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

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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

the properties of the known matrix A and the

distribution of the random vector x.

Positive (Semi-) Definite Matrices

Properties of the Covariance Matrix- Nonsingular
- Symmetric
- Positive Definite

Similar Definitions: Negative (Semi-) Definite, Indefinite

Trace of a Square Matrix

Definition

Properties

Cyclic Permutations

li = eigenvalues of A

Symmetric Idempotent Matrix

li = 1 or 0

Sample Variance

Independence of the Main Effect and Error Sums of Squares

Pairwise Independence of Quadratic formsStatistical 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

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 :

- Verify the ‘Show’ Results

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