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

Axxq

Distributional properties of q depend on boththe properties of the known matrix A and the

distribution of the random vector x.

Multivariate Normal Distribution

e)(2

1f(y)

,N~y

yy2

1

2/1k/2

1

...

............

...

...

, ...

,

y

...

y

y

y

kkk2k1

2k2221

1k1211

n

2

1

n

2

1

y,ycov , y var, yE jiijjiiiiii

Properties of the Covariance Matrix

Nonsingular Symmetric Positive Definite

Positive (Semi-) Definite Matrices

positivestrictly areA of seigenvalue theall r,nonsingula isA :esConsequenc

x nonzero allfor 0Axxq iff definite positive isA

zeroor positive areA of seigenvalue theall singular, isA :esConsequenc

0 xoneleast at for holdingequality with

, x nonzero allfor 0Axxq iff definite-semi positive isA

Similar Definitions: Negative (Semi-) Definite, Indefinite

Distribution of Quadratic Forms in Normal Random Variables

rank(A) andA A )(~Ayyq

I0,N~y 22

A21

)rank(A and AA ),(~Ayyq

,N~y 22

Trace of a Square Matrix

Definition

Properties

Symmetric Idempotent Matrix

aA trn

1iii

BCAtrCABtrABC tr, BAtrAB tr Cyclic Permutations

A trn

1ii

i = eigenvalues of A

ArankA tr i = 1 or 0

BtrAtrBA tr

BtrAtrBA tr

Sample Variance

n1

yys 1n

q

1-2

2

n

1i

2i

2

2

JIA

Ayy

Probability Distribution

1)(n~q

01 JnI 1 2

1

1nJnItrJnIrank , JnIAA

I1,N~y...yyy

2

122

1112

2n21

Total Sum of Squares

)y-(y=TSS a

1=i

r

1=j

2ij

i

Quadratic Form

TSS = y A = I - n JT T-1 y A

Degrees of Freedom

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

ShowShowy = (y y . . . y . . . y y . . . y )11 12 1r a1 a2 ar1 a

Main Effect Sum of Squares

Quadratic Form

Degrees of Freedom

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

ShowShow

)y-y(rSS a

1=i

2iiA

SS = y A = diag(r J , . . . , r J ) - n JA A A 1-1

r a-1

r-1

1 a y A

r1-

a1-

ar1-

a1-

r1-

aAi JrJa IJrJa JrI=A :r qualE

Main Effect Sum of Squares

Probability Distribution

),(~q

JaI 2

r XAX 1 A 1

2

1

1aJrtrJaI trJrJaI tr

AAA

JrJaI 1

A , 1IX

I,X1N~y...yyy

AA2

a1

a2A*AA

*A

22A

r1

a1

ar1

a1

a

A*A

2*A

r1

a1

a2*AraA

2An21

ShowShow

Main Effect Noncentrality Parameter

... 0

2

r

2

r

JaI 2

r

a21

a

1i

2i2

a

1i

2i2

a1

a2A

1 0 JaI

... 0 JaI

0 0 JaI

1arank of tesemidefini positive is JaI

a1

a

a21a1

a

a1

a

a1

a

Error Sum of Squares

Quadratic Form

Degrees of Freedom

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

r1

rar1-

r1-

EEE JrII)Jr , ... ,Jdiag(r - I=A yAy=SS

)y-(ySS a

1=i

r

1j

2iijE

Error Sum of Squares

Probability Distribution

)(~q

0 XAX 1 A 1 2

1

1ra JrI tr I tr JrI I tr

AAA

JrI I 1

A , 1IX

I,X1N~y...yyy

E2

A*EA

*E

22A

r1

rar1

ra

E*E

2*E

r1

ra2*EraA

2An21

ShowShow

Pairwise Independence of Quadratic forms

BAtindependenlly statistica pairwise are q nda q

By yq ,Ay yq

,N~y

BA

BA

JaIAA

JrII A , JrJaI A

I,X1N~y...yyy

a1

a2

EA

r1

raEr1

a1

aA

2An21

Independence of the Main Effect and Error Sums of Squares

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

Jr Ja I =A yAy=SS r1-

a1-

aAAA

r1-

raEEE JrI I=A yAy=SS

Single-Factor Experiment

Response Distribution

y ~ N(1 + XA , 2I) raA 1IX

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

Distributional Properties

SSA & SSE are statistically independent

0 )1r(a ),(~SS

)JaI(2

r 1a ),(~

SS

EEEE2

2E

a1

a2AAAA2

2A

tesemidefini positive is )JaI( :Note a1

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

MS

MSA

E~ ( , )F A E iff 1 = ... = a = 0

a

1i

2i2

a

1i

2.i2A

r2

1

)(r2

1 )(2

r

:Model Means Cella

1i

2.i2A

0 if i

Testing Factor Effects

Simultaneous Test for Main Effects

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

Single-Factor Model

H0: 1 = 2 = ... = a vs. Ha: i j for some (i,j)

0 if i 0H iio

i i

i i i i

a21o ... H

EquivalentEquivalent

Test Statistic

2i2AA

AA2

2

2i

2

2i

2A

2

r 1a

),(~)y-y(rˆrSS

)1r(aan

)(~SS-TSSrSS

E

E2

2A

2

2i

2E

)}1r(a,1a{F~MSMS

HE

Ao

Assignment

Verify the ‘Show’ Results