Simple Linear Regression

Simple Linear Regression: Involves one independent variable and one dependent variable, modeled as:

𝑦=𝛽0 +𝛽1𝑥+𝜖

where :

• y is the dependent variable,

• x is the independent variable,

• 𝛽0 is the intercept,

• β 1 is the slope, and

• ϵ is the error term.

Assumptions

• Linearity: The relationship between the dependent and independent variables should be linear.
• Independence: Observations should be independent of each other.
• Homoscedasticity: The variance of error terms should be constant across all levels of the - independent variable.
• Normality: The residuals (errors) should be normally distributed.

Basic Simple Linear Regression

# Load necessary data or generate sample data
x <- c(10, 20, 30, 40, 50)
y <- c(15, 30, 45, 50, 65)

Fit the linear regression model

model <- lm(y ~ x)

View the summary to understand model performance

summary(model)

## 
## Call:
## lm(formula = y ~ x)
## 
## Residuals:
##          1          2          3          4          5 
## -2.000e+00  1.000e+00  4.000e+00 -3.000e+00 -1.665e-15 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)    5.000      3.317   1.508  0.22878   
## x              1.200      0.100  12.000  0.00125 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.162 on 3 degrees of freedom
## Multiple R-squared:  0.9796,	Adjusted R-squared:  0.9728 
## F-statistic:   144 on 1 and 3 DF,  p-value: 0.001245

Plot and add regression line

plot(x, y, main = "Regression Example", xlab = "X values", ylab = "Y values", pch = 16)
abline(model, col = "red", lwd = 2)

Predict new values

new_data <- data.frame(x = c(25, 35))
predictions <- predict(model, newdata = new_data)
print(predictions)

##  1  2 
## 35 47