Psychology 202b Advanced Psychological Statistics, II

1. Psychology 202bAdvanced Psychological Statistics, II January 18, 2011

2. Overview • What will we do this semester? • Accessing the syllabus. • Review of the 202a final exam. • Introduction to matrices.

3. Matrices • What is a matrix? • First, what is a vector? • A variable for which the meaning is carried by a set of values. • Any type of variable can be vector valued. • A matrix may be thought of as a vector of vectors.

4. Notation • It is conventional to indicate that a variable is a vector or matrix by using bold face type. • Vectors are lower case: a, b. • Matrices are upper case: A, B. • In contrast to scalars, vector and matrix variables are not italicized.

5. Examples of vectors • The set of 40 Peabody scores that we analyzed last semester could be thought of as a vector of Peabody scores. • The Peabody and Raven score for the first subject in our data set could be thought of as a vector of scores associated with that person.

6. Example of a matrix • If we combine those two, the set of 40 pairs of Raven and Peabody scores is a 40-by-2 matrix. • What are the dimensions of this matrix?

7. Classifications of matrices • Various special types of matrices exist. For example, we will consider: • Symmetric matrices • Upper and lower triangular matrices • Diagonal matrices

8. Symmetric matrices • A matrix is symmetric if the elements on both sides of the diagonal that runs from the upper left corner to the lower right corner are reflections of each other:

9. Symmetric matrices (cont.) • Examples of symmetric matrices that we frequently encounter in statistics include • Correlation matrices • Covariance matrices

10. Triangular matrices • A triangular matrix is one that consists solely of zeroes on one side of the diagonal:

11. Triangular matrices (cont.) • That was a lower triangular matrix: The non-zero values were on and below the diagonal. • An upper triangular matrix would have zeroes below the diagonal, and the non-zero values would all be on or above the diagonal.

12. Diagonal matrices • A diagonal matrix is one in which all values that are not on the diagonal are zero:

13. Matrix multiplication • How does matrix multiplication work? • Examples on board. • In-class exercise. • Note that matrix multiplication is not commutative.

14. Matrix multiplication vocabulary • The matrix on the left in matrix multiplication is called the “premultiplier.” • The matrix on the right is called the “postmultiplier.”

15. Matrix transposition • The transpose of a matrix is a matrix where the rows and columns are reversed. • Example:

16. Matrix transposition (cont.) • Sometimes when a multiplication problem does not conform, it will when transposed. • Example on the board.

17. Next time • Matrix division: the inverse matrix. • Manipulating matrices in R. • The relevance of matrices to regression.