1. Statistics 350 Lecture 19

2. Today Last Day: R 2 and Start Chapter 7 Today: Partial sums of squares and related tests

3. Example Consider Example on page 257 Y = Percent Body Fat X 1 = Triceps Skinfold Thickness X 2 = Thigh Circumference X 3 = Midarm Circumference

4. Example Suppose only consider first two explanatory variables Test hypothesis: H o :  1 =0 (assuming X 2 already in model) H A :  1 ≠0 (assuming X 2 already in model) Important lesson is that the presence or absence of X 2 in the model may change depending upon which variables are in the model

5. Extra Sum of Squares Some Notation: SSR(X 1 )= SSR(X 1,X 2 ) = SSR(X 1 |X 2 ) =

6. Extra Sum of Squares Some Notation: SSR(X 1, X 2, X 3 )= SSR(X 3 | X 1, X 2 )= SSR(X 2, X 3 | X 1 )= And so on…

7. Extra Sum of Squares Decomposition:

8. Back to Example Look at the sums of squares for this problem

9. Back to Example Why doesn’t the regression sum of squares for the first two models not sum to that for the third? SSR(X 1,X 2 ) -SSR(X 2 ) =3.47…what is this difference and what does it mean?

10. Back to Example What is SSR(X 2,X 1 ) = In words: SSR(X 2,X 3 | X 1 ) = In words:

11. Back to Example Who cares? For example, Thigh Circumference (X 2 ) and Midarm Circumference (X 3 ) are easily measured with precision using an ordinary tape measure, but Triceps Skinfold Thickness (X 1 ) requires a trained technician and an expensive instrument. Then a natural question is whether we even need to bother with X 1 once we've already measured X 2 and X 3 ?

12. Back to Example Who cares? For example, Thigh Circumference (X 2 ) and Midarm Circumference (X 3 ) are easily measured with precision using an ordinary tape measure, but Triceps Skinfold Thickness (X 1 ) requires a trained technician and an expensive instrument. Then a natural question is whether we even need to bother with X 1 once we've already measured X 2 and X 3 ?