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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 % %#$$ 1 ก!" 6.4 % L &ก "`กI !&I!5$ !" 0.05 m$ %*ก!"กก Van der Leeden (1997) , *Iกก-%*" b**)%,1,*ก%"#.*ก$&ก!"ก-+ก-+-""ก -% !&,5+I*% %#$`& m$ก-""$-%% %#$` !ก% +1) *$+ ก กI%!*&$ -""!&!$, I+ก "ก "! aกb b& !""%.&)ก!" !&!. (total sample size) I+ก " !"#"% -%!! !"+.&)ก!"+Iก%(number of groups) กก !&!. +กกก Mok(1995) Van der Leeden -% Busing. (1994) -% Cohen (1998) *(%กก %*ก! # -&I!-%I+ก "+), *!..&)ก!"+Iก%(number of groups) กก+I"%ก%(number of individuals per group)
-&I-""-%% %#$` ก% %#$%#(residual error) ก !"$ 1 !. L &!$,+-&I) +กกก Busing (1993) -% Van der Leeden -% Busing (1994) L &J*ก+I%-""45(simulation) " ก- L &J* กI%!*&$ -""!&!$,+*$-&I*&กก *& +1) ก+ก.-%*&!"ก- !"ก%!.+-&Iก'#$+Iก%กก 100 ก%., Maas -% Hox (2001) , *ก " #$+Iก% 30 ก% กL &J* RML +*$*
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26
-&I -#$+Iก%*&กq กL & !ก%+*$$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!$,& -% *
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31
µ "!!$ # +%#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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33
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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34
+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 &$, *" *& # $1!-
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36
ก *& +1) -""' (Full Maximum Likelihood) !.ก Iก !. !.$ 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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37
( ) ∑∑
−=
−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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38
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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39
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$, *, *
2 jkθ -% *3 jkθ !ก
MdAV
jkk/222
*2 =θ
)( *
22/3
1*3 kjkjkjk AdAC θθ −= −
-% ( )12
3/3
−+= πτσjkjkAAC
L & /
31
3 jkjkjk ACAIM −−=
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40
!.$ 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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41
∝)],,,|(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 !.
สวพ.
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42
!- !"% 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 -%-$I !!.ก+!&!.. ()*+!&, *&"&" ก FML RML -% EB
สวพ.
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44
#$+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++!&$(%(%! aก&!ก&#
สวพ.
มทร.ส
วรรณภ
ม
45
(%! aก&- !กก กJ*#$ %$"* J&%#()*กก&)*J ก, *-+)+()*ก -r%$&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 !" #$ก%!& %'ก กก!..1กกก 2 !" L &J*L% (Monte Carlo) m$#$,ก+I%-""45 # 1) +Iก%(Number of groups) -"1 3 # 30, 50, -% 100 2) ก% (Size of groups) -"ก1 3
สวพ.
มทร.ส
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# 5 , 30 -% 50 3) ! a!! &J!. (Intraclass Correlation Coefficient) -"ก1 3 # 0.1( %'ก) 0.2 ( ก%) -% 0.3 ( 5) !!.+#$,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
สวพ.
มทร.ส
วรรณภ
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47
% %#$ !"$ 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
สวพ.
มทร.ส
วรรณภ
ม
48
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) -%(%
สวพ.
มทร.ส
วรรณภ
ม
49
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,
สวพ.
มทร.ส
วรรณภ
ม
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
สวพ.
มทร.ส
วรรณภ
ม
51
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
สวพ.
มทร.ส
วรรณภ
ม
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)
สวพ.
มทร.ส
วรรณภ
ม
53
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
สวพ.
มทร.ส
วรรณภ
ม
54
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)
สวพ.
มทร.ส
วรรณภ
ม
4
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
สวพ.
มทร.ส
วรรณภ
ม
56
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
สวพ.
มทร.ส
วรรณภ
ม
57
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
สวพ.
มทร.ส
วรรณภ
ม
58
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 ! .$
สวพ.
มทร.ส
วรรณภ
ม
59
(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
สวพ.
มทร.ส
วรรณภ
ม
60
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$!" 3
สวพ.
มทร.ส
วรรณภ
ม
61
3 '!ก Univariate Tests # ก ก !" #ก$ "!$%ก ก &ก
ก'(ก'(')*)'$+!,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
สวพ.
มทร.ส
วรรณภ
ม
62
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 $'
สวพ.
มทร.ส
วรรณภ
ม
63
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
สวพ.
มทร.ส
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64
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*
สวพ.
มทร.ส
วรรณภ
ม
65
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*
สวพ.
มทร.ส
วรรณภ
ม
66
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 ! .$
สวพ.
มทร.ส
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67
(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
สวพ.
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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$!" 7
สวพ.
มทร.ส
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7 '!ก Univariate Tests # ก ก !" #ก$ "!$%ก ก &ก
ก'(ก'(')'$+!.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$!" 8
สวพ.
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8 '!ก.!) Scheffe # ก ก !" #ก$ "!$%ก ก &ก
ก'(ก'(')'$+!.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 $'
สวพ.
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ก กก !"#$%& "#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
สวพ.
มทร.ส
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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
สวพ.
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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
สวพ.
มทร.ส
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74
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
สวพ.
มทร.ส
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ก
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ก
. (2545). ก. 2. ก: !"#$ %&'(ก)#'*'(*.
'+) #!. (2536). ก !"#$%"&'(&%)ก *+,ก*#%*+-ก!&* : ก!**' / & """ $ ก &0. *'../ .. ( ก'01ก2'). ก: 3)4*'(* %&'(ก)#'*'(*.
0* ก'5%.'. (2548). ก"#. 3. ก: !"#$ %&'(ก)#'*'(*.
/ +ก7. (2542). +. ก: 8'.กก'*1ก. Ayesha Nneka Delpish. (2006). A Comparison of estimators in hierarchical linear
modeling: Restricted maximum likelihood versus Bootstrap via minimum norm quadratic unbiased estimators. A dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The Florida State University.
Bolstad M.W. (2004). Introduction to Bayesian Statistics. New Jersey: John Wiley and Sons.
Browne, W.J. (1998). Applying MCMC methods to multilevel models. University of Bath, UK.
Browne, W.J., and Draper, D. (2000). Implementation and performance issues in the Bayesian and likelihood fitting of multilevel models. Computational Statistics, 15, 391-420.
Browne, W.J., and Draper, D. (2006). A comparison of Bayesian and likelihood-based method for fitting multilevel models. International Society for Bayesian Analysis, 1(3), 473-514.
สวพ.
มทร.ส
วรรณภ
ม
78
Busing, F. (1993). Distribution characteristics of variance estimates in two-level models. Unpublished manuscript. Leiden: Department of Psychometrics and Research Methodology, Leiden University.
Cohen, M.P. (1998). Determining sample sizes for surveys with data analyzed by hierarchical linear models. Journal of Official Statistics, 14, 267-275. Darandari, E.Z.M. (2004). Robustness of parameter estimates under violations of second-
level residual homoskedasticity and independence assumptions. Unpublished doctoral dissertation. The Florida State University.
Dempster,A. P.,Laird,N.M. and Rubin,D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Seires B, 39, 1-8.
De Leeuw, J., and Kreft, I.G.G. (1986). Random coefficient models for multilevel analysis. Journal of Educational Statistics, 11, 57-85.
Elston, R. C., and Grizzle, J.E. (1962). Estimation of time response curves and their confidence bands. Biometrics, 18, 148-159.
Goldstein, H. (1995). Multilevel statistical model. 2nd Ed. New York: John Wiley. Good, P.I. (1999). Resampling methods: a practical guide to data analysis. Boston/Berlin:
Birkhauser. Heck, R.H., and Thomas, S.L. (2000). An introduction to multilevel modeling techniques. Mahwah, NJ: Lawrence Erlbaum Associates. Hedeker, D., and Gibbons, R.D. (1996a). MIXOR: A computer program for mixed effects
ordinal regression analysis. Computer Methods and Programs in Biomedicine, 49, 157-176.
Hox, Joop J. (2002). Multilevel Analysis : Techniques and Applications. New Jersey : Lawrence Erlbaum Associates. Kreft, Ita, G.G. (1996). Are Multilevel Techniques Necessary? An Overview, Including Simulation Studies. California State University, Los Angeles.
สวพ.
มทร.ส
วรรณภ
ม
79
Laird, N.M., and Ware, H. (1982). Random-effects models for longitudinal data. Biometrics, 38, 963-974.
Lauren Terhorst. (2007). A Comparison of estimation methods when an interaction is omitted form multilevel model. Submitted to the Graduate Faculty of the School of Education in partial fulfillment of the requirements for the degree of Doctor of Philosophy. University of Pittsburgh.
Longford, N. (1987). A fast scoring algorithm for maximum likelihood estimation in unbalanced models with nested random effects. Biometrika, 74(4), 817-827.
Longford, N.T. (1990). VARCL. Software for variance component analysis of data with nested random effects (Maximum Likelihood). Princeton, NJ: Educational Testing Service.
Longford, N. (1993). Random coefficient models. Oxford: Clarendon. Maas, C.J.M., and Hox, J.J. (2001). Robustness of multilevel parameter estimates against
non-normality and small sample sizes. In J. Blasius, J. Hox, E. de Leeuw and P.Schmidt (Eds.), Social science methodology in the new millennium. Proceedings of the Fifth International conference on logic and methodology. Opladen, FRG: Leske+Budrich.
Maas, C.J.M., and Hox, J.J. (2004). Robustness of multilevel parameter estimates against small sample sizes. Department of Methodology and Statistics, Utrecht University.
Maas, C.J.M., and Hox, J.J. (2004). Sample Sizes for multilevel Modeling. Department of Methodology and Statistics, Utrecht University.
Maronna R.A. and Martin R.D. and Yohai J.V. (2006). Robust Statistics: Theory and Methods. England: John Wiley and Sons.
Martinez R.A. and Martinez L.W. (2002). Computation Statistics Handbook with MATLAB. United Stated of America : Chapman and Hall/CRC.
Mason, W.M., Wong, G.M., and Entwistle, B. (1983). Contextual analysis through the multilevel linear model. In S. Leinhardt (Ed.), Sociological methodology (pp. 72-103). San Francisco: Jossey-Bass.
สวพ.
มทร.ส
วรรณภ
ม
80
Moerbeek, M., van Breukelen, G.J.P., and Berger, M. (2001). Optimal experimental design for multilevel logistic models. The Statistician, 50, 17-30.
Mok, M. (1995). Sample size requirements for 2-level designs in educational research. Unpublished manuscript. London: Multilevel Models Project, Institute of education, University of London. Available at http://multilevel.ioe.ac.uk.
Newsom T.J., and Nishishiba M. (2002). Nonconvergence and Sample Bias in Hierarchical Linear Modeling of Dyadic Data, Institute on Aging School of Community Health, Portland State University .
Ramberg, J.S., E.J. Dudewicz , P.R. Tadikamalla and E.F. Mykytka. 1979. A probability distribution and its uses in fitting data. Technometrics. 21: 201-214. Rasbash, J., Browne, W., Goldstein, H.,Yang, M., Plewis, I., Healy, M., Woodhouse, G.,
Draper, D., Langford, I., and Lewis, T. (2000). A useres guide to MlwiN. London: Multilevel Models Project, University of London.
Raudenbush, S.W., and Bryk, A.S. (1992). A Hierarchical Linear Model: Applications and Da ta Analysis Methods. California : Sage Publications.
Raudenbush, S.W., Bryk, A.S., Cheong, Y.F., and Congdon, R. (2000). HLM 5. Hierarchical linear and nonlinear modeling. Chicago: Scientific Software International.
Raudenbush, S.W., and Bryk, A.S. (2002). A Hierarchical Linear Model : Applications and Da ta Analysis Methods. 2nd Ed. California : Sage Publications.
Rosenberg, B. (1973). Linear regression with randomly dispersed parameters. Biometrika, 60, 61-75.
Searle, S.R., Casella, G., and McCulloch, C.E. (1992). Variance components. New York: Wiley.
สวพ.
มทร.ส
วรรณภ
ม
81
Sirichai Kanjanawasee. (1989). Alternative Strategies for Policy Analysis: An Assessment of School Effects on Studentse Cognitive Effective Mathematics Outcomes in Lower Secondary School in Thailand. Doctoral Dissertation, University of California, Los Angeles.
Singer, J.D. (1998). Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. Journal of Educational and Behavioral Statistics, 23(4), 323-355.
Snijders, T.A.B., and Bosker, R. (1999). Multilevel analysis. An introduction to basic and advanced multilevel modeling. Thousand Oaks, CA: Sage. Spiegelhalter, D. (1994). BUGS: Bayesian inference Using Gibbs Sampling. MRC
Biostatistics Unit, Cambridge, UK. Available at: http://www.mrc-bsu.cam.ac.uk/bugs. Van der Leeden, R., and Busing, F. (1994). First iteration versus IGLS/RIGLS estimates
in two-level model: A Monte Carlo study with ML 3. Unpublished manuscript. Leiden: Department of Psychometrics and Research Methodology, Leiden University.
Van der Leeden, R., Busing, F., and Meijer, E. (1997). Applications of bootstrap methods for two-level models. Paper, Multilevel Conference, Amsterdam, April 1-2, 1997. Weisner, A.J. (2004). Comparing bootstrap, MINQUE40, and ReML estimates of variance components of a two-level random coefficient model with non- normal errors: A simulation study. Unpublished doctoral dissertation. University of Pittsburgh, Pittsburgh, PA.