## Problem Description:

In SPSS assignment, we examined the factors affecting students' exam scores. We considered two key factors: GPA scores (fixed) and the condition of active learning (random), which can either be having or not having active learning. To analyze these factors, we employed a Linear Mixed Model and fitted it with the Restricted Maximum Likelihood (REML) method. The aim was to determine the influence of GPA and the teaching method on exam scores.

## Solution

**Mathematical Model & Parameters: To predict students' exam scores, we used the following mathematical model:
**

**y_ij =**Exam score of the ith student (i = 1, …… 324) in the jth instructor (j = 1, …. 10)**μ =**Fixed intercept**β_1 =**Effect on condition**x_ij2 =**GPA score of ith student in the jth condition**β_2 =**Effect on GPA score (slope between intake and exam score)**d_j =**Random effect of jth instructor; d_j ~ N(0, σ²d)**e_ij =**Residual/error of ith student in jth instructor e_ij ~ N(0, σ²e)

Linear Mixed Method - GPA and Teaching Method as Predictors: We applied a Linear Mixed Model to assess the impact of GPA as covariates and the teaching method or condition of active learning as a random factor on exam scores. The model's goodness of fit was assessed using Akaike information criterion (AIC) and Schwarz's Bayesian Criterion (BIC).

## Results:

- GPA significantly affects exam scores (F (1, 277.509) = 724.365, p < 0.001).
- The condition of active learning does not significantly affect exam scores (F (1, 8.007) = 1.3589, p = 0.272).

**Prediction of Examination Score:** Using the mathematical model, we predicted exam scores based on GPA and active learning condition. For example, if a student had a GPA of 3.0 and was in an active learning condition, their exam score was estimated to be 71.823 or 72.

**Correlation of Exam Score Between Students Under the Same Instructor: **The correlation of exam scores between students under the same instructor was computed as 0.49, indicating a moderate correlation.

**Comparing Models: **The Schwarz's Bayesian Criterion (BIC) showed that the model using GPA and teaching method as predictors (Model 1) was the better fit with a lower BIC value (1766.509) compared to the model with only the teaching method (Model 2, BIC = 1925.255).

## Question 2: Analyzing Blood Cholesterol Levels

**Problem Description:** In this part of the assignment, we employed binary logistic regression to analyze factors that predict whether a person meets the blood cholesterol level goal (METGOAL). The variables examined include ADHERENCE, SMOKE, and MONTHS.

**Analysis - Binary Logistic Regression:** Since the dependent variable is binary (met the goal or not), binary logistic regression was the appropriate method.

## Results:

- Chi-square is significant at 5% (χ² (3) = 165.272, p < 0.001).
- Nagelkerke R-square indicates that 18.7% of the variation in the probability of meeting the blood cholesterol level goal is explained by ADHERENCE, SMOKE, and MONTHS.
- Hosmer and Lemeshow Test shows that the model fits well with the data.

**Significant Predictors: **The independent variables ADHERENCE, SMOKE, and MONTHS are significant predictors of whether a person meets the blood cholesterol level goal.

**Formula for Probability of Meeting Cholesterol Goal: **We derived a formula to calculate the probability of a patient reaching their blood cholesterol goal based on the given variables.

**Predicting Probability of Meeting Cholesterol Goal: **Using the formula, we calculated the probability of a patient meeting their cholesterol goal under specific conditions.

**Percentage of Correct Predictions:** The classification table showed that 64.5% of predictions were correct, with high accuracy for both categories. This indicates the validity of the binary logistic regression model's findings.