discussion12

First, we will load the mtcars dataset and create the model. We will be using these fields:

• Quadratic term: I(hp^2) represents the quadratic term for horsepower (hp). It captures the non-linear relationship between horsepower and miles per gallon (mpg).
• Dichotomous term: am represents whether the car has an automatic transmission. It’s dichotomous because it takes on two values: 0 for manual transmission and 1 for automatic transmission.
• Interaction term: hp:am represents the interaction between horsepower (hp) and transmission type (am). It captures how the relationship between horsepower and miles per gallon differs depending on whether the car has an automatic or manual transmission.

We will be using these to predict miles per gallon (mpg).

data(mtcars)

# fit the regression model
model <- lm(mpg ~ hp + I(hp^2) + wt + am + hp:am, data = mtcars)

# print the summary
summary(model)

## 
## Call:
## lm(formula = mpg ~ hp + I(hp^2) + wt + am + hp:am, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.9689 -1.7441 -0.4744  1.2910  4.1313 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.854e+01  3.159e+00  12.200  2.9e-12 ***
## hp          -1.142e-01  3.387e-02  -3.371  0.00235 ** 
## I(hp^2)      2.567e-04  9.441e-05   2.719  0.01152 *  
## wt          -2.757e+00  8.527e-01  -3.234  0.00331 ** 
## am           4.349e+00  2.140e+00   2.032  0.05245 .  
## hp:am       -2.498e-02  1.411e-02  -1.770  0.08841 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.299 on 26 degrees of freedom
## Multiple R-squared:  0.878,  Adjusted R-squared:  0.8545 
## F-statistic: 37.43 on 5 and 26 DF,  p-value: 4.476e-11

Interpreting the fields, we can see that:

• Intercept: Represents the estimated mpg when all other predictors are zero. It’s very statistically significant and is positive so this means all vehicles typically start with a certain mpg.
• hp: Represents the estimated change in mpg for a one-unit increase in horsepower, holding all other predictors constant. As this is negative, this means that the higher it is, the lower mpg and vice-versa. We can see that this is statistically significant so it is a relevant predictor.
• wt: Represents the estimated change in mpg for a one-unit increase in weight, holding all other predictors constant. As this is negative, this means that the higher it is, the lower mpg and vice-versa. We can see that this is just as statistically significant as hp so this is also relevant. This also makes sense as the heavier the car, typically this means it’s slower.
• am: Represents the estimated difference in mpg between cars with automatic and manual transmissions, holding all other predictors constant. As this is positive, this means that the higher it is, the higher mpg and vice-versa. this is also statistically significant, but less so than the previous two fields.
• hp:am: Represents how the relationship between horsepower and mpg changes depending on the type of transmission. As it is negative, this means that the higher it is, the lower mpg and vice-versa. We can also see it is statistically significant.
• I(hp^2): Represents the estimated change in mpg for a one-unit increase in the square of horsepower, holding all other predictors constant. This term accounts for the nonlinear relationship between horsepower and mpg, allowing the effect of horsepower on mpg to be nonlinear rather than strictly linear. As it is positive, this means that the higher it is, the higher the mpg and vice versa; this is also statistically signifcant so it’s relevant to the prediction.

# histogram
hist(resid(model))

# residuals vs. each predictor variable
par(mfrow = c(2, 2))
plot(model)

Residual analysis is done to assess whether the assumptions of linear regression are met:

• Linearity: This is checked by plotting residuals vs. fitted values. We can see that it doesn’t look super linear and there aren’t many clear trends here, so the linearity assumption holds.
• Homoscedasticity: This is checked by plotting residuals vs. fitted values. The spread of residuals remains roughly constant across all levels of the fitted values so homoscedasticity holds.
• Normality of residuals: This is checked by plotting a histogram of residuals and a QQ plot. The histogram is not roughly bell-shaped, but the points in the QQ plot lie close to the diagonal line. Since the Q-Q residuals seem to be mainly close to the diagonal line, we can say this assumption holds but with a caveat especially given the histogram isn’t perfectly bell-shaped.
• Independence of residuals: This is usually assumed if the data were collected randomly. Since the residuals do not show a clear pattern over time or across observations, this assumption holds.
• Absence of influential points: There don’t seem to be any influential points that could heavily affect the model’s parameters.

This model passes our residual analysis so the model was appropriate, with a slight caveat for the normality of residuals.