最大概似估計式與最小變方不偏估計式在評估平均生体相等性之比較

239

–

258

Ü

|×–N,lD|ü‰j.R,l

ÊÇ,

ÌÞ•ó45ªœ

˜

C ‘

1

_{r ó ”}

2 1

_{Å « É × ç   ‘ ç û ˝ F Þ Ó $ l}

2

_{Å A Š × ç $ l ç Í  û ˝ F}

¿

b

Ÿ  “ û ê Û b I ‘ Î × ‹ ‰ D Å ‚ I ’

,

F J Ÿ  “ “ g

u

Ý  ú 
& Ÿ  “ ù ‚ ‚ Å (

,

w … “  † ª " ¨ ó ° “ ¹

,

¥  " Î “ ˚ Ñ ç ± “

,

Ä ç ± “ . â Å ‚ û ê I ’

,

F J ç ±

“

g  œ Ÿ  “ Ñ Z í
Ä ¤ ª  ô » W ‘ à í X |
Ñ ü \ ç

± “ í W ^ ¸ é r 4 D Ÿ  “ ó °

,

“

\ À P b °  ¼ . â } p

ç ± “ D Ÿ  “ x   Ì Þ • ó  4 ( n Ž z ,
| × – N ,

l  ˛ \ S à Ê Ç ,  Ì Þ • ó  4

,

O | × – N , l  } ò ,

 Ì Þ • ó  4
F J B b  ‡ S à | ü ‰ j . R , l  Ç , 

Ì Þ • ó  4
B b 1 Ï W ø _ _ Ò û ˝ %

,

J % ð 

I

Ï Ï

D

% ð  ì ‰ V ª œ s  , l  5 i š
B b ? J õ Ò ’ e z p |

ü ‰ j . R , l  Ê Ç , Þ • ó  4 , í @ à

É œ È

:

 Ì Þ • ó  4 _ Ò û ˝ % ð  Ï % ð  ì ‰ ] ˝ – È ¨ Ö 0

1 Å b ç } } é Ø ù

:

3 b

62F03;

Ÿ

b

62–07

1.

‡

k

h “ C Ÿ  “

(innovative drugs)

í û

˝ ¸ ê   Ì ø I ’

_8∼12

h

1 j £

10∼12

 í v

È

,

F J û ê “ Ó uÅ ‚

,

Î

ç ’ À ¸ ò ê Ô í I ’
¢ Ä  ù

‚ ‚ í \

ˆ

,

F J Ÿ  “ í g 

¦  u œ Ñ ú  í

,

k u ' Ö ç ± “  }

 ƒ

Ÿ  “ ù ‚ ‚ Å (

,

² \  T |

 b h “ C ~

(Abbreviated New Drug

Applica-tion, ADNA),

1"Ο“íAM`¨|ó°í“¹
¥é"Γ˚Ñ籓

(generic drugs)

Ñ ü \ ç ± “ í W ^ ¸ é r 4

D Ÿ  “ ó °

,

0 ä ® Å í “ \

À P

,

à

1 Å ë ¹ D “ Ó  Ü

(FDA)

D B Å ¨ Þ  “ \ T Ì b °  ¼ . â

}

p ç ± “

D Ÿ  “ x   Ì Þ • ó  4

(average bioequivalence, ABE)

籓DŸ“Ìó4¹Êªœs6ÌÞ•ªà0

(average

bioavail-ability)

u ´ Ñ ú

(equivalent)

Ê “ Ó  ‰ ç

(pharmacokinetics)

,

,

Þ • ª à

0 í ¥ @ M

(Responses)

¨ Ž v

È

-

¦ ¯ ë

 - í 

( Þ

(area under the curve

of time-plasma concentration, AUC)

£|ò¦¯ë

(maximum concentration,Cmax

)

ñ‡

1Å

FDA,

BŨÞ£0䮓\ÀPÌSà|×–N,

l 

(max-imum likelihood estimator, MLE)

Ç

, Ì Þ • ó

 4
l ø

AUC

£

Cmax

¦ ú

b

ž ²

(logarithmic Transformation)

( Ê ú b



(log-scale)

£  G c

q - l

 ç ± “ D Ÿ  “ 5 ú b  Ì Þ • ª à 0 Ï í

90%

] ˝ –

È
1 y ¦ N b ž

²

( ) Ÿ á



(original-scale)

- ç ± “

D Ÿ  “  Ì ª à 0 ª í

90%

] ˝

–

È
J

90%

] ˝ –

È ê r ¨ Ž Ê

(80.00%, 125.00%)

5 q

,

† Ê

5%



I

Ï Ï

œ 0 -

,

“

\ À P † Ž˚ ç ± “ D Ÿ  “ Ñ  Ì Þ • ó 
Ö Í

MLE

Ê

’ e

¦ ú b (

,

‡

ú F Ç , í ¡ b u . R ,

l M

,

ª uø ï ø w ¦ N b ž ²  Ÿ á



,

â k N b

ž ² 1 . u ( 4 í

,

k u ß Þ R Ï
Ä ¤ … d5 ? $ l , í .

R 4

,

k u 

‡ S à | ü ‰ j . R , l 

(minimum variance unbiased estimator,

MVUE)

V Ç ,  Ì Þ • ó

 4

… d ñ í Ê k û

˝

MLE

¸

MUVE

Ê Ç ,

ABE

í

[ Û Ï æ

,

3

b ‡ ú w õ

,

l M % ð 

I

Ï Ï

(size)

¸

% ð  ì ‰

(power)

D ] ˝ – È í ¨ Ö 0

(prob-ability coverage)

í

¶ } V ª W ª œ

,

Í

7 õ , l M $ l 4 ” í ¶ M ª œ

,

ª

c

k

Liu and Weng (1992)

… d Ê

 ù  2

,

5 ? . ° í > Œ q l ¸ . ° í ! … c q

,

R û

MVUE

1

, 

Ì Þ • ó

 4 í _ Ò t ð 5 ! ‹

;

Ê û  2

,

B b àø _

_2×3

> Œ

q l

í b W

z p

MVUE

5 õÒ @ à
| (

,

Ê

 ü  2

,

T X

ø < n 

2.

| × – N , l 

(MLE)

D | ü ‰ j . R , l 

(MVUE)

‡

ú s å  s ‚

È > Œ q l

_{(two-sequence, two-period crossover design; 2×2}

crossover design), Liu and Weng (1992)

ª

œ | × – N , l  D | ü ‰ j . R ,

l  5 õ , l í . ° $ l 4 ”
… d Ê ò ¼ > Œ q l -

(higher-order crossover

design)

R û |

MVUE

í ,

l  £ ç ± “ D Ÿ  “  Ì Þ • ª à 0 ª 5

_{(1 −}

2α)%

] ˝ –

È

,

D ª œ

MLE

D

MVUE

Ê Ç ,  Ì Þ • ó

 4 í i š

> Œ

q l u ø ø ˇ ¯ ¯ t ð Ñ p D § Î ‘ K í § t 6 Ó œ N » B b _ å



(sequence)

\ N » B å  í § t 6

,

Ê . °

T Ü ‚ È

(treatment period)

€

à ç ± “

(test formulation, T )

C Ÿ  “

(reference formulation, R)

Ñ 7

Ê ¢

“

Ó {

ì ^ @

(carryover effects)

í ê Þ

,

¦  Ê s _ T Ü ‚ È 2 ‹ p À ž ‚ È

(washout period)

[

1

T X

3

_ |  U à í s å  . ° ‚

È í > Œ q l

I

X

ijk

H

[ Ê 

i

å  2



k

_ §

t 6 k 

j

_

T Ü ‚ È 5 “ Ó  ‰ ç ¥

@ M

, k=1,. . . ,n

i

;j=1,...,J;i=1,2

ç

q l Ñ

2×2

> Œ

q l v

,J=2;

Ñ

2×3

> Œ

q

l v

,J=3;

Ñ

_2×4

> Œ

q l v

,J=4

Ê

X

ijk

¦ ú bž ² (

, Chow and Liu (2000)



‡ -  ª ‹ 4 ( 4 _ 

(addi-tive linear models)

,

l  Ì Þ • ó  4

:

Y

ijk

= µ + G

i

+ S

ik

+ π

j

+ τ

f

+ C

j−1,i

+ ε

ijk

(1)

Y

ijk

= ln(X

ijk

); ln

H

[ A Í ú b

,µ

u,  Ì M

,G

i

Ñ



i

_ å  í ì  ^ @

,

π

j

Ñ

j

_ ‚

È í ì  ^ @

, τ

f

Ñ Ê



i

_ å D 

j

_ ‚

È 5 T Ü í

ì  ^ @

(f = T, R), C

_(j−1,i)

Ñ Ê

i

_ å 

D 

j

_ ‚

È 5 ì  í  ø ¼

“

Ó {

ì ^ @

(first-order carryover effect), S

ik

Ñ



i

_ å  2



k

_ §

t 6 5

Ó œ ^ @

(random effect); ε

ijk

Ñ



i

_ å 



k

_ §

t 6 Ê 

j

_ ‚

È ¥ @

M

5

Ó œ Ï Ï

(random error)

B b ° v c

q ì  ^ @ 5 ¸ Ñ

0

S

ik

x  Ö

D ó °

(independently and identically distributed, i.i.d.)

 Ì b Ñ

0

D ‰ j Ñ

σ

2_s

5  G } 0

ε

ijk

Ñ

i.i.d.

 Ì b Ñ

0

D ‰ j Ñ

σ

e2

5  G } 0

σ

s2

H

[ §

[

1

s å  í > Œ

q l

(1) 2×2

> Œ

q l

‚È

å 

I

II

1

T

R

2

R

T

(2) 2×3

> Œ

q l

‚

È

å 

I

II

III

1

T

R

R

2

R

T

T

(3) 2×4

> Œ

q l

‚

È

å 

I

II

III

IV

1

T

R

R

T

2

R

T

T

R

variability)

B b ? c

q

S

ik

D

ε

ijk

ó  Ö

Ê

2×2

> Œ

q l -

,

å  ^ @D

“

Ó {

ì ^ @ u  ó ¹ Æ

(confounded)

7 / J “ Ó { ì ^ @ æ Ê v

,

T Ü ^

@ 5 . R ,

 M . æ Ê
F J B b c q Ê

_2×2

> Œ

q l - % ¬ — D Å í À ž

‚

,

“

Ó {

ì ^ @ . æ Ê

ø

(1)

 ª ‹

( 4 _  ¦ N b ž ² Ñ Ÿ á

 ( w _  Ñ

:

X

ijk

= exp{µ + G

i

+ S

ik

+ π

j

+ τ

f

+ C

_(j−1,i)

+ ε

ijk

}

(2)

Ä B b c

q

S

ik

£

ε

ijk

Ñ  G

,

] _ 

(2)

¢

˚ Ñ ú b ( 4  G _ 

(Log-linear

normal model, Bradu and Mundlak, 1970;Crow and Shimizu, 1988)

Ê Ÿ á

 -

,

ç ± “

D Ÿ  “ 5  Ì Þ • ª à 0 }  Ñ

e

µ+τF

£

_e

µ+τR

F J ñ ‡ 0 ä ® Å “ \ À P Ç ,  Ì Þ •ó

4 5 ¡b

(parameter of interest)

Ñ ç ± “

D Ÿ  “  Ì Þ • ª à 0 5 ª

:

θ =

e

µ+τT

e

µ+τR

= e

τT_−τR

= e

τ

(3)

w 2

τ = τ

T

− τ

R

ú

AUC

£

Cmax

7 k

,

J

δ

Ê

0.8

D

1.25

5

È

,

® Å “ \ À P ¹ ª Ž

˚ ç ±

“

D Ÿ  “ x   Ì Þ • ó  4 7 ª Ž z ç ± “ , ù »
F J Ç ,  Ì Þ

•

ó

 4 5 $ l c z

(Statistical hypothesis)

Ñ

:

H

0

: θ ≤ ∆

L

or

Ha : θ ≥ ∆

U

vs

Ha : ∆

L

< θ < ∆

U

,

(4)

∆

U

=1.25

£

∆

L

=0.8 (=1/1.25)

Ñ Ÿ á

 - 5 ó

 4 ä Ì

(equivalence limit)

c

z

(4)

? ª Ê ú b



[ ý à -

:

H

0

: τ ≤ δ

L

or

Ha : τ ≥ δ

U

vs

Ha : δ

L

< τ < δ

U

(5)

w2

δ

U

D

δ

L

Ñúb

-5ó

4äÌ

, δ

U

= ln(1.25) = 0.2231, δ

L

= ln(0.80) =

−0.2231;

F J Ê ú b

 -

,

, - ó 4 ä Ì u ú ˚ k

0

ñ ‡ 0 ä ® Å “ \ À P Ê c

z

(3)

-

,

Ê Ç , ç ± “

D Ÿ  “ 5  Ì Þ •

ó

 4 v í $ l j ¶ Ñ í l Ê ú b

 - ° )

τ

5

MLE

£ ó ú @

τ

í

90%

] ˝ –

È ( y ø ! ‹ ¦ N b ° )

θ

5

MLE

D

θ

5

90%

] ˝ –È
J

θ

5

90%

]˝–

Èí,-Ìêr¨ÖÊ

(0.8,1.25)

q†Ž˚籓DŸ“xÌÞ

•

ó

 4

(Chow and Liu,2000)

Ê s å  í > Œq l -

,τ

í

MLE

ª ; W Ê ú b

 - _

ñ q ú ª

(intra-subject contrast), d

ik

° )

, k = 1, ..., n

i

; i = 1, 2

I

d

i·

Ñ



i

å  5 _

ñ q ú ª

 X  Ì b

, i = 1, 2

τ

5

MLE

¹ Ñ

s å  5 _ñ q ú ª  Ì b í Ï

:

b

τ = d = d

_1·

_{− d}

_2·

(6)

/

_{d ∼ N(τ, aσ}

2

d

)

(7)

w 2

a =

_n1₁

+

_n1₂

£

σ

_d2

Ñ _ñ q ú ª í ‰ j

(Chow and Liu,2000)

. ° s å 

> Œ

q l í

d

ik

D

d

ª

c k [

2

Ê  G cq -

,τ

í

90%

] ˝ –

È Ñ

:

(L

d

, U

d

) = d ± t(0.05, df)S

2p

r

c(

1

n

1

+

1

n

2

)

(8)

w 2

,t(0.05, df )

Ñ

A â  Ñ

df

2 -

t

} 0 í

_{1−0.05 =0.95}

ì } P M

, (8)

 2 .

° > Œ

q l í

df ,S

2 p

D

c

?

c [

2

; W

(6)

£

(8)

 ¦ N b ( ¹ )

θ

5

MLE

D w

90%

] ˝ –

È à -

:

m = e

bτ

£

(e

Ld

, e

Ud

)

(9)

[

2

Ê . ° í > Œ

q l 2 5 _ ñ q ú ª

> Œ q l dik d a c Sp2 df 2×2 1 2(Yi1k_{− Y}i2k) 1 2  Y11._{− Y}11.
_{− Y}21._{− Y}22.
 _n1 1+ 1 n 2 1 S 2 n1+ n2_{− 2} 2×3 1

4(2Yi1k_{− Y}i2k_{− Y}i3k) 1 4  2Y11._{− Y}12._{− Y}13.
₋ 2Y21._{− Y}22._{− Y}23.
 3 8 ₁ n 1 + 1 n 2  ₃ 8 S 2 2(n1+ n2_{− 2)} 2×4 1

20(6Yi1k− 3Yi2k−

7Yi3k+ 4Yi4k) 1 20  6Y11._{− 3Y}12._{− 7Y}13.+ 4Y14.
− 6Y21._{− 3Y}22._{− 7Y}23.+ 4Y24.
 11 40 ₁ n 1+ 1 n 2  ₁₁ 40 S 2 3(n1+ n2) − 5

s

2

H

[ ‰ j } & [ 2 í § t 6 q Ì j Ï Ï(intra-subject mean squared error);dfH [ A â 

O ú b

ž ² u ø Ý ( 4 ž ²

:

E [m] = E

h

e

bτ

i

= θe

aσ2d/2

]

θ

5

MLE

Ñ

ø  R í , l  7 / w R Ï 0 Ñ £
F J

m

}

ò ,

θ

7 /

w

ò , í ˙  } Ó ‰ j Ó ‹ D š … b - ± 7 Ó ×

; W

Bradu

D

Mundlak (1970)

£

Liu

D

Weng (1992),

[

1

F  ® s å  > Œ

q l -

, θ

5 | ü

‰ j . R , l 

(MVUE)

5

ø O  Ñ

:

T = b

θ Φ

df

[−a · df · s

2

],

(10)

s

2

Ñ ; W

d

ik

l  F ) 5 ¯ 9 š … ‰ j

(Pooled Sample variance),

Φ

df

[−a · df · s

2

] =

∞

X

j=0

Γ[df /2]

Γ[(df /2) + j]j!

[(−a/4)df · s

2

_]

j

_,

₍₁₁₎

w 2

_Γ(·)

Ñ  ; ƒ b

(Gamma function)

7

T

5

‰ j í , l  Ñ

:

d

V ar(T ) = e

2d

_{[Φ

df

(−a · df · s

2

)]

2

− Φ

df

(−4a · df · s

2

)}

Ä

MVUE

5 ü ~ } 0 „ ø

,

B b 

‡ ; W  G } 0 £ 2 -

t

} 0 

θ

5

90%

] ˝ –

È à -

:

(L

T 1

, U

T 1

) = T ± Z(α)

q

d

V ar(T )

£

(L

T 2

, U

T 2

) = T ± t(α, df)

q

d

V ar(T ),

(12)

Z(α)

Ñ ™ Ä  G } 0

_{(1 − α)}

ì } P M

J

(L

T 1

, U

T 1

)

C

(L

T 2

, U

T 2

)

ê r ¨ Ö Ê

(0.8,1.25)

q

,

† Ê

5%

é

O ® Ä -

,

ç

± “

D Ÿ  “ x   Ì Þ • ó  4

3.

_ Ò

û ˝

(Simulation Studies)

B b Ï

W ø _ ×  í _ Ò û ˝ ª œ

MVUE

D

MLE

5 i š

,

Ç

, í

á ñ ¨

Ž

:

 Ì R Ï

(average bias),

 Ì Ï Ï Ì j

(average mean square error),

] ˝ –

È í ¨ Ö œ 0

(coverage probability),

% ð 

I

Ï Ï

(empirical size)

¸

% ð  ì

‰

(empirical power)

_ Ò û

˝ 5 ? [

1

ú  . ° s ß å í > Œ q l
ú © ø 

> Œ

q l B b ª œ

5

 . ° í

θ

M

: 0.8,0.9,1,1.1,1.25

w 2

0.8

D

1.25

u Ê Ç ,

MVUE

D

MLE

5 

I

Ï Ï

,

7

0.9,1

D

1.11

u Ê Ç , s j ¶ 5  ì ‰
B b

?

5 ?

4

_ . ° š … b

,

© å  Ñ

6,8,12,18

§

t 6
¥

4

 š … b H [ ø O Þ

•

ó

 4 t ð | Q £ | ò í š … b
‡ ú u ‰ j ä ³ í ! Z B b 5 ?  ù 

2

S

ik

D

ε

ijk

compound symmetry

c

q -

12

 . ° × ü

σ

s2

D

σ

2e

í

¯
Ç Õ

B b ?

5 ?

6

 . ¯ ¯

compound symmetry

c

q í u ‰ j ä ³ ¯
F J B

b

, u ê A 7

1080

_

_{(3×5×4×18)}

¯ í _ Ò û ˝

‡

ú ©

ø _ ¯ B b U à

SAS version 8.2

5

$ l , ñ ª W _ Ò û ˝

,

1 J

O’Brien (1984)

T | í -  j ¶ V Þ A _ Ò’ e

:

Y

∗

ijk

= c

f

[c

0

Z

i0k

+ (1 − c

20

)

1/2

Z

ijk

]

(13)

w 2

k=1,. . . ,n ; j=1 to J (

H

[ T Ü ‚ È 1 Ó O . ° í > Œ q l Z ‰

); i=1,2 ;

f=T,R ;c

f

¸

c

0

u  b

,

7

Z

i0k

¸

Z

ijk

u Ö

í  G } º
Ê t 

(13)

2

,c

0

¸

Z

i0k

ª − „ §t 6 È í ‰ æ

(1 − c

0

)

1/2

¸

Z

ijk

ª − „ §

t 6 q í ‰ æ

,

7 / .

° í

c

f

M }

. ° í u

‰ æ b ä ³ ! Z
w 2

c

0

£

c

f

í M ª

c k [

3

_ Ò

¬ ˙

,

ª } Ñ -Þ ú _ ¥

:

¥

1:

² Ï ã ì í q l

,

T Ü ì  ^ @

,

u

‰ j ä ³ £ š … b Þ A Ÿ á



’ e
l 

MVUE,MLE

¸ w ú @

θ

í

90%

] ˝ –

È

¥

2:

Ê ó

 ä Ì Ñ

(0.8,1.25)

-

,

 Œ

MVUE

D

MLE90%

5 ] ˝ –

È u ´

ê r r Ê ó

 ä Ì q V Ç ,

ABE

[

3

_ Ò 2 . ° í ¡ b M

u

‰ æ b ä ³ ! Z

, c

0

=

√

0.5

1

2

3

4

5

6

7

8

9

c

T

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

c

R

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

u

‰ æ b ä ³ ! Z

, c

0

=

√

0.5

10

11

12

13

14

15

16

17

18

c

T

2.0

√

0.5

√

2

√

0.5

√

0.2

√

0.2

√

0.5

√

0.5

√

0.2

c

R

2.0

√

0.5

√

2

1.0

√

0.5

√

0.7

√

2

2.0

1.0

ì ^ @

τ

π

µ

1

-0.2231

0

0

2

-0.1054

0

0

3

0

0

0

4

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0

0

5

0.2231

0

0

D

MLE

D v ¯ í

θ

5  Ì Ï æ
 Ì Ï Ï Ì j Ñ

5000

’ e

MVUE

D

MLE

D v ¯ í

θ

5 Ï í  j ¸Î J

5000

] ˝ –È 5 ¨ Ö 0 Ñ

5000

’elF)

5000

_

MVUE

D

MLE

]˝–ȨŽv ¯5

θ

íªW

% ð 

I

Ï Ï

C  ì ‰ Ñ

5000

’ e l  F )

5000

_

MVUE

D

MLE

í

90%

] ˝ –

È , - Ì ê r ¨ Ö Ê

(0.8,1.25)

q 5 ª W

Ä _ Ò

¯ b ¬ k ó ×

,

ú k ©

 q l

,

T Ü ì  ^ @ £ š … b

,

B b c T

X s _ u

‰ j ä ³ í _ Ò ! ‹

,

ø _ ¯ ¯

Compound Symmetry,

Ç

ø _ . ¯ ¯

Compound Symmetry

[

4

×Û¥< ¯-

MVUE

D

MLE

íÌRÏ ÌÏÏÌj£s,l

¾í^4í!‹

MVUE

D

MLE

5^4Ñ

MLE

5ÌÏÏÌj

D

MVUE

5  Ì

‰ j í ª
w F u ‰ j ä ³ ¯ í ! ‹ D [

4

é N
â [

4

2

,

ê Û Ì

 Ê S  ¡ b ¯ -

, MLE

í  Ì R Ï ¸  Ì Ï Ï Ì j

· } ª

MVUE

×

7

/

MLE

¸

MVUE

í  ^ 4 M· } × k

1,

[ ý

MVUE

Ê · H

$ l ,

,

u ª

œ

[

4

s

 , l ¾ R Ï Ì j Ï Ï D  ^ 4 5 ª œ

(a) 2×2

> Œ

q l

u‰ j ä ³ θ š … b MVUE MLE MVUE MLE

Eff (T, m) (ú b ) (Ÿ á ) R Ï R Ï ‰ j Ì j Ï Ï 0.04 0.02 0.8 6 0.0005 0.0018 0.0021 0.0021 1.0047 8 0.0001 0.0011 0.0016 0.0016 1.0062 12 0.0002 0.0009 0.0011 0.0011 1.0000 0.04 18 0.0004 0.0008 0.0007 0.0007 1.0029 0.9 6 0.0008 0.0023 0.0027 0.0028 1.0052 8 −0.0009 0.0003 0.0020 0.0020 1.0100 12 0.0005 0.0013 0.0013 0.0013 1.0028 18 −0.0001 0.0004 0.0009 0.0009 1.0000 1 6 0.0012 0.0029 0.0033 0.0033 1.0052 8 −0.0003 0.0010 0.0025 0.0025 1.0040 12 −0.0001 0.0008 0.0017 0.0017 1.0000 18 −0.0002 0.0004 0.0011 0.0011 1.0092 1.11 6 0.0008 0.0026 0.0040 0.0041 1.0250 8 0.0002 0.0016 0.0031 0.0032 1.0024 12 −0.0005 0.0004 0.0020 0.0021 1.0265 18 0.0006 0.0012 0.0014 0.0014 1.0000 1.25 6 0.0002 0.0023 0.0054 0.0054 1.0041 8 0.0002 0.0018 0.0040 0.0040 1.0013 12 0.0001 0.0011 0.0026 0.0026 1.0038 18 −0.0006 0.0001 0.0017 0.0017 1.0012 0.50 √0.125 0.8 6 −0.0014 0.0253 0.0450 0.0484 1.0755 8 0.0004 0.0205 0.0316 0.0336 1.0620 12 0.0034 0.0168 0.0219 0.0229 1.0463 1.00 18 0.0025 0.0114 0.0145 0.0150 1.0305 0.9 6 0.0001 0.0305 0.0537 0.0582 1.0828 8 −0.0008 0.0219 0.0414 0.0439 1.0605 12 0.0011 0.0160 0.0265 0.0277 1.0420 18 0.0024 0.0124 0.0184 0.0189 1.0296 1 6 −0.0012 0.0320 0.0692 0.0748 1.0803 8 −0.0076 0.0172 0.0486 0.0511 1.0518 12 0.0014 0.0182 0.0328 0.0343 1.0444 18 0.0027 0.0138 0.0223 0.0230 1.0301 1.11 6 −0.0029 0.0339 0.0840 0.0905 1.0772 8 −0.0057 0.0222 0.0624 0.0659 1.0559 12 0.0012 0.0198 0.0404 0.0421 1.0421 18 0.0023 0.0145 0.0273 0.0281 1.0307 1.25 6 0.0088 0.0510 0.1096 0.1192 1.0875 8 −0.0052 0.0259 0.0801 0.0848 1.0586 12 −0.0022 0.0188 0.0531 0.0552 1.0406 18 0.0023 0.0161 0.0353 0.0363 1.0285

MVUE H [ | ü ‰ æ . R , l , MLE H [ | × – N , l , T Ñ MVUE;m Ñ MLE

Ê · H$ l 2

,

ª

œ

MLE

¸

MVUE

í R Ï ¸ Ï Ï Ì j

,

w

  ª â [

4

2

,

(b) 2×3

> Œ

q l

u‰ j ä ³ θ š … b MVUE MLE MVUE MLE

Eff (T, m) (ú b ) (Ÿ á ) R Ï R Ï ‰ j Ì j Ï Ï 1.00 0.50 0.50 0.8 6 0.0012 0.0265 0.0421 0.0454 1.0784 1.00 0.50 8 0.0010 0.0200 0.0314 0.0333 1.0605 1.00 12 0.0010 0.0136 0.0199 0.0207 1.0402 18 −0.0003 0.0080 0.0135 0.0138 1.0222 0.9 6 0.0027 0.0315 0.0533 0.0577 1.0826 8 0.0035 0.0250 0.0387 0.0411 1.0620 12 0.0021 0.0163 0.0268 0.0279 1.0410 18 0.0017 0.0112 0.0173 0.0177 1.0231 1 6 0.0054 0.0376 0.0662 0.0717 1.0831 8 0.0053 0.0289 0.0481 0.0511 1.0624 12 0.0009 0.0167 0.0318 0.0331 1.0409 18 0.0007 0.0112 0.0219 0.0225 1.0274 1.11 6 0.0071 0.0429 0.0817 0.0888 1.0869 8 −0.0011 0.0252 0.0593 0.0627 1.0573 12 −0.0002 0.0172 0.0396 0.0410 1.0354 18 0.0041 0.0157 0.0268 0.0276 1.0299 1.25 6 −0.0008 0.0389 0.0985 0.1062 1.0782 8 −0.0016 0.0281 0.0739 0.0781 1.0568 12 −0.0024 0.0173 0.0484 0.0502 1.0372 18 0.0031 0.0162 0.0342 0.0352 1.0292 0.50 √0.125√0.125 0.8 6 0.0006 0.0201 0.0337 0.0357 1.0594 1.00 0.50 8 0.0017 0.0164 0.0248 0.0260 1.0475 1.00 12 0.0031 0.0129 0.0170 0.0175 1.0327 18 −0.0019 0.0046 0.0110 0.0112 1.0182 0.9 6 0.0045 0.0269 0.0403 0.0430 1.0668 8 0.0045 0.0212 0.0323 0.0340 1.0500 12 −0.0017 0.0093 0.0207 0.0213 1.0274 18 −0.0011 0.0062 0.0133 0.0135 1.0192 1 6 −0.0026 0.0220 0.0499 0.0527 1.0580 8 0.0016 0.0201 0.0382 0.0400 1.0472 12 −0.0018 0.0103 0.0262 0.0269 1.0284 18 0.0017 0.0099 0.0166 0.0170 1.0219 1.11 6 0.0060 0.0334 0.0656 0.0699 1.0658 8 0.0015 0.0222 0.0467 0.0490 1.0485 12 0.0027 0.0163 0.0326 0.0336 1.0330 18 −0.0019 0.0072 0.0203 0.0212 1.0441 1.25 6 −0.0018 0.0286 0.0777 0.0823 1.0594 8 −0.0027 0.0203 0.0587 0.0613 1.0427 12 −0.0033 0.0120 0.0391 0.0402 1.0275 18 −0.0019 0.0083 0.0274 0.0279 1.0183

MVUE H [ | ü ‰ æ . R , l , MLE H [ | × – N , l , T Ñ MVUE; m Ñ MLE;Cs H [

Ê ú˚ ä ³ í ‘ K - F d í R û j ¶

1. MLE

í  Ì R Ï }Ó O , ‰ æ Ó ‹ 7 ‰ ×
7

MVUE

† Ì ¤ Û ï

(c) 2×4

> Œq l

u‰ j ä ³ θ š … b MVUE MLE MVUE MLE

Eff (T, m) (ú b ) (Ÿ á ) R Ï R Ï ‰ j Ì j Ï Ï 1.00 0.50 0.50 0.50 0.8 6 0.0004 0.0197 0.0289 0.0306 1.0600 1.00 0.50 0.50 8 0.0006 0.0149 0.0220 0.0231 1.0465 1.00 0.50 12 0.0005 0.0099 0.0147 0.0151 1.0296 1.00 18 −0.0019 0.0043 0.0099 0.0100 1.0174 0.9 6 −0.0037 0.0179 0.0370 0.0391 1.0560 8 0.0004 0.0164 0.0284 0.0297 1.0445 12 −0.0030 0.0076 0.0186 0.0191 1.0268 18 0.0002 0.0072 0.0127 0.0129 1.0197 1 6 0.0032 0.0271 0.0470 0.0498 1.0607 8 −0.0010 0.0167 0.0348 0.0363 1.0427 12 −0.0001 0.0116 0.0225 0.0232 1.0297 18 0.0013 0.0090 0.0155 0.0158 1.0205 1.11 6 0.0035 0.0302 0.0594 0.0631 1.0634 8 0.0026 0.0224 0.0445 0.0465 1.0451 12 −0.0012 0.0118 0.0288 0.0296 1.0281 18 −0.0003 0.0084 0.0188 0.0191 1.0192 1.25 6 0.0061 0.0363 0.0778 0.0827 1.0632 8 −0.0001 0.0221 0.0517 0.0540 1.0453 12 −0.0060 0.0086 0.0359 0.0368 1.0243 18 −0.0014 0.0083 0.0235 0.0239 1.0182 0.50 √0.125√0.125 0.25 0.8 6 0.0058 0.0216 0.0250 0.0263 1.0537 1.00 0.50 √0.125 8 0.0012 0.0128 0.0196 0.0203 1.0359 1.00 √0.125 12 −0.0011 0.0066 0.0126 0.0128 1.0223 0.50 18 0.0006 0.0057 0.0082 0.0084 1.0169 0.9 6 0.0028 0.0202 0.0316 0.0332 1.0505 8 0.0013 0.0146 0.0232 0.0241 1.0383 12 −0.0001 0.0086 0.0164 0.0168 1.0239 18 0.0017 0.0075 0.0103 0.0105 1.0181 1 6 0.0015 0.0209 0.0392 0.0412 1.0500 8 0.0033 0.0180 0.0288 0.0299 1.0394 12 −0.0002 0.0095 0.0193 0.0198 1.0235 18 0.0016 0.0081 0.0129 0.0132 1.0174 1.11 6 −0.0011 0.0205 0.0493 0.0515 1.0456 8 0.0004 0.0167 0.0358 0.0370 1.0356 12 −0.0013 0.0094 0.0246 0.0252 1.0224 18 −0.0009 0.0062 0.0157 0.0159 1.0152 1.25 6 0.0025 0.0269 0.0621 0.0653 1.0513 8 −0.0001 0.0181 0.0469 0.0486 1.0362 12 −0.0018 0.0103 0.0303 0.0309 1.0224 18 −0.0036 0.0044 0.0210 0.0213 1.0123

MVUE H [ | ü ‰ æ . R , l , MLE H [ | × – N , l , T Ñ MVUE;m Ñ MLE ;Cs H [

Ê ú

˚ ä ³ í ‘ K - F d í R û j ¶

3. MLE

í  Ì R Ï }Ó O š … b Ó ‹ 7 Á ü

5. MLE

¸

MVUE

í  Ì Ï Ï Ì j }

Ó O , ‰ æ Ó ‹ 7 ‰ ×

6. MLE

¸

MVUE

í  Ì Ï Ï Ì j }

Ó O š … b ‰ × 7 ‰ ü

7. MLE

¸

MVUE

í  Ì Ï Ï Ì j }

Ó O

δ

í Ó ‹

7 ‰ ×

Ç

1

× Û

Ê

_2×2

> Œq l -

,

ç

θ=1.00,σ

2_e

=2.00

£ © å  š … b Ñ

6

í

¯

-_ Ò

5000

Ÿ 5

MLE

D

MVUE

5 ò j Ç

MLE

D

MVUE

í ò j Ç é ý s _ ,

l M } Ó Ì Ñ ¬ R

(Skewed to the right)

Ê ¤

ø ¯ -

5000

_

MVUE

5

 X

 Ì M Ñ

1.00

7 ™ Ä Ï Ñ

0.55,

7

MLE

í X  Ì M º Ñ

1.14,

ò ,

θ

í

¾ Ñ

0.14

7

MLE

5 ™ Ä Ï Ñ

0.62

? × k

0.55

Ç 1 Ê 2×2 >Œql2, s,líòjÇ5ªœ

(½ µ 5000 Ÿ, š … b Ñ 6,δ = 1.00,σ

2 e

= 2.00)

[

5

× Û

MVUE

D

MLE

í

% ð 

I

Ï Ï

% ð  ì ‰ ¸ ] ˝ – È ¨ Ö 0

í

! ‹

Ê - Ï H B b J

MLE

H

[ â

MLE

û |

(9)

 5

δ

í

90%

] ˝ –

È

,

J

MVUE (z)

£

MVUE (t)

}  H[ â

MVUE

£  G } 0 £ 2 -

t

} 0 û |

(12)

 5

δ

í

90%

] ˝ –È

,

â

[

5

ª ø

MLE,MVUE (t)

D

MVUE (z)

í ¨ Ö œ 0 Ì

Ê

87%

J ,

,

O

MVUE (z)

5 ¨ Ö œ 0œ

MLE

D

MVUE (t)

í ¨ Ö œ 0 Ñ Q

,

Ê

u

‰ j ä ³ . ¯ ¯

Compound Symmetry

c

q -

, MVUE (t)

5 ¨ Ö œ 0œ

MLE

Ì Q



1-2%

Êò ¼ > Œ q l - D . ¯ ¯

Compound Symmetry

c

q -

,

ú  j

[

5

s

 , l ¾ 

I

Ï ÏD  ì ‰ 5 ª œ

(%)

(a) 2×2

> Œ

q l

u‰ j ä ³ θ š … b  I Ï Ï D  ì ‰ ¨ Ö œ 0

(ú b ) (Ÿ á ) MLE MVUE (t) MVUE (z) MLE MVUE(t) MVUE(z) 0.04 0.02 0.8 6 5.04 4.24 5.78 90.54 90.44 86.88 8 5.26 4.48 5.42 89.52 89.36 87.28 12 5.26 4.50 5.26 89.62 89.68 88.36 0.04 18 5.22 4.82 5.06 90.16 90.02 89.26 0.9 6 60.82 56.16 62.54 89.60 89.40 86.50 8 71.56 67.28 71.88 89.84 89.96 87.60 12 88.50 86.72 88.28 90.32 90.22 88.76 18 96.34 95.78 96.20 90.02 90.06 89.14 1 6 94.62 94.38 96.26 89.74 89.72 87.16 8 98.98 98.90 99.28 90.14 90.54 87.94 12 99.94 99.96 99.96 89.58 89.70 88.02 18 100.00 100.00 100.00 90.22 90.36 89.50 1.11 6 59.34 63.58 69.06 90.50 90.08 87.28 8 72.54 75.78 78.64 89.52 89.36 87.28 12 87.46 88.98 90.48 90.32 90.26 88.76 18 96.42 97.00 97.24 90.54 90.38 89.42 1.25 6 5.00 5.88 7.66 89.72 89.60 86.44 8 4.88 5.80 7.06 89.74 89.68 87.30 12 5.14 5.74 6.60 89.76 90.02 88.32 18 4.80 5.48 5.92 90.36 90.26 89.50 0.50 √0.125 0.8 6 0.00 0.02 0.04 89.56 87.80 84.42 8 0.06 0.10 0.16 90.20 89.20 87.12 12 0.10 0.08 0.14 89.72 89.22 87.86 1.00 18 0.58 0.46 0.66 90.24 89.58 88.72 0.9 6 0.06 0.06 0.12 90.22 88.42 85.48 8 0.08 0.06 0.16 90.32 88.54 86.62 12 0.10 0.02 0.14 89.92 89.38 87.90 18 1.18 1.12 1.64 89.72 89.18 88.10 1 6 0.08 0.02 0.10 89.88 87.38 84.32 8 0.02 0.06 0.12 89.88 88.74 86.78 12 0.20 0.12 0.20 90.76 89.14 87.76 18 1.68 1.26 1.88 90.34 89.54 88.66 1.11 6 0.06 0.04 0.10 90.14 87.62 84.22 8 0.02 0.02 0.10 90.08 87.92 85.96 12 0.18 0.14 0.26 90.34 89.54 88.20 18 1.34 1.34 1.84 90.14 89.80 88.78 1.25 6 0.04 0.04 0.10 90.52 88.42 85.10 8 0.02 0.02 0.06 90.12 87.92 85.80 12 0.06 0.08 0.08 90.02 88.68 87.52 18 0.62 0.66 1.00 89.98 89.32 88.40

MVUE (t) H [ MVUE í } º Ñ t } º;MVUE (z) H [ MVUE í } º Ñ  G } º

 j ¶ í ¨ Ö œ 0 N ˛ . § š … b 5 à

(b) 2×3

> Œ

q l

u‰ j ä ³ θ š … b  I Ï Ï D  ì ‰ ¨ Ö œ 0

(ú b ) (Ÿ á ) MLE MVUE (t) MVUE (z) MLE MVUE(t) MVUE(z) 0.04 0.02 0.02 0.8 6 4.28 3.42 4.14 90.66 90.46 88.86 0.04 0.02 8 5.14 4.22 4.78 90.22 90.38 89.22 0.04 12 5.32 4.58 4.86 89.52 89.64 88.86 18 4.94 4.40 4.64 89.40 89.46 88.92 0.9 6 74.08 70.42 73.30 89.60 89.52 88.02 8 84.36 82.30 83.66 89.78 89.54 88.50 12 94.88 94.24 94.60 89.94 90.00 89.30 18 99.16 99.10 99.12 89.92 90.04 89.58 1 6 99.22 99.16 99.38 90.32 90.06 88.56 8 99.96 99.90 99.96 89.66 89.56 88.40 12 99.98 99.98 99.98 90.04 89.92 89.28 18 100.00 100.00 100.00 89.98 89.90 89.38 1.11 6 73.60 76.54 78.80 90.84 90.70 89.04 8 83.56 85.54 86.74 89.06 89.16 87.68 12 94.32 95.26 95.50 89.52 89.48 88.62 18 99.10 99.28 99.34 90.24 89.78 89.44 1.25 6 4.80 5.90 6.66 90.24 89.98 88.54 8 5.02 5.80 6.26 90.04 90.20 89.06 12 4.98 5.62 5.94 90.60 90.66 89.96 18 5.34 5.86 6.14 89.54 89.48 89.04 0.50 √0.125√0.125 0.8 6 0.02 0.00 0.02 89.56 88.32 86.90 1.00 0.05 8 0.04 0.00 0.00 89.24 87.82 87.00 1.00 12 0.20 0.12 0.22 88.14 88.18 87.46 18 1.98 1.22 1.46 89.04 88.22 87.72 0.9 6 0.00 0.00 0.00 89.80 89.22 87.62 8 0.02 0.02 0.02 89.22 88.58 87.56 12 0.64 0.28 0.42 89.62 88.52 87.86 18 6.30 4.90 5.78 90.24 89.58 89.00 1 6 0.02 0.02 0.06 89.60 87.34 85.84 8 0.04 0.02 0.08 89.60 87.90 86.80 12 0.66 0.44 0.70 88.98 88.32 87.50 18 9.50 8.08 9.28 90.08 89.52 89.02 1.11 6 0.04 0.04 0.06 89.24 87.90 86.28 8 0.02 0.02 0.04 89.40 88.60 87.30 12 0.46 0.44 0.62 88.58 88.60 87.82 18 6.12 6.98 7.80 89.48 88.72 88.20 1.25 6 0.00 0.00 0.00 89.20 87.68 86.18 8 0.00 0.00 0.02 89.68 88.18 87.16 12 0.16 0.14 0.18 89.96 88.46 88.08 18 2.34 2.70 2.98 88.78 87.98 87.60

MVUE (t) H [ MVUE í } º Ñ t } º;MVUE (z) H [ MVUE í } º Ñ  G } º

¶ í

% ð 

I

Ï ÏÑ

ò
7 / Ê

_2×2

> Œq l - Ì ò k

5%,

7

MLE

D

MVUE

(t)

í

% ð 

I

Ï Ï

Ì ª − „ Ê

5%

˝ ¬
Ê

_2×4

> Œq l

MVUE (t)

D

MVUE

(c) 2×4

> Œq l

u‰ j ä ³ θ š … b  I Ï Ï D  ì ‰ ¨ Ö œ 0

(ú b ) (Ÿ á ) MLE MVUE (t) MVUE (z) MLE MVUE(t) MVUE(z) 0.04 0.02 0.02 0.02 0.8 6 4.24 3.42 3.82 91.26 91.48 90.64 0.04 0.02 0.02 8 4.74 3.70 4.06 90.36 91.02 90.32 0.04 0.02 12 4.30 3.78 3.96 90.92 90.96 90.66 0.04 18 4.34 3.86 4.00 91.26 91.42 91.06 0.9 6 84.30 81.42 82.82 89.94 90.30 89.08 8 92.78 91.44 92.02 91.10 91.42 90.72 12 98.44 98.18 98.22 90.32 90.48 89.92 18 99.92 99.88 99.90 90.34 90.54 90.14 1 6 99.98 99.96 99.96 89.84 90.10 89.06 8 100.00 100.00 100.00 89.74 90.30 89.58 12 100.00 100.00 100.00 88.90 89.14 88.66 18 100.00 100.00 100.00 90.58 90.96 90.58 1.11 6 85.66 86.68 87.76 91.00 91.18 90.30 8 93.00 93.44 93.84 90.50 90.64 89.88 12 98.54 98.80 98.90 90.10 90.52 90.08 18 99.96 99.96 99.96 90.78 91.04 90.74 1.25 6 4.16 4.46 4.94 91.64 92.26 91.34 8 4.20 4.74 5.12 92.10 92.28 91.56 12 4.44 4.94 5.24 90.94 90.96 90.44 18 5.36 5.72 5.80 90.00 90.30 90.04 0.50 √0.125√0.125 0.25 0.8 6 0.02 0.00 0.00 88.04 87.60 86.30 1.00 0.05 √0.125 8 0.08 0.02 0.02 89.22 88.16 87.46 1.00 √0.125 12 1.28 0.90 1.08 88.70 88.38 88.00 0.50 18 4.36 2.70 2.82 88.68 88.20 87.90 0.9 6 0.02 0.02 0.06 89.70 89.36 88.42 8 0.24 0.18 0.22 88.78 88.52 87.90 12 3.94 2.80 3.14 89.72 88.58 88.14 18 19.80 15.62 16.34 87.34 87.38 87.14 1 6 0.00 0.00 0.02 88.74 87.94 86.84 8 0.32 0.06 0.18 89.32 88.38 87.62 12 4.56 3.62 4.50 89.12 88.28 87.80 18 29.64 28.14 29.22 89.26 88.42 88.10 1.11 6 0.00 0.02 0.04 89.22 88.06 87.00 8 0.26 0.12 0.16 89.04 88.18 87.24 12 3.80 3.58 4.36 88.94 88.40 87.84 18 20.10 21.70 2.60 88.32 88.10 87.88 1.25 6 0.00 0.00 0.00 89.02 88.10 86.92 8 0.08 0.08 0.12 89.04 88.16 87.36 12 1.08 1.22 1.38 89.56 89.54 89.06 18 4.88 6.58 6.82 88.94 88.32 88.00

MVUE (t) H [ MVUE í } º Ñ t } º;MVUE (z) H [ MVUE í } º Ñ  G } º

Ì

œ \ è

,

7 š … b ý v % ð 

I

Ï Ï

œ š … b × v Ñ Q

,

é ý

ú  j ¶ Ê š

… b ü v Ì

œ \ è . q x "  ø  Ï Ï

,

O Ê

δ=1.25

v

D

_2×2

> Œq l -

,

ç

12

v

,

Ê

5%

é

O ® Ä -

,MVUE (z)

5

% ð 

I

Ï Ï

Ì × k

6%

é ý

MVUE (z)

Ê

_2×2

C

_2×3

> Œ

q l - š … b ü v £ ç

θ

Q

¡ k ó  , Ì v

, MVUE(z)

œ

ñ q x " 

I

˜ Ï

(

¹

ç ± “

D Ÿ  “ . Ñ  Ì Þ • ó 

,

O

\ Ž ˚ Ñ  Ì Þ

•

ó

 5 ˜ Ï

)

O

MLE

D

MVUE (t)

Ê

ú  . ° å  í > Œ q l - w % ð 

I

Ï Ï

Ì ª − „ Ê

5%

ú k u‰ j ä ³ . ¯ ¯

Compound Symmetry

c

q v

,

Ä ¤ ° vb n j ‰ æ

× ü Ê

ø O Þ ñ ó  4 š … b - ú % ð 

I

Ï Ï

D  ì ‰ 5 à
B b Ê [

5

2 × Û ×‰ æ ¯ í ! ‹
â [

5

ª øÎ 7 Ê

_2×4

> Œ

q l D © å  š … b

Ñ

18

A í

¯ Õ

,

ú  j ¶ í % ð 

I

Ï Ï

Ì Ê

0.00%

B

3.00%

5

È

,

]

‰ æ

× v

D . ¯ ¯

Compound Symmetry

c

q v

,

ú  j ¶ Ì œ \ è 7 . q Ž µ 

Ì Þ • ó

 4

[

5

2 T X ç

θ

M Ñ

0.9,1

D

1.11

- í ®

 > Πq l

,

š … b

D u ‰ j ä ³

í

% ð  ì ‰

,

ø O 7 k  ì ‰ Ó š … b í Ó ‹ 7 Ó ‹
Ê ó ° ¯ -

,

ú 

j ¶ í  ì ‰ í Ï æ Ì Ê

5%

5 q

MVUE (z)

5

% ð  ì ‰ ò k w F s  j

¶

,

Ê

θ=1

v

,MVUE (t)

D

MLE

í

% ð  ì ‰ Ì ' Q ¡

,

Ê

θ=0.9

v

, MVUE (t)

5

% ð  ì ‰ ü k

MLE

í

% ð  ì ‰  Ê

4%

5 q

,

O Ê

θ=1.1

v

,MVUE (t)

5

% ð  ì ‰ × k

MLE

í

% ð  ì ‰  Ê

4%

5 q
Ç

2

Ñ Ê

_2×2

> Œ

q l

,

© å  Ñ

12

P §

t 6 ú b

 5 u

‰ b ä ³ Ñ

"

0.04, 0.02

0.04

#

-

MVUE (t)

D

MLE

5% ð  ì ‰  (
Ç

2

é ý

ù  j ¶ % ð  ì ‰  ( I Ñ ¬ R

,

Ê

θ

<₌

₁

v

ù  j ¶ % ð  ì ‰  (  ˛ ½ L

,

O Ê

θ >1

v

MVUE (t)

í

% ð  ì ‰



( ò k

MLE

5

% ð  ì ‰  (
7 Ê ó ° ¡ b D š … b ¯ -

,

ò ¼ > Œ

q l 5 % ð  ì ‰ ò k

_2×2

> Œ

q l í % ð  ì ‰

[

5

2

ú  > Œ q l - u ‰ j ä ³ . ¯ ¯

Compound Symmetry

v

,

Ä w

‰

æ

œ ×

,

] Ê . °

θ

M

- 5% ð  ì ‰ Ì ” Q
Ê

δ=1

v

[

5

2

ú  > Œ q l

. ¯ ¯

Compound Symmetry

u

‰ j ä ³ 5 ¯ -

,

k

® ƒ

80%

 ì ‰ F

Û 5

š … b }  Ñ

_:2×2

> Œ

q l © å 

70

A

_{; 2×3}

> Œ

q l © å 

48

A

_;2×4

>

Œ

q l © å 

42

A
B b ? Ê , H

® ƒ

80%

 ì ‰ í š … b - Ï

W _ Ò û

˝

,

! ‹ é ý ú  j ¶ 5 % ð  ì ‰ Ì Ê

80%

˝ ¬

7 w % ð  ì ‰ í Ï æ Ì

Ê

3%

5 q

Ç

2

| × – N ,  ¸ | ü

‰ j . R , l  í  ì ‰  (

(Power Curve)

4.

b W

/

ø ç ± “  à [

1

2

_2×3

> Œ

q l í Þ • ó  4 t ð ª W Ç , v  F

Þ ß í ç ± “

(T)

D Ÿ  “

(R)

5  Ì ó 4
¤ t ð ø u Ó œ N »

48

P U

ì

A è § t 6

(healthy normal volunteers)

B s å 

,

© å 

24

±

,

Ó œ N » B

 ø å  § t 6 Q § T Ü “ ¹ í ß å Ñ

TRR;

 ù å  † Ñ

RTT

[

6

T X

§

t 6 í ú b

AUC

5 b W Ê

(1)

 - í

‰ j } & [

(analysis of variance table),

â

[

6

2 §

t 6 È ‰ æ 5

F

M Ñ

4.069

D

P-value

Ñ

_5.23084×10

−9

_,

é ý …

t

ð ú b

AUC

5 §

t 6 È ‰ æ × k § t 6 q ‰ æ

,

] @ S à > Œq l ª W T Ü

ª

œ ¢ { ì ^ @ 5

F

M Ñ

1.438

w

P-value

Ñ

0.2335873,

é ý …

t ð Ê Ï W j

Þ
} Ã ã J _ Ì { ì ^ @

,

| (T Ü ^ @ í

F

M

ü k

1,

] ç ± “

D Ÿ  “

Ê

AUC

, Ì é

O Ï æ

,

O Ì é

O Ï æ . H [ ç ± “ D Ÿ  “ í

AUC

Ñ  Ì

Þ • ó


[

7

T X U à

MLE

£

MVUE (t)

Ç

, Þ • ó 4 í ! ‹

θ

5

MLE

D

MVUE

}  Ñ

0.988

£

0.987

MLE

I Ñ

ò ,
7

θ

5

MLE

D

MVUE (t) 90%

] ˝ –

È }  Ñ

(0.911,1.072)

D

(0.907,1.067)

MVUE (t)

j ¶

90%

] ˝ –

È í

O®Ä-

MVUE (t)

D

MLE

sj¶ÌªŽ˚籓DŸ“ÑÞÓó4

[

6

‰ j b } & [

‰ æ V Ä

 j ¸

A â 

Ì j

F

P-value

Inter-subject

15.2236549

47

0.3239076

Sequence

0.8539214

1

0.8539214

Residual

14.3697335

46

0.3123855

4.0689447

_5.23084×10

−9

Intra-subject

7.1855060

96

0.0748490

Period

0.0119008

2

0.0059504

Formulation

0.000105623

1

0.000105623

0.0013758

0.9704922

Carry-over

0.1103785

1

0.1103785

1.4377236

0.2335873

Residual

7.0631211

92

0.0767731

Total

22.4091609

143

[

7

ú b

 ( - Þ í b W } &

,

l ¾

90%

] ˝ –

È

MLE

MVUE

MLE

MVUE(t)

0.988

0.987

(0.911,1.072)

(0.907,1.067)

5.

n  D  ‡

Ê Ç ,  Ì Þ • ó

 4 2

,

â k | × – N ,

l ¾ ˛  ˜ Ë \ 1 Å

FDA,

«

É

D 0 ä “ \ À P F Q § @ à

,

Ö Í

MLE

Ê ú b

ž ² - í } & u ¡ b

τ

í .

R ,

¾

,

ª u

ø ï ¥ ž  Ÿ á

 (

,MLE

º u

ò ,

θ,

k u … d5 ? $ l , í

. R 4

,

S à | ü

‰ æ . R , l ¾ V Ç ,  Ì Þ • ó  4

,

1 Ï

W _ Ò û ˝

,

ø

MVUE

D

MLE

5

% ð 

I

Ï Ï

¸

% ð  ì ‰ ª W ª œ
_ Ò ! ‹ é ý Ö Í

MVUE

í } 0 u ¬ R

,

O U à

MVUE (t)

j ¶ 5

¡ N

_{(1 − 2α)%}

] ˝ –

È í ¨ Ö

œ 0 . O ª

® ƒ

_{(1 − 2α)%}

í ¨ Ö œ 0

7 / D

MLE

í ¨ Ö œ 0

Ý  Q ¡

,

Ç

Õ

,

U à

MVUE (t)

j ¶ Ç ,  Ì Þ • ó

 4 v

,

w

% ð 

I

Ï Ï

D % ð  ì

‰ Ì

D

MLE

ó Ï .±

,

é ý

MVUE (t)

j ¶ . O ª − „ 

I

Ï Ï

7 / w  ì ‰

œ

MLE

F T X í  ì ‰ . ó , -

Ä ú bž ² Ñ Ý ( 4 ž ²

,

F J

MLE

Ê ,



θ

v }

ò ,
ç š … b ü C ‰

æ × v

,

w

ò , í ˙  u ó ç ª h í
Ç Õ

MVUE (t)

Ç

, Þ • ó

 4 w 

I

Ï Ï

D  ì ‰ D

MLE

é N

,

] B b 

‡ Î 7

MLE

j ¶ Õ

,

Þ • ó

 4 ’ e ?

ª S à

MVUE (t)

j ¶

ª W } &

_ á È

:

… û

˝ [ â Å  } l å

(NSC 92-2118-M-006-001)

T X

¶ M ı Œ

¡ 5 d .

Bradu, D. and Mundlak, Y (1970). Estimation in lognormal linear models. Journal

of the American Statistical Association 65, 198-211.

Chow, S.C and Liu, J.P. (2000). Design and Analysis of Bioavailability and

Bioequiv-alence Studies

2nd edtion, Marcel Dekker, New York.

Liu, J.P. and Weng, C.S (1992). Estimation of direct formulation effect under

log-normal distribution in bioavailability/bioequivalence studies. Statistics in Medicine

11, 881-896.

O’Brien, P.C. (1984). Procedure for comparing samples with multiple endpoints.

Biometrics 40, 1079-1087.

U.S. FDA. Guidance for industry on bioavailability and bioequivalence studies for

orally administered drug products - general considerations. Center for Drug

Eval-uation and Research, Food and Drug Administration, Rockville, Maryland, 2003.

COMPARISON OF MLE AND MVUE FOR

EVALUATION OF AVERAGE BIOEQUIVALENCE

Jen-Pei Liu

1

and Fu-Min Xu

2

1

_{Division of Biometry, Department of Agronomy,}

National Taiwan University and

2

_{Department of Statistics, National Chen-Kung University}

ABSTRACT

The research and development of an innovative drug product takes 10-12 years and

800 millions to 1 billion US dollars. Therefore, it is a costly, time-consuming, and

highly risky endeavor. One way to reduce the drug cost is to introduce generic drugs

after the patent of the innovative drugs expires. Currently, most regulatory agencies in

the world only require evidence of average bioequivalence from in vivo bioequivalence

trials to approve the generic drugs. Currently, most of heath regulatory agencies in

the world such as the U.S. Food and Drug Administration and Taiwan Department of

Heath use maximum likelihood estimator (MLE) for evaluation of average

bioequiva-lence. Since MLE is not an unbiased estimator for the relative bioavailability, therefore,

we propose the minimum variance unbiased estimator (MVUE) to assess the average

bioequivalence. We performed a simulation study to compare the bias, mean square

er-ror, empirical size, empirical power and probability coverage between MLE and MVUE

on the various combinations of parameters and sample size under 2×2 crossover design

and higher-order crossover design. The simulation results showed that in addition to

unbiasedness, the size and power of the MVUE is comparable to those of MLE. A

numerical example illustrates the procedure of the MVUE for evaluation of average

bioequivalence.

Key words and phrases:

Average bioequivalence, simulation study, empirical size,

empirical power, probability coverage.