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Statistical Analysis. Professor Lynne Stokes Department of Statistical Science Lecture 6QF Multivariate Normal Distribution, Chi-square Distribution of Quadratic Forms, Testing the Significance of Factor Effectrs. Quadratic Forms. Distributional properties of q depend on both

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Statistical analysis
Statistical Analysis

Professor Lynne Stokes

Department of Statistical Science

Lecture 6QF

Multivariate Normal Distribution,

Chi-square Distribution of Quadratic Forms,

Testing the Significance of Factor Effectrs


Quadratic forms
Quadratic Forms

Distributional properties of q depend on both

the properties of the known matrix A and the

distribution of the random vector x.



Properties of the covariance matrix

Positive (Semi-) Definite Matrices

Properties of the Covariance Matrix

  • Nonsingular

  • Symmetric

  • Positive Definite

Similar Definitions: Negative (Semi-) Definite, Indefinite



Trace of a square matrix
Trace of a Square Matrix

Definition

Properties

Cyclic Permutations

li = eigenvalues of A

Symmetric Idempotent Matrix

li = 1 or 0


Sample variance

Probability Distribution

Sample Variance


Total sum of squares
Total Sum of Squares

Quadratic Form

Show

Degrees of Freedom

n -1 = ar - 1 = rank(AT) = tr(AT)


Main effect sum of squares
Main Effect Sum of Squares

Quadratic Form

Show

Degrees of Freedom

a -1 = rank(AA) = tr(AA)


Main effect sum of squares1
Main Effect Sum of Squares

Probability Distribution

Show



Error sum of squares
Error Sum of Squares

Quadratic Form

Degrees of Freedom

Show

n - a = rank(AE) = tr(AE)


Error sum of squares1
Error Sum of Squares

Probability Distribution

Show


Pairwise independence of quadratic forms

Independence of the Main Effect and Error Sums of Squares

Pairwise Independence of Quadratic forms


Statistical tests for fixed main effects and interactions balanced complete factorials
Statistical Tests for (Fixed) Main Effects and Interactions :Balanced Complete Factorials

Single-Factor Experiment

Response Distribution

y ~ N(m1 + XAa , s2I)


Statistical tests for fixed main effects and interactions balanced complete factorials1
Statistical Tests for (Fixed) Main Effects and Interactions :Balanced Complete Factorials

Distributional Properties

SSA & SSE are statistically independent


Statistical tests for fixed main effects and interactions balanced complete factorials2
Statistical Tests for (Fixed) Main Effects and Interactions :Balanced Complete Factorials

iff a1 = ... = aa = 0


Testing factor effects
Testing Factor Effects :

Single-Factor Model

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

Equivalent

Simultaneous Test for Main Effects

H0: a1 = a2 = ... = aa vs. Ha: aiaj for some (i,j)



Assignment
Assignment :

  • Verify the ‘Show’ Results


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