Regression Analysis

# Regression Analysis

Regression analysis is one of the most widely used statistical techniques for understanding relationships between variables and making predictions.

## Learning Objectives

In this lecture, you will learn:
- The fundamentals of linear regression
- How to interpret regression coefficients
- Multiple regression techniques
- Model assumptions and diagnostics
- Practical applications in R

## Simple Linear Regression

Linear regression models the relationship between a dependent variable (Y) and an independent variable (X):

**Y = β₀ + β₁X + ε**

Where:
- Y is the dependent variable
- X is the independent variable
- β₀ is the intercept
- β₁ is the slope coefficient
- ε is the error term

## Multiple Regression

When we have multiple predictors:

**Y = β₀ + β₁X₁ + β₂X₂ + ... + βₖXₖ + ε**

## Key Assumptions

1. **Linearity**: Relationship between variables is linear
2. **Independence**: Observations are independent
3. **Homoscedasticity**: Constant variance of errors
4. **Normality**: Errors are normally distributed

## Practical Example in R

```r
# Load data
data(mtcars)

# Simple linear regression
model1 <- lm(mpg ~ wt, data = mtcars)
summary(model1)

# Multiple regression
model2 <- lm(mpg ~ wt + hp + cyl, data = mtcars)
summary(model2)

# Model diagnostics
plot(model2)
```

## Applications

Regression analysis is used for:
- Predicting house prices based on features
- Analyzing factors affecting sales performance
- Medical research and treatment effectiveness
- Economic forecasting and policy analysis