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# Statistical Analysis - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about ' Statistical Analysis' - preston-house

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Presentation Transcript

Professor Lynne Stokes

Department of Statistical Science

Lecture 6QF

Multivariate Normal Distribution,

Testing the Significance of Factor Effectrs

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

• Nonsingular

• Symmetric

• Positive Definite

Similar Definitions: Negative (Semi-) Definite, Indefinite

Definition

Properties

Cyclic Permutations

li = eigenvalues of A

Symmetric Idempotent Matrix

li = 1 or 0

Sample Variance

Show

Degrees of Freedom

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

Show

Degrees of Freedom

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

Probability Distribution

Show

Degrees of Freedom

Show

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

Probability Distribution

Show

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 Factorials

Distributional Properties

SSA & SSE are statistically independent

Statistical Tests for (Fixed) Main Effects and Interactions :Balanced Complete Factorials

iff a1 = ... = aa = 0

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)

• Verify the ‘Show’ Results