Statistical Model

1. The Core Framework: Nonlinear Dynamic Latent Class SEM (NDLC-SEM)

The forecasting of critical states in the SAM study required innovative techniques to capture the complex, multi-level nature of student dropout.
The methodological approach relies on the Nonlinear Dynamic Latent Class Structural Equation Model (NDLC-SEM) — a flexible Bayesian framework integrated with a modified Forward Filtering Backward Sampling (FFBS) algorithm for real-time prediction.

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1.1 Key Capabilities

The NDLC-SEM framework combines the capabilities of several dynamic models to simultaneously address four essential data structures for modeling the longitudinal dropout process using Intensive Longitudinal Data (ILD).

It models:

1. Inter-Individual Differences (Traits): Stable characteristics (e.g., cognitive abilities).
2. Intra-Individual Changes (States): Changeable psychological states (e.g., affective and motivational states).
3. Unobserved Heterogeneity of Trajectories: Captured via time-varying latent classes following a hidden Markov process.
   • In this study, these classes represent the discrete latent variable $S_{it}$:
     • $s=1$: Intention to stay
     • $s=2$: Intention to quit
4. Time-Dependent Nonlinearities: Latent within-person variables predict transition probabilities with nonlinear effects.

The model was implemented in JAGS 4.2 and executed via the R2jags package, using a Gibbs sampler for Bayesian estimation.

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2. Model Specification

The model specification outlines how observed variables relate to latent constructs (measurement models) and how these latent constructs evolve and interact across levels (structural models).

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2.1 Measurement Models

Within-Level (States – $\mathbf{\eta}_{1it}$):

Seventeen observed variables ($\mathbf{Y}_{1it}$) operationalize seven continuous latent within-factors ($\mathbf{\eta}_{1its}$), representing affective/cognitive states such as stress, fear of failure, and affect balance.

All observed variables were centered and oriented so that higher values indicate stronger intention to quit.

The within-level measurement model is consistent across latent states:

$$\textbf{Equation (1):} \quad (\mathbf{Y}_{1it} \mid S_{it} = s) = \mathbf{\Lambda}_{10}\mathbf{\eta}_{1its} + \mathbf{\varepsilon}_{1it}$$

Where:

• $\mathbf{Y}_{1it}$ = (17 × 1) vector of observed variables
• $\mathbf{\eta}_{1its}$ = (7 × 1) vector of latent state factors
• $\mathbf{\varepsilon}_{1it}$ = vector of residuals

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Between-Level (Traits – $\mathbf{\eta}_{2i}$):

A single latent construct — cognitive ability (IQ) — was modeled using three CFT-3 test items measured at baseline.

$$\textbf{Equation (2):} \quad \mathbf{Y}_{2i} = \mathbf{\Lambda}_{2}\mathbf{\eta}_{2i} + \mathbf{\varepsilon}_{2i}$$

Where:

• $\mathbf{Y}_{2i}$ = (3 × 1) vector of observed indicators
• $\mathbf{\eta}_{2i}$ = latent IQ factor
• $\mathbf{\varepsilon}_{2i}$ = uncorrelated residuals

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2.2 Structural Dynamics (Within-Level)

The within-level dynamics were modeled using a first-order autoregressive process (AR(1)), specific to the discrete latent class $S_{it}=s$:

$$\textbf{Equation (3):} \quad (\mathbf{\eta}_{1it} \mid S_{it}=s) = \mathbf{\alpha}_{1is} + \mathbf{B}_{1is}\mathbf{\eta}_{1i,t-1} + \mathbf{\zeta}_{1it}$$

Where:

• $\mathbf{\eta}_{1i,t-1}$ = latent states at previous time $t-1$
• $\mathbf{\alpha}_{1is}$ = class-specific intercept vector
• $\mathbf{B}_{1is}$ = diagonal matrix of AR(1) effects
• $\mathbf{\zeta}_{1it}$ = innovation term

Note: Initial tests showed near-zero cross-lagged effects, justifying a simplified AR(1) structure.

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2.3 Structural Models (Between- and Cross-Level Interactions)

The stable latent trait (IQ, $\mathbf{\eta}_{2i}$) influences both the intercepts and autoregressive dynamics of the latent state processes.

Intercept Function:

$$\textbf{Equation (4):} \quad \mathbf{\alpha}_{1is} = \mathbf{\alpha}_{21s} + \mathbf{\beta}_{2s}\eta_{2i} + \mathbf{\zeta}_{2i}$$

AR Coefficients with Cross-Level Moderation:

$$\textbf{Equation (5):} \quad \mathbf{B}_{1is} = \mathbf{B}_{1s} + \mathbf{\Omega}_{2s}\eta_{2i}$$

Here, $\mathbf{\Omega}_{2s}$ allows cognitive ability (IQ) to moderate motivational and self-regulatory state dynamics over time.

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2.4 Markov Switching Model (Transition Probabilities)

The discrete latent state $S_{it}$ evolves according to a hidden Markov process, governed by transition probabilities derived via a logit link function.

Probability of Staying in $s=1$ (No Intention to Quit):

$$\textbf{Equation (6):} \quad \text{P}(S_{it}=1 \mid S_{i,t-1}=1) = \frac{\exp(\nu_{it}^{11})}{\exp(\nu_{it}^{11}) + 1}$$

Logit Function Definition:

$$\textbf{Equation (10):} \quad \nu_{it}^{11} = \gamma_{1} + \gamma_{2}\eta_{2i} + \mathbf{\gamma}_{3}\mathbf{\eta}_{1i,t-1} + \mathbf{\gamma}_{4}\mathbf{\eta}_{1i,t-1}\eta_{2i}$$

Return Probability (P₁₂):
The transition from intention to quit ($s=2$) to intention to stay ($s=1$) is assumed rare and slow:

$$P_{12} \sim \text{unif}(0.0, 0.1)$$

This aligns with the Rubicon model, positing that individuals rarely revert once a quitting intention is formed.

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3. Identification of Latent Discrete States

Identifying the latent states ($S_{it}$) as “intention to drop out” involved a confirmatory modeling strategy:

1. Imposed Constraints:
   Persons in state $s=2$ constrained to show higher scores on all seven negative affect scales.
2. Predictive Link:
   Transition probabilities were regressed on affective scales and their interactions with IQ.
3. Temporal Restrictions:
   Transitions back to $s=1$ restricted to low probability.
4. Partial Observation:
   Dropouts observed during the semester were coded directly as $S_{it}=2$ (manifest dropout).

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4. Forecasting Implementation: Forward Filtering Backward Sampling (FFBS)

Forecasting dynamic latent states is achieved through the Forward Filtering Backward Sampling (FFBS) algorithm — a Bayesian sequential estimation method adapted for hidden Markov models (see West & Harrison, 1997).

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4.1 Key Features of the FFBS Adaptation

• Integrates seamlessly with the Gibbs sampler (forecasting within estimation loop).
• Handles latent time-dependent predictors ($\mathbf{\eta}_{1it}$) driving state transitions.
• Produces real-time posterior forecasts for the latent dropout intention states.

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4.2 Core Algorithmic Steps

Step 1 — Reformulation (Aspect i.)

Reformulate NDLC-SEM into a Dynamic Linear Model (DLM) framework:

• Observation Equation:

  $$\mathbf{\eta}_{1jts} = \mathbf{F}_{jt}\mathbf{\theta}_{jts} + \mathbf{v}_{jts}$$

• System Equation:

  $$\mathbf{\theta}_{jts} = \mathbf{G}_{jts}\mathbf{\theta}_{j,t-1,s} + \mathbf{w}_{jts}$$

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Step 2 — Define Strata (Aspect ii.)

Define four strata $(s, s')$ for every consecutive time pair $(t, t-1)$, covering all possible transitions between “stay” and “quit” states.

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Step 3 — Continuous State Prediction (Aspect iii.)

For each stratum, sample latent factor scores, producing four forecast draws:

$$(\eta_{1jit} \mid (s,s'), D_{t-1})$$

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Step 4 — Marginal Predictive Density (Aspect iv.)

Compute the mixture of predictive densities across the four strata:

$$\text{P}(\mathbf{\eta}_{1jit}\mid D_{t-1}) = \sum_{s=1}^{2}\sum_{s'=1}^{2} \left[ \pi_{i}(s,s')p_{i,t-1}(s') \text{P}(\mathbf{\eta}_{1jit}\mid(s,s'),D_{t-1}) \right]$$

This produces the overall forecast distribution of the continuous latent variable.

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4.3 Posterior Updating and Smoothing

After observing $D_t$, priors are updated and the joint posterior over model combinations is computed:

$$p_{it}(s,s') \propto \pi_{i}(s,s')p_{i,t-1}(s')\text{P}(\mathbf{\eta}_{1jit}\mid M_{it}(s), M_{i,t-1}(s'),D_t)$$

The smoothed posterior for each latent state is then:

$$p_{it}(s) = \sum_{s'=1}^{2} p_{it}(s,s')$$

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4.4 H-Steps-Ahead Forecasting

For future predictions ($t > N_t$):

• Use last observed posteriors ($t = N_t$).
• Recursively define expected values $\mathbf{a}_{jt}$ and covariance matrices $\mathbf{R}_{jt}$.
• Generate the $H$-step-ahead forecast distribution of latent factors $\mathbf{\eta}_{1jt}$.

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Summary:
The integrated NDLC-SEM + FFBS framework enables dynamic, real-time prediction of latent dropout intentions, bridging stable cognitive traits and fluctuating emotional-motivational states.
It provides a powerful approach to modeling nonlinear psychological dynamics in longitudinal educational data.