Chapter 3. Multiple Regression Model 보충
!
Multiple linear regression model
(i) Definition
!" # $% & $'('" &⋯& $*(+ & ,"- " # '- ⋯- .
$%- $'- ⋯- $* : "regression coefficient"
εi
: "random disturbance"
(ii) Basic Assumption (BA) :
εi
～ iid mean zero and variance σ 2
cf. ε i ～ iid N (0,σ 2 )
!
Parameter Estimation
"
Estimation
Least squares method
cf. ML method
.
/ 0$%- $'- ⋯- $* 1 #
20! 3$ 3$ ( 3⋯3$ ( 1
"#'
4
"
%
' '"
* *"
Normal equation :
2
2
2
2
2
2
2
2( !
⋮
6
.( 5 &02 ( ( 15 &02 ( ( 15 &⋯&02 ( 15 # 2 ( !
.5% &0 ('" 15' &0 (4" 154 &⋯&0 (*" 15* # !"
.6
( 5 &0 (4 15 &0 ( ( 15 &⋯&0 ( ( 15 #
' %
'"
'
8 %
4
8" '"
'" 4"
'
4
*" 4"
4
Chapter 3 - 1
'" *"
*
'" "
4
*"
*
*" "
⇔ /''5' & /'454 &⋯& /'*5* # /!'
/'45' & /4454 &⋯& /4*5* # /!4
⋮
/''5' & /'454 &⋯& /'*5* # /!'
5% # 6! 3 5'6(' 3 546(4 3⋯3 5*6(*
S ij and S yi are defined as the sum of squares
"
!"# 5% & 5'('" &⋯& 58(*"
Fitted value : :
"
!"
Residual : ;" # !" 3 :
<matrix notation 소개>
matrix notation
V# 0='- ⋯- =* 1> , W# 0@'- ⋯- @A 1>
; all random
U# 0,"C 1*×*
A# 0F"C 1G ×* ,
'
B= ( b ij ) m ×l
2
; all const.
(i) HV# 0H='- ⋯- H=* 1>
(ii) H,# 0H,"C 1*×*
(iii) =FI0V1 # 0JKL 0="- =C 11*×*
# H MV3 HVNMV3 HVN>
(iv) OKL 0V-W1 # 0OKL 0="- @C 11*×*
# H MV3 HVNMW3 HWN>
(v) H 0AV1# AH 0V1
Chapter 3 - 2
(vi) =FI0AV1 # AH 0V1
T
(vii) OKL 0AV- BW1 # AOKL 0V- W1B
"
Matrix Representation of Model
!' # $% & $' ('' &⋯& $* (*' & ,'
R !' U # R ' ('' ⋯ (*'UR $%U R ,' U
!4
,
' ('4 ⋯ (*4 $'
& 4
⋮
⋮⋮⋮ ⋮
⋮
S !.V S ' ('. ⋯ (*.VS $*V S ,. V
TTWW WT W
⇓
Y
Y # Z$& ,
⇓
Z
R ,' U RH 0,' 1U R % U
,
H 0,4 1
⋮
#
#%
H 0,1# H 4 #
⋮
⋮
⋮
S ,. V SH 0,. 1V S % V
Chapter 3 - 3
⇓
$
⇓
,
=FI0,1# H 0,⋅,′1
R ,4' ,',4 ⋯ ,',.U
TTT W TW WW
,4,' ,44 ⋯ ,4,.
#H
⋮ ⋮
⋮
4
S,.,' ,.,4 ⋯ ,. V
T
W
R H 0,4' 1 H 0,',4 1 ⋯ H 0,',. 1U
#
H 0,44 1
H 0,4,' 1
⋯ H 0,4,. 1
⋮
⋮
⋮
4
SH 0,.,' 1 H 0,.,4 1 ⋯ H 0,. 1 V
T
W
R \4 % ⋯ % U
R ' % ⋯ %U
4
% \ ⋱⋮
% ' ⋱⋮
#
# \4
# \4⋅I
⋮⋱⋱ %
⋮⋱⋱ %
S % ⋯ % 'V
S % ⋯ % \4V
Y # Z $ & ,- E0,1 # %- Var0,1 # \4⋅I.
"
Estimation
(i) LSE
.
/#
2, # ,′⋅,
"#'
4
"
# 0Y 3 Z$1′0Y 3 Z$1
# Y′Y 3 $′Z′Y 3 Y′Z$ & $′Z′Z$
# Y′Y 3 4$′Z′Y & $′Z′Z$
minS
$
g/
⇒ 6 #3 4Z′Y& 4Z′Z$ # %
g$
⇒ 3 4Z′Y& 4Z′Z5# %
⇒ Z′Z5# Z′Y 0Normal equation1
3
5# 0Z′Z 1 'Z′Y: LSE
Chapter 3 - 4
(ii) MLE
Need
,∼ p 0%- \4⋅I1
q0, r $- \4 1 #
#
#
0
2
1
,4"
'
⋅exp 3 6
6
04+\4 1.s4
4\4
'
,′⋅,
⋅exp 3 6
6
4 .s4
04+\ 1
4\4
0Y 3 Z$1′0Y 3 Z$1
'
exp
3
⋅
6
6
04+\4 1.s4
4\4
0
0
1
.
.
v # ln q0,w$- \4 1 #3 6 ln04+1 3 6 ln\4
4
4
'
3 64 0Y 3 Z$1′0Y 3 Z$1
4\
gv
'
# 64 0Z′Y 3 Z′Z$1
6
g$ \
gv
.
'
'
#3
&
⋅
64
6
6 0Y 3 Z$1′0Y 3 Z$1
4 6
g\
\4 g\x
⇒ Z′Zy
5 # Z′Y
y4 '
\ # 6 0Y 3 Zy
5 1′0Y 3 Zy
51
.
⇒y
5 # 0Z′Z 1 'Z′Y : LSE
3
Chapter 3 - 5
1
!
Interpretation of Regression Coefficient
"
Model:
!" # $% & $'('" &⋯& $8(+ & ,"- " # '- ⋯- .
$'- ⋯- $8 : "regression coefficient"
"
How to interpretate $C- C # '- ⋯- 8 ?
① change in Y corresponding to a unit change in
xj
after all other
independent variables are held fixed
② contribution of x j to Y after adjusting for all other independent
variables
"
Illustration when 8 # 4
Regress Y on Z'- Z4
⇔
① Regress Y
on Z'
② Regress Z4 on Z'
③ Regress ;Y⋅Z ' on ;Z 4⋅Z '
"
Geometrical interpretation
Chapter 3 - 6
!
Properties of LSE
1. Under (BA)
(i) H 05" 1 # $" w " # %- '- ⋯- *
3'
(ii) =FI05 1 # 0Z >Z 1 \4
(iii)
R=FI05' 1 OKL 05'- 54 1 ⋯ OKL 05'- 5* 1U
=FI054 1 ⋯ OKL 054- 5* 1
⋱
⋮
=FI05* 1 V
S 0z!GG1
T WW
z'4 ⋯ z'*U3 '
z44 ⋯ z4*
#
\4
⋱⋮
z**V
S0z!GG1
R
z''
i.e., =FI05" 1 # J""\4
w " # '- ⋯- *
OKL 05"- 5C 1 # J"C\4
w "- C # '- ⋯- *- " ≠C
when c ij is the ( i, j)th element of
z'4 ⋯ z'*U3'
z44 ⋯ z4*
⋱⋮
z**V
S0z!GG1
R
z''
And
=FI05% 1 #
0
1
*
*
'
4
6
&
0(" 1 J"" & 4 6(" 6(C J"C \4
6
. "#'
" |C
2
# J%%\4
Chapter 3 - 7
2
(iv) Gauss-Markov
b i : Best Linear Unbiased Estimator (BLUE)
in the sense that
=FI0:
$ 1 ≧ =FI05 1
"
"
w " # %- '- ⋯- *
for any linear unbiased estimator
(v) OKL 06! - 5" 1 # %
(vi)
β̂ i
w " # '- ⋯- *
//H
z4 # 6 # ~/H
0. 3 * 3 '1
⇒ H 0z4 1 # \4
Proof
(i)
H 05 1 #H 00Z′Z 13 'Z′Y 1
# 0Z′Z 13 'Z′H 0Y 1 # 0Z′Z 13 '⋅Z′$ # $
(ii)
=FI05 1 #=FI00Z′Z 13 'Z′Y 1
# 0Z′Z 13 'Z′ =FI0Y 1 Z 0Z′Z 13 '
# 0Z′Z 13 'Z′⋅0• \4 1 Z 0Z′Z 13'
0∵=FI0Y 1 # =FI0Z$ & ,1 # =FI0,1 # • \4 1
# 0Z′Z 13 '\4
(iii)
5% # 6!3 5'6
('3⋯3 5*6
(*
Chapter 3 - 8
=FI05% 1 # =FI06! 1 &
*
206( 1 =FI05 1
4
"
"#'
"
*
2 6( 6( OKL05 - 5 1
&4
" |C
0
"
C
"
C
1
*
*
'
4
6
# 6&
0(" 1 J"" & 4 6(" 6(C J"C \4
. "#'
"|C
2
2
from (v).
(iv)
Consider a linear unbiased estimator :
:
$# vY
0v r 08 & '1 ×. 1 with v r vZ # •
H 0:
$1 # H 0vY 1 # v H 0Y 1 # v 0Z$1 # $
Then,
=FI0:
$ 1 # v=FI0Y 1v > # vv >\4
# •0Z >Z 13 'Z > & v 30Z >Z 13 'Z >‚
•0Z >Z 13'Z > & v 30Z >Z 13'Z >‚>\4
# 0Z >Z 13'
& •v 30Z >Z 13 'Z >‚•v 30Z >Z 13'Z >‚\4
>
&0Z >Z 13 'Z >•v 30Z >Z 13 'Z >‚ \4
>
& •v 30Z >Z 13 'Z >‚•0Z >Z 13 'Z >‚ \4
Chapter 3 - 9
# 0Z >Z 13'\4
>
& •v 30Z >Z 13 'Z >‚•v 30Z >Z 13'Z >‚ \4
3
≧ 0Z >Z 1 '\4 # =FI05 1
≥ in the sense of nonnegative definiteness
(v)
'
OKL 06!- 5 1 # OKL 0 6 '>Y-0Z >Z 13'Z >Y 1
.
'
# 6 '>=FI0Y 1Z 0Z >Z 13 '
.
'
# 6 '>Z 0Z >Z 13 '\4
.
'
# 6 \4•MZ >Z 0Z >Z 13 ' N의 첫번째 IKƒ‚
.
0
1
'
# 6 \4- %- ⋯- %
.
>
>
3'
• # Z Z 0Z Z 1
0 1
0
'>
3'
>
#
> Z 0Z Z 1
Z%
1
3'
'>Z 0Z >Z 1
#
3
Z%>Z 0Z >Z 1 '
R 0' % % ⋯%1 U
R% ' % ⋯ % U
#
%'⋯ %
⋱⋮
SS
% ' VV
T
W
Chapter 3 - 10
(v)
.
//H #
20! 3:! 1
4
"#'
>
"
"
#; ;
# 0Y 3 Z 0Z′Z 13'Z′Y 1′0Y 3 Z 0Z′Z 13'Z′Y 1
# Y′Y 3 Y′Z 0Z′Z 13 'Z′Y
# Y′0• 3 Z′0Z′Z 13'Z′1Y
# Y′0• 3 „ 1Y
3
where „ # Z 0Z >Z 1 'Z >
H MYY > N # H M0Y 3 HY 10Y 3 HY 1> & YH 0Y 1>
& H 0Y 1Y > 3 H 0Y 1H 0Y 1> N
# =FI0!1 & H 0Y 1H 0Y 1>
H MY >0• 3 „ 1Y N # …IMH M0• 3 „ 1YY > NN
# …IM0• 3 „ 1•=FI0Y 1 & H 0Y 1H 0Y 1>‚N
# …IM0• 3 „ 1=FI0Y 1N
& H 0Y 1> 0• 3 „ 1H 0Y 1
# …IM0• 3 „ 1N\4 & $>Z >0• 3 „ 1Z$
# 0. 3 * 3 '1\4
참고.
† # 0F"C 1G×. ‡ # 05*A 1.×G
.
†‡ # 0
2F 5 1
G
2
5"AFAC 1.×.
"* *C G×. ‡† # 0
*#'
A#'
Chapter 3 - 11
G
…I0†‡ 1 #
.
22F 5
" # '* # '
.
…I0‡† 1 #
"* *"
G
225 F
" # 'A # '
"A A"
H 0Y′†Y 1 # …I0†= 1 & ˆ′† ˆ
2. Under the normality assumption for the error terms (NA) i.e. ε i ～ iid
2
N (0,σ )
3
(i) 5 ∼ p*&' 0$-0Z′Z 1 ' \4 1
3'
(ii) J5 ∼ p 0J$- J′0Z′Z 1 J \4 1- for '× 0* & '1 vector J
5" ∼ p 0$"- J""\4 1
" # %- '- ⋯- * , where c ii is the ( i +1)th diagonal
3
element of 0Z′Z 1 ' matrix
(iii)
O 5 ∼ p 0O$- O′0Z′Z 13'O \4 1- ‹× 0* & '1 matrix O
//H
(iv) 6
～ Œ4 0. 3 * 3 '1
4
\
R $'U
//•
4
～ Œ 0*1 when ⋮ # %
6
\4
S $kV
TW
(v) 5" and //H 0z4 1 are independent
5" 3 $"
(vi) 6
～ …0. 3 * 3 '1 " # %- '- ⋯- *
J"" z
‘6
Chapter 3 - 12
!
Statistical inference under normality assumption
(i) Test of hypotheses ’% r $" # $%"
5" 3 $%"
…#6
∼ …0. 3 * 3 '1 H%
6
J z
‘6
""
(ii) Confidence interval for β i
estimator
± percentile SE
5" ± …–s4 0. 3 * 3 '1 ‘6
J"" z
(iii) Prediction interval and hypothesis testing for the mean and an
individual at ( # (%
(iv) Hypothesis testing of O$ # %
"
Valid for large sample size data without normality
Review on quadratic forms
① Y ∼ p 0ˆ- = 1
Y′†Y- Y′‡Y are independent iff †=‡ # % or ‡=† # %
② Y ∼ p 0ˆ- = 1
'
Y′†Y ∼Œ4′ 0I0† 1- 6 ˆ′† ˆ1 iff †= is idempotent
4
③
H 0Y′†Y 1 # …I0†= 1 & ˆ′† ˆ
Chapter 3 - 13
Quadratic forms
․
//> #
#
#
․
//• #
#
#
#
#
․
//H #
#
#
20! 3 6!1 # 2! 3.06!1
4
"
"
4
"
4
—
Y′Y 3 Y′ 6 Y - 0— # '′'- '′ # 0' ' ⋯ '1
.
—
Y′0• 3 6 1Y
.
20:!3 6!1 # 2:!3.06!1
4
"
4
"
4
—
6
:
Y′:
Y3 .!4# 5′Z′Z5 3 Y′ 6 Y
.
—
MY′Z 0Z′Z 13' NZ′Z M0Z′Z 13 'Z′Y N 3 Y′ 6 Y
.
—
Y′Z 0Z′Z 13'Z′Y 3 Y′ 6 Y
.
—
Y′0„ 3 6 1Y- 0„ # Z 0Z′Z 13'Z′1
.
//> 3 //•
—
—
Y′0• 3 6 1Y 3 Y′0„ 3 6 1Y
.
.
Y′0• 3 „ 1Y
Thm. In a multiple regression model, Y ∼ p. 0Z$- •\4 1
//H
∼ Œ4 0. 3 * 3 '1
1. 6
4
\
0
1
//•
'
—
4′
where
0
1#
3
∼
Œ
*
˜
˜
$′
Z
′
„
Z$
2. 6
6
6
4
.
\4
3. //H and //• are independent
Chapter 3 - 14
~/•
//•s*⋅\4
4. ™% # 6 # 64 ∼ ™ 0*- . 3 * 3 '- š1,
~/H //Hs0. 3 * 3 '1\
0
1
'
—
where š # 64 $′Z′ • 3 6 Z$
.
4\
proof.
1. Note that
//H
4
where
#
#
0
3
1s
Y
′
†
Y
†
•
„
\
.
H
H
6
\4
Sufficient to show that
# †H =FI0Y 1 .
0†H =FI0Y 11 0†H =FI0Y 11
0 ⇔ 0• 3 „ 1 0• 3 „ 1
# • 3„ 1
Also,
IF.*0†H 1 # …IFJ;0†H 1 # . 30* & '1 and
'
'
˜ # 6 ˆ′†H ˆ # 64 $′Z′0• 3 „ 1Z$ # %.
4
4\
2. Note that
//•
—
where
#
#
0
3
1s\4.
Y
′
†
Y
†
„
•
•
6
6
4
.
\
Sufficient to show that
# †•=FI0Y 1 .
0†•=FI0Y 11 0†•=FI0Y 11
0
—
—
⇔ 0„ 3 6 1 0„ 3 6 1
.
.
—
# „36
.
1
Also,
IF.*0†• 1 # …IFJ;0†• 1 # * and
'
'
—
˜ # 6 ˆ′†• ˆ # 64 $′Z′0„ 3 6 1Z$.
4
.
4\
Chapter 3 - 15
3.
—
0• 3 „ 1⋅•\4⋅0„ 3 6 1
.
—
# 0• 3 „ 10„ 3 6 1
.
—
—
# „ 3 „ ⋅„ 3 6 & „ ⋅6 # %
.
.
Analysis of Variance
(1) Decomposition of sum of squares
//> # //• & //H
›q
—
—
Y′0• 3 6 1Y # Y′0„ 3 6 1Y & Y′0• 3 „ 1Y
.
.
#
. 3'
* & 0.—* 3 '1
(2) ANOVA
ANOVA for Multiple Regression
/KœIJ;
//
~/
™%
//•
™% # ~/•s~/H
Model //•
*
~/• # 6
*
//H
Error //H . 3 * 3 ' ~/H # 6
. 3* 3'
›q
(3) Testing
"
Hypothesis
’% r $' # $4 #⋯# $* # % vs ’' r $C ≠%, for at least one j
"
Test statistic
Chapter 3 - 16
~/•
™% # 6 ∼ ™ 0*- . 3 * 3 '1
~/H
Reject ’% if ™% Ÿ ™– 0*- . 3 * 3 ' 1
!
Multiple Correlation Coefficient
Multiple correlation coefficient
.
20! 3 6! 10:!3 :6! 1
"
"#'
"
•#6
's4
.
.
6
4
4
:
0!" 3 6! 1
0!"3 :
!1
•2
"#'
"
"
‚
2
"#'
Measures the relationship between a variable Y and a set of variables
('- ⋯- (*
"
6
• # ‘• 4 , where
.
//•
•4 # 6
//>
20:!3 6! 1
" #'
4
"
#6
.
0!" 3 6! 14
2
"#'
.
//H
# '3 6
//•
20! 3:! 1
4
"#'
"
"
# '3 6
.
0!" 3 6! 14
2
"#'
Chapter 3 - 17
"
Geometric interpretation :
0!' 3 6!- ⋯- !. 3 6! 1
`
θ
span of 0('' 3 6
('- ⋯- ('. 3 6
(' 1⋯-0( 3 6
( - ⋯- ( 3 6
( 1
*'
"
*
*.
*
• 4 # cos4¡
Remarks
① • 4 ↑ whenever an additional explanatory variable is added
Chapter 3 - 18
② Not good for comparing goodness of fit of models with different
number of explanatory variables
Adjusted • 4
.
20! 3:! 1 s0.—* 3'1
4
"
"
"
//HF
"#'
•F4 # ' 3 6 # ' 3 6
.
//>F
0!" 3 6! 14s0. 3 '1
2
" #'
"
meaning
!
Tests of Hypotheses in the Linear Model
"
The basic starting model
Full Model (FM):
!" # $% & $'('" &⋯& $* (*" & ," ; ε i ～ iid p 0%- \4 1
"
Three types of hypotheses:
(i) All coefficients are zero
’% r $' #⋯# $* # %
(ii) Some coefficients are zero
’% r $*3 ' # $* # %
(iii) Some coefficients are equal
’% r $' # $4
(iv) Some constraints
Unified approach
"
Full Model (FM)
Chapter 3 - 19
"
Reduced Model (RM)
Model which accommodates the null hypothesis
(i) !" # $%′ & ,", ," ∼"¢"¢›¢ p 0%- \4 1
(ii) !" # $%′ & $'′('" &⋯& $* 3 4′(* 3 4- " & ,"
(iii) !" # $%′ & $'′ 0('" & (4" 1& $£′(£" &⋯& $*′(*" & ,"
"
F-test statics
//H 0•~ 1 3 //H 0™~ 1 //H 0™~ 1
① ™ # 6 s6 ,
0. 3 *′ 3 '1 30. 3 * 3 '1 . 3 * 3 '
where *′0| * 1 is the number of parameters in the RM.
0. 3 *′ 3 '1 30. 3 * 3 '1 # * 3 *′ : RM과 FM에서 모수개수의 차이
② Meaing
③ Under H 0 i.e. if the reduced model is correct,
™ ∼ ™ 0* 3 *′- . 3 * 3 ' 1
Reject H 0 when ™ Ÿ ™– 0* 3 *′- . 3 * 3 '1
④ (//H 0•~ 1 Ÿ //H 0™~ 1 설명
.
min$ - $ - $
%
'
4
20! 3$ 3$ ( 3$ ( 1
" #'
4
"
%
' '"
4 4"
.
20! 3$ 3$ ( 1
≦ min$%-$'
"#'
4
"
%
' '"
Note that • 4 ↑ as including new predictors
Equivalent testing procedure I
(eg) !" # $% & $'('" &⋯& $*(*" & ,", ," ∼"¢"¢›¢ p 0 %- \4 1
Chapter 3 - 20
(i) ’% : $A # %
5A
5A
#
#
… 6 6
/H 05A 1 ‘6
OA&'-A&' z
OAA′ is the 0A- A′1 element of ( X T X ) - 1
⇔
//H 0•~ 1 3 //H 0™~ 1 //H 0™~ 1
™ # 6 s6
'
. 3* 3'
when RM :
!" # $%′ & $'′('" &⋯& $A 3 '′(A 3 '- "
& $A&'′(A&'-" &⋯& $*′(*" & ,"
(ii) H 0 : $A 3 $A′ # % (" ≠C )
5A 3 5A′
…#6
/H 05A 3 5A′ 1
5A 3 5A′
#6
6
=FI05A 1 &:
=FI05A′ 1 3 4:
OKL05A- 5A′ 1
‘:
5A 3 5A′
#6
0O
&O
3 4O
1z
‘6
A&'-A &'
A′ &'-A′ &'
A&'-A′ &'
⇔
//H 0•~ 1 3 //H 0™~ 1 //H 0™~ 1
™ # 6 s6
'
. 3* 3'
where RM :
!" # $%′ & $'('" &⋯
& $A′0(A" & (A′" 1 &⋯& $*′(*" & ,"
™ test statistics in terms of • 4 :
Chapter 3 - 21
•*4 : • 4 of FM with * explanatory variables
•*′4 : • 4 of RM with *′ explanatory variables
//H 0•~ 1 3 //H 0™~ 1 //H 0™~ 1
™ # 6 s6
* 3 *′
. 3*3'
•*43 •*′4 ' 3 •*4
# 6 s6
* 3 *′
. 3* 3'
Equivalent testing procedure II
O$ # %
O is a ‹× 0* & '1 contrast matrix.
"
Three types of hypotheses:
(i) All coefficients are zero
’% : $' #⋯# $* # % ⇔ O$ # %,
¥ % ' % ⋯ %¨
% % ' ⋯%
O#
%%
¦% ⋯ '
©
§
ª
(ii) Some coefficients are zero
’% : $*3' # $* # % ⇔ O$ # %
¥% ⋯% ' %ª¨
O#§
¦% ⋯% % '©
(iii) Some coefficients are equal
’% r $' # $4 ⇔ O$ # %
O # M % ' 3 ' % ⋯% N
Chapter 3 - 22
3
5 ∼ p* & ' 0$- 0Z′Z 1 '\4 1
3
⇒ O 5 ∼ p‹ 0O$- O 0Z′Z 1 'O′\4 1
⇒ 0O 53 O $1′ M O 0Z′Z 1 'O′\4 N
3
3'
0O 53 O $1′ ∼ Œ4 0‹1
3'
0O 51′ M O 0Z′Z 13 'O′ N 0O 51′
™% # 6 ∼ ™ 0‹- . 3 * 3 '1- under ’%
//Hs0. 3 * 3 '1
!
Predictions
1. Confidence intervals for the mean response at 0(%'- ⋯- (%* 1
ˆ% # $% & $'(%' &⋯& $*(%*
:̂ # 5 & 5 (% &⋯& 5 (%
%
%
' '
* *
# 6! & 5' 0(%' 3 6
(' 1 &⋯& 5* 0(%* 3 6
(* 1
① Under (BA)
H 0:̂% 1 # ˆ%
• 2
‚
*
* *
'
%
4
:̂
6
=FI0 % 1 # 6 & J"" 0(" 3 (" 1 &4
J"C 0(%" 3 6
(" 10(%C 3 6
(C 1 \4
. "#'
" |C
22
② Under (NA)
:̂3 ˆ
%
%
～ …0. 3 * 3 '1
6
:̂
/H 0 1
%
Chapter 3 - 23
where
• 2
22
:̂ ± …
:̂
–s4 0. 3 * 3 '1/H 0 % 1
%
2. Prediction intervals for an individual !% at 0(%'- ⋯(%* 1
:
!%# :̂%
① Under (BA)
H 0!% 3:
!% 1 # %
=FI0!% 3:
!% 1 # =FI0!% 1 & =FI0:
!% 1
•
by independence
‚
*
* *
'
%
4
6
# ' & 6 & J"" 0(" 3 (" 1 & 4
J"C 0(%" 3 6
(" 10(%C 3 6
(C 1 \4
. "#'
"|C
2
22
② Under (NA)
!% 3:
!%
∼…0. 3 * 3 '1
6
/H 0! 3:
!1
%
%
:
!% ± …–s4 0. 3 * 3 '1 /H 0!% 3:
!% 1
3. Use of vector and matrix notation
Chapter 3 - 24
's4
‚
8
8 8
'
%
4
:̂
6
/H 0 % 1 # 6 & J"" 0(" 3 (" 1 & 4
J"C 0(%" 3 6
(" 10(%C 3 6
(C 1
. " #'
" |C
z
ˆ% # $% & $'(%' &⋯& $*(%*
# (% $(%# 0'- (%'- ⋯- (%* 1′
:̂% # (% 5=FI0:̂% 1 # (%=FI05 1(%′
# (% 0Z′Z 13'(%\4
Chapter 3 - 25