Simple linear regression is a statistical method used to model the relationship between a dependent variable \(y\) and one independent variable \(x\). We use this method to predict the outcome of \(y\) based on the value of \(x\).

The linear regression model assumes a relationship of the form:

\[ y = \beta_0 + \beta_1 x + \epsilon \]

Where: - \(\beta_0\): Intercept
- \(\beta_1\): Slope
- \(\epsilon\): Error term

This equation forms the basis of linear regression.

We will use the built-in mtcars dataset to explore the relationship between car weight (wt) and miles per gallon (mpg).

head(mtcars[, c("wt", "mpg")])

##                      wt  mpg
## Mazda RX4         2.620 21.0
## Mazda RX4 Wag     2.875 21.0
## Datsun 710        2.320 22.8
## Hornet 4 Drive    3.215 21.4
## Hornet Sportabout 3.440 18.7
## Valiant           3.460 18.1

This provides a quick look at the data we’ll model.

This plot shows a negative linear relationship between weight and mpg.

The lm() function in R is used to fit a linear regression model. Here is a summary of the results:

summary(model)

## 
## Call:
## lm(formula = mpg ~ wt, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5432 -2.3647 -0.1252  1.4096  6.8727 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
## wt           -5.3445     0.5591  -9.559 1.29e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.046 on 30 degrees of freedom
## Multiple R-squared:  0.7528, Adjusted R-squared:  0.7446 
## F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

This plot helps assess the fit and check for violations of regression assumptions:

This 3D plot gives another perspective on how residuals vary across predictor and response variables:

From the fitted model:

\[ \hat{mpg} = \hat{\beta}_0 + \hat{\beta}_1 wt \]

• The intercept \(\hat{\beta}_0\) represents the expected mpg when weight is 0.
  
• The slope \(\hat{\beta}_1\) shows the change in mpg for each unit increase in weight.
  
• A negative slope confirms an inverse relationship.

• A strong linear trend exists between weight and mpg.
  
• The model explains a large proportion of the variance in mpg.
  
• Diagnostic plots support that linear regression is appropriate.
  
• We can now predict mpg from weight with reasonable confidence.

Simple linear regression is a powerful tool for prediction and understanding relationships between variables.