3 Interpreting the Model Parameters

\(\beta_0\) (intercept) and \(\beta_1\) (slope), estimated by \(b_0\) and \(b_1\), interpreted as…

The interpretation of \(\beta_0\) is only meaningful if \(X=0\) is in the scope of the model. If \(X=0\) is in the scope of the model, then the intercept is interpreted as the average y-value, denoted \(E\{Y\}\), when \(X=0\).

The interpretation of \(\beta_1\) is the amount of increase (or decrease) in the average y-value, denoted \(E\{Y\}\), per unit change in \(X\). It is often misunderstood to be the “average change in y” or just “the change in y” but it is more correctly referred to as the “change in the average y”.

To better see this, consider the three graphics shown below.

par(mfrow=c(1,3))
hist(mtcars$mpg, main="Gas Mileage of mtcars Vehicles", ylab="Number of Vehicles", xlab="Gas Mileage (mpg)", col="skyblue")
boxplot(mpg ~ cyl, data=mtcars, border="skyblue", boxwex=0.5, main="Gas Mileage of mtcars Vehicles", ylab="Gas Mileage (mpg)", xlab="Number of Cylinders of Engine (cyl)")
plot(mpg ~ qsec, data=subset(mtcars, am==0), pch=16, col="skyblue", main="Gas Mileage of mtcars Vehicles", ylab="Gas Mileage (mpg)", xlab="Quarter Mile Time (qsec)")
abline(lm(mpg ~ qsec, data=subset(mtcars, am==0)), col="darkgray")
mtext(side=3, text="Automatic Transmissions Only (am==0)", cex=0.5)
abline(v = seq(16,22,2), h=seq(10,30,5), lty=3, col="gray")

The Histogram	The Boxplot	The Scatterplot
The histogram on the left shows gas mileages of vehicles from the mtcars data set. The average gas mileage is 20.09.	The boxplot in the middle shows that if we look at gas mileage for 4, 6, and 8 cylinder vehicles separately, we find the means to be 26.66, 19.74, and 15.1, respectively. If we wanted to, we could talk about the change in the means across cylinders, and would see that the mean is decreasing, first by \(26.66 - 19.74 = 6.92\) mpg, then by \(19.74 - 15.1 = 4.64\) mpg.	The scatterplot on the right shows that the average gas mileage (for just automatic transmission vehicles) increases by a slope of 1.44 for each 1 second increase in quarter mile time. In other words, the line gives the average y-value for any x-value. Thus, the slope of the line is the change in the average y-value.