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á standard LRT method, LRT-STP¼ ² u LRT method with scale purification, LRT-SP[\ pure anchor ! " "# LRT method with pure anchor, LRT-PA$% & © ' ( ) * + likelihood ratio test, LRT#~$, - . °[. / ¤ 0 1 ¶"2 4 differential item functioning, DIF+ 3 4 4 * 5 5. / 67 8 9 % : B ; < = é > ? ¶@ 4 ;é A ability differencePB U DIF¶"C é * DIF percentage[\DIFD E DIF patternFG < = D H0 Type I error£+ ;power5
Ø Ù I J ~ë & . / ð K °% & ¤ #DD H0 L 1 M N ~O P
Q R S 5+ ;¤ ã % & ¤ #~DIF + 3 + 4 4 T é U æ V L q DIF ¶"C é * W ¼ =°X Y Z 5 é [ ~DIF D E constantV \] 4 ;ó $ð ^ °LRT-SP#D_ü ` a 9 LRT-PA#= DIF¶"C é * Ô W ¤ #D+ ;L 1 ` b9 LRT-ST#Fc\] 4 ;© ó $V DIF ¶ "C é * W ¼ d 30%<LRT-SP#e f ü Íg b+ ;5¤ ã cDIF D E balanced<© 7 \] 4 ;ó $£h Ø Ù i ü % & ¤ #L 1 j ü k l + 3 4 4 5 m n Ø Ù N Í~GRM. °o ( ) * + # DIF+ 3 R S ~+ 3 ¼ ² u p DFTDq ` r s ~bDIFC é * B U 1 t u v+ 3 4 4 W ¼ + 3 Ø Ù w x 5 y z Ñ {0 1 ¶"2 4 P( ) * + P² u P$, - .
A Comparison of Three Likelihood Ratio Test Method in
Assessing Differential Item Functioning for Polytomous Items
Abstract
Three different DIF assessment methods, “LRT -ST”, “LRT -SP” and “LRT -PA” was compared in assessing DIF under the graded response model. Three independent variables were manipulated in the simulation study , including the ability difference of subjects, the percentage of DIF items in the test and the DIF patterns. The dependent variables were Type I error and power of DIF assessment.
The results showed that in a variety of simulated situations, Type I error rates of three methods were well-controlled. The difference in power between three methods was pretty small, and power rates decreased while the DIF percentage increased. Furthermore, when the DIF pattern was constant and the mean abilities of the two groups were equal, the LRT-SP performed better than LRT-PA. With the percentage of DIF items in the test increased, the other two methods yielded slightly higher power than LRT-ST. When the abilities of the two groups were unequal and the DIF
percentage increased up to 30%, LRT-SP method showed higher power than other two methods. However, when the DIF pattern was balanced, all three methods performed similarly under every conditions in this study. It was recommended that the scale purification or DFTD strategy should be included into the LRT method when assessing DIF for polytomous items.
Keywords: differential item functioning, likelihood ratio test, scale purification, graded response model
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U r TIMSSThe Trends in International Mathematics and Science StudyP
NAEPThe National Assessment of Educational Progress$æ ¦ ÷ ö r "P¿ "$R - ¶"56r ¼ ~ é "p / é
"D B U + 3 0 1 ¶"2 4 I · 7 5
~ qy é "DIF+ 3 ¤ # 1 é / IRT [\IRT 67 á IRT ã é 5õ á DIRT DIF+ 3 ¤ #q( ) * + # likelihood ratio test, LRT; Thissen, Steinbreg, & Wainer, 1988\¶"£0 1 2 4 + B #differential functioning of items and tests, DFIT; Raju, van der Linden, &
Fleer, 19955 ( ) * + #1 ~ é "£ é "DIF+ 3 5á 8 D 1 i ü ± ~ é "DIF+ 3 ( ) * + # ð ß ó c Bolt, 2002; Cohen, Kim, & Wollack, 1996; Kim & Cohen, 1998; Stark, Chernyshenko, Oleksandr & Drasgow, 2006; Wang, 2004; Wang & Yeh 200367 LRT#1 [ + 3 © '& DIFÂ | x DIF\/ | x DIF[ ° 5 ~ é " ¤ ã @ º i ü ~$, - .
° ( ) * + #+ 3 0 1 ¶"2 4 <~D H0 M N + ó ck l
Ankenmann, Witt, & Dunbar, 1999; Kim & Cohen, 1998; Bolt, 20026 ö Thissen$3 ~1988§ u Í( ) * + #¦ é "DIF+ 3 5
§ 9 ~ DIFé [ <äB U qDIF¶" 1 4 O | $ ! HV DIF¶"* "bP# \] q$ "b! Hð ß "¹ · = ¨ © DIF+ 3 Ø Ù Lord, 198056 cB U DIF¶"* % 10% < DIF+ 3 ¤ #DD H0 Type I errorù i $ & ' inflationü (~D H0 & ' ù M <DIF+ ;powerâ © 1 ¯ 5 ) 6* " q @ u Ͳ u scale purification ( Candell & Drasgow, 1988;
Clauser, Mazor, & Hambleton, 1993; French & Maller, 2007; Hidalgo-Montesinos & Gómez-Benito, 2003; Holland & Thayer, 1988; Lord, 1980; Park & Lautenschlager, 1990; Miller & Oshima, 1992; Navas-Ara & Gómez-Benito, 2002; Wang & Su,
2004V á . / i ü ~DIF¶"© % 20%ð ß °² u ~© ' ¤ #+ 1 [E % q4 M N D H0 4 Ù 5© cB U DIF¶"* % 20%<Â ² u D H0 G + i $ & ' =ù M ð ß 5 6 "#constant-item method, CI ;Thissen, Steinberg, & Wainer, 1988; Wang & Yeh, 2003( u ÍÓ V i ü ~ 6¤ # DIF+ 3 <ä4 w , qDIFDIF-free¶"- ! "anchor item 1 q4 M N DIF+ 3 D H0 W ¼ + 3 4 4 6Â Wang2008§ u Ía ! ´+ 3 DIF-free-then-DIF¿ à DFTDq ` 5á . / WangP SuniShih2010 i ü DFTDq ` î q4 M N DIF+ 3 D H0 Ó 1 ` . u v+ ;5
m n + Ø é [ / IRT ( ) * + # á ~ é . ° ² u q ` Ö h 4 r [³ ~ é § i ü 1 [q4 M N D H0 / cB U DIF¶"* ® qbC é * <² u 4 4 Ö h º 4 q4 i 0 5 î ¼ ² u ( ) * + #~
º ¼ ( ) * + #[\ DIF-free¶"- ! " "#[6% & ¤ # . / Ó * 5 6% & ¤ #~ é . °
cG B U 1 : \] ? ¶@ f B V @ * 5 © '\] r x >P& Ê P Ì Í Î P2 3 4 $? ¶@ ~6G B U + _ü Ö h q§ 0 1 < @ Ã Ä a w Ï Ð B U § B ² % Á 5 p( \] =À Ö h ó '5 c? ¶@ ¥ 9 \] p ñ 6 © '~¶"+ - + $ 5 qp5 © ð ^ e Ð ¶"â ® q§ ½ 0 1 ¶"2 4 5 é "=À Ö ¶"s 7 é £? ¶@ 8 ~4 ;D9 y : c4 ;ó 'A Ý É 9 © ';] \] ~¶"s 7 é + < © ó $<J Ã#6\] ? ¶@ B U Íó '8 ~< = =6¶"Â = ® qDIF ü (Cohen, Kim, & Baker, 1993; Kim & Cohen, 19985
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? ¶@ ~'Þ ¶"+ < 5ä ? \] £B Þ \] D- - é > Ó [G µ \] DICC 9 ' ã + e ® q| x 0 1 ¶"2 4 ¶"1 _J r 15 1 | x 0 1 ¶"2 4 § 11 ü ͧ 9 'È 7 ~6¶"E % 0.5 < B Þ \] § è 4 ; 5 bI J 6¶"9 B Þ \] ? ¶@ Ö 5 H I º  9 \] ? ¶@ =À 6¶"bI £ Ú I q§ © '5 F J £XK L M £\] ICC ë C ó N § < é >~0.7£0.5O P Q _ Á 5 Ö 9 ó '4 ;6R 4 ; S 0.55? ¶@ =À ? ;] * B Þ ; ] ~Ð "+ 0.2 < 6Â Ø D2 9 qÍ Î 5V © 7 ? ¶@ 4 ;bA q\] L ? \] 6<e à 6¶"® quniform DIF5
¤ ã r Ù 9 b4 ;£A 4 ;? ¶@ =À 2 9 qÍ Î \] © 'º Â r °2§ J \] ICC $ N T Dð ß J Ö aUV >£ W q0 1 6<q\] $ £? ¶@ 4 ; qy =/ r 'Ø | uniform DIF 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0 - 3 .0 - 2 .0 - 1 .0 0.0 1 .0 2 .0 3 .0
x q9 # \] <Â à Ð ¶"® qnonuniform DIF5 2 / | x 0 1 ¶"2 4
é . ~c? ¶@ - ï ð ÷ Ã#X ¿ ø Yp 'Á © 'Á ¦ j ü =Ö q & 1 4 x - B U p² _ r ¿ "PZ 7 "$[ \ - "D [\] ^ _ ² _Likert-type scaleP3 ` ² _ $E £ ² _5 ~ Ö á é > © 'é p £ © '$, ¦ Ï q? ¶@ - - @ a x 5æ 2 T ¦ b ÷ i f Í & é . ¦ é [ © '
é ï ð õ c é . q$, - . Graded Response Model,
GRM; Samejima,1969P» ² . Rating Scale Model, RSM; Andrich, 1978Pã G é . Partial Credit Model, PCM; Masters, 1982[\ u ã é é . Generalized Partial Credit Model, GPCM; Muraki, 19935 & . ¿ ø r °{ P $, - . nonuniform DIF
§Samejima1969§u ÍdC é . 5 9 é [ q ? x - - ï ð r] ^ ² _ pÖ e C P k ; » ² Reise
& Yu, 19905 ~ ? ¶@ - ñ ò < è 7 f º = Ö 9 U indirectIRT. Embretson & Reise, 20005
a Í4 ; θ? ¶@ ~ > g βij° 8 - Pix∗(θ)5 i ij i ij i ix a x j m a P 1,..., )] ( exp[ 1 )] ( exp[ ) ( * = = − + − = β θ β θ θ 1
6 8 - operating characteristic curves, OCC ∗(θ)
ix
P Â h i <
º _ J 4 ; θ? ¶@ ~ i" % x é [ + < 5 ai i" V >
£ βij i" j : = g mi g: mi+1 - > :
r{q j : - = Â q : I £ 5
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² _  ? ¶@ ~ ë» é Þ + 0 1 §q " ~ = À ù Ö ó ' 5 o p 6 q ë¶" r Õó ' g Vs N g D i 05 [ 4 ; θ? ¶@ = À ~ i" + x é < 1 §° t ° 4 _ J {
∑
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= = = = − + = ∑ − + ∑ − + = 0 0 0 0 0 [ ( )] 0 ]} ) ( [ {exp } )] ( [ exp{ ) ( j i j m x x j i j x j i j x P θ λ δ δ λ θ δ λ θ θ 4 λ i i" I £ δje Ö ² _ i" j : g 5 %P ã G é . ã G é . Ö §Masters1982u Í a ~ Ö u o ø pø YF é # © ° ü ( w ã é ¶" ? ¶@ 1 q ã é é 5 A i f d ~ v Õ 7 Ö " ~ » é Þ q ? ( % A é 5 « % b é 5 H I < 4 9 6 & . 5 4 ; θ? ¶@ ~ i" + x é < 1 §° t ° 5_ J {∑
∑
= = = = − ≡ ∑ − ∑ − = 0 0 0 0 0 ( ) 0 ] ) ( [exp )] ( exp[ ) ( j ij m r r j ij x j ij ix i P θ δ δ θ δ θ θ 5 δij i" j : f I £ 5 P u ã é é . 6 . Muraki1992 dPCM. = ¦ 5 w ' G ² ë¶" q © ' V > £ x ~ 5GPCMÖ & ® q y x . c B U ë¶ " a ó ' < , PCM. Fä ¥ s N ë¶" r ' ' Õg < RSM. 5GPCM. 1 §° t ° 6_ J {( ) 0 )] ( [exp )] ( exp[ ) ( 0 0 0 0 0 − = − ∑ − ∑ =
∑
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= = = = j i ij m k i ij k j ij i x j ix a b b a b a P θ θ θ θ 6 aiÖ N i" V > £ bij i" j : I £ 5" 90 Q z » ² ¤ » > u < B U " D © ¥ Õ Ö ø ) é " ß [\pR -»²{®[| 6 5 »²{®=À ¯ 4 £P° x $ N G Ç · 7 5ú T +ü _ü»²¤ *z ± u "DBU J « Í üq! H * "56~ é PR -¶ "» ²+BDIF I · 7 5 [³ ~ é . î i f Í ó c DIF+3 ¤ # J h i , ÙSwaminathan & Rogers, 1990; Zwick, Donoghue, & Grima, 1993;
Clauser,Mazor, & Hambleton, 1993; Rogers & Swaminathan, 1993; Fidalgo,
Mellenbergh, &Muñiz, 20005æ ¦2 & º i f V ~ é "DIF +3 +5~ é "DIF+3 ¤ #ä | +1é & {/IRT [\IRT 5 @M U [F J é - A<= matching variable é ó ' ?¶ @ 4;ó ' F´ @eG ?¶ @4; - A<= +3 4;ó ' ?
¶ @~' Þ ¶ "+ < p¶ " Ö h ó $ 6Ã Ä a ¶ "
\4; 5¶ Ç IRT ¤ #} \¶ " £4; ~ +5/IRT ¤ #¦ ~ <A Ò £5b6ê @ § ö 5
Mantel-Haenszel#GMH; Mantel & Haenszel, 1959Ppoly-SIBTEST#Shealy & Stout, 1993P 4 l é [#logistic discriminant function analysis,LDFA;
Miller & Spray, 1993\ordinal logistic regressionOLR; Zumbo, 1999F=IRT
+3 ¤ #q( ) *+ #likelihood ratio test, LRT; Thissen, Steinbreg, &
Wainer, 1988\¶ "£012 4+B#differential functioning of items and tests, DFIT; Raju, van der Linden, & Fleer, 19955" 8 ~$ , -. ° ( ) *+ #+3 01¶ "2 4<~DH 0M N+ó ck l Ankenmann, Witt, & Dunbar, 1999; Kim & Cohen, 1998; Bolt, 20026 ö ( ) *+ #¦ é "DIF+3 5
( ) *+ #³ Ö § Thissen$ 3Thissen, Steinberg, & Gerrand, 1986;
Thissen, Steinberg, & Wainer, 1988, 1993 9 DIF+3 +5[% IRT. LRT# ¶ "DIF+B<6 7 [°%: f {
P \ ] =À s NBU § q¶ " Ló ' º Âq § q¶
" , qDIF6§ ½ . compact model5D´ G °t ° 7 k . likelihood devianceÓ[GC2_J D{
GC2 =−2×log(likelihoodcompact) 7
P é +3 D¶ "studied item î ¶ " \ ] ó $ Ds N º « Ð ¶ " \ ] ?¶ @ ©ó ' õ N a pb 6 § ½ d . augmented modelU ' È k likelihood devianceÓ [G _J D{A2
G2A =−2×log(likelihoodaugmented) 8
%P k d . £ . Dlikelihood deviance01Ó[G _J D2
º ÂG2 G2 G2
A
: D05rr~ f X « ¶ "b ó $ eC § £ 1Fä' < « ¶ "a £b \ ] ©' eC § £ 25äG ä 9 DC § £2
¤ eLRT# +3 ¶ "®qDIF5
[G ®q30Þ ¶ "V£ . 5 ABU 6<ä7 +B1 "Ö h ®quniform DIFe` aR; . s NBU § q¶ " \ ] =À Ló ' º Âa G BU ÓÃDIF¶ "Ó k G FU ~d . C2 î 1"I £ \ ] Ló ' Ds NAêq V > £ \ ] = À Ö ó ' 6< k G F³ ´ k2A 2 C G £ 2 A G D0Â G § 9 +Ø . 2 X ó 0: ¶ " 1"I £ G2 " C § £ 1 ¤ é A =) ª 2 3.84 ) 1 ( = X ä k ´ DG ä 92 3.84e 1"®quniform
DIF5r +3 ¶ "~V > £ +Ö h q§ 01º ÂÖ h ®qnonuniform DIF <Õ è 7 ~ f q ¶ " \ ] =À V > £ ©' AI £ ó ' eG ' È " C § £ 2 1 ¤ é A) ª º 2 3.84 ) 1 ( = X Fä @
Õ +3 ¶ "Ö h ®qDIFA©~Á Ö ¼ & DIF<eX è ~ f q 1 "I £ \V > £ \ ] L©' 6<G " C § £ 2 2 ¤ é A) ª 2 5.99 ) 2 ( = X äG ä 92 5.99e Ð ¶ "®qDIFd9 Ð "Ö
uniform DIF pÖ nonuniform DIFe é > é a \b +3 5
DIF
~ DIF+3 © IRT p/ IRT +3 ¤ #LÄ á R; r ' ² common metric [©' \ ] ¶ "-> *55 r ' ² ÂÖ § ½ A<= matching variable~ DIF+3 <Ã Ä w
Ï A<= Ö P©®DIFü( ¶ "äBU q DIF¶ "e ² ? contamination =O | 4; ! H [\© w +BØ Ù5=[³ LN Í cBU ®qbC é *DIF¶ "<DH 0Â « i $ & ' ù M ü( 6<+BÍ ØÙâ ©1¯ 5 ) 6* "q
@u Í ² u scale purification ( Candell & Drasgow, 1988; Clauser, Mazor, & Hambleton, 1993; French & Maller, 2007; Hidalgo-Montesinos & Gómez-Benito, 2003; Holland & Thayer, 1988; Lord, 1980; Park & Lautenschlager, 1990; Miller & Oshima, 1992; Navas-Ara & Gómez-Benito, 2002; Wang & Su,
20045~ Ö BU ®qDIF ü( ¶ "C A<= î ¥ [F t à DIF ¶ "Õ , DA<= BU ¶ " DIF+3 5
± 8 LI J ~DIF+3 ¼ ² u 1q4 X A DH 0Óu v+3 4 4rFidalogPMellenberghiMuniz2000~3PLM° i ü~DIF¶ "C é * 10%[°<MH#1qP _üAcDIF¶ "C é *W ¼ 15%< ² u MH-2£ Q ² u MH-iê4q©Y _ ü=ä¥ DIF¶ "C é *u vd30%<Õ °MH-iê4 $ P ØÙ5= ~SuiWang2005 ' È J i üMH-2PMH-i4 Ùa 9 MH5 ¤ ã ~ é "DIF+3 +Wang iSu2004b N Í ~Mantel#\GMH# ¼ u J q' È 4 Ù ² u Mantel-2£ Q ² u
Mantel-i4 Ùa 9 Mantel# ² u GMH-2£ Q ² u GMH-i 4 Ùa 9 GMH#5 J q N Í LR#+B é "<~ä È \DIF ¶ " ²bð K °¼ u ' È 1[ q4 Í +BÍ uniform DIF
Hidalgo-Montesions & Gómez-Benito, 2003; Su & Wang, 20055
Ç =T +² u X ~DIF ¶ "C é *5A <4q5P 4 Ùc
G BU q DIF¶ "<Ââ ¼ u DH 0ê i $ & ' ù M ü( 5 u o 6ü( "#constant-item method ,CI; Thissen, Steinberg,
& Wainer, 1988; Wang & Yeh, 2003( u Í ± º ÷ T 6#*ó $ I £#equal-mean-difficulty, EMD\ "#all-other-item, AOI+ B4 Ù Wang, 2004; Wang & Shih, 2010; Wang & Yeh, 20035Vc " # DIF+3 <ä4a w Õ , q DIFDIF-free¶ "- ! "anchor item¥ é BU ¶ " +3 1q4 M NDH 0 W¼ +3 4 4Candell & Drasgow, 1988; Cheung & Rensvold, 1999; Stark et al.,
2006; Thissen et al., 1988; Wang & Yeh, 200356ÂWang2008§ u Í a ! ´ +3 DIF-free-then-DIF¿ à DFTDq ` DFTD-#â Ö á Í ³ ©14®qDIF ¶ "- DIF-free¶ "[6 ! " "#DIF+B ! " RS ö Þ ¶ "Wang & Yeh, 2003; Shih &
Wang, 20095á . / Wang$ 32010i üDFTDq ` î q4 M NDIF+3 DH 0Ó1` . u v+ ;5
6 7 ~ á ~ GRM. ° IRT ( ) *+ #á ¼ ² u q ` Ö h 4~®qDIF ¶ "BU q4 M NDH 05 î ¼ ² u ( ) *+ #LRT method with scale purification, [ °¿ à LRT-SP# º ¼ ( ) *+ #standard
LRT method, [°¿ à LRT-ST#[\ pure anchor DIF-free¶ " ~. /ï ð ÷ ÃDIFü( D¶ "- ! " "#LRT method
with pure anchor, [°¿ à LRT-PA#§ 9 LRT-PA#Ö [w = ÃDIF ü ( D¶ "- ! "r s DIF+3 DDH 0M N q³ k l 4 Ù 6 ØÙ1- *5 á ¤ #+3 4 4D 5 [+%& ¤ # . / Ó*5 á 6%& ¤ #~GRM . °DDIF+3 4 45 [°¿ ø LRT-SP#[\LRT-PA#D+3 f 5 LRT-SP#+3 f r°{ 1BU § q¶ " DIF+3 F 2 +3 Í DIFD¶ "C A<= î F 3· BU ë ¶ " DIF+3 F 4· f 2£ f 3M % ó ?DIF+3 ØÙó ' v 5 LRT-PA#6 7 Ö ~. /ï ð [" Ã DIFü( ¶ "- ! "G ( ) *+ # "# ¶ " DIF+3 5
t 9 _ 1§ -@ Matlab $ GRM. D. /ï ð ´ ¥ [IRTLRDIF DIF +3 5
6 7 89 %: B ; <= é > ?¶ @4;é Aability
differencePBU DIF¶ "C é *DIF percentage[\DIFDE DIF pattern5 P ?¶ @4;é A 6ã é 6 7 89 & ð ß é > \ ] 4;ó $ £ \ ] 4;©ó $ 5~ \ ] 4;ó $ ð ß ° ? \ ] £B Þ \ ] 4;L89 0< 1 1 õ E é AF~ó y L B Þ \ ] Z \ ] 6~ \ ] 4;©ó $ ð ß °e ? ; ] 4;£+Ø ó ' =B Þ \ ] 4;89 -1<1 1 õ E é A5 P BU DIF¶ "C é * ~ r 89 j & ©' £DIF¶ "C é *é > 0%P10%P20%P 30%[\40%_J 89 DIF "G 0P2P4P6[\8"m [³ 1i üDIF ¶ ""W¼ « , DH 0& ' ù M ü( Finch,
2005; Wang & Yeh, 200356 /á 89 bE40%DIFC é *[é > LRT-ST#PLRT-SP#\LRT-PA#~bC é *DIF¶ "BU DH
0M N4 45
%P DIFDE
6 7 89 & ©' DIFDE {@ ø $ constant\ ¡ $
balanced5constantDE  § qDIF¶ " ? \ ] q =balanced DE eÖ ¢ DIF¶ " ? \ ] q ¢ e B Þ \ ] q 5 £ ~¬ T ð K °DIF+3 4 4_üQ R 89 ³ ¹ · \³ . & DIFDE 5§ 9 j Þ ² q: £ ¤ n ¬ T ð K º ~© ' £+§ , DIFð ß Ö h ¨©+3 4 4 ~ & DE °é > 89
"Â qI £+01 "e qI £+01=ä89
DIF"W¼ "%"e %qI £+01"e ùqI £+01[6-¥ Ñ 5 ~balancedDIFDE °ä89 DIF " "<e ; ] ë 89 "Vé > £ qI £+015
î +Ø 89 <= ~ ¤ ã § È ? \ ]
10003B Þ \ ] 5003BU - £ 20"5=~ BU uniform DIF 5 õ c DIF D6~ X á uniform DIFð ß ~DIF¶ "I £8 9 + \ ] ~DIF¶ "I £01L " 0.3\ 0 0.05õ E é A $ £DIFü( V§ qð K °ï ð L· . /100?5
G <= DH 0Type I error£+ ;powerDH 0Ö ÃDIF¶ "Y H N = , DIF¶ "ð ^ =+ ;e w N = Í
DIF¶ "ð ^ 5 § I ¦ .05Ì § = é A k ~100 ?· T U 1« 0d.0927 < ÃDIF¶ "H DIF¶ "6 n k 1U ?4 9 Q R äDH 0 9 & ' % Í Q R D 6<Ââ q¥ b+
;J ©®qÁ Ö 5
~ ± DIF¶ "C é * £õ § ¨ DH 0ù M ¨ ©+3 4 46 A~WangiSu2004a e= average signed area
[°¿ à ASAä © ª Ö , DIF+ #ù M Ã 7 5ASA \ ] ~ s 7 é é A+expected score distribution§ $ ã i 5 ° r°{
SAi =(1−ci)(biF −biR) 9
SAi i"signed areaci i"« B£ biFibiReé > B Þ \ ] i ? \ ] ~i"+I £ 5ASAÂ SA ° {
ASA SA I ci biF biR I I i i I i∑1 / =∑1(1− )( − )/ = = = 10
I BU "5cci 0<6< . °ASA° r°{ iF iR F R I i b b I b b ASA=∑ − = − =1( )/ 11 " ° 111 ~ . °ASA \ ] I £D056 cASA 0<Q _ \ ] I £ó ' DIF¶ " \ ] =À Ó¬ ß , q p@© ð ß º _J ¶ qDIF¶ "Aa ] =À \ ] ¨©Ö ° 5 -DcASAä 9 0<Q _B Þ \ ] I £ä 9 ? \ ] I £ Â_J DIF¶ " ? \ ] ß , 5q ð ^ 5~Wang2001 N Í ~
balancedDIFDE § k Í ASAT é U æ 9 0F=~constantDIF DE cDIF¶ "C é * 20%ASA 0.09< "# DIF+3 DDH 0@ 1M NO P Ç =ä~ó ' ð K °ASAu bd0.18< e O | DH 0 $ & ' ù M ð ß F ¤ ã cASA 0<Ââ BU DIF¶ "C é *u bd50%DH 0ê1Ï ¯ ~M NQ R DS5=Wang iSu2004a º = cASA© 0<Mantel#iGMH#~+BDIF
¶ "+DH 0 qù M ü( 5§ 61c î DIF¶ "C é * £
ASA Ö ¨©DIF+3 · 7 N 56 /~± & ð K °é > k
_1 ¶ " Item a b1 b2 b3 b4 1 1.57 -0.38 0.49 1.04 1.67 2 1.31 -0.61 0.63 1.37 1.82 3 1.63 0.01 0.67 1.33 1.91 4 0.97 -0.23 0.31 0.98 1.62 5 1.53 -0.31 0.60 1.27 1.79 6 0.99 0.06 0.99 1.50 2.02 7 1.79 -0.10 0.35 1.01 1.72 8 1.39 -0.02 0.63 1.28 1.87 9 0.85 -0.35 0.67 0.97 1.24 10 1.20 0.18 0.70 1.29 1.42 11 1.05 -0.37 0.03 0.54 1.01 12 1.42 -0.56 -0.13 0.67 1.54 13 1.23 -0.36 0.53 1.20 1.52 14 1.34 -0.52 0.39 1.54 1.89 15 1.63 -0.53 -0.12 1.27 1.81 16 1.45 -0.20 0.49 1.00 1.59 17 1.71 0.02 0.56 0.93 1.48 18 1.28 -0.44 0.18 0.72 1.31 19 1.43 -0.01 0.39 1.36 1.69 20 1.10 0.10 1.06 1.61 2.01
IRTLRDIFÖ § ® ¦¯ ä University of North Carolina at Chapel HillDavid Thissen m § 6 ] [( ) *+ # k 7 ¥ ° á d . 89 IRTLRDIF1 [ uniform\nonuniformDIFé [± [
IRTLRDIFÖ ² ³ ~ ] 1w qÁ ´ DIF+3 C @á µ * °Û ¼ + ¶ · ó c 69 C @~ ] \ + ó c¤ â 5
IRTLRDIF DIF +3 <1 "#p "#5 ] r ¸ Ö ö "#º Âî +3 ¶ " BU ¶ "Lq Ó¬ ®qDIF5 Ç =~WangiYeh2003 i ücBU ®q5bC é *DIF ¶ " <¨©6#D+ ;56º q @ ¹ "#î ~DH 0 M N+1[y 5 4 Ù ! "W¼ + ;J Du v5 Ç = ²üT ð ^ ö ! "W¼ ! " DIF¶ " < 6 =À ¸ RS ö 4Þ ¶ "- ! "Thissen et al., 1988; Shih & Wang,
2009; Wang & Yeh, 20035
IRTLRDIFÕ 4 ¦+3 ¤ n é p% . [\ é GRM. ï ð Ã# 9 Rasch. p é . 6~ +è 7 g¼ º Á 5
DØÙé > t 9 _ 2d_5_ j ü%& ¤ #~©' ð K ° DH 0£+ ;[°ØÙé DIFDE Ù Ú5 PDIFDE constant _2 \ ] 4;ó $ ØÙ~ë DIF¶ "C é *ð ^ °%& ¤ # DDH 0L1?% O P M NLRT-ST #~bDIFC é *<g. +vd.04 =LRT-SP#£LRT-PA#eÔ ¯ ~.02 d.03 D9 5 + ;¦Ù Â ~bDIF C é *ð ^ °%& ¤ #_üL~.84[+=DIF¶ "C é *W¼ % & ¤ #D+ ;ùqggÔ Y Z 5a ] +=À LRT-SP#£LRT-PA#_ üT é U æ V~b DIF¶ "C é *ð ^ °ù5 LRT-ST#¦ P t 5
_2 \ ] 4;ó $ VDIFDE constant
DIF% ASA
Type I error Power
L-ST L-SP L-PA L-ST L-SP L-PA 0% 0.00 .03 .02 .02 10% 0.04 .02 .02 .02 .99 1.00 .97 20% 0.15 .03 .02 .02 .85 .86 .84 30% 0.19 .03 .03 .03 .87 .91 .90 40% 0.30 .04 .02 .03 .84 .90 .90 _3 \ ] 4;©ó $ ØÙ%& ¤ #DDH 0~ë DIF¶ "C é *ð ^ °ê1Ô ¯ ~.03d.04D9 ¶ Ç 5 \ ] 4;ó $ ð ^ °¦ bAG Ç LM N~1U ?Q R S5
§ _2£_ 3º 1F J % cDIFDE constant<ASA DIF ¶ "C é *ÔW=W¼ Ç =~ T U ð K °Â ASAbE0.3
6Â_J 9 \ ?¶ @=À ± I £01S 0.3%& ¤ #DD H 0êÇ _üO P ÓÃ& ' ù M ð ß 5 ~+ ;¤ ã %& ¤ #~6ð
K °+ ;ÚI 5 \ ] 4;ó $ <¦A § 61c \ ] 4;+0
1w + ; , ¨©5~%& ¤ #LRT-ST#~DIF ¶ "C é *5A < _ü*LRT-SP#£LRT-PA#k l =DIFC é ÔWLRT-SP#ef üÍ gb+ ;5
_3 \ ] 4;©ó $ VDIFDE constant DIF% ASA
Type I error Power
L-ST L-SP L-PA L-ST L-SP L-PA 0% 0.00 .04 .03 .03 10% 0.04 .04 .03 .04 .93 .84 .89 20% 0.15 .04 .03 .04 .76 .71 .69 30% 0.19 .04 .04 .03 .73 .77 .72 40% 0.30 .04 .03 .03 .70 .71 .70 PDIFDE balanced _4 \ ] 4;ó $ ØÙ~DIFDE balanced<_J BU \ ] =À Ö ° ð ^ ~Wang2001 N Í 6<§ k Í ASA T é U æ 9 0V~ë & DIF¶ "C é *DH 01M N~U ?Q R DS5 =? º 1F J % 6ð K °§ k Í ASAL 0%& ¤ #DD H 0º L?% ó cO P M N5 ~+ ;¤ ã LRT-ST#÷f ü³ + 3 4 Ù² u #£pure anchor#~6ð K °ÓÃ#i 0 ÚI 4 45
_5 \ ] 4;©ó $ ØÙ~ë & DIF¶ "C é *°6%& ¤ #DDH 0êÇ M NO P » ?% ?¶ @ 4;01D¨© t W¼
.015~+ ;¤ ã º ?% \ ] 4;01¨©%& ¤ #D+ ;L*z \ ] 4;ó $ <X A V~DIF ¶ "C é *" b<0n " ä 5=
DIF¶ "C é *W¼ %& ¤ #+ ;LqÔ Y Z ó 5D°LRT-ST
#_üg LRT-SP#£ LRT-PA#eÃ#f ü5 4 45
_ 4 \ ] 4;ó $ VDIFDE balanced DIF% ASA
Type I error Power
L-ST L-SP L-PA L-ST L-SP L-PA 0% 0.00 .02 .02 .02 10% 0.00 .02 .02 .02 .97 .98 .95 20% 0.00 .02 .03 .02 .99 .98 .97 30% 0.00 .03 .03 .03 .93 .92 .92 40% 0.00 .03 .03 .03 .93 .91 .89 _5 \ ] 4;©ó $ V DIFDE balanced DIF% ASA
Type I error Power
L-ST L-SP L-PA L-ST L-SP L-PA 0% 0.00 .03 .04 .04 10% 0.00 .04 .03 .04 .89 .86 .82 20% 0.00 .04 .03 .04 .91 .88 .88 30% 0.00 .04 .03 .03 .78 .79 .77 40% 0.00 .03 .03 .03 .78 .74 .70 %P Ø
m n [+é [%& ¤ #~& ð K D+B°Â DIF¶ "C é *bE
40%PASAbE0.3DH 0L4M N~1U ?Q R SÓÃù M ð ß 5 ä | +=À DIF¶ "C é *ÔW%& ¤ #D+ ;LqÔ DY Z 5 =~ & ©' DIFDE °L1c \ ] 4;+01+ ;§ , I ¨©5 ~ DIFDE constantV \ ] 4;ó $ ð K °LRT-SP#£ LRT-PA#_üó 0©ä VDIF¶ "C é *ÔW ¤ #D+ ;1 ` b9 LRT-ST#5=~ \ ] 4;©ó $ ð ^ °1F J % c DIF¶ "
C é *© 9 30%<LRT-ST#_ü` ¼ 9 LRT-SP#£LRT-PA#=
DIF¶ "C é *W¼ d30%<LRT-SP#ef üÍ gb+ ;5 ~DIF DE balancedð K °©7 \ ] 4;ó $ £h LRT-ST#ù÷j ü ³ +3 4 ÙLRT-SP#£ LRT-PA#ÓÃ#i 0 ÚI u o 4 45a ] + =À ~ T U ð K °%& ¤ #~+ ; +01ó c. © º Â
_J %& ¤ #~GRM. °D DIF+3 4 4T é ó æ 5
%& ¤ #~©' DIF++3 ð ß 1á _6d_9¦Ù Ú5~& ©' ð K °c DIFð ^ £<%& ¤ #D+ ;L* z a ] + ;¦b=c DIFð ^ W¼ %<%& ¤ #D+ ;Âg. X A A£a ] + ;ØÙó 0©ä FÇ =~ù qDIF ð ^ °1F J % %& ¤ #D+ ;eÚI °X VA 9 a ] + ; Ñ 7 14Ö § 9
~¶ " +I £ ÷b6 ~6I £+¥ W¼ 0.3DIF ²DIF+3 4 Ùº , q½ ä ¨©5 _6 \ ] 4;ó $ V DIFDE constant1~4°D+ ; L-ST L-SP L-PA DIF% ASA 1 2 3 4 1 2 3 4 1 2 3 4 10% 0.04 .99 .99 1.00 1.00 .96 .98 20% 0.15 .99 .98 .90 .53 1.00 .96 .97 .52 1.00 .97 .93 .46 30% 0.19 .97 .95 .93 .48 .99 .97 .95 .58 .99 .97 .90 .55 40% 0.30 .98 .95 .91 .53 .98 .95 .97 .70 .99 .97 .94 .69 _ 7 \ ] 4;©ó $ VDIFDE constant1~4°D+ ; L-ST L-SP L-PA DIF% ASA 1 2 3 4 1 2 3 4 1 2 3 4 10% 0.04 .94 .92 .92 .75 .89 .88 20% 0.15 .97 .89 .70 .46 .90 .88 .62 .43 .97 .85 .62 .30 30% 0.19 .87 .80 .65 .39 .89 .85 .75 .41 .86 .80 .63 .35 40% 0.30 .89 .71 .76 .44 .90 .73 .77 .44 .89 .76 .73 .42
_8 \ ] 4;ó $ VDIFDE balanced1~4°D+ ; L-ST L-SP L-PA DIF% ASA 1 2 3 4 1 2 3 4 1 2 3 4 10% 0.00 .97 .98 .95 20% 0.00 .99 .98 .98 .98 .98 .97 30% 0.00 .97 1.00 .81 .99 .99 .78 .99 .98 .78 40% 0.00 1.00 .97 .99 .79 1.00 .98 .97 .71 1.00 .97 .96 .63 _9 \ ] 4;©ó $ VDIFDE balanced1~4°D+ ; L-ST L-SP L-PA DIF% ASA 1 2 3 4 1 2 3 4 1 2 3 4 10% 0.00 .89 .86 .82 20% 0.00 .96 .87 .90 .86 .90 .86 30% 0.00 .93 .86 .55 .93 .85 .61 .93 .83 .56 40% 0.00 .88 .80 .85 .58 .87 .78 .80 .53 .88 .79 .75 .38 á [°¶ " º 1ü Í DIF~£ð ^ °D+ ;0 153d5é > ÃDIFP~+ qDIF[\~+ q DIFDð K °§ G µ D ICC5 P0dP4ë Q _?¶ @~6¶ "+0é d 4é < 5*53£ 41F J Í DÚI <u 6Ñ 7 ~DIFð ^ °DIF Í < 1u b+ ;+vF ¤ ã " 3£51ü Í j ñ î a ] ³ P ! Óà ä z ¾ u <6Ñ 7 ~ DIF > + I 5H I + ;C Ç X A 5
3 ÃDIF¶ " 4 DIF¶ "
5 DIF¶ "
Ü £ ë B <= +3 ØÙDDH 0\+ ;§ , ¨ © %& ¤ #DØÙ <1é [Analysis of VarianceÓ[¿ ~ # Scheffé methed ú ´ *5[°é [ØÙé , DH 0£+ ; ã é Ù Ú5 DH 0 " _101ü Í ¨©DH 0D³ 6 7 <= G \ ] 4;0 F1,30=78.035P η2=0.722[\¤ #0F2,30=5.462P η2=0.2675" ØÙº I J ³ I N ò - ¨©G \ ] 4;0£DIFC é *F4,30=6.286P η2 η2
~%& ¤ #DH 0L?% \ ] 4;0¨©c \ ] 4;©ó $ <LRT-ST#DDH 0 q.01d.02W¼ À £LRT-SP#£LRT-PA#e t W¼ .01A§ 9 W¼ À £ó c© %& ¤ #DDH 0êM N~O P Q R S5 _ 10 DH 0<1é [ØÙ F Eta Abilitydifference 1 0.001 78.035 0.000 0.722 DIFpattern 1 < 0.001 0.347 0.560 0.011 Percentage 4 < 0.001 1.951 0.128 0.206 method 2 < 0.001 5.462 0.009 0.267 STSP abilitydifference * DIFpattern 1 < 0.001 0.000 1.000 0.000 abilitydifference * percentage 4 < 0.001 6.286 0.001 0.456 abilitydifference * method 2 < 0.001 0.780 0.467 0.049 DIFpattern * percentage 4 < 0.001 0.130 0.970 0.017 DIFpattern * method 2 < 0.001 3.728 0.036 0.199 percentage * method 8 < 0.001 0.585 0.782 0.135 30 < 0.001 60 a R ¤ = .990 (ô ´ R ¤ = .980) + ; " _111ü Í ¨©+ ;D³ 6 7 <= G \ ] 4;0 F1,23=525.271P η2=0.958PDIFC é *F3,23=72.073P η2=0.904PDIFDE F1,23=68.169P η2=0.748[\¤ #0F2,23=4.717P η2=0.2915" ØÙº I J ³ I N ò - ¨©G DIFDE £DIFC é *F3,23=39.191P η2=0.836P \ ] 4;0£DIFC é *F 3,23=7.707P η2=0.501[\ \ ] 4 ;0£¤ #0F2,23=3.835P η2=0.2505
] 4;§ ©ó $ Á ó $ <+ ;ÚI W¼ VcDIFC é *ÔW<+ ; W¼ À £º D¼ ä 5~LRT-ST#+ ; 1W¼ 14%DIFC é *W¼ d 1W¼ 20%F=LRT-SP#£LRT-PA#_üT é U æ c \ ] 4 ;ó $ <+ ; 1W¼ 21%d e1W¼ 26%£28%5 _11 + ;<1é [ØÙ F Eta abilitydifference 1 0.216 525.271 0.000 0.958
DIFpattern 1 0.028 68.169 0.000 0.748 balanced constant
percentage 3 0.030 72.073 0.000 0.904 10%20%30%40% method 2 0.002 4.717 0.019 0.291 STPA abilitydifference * DIFpattern 1 < 0.001 0.993 0.329 0.041 abilitydifference * percentage 3 0.003 7.707 0.001 0.501 abilitydifference * method 2 0.002 3.835 0.037 0.250 DIFpattern * percentage 3 0.016 39.191 0.000 0.836 DIFpattern * method 2 0.001 1.920 0.169 0.143 percentage * method 6 < 0.001 1.204 0.340 0.239 23 < 0.001 48 a R ¤ = 1.000 (ô ´ R ¤ = .999)
6 7 %& ©' ( ) *+ # ~GRM . Ó*5 LRT-ST
#PLRT-SP#[\ LRT-PA#~DIF+3 +DH 0\+ ;5 ØÙI J %& ¤ #DDH 0L \ ] ?¶ @ 4;D01=` vbAÂ ~
\ ] 4;©' [\ASAbE 0.3ð ^ °G Ç 1M N~O P Q R S561Ù DH 0_ü£Woods2009( ) *+ # ~ $ , -. ï ð + ØÙ| 5 + ;¤ ã ~ & ©' DIFDE ?¶ @ 4;+01' È I ¨©%& ¤ #D+3 4 4 + ;L \ ] ?¶ @ 4;D01=Ú I °X 6i ü£Â 20109 é ï ð D ØÙó ' 5Ì § F J LRT-ST#PLRT-SP#[\LRT-PA#+3 ØÙD01ó c. © V% @D+ ;LqDIF¶ "C é *W¼ =°X Y Z 61Ù %& ¤ #~
DIF+3 +4 4T é U æ 5ä é [~DIFDE constantV \ ] 4;ó $ ð ^ °LRT-SP#D+ ;~ ð K ` b9 LRT-PA#= DIF¶ "C é *ÔW ¤ #D+ ;L1` b9 LRT-ST#F=c \ ] 4;©ó $ VDIF¶ "C é *© 9 30%<LRT-ST#_ü` ¼ 9 LRT-SP #£LRT-PA #Ç =DIF¶ "C é *W¼ d 30%<LRT-SP#ef üÍ gb+ ;5 ¤ ã cDIFDE balanced<©7 \ ] 4;ó $ £h ØÙi ü LRT-ST#+3 4 4` a 9 & ¤ #A01à © 5
m n [+é [1c ~GRM . °o %& ©' ( ) *+ # DIF +3 <DDH 06 7 ?% \ ] 4;01D¨©=+ ;¤ ã 6 7 e?
% \ ] 4;01PDIF¶ "C é *[\DIFDE D¨©5-@á T U o ³ ~ ] IRTLRDIF~GRM. °[( ) *+ #¦ DIF+3 i ü6 ] ~ ð K °÷Ï q +3 4 4Ç =~T ï ð \ ] ?¶ @~ 4;£ I £+01[\BU DIF¶ "C é * £LÉ @Ã#8M £r B6 6RS ~+3 ¼ ² u p DFTDq ` r s 1t u v+3 4 4W¼ +3 ØÙ w x 5
á IRTLRDIF( ) *+ # ~GRM . ØÙN Í LRT-ST#PLRT-SP#[\ LRT-PA#%@9 DIF¶ "+3 4 401©ä J I J Í IRTLRDIF ] ~ é "DIF +3 +T x 5=~´ +RS 1 u < ð K W¼ B <= 89 º p é [ ] $ [ Ò w á ( ) *+ #~GRM. °+3 4 45 ØÙº i ü~ GRM. °89 DIF ¶ "<DIF
¶ "+3 Í DIF < ÚI b9 DIF¶ "6ð ß I ¨©DIF¶ " a ] + ;56~ é . ¶ "I £01~ë I £+ ĺ 1, ¬ ¦ ¯ á ¤ 5 =~ Süq01¶ "2 4+3 ó y 8 6 "ê É9 é . ï ð 6RS ¬ ¦ @1 ² u [\DFTDq ` ~ é "01¶ "2 4+3 ¤ #+ ë é . ï ð +3 é [[ á ² u [\DFTDq ` ~ é "+3 ¤ #+DIF+3 4 45
 (2010)5 ; Å Æ ä Æ BU § k 7 8 ¬ Í Ç Å È 5
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