2. Matrix Multiplication

3. Rank and Dependency n Columns of a matrix: n Matrix A:

4. Linear Combinations n Columns: n Linear Combination: “Nontrivial” if at least one is not 0 ** Columns are dependent if there is a nontrivial linear combination for which:

5. Rank n The RANK of a matrix is the maximum number of linearly independent columns that can be selected from the columns of the matrix. n Rank of X is same as rank of X’X. n A matrix is invertible only if u It is square and u It is of FULL RANK

6. Example - Look for Dependency DEPENDENT? MAYBE, MAYBE NOT!

7. Same Example (con.t) DEPENDENT? YES !!!!

8. * * * * * * * * * * * * * * * * * X Y  r i 

9. Class Variables  Data Set X MATRIX

13. Treatment 1: Treatment 2: Treatment 3:

14. X YA 2 20 A 5 24 A 8 22 A 11 28 A 14 26 A 17 25 X YB 2 15 B 6 19 B 7 20 B 12 26 B 14 30 B 17 28 X YC 1 2 C 5 8 C 9 12 C 11 10 C 14 18 C 15 20

15. A 2 20 A 5 24 A 8 22 A 11 28 A 14 26 A 17 25 B 2 15 B 6 19 B 7 20 B 12 26 B 14 30 B 17 28 C 1 2 C 5 8 C 9 12 C 11 10 C 14 18 C 15 20 X= 2 5 8 11 14 17 2 6 7 12 14 17 1 5 9 11 14 15 <== One column One slope

16. A 2 20 A 5 24 A 8 22 A 11 28 A 14 26 A 17 25 B 2 15 B 6 19 B 7 20 B 12 26 B 14 30 B 17 28 C 1 2 C 5 8 C 9 12 C 11 10 C 14 18 C 15 20 X = 111111111111111111111111111111111111 1 0 0 1 0 2 5 8 11 14 17 2 6 7 12 14 17 1 5 9 11 14 15 <== Add TRT*X Interaction 2 0 5 0 8 0 11 0 14 0 17 0 0 2 0 6 0 7 0 12 0 14 0 17 0 0 000000000000111111000000000000111111 0 1 5 9 11 14 15

17. PROC GLM; CLASS TRT; MODEL Y = TRT X TRT*X; F test to delete TRT*X What is being tested? Key => What is REDUCED MODEL? Y = TRT X ==> Single slope Testing …… H0: Arbitrary lines H1: Parallel lines

18. PROC GLM; CLASS TRT; MODEL Y = TRT TRT*X; One dependency in TRT NO dependencies in TRT*X F for TRT*X What is it testing NOW? Reduced model = ? MODEL Y=TRT All slopes 0 H0: All slopes 0 (not just equal) H1: Arbitrary slopes

19. PROC GLM; CLASS TRT; MODEL Y = TRT TRT*X; One dependency in TRT

20. Parameter Estimate Estimate INTERCEPT 4.0123 B TRT A 12.2217 B B 10.9158 B C 0.0000 B X 0.8350 Covariance adjusted means NOTE: The X'X matrix has been found to be singular... <=

21. grade IQ study b 1 b 2 b 0

22. grade IQ study “Interaction”!

23. IQ=100 100 -13.00+5.30 IQ=110 110 -14.300+5.83