11.60.* Software reports four types of sums of squares in multiple regression models. The Type I sum of squares, sometimes called sequential SS, represents the variability explained by a variable, controlling for variables previously entered into the model. The Type III sum of squares, sometimes called partial SS, represents the variability explained by that variable, controlling for all other variables in the model.

(a) For any multiple regression model, explain why the Type I sum of squares for x1 is the regression sum of squares for the bivariate model with x1 as the explanatory variable, whereas the Type I sum of squares for x2 equals the amount by which SSE decreases when x2 is added to the model.

(b) Explain why the Type I sum of squares for the last variable entered into a model is the same as the Type III sum of squares for that variable.