unit 4

Statistical Models in R**

Dataset: Students’ Marks in Different Subjects

# Create a sample dataset
students <- data.frame(
  Name = c("Alice", "Bob", "Charlie", "David", "Ella"),
  Maths = c(85, 70, 90, 65, 95),
  Science = c(88, 75, 92, 60, 98),
  English = c(80, 68, 85, 70, 90),
  StudyHours = c(5, 3, 6, 2, 7)
)
 
students

Output:

Name	Maths	Science	English	StudyHours
Alice	85	88	80	5
Bob	70	75	68	3
Charlie	90	92	85	6
David	65	60	70	2
Ella	95	98	90	7

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1️⃣ Linear Regression (Simple)

Objective: Predict Science marks based on Maths marks.

# Linear regression model
linear_model <- lm(Science ~ Maths, data = students)
summary(linear_model)

Explanation:

• Science is the dependent variable, Maths is independent.

• R will estimate coefficients (\beta_0) and (\beta_1).

• Can be used to predict Science marks for a given Maths score.

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2️⃣ Multiple Linear Regression

Objective: Predict Science marks based on Maths and English marks.

# Multiple linear regression
multi_model <- lm(Science ~ Maths + English, data = students)
summary(multi_model)

Explanation:

• Combines multiple predictors.

• Useful to see combined effect of Maths and English on Science.

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3️⃣ Logistic Regression

Objective: Predict whether a student passed Science (pass ≥ 75) based on StudyHours.

# Create a binary variable: pass/fail
students$PassScience <- ifelse(students$Science >= 75, 1, 0)

students$PassScience <- ifelse(students$Science >= 75, 1, 0)
 
for (i in 1: length(students$Names}))
{
	if (students[i]$Science >= 75){
students$PassedScience = 1
}
else{
students[i]$PassedScience = 0
}
}
 

# Logistic regression
log_model <- glm(PassScience ~ StudyHours, data = students, family = binomial)
summary(log_model)

Explanation:

• PassScience is binary (1 = Pass, 0 = Fail).
• StudyHours is the predictor.
• Can estimate probability of passing based on hours studied.

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4️⃣ ANOVA (Analysis of Variance)

Objective: Test if the average Maths marks differ across groups based on StudyHours.

# Categorize students into groups
students$StudyGroup <- cut(students$StudyHours, breaks=c(0,3,5,7), labels=c("Low","Medium","High"))
 
# ANOVA model
anova_model <- aov(Maths ~ StudyGroup, data = students)
summary(anova_model)

Explanation:

• Compares mean Maths marks between Low, Medium, High study groups.

• Tests the hypothesis: “Do study groups differ in Maths performance?”

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5️⃣ Generalized Linear Model (GLM) - Poisson Example

Objective: Model the number of subjects scored above 85 (count data) based on StudyHours.

# Count subjects scored above 85
students$HighScores <- rowSums(students[,c("Maths","Science","English")] > 85)
 
# Poisson regression
glm_model <- glm(HighScores ~ StudyHours, data = students, family = poisson)
summary(glm_model)

Explanation:

• Response: Count of high scores per student.

• Predictor: StudyHours.

• GLM allows modeling non-normal response variables like counts.

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✅ Summary Table of Models

Model	Dependent	Independent	Type of Response	R Function
Linear Regression	Science	Maths	Continuous	lm()
Multiple Linear Regression	Science	Maths + English	Continuous	lm()
Logistic Regression	PassScience	StudyHours	Binary	glm(family=binomial)
ANOVA	Maths	StudyGroup	Continuous (grouped)	aov()
GLM (Poisson)	HighScores	StudyHours	Count	glm(family=poisson)