1. 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

2. Quadratic Forms Distributional properties of q depend on both
the properties of the known matrix A and the
distribution of the random vector x.

3. Multivariate Normal Distribution

4. Properties of the Covariance Matrix
Nonsingular
Symmetric
Positive Definite
Positive (Semi-) Definite Matrices
Similar Definitions: Negative (Semi-) Definite, Indefinite

5. Distribution of Quadratic Forms in Normal Random Variables

6. Trace of a Square Matrix
Definition
Properties
Cyclic Permutations
li = eigenvalues of A
Symmetric Idempotent Matrix
li = 1 or 0

7. Sample Variance
Probability Distribution

8. Total Sum of Squares Quadratic Form Show Degrees of Freedom
n -1 = ar - 1 = rank(AT) = tr(AT)

9. Main Effect Sum of Squares
Quadratic Form
Show
Degrees of Freedom
a -1 = rank(AA) = tr(AA)

10. Main Effect Sum of Squares
Probability Distribution
Show

11. Main Effect Noncentrality Parameter

12. Error Sum of Squares Quadratic Form Degrees of Freedom Show
n - a = rank(AE) = tr(AE)

13. Error Sum of Squares
Probability Distribution
Show

14. Pairwise Independence of Quadratic forms
Independence of the Main Effect and Error Sums of Squares

15. Statistical Tests for (Fixed) Main Effects and Interactions : Balanced Complete Factorials
Single-Factor Experiment
Response Distribution
y ~ N(m1 + XAa , s2I)

16. Statistical Tests for (Fixed) Main Effects and Interactions : Balanced Complete Factorials
Distributional Properties
SSA & SSE are statistically independent

17. Statistical Tests for (Fixed) Main Effects and Interactions : Balanced Complete Factorials
iff a1 = ... = aa = 0

18. 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: ai aj for some (i,j)

19. Test Statistic

20. Assignment
Verify the ‘Show’ Results