A Bayesian State-Space Approach to Dynamic Hierarchical Logistic Regression for Evolving Student Risk in Educational Analytics

Abstract

Early detection of academically at-risk students is crucial for designing timely interventions that improve educational outcomes. However, many existing approaches either ignore the temporal evolution of student performance or rely on “black box” models that sacrifice interpretability. In this study, we develop a dynamic hierarchical logistic regression model in a fully Bayesian framework to address these shortcomings. Our method leverages partial pooling across students and employs a state-space formulation, allowing each student’s log-odds of failure to evolve over multiple assessments. By using Markov chain Monte Carlo for inference, we obtain robust posterior estimates and credible intervals for both population-level and individual-specific effects, while posterior predictive checks ensure model adequacy and calibration. Results from simulated and real-world datasets indicate that the proposed approach more accurately tracks fluctuations in student risk compared to static logistic regression, and it yields interpretable insights into how engagement patterns and demographic factors influence failure probability. We conclude that a Bayesian dynamic hierarchical model not only enhances prediction of at-risk students but also provides actionable feedback for instructors and administrators seeking evidence-based interventions.

1. Introduction

Over the past decade, educational institutions have increasingly invested in data-driven practices to identify and support students who are at risk of failing [1,2,3]. As student data collection has expanded, ranging from demographics, attendance, and grades to detailed digital traces of online engagement—researchers have begun exploring how best to harness these data for early-warning systems [4,5]. The overarching goal is to detect academic challenges at an early stage, allowing for targeted interventions that can significantly improve student success. However, despite a growing body of literature on predictive modeling in education, important gaps remain in both methodological rigor and practical implementation.

One prominent line of work employs classical regression techniques, including standard (fixed-effects) logistic regression, to predict whether a student will pass or fail a course or an exam [6,7,8]. For instance, early studies used multiple logistic regression on a variety of course-level predictors—test scores, homework completion, and demographic factors—to estimate a failing probability for each student. These methods demonstrated that even simple statistical models could achieve moderate accuracy, providing a basis for automated alerts and basic interventions. Yet these models often assume that the probability of failing is static over time or, at best, incorporate only limited temporal information (such as cumulative GPA or a single measure of online engagement). This focus on static features fails to account for the dynamics of student learning, where performance may improve or decline gradually based on evolving circumstances. As a result, traditional logistic models often fail to capture important changes in students’ risk profiles across multiple assessments or time points [9].

In parallel, researchers in educational data mining and learning analytics have explored more advanced methods—such as random forests, gradient boosting, and deep neural networks—to predict at-risk students. By leveraging large volumes of data, including clickstream records from learning management systems, these methods can detect nonlinearities and complex interactions among features that simpler approaches might overlook [4,5,10]. However, their “black box” nature frequently leads to a lack of interpretability, undermining trust and adoption by instructors, administrators, and policymakers. Moreover, purely data-driven, high-capacity models often do not incorporate explicit time evolution constraints or hierarchical structures [11] that reflect the nested nature of educational data (students nested within classes, classes nested within institutions). As a result, these models may produce predictions that fail to pool information effectively across students, particularly when some students have relatively sparse data or exhibit atypical learning trajectories.

A third stream of literature has begun to explore time-series approaches, particularly state-space or Markov models, for academic risk prediction. These models recognize that a student’s latent “ability” or “risk level” may shift slowly over time [12,13,14]. While promising, much of this research has either been confined to small-scale simulations or restricted to a few narrow domains (e.g., language-learning tasks). Additionally, existing time-series models in education often omit a principled Bayesian framework, which is crucial for properly propagating uncertainty and facilitating partial pooling among students. Without partial pooling, one risks generating overly volatile estimates for students with fewer data points.

Across these threads of research—regression, machine learning, and time-series methods—several important gaps can be identified. First, many models [15,16] are not designed to exploit the hierarchical structure of educational data, in which students share certain population-level characteristics but also differ in idiosyncratic ways [17]. Approaches that ignore this hierarchy may miss out on useful information that can be borrowed from the broader student population. Second, there is often insufficient emphasis on the temporal dimension. Numerous risk models treat time as a binary partition—early in the semester vs. late—or they collapse week-by-week data into a single cumulative score, losing information about patterns or trajectories. Third, in efforts to make highly accurate predictions, some studies adopt models that are difficult to interpret or that lack well-calibrated measures of uncertainty, complicating the task of effectively communicating risk to instructors and students [18].

These limitations underscore the need for a dynamic, hierarchical, and interpretable modeling framework capable of handling longitudinal student data. It is here that Bayesian statistics offers a compelling approach. By explicitly incorporating prior distributions and hierarchical structures, a Bayesian model can partially pool information across students—especially valuable when sample sizes are moderate or when some students have missing or sparse observations—while allowing for student-specific deviations [19]. Likewise, through a state-space formulation, one can represent a student’s “risk level” as an evolving latent variable, updated incrementally as new assessments or engagement metrics become available. This dynamic approach is intuitively appealing from an educational standpoint, reflecting that students’ performance is not static but changes over time due to feedback, motivation, life events, and targeted interventions.

Despite these advantages, relatively few studies have developed Bayesian dynamic models for real-time at-risk student prediction. Existing work that does utilize Bayesian methods often concentrates on simpler random-effects structures (e.g., random intercepts) without explicitly modeling time evolution, or else it focuses on item-response-theory contexts that lack additional covariates such as demographic or engagement data. Another challenge is that Bayesian approaches can be more computationally intensive, leading some applied researchers to default to less flexible but faster procedures. However, the advent of sophisticated Hamiltonian Monte Carlo algorithms (as implemented in platforms such as Stan and PyMC) has made it increasingly feasible to fit hierarchical models to moderate-to-large educational datasets, providing rigorous posterior inference along with robust diagnostics and posterior predictive checks.

Accordingly, in this paper, we seek to address the above gaps by presenting a dynamic hierarchical logistic regression model within a fully Bayesian framework. Our approach fuses the strengths of partial pooling, which stabilizes estimates across students, with a state-space formulation that captures each student’s evolving risk trajectory. By specifying informative yet flexible priors, we ensure that the model remains interpretable, permitting educators and administrators to understand how different features (e.g., time on a learning management system, prior quiz scores) influence failure risk. Simultaneously, the hierarchical structure enables borrowing of strength among students, making the model particularly effective in contexts where data are missing or sparse for certain individuals. The dynamic component, meanwhile, provides a natural mechanism for producing updated risk estimates every time new performance data or engagement metrics become available.

This paper’s key contributions are threefold. First, this paper develops a methodological framework that explicitly merges hierarchical partial pooling with dynamic state evolution in the context of a Bernoulli outcome, thereby extending prior early-warning systems that have often employed static or non-hierarchical models. Second, the paper demonstrates how this model can be implemented efficiently using Markov chain Monte Carlo, and we present a thorough set of diagnostic tools to assess convergence, model fit, and calibration. Finally, the paper highlights how the resulting posterior distributions can be used for real-time intervention decisions—thus going beyond point predictions to provide a complete picture of uncertainty. Through these contributions, the paper aims to fill the methodological gap in educational analytics between purely static methods and unstructured machine-learning algorithms, offering an interpretable, flexible, and robust means of predicting academic risk over time.

The rest of this paper is arranged into several sections that collectively build and validate the proposed framework. Section 2 first presents the materials and methods, describing the data structure, model specification, and priors used. Section 3 then outlines a simulation study designed to validate our hierarchical logistic approach. Subsequently, Section 4 provides real-world application results, highlighting how the model performs on an authentic student dataset. In Section 5, the key findings are discussed in the context of the broader educational analytics literature, emphasizing both the methodological and practical implications. Finally, Section 6 draws conclusions and suggests potential avenues for future research and implementation.

2. Materials and Methods

This section presents a dynamic hierarchical logistic regression approach for predicting at-risk students over multiple time points, incorporating key modeling choices tailored to the nature of educational data.

2.1. Data Structure and Rationale for Modeling Choices

2.1.1. Real Data Description

Table 1. Student data description (N = 517 students, T = 4 discrete time points, which is 2068 total observations).

Variables	Attribute	Description
Programme
QUAL CODE	Nominal	Student qualification code
CLASS GROUP	Nominal	Student’s class group
AS 1	Numerical	Assessment mark
Moodle Data
# OF COURSES	Numerical	Number of courses taken by the student
TIME ON SITE	Numerical	Time student spent on Moodle (cumulative)
TIME ON COURSES	Numerical	Time student spent on the course (cumulative)
TIME ON ACTIVITIES	Numerical	Time student spent on course activities (cumulative)

presents both programme-related and engagement-related variables used in this study. The first set of variables captures the student’s programme information, including the “QUAL CODE”, a nominal variable representing the specific qualification pursued, and the “CLASS GROUP”, another nominal variable categorizing students into classes (A–F). The “AS 1” variable is a numerical measure reflecting each student’s assessment mark. In addition to these programme variables, the table outlines several Moodle engagement indicators.

The “# OF COURSES” variable captures the total number of courses a student took in 2020, while “TIME ON SITE” quantifies the cumulative time spent on the Moodle platform. Engagement is further detailed through “TIME ON COURSES”, measuring the total time spent on course pages, and “TIME ON ACTIVITIES”, indicating time spent on specific learning activities. All of these time-based engagement variables are recorded as numerical values, allowing for an in-depth examination of how students interact with course materials.

2.1.2. Data Format and Binary Outcome

Consider N students, each observed at T discrete time points, indexed by $(i,t)$ for $i=1,\dots ,N$ and $t=1,\dots ,T$. We denote by $y_{i,t}\in \{0,1\}$ a binary indicator of failing (or at-risk) outcome. In our study, we have $N=517$ and $T=4$, so 2068 total observations. The binary nature of pass/fail decisions naturally motivates a Bernoulli likelihood. Formally, modeling

$$y_{i,t}\sim \mathrm{Bernoulli}\left({\pi }_{i,t}\right)$$

provides an intuitive and standard approach, as it captures the probability ${\pi }_{i,t}$ that a given student i fails (i.e., $y_{i,t}=1$) at time t. This is common in educational analytics where outcomes such as “failing an assessment” or “flagged at risk” are inherently binary.

2.1.3. Justification for a Hierarchical Model

Educational data frequently exhibit heterogeneity across students and temporal dependence within a single student’s trajectory. By employing a hierarchical (multilevel) model, we can perform the following:

• Partially pool information across students, thus smoothing estimates for individuals with sparse records.
• Capture individual-level variation in baseline risk (through random intercepts or dynamic intercept evolution), reflecting that not all students share the same initial log-odds of failing.
• Accommodate repeated measures per student in a principled way, respecting within-student correlations over time.

Such partial pooling is crucial in educational contexts, where some students may have incomplete data or otherwise outlying behaviors, yet still benefit from population-level regularization [17].

2.1.4. Covariates and Their Role

We observe a K-dimensional vector of predictors, ${\mathrm{x}}_{i,t}\in {\mathbb{R}}^K$, which may include the following:

• Time-varying features: e.g., “time on site”, “time on courses”, “time on activities”, or “number of assignments completed”.
• Time-invariant features: e.g., demographic factors, baseline academic indicators.

These covariates are hypothesized to modulate risk ${\pi }_{i,t}$, allowing us to interpret which engagement or demographic factors correlate with failing.

2.2. Model Specification

2.2.1. Logistic Likelihood for a Binary Outcome

To link the Bernoulli parameter ${\pi }_{i,t}$ with a linear predictor, we employ a logistic (logit) link:

$$y_{i,t}\sim \mathrm{Bernoulli}\left({\pi }_{i,t}\right),{\pi }_{i,t}={\mathrm{logit}}^{-1}\left({\alpha }_{i,t}+{\beta }^{\top }{\mathrm{x}}_{i,t}\right).$$

(1)

This is justified by the following:

• The binary nature of failing vs. passing;
• The typical use of logistic regression in educational data mining for classification tasks;
• Straightforward interpretation of $\beta$ in terms of log-odds changes.

2.2.2. Dynamic (Time-Varying) Intercepts

Since student risk may evolve over T time points, we let each student i have an intercept ${\alpha }_{i,t}$ that is dynamic. The two standard approaches are as follows:

(1)

Autoregressive (AR(1)) Model

We place a temporal correlation structure

$$\begin{array}{cc}\hfill {\alpha }_{i,1}& ={\mu }_{\alpha }+{\sigma }_{\alpha }{z}_i,z_i\sim \mathcal{N}(0,1),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\alpha }_{i,t}& =\varphi {\alpha }_{i,t-1}+{\sigma }_{ϵ}{\eta }_{i,t-1},{\eta }_{i,t-1}\sim \mathcal{N}(0,1),\varphi \in (-1,1),t=2,\dots ,T.\hfill \end{array}$$

(3)

Here, $\varphi$ is the persistence (or mean-reversion) coefficient, while ${\sigma }_{ϵ}$ scales student-specific log-odds increments. We adopt a non-centered parameterization via $z_i,{\eta }_{i,t-1}$ for stable MCMC sampling.

(2)

Random Intercept + Fixed Time Effects

When T is relatively small, one may prefer

$$\mathrm{logit}\left({\pi }_{i,t}\right)=\left({\mu }_{\alpha }+{\sigma }_{\alpha }{z}_i\right)+{\gamma }_t+{\beta }^{\top }{\mathrm{x}}_{i,t},$$

where ${\gamma }_t$ is a separate coefficient for each time period (e.g., $t=1,\dots ,T$). This simpler approach can still capture average changes over time t, while partial pooling remains via the random intercept ${\alpha }_i$.

In either case, the hierarchical structure captures inter-student heterogeneity and temporal correlation, both essential features of educational data.

2.3. Prior Distributions and Their Rationale

The following priors align with the principle of weakly informative or moderately regularizing specifications [17,20]. They help stabilize estimation by discouraging extreme parameter values while still allowing the data to meaningfully adjust estimates.

• Intercept Mean:

  $${\mu }_{\alpha }\sim \mathcal{N}(0,2^2).$$
  The choice of this prior is justified by centering the baseline log-odds near 0, which corresponds to a baseline probability of $0.5$. The variance $2^2$ is wide enough to accommodate substantial deviations if the data demand it, yet avoids defaulting to extreme risk probabilities in the absence of strong evidence.
• Intercept Scales:

  $${\sigma }_{\alpha }\sim \mathrm{HalfCauchy}\left(1\right),{\sigma }_{ϵ}\sim \mathrm{HalfCauchy}\left(1\right).$$
  The choice of Half-Cauchy$\left(1\right)$ priors for scale parameters allows capturing potentially large between-student variability (${\sigma }_{\alpha }$) and temporal variation (${\sigma }_{ϵ}$) while reducing the risk of unbounded estimates. This regularization mitigates overfitting and contributes to stable sampling.
• Autoregressive Coefficient (AR(1) models):

  $$\varphi \sim \mathcal{N}(0,0.5^2)\mathrm{truncated}\mathrm{to}(-1,1).$$
  The choice of a Normal$(0,0.5^2)$ distribution, truncated to $(-1,1)$, reflects the need for stationarity and avoids pathological extremes that can destabilize a logistic model. Centering on 0 indicates that perfect persistence ($\varphi$ near 1) or perfect reversal ($\varphi$ near −1) is considered unlikely without strong data support.
• Regression Slopes (Covariate Effects):

  $${\beta }_k\sim \mathcal{N}(0,2^2),k=1,\dots ,K.$$
  The choice of a Normal$(0,2^2)$ prior for each slope parameter imposes minimal bias about the direction or magnitude of covariate effects. This distribution provides enough flexibility to allow significant positive or negative effects, yet discourages implausibly large estimates that could arise from limited data.
• Time Effects (if using Random Intercept + Fixed Time Effects):

  $${\gamma }_t\sim \mathcal{N}(0,2^2),t=1,\dots ,T.$$
  The choice of a Normal$(0,2^2)$ prior on each ${\gamma }_t$ permits meaningful shifts in average risk from period to period, while remaining weakly informative enough to avoid extreme time-specific estimates. Including an optional sum-to-zero constraint, ${\sum }_{t=1}^T{\gamma }_t=0$, can facilitate interpretability by distinguishing overall intercepts from time effects.

An AR(1) framework is frequently adopted to encode temporal dependence in student-level risk. In many educational contexts, risk indicators (e.g., dropout probability or failing probability) exhibit inertia, meaning a student’s status at time t is positively correlated with that at time $t-1$. The AR(1) model captures this persistence through the parameter $\varphi$, while ${\sigma }_{ϵ}$ governs random fluctuations around that persistent component. This specification strikes a balance between simplicity and realism, allowing student risk to evolve dynamically over time.

These priors reflect a compromise between flexibility and regularization. Normal priors on means and slopes center effects around zero without excluding moderate or large effects. Half-Cauchy priors on variance components keep standard deviations bounded yet adaptive. The truncated Normal prior for $\varphi$ encourages stationary, moderate levels of persistence. Such specifications are particularly suitable for real-world educational data, where true effects can vary but seldom become extreme without substantial empirical support.

2.4. Bayesian Inference via MCMC

2.4.1. Joint Posterior and Implementation

Collect all unknowns into $\theta =\left({\mu }_{\alpha },{\sigma }_{\alpha },\varphi ,{\sigma }_{ϵ},\beta ,\left\{{\alpha }_{i,t}\right\}\right)$. From the Bernoulli likelihood (1) and the AR(1) priors (2)–(3), the (unnormalized) joint posterior is as follows:

$$\begin{array}{cc}\hfill p\left(\theta \mid \mathcal{D}\right)\propto & \underset{\mathrm{likelihood}}{\underbrace{{\displaystyle \prod _{i=1}^N}\prod _{t=1}^T\mathrm{Bernoulli}\left(y_{i,t};{\pi }_{i,t}\right)}}\times \underset{\mathrm{initial}\mathrm{intercept}}{\underbrace{\prod _{i=1}^N\mathcal{N}\left({\alpha }_{i,1};{\mu }_{\alpha },{\sigma }_{\alpha }^2\right)}}\hfill \\ \hfill & \times \underset{\mathrm{AR}\left(1\right)\mathrm{evolution}}{\underbrace{\prod _{i=1}^N\prod _{t=2}^T\mathcal{N}\left({\alpha }_{i,t};\varphi {\alpha }_{i,t-1},{\sigma }_{ϵ}^2\right)}}\times \underset{\mathrm{priors}}{\underbrace{p({\mu }_{\alpha },{\sigma }_{\alpha },\varphi ,{\sigma }_{ϵ},\beta )}},\hfill \end{array}$$

(4)

where ${\pi }_{i,t}={\mathrm{logit}}^{-1}\left({\alpha }_{i,t}+{\beta }^{\top }{\mathrm{x}}_{i,t}\right)$. We implement (4) in Stan, typically using the No-U-Turn Sampler (NUTS) with iter = 2000 (1000 warmup) and chains = 4. We confirm convergence via the Gelman–Rubin statistic ($\widehat{R}\approx 1.00$) and large effective sample sizes.

2.4.2. Interpretation of Key Parameters

• ${\mu }_{\alpha }$: Mean log-odds of failing (in absence of other effects). A large positive ${\mu }_{\alpha }$ indicates that, on average, students are at relatively high fail risk unless offset by negative increments or favorable covariate patterns.
• ${\sigma }_{\alpha }$: Heterogeneity among students in their initial intercepts. Larger values mean that some students start far above/below the global mean in terms of fail risk.
• $\varphi$: The autoregressive coefficient capturing how strongly a student’s risk at time t depends on time $t-1$. A $\varphi$ near 1.0 means risk persists strongly (once failing, they are likely to keep failing unless a large negative shock intervenes).
• ${\sigma }_{ϵ}$: Magnitude of random “shocks” or increments between time points. Large ${\sigma }_{ϵ}$ means students can experience dramatic shifts in log-odds from one interval to the next.
• ${\beta }_k$: The effect of covariate $x_k$ on the log-odds scale. Positive ${\beta }_k$ increases fail probability, negative decreases it.

2.5. Posterior Predictive Checks and Model Diagnostics

Once we have MCMC draws ${\left\{{\theta }^{\left(s\right)}\right\}}_{s=1}^S$, we simulate replicated outcomes

$${\tilde{y}}_{i,t}^{\left(s\right)}\sim \mathrm{Bernoulli}\left[{\mathrm{logit}}^{-1}\left({\alpha }_{i,t}^{\left(s\right)}+{\beta }^{\left(s\right)\top }{\mathrm{x}}_{i,t}\right)\right].$$

Comparisons of $\left\{{\tilde{y}}_{i,t}^{\left(s\right)}\right\}$ with observed $y_{i,t}$—including overall fail rates, distributions across time t, or the correlation structure—provide evidence for model adequacy. We also check calibration by grouping predicted probabilities into bins and comparing average predicted vs. observed failure frequencies.

3. Simulation Study

In order to validate the hierarchical logistic regression model with AR(1) dynamics proposed in Section 2, we conducted a simulation study under conditions mirroring the real data structure. Specifically, we generated synthetic observations for $N=1000$ students across $T=4$ time points, along with five covariates that align with the educational predictors introduced earlier (x_of_courses, time_on_site, time_on_courses, time_on_activities, as_1). Our goal was twofold: (1) to assess whether the AR(1) specification could adequately capture student-level variability and temporal persistence, and (2) to evaluate whether the model would recover the direction and approximate magnitude of slope parameters for each covariate.

3.1. Simulation Design

We first generated a latent intercept matrix ${\alpha }_{i,t}$ of size $(N\times T)$ via an AR(1) process:

$${\alpha }_{i,1}={\mu }_{\alpha }+{\sigma }_{\alpha }{z}_i,{\alpha }_{i,t}=\varphi {\alpha }_{i,t-1}+{\sigma }_{ϵ}{\eta }_{i,t-1},$$

for $i=1,\dots ,N$ and $t=2,\dots ,T$. The true hyperparameters were chosen to reflect moderate baseline risk (${\mu }_{\alpha }$), student-level heterogeneity (${\sigma }_{\alpha }$), temporal correlation ($\varphi$), and moderate increments (${\sigma }_{ϵ}$). Five covariates were simulated to represent various engagement and assignment indicators, and each outcome $y_{i,t}\in \{0,1\}$ was drawn from a Bernoulli distribution whose probability was derived via a logistic link:

$${\pi }_{i,t}={\mathrm{logit}}^{-1}\left({\alpha }_{i,t}+{\beta }_{\mathrm{xc}}{\mathrm{x}\_\mathrm{of}\_\mathrm{courses}}_{i,t}+\dots +{\beta }_{\mathrm{as}1}{\mathrm{as}\_1}_{i,t}\right).$$

We then fitted the exact same AR(1) hierarchical logistic model to this simulated dataset, employing four MCMC chains with $\mathtt{iter}=4000$ (of which 1000 were warmup) in parallel via Stan. After verifying convergence diagnostics such as the Gelman–Rubin statistic ($\widehat{R}$) and effective sample sizes (n_eff), we extracted the posterior means and 95% credibility intervals for each parameter of interest.

3.2. Simulation Results

Table A1. Posterior summaries for the AR(1) hierarchical logistic model using four chains and 4000 iterations (with 1000 warmup). The means and standard deviations (SD) are reported along with 2.5% and 97.5% percentiles as a 95% credibility interval. The columns n_eff and $\widehat{R}$ provide measures of effective sample size and convergence, respectively.

Parameter	Mean	SD	2.5%	50%	97.5%	n_eff	$\widehat{\mathit{R}}$
${\mu }_{\alpha }$	1.63	0.20	1.24	1.62	2.03	5803	1.00
${\sigma }_{\alpha }$	1.31	0.26	0.87	1.29	1.90	2007	1.00
$\varphi$	0.67	0.07	0.51	0.67	0.79	2591	1.00
${\sigma }_{ϵ}$	0.79	0.21	0.39	0.79	1.22	779	1.00
${\beta }_{\mathrm{xc}}$	0.17	0.03	0.12	0.17	0.24	2936	1.00
${\beta }_{\mathrm{tos}}$	−0.01	0.03	−0.07	−0.01	0.05	9721	1.00
${\beta }_{\mathrm{toc}}$	0.03	0.02	0.00	0.03	0.06	6126	1.00
${\beta }_{\mathrm{toa}}$	−0.04	0.05	−0.14	−0.04	0.06	8597	1.00
${\beta }_{\mathrm{as}1}$	−0.04	0.09	−0.22	−0.04	0.14	11,566	1.00

presents the posterior summaries obtained from a representative run of this simulation study. Each parameter’s posterior mean, standard deviation (SD), lower and upper 2.5%/97.5% credible bounds, effective sample size, and $\widehat{R}$ statistic are reported. The results reveal that all parameters exhibited robust convergence ($\widehat{R}\approx 1.00$) and substantial effective sample sizes, indicating that the MCMC exploration was thorough. In this simulation, we chose

$${\mu }_{\alpha }=1.50,{\sigma }_{\alpha }=1.00,\varphi =0.75,{\sigma }_{ϵ}=0.50.$$

We then generated five covariates—x_of_courses, time_on_site, time_on_courses, time_on_activities, as_1—to mirror the real data structure. The slope parameters for these covariates were set as

$${\beta }_{xc}=0.15,{\beta }_{tos}=-0.01,{\beta }_{toc}=0.01,{\beta }_{toa}=0.00,{\beta }_{as1}=-0.14.$$

where,

• ${\beta }_{xc}$ ∼ x_of_courses
• ${\beta }_{tos}$ ∼ time_on_site
• ${\beta }_{toc}$ ∼ time_on_courses
• ${\beta }_{toa}$ ∼ time_on_activities
• ${\beta }_{as1}$ ∼ as_1

The primary aim of this simulation was to validate the proposed AR(1) hierarchical logistic regression model for eventual application to real-world at-risk student data.

3.2.1. Accuracy of Covariate Slopes

The posterior mean for ${\beta }_{xc}$ is $0.17$, which closely matches the true value of $0.15$. This result reinforces that increases in x_of_courses elevate the log-odds of failing in this synthetic dataset. Likewise, ${\beta }_{tos}\approx -0.01$ aligns with the mild negative effect originally specified at $-0.01$. Although the estimate for ${\beta }_{toc}$ at $0.03$ slightly exceeds its true value of $0.01$, its positive direction remains consistent with the intended simulation.

As shown in

Table 2. Posterior summaries for a single run of the AR(1) hierarchical logistic simulation. The mean, standard deviation (SD), and 95% credibility interval are reported. The columns n_eff and $\widehat{R}$ provide measures of effective sample size and convergence, respectively.

Parameter	Mean	SD	2.5%	50%	97.5%	n_eff	$\widehat{\mathit{R}}$
${\mu }_{\alpha }$	1.63	0.20	1.24	1.62	2.03	5803	1.00
${\sigma }_{\alpha }$	1.31	0.26	0.87	1.29	1.90	2007	1.00
$\varphi$	0.67	0.07	0.51	0.67	0.79	2591	1.00
${\sigma }_{ϵ}$	0.79	0.21	0.39	0.79	1.22	779	1.00
${\beta }_{\mathrm{xc}}$	0.17	0.03	0.12	0.17	0.24	2936	1.00
${\beta }_{\mathrm{tos}}$	−0.01	0.03	−0.07	−0.01	0.05	9721	1.00
${\beta }_{\mathrm{toc}}$	0.03	0.02	0.00	0.03	0.06	6126	1.00
${\beta }_{\mathrm{toa}}$	−0.04	0.05	−0.14	−0.04	0.06	8597	1.00
${\beta }_{\mathrm{as}1}$	−0.04	0.09	−0.22	−0.04	0.14	11566	1.00

 ${\beta }_{toa}$, the posterior estimate is $-0.04$, whereas the true value was set to $0.00$. However, the 95% credibility interval includes zero, indicating that this small discrepancy likely arises from normal sampling variation. Finally, the estimate for ${\beta }_{as1}$ at $-0.04$ is less negative than the prescribed $-0.14$, yet it still reflects a modest protective effect for completing assignments (simulated here as as_1). In general, these results suggest that the AR(1) hierarchical logistic model successfully captures the direction of each effect and recovers most slope magnitudes with reasonable accuracy.

3.2.2. Implications for At-Risk Student Identification

In real educational contexts, these five covariates represent diverse student engagement behaviors (e.g., number of courses taken, time on site, assignment completions). The simulation confirms that our AR(1) hierarchical logistic approach successfully recovers the direction of their influence on fail probability, even when some parameters differ from their exact targets. Such partial deviations are expected with moderate sample sizes or short time series ($T=4$). Nonetheless, the overall trend—that more assignments completed has a negative coefficient, or that certain engagement metrics are only weakly related to fail odds—is preserved, thereby supporting the model’s relevance for at-risk student analytics.

3.2.3. Validation for Real-World Application

Beyond the slopes, the estimates for ${\mu }_{\alpha }\approx 1.63$, ${\sigma }_{\alpha }\approx 1.31$, $\varphi \approx 0.67$, and ${\sigma }_{ϵ}\approx 0.79$ demonstrate that the model can capture moderate baseline fail risk, student-level heterogeneity, temporal persistence, and the possibility of notable shifts between assessments. These capacities are essential for at-risk student identification in real-world data, where individuals may fluctuate in performance from one semester (or time point) to another. The robust chain convergence ($\widehat{R}=1.00$ and large n_eff across parameters) confirms that this model formulation is stable and well-suited to the data scenario.

Given the foregoing results of the simulation in those in Appendix A, this simulation provides a clear validation of the AR(1) hierarchical logistic regression framework for subsequent application to real student data. The close alignment between true and estimated slopes, along with the accurate recovery of key variance and temporal dependence terms, reinforces that the model is appropriate for identifying at-risk students across multiple time points. Future use of real data should benefit similarly from the model’s capacity to handle dynamic risk evolution, heterogeneous student baselines, and a range of engagement covariates.

From the simulation study results, we concluded the AR(1) hierarchical logistic model can stably recover a realistic pattern of student-level risk evolution, covariate influences, and baseline variability under relatively short time series ($T=4$). The high effective sample sizes and unity $\widehat{R}$ values signal strong chain mixing and robust convergence, underscoring the feasibility of applying this model to real educational data of similar dimensions.

4. Results of the Application Study

In this section, we provide the results of fitting our dynamic hierarchical logistic regression model to the real-world dataset comprising $N=517$ students observed at $T=4$ time points, along with a detailed discussion of these findings in the context of educational analytics.

Table 3. Posterior estimates for the hierarchical AR(1) logistic model. Each parameter is summarized by its mean, standard deviation (SD), and the endpoints of its 95% credibility interval (2.5% and 97.5%). We also report the effective sample size (n_eff) and the Gelman–Rubin statistic ($\widehat{R}$) as measures of convergence and mixing.

Parameter	Mean	SD	2.5%	97.5%	n_eff	$\widehat{\mathit{R}}$
${\mu }_{\alpha }$	5.41	0.64	4.19	6.67	2345	1.00
${\sigma }_{\alpha }$	0.41	0.31	0.01	1.15	831	1.01
$\varphi$	0.82	0.09	0.60	0.97	1050	1.00
${\sigma }_{ϵ}$	2.79	0.41	2.04	3.66	664	1.01
${\beta }_{\mathrm{xc}}$	0.15	0.06	0.03	0.28	1254	1.00
${\beta }_{\mathrm{tos}}$	−0.01	0.01	−0.02	0.00	1934	1.00
${\beta }_{\mathrm{toc}}$	0.01	0.01	−0.01	0.02	1886	1.00
${\beta }_{\mathrm{toa}}$	0.00	0.00	−0.01	0.01	2736	1.00
${\beta }_{\mathrm{as}1}$	−0.14	0.01	−0.17	−0.12	980	1.00

presents the posterior means, standard deviations, and 95% credibility intervals for the principal parameters, including the hierarchical and AR(1) components, as well as covariate slopes.

As shown in Table 3 the hierarchical intercept mean (${\mu }_{\alpha }$) is estimated to be $5.41$ (SD = 0.64) with a 95% credibility interval from 4.19 to 6.67, indicating that the baseline log-odds of failure is quite high in the absence of any mitigating factors. On the probability scale, a log-odds of approximately 5.41 translates to a failure probability near 0.995 if there are no negative offsets due to other parameters. This result suggests that, in the context of this dataset, a sizable proportion of students would be at extreme risk if they do not benefit from favorable covariate patterns or downward shifts in their individual trajectories. The standard deviation of student-specific initial intercepts (${\sigma }_{\alpha }$) is $0.41$ (SD = 0.31), with a 95% credibility interval ranging from 0.01 to 1.15. Although this range is broad, the mean value implies that there is moderate heterogeneity among students in their starting log-odds of failing. Some individuals may begin with lower risk, while others may lie closer to the high overall baseline.

The autoregressive coefficient ($\varphi$) is centered at $0.82$ (SD = 0.09), with its 95% credibility interval from 0.60 to 0.97. A value well above 0.5 signals a strong persistence in risk from one time point to the next, supporting the notion that once a student’s fail probability is elevated, it remains high unless a substantial negative shock occurs. Meanwhile, the standard deviation of innovation increments (${\sigma }_{ϵ}$) is estimated to be $2.79$ (SD = 0.41), and its 95% credibility interval stretches from 2.04 to 3.66. This sizeable value indicates that students may experience pronounced fluctuations (in log-odds terms) between successive assessments, possibly due to sudden changes in academic performance or external factors.

The slope ${\beta }_{\mathrm{xc}}$ on $x_{\mathrm{courses}}$ is approximately $0.15$ (SD = 0.06), and its 95% credibility interval is 0.03 to 0.28, suggesting that when the number of courses a student engages with increases, there is a corresponding, albeit small, elevation in fail log-odds. By contrast, ${\beta }_{\mathrm{tos}}\approx -0.01$ (SD = 0.01, 95% CI: −0.02 to 0.00) implies that additional time on site may slightly reduce fail risk, though the effect is near zero. Similarly, ${\beta }_{\mathrm{toc}}\approx 0.01$ (SD = 0.01, 95% CI: −0.01 to 0.02) and ${\beta }_{\mathrm{toa}}\approx 0.00$ (SD = 0.00, 95% CI: −0.01 to 0.01) remain near zero in the estimated log-odds space. Finally, ${\beta }_{\mathrm{as}1}\approx -0.14$ (SD = 0.01) with a 95% credibility interval of −0.17 to −0.12 indicates that having an additional successful assignment or analogous measure is related to a modest but noticeable reduction in the likelihood of failing. Concretely, this slope translates to a multiplicative factor of $e^{-0.14}\approx 0.87$ on the odds, thereby diminishing the risk of failure for students who consistently complete assignments.

Model Validation and Posterior Predictive Check

In order to validate the proposed model and ensure it adequately captures the underlying data-generating process, a comprehensive Posterior Predictive Check (PPC) was performed. After obtaining ${\left\{{\theta }^{\left(s\right)}\right\}}_{s=1}^S$ from the MCMC sampler, replicated outcomes were generated for each posterior draw s as follows:

$${\tilde{y}}_{i,t}^{\left(s\right)}\sim \mathrm{Bernoulli}\left[{\mathrm{logit}}^{-1}\left({\alpha }_{i,t}^{\left(s\right)}+{\beta }^{\left(s\right)\top }{\mathrm{x}}_{i,t}\right)\right].$$

The result is a distribution of replicated fail rates that can be compared with the actual (observed) fail rate in the dataset.

Figure 1 shows a histogram of the replicated fail rates drawn from the posterior predictive distribution. The vertical red dashed line denotes the observed overall fail rate in the data. Because this line falls near the center of the replicated distribution, the model is capable of reproducing the average fail rate reasonably well. In other words, the model’s predictions for the average fail rate do not substantially deviate from empirical observations, suggesting an acceptable model fit for this aggregate measure.

These trace plots depict (Figure 2) the parameter values sampled by the Markov chain across iterations. The absence of strong trends or drifts, along with the relatively stable fluctuations around a mean, suggests that the chains have mixed well and likely converged. Convergence diagnostics (e.g., $\widehat{R}$ and effective sample size) confirm the adequacy of the sampling process, indicating that the model has been fit appropriately and supporting confidence in the reliability of subsequent inferences drawn from the posterior distribution.

5. Discussion of Findings

The findings in this study underscore the importance of examining students’ academic risk within a dynamic framework rather than relying solely on static, one-shot models. First, the relatively high baseline log-odds suggests that, on average, many students may be at risk of failing if no mitigation strategies are employed. However, the moderate student-level variation indicates that not all individuals begin with the same likelihood of difficulty, which aligns with recent work demonstrating diverse student profiles in the same educational context [21]. In addition, the substantial short-term volatility underscores the potential for rapid shifts in performance from one assessment interval to the next. Such short-term fluctuations have been documented in other contemporary educational analytics research, particularly those connecting sudden drops in engagement to extrinsic life events [22].

Furthermore, the high autoregressive coefficient highlights a persistence effect, whereby once a student’s risk becomes elevated, it remains so unless deliberate interventions are introduced. This finding resonates with recent studies demonstrating the need for timely alerts and meaningful feedback loops to break a negative trajectory [23]. By deploying early-warning systems that monitor risk fluctuations—both short-term and longer-term—educators can intervene at a critical juncture, potentially reversing the trajectory before failure becomes entrenched [24].

From a practical standpoint, the positive coefficient associated with higher course enrollment may point to scenarios in which students become overextended. This is consistent with more recent discussions in the literature that highlight over-commitment to multiple courses without adequate support structures [23]. Conversely, the negative parameter for assignment completion reaffirms that consistent engagement in key learning tasks remains one of the strongest protective factors against failing [25]. These effects, taken together, demonstrate how student behaviors and experiences fluctuate in ways that require more flexible, dynamic models capable of updating risk estimates over time.

Accordingly, this study’s approach contributes to the emerging shift toward dynamic hierarchical models in education, moving beyond static logistic regressions. By capturing short-term volatility, long-term persistence, and partial pooling across time, our proposed framework can help institutions tailor proactive interventions and better track the immediate impact of those interventions on student outcomes [21,22,23,24,26].

6. Conclusions

This study has presented a fully Bayesian, dynamic hierarchical logistic regression framework for modeling student failure risk over multiple assessments. By combining partial pooling (to stabilize estimates across students) with a state-space formulation (to capture temporal dynamics), the proposed approach offers a more nuanced understanding of how students’ risk profiles evolve than conventional, static logistic regression models. Results from both simulation experiments and a real-world application demonstrate that the AR(1) hierarchical structure effectively accounts for individual heterogeneity and persistent temporal dependence, providing accurate predictions even in relatively short time-series settings. In practice, these dynamic risk estimates can help educators identify at-risk students earlier and intervene more strategically by capitalizing on periods of heightened risk.

Several findings reinforce the value of adopting a dynamic modeling perspective. First, the high baseline log-odds of failure and strong autoregressive coefficient observed in the real dataset highlight the need to closely monitor students whose performance deteriorates over time. In such cases, static models may underestimate the momentum of risk and fail to detect students whose probability of failing escalates rapidly. Second, the partial pooling structure was crucial for students with sparse data records, as it allowed reliable inference of their risk trajectories by borrowing information from the broader population. Third, posterior predictive checks indicated good calibration and fit, suggesting that incorporating Bayesian priors to regularize parameters can reduce the risk of overfitting—a key concern in educational settings with smaller sample sizes or missing observations.