> ## Documentation Index
> Fetch the complete documentation index at: https://methodscenter.mintlify.app/llms.txt
> Use this file to discover all available pages before exploring further.

# 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.

***

### 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.

***

## 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**).

***

### 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

***

**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

***

### 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.

***

### 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.

***

### 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.

***

## 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).

***

## 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).

***

### 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.

***

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

***

#### 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.

***

#### Step 3 — Continuous State Prediction (Aspect iii.)

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

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

***

#### 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.

***

### 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:

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

***

### 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 $\mathbf{\eta}_{1jt}$.

***

> **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.