Multiple linear regression models the relationship between one response variable and two or more explanatory variables.
For this example, the goal is to predict a car’s fuel economy using:
The mtcars dataset contains measurements for 32 automobiles.
A multiple linear regression model can be written as
\[ Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \epsilon \]
For this analysis:
\[ MPG = \beta_0 + \beta_1(Weight) + \beta_2(Horsepower) + \epsilon \]
where \(\epsilon\) represents variation that is not explained by the model.
Heavier vehicles generally have lower fuel economy.
Cars with greater horsepower also tend to have lower MPG.
The model can be created in R using the lm() function.
model <- lm(mpg ~ wt + hp, data = mtcars) coef(model)
## (Intercept) wt hp ## 37.22727012 -3.87783074 -0.03177295
The estimated coefficients show how MPG changes with weight and horsepower while holding the other predictor constant.
Using the fitted model, the estimated regression equation is approximately
\[ \widehat{MPG} = 37.23 + (-3.88)(Weight) + (-0.03)(Horsepower) \]
The fitted coefficients are approximately:
Because both predictor coefficients are negative, increases in weight or horsepower are associated with decreases in predicted MPG.
The interactive graph shows the relationship between vehicle weight and MPG, while marker color represents horsepower.
## R_squared Adjusted_R_squared ## 1 0.8267855 0.8148396
The coefficient of determination is
\[ R^2 = 0.827 \]
This means that approximately 82.7% of the variation in MPG is explained by vehicle weight and horsepower together.
Points close to the dashed line represent vehicles for which the model’s prediction is close to the actual MPG.
Multiple linear regression lets us examine several predictors at the same time.
In the mtcars data:
Overall, heavier and more powerful vehicles tend to have lower fuel economy.