechanisms that may be influenced by maternal nutrition. [25] conducted a comparative gene expression analysis to enrich polymorphonuclear leukocytes and gastrointestinal epithelial cells from fecal RNA, emphasizing the utility of fecal samples in studying neonatal immune responses. [26] identified a polymorphism in the FABP2 gene that increases fat absorption in human intestines, which could be linked to variations in nutrient absorption efficiency in young animals. [27] investigated the expression of mucin genes in piglets susceptible to enterotoxigenic Escherichia coli, connecting gut barrier function with susceptibility to infections. [28] identified biomarkers for gut barrier failure in broiler chickens, further underscoring the relevance of gut health in early life stages.
[29] identified molecular markers for epithelial cells across gastrointestinal tissues in preweaning dairy calves, which aligns with our focus on early immune and gut development. [30] demonstrated that the exfoliated transcriptome reflects tissue-level gene expression in a model of NSAID enteropathy, offering a non-invasive approach that could be applied to similar studies in young animals. [31] developed a method for purifying RNA from human stool samples, providing a foundation for non-invasive gastrointestinal studies. [32] highlighted the discrepancies among commercially available kits for reverse transcription quantitative PCR, which is critical for ensuring the reliability of gene expression studies, particularly in sensitive contexts such as early development.
[33] examines the complex interplay between gut health, stress, and immunity in neonatal dairy calves, focusing on the host side of host-pathogen interactions. The study highlights how perinatal events, such as maternal stressors during late pregnancy, can have long-lasting effects on calves’ intestinal development and immune function, potentially predisposing them to enteropathogenic infections like Escherichia coli. This work is directly relevant to our research as it underscores the critical importance of maternal factors in shaping the neonatal immune system and gut health, providing a foundation for exploring the impacts of prenatal and postnatal factors on growth efficiency and immune responses in young animals.
Building on these findings, this paper presents a Bayesian hierarchical model that integrates dietary, environmental, and microbiota-related factors to assess their combined effects on mucosal immune responses and growth efficiency in young animals. The model incorporates individual-level variability and group-level effects, providing a comprehensive framework for understanding how these factors interact. A novel aspect of this research is using data-dependent empirical priors, which enhance the model’s accuracy and interpretability by allowing the priors to be informed by the data. This approach improves the robustness of the model and offers a more nuanced understanding of the biological processes at play.
By focusing on the interactions between diet, stress, microbiota diversity, and immune markers, this study aims to provide insights that can inform strategies for improving animal health and productivity. The Bayesian hierarchical model presented here represents a significant advancement in analyzing complex biological data, offering a powerful tool for researchers and practitioners in animal science.
2 The need for a Bayesian model
Traditional frequentist models often face limitations in analyzing complex biological data, such as mucosal immune responses and growth efficiency in young animals, due to challenges in capturing uncertainty, accommodating hierarchical structures, and leveraging prior knowledge. Bayesian models overcome these issues by incorporating prior information, enabling hierarchical modeling to account for variability at multiple levels, and providing full posterior distributions for rigorous uncertainty quantification. They are highly adaptable, leveraging empirical priors to fit specific data characteristics, and often demonstrate superior fit and predictive performance. These advantages make Bayesian hierarchical models a powerful tool for analyzing the multifaceted interactions of dietary, environmental, and genetic factors, offering a comprehensive understanding of complex biological processes.
3 Objective and novelty of this research
The primary objective of this research is to develop and validate a novel Bayesian hierarchical model that thoroughly evaluates the impact of dietary and environmental stressors on mucosal immune responses and growth efficiency in young animals. Unlike usual notions that treat dietary and environmental factors independently, this approach captures the intricate and interdependent relationships between these variables and their combined effects on immune responses. The hierarchical Bayesian framework employed here enables data integration across multiple levels, accounting for individual animal variability and group-level environmental effects. This method enhances the precision of the estimates and provides a more comprehensive understanding of the biological processes involved, thereby filling a critical gap in the existing literature.
4 Methodology 1 (for simpler model & without explicit mucosal immune responses)
4.1 Hierarchical Bayesian framework
The proposed model adopts a hierarchical Bayesian framework, allowing for incorporating fixed and random effects at multiple levels. The model is structured as follows:
4.1.1 Level 1: Individual animal data.
At the individual level, let yij represent the observed response variable for the i-th animal in the j-th group. The observations include growth efficiency, immune markers, and microbiota composition. The model at this level is expressed as:
where:
is the design matrix for the i-th animal in the j-th group, including covariates such as dietary composition, environmental stressors, and genetic background.
is the vector of fixed effect coefficients.
represents the random effect for individual animal variability.
is the residual error term.
4.1.2 Level 2: Group/environmental effects.
At the group level, we model the effects of dietary groups and environmental conditions. Let represent the group-level effect for the j-th group, modeled as:
where:
is the design matrix for the group-level covariates, such as dietary group and environmental conditions.
is the vector of group-level fixed effect coefficients.
represents the group-level random effect.
4.2 Priors and posterior inference
4.2.1 Priors.
The prior distributions for the fixed effects and
are specified as:
where are prior means and
are covariance matrices.
The prior distributions for the variance components are modeled as follows:
where are hyperparameters.
4.2.2 Posterior inference.
The posterior distribution is obtained by combining the likelihood function with the prior distributions proportional to:
Markov Chain Monte Carlo (MCMC) methods generate samples from the posterior distribution, allowing uncertainty quantification of the model parameters.
4.3 Response variables and covariates
4.4 Interaction terms and model validation
Interaction terms between diet, environmental stressors, and microbiota composition are introduced to capture the synergistic effects on immune response and growth efficiency:
where is the vector of interaction coefficients.
4.5 Sensitivity analysis
Sensitivity analysis is conducted by varying the hyperparameters to evaluate the robustness of the posterior estimates. The sensitivity of the primary and secondary outcomes to these variations provides insights into the model’s robustness and the key drivers of mucosal immune responses.
5 Methodology 2: Advanced Bayesian hierarchical model for mucosal immune responses
In analyzing growth efficiency in young animals, it is imperative to consider the multifaceted biological processes that influence this outcome. The novelty of this model lies in its integration of sophisticated, data-dependent (empirical) priors, which are tailored to the complexities of the biological processes under investigation. By employing these empirical priors, the model is robust and sensitive to the nuances of the data, thereby improving the accuracy and interpretability of the results. Using empirical priors, as opposed to traditional Jeffreys priors, allows the model to be more flexible and data-driven, particularly in addressing the newly introduced variables [2,3,15,16].
5.1 Mathematical formulation of the enhanced model
Let Yi denote the growth efficiency of animal i. The model is specified as:
where the expected growth efficiency, , is modeled by:
Here:
,
,
are the dietary, stress, and genetic covariates, respectively.
represents the diversity of the gut microbiota.
measures cytokine levels, reflecting immune response.
and
represent villus height and crypt depth, respectively, indicators of gut health.
is the animal-specific random effect.
is the group-specific random effect [14].
The coefficients ,
,
, and
are of particular interest as they quantify the effects of microbiota diversity, cytokine levels, villus height, and crypt depth on growth efficiency. The empirical Bayes method is employed to estimate these priors directly from the data, ensuring that the model adapts to the specific characteristics of the dataset.
5.2 Empirical priors and posterior inference
Given the complexity of the biological interactions in this study, we replace the previously proposed Jeffreys priors with sophisticated empirical priors. These priors are derived from the data, allowing the model to better reflect the underlying processes:
where:
is the empirical mean of the least squares estimates for each
.
represents the empirical variance of these estimates.
The empirical Bayes approach provides a data-driven mechanism to set the prior distributions, enhancing the model’s flexibility and responsiveness to the actual data. This choice of priors allows the model to remain non-informative while adapting to the scale and variability of the studied biological processes.
The posterior distribution for the parameters is obtained via Markov Chain Monte Carlo (MCMC) methods, with the likelihood function given by:
With its empirical priors, this enhanced model allows for a more accurate and nuanced understanding of the complex relationships between the mucosal immune environment and growth efficiency in young animals. The use of data-dependent priors ensures that the posterior distributions are more reflective of the true underlying biological processes, providing a robust framework for future research in this area.
The advanced Bayesian hierarchical model proposed in this section introduces a novel approach to incorporating key biological factors—microbiota diversity, cytokine levels, villus height, and crypt depth—into the analysis of growth efficiency in young animals. The model maintains flexibility and adaptability by leveraging empirical priors while accurately reflecting biological systems’ complex, multi-layered interactions.
6 Simulation result with synthetic dataset 1
In this section, we present the results of a Bayesian hierarchical analysis conducted on a synthetic dataset, which was generated to study the impact of diet, stress, and genetic factors on the growth efficiency of animals. The model’s purpose was to accurately estimate the effects of these variables while accounting for group and individual-level variability.
6.1 Synthetic data generation
We generated a synthetic dataset of n = 100 animals, categorized into ng = 5 groups. The data included three continuous covariates: diet (x1), stress (x2), and genetic factors (x3), each drawn from a standard normal distribution, i.e., . The group effects uj were modeled as
for
, and the individual random effects
were modeled as
for
. The outcome variable (growth efficiency yi) was then generated using the following linear model:
where ,
, and
are the true coefficients for diet, stress, and genetic factors, respectively. The noise term
was drawn from
.
The first few rows of the generated dataset are presented in Table 1.
- animal_id: A unique identifier for each animal in the study.
- group_id: The identifier for the group to which each animal belongs, representing different environmental or dietary conditions.
- diet: A continuous variable representing the dietary condition or treatment applied to each animal.
- stress: A continuous variable representing the stress level experienced by each animal, which various environmental factors could induce.
- genetic: A continuous variable capturing each animal’s genetic factor or predisposition.
- interaction: The interaction term between diet and stress is included to account for potential synergistic effects [4,5,7,10].
- growth_efficiency: The response variable of interest, representing the growth efficiency of each animal, is measured as the output of the hierarchical Bayesian model.
6.2 Bayesian hierarchical model fitting
We employed the brms package in R to fit a Bayesian hierarchical model to the synthetic dataset. The model was specified as [6,8,9,11,13]:
with
The priors were set as follows: ,
for
, and
. We ran 4 chains with 10,000 iterations each (5,000 for warmup) and assessed convergence using the potential scale reduction factor (
), which was close to 1 for all parameters, indicating good convergence.
Table 2 shows the posterior estimates for the random effects’ regression coefficients and standard deviations [12].
Abbreviation: l-95-CI = lower bound of the 95% credible interval; u-95-CI = upper bound of the 95% credible interval; Rhat = Gelman & Rubin’s potential scale reduction factor (convergence diagnostic); Bulk-ESS = effective sample size for the bulk of the posterior distribution; Tail-ESS = effective sample size for the tails of the posterior distribution.
6.3 Posterior analysis and diagnostic plots
The posterior distributions of the parameters (Diet),
(Stress), and
(Genetic) were analyzed with an adjusted kernel bandwidth to detect any potential multimodal or bivariate peaks. Fig 1 shows the posterior density plots for these parameters with their respective truth lines. The adjusted bandwidth highlights intricate features in the posterior distributions, providing deeper insight into the data’s structure.
Fig 2 presents the plot of observed vs. predicted growth efficiency values. The red diagonal line represents perfect prediction, where observed and predicted values are equal. The close alignment of the points with this line indicates the model’s good predictive accuracy.
Red diagonal line represents perfect prediction, where observed and predicted values are equal. The value is 0.865, showing effective prediction.
Fig 3 shows the posterior predictive check plot, which compares the observed data to the model’s predicted data. The overlay indicates that the model’s predictions align well with the observed data distribution, suggesting an adequate model fit.
Here, y denotes the observed growth efficiency, and yrep represents the replicated values drawn from the posterior predictive distribution.
Lastly, the trace plots for the MCMC chains in Fig 4 demonstrate that the chains mixed well and converged. No significant autocorrelation was observed, indicating that the model fit was stable and reliable.
6.4 Interpretation of results
The results of the Bayesian hierarchical model applied to the synthetic dataset provide several important insights:
- Parameter Estimation: The posterior distributions for the diet, stress, and genetic factors indicate that the model successfully recovered the true parameter values used in the synthetic data generation. The posterior mean estimates are close to the true values, and the credible intervals cover these values.
- Random Effects: The random effect standard deviations for both group and individual levels indicate the model’s ability to capture variability at different hierarchical levels. The relatively larger variability at the individual level (
) compared to the group level (
) suggests significant heterogeneity among individual animals within groups.
- Model Diagnostics: The posterior predictive checks, trace plots, and observed versus predicted plots confirm that the fit of the model is satisfactory. The convergence diagnostics show that the chains have mixed well, and the posterior predictive checks suggest that the model adequately captures the data’s underlying structure.
- Predictive Accuracy: The predicted vs. observed plot demonstrates the model has good predictive accuracy, with most points lying close to the identity line. This indicates that the model’s predictions align well with the observed data.
In conclusion, this simulation study with Synthetic Dataset 1 demonstrates that the Bayesian hierarchical model is robust and capable of accurately estimating parameters and predicting outcomes in the context of growth efficiency influenced by diet, stress, and genetic factors. The model’s ability to account for both group and individual variability makes it a powerful tool for analyzing complex datasets in biological studies.
7 Simulation results with synthetic dataset 2 using Bayesian hierarchical model with empirical priors
7.1 Introduction and model motivation
Understanding the factors influencing growth efficiency in young animals is pivotal for optimizing health and productivity in agricultural and research settings. Growth efficiency, a complex trait, is affected by many factors, including dietary intake, stress levels, genetic predispositions, and various biological parameters such as gut microbiota diversity, cytokine levels, villus height, and crypt depth. To capture the intricate relationships among these variables, we employ a Bayesian hierarchical model with empirical priors, specifically the horseshoe prior, which is adept at handling sparsity in high-dimensional data.
The hierarchical structure allows for incorporating random effects at multiple levels, accounting for individual animal and group variability. Empirical priors provide flexibility by allowing the data to inform the prior distributions, enhancing the model’s adaptability and robustness.
7.2 Synthetic data generation and model specification
7.2.1 Data generation.
We generated a synthetic dataset to simulate the growth efficiency of young animals under varying conditions. The dataset comprises:
- Number of Animals:
- Number of Groups:
- Predictors:
- Dietary factors (
)
- Stress levels (
)
- Genetic factors (
)
- Microbiota diversity (
)
- Cytokine levels (
)
- Villus height (
)
- Crypt depth (
)
- Dietary factors (
- Outcome Variable: Growth efficiency (Y)
The synthetic data was generated based on the following hierarchical model:
where:
is the animal-specific random effect.
is the group-specific random effect.
is the residual error.
The true coefficient values used for data generation were
corresponding to the intercept and the seven predictors, respectively. The random effects were generated with standard deviations
and
. The residual error standard deviation was set to
.
7.3 Empirical priors and posterior inference
7.3.1 Prior specification.
The horseshoe prior was chosen for the regression coefficients () to allow for strong shrinkage of irrelevant coefficients while retaining the flexibility to detect significant effects. The hierarchical structure of the horseshoe prior is defined as:
This prior is particularly effective in high-dimensional settings where sparsity is expected.
7.3.2 MCMC sampling.
The posterior distributions were estimated using Markov Chain Monte Carlo (MCMC) sampling with the following settings:
- Number of Chains: 4
- Iterations per Chain: 20,000
- Warm-up Iterations: 5,000
- Seed: 123
Convergence diagnostics were assessed using trace plots and the Gelman-Rubin statistic (), ensuring that all parameters achieved satisfactory convergence.
7.4 Results and interpretation
7.4.1 Posterior estimates of model coefficients.
Table 3 presents the regression coefficients’ posterior means and standard errors, random effects standard deviations, residual standard deviation, and hyperparameters from the original model.
7.4.2 Sensitivity analysis.
A sensitivity analysis was conducted to assess the robustness of the model by slightly altering the initial values and rerunning the sampler. Table 4 compares the posterior means of the original model with those from the sensitivity analysis.
The comparison indicates minimal variations between the original and sensitivity analyses, underscoring the stability and robustness of the model’s posterior estimates against slight perturbations in the initial values.
7.4.3 Goodness-of-fit and model assessment.
Predicted vs. observed growth efficiency.
Fig 5 illustrates the predicted and observed growth efficiency relationship. The R-squared value quantifies the proportion of variance in the observed data explained by the model, providing a measure of the model’s predictive performance.
Red diagonal line represents perfect prediction, where observed and predicted values are equal. The value reaches 0.987, highlighting superior fit using the enhanced model and horseshoe prior.
The R-squared value of 0.987 indicates a strong predictive capability of the model, with approx (rounded) 99% of the variability in growth efficiency being explained by the predictors and random effects included in the model.
Posterior predictive checks. Posterior predictive checks (PPC) assess the model’s ability to replicate the observed data. Fig 6 displays the PPC plots, which compare replicated data distribution from the posterior predictive distribution to the actual observed data.
Here, y denotes the observed growth efficiency values, and yrep represents replicated values generated from the posterior predictive distribution.
The PPC plots demonstrate that the model adequately captures the central tendency and variability of the observed growth efficiency data, as the observed data points fall within the range of the posterior predictive distributions.
Traceplots for MCMC convergence. Traceplots are essential for diagnosing the convergence of MCMC chains. Fig 7 presents the trace plots for selected parameters, including regression coefficients and random effects.
The trace plots exhibit good mixing and lack of apparent trends or autocorrelation, indicating that the MCMC chains have converged to the posterior distributions.
7.4.4 Posterior density plots.
Posterior density plots visually represent the posterior distributions of the model parameters. Each plot includes a vertical dashed line indicating the true parameter value used in data generation, facilitating a comparison between the estimated and true values.
The posterior density plots presented in Figs 8–14 provide a comprehensive visualization of the estimated posterior distributions for the model parameters. Each plot is critical in understanding how well the Bayesian hierarchical model with empirical priors has captured the underlying true values of the parameters used in the synthetic data generation.
Fig 8 illustrates the posterior distribution for the dietary effect (b_diet) on growth efficiency. The proximity of the estimated distribution to the true parameter value, indicated by the vertical dashed line, suggests that the model has accurately inferred this effect.
In Fig 9 the posterior distribution for the stress parameter (b_stress) is shown. The concentration of the posterior around the true value reflects the model’s robustness in estimating the influence of stress on growth efficiency.
Fig 10 provides insight into the genetic effect (b_genetic). The posterior density is tightly centered around the true value, indicating that the model has effectively captured the genetic contribution to growth efficiency.
The posterior distribution for gut microbiota diversity (b_microbiota_diversity), as shown in Fig 11, emphasizes the significant role that this factor plays in the model. The closeness of the posterior mode to the true value reinforces the model’s ability to accurately represent this biological factor.
In Fig 12, the posterior density for cytokine levels (b_cytokine_levels) is depicted. The alignment of the posterior with the true value underlines the model’s precision in capturing the immune response’s impact on growth efficiency.
Figs 13 and 14 present the posterior densities for villus height (b_villus_height) and crypt depth (b_crypt_depth), respectively. These plots reveal how the model has successfully integrated these indicators of gut health into the overall analysis. The closeness of the estimated posteriors to the true values demonstrates the model’s competence in incorporating complex biological factors.
Overall, these posterior density plots collectively validate the effectiveness of the Bayesian hierarchical model with empirical priors in accurately estimating the effects of multiple interacting biological factors on growth efficiency. Each plot corroborates the model’s reliability and provides a clear visual comparison of the estimated and true parameter values, thus confirming the robustness of the inference process.
8 Model diagnostics and performance evaluation
8.1 K-fold cross-validation
K-fold cross-validation is a robust technique to assess the predictive performance of our Bayesian hierarchical model. We performed 10-fold cross-validation, partitioning the dataset into 10 equal subsets. Each iteration used one subset as the validation set while the remaining nine subsets were used for training. This process was repeated 10 times to ensure that each subset was used for validation exactly once.
The primary metric of interest is the expected log predictive density (ELPD), which estimates the model’s predictive accuracy. The ELPD is derived from the log-likelihood and is averaged across all data points in the validation set. Additionally, the K-fold information criterion (Kfoldic) is calculated as , serving as a penalized measure of model fit, where lower values indicate better predictive performance.
The results of the 10-fold cross-validation are presented in Table 5.
Significance: The ELPD measures the model’s ability to predict unseen data. The value of indicates the penalized model fit, where lower values suggest a better fit. The relatively low standard errors (SE) indicate these estimates’ stability across different cross-validation folds.
8.2 Effective Sample Size (ESS) and Monte Carlo Standard Error (MCSE)
In the context of MCMC sampling, the Effective Sample Size (ESS) measures the number of independent samples in the chain, which accounts for autocorrelation. A higher ESS indicates that the chain has better mixing and is more reliable for inference. Monte Carlo Standard Error (MCSE) estimates the standard error associated with the MCMC estimates, providing insight into the precision of these estimates.
The ESS and MCSE values for the model parameters are summarized in Table 6.
Significance: The high ESS values for most parameters suggest that the MCMC chains have mixed well and that the samples are reliable for drawing inferences. The low MCSE values indicate high precision in the estimation of these parameters. The consistency between the ESS and MCSE further supports the robustness of the model’s posterior estimates.
These diagnostic metrics ensure the reliability of our Bayesian hierarchical model and validate its use for predictive modeling in the context of mucosal immune responses and growth efficiency in young animals.
9 Discussion
The Bayesian hierarchical model presented in this study offers a comprehensive framework for analyzing the complex interplay between dietary factors, environmental stressors, and biological markers on mucosal immune responses and growth efficiency in young animals. By integrating data at individual and group levels, the model provides nuanced insights into the underlying mechanisms that influence these outcomes.
One of the key strengths of this approach lies in its ability to incorporate empirical priors, which are informed directly by the data. This enhances the model’s adaptability and robustness and allows for more accurate estimates of the effects of various predictors. Hierarchical modeling further enables the partitioning of variability at different levels, thereby capturing the heterogeneity inherent in biological systems.
The results of the k-fold cross-validation demonstrate the model’s strong predictive accuracy, as indicated by the ELPD and Kfoldic metrics. These findings suggest that the model is well-calibrated and generalizes effectively to new data. The high Effective Sample Size (ESS) and low Monte Carlo Standard Error (MCSE) values further corroborate the reliability of the MCMC estimates, ensuring that the posterior distributions are well-estimated.
Moreover, the interaction terms introduced in the model capture the synergistic effects between diet, stress, and microbiota diversity, offering deeper insights into how these factors jointly influence immune responses and growth efficiency. This approach marks a significant advancement over traditional models, often treating these factors in isolation, thereby missing the complex interactions crucial for understanding biological processes.
However, it is essential to recognize the limitations of this study. While the model demonstrates strong predictive performance and robustness, the synthetic nature of the dataset means that real-world applications may present additional challenges. Biological data can be highly variable and subject to measurement errors, which may require further refinement of the model’s assumptions and structure. Additionally, the reliance on empirical priors, while advantageous in many respects, necessitates careful consideration of prior information to avoid potential biases.
10 Conclusion
This research has successfully developed and validated a powerful Bayesian hierarchical model for analyzing the multifaceted factors influencing mucosal immune responses and growth efficiency in young animals. By leveraging empirical priors and hierarchical modeling, the study has demonstrated the model’s ability to accurately estimate the effects of dietary and environmental factors while accounting for individual and group-level variability.
The findings highlight the importance of considering interactions between diet, stress, and microbiota diversity in understanding immune responses and growth outcomes. The model’s robustness, as evidenced by strong predictive accuracy and reliable MCMC estimates, underscores its potential for application in animal health research.
This study contributes to the field of animal science by providing a novel methodological framework that integrates complex biological data in a statistically rigorous manner. The model’s ability to capture individual variability and group-level effects offers a more nuanced understanding of the factors influencing animal health and growth, paving the way for more targeted and effective interventions.
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