28
Stat 200b. Chapter 8. Linear regression models. T j j j T j j jp p j j j j j x y E x x x y p n j E s x x y x } { ... ies explanator ,..., 1 , 0 } { fixed, ' 8.1 Example , 1 ies explanator two - level sea Venice 1 0 1 0

Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

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Page 1: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Stat 200b. Chapter 8. Linear regression models.

T

jj

j

T

j

jjppjj

jjj

xyE

x

xxy

p

njEsx

xy

x

}{

...

iesexplanator

,...,1 ,0}{ fixed, '

8.1 Example

,1 iesexplanator two- level sea Venice

10

10

Page 2: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

nnn x

x

x

y

y

y

.

.

.

1

.

.

.

1

1

.

.

.

8.1 Example

model regressionlinear

matrixdesign :

rowsstack :formmatrix

2

1

1

2

1

2

1

0 Xy

X

Xy

n by 1, n by 2, 2 by 1, n by 1

Page 3: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

s'unknown in linear Note.

1by p pby n

.

.

.

... 1

.

.

.

... 1

...y

regression polynomial single 8.2 Example

1

0

1

1

11

1

1

2

210j

pp

nn

p

j

p

jpjj

xx

xx

xxx

x

X

Page 4: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

12 8 68 10 1

.

.

.

52 15 29 1 1

60 6 26 7 1

...

Model

CaOAl 3

clinkers weight % :,...,

gram)(calories/ evolvedheat :

dataCement 8.3 Example

44110

32

41

X

jjjj xxy

O

xx

y

13 by 5

Page 5: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1
Page 6: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

- psi 60 40, pressure tyre

-on off, dynamo

- 30" ,26" height seat

levels 2 with factors Three

hill up ride to time:

data cycling 8.4 Example

y

Page 7: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1
Page 8: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

1by 4 4by 16

1 1 1 1

1 1 1 1

1 1 1- 1

1 1- 1 1

1 1- 1 1

1 1- 1- 1

1 1- 1- 1

1- 1 1 1

1- 1 1 1

1- 1 1- 1

1- 1 1- 1

1- 1- 1 1

1- 1- 1 1

1- 1- 1- 1

1- 1- 1- 1

3

2

1

0

X

Effect of increasing seat height is 21

Page 9: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Some matrix review

transpose

multiplication

inverse

derivatives

Page 10: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Normal linear model. Estimation

exists inverse if )(ˆ

)ˆ(

equations Normal

p1,...,r ,0ˆ)(2/)(

)()()()(

2/})(1

log{),(

),(

),(~

11

2

1

2

12

22

2

2

yXXX

0XyX

XyXy

T

T

T

j

T

jjrr

TT

j

n

jj

T

j

n

jj

T

jj

xyxSS

xySS

xynl

L

xINy

Page 11: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Profile log likelihood

)/()ˆ( S

/)ˆ()ˆ(ˆ

),(max)(

22

2

22

pnxy

n

ll

T

jj

p

XyXy T

Page 12: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Straight-line/simple regression.

2

1

0

1

1

0

11

)( )(

)(

ˆ

ˆˆ

.

.

.

1

.

.

.

1

.

.

.

xxxx

xxn

xx

xx

y

y

jj

j

nnn

XXT

Page 13: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

)2/()ˆ()ˆ( :estimate variance

ˆ:residuals

)(

)(/1 0

0 /1)(

2

n

yxx

y

xx

n

jj

j

j

XyXy

Xye

yX

XX

T

T

1T

Page 14: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Fitted values.

ˆ

)(I

ˆ

:Residuals

)ˆ()ˆ()ˆ(

matrixhat ,

)(ˆ

yH

Hyy

Xye

yyyy

Hy

yXXXXXy

T

T1T

n

SS

NB. Assuming matrix inverse exists

Page 15: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Weighted least squares.

WyXWXX

W

XyWXy

T1T

2T

)(ˆ

)()(

][

2/}/)()(log{),(

known /var

2

22

2

j

T

jj

j

jjj

wxySS

wdiag

nl

wwy

inverse existing

Page 16: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Example 8.8. Cycling data

Page 17: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

up pressure ,563.1

on dynamo ,563.1

upseat ,437.5

average overall ,19.47

)(ˆ

/16)(

16

1)by 1,16by 4,4by 1,16by (16

(8.2) Model

4

1

4

yXXX

IXX

IXX

Xy

T1T

T

T

Page 18: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

55psi pressure dynamo, no seat,high :mefastest ti

ec2(-1.563)s change fitted :40not 55psi, pressure

c2(1.563)se change fitted :offnot on, dynamo

ec2(-5.437)s change fitted 26":not ,30"seat

563.1563.1437.519.47ˆ

1,,

321

321

xxxy

xxx

Page 19: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Example 8.10. Maize data.

22

0

1

0

12

012

202

101

)( 2/ˆ )2,...,2,2(

1by 1m and 1mby m

.

.

1...100

1...00 1

.

.

0...01 1

0...101

0...10 1

0...0 1 1

2}{

15,...,1

ddSdmdiag

yyd

yyE

mj

y

y

j

m

jjj

jj

jjj

jjj

XX

X

T

Page 20: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Likelihood quantities.

})(21

{)(

)(1

1

2

6422

2

42

2

2

2

T

jj

T

jjjr

r

jsjr

sr

xyl

xyxl

xxl

Page 21: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Take expected values

)/2

)(;( asymp

ˆ

ˆ

/2

)(),(

2/

),(

4

2

222

4

12

12

4

2

2

nN

nI

nI

0

0XX

0

0XX

0

0XX

1T

T

T

Page 22: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1
Page 23: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Normal distribution theory. Full rank case

1T2

1T

T1T

2

XXVarI

XX

yXXX

)(ˆ, if

ˆ,0 If

case normalNon

pby 1 and 1by p )ˆ)(ˆ(ˆVar

~)ˆ(

tlyindependen

))(,(~

)(ˆ

),cov( ,0

),0(~

2

22

2

2

T

T

pn

n

j

E

EE

E

SS

N

IE

IN

Page 24: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

A useful decomposition.

22 )(

valuesexpected Taking

)ˆ()ˆ(

)ˆˆ()ˆˆ(

)()(

pEn

ee

XXee

XXXyXXXy

XyXy

T

TTT

T

TT

Page 25: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Confidence interval.

2/1

}

12

11

2

)(1{

ˆ

for interval Prediction

)(

pivotal ~)ˆ(

xXXx

x

XXT

TT

T

rs

pn

rr

rr

S

y

y

tS

Page 26: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Gauss-Markov Theorem. page 374

HHHIBBB

AHIA

AXXXAXAA

XXIA

IAX

AX

XXX

XXyvarXyE

2

n

T

T

TT1TT

T

n

p

TT

T2

noting , where0

)(

})({

)(ˆvar~

var

so

allfor ~

E Proof.

ˆvar~

var meansBest

~E means Unbiased

A somefor ~

meansLinear

(BLUE)

of estimate unbiasedlinear best theis ˆthen

ˆ

rnonsingula ,)(,)( Suppose

2

2

2

12

n

n

Ay

y

I

Page 27: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

There is a generalized inverse variant

Example.

ˆvar~

var,ˆ

ˆ is BLUE

1by p and pby 1 ,Consider

PPPEP

P

P

Page 28: Stat 200b. Chapter 8. Linear regression models.. n by 1, n by 2, 2 by 1, n by 1

Eg. Teaching methods data, p. 427

Method average

Usual 17 14 24 20 24 … 24 19.67

Praised 28 30 29 24 27 … 23 27.41

2121

2121

- BLUE, - in interested beMight

] [ BLUE], [in interested beMight

1,...,9r 1,2 t Model

yy

yy

y trttr