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Statistical Analysis

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Professor Lynne Stokes

Department of Statistical Science

Lecture 6QF

Matrix Solutions to Normal Equations

(Left) Direct Product

Some Properties

assuming all operations are valid

Single-Factor, Balanced Experiment

yij = m + ai + eij i = 1, ..., a; j = 1, ..., r

Matrix Formulation

y = Xb + e

y = (y11 y12 ... y1r ... ya1 ya2 ... yar)’

Show

- Symmetric
- Same rank as X
- Has an inverse (X’X)-1 iff X has full column rank

Residuals

Least Squares

Solution: Solve the Normal Equations

Normal Equations

Problem: X’X is Singular, has no inverse

Show

A

Definition

AGA = A

not unique

Moore-Penrose Generalized Inverse

- AGA = A
- GAG = G
- AG is symmetric
- GA is symmetric

unique

Some Properties

if A has full row rank, G = A’(AA’)-1

if A has full column rank, G = (A’A)–1A’

Common Notation

A

Theorem: For any , X’X X’ = X’

(X’X)

(X’X)

Normal Equations

Solutions

- Every solution to the normal equations corresponds to a generalized inverse of X’X
- Every generalized inverse of X’X solves the normal equations

One Generalized Inverse

Verification

Corresponds to the solution to the normal equations

with the constraint m = 0 imposed

- Find another generalized inverse for X’X in a one-factor balanced experiment
- Verify that it is a generalized inverse
- Solve the normal equations using the generalized inverse
- Determine what constraint on the model parameters correspond to the generalized inverse