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Title A STUDY EFFECTIVENESS OF THE METHOD OF PARAMITER ESTIMATIONS IN MULTILEVEL ANALYSIS FOR SMALL SAMPLE SIZES Author Montri Songthong Capital of research Budget of 2014 Keywords Multilevel analysis, Hierarchical Linear Models

ABSTRACT

This research is aimed to study the effectiveness of the method of parameter estimations in multilevel analysis for small sample sizes and also compare 3 methods of effectiveness of parameter estimation in multilevel analysis for small sample sizes. They were FML, RML and EB. This research study was modeled the simulations by using the Monte Carlo by the programming language R which consisted of the following simulation conditions: 1) the population distributions were left skewness and kurtosis lower than normal and right skewness and kurtosis higher than normal ; 2) 1 Independent variable each level; 3) Intraclass Correlation Coefficient were 0.01 and 0.20; and 4) sample sizes, 3 for each level which were 5, 15, and 25. Each situation was modeled by 1,000 series of information and compare statistic of effectiveness was One-way Multivariate Analysis of Variance (One-way MANOVA). The results were found :

FML and RML methods most effective in estimate of fixed effects and EB method is most effective in estimate of random effects, The test at significance level 0.01. for the FML with RML methods are effective in estimate of fixed effects and random effects, differences are statistically non-significant at the 0.01. for population distributions were left skewness and kurtosis lower than normal and right skewness and kurtosis higher than normal.

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-I!"J#$$กI,J*&-%& # M N (Hierarchical Linear Models) #$+ก- *'b%!กL**)%$m*ก!&) L &4++"!ก !".ก', *กI,&กJ*&ก*,+1ก$&ก!"ก!Gกกก(studies of growth) %ก (organizational effects) -%ก!+!& (research synthesis) 1* (Raudenbush and Bryk. 2002) #$+กก !", *กI,J*&-%& +, *ก!G *q L &rก -%!b$J*ก "`&#$-ก'*+Iก! ก$&ก!"m!"m*กI -%#$กI*J&ก+ก!GL-กI'+) !.$J*I!" !"L &r J L-ก HLM m$bJ*ก, *!.)-""J*-%,J*J* (Raudenbush et al. 2000) L-ก MIXOR (Hedeker and Gibbons. 1996) L-ก MLWIN (Rasbash et al. 2000) L-ก VARCL (Longford. 1988) -%L-ก BUGS (Spiegelhalter et al. 1994) 1* I!"L-กI'+)b!$,$I!$$J*ก !" # L-ก SAS Proc Mixed (Littell et al. 1996) -%L-ก SPSS m$!G.&!,กzกI,J*-%&ก!ก (Hox. 2002) 2. $กกก !" #$ 4++"!ก กb b&-" " !" 1$ )* +! ก& -%& !ก+!&&*ก&กJ#$$-กก! J L %! a-""(random coefficient models) (de Leeuw and Kreft. 1986 ; Longford. 1993) L %ก"-(variance component models) (Longford. 1987) (hierarchical linear model) (Raudenbush and Bryk. 2002) L % %-""( (mixed model) (Littell, Milliken, Stroup and Wolfinger. 1996) -%L % !" (Multilevel Model) (Snijders and Bosker. 1999 ; Heck and Thomas. 2000 ; Hox. 2002) )-""$ก%!. -*$+-%*ก'# ก*)%$L*JJ!.!$ I!"กI*+ก$&ก!"ก !"+#.`+กกกb b& (Regression Analysis) -%ก- (Analysis of Variance) L &4++"!.ก !"b*-"L**)%-"ก1 2 # ก !" ก 2 !" -%ก !" ก 3 !" L &&%& !.

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2.1 ก !" #$ ก% 2 $ ก !"+$*+กก+!! ก*กก

!- !" & ()*+!&+I&!- 1 !- # (%! aก&J -%!- # b`ก+!ก& m$+ 1 L& L &b&กb b&, * !.

iii rXY ++= 10 ββ L & iY # (%! aก&J 0β # + $*ก! ก!"-ก Y m$ก'# (%! ar%$& !ก&$ i #$ iX 1 0 1β # J! (Slope) 1$ "&b!-( iX ) #$%$&, 1 & +I* iY %$&, 1β & ir # % %#$(residual error) m$*ก%"#.**ก-+ก-+-""ก r%$&ก!" 0 -%-ก!" 2σ L &b&ก- !! (%! aก&Jก!"b`ก+, * ! 1

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1β

&" 1 ก' $"(! )*+(,ก-%.!ก /.01กก2 1 3

b*!"ก%!- +J&I* 0β (intercept) &&$.ก %"!-(b`ก+) *&r%$&b`ก+ -% : XiX − -%Iก&ก, * ! 2

0β

(%! aก&

b`ก+

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1β -2 -1 0 1 2

&" 2 ก' $"(! )*+(,ก-%.!ก /.01กก3$ก$)4/.01ก(SES )

ก$Iกกb b& 2 L& m$+กb b& 2 ก #$I

*b b&-%L&&ก,* &ก! ! 3 +I* 0β (intercept) # r%$&(%! aก&J&ก&$. #$J! (Slope) &!,%$&-%

0β

(%! aก&

b`ก+

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111011 )( iii rXXY +−+= ββ

212022 )( iii rXXY +−+= ββ 12β -2 -1 0 1 2

&" 3 ก' $"(! )*+(,ก-%.!ก /.01กก2 2 3

+ก 3 " L&$ 1 -%L&$ 2 $$-กก!&) 2 ' # 1) L&$ 1 +r%$&(%! aก&J)กL&$ 2 L &+, *

11β

02β

(%! aก&

b`ก+

01β

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+ก 0201 ββ > 2) b`ก++ %(%! aก&JL&$ 1 $IกL&$ 2 L &+, *+ก 1211 ββ < -b*Iกก!! (%! aก&Jก!"b`ก+Jก J L& L & j L& +, *กb b&!. j ก #

111011 )( iii rXXY +−+= ββ

212022 )( iii rXXY +−+= ββ . . .

ijijjij rXXY +−+= )(10 ββ L &+กก ijr +ก-+ก-+-""ก--%L& ก! ( ),0(~ 2σNrij ) b*++กก " -%L&+ j0β -% j1β L &ก !" !"$ 2 !. +I.,1!- m$+!- !"$ 2 1!&ก !"$ 2 !- 1 ! # !ก! L& - *& jW L &

*1 1 b*1L&!ก! %ก(catholic) -%1 0 #$1L&!ก! !` (public) m$b*J!1"ก- r%$&(%! aก&JL&!ก! %ก+)กL&!ก! !` -b*J!1%" - (%! aก&JL& %ก+$IกL&!ก! !` m$b&กb b& !"$ 2 , * !. jjj uW 001000 ++= γγβ jjj uW 111101 ++= γγβ #$ 00γ # r%$&(%! aก&JL&

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!ก! !`(public) 01γ # -ก(%! aก&J

L &r%$&L&!ก! %ก(catholic) -%L& !ก! !`(public)

10γ # r%$&b`ก+$(%! aก& JL&!ก! !`(public)

11γ # -กb`ก+$(%! aก &JL &r%$&L&!ก! %ก (catholic) -%L&!ก! !`(public)

ju0 # % %#$(residual error) # % j0β

ju1 # % %#$(residual error) # % j1β

!!. b&กก !" ก 2 !" , * !.

ijijjij rXY ++= 10 ββ

jjj uW 001000 ++= γγβ jjj uW 111101 ++= γγβ b*Iก !ก%&ก!, * !.

ijijjjjjij rXuWuWY ++++++= )( 1111000100 γγγγ

ijjjijjijijj ruuXWXXW ++++++= 0111100100 γγγγ

+กกb*,!-!. !"$ 1 -% !"$ 2 b&ก, * !.

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ijj000ij ruY ++γ= m$ก.&ก)-"",#$,&")# Null Model L &ก !"+$*+กก Null Model #$$+, *"-% !"-ก$+!-Iก&ก#, L &! -&J!. b+, *+ก! a!! &J!.(Intraclass correlation coefficient : ρ ) m$I+กก-&ก- ijY !.

)( ijYV = )( 000 ijj ruV ++γ

)()()( 000 ijj rVuVV ++= γ

22 ijroju

σσ +=

+กก-&ก- ijY +ก" *&- !"$ 1 -%

- !"$ 2 b- ! - !"$ 1 -% !"$ 2 , * #

2

r

2

u

2

r

1

ijoj

ij

σ+σ

σ=ρ

2

r

2

u

2

u

2

ijoj

oj

σ+σ

σ=ρ

I!"ก-"",#$,&") (Fully Unconditional Model) 1ก#$"(!-!. 2 !" # !"!ก& -% !"L& (!--% !"b "&, *L &! a!! &J!. -I!")-""ก&#$, (Conditional Model) +กI!-*,ก#$+! -$b "&, *L &!- !ก% L & !"$ 2 +I+ ! -ก

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(Intercept) -%J! (Slope) !"$ 1 ,1!- -%I!-*,)-""#$ "&! aกb b& !ก%

2.2 ก !" #$ ก% 3 $ ก !" ก 3 !"!. 1ก&&- ก$&ก!"ก !"*"%L**)%#$% ( % $ก +กก(%* !"##$* %*ก!"45+!& -ก !" ก 3 !"!. +m!"m*กก #$+กI+ ! -ก (intercept) -%J! (slope) !"$ 2 1!- !"$ 3 I!"!.ก+#ก!"ก 2 !" # !.-ก+)-"",#$,&") (Fully Unconditional Model) #$ )(!-- !" -%!.$ 2 +)-""#$, (Conditional Model) # กI!-*,)-"" L &&%& !. 8 1 9 :);:)9%! (Fully Unconditional Model) )-""ก 3 !"&& # ก-"",#$,&")#ก$&!,, *I!-*,)-"" m$&กก& ก Null Model *กก4++!&$(%(%! aก&L &J*ก !" ก 3 !" L & !"$ 1 # !ก& !"$ 2 # *& -% !"$ 3 # L& 9 $ก(Child-Level Model) m$1)-""(%! aก&!ก&-%L &ก" *&r%$&*&"กก!"% %#$-"" L &&1ก, * !. ijkjkijkY επ += 0 #$ ijkY # (%! aก&!ก&$ i *&$ j -%

&L&$ k jk0π # r%$&(%! aก&*& j &L& k ijkε # %!ก&#"$&"-!ก&

$ i *& j &L& k ก!"r%$&*&

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m$ก'#% %#$!$ L &*ก%"#.** ก-+ก-+-""ก r%$& ก!" 0 -%- ก!" 2σ

L & i # !ก&, j # *& -% k # L& i = 1,2,, jkn (!ก&*& j &L& k)

j = 1,2,, kJ ( *&&L& k) k = 1,2,,K (L&) 9 $ (Classroom-Level Model) m$+I r%$&-% *&1!- ( jk0π ) !"$ 2 L &&1ก, * !. jkkjk r0000 += βπ

#$ k00β # r%$&(%! aก&L& k

jkr0 # %*&#"$&"r%$& !" *&$"$&",+กr%$&L& m$ก' # % %#$!$ L &*ก%"#.**ก -+ก-+-""ก r%$& ก!" 0 -%-ก!"

πτ 9 $3(School-Level Model) m$)-"" !"$ 3 .1ก- (!-L& L &+Ir%$&-%L&( k00β ) 1!- !"$ 3 b&1ก, * !.

kk u0000000 += γβ

#$ 000γ # r%$&(%! aก&กL& (grand mean)

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ooku # % !"L&#"$&"r%$&-% L&$"$&",+กr%$&(grand mean) m$ก' # % %#$!$ L &*ก%"#.**ก -+ก-+-""ก r%$& ก!" 0 -%-ก!"

βτ ก ก (Variance Partitioning) ก-&ก- 3 !" # ก-"-!- )( ijkY ก1 3 # - !"$ 1 #-!ก&&*&

)( 2σ - !"$ 2 #-*&&L& )( πτ -%- !"$ 3 #-L& )( βτ #$I,! &*& *&L&!. -%L& &ก ! a!! &J!. (Intraclass correlation coefficient : ρ ) L &bI, * !.

βπ ττσ

σρ

++=

2

2

1 # ! -&*&

βπ

π

ττσ

τρ

++=

22 # ! -*&

&L&

βπ

β

ττσ

τρ

++=

23 # ! -L&

8 2 9 );: (Conditional Model)

ก-"",#$,&") (Fully Unconditional Model) 1ก#$"(!-!. 3 !" # !"!ก& !"*& -% !"L& L & (!--% !"b "&, *L &! a!! &J!. -I!")-""ก&#$, (Conditional Model) +กI!-*,ก#$+

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! -$b "&, *L &!- !ก% I!" !"$ 2 -% !"$ 3 +I+ ! -ก(Intercept) -%J! (Slope) ,1!--%I!-*,)-""#$ "&! aกb b& !ก% !!.b)-""&#$,-% !", * !. 9 :2$ 1 &-%*&!.4กJ!(%! aก&+ก" *& !- !"!ก&"กก!"% %#$ b&1 ก, * !. ijkpijkpjkijkjkijkjkjkijk aaaY εππππ +++++= ...22110 #$ ijkY # (%! aก&!ก&$ i *& j -%

&L& k jk0π # + ! -ก(Intercept) *&$ j &L& k pijka # !- !"$ 1 $J*ก "&(%! a

ก& L &!- p ! ; p = 1,2,,P pjkπ # J!(Slope) #! aกb b& !"$ 1

m$- b %!-(%! aก &!ก&*&$ j &L& k

ijkε # %-"" !"$ 1 #"$&"- !ก&$ i *&$ j &L&$ k $"$&", +ก-&ก m$ก'#% %#$!$ L & *ก%"#.**ก-+ก-+-""ก r%$& ก!" 0 -%-ก!" 2σ

9 :2$ 2 +I! aกb b&)-"" !"!ก& ( !"$ 1) ก" *&! -ก (Intercept) -%J! (Slope) ,1!- !"$ 2 # !"*& L &b&ก, * !.

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pjkqjk

pQ

Qpqkkppjk rX ++= ∑

=10 ββπ

#$ kp0β # ! -ก (Intercept) L& k #r%$&-

(%! a-%L&

qjkX # !- !"$ 2 $J*ก "& %

*& )( pjkπ L &!- q ! ; Q = 1,2,, pQ

pqkβ # J!(Slope) #! aกb b& !"$ 2 m$- b %!- % *& )( pjkπ

pjkr # %-"" !"$ 2 #"$&"! a *&$ j &L&$ k $"$&",+ก&ก )-"" !"*& m$ก'#% %#$!$ L &*ก%"#.**ก-+ก-+-""ก%&!- (Multivariate Normal) r%$& ก!" 0 -%- -%-ก!

9 :2$ 3 ก !" !"$ 3 +I! aกb b&)-"" !"*& ( !"$ 2) ก" *&! -ก (Intercept) -%J! (Slope) .,1!- !"$ 3 # !"L& L &b&ก, * !.

pqksk

pqs

spqspqpqk uW ++= ∑

=10 γγβ

#$ 0pqγ # ! -ก (Intercept) กL&#r%$&

-(%! aกL&(grand mean)

skW # !- !"$ 3 $J*ก "& %

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L& )( pqkβ L &!- s ! ; s = 1,2,, pqS

pqsγ # J!(Slope) #! aกb b& !"$ 3 m$- b %!- % L& )( pqkβ

pjku # %-"" !"$ 3 #"$&"! a

L&$ k $"$&",+ก&ก)-"" !" L& m$ก'#% %#$!$ L &*ก%"#.* *ก-+ก-+-""ก%&!- (Multivariate Normal) r%$& ก!" 0 -%--%- ก!

3. VW กกก !" #$ 3.1 VW กก(ก%)"!

ก! aกb b&ก !"4++"! # +1) (Maximum Likelihood: ML) (Hox. 2002) -ก' ก #$ก J กI%!*&$ -""!&!$,(Generalized Least Squares: GLS) -""ก!$,(Generalized Estimating Equations: GEE) -"" Bootstrapping -% ก-"""& (Bayesian methods) m$" ก1&กI- -&!,กI,+"#&กJ*L-กI'+) L &-% ก'* -%*+Iก! -กก!, L &&%& !.

1. +1) (Maximum Likelihood: ML) +1) 1 ก$กI,J*L-กI'+)ก$&ก!"ก !"ก$ !.#$+ก +1) "!ก1!$ # -ก(robust) (efficient) -%-"!&(consistent)-ก%!&* 5 +1) +-กกb)ก%%&*ก%"#.*ก$&ก!"ก-+ก-+% %#$ กL & .+J*4กJ!+1) (likelihood function) L & +1) $J*ก !" +-"ก1 2 # Full Maximum Likelihood (FML) m$ .+

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! aกb b&-%ก"-(variance components) *ก! L &J*4กJ!+1) (likelihood function) Restricted Maximum Likelihood (RML) +ก"-(variance components)กL &J*4กJ!+1) (likelihood function)-%+Iก! aกb b& RML +&$Iก FML (Longford. 1993) L &rก$Jก-%ก%ก! (equal group sizes) กL & RML +*$-&Iกก FML (Searle, Casella and McCulloch. 1992) -"!#$*)%L &J*L-กI'+)5-%* !"$ 2 +-กก!*&ก&&I-$(Browne. 1998) L & FML +*, *&"กI,J* # &&กกI*&ก -%! aกb b&+J*4กJ!+1) (likelihood function) !. I!"กL & +1) +$*+ก*L-กกI $* ! !",+Iก!"$*L &Iกm.I -%I$, *&"&"ก!"$I!.$(b*ก ก%$&-%*&ก - !.#%)*)ก1$ -ก !"L &J*L-กI'+)+45#$+I"กm.Im$กก$กI ,*-%*&!,, *$ -%45ก&$ # กL &J* +1) (Maximum Likelihood: ML) !&* 5 (Hox. 2002)

2. กI%!*&$ -""!&!$, (Generalized Least Squares: GLS) กL &J* กI%!*&$ -""!&!$, (Generalized Least Squares: GLS) #.`+ก +1) L &กI +Iกm.I m$ ก *& กI%!*&$ -""!&!$,+ก%*&ก!"ก *& +1) m$+*$b)ก*ก'#$!& 5ก "! .&!,J $ $ -bI, * 'ก FML L &b$+J*$, *+ก +1) 1+ $*กI J กL & +1) -%*&!,, *$#$+ก+I"กm.I,& b$+I .,J&ก#$-ก*45 !ก% -+กกก+!&L &J*ก+I%-""45 " ก *&กI%!*&$ -""!&!$,+

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$Iก +1) -%% %#$`$IL & กI%!*&$ -""!&!$,1#.`*"$&",,, **$-*+ (Kreft. 1996) m$L &!$, +1) +*$b)ก*กก 3. ก-""!&!$, (Generalized Estimating Equations) กL &J*ก-""!&!$,(Generalized Estimating Equations) L &+$*--%- %L &+ก% %#$ (residuals) m$bI, * 'ก FML L & ก-""!&!$,%!+ก$, *I--%--%*+J* กI%!*&$ -""!&!$,ก! aกb b& L & ก-""!$,+กI,J*L-ก HLM (Bryk, Raudenbush and Congdon. 1996) ก L &J*ก-""!&!$,+ -ก+ก ก-""+1) #$)-""!! ,1*(nonlinear) L &+กกก Goldstein (1995) #$&"&" ก-""!&!$,ก!" ก-""+1) " ก-""!&!$,+ $Iกก-""+1) -+*ก%"#.**&กก$&ก!"L* %)-"" !" b*)-"" %b)ก* ก *& +1) + )ก -%% %#$`+$Iก ก-""!&!$, -b*)-"" %,b)ก*กL &J* ก-""!&!$, +1!$-"!& !..!&* 5 4. ก-""")- (Bootstraping) ก-""")-(Bootstraping)!. +$*+กก*)% "Iก-%# (with replacement)%, -%Iก.ก#$#ก!"!.-ก L &ก-%!.+L &J* $++1 FML RML # กI%!*&$ (Least Square Method) m$ ก-""")- bJ*ก!"ก-%% %#$`*b)ก*&$. .+I&&กก #$q -&!I&ก ก-""" ก *& .ก!" FML +*$

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ก%*&ก! - ก-""")-+*$b)ก*#$ !&%'ก, * ก +1) -""' (Good. 1999) I!"+I"กIm.I Efron and Tibshirani (1986) ก%,*ก *& ก-"" ")-#$* Im.I,$Iก 400 " m$ %*ก!"กก Ayesha Nneka Delpish (2006) ก% ก *& ก-""")-,$Iก 400 " I**ก%*$-*+ 5. ก-""" (Bayesian method) ก-"""1ก$I%!กก% ,-กJ* L &Iก-+ก-+"#.* (prior distribution) *)%1#.`กก!"+1*)%ก-+ก-+&%! (posterior distribution) L &!$,-%*-ก-+ก-+&%!+$Iกก-+-+ก"#.* &ก*)%"#.*+J&*กb)ก*&$.+I*!-% % I!"!&ก-+ก-+-""&%! (posterior distribution) J ก-+ก-+-""ก (normal distribution) m$b*ก-""+ (point estimator) -%ก-""J (interval estimator) L &!&4กJ!ก-+ก-+ &,ก'b*1กก-+ก-+-""%&!- (multivariate distribution) +m!"m*-%&&กกIก-+ก-+&%! (posterior distribution) - ก-"""*$+, *!$-&Iqก!" +1) -+m!"m*กIกก #$ (Goldstein. 1995)

3.2 VW กก$) -&I-""$ -%% %#$`

ก! aกb b& L &!$,+"! 1!$ ,& ,+ *& กI%!*&$ -"" (Ordinary Least Squares: OLS) กI%!*&$ -""!&!$,(Generalized Least Squares : GLS) -% +1) (Maximum Likelihood: ML) (Van der Leeden, Busing and Meijer. 1997 ; Maas and Hox. 2001) -ก *& กI%!*&$ -"" -*+1!$,

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&-+ $Iก #$-) #$&"ก!" #$q + 90% (Kreft. 1996) I!"ก !"4++"!, *L-กI'+)%&L-กJ&ก m$L-ก%.ก " %-""$ (fixed effect) !.*ก%"#.*ก$&ก!"ก%!&* 5 -b*,1,*ก%"#.*+(%I*ก " !ก%,b"% %#$$ 1 , * !Jกก Maas -% Hox (2001) , *กก$&ก!"% %#$`-""$ m$ *& RML (%กก " +&ก .b*+Iก%*&ก 50 ก% ก$+Iก%ก!" 30 % %#$$ 1 ก!" 6.4 % L &ก "`กI !&I!5$ !" 0.05 m$ %*ก!"กก Van der Leeden (1997) , *Iกก-%*" b**)%,1,*ก%"#.*ก$&ก!"ก-+ก-+-""ก -% !&,5+I*% %#$`& m$ก-""$-%% %#$` !ก% +1) *$+ ก กI%!*&$ -""!&!$, I+ก "ก "! aกb b& !""%.&)ก!" !&!. (total sample size) I+ก " !"#"% -%!! !"+.&)ก!"+Iก%(number of groups) กก !&!. +กกก Mok(1995) Van der Leeden -% Busing. (1994) -% Cohen (1998) *(%กก %*ก! # -&I!-%I+ก "+), *!..&)ก!"+Iก%(number of groups) กก+I"%ก%(number of individuals per group)

-&I-""-%% %#$` ก% %#$%#(residual error) ก !"$ 1 !. L &!$,+-&I) +กกก Busing (1993) -% Van der Leeden -% Busing (1994) L &J*ก+I%-""45(simulation) " ก- L &J* กI%!*&$ -""!&!$,+*$-&I*&กก *& +1) ก+ก.-%*&!"ก- !"ก%!.+-&Iก'#$+Iก%กก 100 ก%., Maas -% Hox (2001) , *ก " #$+Iก% 30 ก% กL &J* RML +*$*

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-&I -#$+Iก%*&กq กL & !ก%+*$$Iกก$1+ I!"ก "ก"- (variance components) !. b*ก%!& *&ก 100 ก% " ,b"% %#$$ 1 , * !กก Browne -% Draper (2006) -% Maas -% Hox(2001) L &J*ก+I%-""45 m$*(%กก %*ก! # b*+Iก% 24-30 ก% ก "`!.+ก % %#$$ 1 ก!" 0.09 -b*+Iก% 48-50 ก% +ก % %#$$ 1 0.08 -%b*+Iก% 100 ก% +ก % %#$$ 1 0.06 m$กก!.. "`$ !" 0.05

-&I-% !& L &ก-%*ก "`bก !"b*ก$ !&ก !" ! -%% %#$`ก'+b)ก*). +กกก Kreft (1996) ก$&ก!"กกI !& -%, *I1กz&&(rule of thumb) I!"ก !" ก 2 !" # กz 30/30 L & Kreft (1996) ก% #$*ก -&Iก !"+Iก%(number of groups) #&!&$J*ก !"$ 2 &*& 30 ก%., -%*+I"%ก%(number of individuals per group) #&!& !"$ 1 1 ก%*Jก&*& 30 ., -%+กกก!ก!G#$L &J*ก+I%-""45 , **$+กก$&ก!"ก-""$ L &ก!"กz&& (rule of thumb) *%!กr!b$+ก L &b*+!! * !"(cross-level interaction) +Iก%กก+I"%ก%กz 50/20 L &+Iก%ก!" 50 -%+I"%ก%ก!" 20 b*+ก$&ก!" %-""m$ก$&ก!"ก--- -%% %#$` +Iก%$+5ก ก%กz 100/10 # +Iก%!.- 100 ก%., -% ก%!.- 10 & ., (Hox. 2002) กz !ก%1ก+ !&I+b -I!"ก$ # *J*+&$J*กก'""*)% L & Moerbeek, van Breukelen -% Berger (2001) , *&ก$&ก!"45ก%#ก !&I!"ก !" ก 2 !" L &+

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ก$&ก!"I+b -%*J*+&$J*กก'""*)% L &" *$J*กก'""*)%+.&)ก!" กก'""*)% 4. #%(%)"! #$*กก JJก$ ()*+!&*"+J*!- - JJก!. ()*+!&+J*!- X ก- "!Jก X ก.ก-+ก-++1 X I!5 I,) J#"!Jก, * J I*"r%$&-%-Jก, * L &!$,-%*4กJ!ก-+ก-++1!-!ก.&)ก!"!$#กก !ก%!ก1!," !!.ก+1$$ก #$, *-%* +I*()*+!&b!#$q $+, *ก J b*!- X 4กJ!ก-+ก-+.&)ก!" θ $$- "!"กJก!ก14กJ! θ *& J r%$&Jก# & )(XE=µ -%-Jก 22 )( µσ −= XE +14กJ! θ ก θ +I*()*+!&b µ -% 2σ , *ก *& ก!. *J**)%#!ก+ก!& ก%# +J*b"!1!$+ ก.+*J*!& #$+ก+I*, **)%L &,J*)*ก ' J" #&$+()*+!& #()*ก'""*)% b! !"% %#$, * ก.zb$J*ก*)%%*#.`+กกJ*!&!.. !!.กJ**)%+ก!&$,J*!& +I*()*+!&,bJ* กb$()*!G.-%*&b)ก*, * ,b! !"% %#$ก (%กก+,ก%*&1+ ก(Estimation) # กJ**)%+ก!&)b#$ก (θ ) #&)J ก*!$J*ก m$ก 2 # 1) ก-""+ (Point estimation) &b กก1 &#+ &

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2) ก-""J(Interval estimation)&bก ก%&!ก1J &ก JJ#$!$(Confidence interval) JJ#$!$ !ก%, *L &!&! ),...,(ˆˆ

1 nXXθθ = θ กJ*ก-+ก-++1 ),...,(ˆ 1 nXXθ ก*JJ#$!$ #%%$ ก-""+ 1กJ*4กJ! ),...,(ˆˆ

1 nXXθθ = !&

nX1X ,..., L &+J*4กJ!$"!"ก1! θ ก+"! θ +!&ก-+ก-+!&%&ก+ θ $+J*, *%&!#%&4กJ! +%!กก$J*+"!!$ #$%#กJ*!* -%!กก$J*%#ก!$&)%&%!กก !!$+"!1!$ , *%&ก *& !กb ก!$&)%& !++, "!.กJ*!5Iก+I,)!$"!$b"ก *&m.I J ก -+ก-+-"กJ !กJ*r%$&!&1!r%$&Jก -%r%$&!ก"!$ %&ก %&!. ก!r%$&Jกก'*(%1r%$&!& *& &,ก'&!!#$q r%$&Jกก J +J*! &`!& ก$!&!& `&% กr%$&Jก +%#กJ*!* -%"!$ "ก "!$ !$+J*1กกก%#กJ**%&ก # ,&(Unbiasedness) *(Consistency) -$I (Minimum Variance) (Efficiency) -%% %#$กI%!r%$&$I (Minimum mean squared error) I!"&%& !. 1. %:) ! ),...,(ˆˆ

1 nXXθθ = 1!$,&(Unbiased estimator) θ ก'#$

θθ =)ˆ(E

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θ 1!$&(Biased estimator) θ ก'#$ θθ ≠)ˆ(E -%&ก(% θθ −)ˆ(E &(Bias) θ ก θ

ก!$,&I, *%& # 1) J*!5Iก L &%#ก! θ θ !$ -%* &!

!. b* baE += θθ )ˆ( #$ a,b 1$-% 0a ≠ L &b!"-ก*, * )ˆ(1ˆ

1 ba

−= θθ

1!$,& θ 2) J*"!! θ $, *+ก ก$)-""$1!$,& θ J !-""J*&& Y " X $ Y = εβα ++ X &**` !-""กI%!$I (Least squares estimators) α -% β 1!$,& 3. %#ก! θ θ !$ -%* )ˆ(θE b*, * θθ =)ˆ(E - θ 1!$,& θ -b* θθ ≠)ˆ(E &&!,&& θθθ −= )ˆ()( Eb I!"-ก* θ +, *!,& θ 4. ก% & θ % J J* -+, (Jackknife) -"")%%(Quenouille) I !" "! ก$ & ก!" ก 1! $ , & . ! $ , & θ ++"!., *%&! 2. % ! nθ +1!$* θ ก'#$ nθ %)*+1 (Converge in probability or converge stochastically) , θ #$ !&*ก%*! - , * !ก 0]ˆ[lim =≥−

∞→εθθn

nP #

1]ˆ[lim =<−

∞→εθθn

nP

"!!$*

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1) !$* θ ,+I1+*1!$,& θ 2) !$*+, *%&! 3) !$*+1!$&#,&ก', * !!. "!ก$&ก!"ก1!$* # #$!& !&5*ก%*! !-%*ก! 3. % Y#$ ! θ θ 1!$-$I (Minimum variance estimator : MVE) #!$ $ (Best estimator) θ ก'#$ θ -,กก-!#$ θ ก&"&"-! θ &"&"ก!"-! θ $"!#$q #q ก! $&+ก!ก , *-ก -!$,& θ #!$14กJ!J*!ก-%,& θ !!. ! θ θ 1!$,&$-$I (Minimum variance unbiased estimator : MVUE) #!$,& $ (Best unbiased estimator) θ ก'#$ θ 1!$,& θ $-$I " !$,& θ *&ก! ก+ก.-%* ! θ 1!J* $ ,& (Best linear unbiased estimator : BLUE) θ ก'#$ θ 14กJ!กI%!$!&$,& θ -%-$I " !$14กJ!กI%!$!&-%,& θ *&ก! ก%L &-%*"!!ก$&ก!"-$I # กI!$"!$ "ก&)-%* J 1!$,& m$%&!&"&"-ก!#$!$ $ !$ 4. %(&"9 +ก!$"!,&-%* +" r%$&!& )(X -%! &` )

~(X ก'1!$,& -% *

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µ "!!$ # +%#ก!$-*&$ Cramer , *J*I M!$ N &b 1!$,& -%-*&$ (Minimum variance unbiased estimator : MVUE) "!.&ก !$ $ (Best estimator) L &"!!$ ) b, * !. 1) θθ =)ˆ(E # θ 1!$,& 2) )ˆ(θV +**&ก-!#$q $ %*ก!"* 1 # )ˆ(θV +**&ก-!-"",&!#$q b* 21

ˆ,ˆ θθ ก'1!$"!,& θ 1θ + )ก 2θ b* )ˆ()ˆ( 21 θθ VV < #

1)ˆ(

)ˆ(0

2

1 <<θ

θ

V

V

L &&ก )ˆ(

)ˆ(

2

1

θ

θ

V

V !! (Relative Efficiency)

L &!$," !!.%&$!G. กI *1 )ˆ,...,ˆ,ˆ( 21 kθθθ m$"!1!$,& θ !$# )ˆ( iE θ = θ , i = 1,2,,k ก$+%#ก iθ ! !$1! θ L &J*"!#$ L &J* ก&q # %#ก jθ m$ )ˆ()ˆ( ij VV θθ ≤ #$ i, j = 1,2,,k L & ji ≠ 5. %$;กY4Y#$ ! θ θ 1!$% %#$กI%!r%$&$I (Minimum mean squared error estimator ) ก'#$

2*2 )ˆ()ˆ( θθθθ −≤− EE , *θ +1! q θ #$ Ωθε

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&ก 2)ˆ( θθ −E % %#$กI%!r%$& θ ก θ !ก- *& )ˆ(θMSE b* θ 1!$,& θ +', *! )ˆ()ˆ( 2 θθθ VE =− m$bJ* 2)ˆ( θθ −E -% )ˆ(θV 1#$! % %#$ θ ก θ 5. (ก%)"!2ก !" #$

กก !"!. ก$, *!"&กI,&I!$L-ก # +1) (Maximum Likelihood) m$ !ก%., *ก!G&#$ L &-"1 2 # +1) -""+Iก! (Restricted Maximum Likelihood ) -% +1) -""' (Full Maximum Likelihood ) L &J*ก`กI *& EM (Expectation Maximization) I!"L-กI'+)b$J*ก !"กJ* ก !ก%1%!ก # L-ก HLM -%L-ก SPSS 1* ก+ก., *ก!G- ก$&ก!" ก-""" (Bayesian) L &&%& !.(Randenbush and Bryk. 1992 :230-248, 2002 : 399-444) ก%)"!2ก !" #$ ก% 2 $ กก !"ก 2 !" b&ก)-""!$, # ก-""!$, !"$ 1

rXY += β ),0(~ ψNr

L & Y # ก!- X # ก!- !"$ 1 β # ก! a !"$ 1 r # ก% %#$ !"$ 1

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ก-""!$, !"$ 2

uW += γβ , ),0(~ τNu

L & β # ก!- !"$ 2 W # ก!- !"$ 2 γ # ก! a !"$ 2 u # ก% %#$ !"$ 2

#$-ก$ 2 ก$ 1 +, *ก !.

rXuXWY ++= γ #$Iก$ 3 +! )%!ก)-""( (Mixed Model) , *ก !.

rAAY ++= 2211 θθ #$ XWA =1 , ,2 XA = γθ =1 -% u=2θ

L & Y # ก!- 1θ # ก %-""$ (Fixed Effect) m$," 2θ # ก %-"" (Random Effect) !"$ 2

1A -% 2A # ก!- r # ก %-"" (Random Effect) !"$ 1

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+กก)-""( (Mixed Model) m$+1ก#.`กI, *& +1) -""+Iก! (Restricted Maximum Likelihood ) -% +1) -""' (Full Maximum Likelihood )

ก

ก *& +1) -""+Iก! (Restricted Maximum Likelihood) !.ก Iก !. !.$ 1 $* (Starting Estimates) 2σ -% τ !.$ 2 I 2σ -% τ , *

1θ -% *2θ

][ 2

112111

*1 ∑ ∑ −−= yjjjyj SCSSVθ

112*

11 VD σ= )( *

12121*

2 θθ jyjjj SSC −= −

jVD 22

2*22 σ=

-% 1

12*11

*12

−−= jjj CSDD

L & 12

22−+= τσjj SC

[ ] 1

211

121111

−−∑ ∑−= jjjj SCSSV

-%

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35

1121121

1122

−−− += jjjjjj CSVSCCV

!.$ 3 I 2σ , τ , *

1θ -% *2 jθ ,- E(CDSS)

( )∑ τσ ,,| 2/ YrrE jj = ( ) ( )[ ]∑∑ +− −

jjjjj SVtrSCSVtrF 2222211

12112σ

( )∑ τσθθ ,,| 2/

22 YEjj = ∑∑ + jjj V22

2/*2

*2 σθθ

!.$ 4 I$, *!.$ 3 ,-ก ∑= Nrr jj /ˆ /2σ

∑−= /

221ˆ

jjJ θθτ

!.$ 5 I$, *!.$ 4 -!.$ 2 +, * *

1θ -% *2θ

!.$ 6 I$, *!.$ 4 -% 5 ,-4กJ!+1 (Likelihood Function) ก

∑ ∑−++−−−−∝ − *111

22 loglogˆlog)ˆlog()()],|(log[ jj QCVJFJRNYf τστσ

#$ 2*

22*11

'* /)ˆˆ( σθθ jjjjjj AAYYQ −−= !.$ 7 $, *!.$ 4 -% 5 +b)กI,J*!., L &+& กก'#$4กJ!+1 (Likelihood Function) %$&-%,ก 6 (log 0.000001) #ก 100 " L &$, *" *& # $1!-

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ก *& +1) -""' (Full Maximum Likelihood) !.ก Iก !. !.$ 1 $* (Starting Estimates) 2σ , τ -% 1θ !.$ 2 I$, *, *

2 jθ , * !ก )( 11

/2

1*2 θθ jjjj AYAC −= −

-%

12*22

−= CD σ

L & ( ) 1122

/2

1 −−− += τσAAC !.$ 3 I 2σ , τ -% *

2 jθ ,- E(CDSS) ก$ ( )∑ τσθθ ,,,| 2

122/1

YAAE jjj= *

22/1 jjj

AA θ∑

( )∑ τσθθθ ,,,| 2

1/22 YEjj = ∑∑ −+ 12/*

2*2 jjj Cσθθ

( )∑ τσθ ,,,| 2

1/ YrrE jj = ∑ ∑ −+ jjjjj AACtrrr 2

/2

1*/*

L & *

2211*

jjjjj AAYr θθ −−= !.$ 4 I$, *!.$ 3 ,-ก

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( ) ∑∑

−=

−jjjjjj

AYAAA22

/1

1

1/11

ˆ θθ

∑−= /22

1ˆjjJ θθτ

∑= Nrr jj /ˆ /2σ

!.$ 5 I$, *!.$ 4 ,-4กJ!+1 (Likelihood Function)

∝)],,|(log[ 12 θτσYf ∑ −+−−− 12 logˆlog)ˆlog()( jCJJRN τσ

( ) ( ) 2*2211

/11 ˆ/ˆˆ σθθθ jjjjjj AAYAY −−−−∑

!.$ 6 $, *!.$ 4 +b)กI,J*!., L &+& กก'#$4กJ!+1 (Likelihood Function) %$&-%,ก 6 (log 0.000001) #ก 100 " L &$, *" *& # $1!- ก%)"!2ก !" #$ ก% 3 $ กก !" ก 3 !"!.*m!"m* m$L-ก$J*ก !" J L-ก HLM -%L-ก SPSS 1* +Iก *& +1) -""' L &J*ก`กI *& EM (Expectation Maximization) m$&%& !. ก-""!$, !"$ 1

επ += aY , ),0(~ ψε N

L & Y # ก!-

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a # ก!- !"$ 1 π # ก! a !"$ 1 ε # ก% %#$ !"$ 1

ก-""!$, !"$ 2

rX += βπ , ),0(~ πτNr

L & π # ก!- !"$ 2 X # ก!- !"$ 2 β # ก! a !"$ 2 r # ก% %#$ !"$ 2

ก-""!$, !"$ 3

uW += γβ , ),0(~ πτNu

L & β # ก!- !"$ 3 W # ก!- !"$ 3 γ # ก! a !"$ 3 u # ก% %#$ !"$ 3

#$-ก-""!$, !"$ 3 -% !"$ 2 ก-""!$, !"$ 1 +, *ก !.

εγ +++= araXuaXWY #$Iก$ 3 +! )%!ก)-""( (Mixed Model) , *ก !.

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εθθθ +++= 332211 AAAY #$ aXWA =1 , ,2 aXA = ,3 aA = ,1 γθ = u=2θ -% r=3θ

L & Y # ก!- 1θ # ก %-""$ (Fixed Effect) m$," 2θ # ก %-"" (Random Effect) !"$ 3 3θ # ก %-"" (Random Effect) !"$ 2

1A 2A -% 3A # ก!- ε # ก %-"" (Random Effect) !"$ 1

ก *& +1) -""' (Full Maximum Likelihood) I!"ก !" ก 3 !" !.ก Iก !. !.$ 1 $* (Starting Estimates) 2σ , βπ ττ , -% 1θ !.$ 2 I$, *, *

2 jkθ -% *3 jkθ !ก

MdAV

jkk/222

*2 =θ

)( *

22/3

1*3 kjkjkjk AdAC θθ −= −

-% ( )12

3/3

−+= πτσjkjkAAC

L & /

31

3 jkjkjk ACAIM −−=

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!.$ 3 I 2σ , βπ ττ , -% 1θ ,- E(CDSS) ก ( )∑∑ βπ ττσθεε ,,,| 2

1/

jkjkE = ∑ */*

jkjkεε

( )[ ]∑ ∑+ kjkjkjkVAMAtr 222

2/2

2σ

( )∑∑ −+ 13

/3

2jkjkjk

CAAtrσ

L & *

33*2211

*jkjkkjkjkjkjk AAAY θθθε −−−=

( )∑∑ βπ ττσθθθ ,,,,| 2

1/33 YEjkjk = ∑∑∑∑ + jkjkjk V33

2/*3

*3 σθθ

( )∑ βπ ττσθθθ ,,,,| 2

1/22 YEkk = ∑∑ + kkk V22

2/*2

*2 σθθ

!.$ 4 I$, *!.$ 3 ,-ก

( ) ( )∑∑∑∑ −−=− *

33*22

/1

1

1/11

ˆjkjkkjkjkjkjkjk

AAYAAA θθθ

2σ = Tjkjk

// εε∑∑

πτ = ∑∑− /

331

jkjkJ θθ

βτ = /

221

kkK θθ∑−

!.$ 5 I$, *!.$ 4 ,-4กJ!+1 (Likelihood Function) ก

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∝)],,,|(log[ 12 θττσ βπYf βτσ ˆlog)ˆlog()( 2

32 KJRKRT −−−−

∑∑∑ −++− 122 loglogˆlog

jkk CVJ πτ

∑∑− 2* ˆ/σjkQ L & ( )*

3*22

/*jkjkkjkjkjkjk AAddQ θθ −−=

-% 11 θjkjkjk AYd −= !.$ 6 $, *!.$ 4 +b)กI,J*!., L &+& กก'#$4กJ!+1 (Likelihood Function) %$&-%,ก 6 (log 0.000001) #ก 100 " L &$, *" *& # $1!-

(ก%) !-ก0! (Empirical Bayes)

ก-"""J+!ก (Empirical Bayes) $I&กJ*ก !". &ก Empirical Bayes # Shrinkage Estimator 1 $!"!*-&I. L &J* กb.I!ก *&$& (Reliability) !!.ก *& .+*$-&I -%-$Iก กI%!*&$ -"" (Ordinary Least Square: OLS) กก !" *& ก-"""J+!ก (Empirical Bayes) , *- &%& กI-"ก1 2 ก !. (Raudenbush and Bryk. 2002) 1. ก-"""J+!ก (Empirical Bayes) ก !" ก 2 !" !.$ 1 L & OLS -%ก"-(Variance-Component) *& กI%!*&$ -"" (Ordinary Least Square: OLS) !.$ 2 I$, *!.$ 1 , *& Empirical Bayes # Shrinkage Estimator !.

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!- !"% 1 !- ( )( )jjjj

SE

j W010000ˆˆ1ˆˆ γγλβλβ +−+=

#$ ( ) [ ]jjjn

yreliabilit20 )ˆ(

στ

τβλ

+=

!- !"% 2 !-

( )( )

jjjjj

SE

j WW 2021010000ˆˆˆ1ˆˆ γγγλβλβ ++−+=

#$ ( ) [ ]jjjkn

yreliabilit20 )ˆ(

στ

τβλ

+=

!.$ 3 I$, *!.$ 2 ,1!- !"$ 2 -% %-""$ (Fixed Effect) -% %-"" (Random Effect) 2. ก-"""J+!ก (Empirical Bayes) ก !" ก 3 !" !.$ 1 L & OLS -%ก"-(Variance-Component) *& กI%!*&$ -"" (Ordinary Least Square: OLS) !.$ 2 I$, *!.$ 1 , *& Empirical Bayes # Shrinkage Estimator !. !- !"% 1 !- ( )( )

jkkkjkjkjk

SE

jk X100000ˆˆ1ˆˆ ββλπλπ +−+=

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43

#$ ( ) [ ]jk

jkjkn

yreliabilit20 )ˆ(

στ

τπλ

π

π

+=

( )( )kkokk

SE

k W100000000ˆˆ1ˆˆ γγλβλβ +−+=

#$ ( )( )

++

=−−

∑1

12

00 )ˆ(

jk

kk

n

yreliabilit

σττ

τβλ

πβ

β

!- !"% 2 !- ( )( )jkkjkkkjkjkjk

SE

jk XX 2201100000ˆˆˆ1ˆˆ βββλπλπ ++−+=

#$ ( ) [ ]jk

jkjkn

yreliabilit20 )ˆ(

στ

τπλ

π

π

+=

( )( )kkkokk

SE

k WW 2020010100000000ˆˆˆ1ˆˆ γγγλβλβ ++−+=

#$ ( )( )

++

=−−

∑1

12

00 )ˆ(

jk

kk

n

yreliabilit

σττ

τβλ

πβ

β

!.$ 3 I$, *!.$ 2 ,1!- !"$ 2 -% !"$ 3 -% %-""$ (Fixed Effect) -% %-"" (Random Effect) +ก$ก% ก$&I,J*L-กI'+)bก !" # FML -% RML ก+ก.-%*, *ก!G- กI ก-""" (Bayesian) $&ก Empirical Bayes # Shrinkage Estimator J*ก!"!#$*-&I -%-$I !!.ก+!&!.. ()*+!&, *&"&" ก FML RML -% EB

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#$+ก !ก%, *!"ก&!"-%กI,J*ก!L &!$,L-กI'+)b J L-ก HLM L-ก R -%L-ก SPSS 1*

6. ก L (2536) , *&"&"(% !" *& OLS Separate Equation ก!"L-กJ-%' กก4++!&$ %(%! aก&!ก& !"! &ก%& ก%!&1!ก&J!.! &ก$ 4 +I 649 ) 21 (%กก ก !" *& OLS Separate Equation " !- !"!ก& , *-ก J455 + -%-+)+! a %(%! aก&&!&I!5"*&-L &r%$&ก*&,!&I!5b(%! aก& m$-กก!"ก *&L-กJ-%' $" J455 -%+ %(%! aก&&!&I!5 !- !"J!.&$ %$&!&I!5!. #ก! # "กก) -% L& ก&"&" ก " ก *& OLS Separate Equation +*+! I-*)%#$J*ก 2 !. !"$-" ก *&J-%' Iก+! I-*)%!. & -%J-%' b+"!&I!5(!-!-$+กก-% !" m$ OLS Separate Equation ,bI, * + .1*&#&!ก!.J-%' ก OLS Separate Equation Kanjanawasee (1989) , *ก %L&$(%! aก& * !&-%+J!ก&J!.! &ก*,& L &&"&"ก *& Traditional -""q # -"" Variance Component Analysis -"" Standard Regression Analysis -%-"" Hierarchical Analysis of Covariance ก!"ก !" -"" OLS (Ordinary Least Square) Single Equation -"" OLS Separate Equation -%-"" HLM (Hierarchical Linear Model) J**)%+ก The Second International Mathematics Study (SIMS) ,& ก%!&1!ก&J!.! &ก$ 2 +I 4,030 )-%()*"L&+ก 99 L& 3 !" # !"!ก& !"J!.& -% !"L& (%กก" 4++!&$(%(%! aก&!ก&#

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(%! aก&- !กก กJ*#$ %$"* J&%#()*กก&)*J ก, *-+)+()*ก -r%$&J !"J!.& J!.& "กก) !+I!ก&) 1 -%!G)()*J (%ก&"&" ก " ก !"1ก$กก *& -"" Traditional ก&"&"ก !"-""q " ก-"" OLS Single Equation 1ก$% %#$`ก*ก$( % (overestimate) ก!- !"J!.&-% !"L&&) !"!ก& I*ก% %#$`$I I*!-!&I!5bกก$+1 ก -""J-%' *ก Mean Square Error b)ก*กก-"" OLS !.-""

Newsom and Nishishiba (2002) , *กก$&ก!"&$ก +ก !&-%ก,%)*กก !"ก*)%1) m$L & RML L &+I%*)% *&L% (Monte Carlo) m$, *ก %! a!! &J!. (Intraclass Correlation Coefficient) -% !&(Sample Size) $(%&ก-%% %#$` L &ก! a!! &J!. 4 !" # 0.05 , 0.10 , 0.20 -% 0.30 -% !& 5 !" # 50 , 100 , 200 , 500 -% 1,000 L &Iกกก 2 !" L &+ก %+Iก% (Number of Groups) (%กก" +Iก% (Number of Groups) -%! a!! &J!.(%I*ก &-%% %#$` -%I*ก 45ก+ ! -ก(Intercept) -%J! (Slope) -%L &!$, %-""$ -%% %#$`+&$I -I!" %-"" -%% %#$`+&) ()*+!&+, *- ก-% "` %-"""!J*!& 5b**)%1) Maas and Hox (2004) , *กก$&ก!"-กกก !" #$ก%!& %'ก กก!..1กกก 2 !" L &J*L% (Monte Carlo) m$#$,ก+I%-""45 # 1) +Iก%(Number of groups) -"1 3 # 30, 50, -% 100 2) ก% (Size of groups) -"ก1 3

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# 5 , 30 -% 50 3) ! a!! &J!. (Intraclass Correlation Coefficient) -"ก1 3 # 0.1( %'ก) 0.2 ( ก%) -% 0.3 ( 5) !!.+#$,ก+I%-""45!. 27 #$, L &Iก+I%-""45-%bก+I 1,000 " L &!- !"$ 1 -% !"$ 2 !"% 1 !- I!"ก -%% %#$`J* กI%!*&$ !&!$,m.I-""+Iก! (Restricted Iterative Generalized Least Square: RIGLS) (%กก" +Iก%(Number of groups) ก% (Size of groups) -%! a!! &J!.(%ก"I*ก &ก %-""$ -%I*ก*JJ#$!$-%ก "!&I!5b %-"" !"$ 2 !.,b)ก* #$+Iก%(Number of groups) $Iก 30 ก+I%$+Iก%$I # 30 !. b 9% $ก-"" !"$ 2 &)กJJ#$!$$ 95% !!.กz&& (rule of thumb) Kreft (1996) !.+ก %-""$ -% %-""r !"$ 1 ,, *Ibก "!&I!5 %-"" !"$ 2 Maas and Hox (2004) , *กก$&ก!" !&I!"ก !" กก!..1กกก 2 !" L &J*L% (Monte Carlo) m$#$,ก+I%-""45 # 1) +Iก%(Number of groups) -"1 3 # 30, 50, -% 100 2) ก% (Size of groups) -"ก1 3 # 5 , 30 -% 50 3) ! a!! &J!. (Intraclass Correlation Coefficient) -"ก1 3 # 0.1( %'ก) 0.2 ( ก%) -% 0.3 ( 5) L &!- !"$ 1 -% !"$ 2 !"% 1 !- I!"ก -%% %#$`J* RML (%กก" #$ !& !"$ 2 ,ก 50 ก-%% %#$` !"$ 2 +1!$& -I!" !"$ 1 ก-%% %#$`+1!$,&-%b)ก* Eqbal Z. M. Darandari (2004) , *ก-กกL %J*% %!$&*ก%%&*ก%"#.*ก$&ก!"$--%1% %#$ !"$ 2 L &1กก)-""*+ ! -ก1!- (Random Intercept as Outcomes) m$กก, *กI #$,ก+I%-""45 # 1) กI -% %#$ !"$ 2 +I 3 !" 2) กI 1

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% %#$ !"$ 2 +I 3 !" 3) กI !! !- +I 3 4) กI !& !"$ 2 +I 2 # 50 -% 300 -% 5) กI !& !"$ 1 +I 2 # 30 -% 150 m$ก+I%-""45!..bก!. +I 108 bก -%bกIm.I+I 100 J L &ก!.J* FML (%กก " ก -%$&+-ก-กก!#$ก%%&*ก%"#.*&) !"ก%-% !"ก m$#$*ก%"#.*ก$&ก!"1% %#$b)ก%%& ก-%-+ก & L & %ก%%&*ก%"#.* !ก%+กzJ! +#$ !&% %L &r !& !"$ 2 m$ %.+.&)ก!"$+ -%!! !- ก+ก!.-%*&!I* !! -%!! ( ,+ก)-"" *& Browne and Draper (2006) , *ก&"&" ก-"""-% ก$J*+1) 1`$ก!"L % !" L &!b#$&"&" -"""-% ก$J*+1) 1`กก"- (variance component) -%)-"" %กb b&-""L%+ก ก 2 !" L &ก+I%-""45 กI !& !"$ 1 +I 5-61 & -% !"$ 2 +I 6-36 & (%กก" 1) L %ก"- !"กL & -"""-% ก$J*+1) 1`1!$,& 2) ก!. 2 &&ก-%45ก-#$!& %'ก (small sample size) -%( ก (extreme value) 3) ก-"" quasi-likelihood )-""กกb b&L%+ก-"" !" " ก- %-""!., $& -% 4) ก-"""&&กกJก"#.* -%*ก!"!-""+ -%-""J L &-%* ก-""+1) (Maximum Likelihood : ML) 1 $กก-%bI, * 'ก -""" Ayesha Nneka Delpish (2006) , *ก&"&" กL %J*% %!$ +1) -""+Iก! ก!" ")- L &+I%-""ก

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2 !" L &กI #$,ก+I%45 # กI ก-+ก-+ 2 %!ก # ก-+ก-+-""ก -%ก-+ก-+-"",-$-ก!" 1 กI !-$ก !"% 1 !- กI !& !"$ 1 # 30 & -% !"$ 2 # 30 -% 100 & กI ! a!! &J!. 2 # 0.01 -% 0.20 m$ก+I%-""45!..bก!. +I 8 bก -%กก-%bกIm.I+I 500 " (%กก " +Iก%-%! a!! &J!.(%b)ก*ก L &#$+Iก%).I*&ก% % m$I*!&I!5b&ก#$J* ก-""+1) -""+Iก! -%#$! a&J!.).I* ก!. 2 % % ก+ก.+กกก " ก !ก%b %-""$, *b)ก* I!"ก-+ก-+% %#$(% ก *& !ก% L &r#$% %#$ก-+ก-+-"",- ก!. 2 + % % L & ก-""")-+&$Iก +1) -""+Iก! Lauren Terhorst (2007) , *ก&"&" !! L % !" L &, *Iกกก 2 !" I!" ก$Iกก" *& FML RML %-""$ (Fixed Effect) กI%!*&$ b.I!ก-""$ 1 (Weight Least Square 1 ) กI%!*&$ b.I!ก-""$ 2 (Weight Least Square 2 ) กI%!*&$ b.I!ก-""$ 3 (Weight Least Square 3 ) L &กI !& !"$ 2 +I 3 # 20, 50 -% 100 กI ! a!! &J!. +I 2 # 0.10 -% 0.20 L & !"$ 1 กI !-+I 1 !- -% !"$ 2 +I 3 !- (%กก " FML RML 1 ก$&*&$ L & FML -% RML 1 ก$ RMSD $I$ #$&"&"ก!!. 6 +กก$()*+!&, *กก-%+!&$ก$&* " 45ก !"!.+#$+กก%!& %'ก m$+,(%I*ก&!. %-""$ (fixed effects) -% %-"" (random effects) -%(%

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bก"% %#$$ 1 ก "`b ก+ก.กz&& (rule of thumb) $IL & Kreft (1996) กกI !&ก !" ก 2 !" &!,$ m$ Mass and Hox (2004) ก% กz&& (rule of thumb)!. *I!5ก %-""$ (fixed effects) -% %-"" (random effects) !"$ 1 ,, *Ibก "!&I!5 %-"" (random effects) !"$ 2 +I*,b"% %#$$ 1 !"$ 2 , *ก "` +ก45$ก%m$+', *กก5กI ก-""q J*กก -&!,, *กI&"&"ก!L &#$,ก+I%-""45 &ก!ก$ก%!& %'ก +I*()*+!&+$+ก กก !" #$ก%!& %'ก -%Jก,, *ก-+ก-+-""ก #$, *ก$&ก!" ก$ #$ก%!& %'ก,

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3

ก ก

ก กก !"#$%& "#ก '( R !*+ก(,-&.'!/.ก, 01! "#$ ก,#!2, ก21 "1 / 3ก ,

กก

$ ก ก&ก -4 ก0#"15 &&ก*+ก( 1. ,%ก&*+ก(,ก"& กก 1ก" 2 ก(0, ,%ก& กก9, 1"ก1ก" ,,%ก& กก/, 1-:ก1ก" ก 1 (skewness) ,1 1(kurtosis)

2.60 4.00 -0.50 (-0.50, 2.60) - 0.50 - (0.50, 4.00)

2. ก "-, ,, 1 "$ &2ก, 3. ก 1- ,-&.N-- ! .#'$ % (Intraclass Correlation Coefficient) 2 / 0.01 , 0.20 4. ก / ก21 "1

,& 1 3 / 5, 15 , 25 ,& 2 3 / 5, 15 , 25 ,& 3 3 / 5, 15 , 25

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5. .ก, 01! "#& *+ก(&ก ,ก . FML . RML ,. EB 6. $ "1,-4 ก0#%2/ : 1,000 %2 $%& " # (Monte Carlo Technique)

7. !0,-&.'!/., 01! "#$ ก,#!2, ก21 "1 / 3ก -&"!0 1"15 / θ "ก"1ก1! "#! & !,41-1 $1/ θ "ก"1ก1 θ ก3-1ก-&1, 0,"ก:1$ก1 θ 1 , ก ", 0& ก(0,%1 1 ,ก1", 0& 1&^ ก,กก1&"ก, 0 ก ก0_#&$%&20'!$ ก(0, 1 (Bias) ,", 0&&-2 ", 0& 1 "&-2 ก0_#ก"- ,-&.'! 0ก- ก "1

1 (Bias) /", 01

θ

θ

θ −=∑=

000,1

ˆ

)ˆ(

000,1

1t

t

Bias

θ 1! "#&, 0

tθ ", 01! "# θ $ ก&9& t t /ก&9

8. ก&1 /.ก, 01! "# ก&1 /&.!&(FB) ,&.!-21 (RB) 9+ -:"$ ก 0

P

Bias

FB

p

i

i∑== 1

)ˆ(θ

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52

)ˆ( iBias θ 1 /&.!& P ! "#&, 0/&.!&

P

Bias

RB

p

i

i∑== 1

)ˆ(θ

)ˆ( iBias θ 1 /&.!-21 P ! "#&, 0/&.!-21

--4"&$%$ ก&,-&.'!/.ก, 01! "#$ ก,#!2, ก,# !2:0& (One way MANOVA ) ! -&-4"&-"1 Univariate Tests ,. Scheffe & &-&, - 0.01 !ก ก

ก ก& $% !"#%1$ ก ก& ก& / "

1. -"-21 & กก 1ก" 2 ก(0, $%./ Ramberg ,0,9+- .ก-"-21 &/+ :1ก (Skewness) , 1 (Kutosis) 9+"-21 4:กก ก1! "# 4 1

( )

2

43

1

1

λ

−−+λ=

λλ ]RR[x

& R "/-21 & กกก:$ %1 (0 , 1)

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1λ ! "#&ก " 1 (Location parameter) 2λ ^ ! "#&ก -ก (Scale parameter) 43 λλ , ^ ! "#&ก :1 (Shape parameter) 9+/+ ก1 , 1&ก 4กก - " ,1 43 λ=λ -1 321 λλλ ,, , 4λ 1 ก"กก/ Ramberg et al. (1979) " 1 ,1 1&ก &1 21 λλ , & 1s&1ก 0 , &1ก 1 "141s&1ก µ , &1ก 2σ ,"1 21 λλ , ก"

1λ (µ , 2σ ) = 1λ (0,1) σ + µ 2λ (µ , 2σ ) = 2λ (0,1) /σ

$ กก 1s , a 1s&1ก 7 &1ก 3 X 1s&1ก 10 &1ก 3 W 1s&1ก 13 &1ก 3 ε 1s&1ก 0 &1ก 3 r 1s&1ก 0 &1ก 3 u 1s&1ก 0 / :1ก1 ICC

2. -/ :"" (Y) $ ก(0,:1$ :%- (Linear Model)

ijkY = kjkijkijkjkk uraXW 000100010001000 ++++++ επβγγ

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ก 1- ,-&.Nก44 6,5,3 010001000 === βγγ , 7100 =π

3. ก / ก21 "1 ,& 1 3 / 5, 15 , 25 ,& 2 3 / 5, 15 , 25 ,& 3 3 / 5, 15 , 25

4. .ก, 01! "#& *+ก(&ก ,ก . FML . RML ,. EB 5. $ "1,-4 ก0#%2/ : 1,000 %2 $%& " # (Monte Carlo Technique)

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ก

กก กก ! #$ก% &%'ก ()*+ &,*-!(%ก%ก. 2 0

1 ก ก ก !" #ก$ "!$%ก ก &กก'(ก'(')*)'$+!,ก ก

2 ก ก ก !" #ก$ "!$%ก ก &กก'(ก'(')'$+!.ก ก +!ก(""// 0.)("1!)ก,!"2$"ก3 !"/ FML # ก ( 7. !'% (Full Maximum Likelihood) RML # ก ( 7. !'(,ก"! (Restricted Maximum Likelihood) EB # ก '& ("ก3 (Shrinkage Estimator: SE) FB # $' RB # $'

ICC # " R""&"/ (Intraclass Correlation Coefficient) 1 ก ก !"ก#!#$% &

ก$ % 'ก ก(กก)*ก)*)+,+)- .ก ก

/(,0$ก!$ $!!"/ 1. '!ก ก !" #ก$

"!$%ก ก &กก'(ก'(')*)'$+!,ก ก 2. ก ก !"

#ก$ "!$%ก ก &กก'(ก'(')*)'$+!,ก ก

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1. ) ก !"ก#!#$% & ก$

% 'ก ก(กก)*ก)*)+,+)- .ก ก $! กW!" 1

1 ก ก !" #ก$ " !$%ก ก &กก'(ก'(')*)'$+!,ก ก

n.

level 3

n.

level 2

n.

level 1 ICC

FML RML EB

FB RB FB RB FB RB

5 5

5 0.01 -0.0573 -1.0404 -0.0561* -0.8927 -0.3650 -0.3860*

0.20 0.0628 -0.8332 0.0625* -0.5468 -0.2352 0.1718*

15 0.01 -0.0010* -1.0157 0.0010* -0.8865 -0.1349 -0.2185*

0.20 0.0811 -0.7885 0.0784 -0.5134 -0.0411* 0.5001*

25 0.01 0.0019* -1.0233 0.0047 -0.9016 -0.0853 -0.1958*

0.20 -0.0036 -0.8034 -0.0031* -0.5363* -0.0875 0.5405

5 15

5 0.01 0.0209* -0.9568 0.0212 -0.9044 -0.2888 -0.3694*

0.20 0.1057 -0.7337 0.1053* -0.5354 -0.1777 0.2184*

15 0.01 0.0331 -0.9595 0.0326* -0.9143 -0.1057 -0.2464*

0.20 -0.0046 -0.7386 -0.0043* -0.5460 -0.1256 0.4660*

25 0.01 -0.0106* -0.9587 -0.0108 -0.9141 -0.0977 -0.1976*

0.20 0.0604 -0.7444 0.0601 -0.5582 -0.0211* 0.5194*

5 25

5 0.01 -0.0107* -0.9565 -0.0112 -0.9233 -0.3204 -0.3944*

0.20 -0.0331* -0.7158 -0.0333 -0.5304 -0.3251 0.2310*

15 0.01 0.0102 -0.9487 0.0097* -0.9190 -0.1278 -0.2494*

0.20 0.0679 -0.7102 0.0677 -0.5284 -0.0557* 0.5057*

25 0.01 0.0076* -0.9425 0.0078 -0.9133 -0.0804 -0.1926*

0.20 -0.0433 -0.7194 -0.0430* -0.5409* -0.1335 0.5614

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1 ()

n.

level 3

n.

level 2

n.

level 1 ICC

FML RML EB

FB RB FB RB FB RB

15 5

5 0.01 0.0090* -0.9727 0.0093 -0.9284 -0.3029 -0.3346*

0.20 0.0001* -0.6271 -0.0002 -0.5321 -0.2778 0.4750*

15 0.01 -0.0069 -0.9560 -0.0064* -0.9172 -0.1462 -0.1533*

0.20 -0.0098 -0.6534 -0.0092* -0.5675* -0.1372 0.7027

25 0.01 -0.0009 -0.9563 -0.0005* -0.9187 -0.0909 -0.1056*

0.20 -0.0539* -0.6370 -0.0539* -0.5513* -0.1312 0.8256

15 15

5 0.01 -0.0008 -0.9331 -0.0007* -0.9168 -0.3119 -0.3518*

0.20 -0.0206 -0.6145 -0.0206* -0.5491 -0.3091 0.4050*

15 0.01 0.0073 -0.9340 0.0071* -0.9200 -0.1309 -0.2121*

0.20 0.0019* -0.6002 0.0020 -0.5349* -0.1215 0.7285

25 0.01 -0.0021* -0.9325 -0.0022 -0.9190 -0.0898 -0.1645*

0.20 -0.0003* -0.6080 -0.0003* -0.5446* -0.0741 0.7897

15 25

5 0.01 -0.0001* -0.9312 -0.0002 -0.9207 -0.3105 -0.3653*

0.20 0.0206* -0.6015 0.0206* -0.5407 -0.2655 0.4142*

15 0.01 -0.0096* -0.9245 -0.0097 -0.9153 -0.1471 -0.2135*

0.20 0.0010* -0.6030 0.0010* -0.5439* -0.1215 0.6948

25 0.01 0.0062 -0.9302 0.0061* -0.9214 -0.0819 -0.1802*

0.20 0.0358* -0.6054 0.0358* -0.5472* -0.0386 0.7706

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1 ()

n.

level 3

n.

level 2

n.

level 1 ICC

FML RML EB

FB RB FB RB FB RB

25 5

5 0.01 0.0133* -0.9347 0.0135 -0.9079 -0.2988 -0.2792*

0.20 0.0025* -0.6088 0.0026 -0.5531 -0.2789 0.4885*

15 0.01 -0.0042 -0.9437 -0.0040* -0.9208 -0.1437 -0.1373*

0.20 0.0072* -0.6015 0.0073 -0.5490* -0.1130 0.8064

25 0.01 0.0178* -0.9389 0.0178* -0.9168 -0.0723 -0.0795*

0.20 0.0029 -0.6057 0.0028* -0.5543* -0.0734 0.8823

25 15

5 0.01 -0.0068* -0.9300 -0.0068* -0.9206 -0.3194 -0.3507*

0.20 -0.0057* -0.5862 -0.0057* -0.5470 -0.2924 0.4508*

15 0.01 0.0004* -0.9275 0.0004* -0.9191 -0.1377 -0.2001*

0.20 -0.0073* -0.5835 -0.0073* -0.5453* -0.1265 0.7483

25 0.01 0.0036* -0.9269 0.0036* -0.9188 -0.0846 -0.1540*

0.20 0.0126* -0.5770 0.0126* -0.5386* -0.0661 0.8523

25 25

5 0.01 -0.0013* -0.9234 -0.0013* -0.9173 -0.3120 -0.3543*

0.20 -0.0241* -0.5792 -0.0241* -0.5431 -0.3119 0.4478*

15 0.01 -0.0091* -0.9259 -0.0092 -0.9205 -0.1475 -0.2177*

0.20 -0.0201* -0.5821 -0.0201* -0.5469* -0.1397 0.7312

25 0.01 0.0018* -0.9253 0.0018* -0.9200 -0.0870 -0.1709*

0.20 -0.0040* -0.5830 -0.0040* -0.5478* -0.0827 0.8165

* ก ! .$

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(ก 1 $' (Fixed Effects) ก #ก$ "!" 3 ก" 5, 15 '$ 25 ก$ "!" 2 ก" 5, 15 '$ 25 '$ก$ "!" 1 ก" 5, 15 '$ 25 '$" R""&"/ (ICC) ก" 0.01 '$ 0.20 *`"/! 54 #1 FML '$ RML $', ! (, 35 #1 ก" '$ EB $', ! (, 3 #1 ," FML $', !ก" RML (, 19 #1

$' (Random Effects) ก #ก$ "!" 3 ก" 5, 15 '$ 25 ก$ "!" 2 ก" 5, 15 '$ 25 '$ก$ "!" 1 ก" 5, 15 '$ 25 '$" R""&"/ (ICC) ก" 0.01 '$ 0.20 *`"/! 54 #1 EB $' , ! (, 40 #1 '$ RML $' , ! (, 14 #1

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2. ก !"ก#!#$%

& ก$ % 'ก ก(กก)*ก)*)+,+)- .ก ก

กF% 2-4

2 '!ก' .'!# ก ก !" #ก$ "!$%ก

ก &กก'(ก'(')*)'$+!,ก ก

Dependent

Var.

Inependent

Var.

Pillai's

Trace F

Hypothesis

df

Error

df Sig.

FB,RB Method 0.6671 39.7901** 4 318 0.0000

**%.%[\% 0.01

(ก 2 ก "/ 3 ก $' '$$' 'กก"","2b!" 0.01 !) Univariate Tests กW0$!" 3

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61

3 '!ก Univariate Tests # ก ก !" #ก$ "!$%ก ก &ก

ก'(ก'(')*)'$+!,ก ก

Dependent

Var. Source Sum of Squares df Mean Square F Sig.

FB Contrast 0.7979 2 0.3990 111.3796** 0.0000 Error 0.5695 159 0.0036

RB Contrast 4.6539 2 2.3270 60.3820** 0.0000

Error 6.1275 159 0.0385

**","2b!" 0.01 (ก 3 0$ก Univariate Tests ก "/ 3 ก $' '$$' 'กก"","2b!" 0.01 !) Scheffe กW0$!" 4

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4 '!ก.!) Scheffe # ก ก !" #ก$ "!$%ก ก &ก

ก'(ก'(')*)'$+!,ก ก

Dependent

Var.

InependentVar.

(Method)

FML RML EB

0.0175 0.0175 0.1664

FB

FML 0.0175 - 0.0000 -0.1489**

RML 0.0175 - -0.1489** EB 0.1664 -

0.8017 0.7293 0.4115

RB

FML 0.8017 - 0.0724 -0.3902**

RML 0.7293 - -0.3178**

EB 0.4115 -

**","2b!" 0.01

(ก 4 FML ก" RML ก $',ก EB ","2b!" 0.01 '$ FML ก" RML ก $''กก"1","2b!" 0.01

EB ก $' ,ก FML ก" RML ","2b!" 0.01 '$ FML ก" RML ก $' 'กก"1","2b!" 0.01

(ก0$ก).$ '! ก &กก'(ก'(')*)'$+! ,ก ก FML '$ RML . !ก $' '$ EB . !ก $'

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2 ก ก !"ก#!#$% &

ก$ % 'ก ก(กก)*ก)*)+)- bก ก /(,0$ก!$ $!!"/

1. '!ก ก !" #ก$ "!$%ก ก &กก'(ก'(')'$+!.ก ก

2. ก ก !" #ก$ "!$%ก ก &กก'(ก'(')'$+!.ก ก

1. ) ก !"ก#!#$% & ก$

% 'ก ก(กก)*ก)*)+)- bก ก

กF% 5

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5 ก ก !" #ก$ " !$%ก ก &กก'(ก'(')'$+!.ก ก

n.

level 3

n.

level 2

n.

level 1 ICC

FML RML EB

FB RB FB RB FB RB

5 5

5 0.01 0.0785* -1.3288 0.0794 -1.2107 -0.2404 -0.8011*

0.20 0.0221 -1.1597 0.0210* -0.9328 -0.2785 -0.3617*

15 0.01 0.0043 -1.2723 0.0040* -1.2361 -0.3116 -0.7536*

0.20 0.0048* -1.1416 0.0084 -0.9225 -0.1263 -0.1234*

25 0.01 0.0807 -1.3000 0.0812 -1.1990 -0.0094* -0.6206*

0.20 0.0674 -1.1392 0.0681 -0.9219 -0.0290* -0.0572*

5 15

5 0.01 -0.0316* -1.2673 -0.0322 -1.2249 -0.3401 -0.7963*

0.20 0.0390* -1.0944 0.0400 -0.9386 -0.2645 -0.3400*

15 0.01 0.0100 -1.2639 0.0097* -1.2272 -0.1296 -0.6870*

0.20 -0.0250 -1.0720 -0.0243* -0.9141 -0.1554 -0.0830*

25 0.01 -0.0186* -1.2581 -0.0191 -1.2225 -0.1072 -0.6435*

0.20 -0.0202 -1.0747 -0.0200* -0.9138 -0.0954 -0.0219*

5 25

5 0.01 0.0204* -1.2565 0.0204* -1.2295 -0.2912 -0.8008*

0.20 0.0318 -1.0919 0.0312* -0.9511 -0.2609 -0.3576*

15 0.01 0.0272* -1.2494 0.0273 -1.2248 -0.1116 -0.6793*

0.20 -0.0150* -1.0765 -0.0154 -0.9370 -0.1379 -0.1226*

25 0.01 0.0413 -1.2541 0.0408* -1.2311 -0.0469 -0.6568*

0.20 0.0385* -1.0601 0.0388 -0.9139 -0.0427 -0.0166*

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5 ()

n.

level 3

n.

level 2

n.

level 1 ICC

FML RML EB

FB RB FB RB FB RB

15 5

5 0.01 0.0043 -1.2723 0.0040* -1.2361 -0.3116 -0.7536*

0.20 -0.0098* -1.0075 -0.0101 -0.9326 -0.2988 -0.1354*

15 0.01 0.0018 -1.2625 0.0014* -1.2313 -0.1370 -0.6190*

0.20 -0.0267 -1.0070 -0.0262* -0.9375 -0.1515 0.0933*

25 0.01 0.0309* -1.2617 0.0311 -1.2316 -0.0591 -0.5801*

0.20 0.0048 -0.9961 0.0044* -0.9262 -0.0672 0.1916*

15 15

5 0.01 -0.0001 -1.2472 0.0000* -1.2342 -0.3124 -0.7830*

0.20 0.0229 -0.9848 0.0228* -0.9318 -0.2696 -0.1602*

15 0.01 -0.0010 -1.2508 -0.0008* -1.2395 -0.1391 -0.6766*

0.20 0.0154* -0.9895 0.0155 -0.9394 -0.1088 0.0511*

25 0.01 0.0079* -1.2466 0.0080 -1.2356 -0.0807 -0.6320*

0.20 0.0023* -0.9840 0.0024 -0.9330 -0.0800 0.1384*

15 25

5 0.01 0.0031 -1.2399 0.0030* -1.2317 -0.3084 -0.7858*

0.20 -0.0154* -0.9871 -0.0154* -0.9394 -0.3035 -0.1819*

15 0.01 0.0018* -1.2383 0.0018* -1.2310 -0.1359 -0.6697*

0.20 -0.0398* -0.9858 -0.0399 -0.9389 -0.1574 0.0467*

25 0.01 -0.0100* -1.2417 -0.0100* -1.2346 -0.0982 -0.6405*

0.20 0.0061* -0.9804 0.0061* -0.9338 -0.0728 0.1228*

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5 ()

n.

level 3

n.

level 2

n.

level 1 ICC

FML RML EB

FB RB FB RB FB RB

25 5

5 0.01 0.0087* -1.2550 0.0089 -1.2336 -0.3067 -0.7335*

0.20 0.0034* -0.9808 0.0034* -0.9350 -0.2831 -0.0870*

15 0.01 0.0136 -1.2486 0.0135* -1.2302 -0.1236 -0.5977*

0.20 -0.0240 -0.9804 -0.0239* -0.9388 -0.1466 0.1415*

25 0.01 0.0104* -1.2548 0.0104* -1.2371 -0.0790 -0.5697*

0.20 0.0277 -0.9697 0.0276* -0.9276 -0.0469 0.2447*

25 15

5 0.01 0.0147* -1.2392 0.0147* -1.2315 -0.2975 -0.7724*

0.20 0.0034* -0.9601 0.0034* -0.9282 -0.2848 -0.1214*

15 0.01 -0.0009* -1.2388 -0.0009* -1.2321 -0.1400 -0.6553*

0.20 -0.0295* -0.9625 -0.0296 -0.9318 -0.1486 0.1085*

25 0.01 0.0003* -1.2407 0.0003* -1.2342 -0.0876 -0.6223*

0.20 -0.0068* -0.9662 -0.0068* -0.9358 -0.0890 0.1693*

25 25

5 0.01 0.0060* -1.2383 0.0061 -1.2334 -0.3066 -0.7846*

0.20 0.0175 -0.9748 0.0174* -0.9466 -0.2672 -0.1654*

15 0.01 0.0005* -1.2362 0.0005* -1.2319 -0.1378 -0.6659*

0.20 0.0017 -0.9548 0.0016* -0.9260 -0.1172 0.1111*

25 0.01 0.0098* -1.2400 0.0098* -1.2358 -0.0791 -0.6376*

0.20 -0.0022* -0.9663 -0.0022* -0.9383 -0.0803 0.1498*

* ก ! .$

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(ก 5 $' (Fixed Effects) ก #ก$ "!" 3 ก" 5, 15 '$ 25 ก$ "!" 2 ก" 5, 15 '$ 25 '$ก$ "!" 1 ก" 5, 15 '$ 25 '$" R""&"/ (ICC) ก" 0.01 '$ 0.20 *`"/! 54 #1 RML $', ! (, 35 #1 $ FML $', ! (, 32 #1 '$ EB $', ! (, 2 #1 ," FML $', !ก" RML (, 15 #1

$' (Random Effects) ก #ก$ "!" 3 ก" 5, 15 '$ 25 ก$ "!" 2 ก" 5, 15 '$ 25 '$ก$ "!" 1 ก" 5, 15 '$ 25 '$" R""&"/ (ICC) ก" 0.01 '$ 0.20 *`"/! 54 #1 EB $' , ! ก#1

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2. ก !"ก#!#$%

& ก$ % 'ก ก(กก)*ก)*)+)- bก ก

กF% 6-8

6 '!ก' .'!# ก ก !" #ก$ "!$%ก

ก &กก'(ก'(')'$+!.ก ก

Dependent

Var.

Inependent

Var.

Pillai's

Trace F

Hypothesis

df

Error

df Sig.

FB,RB Method 0.8425 57.8659** 4 318 0.0000

**%.%[\% 0.01 (ก 6 ก "/ 3 ก $' '$$' 'กก"","2b!" 0.01 !) Univariate Tests กW0$!" 7

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7 '!ก Univariate Tests # ก ก !" #ก$ "!$%ก ก &ก

ก'(ก'(')'$+!.ก ก

Dependent

Var. Source Sum of Squares df Mean Square F Sig.

FB Contrast 0.8158 2 0.4079 119.3309** 0.0000 Error 0.5435 159 0.0034

RB Contrast 17.3486 2 8.6743 214.3694** 0.0000

Error 6.4338 159 0.0405

**","2b!" 0.01 (ก 7 0$ก Univariate Tests ก "/ 3 ก $' '$$' 'กก"","2b!" 0.01 !) Scheffe กW0$!" 8

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8 '!ก.!) Scheffe # ก ก !" #ก$ "!$%ก ก &ก

ก'(ก'(')'$+!.ก ก

Dependent

Var.

InependentVar.

(Method)

FML RML EB

0.0178 0.0179 0.1684

FB

FML 0.0178 - -0.0001 -0.1506** RML 0.0179 - -0.1505** EB 0.1684 -

1.1380 1.0811 0.4171

RB

FML 1.1380 - 0.0569 -0.7209**

RML 1.0811 - -0.6640**

EB 0.4171 -

**","2b!" 0.01

(ก 8 FML ก" RML ก $',ก EB ","2b!" 0.01 '$ FML ก" RML ก $''กก"1","2b!" 0.01

EB ก $' ,ก FML ก" RML ","2b!" 0.01 '$ FML ก" RML ก $' 'กก"1","2b!" 0.01

(ก0$ก).$ '! ก &กก'(ก'(')'$+!

.ก ก FML '$ RML . !ก $' '$ EB . !ก $'

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ก กก !"#$%& "#ก '( R "*+,-#!./ก(,-&0'!10ก, 23! "#$ ก,#!+, ก+3 "3 1 4ก ,!&,-&0'!10ก, 23! "#$ ก,#!+, ก+3 "3 1 4ก ,ก-+ ' ,1- , ก

ก./ก(,-&0'!10ก, 23! "#$ ก,#!+, ก+3 "3 1 4ก -+ก 1. ก./ก(,-&0'!10ก, 23! "#$ ก,#!+, ก+3 "3 1 4ก , 1.1 ก2,%ก กก7, 3"ก3ก" 3 1&0!& (Fixed Effects) $ ก, 23! "# ก+3 "3$ ,& 3 &3ก 5, 15 , 25 ก+3 "3$ ,& 2 &3ก 5, 15 , 25 ,ก+3 "3$ ,& 1 &3ก 5, 15 , 25 ,3- ,-&0H-- ! 0#'$ % (ICC) &3ก 0.01 , 0.20 7/ & 54 1 !3 0 FML ,0 RML 3 1&0!&"-+ 35 1 &3ก ,0 EB 3 1&0!&"-+ 3 1 -0 FML 3 1&0!&"-+&3ก0 RML 19 1

3 1&0!-+3 (Random Effects) $ ก, 23! "# ก+3 "3$ ,& 3 &3ก 5, 15 , 25 ก+3 "3$ ,& 2 &3ก 5, 15 , 25 ,ก+3 "3$ ,& 1 &3ก 5, 15 , 25 ,3- ,-&0H-- ! 0#'$ % (ICC) &3ก 0.01 , 0.20 7/ & 54 1 !3 0 EB 3 1&0!-+3 "-+ 40 1 ,0 RML 3 1&0!-+3 "-+ 14 1

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1.2 ก2,%ก กก1, 3-Vก3ก"3 1&0!& (Fixed Effects) $ ก, 23! "# ก+3 "3$ ,& 3 &3ก 5, 15 , 25 ก+3 "3$ ,& 2 &3ก 5, 15 , 25 ,ก+3 "3$ ,& 1 &3ก 5, 15 , 25 ,3- ,-&0H-- ! 0#'$ % (ICC) &3ก 0.01 , 0.20 7/ & 54 1 !3 0 RML 3 1&0!&"-+ 35 1 0 FML 3 1&0!&"-+ 32 1 ,0 EB 3 1&0!&"-+ 2 1 -0 FML 3 1&0!&"-+&3ก0 RML 15 1

3 1&0!-+3 (Random Effects) $ ก, 23! "# ก+3 "3$ ,& 3 &3ก 5, 15 , 25 ก+3 "3$ ,& 2 &3ก 5, 15 , 25 ,ก+3 "3$ ,& 1 &3ก 5, 15 , 25 ,3- ,-&0H-- ! 0#'$ % (ICC) &3ก 0.01 , 0.20 7/ & 54 1 !3 0 EB 3 1&0!-+3 "-+&+ก 1 2. ก&,-&0'!10ก, 23! "#$ ก,#!+, ก+3 "3 1 4ก ,

2.1 ก&,-&0'!10ก, 23! "#$ ก,#!+, ก+3 "3 1 4ก ก2,%ก กก7, 3"ก3ก" !3 0 FML ,0 RML ,-&0'!-V-+$ ก, 23&0!& ,0 EB ,-&0'!-V-+$ ก, 23&0!-+3 & &-&, - 0.01

2.2 ก&,-&0'!10ก, 23! "#$ ก,#!+, ก+3 "3 1 4ก ก2,%ก กก1, 3-Vก3ก" !3 0 FML ,0 RML ,-&0'!-V-+$ ก, 23&0!& ,0 EB ,-&0'!-V-+$ ก, 23&0!-+3 & &-&, - 0.01 - *-+0ก, 23! "#& ,- $ "3,-* ก2# ,กW" 9

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" 9 -0ก, 23! "#& ,- $ "3,-* ก2#

กก !" !#$

7, 3"ก3ก" 0 FML, 0 RML 0 EB 1, 3-Vก3ก" 0 FML, 0 RML 0 EB

ก./ก(,-&0'!10ก, 23! "#$ ก,#!+, ก+3 "3 1 4ก ,4 & 3- $'

ก&,-&0'!10ก, 23! "#$ ก,#!+, ก+3 "3 1 4ก ก2,%ก กก7, 3"ก3ก" ก-กก2,%ก กก1, 3-Vก3ก" 0 FML ก0 RML 3 $ ก, 231&0!&"ก30 EB 3 -&-*"&, 0.01 ,0 EB 3 $ ก, 231&0!-+3 "ก30 FML ก0 RML 3 -&-*"&, 0.01 -0 FML ก0 RML 3 $ ก, 231&0!&,&0!-+3 "ก"3ก 3 3 -&-*"&, 0.01 & กก, 23&0!& (Fixed effects) 0 FML ,0 RML กX ก 2ก / ,-&0'!$กก 7/-ก 1 Longford. 1993 !3 RML , "ก30 FML [!,ก2&- %ก$ "3,ก+3 &3ก (equal group sizes) ก, 230 RML ,$3, 2& 3 กก30 FML (Searle, Casella and McCulloch. 1992) "3$ &c" ,#1 V$%ก -4V-3 $33, 2$ ,& 2 ,"ก"3ก ก!&. $ " 3&-(Browne. 1998) 0 FML , 1$ ก $% +3ก$ ก 2 ก3 ,3- ,-&0Hก**,$%fก#% 3,g -V-+(likelihood function) & กก ก&0 FML ,0 RML ,-&0'!$ ก, 23&0!&-V-+ -ก 1 Lauren Terhorst (2007) ./ก(&0, 23! "#1c- ! 0#$ !+, &ก./ก($ ก2 2 , -0ก, 23! "#& ./ก(,ก 0 FML 0 RML 0, 23&0!& (Fixed

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Effect) 0ก- &-+*3 ก& 1 (Weight Least Square 1 )0ก- &-+*3 ก& 2 (Weight Least Square 2 )0ก- &-+*3 ก& 3 (Weight Least Square 3 ) ก 1 "3$ ,& 2 3 1 20, 50 , 100 ก 3- ,-&0H-- ! 0#'$ % 2 1 0.10 , 0.20 ,& 1 ก "-, 1 " ,,& 2 3 " ก./ก( !3 0 FML 0 RML g 0ก, 23& &-+ 0 FML , RML g 0ก, 23! "#& 3 RMSD "&-+ &ก & 6 0 -ก, 23&0!-+3 (Random effect) 0 EB 3 $ ก, 231&0!-+3 (RB) "ก30 FML ,0 RML ก0 EB $ ก, 23! "# ก*3 ก3! "#!$ & (Reliability) 7/g 0ก&- *$%$ ก+ก, 23! "#, "X $ *Vก"1/ / $ 3 "ก30

%&ก"''

1. กก./ก(,-&0'!10ก, 23! "#$ ก,#!+, ก+3 "3 1 4ก ก2,%ก กก7, 3"ก3ก" ,ก2,%ก กก1, 3-Vก3ก" !3 0 FML ,0 RML ,-&0'!-V-+$ ก, 23&0!& ,0 EB ,-&0'!-V-+$ ก, 23&0!-+3

$ ก,#!+, ก+3 "3 1 4ก ,,%ก กก 3ก" !2"*+,-#1ก!ก$%0$ ก, 23! "#3 ,-

2. ก./ก(! ก2&1 Vg - "3 1กกก1 V$ ,-V &ก./ก(.Vก%- - %3 ก21 Vg - 3 , ,& 3 1 V $%0ก%+2'! -1 V$ ,& 1 , 2 7/ 1 "33 1-V- *$%& ก,# !+,$ ก ก3

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%&ก"'(# )

1. ก./ก(,-&0'!10ก, 23! "#$ V 13- V2# (Fully Conditional Model) 2. ก./ก(,-&0'!10ก, 23! "# ก""-- ! 0# (Autocorrelation) 3. ./ก(0ก, 23&0!-+3 &&$ 3 "-+ &ก0ก, 230 p

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0* ก'5%.'. (2548). ก"#. 3. ก: !"#$ %&'(ก)#'*'(*.

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