psychology 202b advanced psychological statistics, ii
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Psychology 202bAdvanced Psychological
Statistics, II
January 18, 2011
Overview
• What will we do this semester?• Accessing the syllabus.• Review of the 202a final exam.• Introduction to matrices.
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.
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.
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.
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?
Classifications of matrices
• Various special types of matrices exist. For example, we will consider:– Symmetric matrices– Upper and lower triangular matrices– Diagonal matrices
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:
397924741
Symmetric matrices (cont.)
• Examples of symmetric matrices that we frequently encounter in statistics include– Correlation matrices– Covariance matrices
Triangular matrices
• A triangular matrix is one that consists solely of zeroes on one side of the diagonal:
11663091100320004
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.
Diagonal matrices
• A diagonal matrix is one in which all values that are not on the diagonal are zero:
300020001
Matrix multiplication
• How does matrix multiplication work?– Examples on board.– In-class exercise.
• Note that matrix multiplication is not commutative.
Matrix multiplication vocabulary
• The matrix on the left in matrix multiplication is called the “premultiplier.”
• The matrix on the right is called the “postmultiplier.”
Matrix transposition
• The transpose of a matrix is a matrix where the rows and columns are reversed.
• Example:
101102313121221
122031112132011
Matrix transposition (cont.)
• Sometimes when a multiplication problem does not conform, it will when transposed.
• Example on the board.
Next time
• Matrix division: the inverse matrix.• Manipulating matrices in R.• The relevance of matrices to regression.
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