> ## 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.

# Empirical Results & Intervention

## 1. Characteristics of the Latent Dropout States

The application of the NDLC-SEM allowed for the separation of two **discrete latent states**,\
$s = 1$ (no intention to quit) and $s = 2$ (intention to quit), based on distinct response patterns across seven continuous affective and cognitive scales.

### Intention to Quit ($s = 2$) Profile

Students who transitioned to the latent state $s = 2$ exhibited **consistently higher values** across all seven affective/cognitive scales.\
This profile characterizes students with:

* Higher **stress levels**
* Stronger **fear of failure**
* Perceived **overinvestment of time**
* Pronounced **negative affective states (PAN)**

These patterns clearly align with an emerging **intention to quit**.

### Persistent Dynamics in $s = 2$

The autoregressive coefficients ($\mathbf{B}_{1is}$) were **larger under $s = 2$** than under $s = 1$ (except for the “time investment” scale).\
This indicates that students intending to quit display **stronger, more self-reinforcing autoregressive patterns**, suggesting persistent negative feedback loops in affective and cognitive processes.

### Role of Cognitive Skills (IQ)

Baseline cognitive ability ($\eta_{2i}$) showed strong predictive power for within-level scales when students were in state $s = 1$.\
Higher IQ scores corresponded to **lower stress** and **more stable motivational states**.

To reflect the *Rubicon model* of action, the **return probability** from the “intention to quit” state back to “no intention to quit” was **constrained**:

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

The estimated mean value was:

$$
P_{12} = 0.097
$$

indicating that individuals rarely and only **slowly return** to the no-intention state once the quitting intention is formed.

***

## 2. Predictive Factors and the Markov Switching Model

The parameters of the **Markov switching model** identified critical predictors for the transition from $s = 1$ to $s = 2$.

### Primary Predictors ($\gamma_3$)

The transition was primarily driven by:

* **Negative Affect (PAN)**
* **Fear of Failure**

Thus, the **dynamic affective states** and **expectation of failure** are the key determinants of entering the dropout-intention state.

### Cross-Level Effects ($\gamma_4$)

The interaction between baseline cognitive skills (IQ, $\eta_{2i}$) and within-level states ($\mathbf{\eta}_{1i,t-1}$) was **negligible** ($\gamma_4 \approx 0$).\
This shows that while IQ stabilizes affect in $s = 1$, it **does not directly prevent** the switch to $s = 2$, which is primarily emotion-driven.

***

## 3. Establishing the Intervention Window

The most practically relevant finding was the identification of an **intervention window** that precedes actual dropout by several weeks.

* **Critical State Timing:**\
  The transition to “intention to quit” ($s = 2$) occurred, on average, at

  $$
  t = 22.2 \; (\text{SD} = 6.5)
  $$

  corresponding to approximately the **8th week** of the semester.

* **Actual Dropout Timing:**\
  Dropout behavior occurred on average at

  $$
  t = 45.0 \; (\text{SD} = 11.9)
  $$

  corresponding to the **16th week**.

* **Lead Time for Intervention:**\
  The **forecasted state switch occurred about 8 weeks before** the actual dropout behavior.

* **Intervention Implication:**\
  The period around **$t = 22.2$** represents a **critical monitoring point** for identifying and supporting at-risk students.

***

## 4. Overall Student Status and Forecasting Extent

At the end of the observation period ($t = 50$):

* **36.1%** of students had **actually dropped out**.
* The model classified **73.8%** of students as either having dropped out or belonging to the latent class $s = 2$.
* When the forecast was **extended by 5 additional time points**, the proportion predicted to develop dropout intentions rose to **40.2%**, suggesting a **continued increase in risk** beyond the observed period.

***

## 5. Robustness and Performance (Simulation Study)

A **simulation study** was conducted to assess the robustness of the **FFBS forecasting procedure** across varying conditions of sample size ($N_1$) and number of measurement occasions ($N_t$).

### Sensitivity

The model achieved **high sensitivity (> 0.91)** across all conditions.\
This means that individuals who switched states were reliably detected — even for forecasted time points.

### Specificity

Specificity was **lower** in the forecast period, ranging between **0.53–0.70**, indicating a **slight over-prediction** of dropout risk (progressive classification).\
However, **specificity improved significantly** when the sample size increased from $N_1 = 25$ to $N_1 = 50$.

### Forecast Precision

Forecast precision was strongly dependent on the **number of measurement occasions ($N_t$)**:

* With $N_t = 50$, the model achieved **lower quadratic score values** ($\delta_h$), indicating **higher precision**.
* With $N_t = 25$, forecast variance and uncertainty increased.

### Forecast Interval Width

Forecast interval (FI) width **increased with longer forecasting horizons**, forming a **megaphone pattern**.\
However, **increasing both $N_t$ and $N_1$** reduced FI width substantially.

### Coverage Rates

The **95% forecast interval coverage** was slightly below nominal, at **88–90%**.\
This deviation is attributed to the model’s **tendency to over-classify** students into the risk state ($s=2$).

***

> **Summary:**\
> The empirical findings from the SAM study confirm the NDLC-SEM + FFBS framework’s ability to:
>
> * Accurately forecast **latent psychological state transitions**
> * Identify **precise intervention windows** for dropout prevention
> * Achieve robust predictive performance across different sampling and time configurations
>
> Together, these results provide a foundation for **data-driven, early-warning systems** in higher education, capable of detecting and addressing dropout risk dynamically.