Linear vs. average

The $R^2$ gives us a numerical measurement of the strength of fit relative to a null model based on the average of the response variable: $$ \hat{y}_{null} = \bar{y} $$

This model has an $R^2$ of zero because $SSE = SST$. That is, since the fitted values ($\hat{y}_{null}$) are all equal to the average ($\bar{y}$), the residual for each observation is the distance between that observation and the mean of the response. Since we can always fit the null model, it serves as a baseline against which all other models will be compared.

In the graphic, we visualize the residuals for the null model (mod_null at left) vs. the simple linear regression model (mod_hgt at right) with height as a single explanatory variable. Try to convince yourself that, if you squared the lengths of the grey arrows on the left and summed them up, you would get a larger value than if you performed the same operation on the grey arrows on the right.

It may be useful to preview these augment()-ed data frames with glimpse():

glimpse(mod_null)
glimpse(mod_hgt)

Compute the sum of the squared residuals (SSE) for the null model mod_null.
Compute the sum of the squared residuals (SSE) for the regression model mod_hgt.

Exercise

Linear vs. average

Instructions