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1 1 9 11 9 12 10 13 11 14 14 15 15 16 19 2 21 21 21 22 23 221 23 222 24 223 27 23 30 231 2PLM 31 232 CCM 31 233 2PLCM 33 234 34 235 35 3 51 31 51 32 53 321 53 322 54 323 2 BTM 54

2 324 55 325 MCMC 57 33 59 331 59 332 63 333 65 334 66 4 96 41 96 42 97 421 98 422 98 423 99 424 100 425 MCMC 102 43 103 431 104 432 2 BTM 104 433 2PLM 104 434 106 5 118 51 118 52 120 521 120 522 120 523 121 524 123 53 125 531 2PLM 125 532 GRM 125 533 2 BTM 127 534 CCM 129

3 535 CCM Bias 131 536 132 537 132 6 165 61 165 62 166

4 11 N J 9 21 CCM (1,1),(1,0),(0,1),(0,0) (100 100 ) 25 22 2PLCM (1,1),(1,0),(0,1),(0,0) (100 100 ) 25 23 2PLM Bias,RMSE,cor(θ, ˆθ) 36 24 CCM Bias,RMSE,cor(θ, ˆθ) 37 25 CCM Bias, RMSE, cor(θ, ˆθ) 37 26 2PLCM Bias,RMSE,cor(θ, ˆθ) 38 27 2PLCM Bias, RMSE, cor(θ, ˆθ) 38 31 2 BTM Bias,RMSE,cor(a,â) 68 32 2PLM Bias,RMSE,cor(a,â) 69 33 Bias, RMSE, cor(a,â) 70 34 Bias/σ a j RMSE/σ a j 71 35 Bias,RMSE,cor(a,â) 72 36 Bias, RMSE, cor(a,â) 73 37 Bias/σ a j RMSE/σ a j 74 38 2 BTM Bias,RMSE,cor(b,ˆb) 75

5 39 2PLM Bias,RMSE,cor(b,ˆb) 76 310 Bias, RMSE, cor(b,ˆb) 77 41 I 107 42 2 BTM Bias,RMSE 108 43 2PLM Bias,RMSE 109 44 2PLM Bias, RMSE 110 45 2PLM Bias/I, RMSE/I 111 51 2PLM Bias,RMSE,cor(θ, ˆθ) 134 52 GRM Bias,RMSE,cor(θ, ˆθ) 135 53 GRM Bias, RMSE, cor(θ, ˆθ) 136 54 BTM Bias,RMSE,cor(θ, ˆθ) 137 55 BTM Bias, RMSE, cor(θ, ˆθ) 138 56 CCM Bias,RMSE,cor(θ, ˆθ) 139 57 CCM Bias, RMSE, cor(θ, ˆθ) 139 58 σγ 2 d(j) = 14075 BTM (1,1),(1,0),(0,1),(0,0) (100 ) 140 59 b jk = 1714 CCM (1,1),(1,0),(0,1),(0,0) (100 ) 140 510 σγ 2 d(j) = 0155 BTM (1,1),(1,0),(0,1),(0,0) (100 ) 140 511 b jk = 0612 CCM (1,1),(1,0),(0,1),(0,0) (100 ) 140

6 21 CCM Bias 39 22 CCM RMSE 40 23 CCM cor(θ, ˆθ) 41 24 CCM Bias 42 25 CCM RMSE 43 26 CCM cor(θ, ˆθ) 44 27 2PLCM Bias 45 28 2PLCM RMSE 46 29 2PLCM cor(θ, ˆθ) 47 210 2PLCM Bias 48 211 2PLCM RMSE 49 212 2PLCM cor(θ, ˆθ) 50 31 Bias/σ a j 78 32 RMSE/σ a j 79 33 cor(a,â) 80 34 Bias/σ a j 81 35 RMSE/σ a j 82 36 cor(a,â) 83 37 Bias/σ a j 4 84 38 RMSE/σ a j 4 85 39 cor(a,â) 4 86 310 Bias 87 311 RMSE 88 312 cor(b,ˆb) 89 313 Bias 90

7 314 RMSE 91 315 cor(b,ˆb) 92 316 Bias 4 93 317 RMSE 4 94 318 cor(b,ˆb) 4 95 41 Bias/I 112 42 RMSE/I 113 43 Bias/I 114 44 RMSE/I 115 45 Bias/I 4 116 46 RMSE/I 4 117 51 GRM Bias 141 52 GRM RMSE 142 53 GRM cor(θ, ˆθ) 143 54 GRM Bias 144 55 GRM RMSE 145 56 GRM cor(θ, ˆθ) 146 57 GRM Bias 4 147 58 GRM RMSE 4 148 59 GRM cor(θ, ˆθ) 4 149 510 BTM Bias 150 511 BTM RMSE 151 512 BTM cor(θ, ˆθ) 152 513 BTM Bias 153 514 BTM RMSE 154

8 515 BTM cor(θ, ˆθ) 155 516 BTM Bias 4 156 517 BTM RMSE 4 157 518 BTM cor(θ, ˆθ) 4 158 519 CCM Bias 159 520 CCM RMSE 160 521 CCM cor(θ, ˆθ) 161 522 CCM Bias 4 162 523 CCM RMSE 163 524 CCM cor(θ, ˆθ) 4 164

9 1 11 J ( A), N, ( 11) 11 N J 1 2 J 1 1 1 0 2 0 1 1 N 1 1 1, 11 1, 0 (item response theory, IRT),,,,, A,, A,,,,, ( )

1 10,,,,,,,,,,, (eg, Kan, van der Ven, Breteler, & Zitman, 2001; Simms, Goldberg, Roberts Watson, Welte, & Rotterman, 2011) 12 j θ θ j (item characteristic function, ICF) 11,,, 2 (two-parameter logistic model, 2PLM) P j (θ i ) = 1 1+exp[ 17a j (θ i b j )] (11), (11) P j (θ i ) θ θ i j, a j,b j j a j θ i = b j (11), j, a j j, (11) θ i = b j P j (θ i ) = 05, j 05 θ, b j j, (11) a j 1 (one-parameter logistic model, 1PLM) (11) c j P j (θ i ) P j (θ i ) = c j + 1 c j 1+exp[ 17a j (θ i b j )] (12)

1 11 3 (three-parameter logistic model, 3PLM),, (12) θ i θ i =, P j (θ i ) = c j, c j j, j, A B, A, A 100%, B 0%, i j ( ) π ij,, θ i π ij, Lord (1980, pp227-228), P j (θ i ), 1 j, θ i 2 θ i j 3 j, θ i 3, (1984), P j (θ i ) π ij,, θ i π ij, P j (θ i ),, P j (θ i ) = E i θi [π ij ] (13) 13 (local independence) (Lord & Novick, 1968, p360), 2 j k, θ j k u j,u k,, U j, j U j = 1, j U j = 0,

1 12, j k Prob(U j = u j,u k = u k θ i ) = Prob(U j = u j θ i )Prob(U k = u k θ i ) = P j (θ i ) u j (1 P j (θ i )) 1 u j P k (θ i ) u k (1 P k (θ i )) 1 u k (14), (14) Prob(U j = u j,u k = u k θ i ) θ θ i U j,u k u j,u k, Prob(U j = u j θ i ) Prob(U k = u k θ i ) θ i U j,u k u j,u k j k, (14), θ i u k u j Prob(u j u k,θ i ) Prob(u j u k,θ i ) = Prob(u k,u j θ i ) Prob(u k θ i ) = Prob(u j θ i )Prob(u k θ i ) Prob(u k θ i ) = Prob(u j θ i ) (15), j k, θ u j u k, (experimental independence) (Lord & Novick, 1968, p 44), j,k i ( ) u ij,u ik, u ij u ik Prob(u ij,u ik ), Prob(u ij ),Prob(u ik ), j, k, Prob(u ij,u ik ) = Prob(u ij )Prob(u ik ) = π u ij ij (1 π ij ) 1 u ij π u ik ik (1 π ik) 1 u ik (16), i θ i, θ i *1, P j (θ i ) = π ij (17) P k (θ i ) = π ik (18) *1 Lord & Novick (1968, p 539)

1 13, θ i j,k u j,u k Prob(u j,u k θ i ) Prob(u j,u k θ i ) = P j (θ i ) u j (1 P j (θ i )) 1 u j P k (θ i ) u k (1 P k (θ i )) 1 u k (19), j k,,, j k, 12,, j k, (2000), j,k, 2,, π ij,π ik Π j,π k, θ i Π j,π k π j,π k Prob(Π j = π j,π k = π k θ i ) θ i Π j,π k π j,π k Prob(Π j = π j θ i ),Prob(Π k = π k θ i ),, Prob(Π j = π j,π k = π k θ i ) = Prob(Π j = π j θ i )Prob(Π k = π k θ i ) (110) j,k, (2000), (,, ), (14), θ i 1 π i1, i θ,, P 1 (θ), 2 1, 1 π i1, (110), 2 π i2,,, 1, 2, 2 1 P 2 (θ),, 1, 2, P 1 (θ)p 2 (θ) 3 θ i

1 14 14,, Ferrara, Huynh, & Michaels (1999), Hoskens & De Boeck (1997), Kreiner & Christensen (2004), Yen (1993), (local dependence),, 3 ( I) ( II) ( III) I,,, 1 X 2 X, X N x 1,,x N, 2 X X M X S 2 X S 2 X = 1 N N (x i M X ) 2 (111) i=1,, 1 2, I, 13, II, 1 ( ),, j k, ( ) j k ( ), θ i

1 15 j k j k, II, 13, III, θ θ,,,,, *2, j k θ θ, θ θ j,k, θ i π ij π ik, Π j Π k, III, 13, 15 14,,,, 3 1 ( a) ( b) ( c) a, J J d(j), a, (graded response model, GRM) *2,, θ i, (Differential Item Functioning, DIF)

1 16 (Samejima, 1969) P d(j) (r θ i ) = P d(j) (r θ i) P d(j) (r+1 θ i) (112), P d(j) (r θ i ) θ θ i d(j) r, Pd(j) (r θ i) θ θ i d(j) r, Pd(j) (r θ i) 2PLM ((11) ), (112) P d(j) (r θ i ) P d(j) (r θ i ) = 1 ] 1+exp [ 17a d(j) (θ i b r ) 1 ] (113) 1+exp [ 17a d(j) (θ i b r+1 ), (113) b r,b r+1, b r b r+1, d(j),,, a, a,, (Bock, 1972) (Muraki, 1992),, Sireci, Thissen & Wainer (1991), b, 2PLM, b, 2 (Baeysian testlet model, BTM) (Bradlow, Wainer, & Wang, 1999) P j d(j) (θ i ) = 1 1+exp [ 17a j d(j) (θ i b j d(j) γ id(j) ) ] (114), P j d(j) (θ i ) θ i d(j) j, a j d(j),b j d(j), 2PLM, j, a j d(j) γ id(j) 0 θ i = b j d(j) (114), b j d(j) γ id(j) 0 P j d(j) (θ i ) 05 θ, (114) exp 17a j d(j) (θ i b j d(j) γ id(j) ) = 17a j d(j) ( (θi γ id(j) ) b j d(j) ) (115), θ i γ id(j) j,

1 17 (114) γ id(j), θ i i d(j) γ id(j) N(0,σγ 2 d(j) ) *3, σγ 2 d(j) 0, d(j), σγ 2 d(j) 0, d(j), σγ 2 d(j) d(j), d(j),, θ d(j), θ i,,, b, b, a, 2 BTM, 3 (three parameter Bayesian testlet model, 3PBTM) (Wang, Bradlow, & Wainer, 2002), (multidimensional item response model, MIRM),, Nandakumar (1990) Li, Bolt & Fu (2006), MIRM MIRM,, c, 2PLM, c,, (constant combination model, CCM) (Hoskens & De Boeck, 1997) exp[u j Z j +u k Z k u j u k b jk ] P(u j,u k θ i ) = 1+exp[Z j ]+exp[z k ]+exp[z j +Z k b jk ] (116) Z j = 17a j d(j) (θ i b j d(j) ) (117) Z k = 17a k d(j) (θ i b k d(j) ) (118), P(u j,u k θ i ) θ i j k u j,u k, (117) a j d(j),b j d(j) j, (118) a k d(j),b k d(j) k, a j d(j) a k d(j), b jk 0, θ i = b j d(j) θ i = b k d(j), b j d(j) *3, 223

1 18 b k d(j), b jk 0, 05 θ (116), θ i j k ω jk, ω jk = ln ( / ) P(Uj = 1,U k = 1 θ i ) P(Uj = 0,U k = 1 θ i ) P(U j = 1,U k = 0 θ i ) P(U j = 0,U k = 0 θ i ) = ln P(U j = 1,U k = 1 θ i )P(U j = 0,U k = 0 θ i ) P(U j = 1,U k = 0 θ i )P(U j = 0,U k = 1 θ i ) = b jk (119) j k ω jk 0, (116) b jk j k, b jk j,k ( ), j,k, c,, 2 (two-parameter logistic copula model, 2PLCM) 2PLCM Braeken, Tuerlinckx, & DeBoeck (2007) (Rasch copula model) 2PLM, 2PLCM, 2 j,k u j,u k P(u j,u k θ i ) =u j u k +( 1) 2 u j u k Q j (θ i )+( 1) 2 u k u j Q k (θ i ) +( 1) u j+u k C(Q j (θ i ),Q k (θ i )) (120), (120) Q j (θ i ), θ i j,, Q j (θ i ) = 1 1 1+exp[ 17a j (θ i b j )] (121), C(Q j (θ i ),Q k (θ i )) Q j (θ i ),Q k (θ i ) ( ),, C(Q j (θ i ),Q k (θ i )) = 1 [ log 1 W (Q ] j(θ j ))W (Q j (θ k )) δ jk W(1) (122) W(x) = 1 exp[ δ jk x] (123) (Frank, 1979), (122) (123) δ jk, θ i j k

1 19, δ jk, θ i j k, δ jk 0 ω jk 0, δ jk j k, d(j),,, θ i, c, c,, 3 (three parameter constant combination model, 3PCCM, Chen & Wang, 2007) hybrid kernel (Ip, 2002), (locally dependent linear logistic test model, LDLLTM, Ip, Smits, & De Boeck, 2009), (conjunctive item response model, CIRM, Jannarone, 1986),, (2005), CIRM 16, (, 2005;, 1992;, 2001),,, (eg, ),,,,,, 14,,, 15,,,,,, (eg, Yang & Gao, 2008 ),,,,

1 20,,,,,,,,,,,,,,, 2,, 3,,, 4,, 5,,,,,, 6,

21 2 21,, *1,, Bradlow et al (1999), 2 BTM, 2PLM,,,,, 95% (mean 95% posterior interval width, M95%PIW),, M95%PIW,, *1, (2010),

2 22, Junker (1991),,, (2013),, 2PLM,,,,,,,,,,,,, Bradlow et al (1999), 1000, 60, (2013), 1000, 12,,,,,,,, ( ),,,,,,,,,,,

2 23 22,, 2,,, (Bias(ˆθ i )), (RMSE(ˆθ i )),, (cor(θ, ˆθ)), 2 221,, (11) 2PLM ( ) 2PLM ( ) P j (θ i ) = 1 1+exp[ 17a j (θ i b j )] (21),,,, 15 c, (116) CCM (120) 2PLCM (CCM, 2PLCM ) CCM ( ) P(U j,u k θ i ) = exp[u j Z j +U k Z k U j U k b jk ] 1+exp[Z j ]+exp[z k ]+exp[z j +Z k b jk ] Z j = 17a j d(j) (θ i b j d(j) ) Z k = 17a k d(j) (θ i b k d(j) ) (22)

2 24 2PLCM ( ) P(U j,u k θ i ) =U j U k +( 1) 2 U j U k Q j (θ i )+( 1) 2 U k U j Q k (θ i ) +( 1) U j+u k C(Q j (θ i ),Q k (θ i )) (23) 1 Q j (θ i ) =1 1+exp[ 17a j (θ i b j )] C(Q j (θ i ),Q k (θ i )) = 1 δ jk log W(x) = 1 exp[ δ jk x] [ 1 W (Q j(θ j ))W (Q j (θ k )) W(1) ] 222,,,, 100, 300, 500, 1000 4, 10, 30, 50 3, 12,, 2j 1 2j (j = 1,2,,J/2, J ), j,k, CCM b jk = 2, 2PLCM δ jk = 30 b jk δ jk,,,,, 2 j,k (1,1),(0,0),, 100, 10, 2 1,2, 100 100, 21 22, CCM 1,2 (1,1),(0,0) (100 ) 84% (73%+11%), 2PLCM 88% (52%+36%)

2 25 21 CCM (1,1),(1,0),(0,1),(0,0) (100 100 ) U 2 = 1 U 2 = 0 U 1 = 1 73% 10% U 1 = 0 6% 11% 22 2PLCM (1,1),(1,0),(0,1),(0,0) (100 100 ) U 2 = 1 U 2 = 0 U 1 = 1 52% 12% U 1 = 0 0% 36%, Hoskens & De Boeck (1997), b jk 2-2,, b jk -2, δ jk 30,, 221 3, 2PLM, (Markov chain Monte Carlo, MCMC) *2, R 1 θ i (i = 1,2,,N) j (j = 1,2,,J) a j b j, N(0,1), U(03,15) U( 20,20) *3 2, 1, 2PLM *2 MCMC 223 *3, R

2 26 ((21) ), N J A 3 U(0,1) NJ, N J B 4 A,B, U, a ij b ij u ij = 1, a ij < b ij u ij = 0 5 j,k (0,0),(1,0),(0,1),(1,1), 1, CCM ((22) ), N (J/2) D,E,F,G 6 U(0,1) NJ/2, N (J/2) H 7 D,E,F,G H, N J U, d ij h ij (u i(2j 1),u i(2j) ) = (0,0), e ij+d ij h ij > d ij (u i(2j 1),u i(2j) ) = (1,0), 8 2PLCM 5 7, N J U 9 U,U,U 2PLM, MCMC 10 1 9 100 11 10 100, Bias(ˆθ i ) = 1 100 RMSE(ˆθ i ) = 100 r=1 1 100 cor(θ, ˆθ) = 1 100 ˆθ ir θ i (24) 100 ) 2 (ˆθir θ i (25) r=1 100 cor(θ, ˆθ r ) (26) r=1, θ i 1, ˆθ i θ i, ˆθ ir 10 θ i r, θ N θ i, ˆθ θ, ˆθ r 10 θ r, (26) cor(θ, ˆθ r ) θ ˆθ r, Bias(ˆθ i )

2 27 ˆθ i, RMSE(ˆθ i ) ˆθ i, cor(θ, ˆθ) ˆθ θ 223 222,, 2PLM, MCMC, MCMC (2008),, t(t = 0,1,2, ), t X t, x t,, t Prob(X t+1 = x t+1 X 0 = x 0,X 1 = x 1,,X t = x t ) = Prob(X t+1 = x t+1 X t = x t ) (27), X t Ω, X t (t = 0,1,2, ) Ω, Prob(X t+1 = x t+1 X t = x t ), p x t+1 x t, t, p x t+1 x t xt x t+1 P, t X t x t, Ω x t+1 π t+1, X t+1 π t+1, π t+1 = π t P (28),, π t+1 = π 0 P t+1 (29), t+1 X t+1 π t+1 t = 0, 1 2 ω(ω Ω),

2 28, π 0, t π t+1 π, (28), π = πp (210),, π, λ, λ ((211) ) λ = (θ 1,θ 2,,θ N,a 1,a 2,,a J,b 1,b 2,,b J ) (211) λ U,, *4 λ Prob(λ U), (MAP ) (EAP ), λ ((212) ) Prob(λ U) = L(U λ)prob(λ) Prob(U) = L(U λ)prob(λ) (212) L(U λ)prob(λ)dλ, (212) L(U λ), λ, U (λ ),, L(U λ) = N i=1 j=1 J P j (θ i ) u ij Q j (θ i ) 1 u ij (213), U λ, (212) λ,,, MCMC, U λ, *4, A B Prob(B A) = Prob(A B)Prob(B) Prob(A), Prob(B A), A B, B, Prob(B), A B, B

2 29 λ, MAP EAP λ MCMC, (Gibbs sampler, Geman & Geman, 1984), (data augmentation and Gibbs sampling, Tanner & Wong, 1987), (Metropolis-Hastings algorithm, Hastings, 1970),, Patz & Junker (1999) - (Metropolis-Hastings within Gibbs algorithm), 222 9, N J U λ, λ, λ (EAP ) (U,U ), MCMC,, θ i,a j,b j, N(0,1), N(1,025), N(0,1) MCMC 1 U z i r j, p j, z i,r j, 1 p j θ i,a j,b j θi 0,a0 j,b0 j, λ λ0 λ 0 = (λ 0 1,λ0 2,,λ0 N+2J ) = (θ 0 1,θ 0 2,,θ 0 N,a 0 1,a 0 2,,a 0 J,b 0 1,b 0 2,,b 0 J) = (z 1,z 2,,z N,r 1,r 2,,r J,1 p 1,1 p 2,,1 p J ) (214) 2 λ 1 1 λ 1 ((215) ) h(λ 1 λ 0 1) = [ ] 1 exp (λ 1 λ0 1 )2 2πσ 2 2σ 2 (215) 3 λ 1 λ0 1, ((216) ) ( Prob(λ α(λ 1 λ 0 1) = min 1 λ 0 1,U)h(λ0 1 λ 1 ) ) Prob(λ 0 1 λ0 1,U)h(λ 1 λ0 1 ),1 (216), (216) λ 0 1 λ0 λ 0 1 λ 0 1 = (λ0 2,,λ0 N+2J ) (217)

2 30 4 U(0,1) (216) λ 1 1 = λ 1, λ 1 1 = λ0 1 5 λ 0 2,,λ 0 N+2J 2 4, λ 1 = (λ 1 1,λ 1 2,,λ 1 N+2J) (218) 6 λ 1 2 5, λ 2 7 6 20000 8 7 20000 λ, (215) σ 2, 4 (l = 1,,N +2J) λ l,,, λ l 25% 50% σ 2,, 7 20000, 1 (burn-in ), 20000,,, 3000 burn-in, 17000 λ 23,, 2PLM CCM, 2PLCM Bias,RMSE,cor(θ, ˆθ), Bias Bias(ˆθ i ) N, 222,, RMSE RMSE(ˆθ i ) N, 222,,,, 222, cor(θ, ˆθ), 2PLM CCM, 2PLCM Bias

2 31 RMSE, θ i 222,, θ i N(0,1), θ i 1, Bias RMSE,, θ i 231 2PLM, 2PLM Bias,RMSE,cor(θ, ˆθ),, 23 232 CCM, CCM Bias,RMSE,cor(θ, ˆθ),, 24, CCM, 2PLM, Bias,RMSE,cor(θ, ˆθ),, 25, Bias, RMSE, cor(θ, ˆθ), Bias,RMSE,cor(θ, ˆθ) 2PLM, CCM Bias, RMSE, cor(θ, ˆθ), 21, 22, 23, CCM Bias, RMSE, cor(θ, ˆθ), 24, 25, 26, CCM,,, 25, N = 100,, 2PLM, CCM Bias 01 (θ i 10%),, N = 300 N = 500, ( 10 ), CCM Bias 01, 21 24,,, Bias

2 32,,, Bias, Bias, 25, N = 100 J = 50, 2PLM, RMSE 015 (θ i 15%),, 22, N = 100, 2PLM RMSE, 22, N 300, J = 10 J = 30 RMSE, J = 30 J = 50 RMSE, RMSE 2,, J = 10 J = 30, J = 30 J = 50 25,, CCM, 2PLM,,,,, 26,, J = 10 J = 30 cor(θ, ˆθ), J = 30 J = 50 cor(θ, ˆθ),, 3,,,, 11,,,,,

2 33,,,, N = 100,,,,,,,,,, 233 2PLCM, 2PLCM Bias,RMSE,cor(θ, ˆθ),, 26, 2PLCM Bias, RMSE, cor(θ, ˆθ),, 27, 2PLCM Bias, RMSE, cor(θ, ˆθ), 27, 28, 29, 2PLCM Bias, RMSE, cor(θ, ˆθ), 210, 211, 212, 2PLCM,,, 27,, Bias 2PLM 27,, 2PLM, RMSE, J = 10, RMSE 010 (θ i 10%),,,

2 34 27,, 2PLM, cor(θ, ˆθ),, J = 10,, 010 cor(θ, ˆθ), 212,, cor(θ, ˆθ) 2PLM, 29, cor(θ, ˆθ), J = 10, N = 1000, 2PLM cor(θ, ˆθ),,,,,,,,,,,,, 234 22,,,,,,, Bradlow et al (1999) 2, 2 BTM

2 35 Bradlow et al (1999),, CCM,,, Bias,, M95%PIW,,,, M95%PIW, M95%PIW,, 3,, Junker (1991),, CCM, 2PLCM,,, 235, ( ),, Bradlow et al (1999),,, CCM, 2PLCM, (2013),,,, Bias,,, RMSE, cor(θ, ˆθ)

2 36, 3,, CCM, 2,, 3, 4,,, (eg,, ),,,,,, N = 100,,,, 23 2PLM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 100,J = 10 0134 0638 0768 N = 100,J = 30 0083 0438 0900 N = 100,J = 50 0017 0356 0936 N = 300,J = 10 0059 0625 0779 N = 300,J = 30 0042 0431 0905 N = 300,J = 50 0028 0349 0940 N = 500,J = 10 0055 0618 0773 N = 500,J = 30 0044 0428 0902 N = 500,J = 50 0056 0342 0941 N = 1000, J = 10 0022 0615 0779 N = 1000, J = 30 0018 0426 0904 N = 1000, J = 50 0019 0347 0938

2 37 24 CCM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 100,J = 10 0237 0683 0764 N = 100,J = 30 0339 0530 0912 N = 100,J = 50 0388 0512 0943 N = 300,J = 10 0096 0649 0770 N = 300,J = 30 0145 0446 0909 N = 300,J = 50 0196 0440 0940 N = 500,J = 10 0078 0646 0759 N = 500,J = 30 0110 0436 0904 N = 500,J = 50 0156 0382 0936 N = 1000, J = 10 0034 0643 0764 N = 1000, J = 30 0052 0421 0907 N = 1000, J = 50 0076 0360 0935 25 CCM Bias, RMSE, cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 100, J = 10 0103 0045-0004 N = 100,J = 30 0256 0092 0012 N = 100,J = 50 0372 0156 0008 N = 300, J = 10 0037 0023-0008 N = 300,J = 30 0103 0015 0004 N = 300, J = 50 0168 0051-0001 N = 500, J = 10 0024 0029-0014 N = 500,J = 30 0066 0008 0002 N = 500, J = 50 0100 0040-0005 N = 1000, J = 10 0012 0028-0015 N = 1000, J = 30 0034-0005 0003 N = 1000, J = 50 0057 0013-0003

2 38 26 2PLCM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 100,J = 10 0133 0758 0678 N = 100,J = 30 0082 0513 0859 N = 100,J = 50 0017 0435 0899 N = 300,J = 10 0059 0752 0685 N = 300,J = 30 0043 0509 0865 N = 300,J = 50 0029 0417 0912 N = 500,J = 10 0055 0756 0670 N = 500,J = 30 0045 0510 0859 N = 500,J = 50 0055 0410 0912 N = 1000, J = 10 0022 0781 0653 N = 1000, J = 30 0018 0509 0861 N = 1000, J = 50 0019 0414 0911 27 2PLCM Bias, RMSE, cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 100, J = 10-0001 0120-0089 N = 100, J = 30-0001 0075-0041 N = 100, J = 50 0000 0079-0037 N = 300, J = 10 0000 0127-0093 N = 300, J = 30 0001 0077-0040 N = 300, J = 50 0001 0068-0028 N = 500, J = 10 0000 0138-0103 N = 500, J = 30 0001 0082-0043 N = 500, J = 50-0001 0068-0028 N = 1000, J = 10 0000 0166-0125 N = 1000, J = 30 0000 0084-0043 N = 1000, J = 50 0000 0067-0027

2 39 21 CCM Bias

2 40 22 CCM RMSE

2 41 23 CCM cor(θ, ˆθ)

2 42 24 CCM Bias

2 43 25 CCM RMSE

2 44 26 CCM cor(θ, ˆθ)

2 45 27 2PLCM Bias

2 46 28 2PLCM RMSE

2 47 29 2PLCM cor(θ, ˆθ)

2 48 210 2PLCM Bias

2 49 211 2PLCM RMSE

2 50 212 2PLCM cor(θ, ˆθ)

51 3 31,,, *1,, Bradlow et al (1999), 2 BTM, 2PLM BTM,,, 2PLM BTM 2PLM BTM, M95%PIW,, 2PLM, BTM, M95%PIW,, 2PLM, BTM, 2PLM,, Chen & Wang (2007), 3PCCM, 3PLM, *1, (2012b),

3 52,,,,, Jiao Kamatani, Wang, & Jin (2012),,, (Rasch, 1960),,, Looney & Spray (1992),,,, Tuerlinckx & De Boeck (2001), CCM, 2PLM,,,, Wainer & Wang (2000), TOEFL, 3PLM 3PBTM,,, (2012), 2PLM GRM, 2PLM GRM,,,,,,,, (Braeken, 2011; Braeken, et al, 2007; Ip, 2010; Ip, Smits, & De Boeck, 2009), Bradlow et al (1999), 2 BTM, 2PLM BTM, 15, BTM 2PLM,,, Chen & Wang (2007) Tuerlinckx & De

3 53 Boeck (2001), CCM, 2PLM 3PLM, 12 15,,,,,,,, Ip (2010), 2 BTM 2PLM,,,,,,,,,, 2 BTM, 2PLM, Ip (2010),,,,, 32,,,, 321, 31, (114) 2 BTM ( ) P j d(j) (θ i ) = 1 1+exp [ 17a j d(j) (θ i b j d(j) γ ad(j) ] (31)

3 54 322 2, 31, (11) 2PLM ( ) P j (θ i ) = 1 1+exp[ 17a j (θ i b j )] (32) 2 BTM ( (31) ) 323 2 BTM 15, (32) a j θ i = b j (32), (32) b j P j (θ i ) = 05 θ i, (31) a j d(j) γ ad(j) 0 θ i = b j d(j) (31), b j d(j) γ ad(j) 0 P j d(j) (θ i ) = 05 θ i, 2PLM 2 BTM, 31 Ip (2010), (31) γ ad(j) 2 BTM 2PLM, (31) 2 BTM γ ad(j) (marginalized testlet item response function, MIRF), MIRF

3 55, (31) a j d(j) a j = τa j d(j) (33) τ = 1 κ 2 (17a j d(j) ) 2 σγ 2 d(j) +1 (34) κ = 16 3 15π (35) b j = b j d(j) (36), 2 BTM 2PLM a j,b j *2, 31, (33) (36) a j d(j),b j d(j) a j,b j,,,, 324,, 4, 5, 31,,,,,, 4,, 300, 1000 2,, 3, 5 (J d(j) = 5) 3 (J d(j) = 3) 2 (J d(j) = 2), 4, 4 5, 4 ( d(j) 4) 4 3 ( d(j) 3) 4 2 ( d(j) 2) *2 MIRF, Ip (2010)

3 56 4 1 ( d(j) 1) 4 ( d(j) 0),, 2 BTM Bradlow et al (1999) 2 BTM 3 (Bradlow et al, 1999; Li, Bolt, & Fu, 2005; Li, Bolt, & Fu, 2006), 3 17 σγ 2 d(j) ˆσ γ 2 d(j) 4 (14075) σγ 2 d(j), 4 (0155) σγ 2 d(j),, (2 ) (3 ) (5 ) 30, *3 1 θ i (i = 1,2,,N) γ ad(j) (a = 1,2,,N; d(j) = 1,2,3,4) a j d(j) b j d(j) (j = 1,2,,20) N(0,1), N(0,σγ 2 d(j) ), U(05,15), N(0,1) *4 2 1 (31), N 20 A 3 U(0,1) N 20 N 20 B 4 A,B U, a ij b ij u ij = 1, a ij < b ij u ij = 0 5 U 322 2 6 3 5 R 7 323 Ip (2010), 2 BTM 2PLM 8 7 R *3 R *4, R

3 57 Bias(ˆλ j ) = 1 R ˆλ jr λ j (37) R r=1 RMSE(ˆλ j ) = 1 R ) 2 (ˆλjr λ j (38) R cor(λ,ˆλ) = 1 R r=1 R cor(λ, ˆλ r ) (39) r=1 9 8 20 Bias(ˆλ j ) RMSE(ˆλ j ), Bias(ˆλ j ) RMSE(ˆλ j ) Bias,RMSE, λ j j, ˆλ j λ j, ˆλ jr ˆλ j r, λ 20 a j b j, ˆλ, ˆλ r r, 6 R, ˆλ ir,, 1000 R = 50 300 R = 100 325 MCMC MCMC (MCMC 223 ) MCMC WinBUGS 14 (Spiegelhalter, Thomas, & Best, 2003), MCMC (slice sampling) (Neal, 1997),, 2PLM θ i N(0,1) a j N(1,025) b j N(0,1) 2 BTM

3 58 θ i N(0,1) a j d(j) N(1,025) b j d(j) N(0,1) γ ad(j) N(0,σγ 2 d(j) ) σγ 2 d(j) Γ(3, 1), MCMC,, burn-in 2PLM 1000 burn-in 2 BTM 1000 burn-in, MCMC, WinBUGS User Manual (Spiegelhalter, Thomas, Best, & Lunn, 2003), 5% 2PLM 4000 2 BTM 5000, MCMC, θ i 0 a j,a j d(j) 1 b j,b j d(j) 0 γ ad(j) 0 σγ 2 d(j) 3

3 59 33 331,,,, Bias,RMSE,cor(a,â),, 2PLM 2 BTM Bias RMSE, a j, 324, a j d(j) U(05,15), a j a j d(j) 323 (33) (35), a j,, a j a j d(j) U(05,15), a j 0289, Bias RMSE, 0289, a j 2 BTM, 2 BTM Bias,RMSE,cor(a,â),, 31 2PLM, 2PLM Bias,RMSE,cor(a,â),, 32, 2PLM, 2 BTM Bias,RMSE,cor(a,â),, 33, Bias, RMSE, cor(a,â), 2PLM Bias,RMSE,cor(a,â) 2 BTM, 2PLM, Bias RMSE a j,, 34, Bias/σ a j RMSE/σ a j,, Bias RMSE a j, 2PLM Bias/σ a j, RMSE/σ a j, cor(a,â)

3 60, 31, 32, 33, 2PLM Bias/σ a j, RMSE/σ a j, cor(a,â), 34, 35, 36, 2PLM Bias/σ a j, RMSE/σ a j, cor(a,â) 4, 37, 38, 39, 2PLM,,, 34,, 2PLM Bias σ a j 01 (a [ j ] 10% ), 2 BTM Bias, 34,,, Bias/σ a j,, 2PLM Bias, 2 BTM, 34,, 2PLM RMSE BTM RMSE,, N = 300, σ a j 01 (a j 10% ), 32,, RMSE/σ a j, 2PLM, 2PLM RMSE, 2 BTM, 2PLM,, 35,,, RMSE/σ a j, 2PLM RMSE, BTM

3 61 33,, 2PLM cor(a,â) BTM,, 5,,, cor(a,â) -015, 36,,, cor(a,â),,,, 2PLM cor(a,â), BTM,, 39, 5,, cor(a,â) 0,, cor(a,â), 4,, 2PLM, BTM, 4,, 2PLM, 5,,,,,,,,,,,,,,,,, 2PLM, a j d(j) a j, Bias,RMSE,cor(a,â)

3 62,, 35,, Bias, RMSE, cor(a,â),, 36, Bias/σ a j RMSE/σ a j,, 37,,,, 37,,, 2PLM Bias BTM Bias,, 37,,, 2PLM RMSE BTM RMSE,, N = 300, σ a j 01 (a j 10% ),,, 36,,, cor(a,â),, 3 2, cor(a,â) 01,,,

3 63,,,,,, 332,,,, Bias,RMSE,cor(b,ˆb),, 2PLM 2 BTM Bias RMSE, b j, 324, b j d(j) N(0,1), 323 (36), b j N(0,1), Bias,RMSE,, b j 2 BTM, 2 BTM Bias,RMSE,cor(b,ˆb),, 38 2PLM, 2PLM Bias,RMSE,cor(b,ˆb),, 39, 2PLM, Bias, RMSE, cor(b,ˆb),, 310, 2PLM, Bias, RMSE, cor(b,ˆb), 310, 311, 312, 2PLM, Bias, RMSE, cor(b,ˆb), 313, 314, 315, 2PLM,

3 64 Bias, RMSE, cor(b,ˆb) 4, 316, 317, 318, 2PLM,,, 310,, Bias, 2PLM 2 BTM 310,, 2PLM 2 BTM, RMSE, J d(j) = 5, d(j) 4,3,2, 2PLM 2 BTM, RMSE 005 (b j 5% ), 314,, RMSE,, 2PLM RMSE, 2 BTM, 317,, RMSE, 2PLM RMSE, BTM 310,, cor(b,ˆb), 2PLM BTM, 318, 5, cor(b,ˆb) 0,, cor(b,ˆb), 4,,2PLM, BTM,,,, 5,

3 65,,,,,,, J d(j) = 5, d(j) 4,3,2,,,,, 2PLM, 323 (36), b j d(j) b j b j b j,, 333,, Lord & Novick (1968), 2PLM a j,, j u j s r bj a j = r bj 1 r 2 bj (310), 2 BTM θ i d(j) γ id(j), γ id(j) θ i, 2PLM, θ i,, r bj

3 66, (310), a j r bj, 2 BTM 2PLM, â j, 2PL M Bias, 2, 2PLM RMSE,, â j 334 31,,,, 2 BTM, 2PLM, BTM, Ip (2010),, 2 BTM 2PLM Bradlow et al (1999),,,, Bradlow et al (1999),,,,,, M95%PIW,,, Chen & Wang (2007) Jiao et al (2012),,,,

3 67,,,

3 68 31 2 BTM Bias,RMSE,cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) 4 0094 0118 0715 N = 300,J d(j) = 5, d(j) 3 0072 0117 0877 N = 300,J d(j) = 5, d(j) 2 0060 0119 0893 N = 300,J d(j) = 5, d(j) 1 0048 0119 0872 N = 300,J d(j) = 5, d(j) 0 0026 0119 0848 N = 300,J d(j) = 3, d(j) 4 0112 0134 0787 N = 300,J d(j) = 3, d(j) 3 0085 0135 0890 N = 300,J d(j) = 3, d(j) 2 0066 0131 0901 N = 300,J d(j) = 3, d(j) 1 0050 0129 0884 N = 300,J d(j) = 3, d(j) 0 0024 0122 0880 N = 300,J d(j) = 2, d(j) 4 0142 0162 0626 N = 300,J d(j) = 2, d(j) 3 0096 0155 0872 N = 300,J d(j) = 2, d(j) 2 0075 0149 0882 N = 300,J d(j) = 2, d(j) 1 0048 0138 0853 N = 300,J d(j) = 2, d(j) 0 0015 0125 0837 N = 1000,J d(j) = 5, d(j) 4 0033 0060 0827 N = 1000,J d(j) = 5, d(j) 3 0020 0056 0963 N = 1000,J d(j) = 5, d(j) 2 0017 0062 0964 N = 1000,J d(j) = 5, d(j) 1 0017 0069 0953 N = 1000,J d(j) = 5, d(j) 0 0007 0071 0948 N = 1000,J d(j) = 3, d(j) 4 0034 0060 0874 N = 1000,J d(j) = 3, d(j) 3 0022 0065 0965 N = 1000,J d(j) = 3, d(j) 2 0018 0069 0968 N = 1000,J d(j) = 3, d(j) 1 0019 0076 0958 N = 1000,J d(j) = 3, d(j) 0 0007 0075 0959 N = 1000,J d(j) = 2, d(j) 4 0039 0065 0805 N = 1000,J d(j) = 2, d(j) 3 0024 0072 0962 N = 1000,J d(j) = 2, d(j) 2 0021 0078 0962 N = 1000,J d(j) = 2, d(j) 1 0019 0085 0945 N = 1000,J d(j) = 2, d(j) 0 0002 0083 0940

3 69 32 2PLM Bias,RMSE,cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) 4 0227 0257 0647 N = 300,J d(j) = 5, d(j) 3 0191 0248 0566 N = 300,J d(j) = 5, d(j) 2 0163 0223 0682 N = 300,J d(j) = 5, d(j) 1 0144 0207 0708 N = 300,J d(j) = 5, d(j) 0 0118 0181 0837 N = 300,J d(j) = 3, d(j) 4 0174 0206 0735 N = 300,J d(j) = 3, d(j) 3 0148 0194 0849 N = 300,J d(j) = 3, d(j) 2 0135 0188 0873 N = 300,J d(j) = 3, d(j) 1 0122 0182 0877 N = 300,J d(j) = 3, d(j) 0 0107 0175 0876 N = 300,J d(j) = 2, d(j) 4 0173 0204 0607 N = 300,J d(j) = 2, d(j) 3 0141 0189 0872 N = 300,J d(j) = 2, d(j) 2 0135 0192 0881 N = 300,J d(j) = 2, d(j) 1 0123 0189 0861 N = 300,J d(j) = 2, d(j) 0 0111 0186 0823 N = 1000,J d(j) = 5, d(j) 4 0155 0169 0797 N = 1000,J d(j) = 5, d(j) 3 0116 0175 0658 N = 1000,J d(j) = 5, d(j) 2 0091 0153 0723 N = 1000,J d(j) = 5, d(j) 1 0082 0136 0726 N = 1000,J d(j) = 5, d(j) 0 0060 0099 0943 N = 1000,J d(j) = 3, d(j) 4 0096 0114 0843 N = 1000,J d(j) = 3, d(j) 3 0076 0107 0949 N = 1000,J d(j) = 3, d(j) 2 0068 0104 0951 N = 1000,J d(j) = 3, d(j) 1 0067 0108 0942 N = 1000,J d(j) = 3, d(j) 0 0055 0101 0953 N = 1000,J d(j) = 2, d(j) 4 0074 0096 0767 N = 1000,J d(j) = 2, d(j) 3 0063 0098 0964 N = 1000,J d(j) = 2, d(j) 2 0062 0104 0959 N = 1000,J d(j) = 2, d(j) 1 0065 0111 0939 N = 1000,J d(j) = 2, d(j) 0 0057 0111 0930

3 70 33 Bias, RMSE, cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) 4 0132 0139-0068 N = 300,J d(j) = 5, d(j) 3 0119 0131-0311 N = 300,J d(j) = 5, d(j) 2 0103 0104-0210 N = 300,J d(j) = 5, d(j) 1 0096 0088-0164 N = 300,J d(j) = 5, d(j) 0 0092 0062-0011 N = 300,J d(j) = 3, d(j) 4 0062 0071-0052 N = 300,J d(j) = 3, d(j) 3 0063 0059-0041 N = 300,J d(j) = 3, d(j) 2 0068 0057-0028 N = 300,J d(j) = 3, d(j) 1 0072 0053-0008 N = 300,J d(j) = 3, d(j) 0 0084 0053-0004 N = 300,J d(j) = 2, d(j) 4 0031 0042-0018 N = 300,J d(j) = 2, d(j) 3 0045 0034 0000 N = 300,J d(j) = 2, d(j) 2 0060 0043-0001 N = 300,J d(j) = 2, d(j) 1 0075 0051 0009 N = 300,J d(j) = 2, d(j) 0 0095 0061-0015 N = 1000,J d(j) = 5, d(j) 4 0122 0109-0030 N = 1000,J d(j) = 5, d(j) 3 0096 0119-0305 N = 1000,J d(j) = 5, d(j) 2 0074 0091-0241 N = 1000,J d(j) = 5, d(j) 1 0065 0068-0228 N = 1000,J d(j) = 5, d(j) 0 0054 0028-0004 N = 1000,J d(j) = 3, d(j) 4 0062 0054-0031 N = 1000,J d(j) = 3, d(j) 3 0054 0042-0015 N = 1000,J d(j) = 3, d(j) 2 0049 0035-0016 N = 1000,J d(j) = 3, d(j) 1 0048 0032-0017 N = 1000,J d(j) = 3, d(j) 0 0049 0026-0006 N = 1000,J d(j) = 2, d(j) 4 0035 0032-0038 N = 1000,J d(j) = 2, d(j) 3 0039 0026 0002 N = 1000,J d(j) = 2, d(j) 2 0042 0026-0003 N = 1000,J d(j) = 2, d(j) 1 0047 0026-0006 N = 1000,J d(j) = 2, d(j) 0 0054 0029-0009

3 71 34 Bias/σ a j RMSE/σ a j Bias/σ a j RMSE/σ a j N = 300,J d(j) = 5, d(j) 4 0459 0481 N = 300,J d(j) = 5, d(j) 3 0412 0454 N = 300,J d(j) = 5, d(j) 2 0359 0361 N = 300,J d(j) = 5, d(j) 1 0334 0305 N = 300,J d(j) = 5, d(j) 0 0317 0214 N = 300,J d(j) = 3, d(j) 4 0214 0247 N = 300,J d(j) = 3, d(j) 3 0217 0204 N = 300,J d(j) = 3, d(j) 2 0237 0198 N = 300,J d(j) = 3, d(j) 1 0250 0184 N = 300,J d(j) = 3, d(j) 0 0290 0183 N = 300,J d(j) = 2, d(j) 4 0108 0146 N = 300,J d(j) = 2, d(j) 3 0154 0119 N = 300,J d(j) = 2, d(j) 2 0207 0148 N = 300,J d(j) = 2, d(j) 1 0259 0176 N = 300,J d(j) = 2, d(j) 0 0331 0210 N = 1000,J d(j) = 5, d(j) 4 0423 0378 N = 1000,J d(j) = 5, d(j) 3 0333 0412 N = 1000,J d(j) = 5, d(j) 2 0256 0314 N = 1000,J d(j) = 5, d(j) 1 0224 0235 N = 1000,J d(j) = 5, d(j) 0 0186 0099 N = 1000,J d(j) = 3, d(j) 4 0214 0186 N = 1000,J d(j) = 3, d(j) 3 0187 0145 N = 1000,J d(j) = 3, d(j) 2 0171 0122 N = 1000,J d(j) = 3, d(j) 1 0167 0110 N = 1000,J d(j) = 3, d(j) 0 0169 0092 N = 1000,J d(j) = 2, d(j) 4 0121 0109 N = 1000,J d(j) = 2, d(j) 3 0134 0090 N = 1000,J d(j) = 2, d(j) 2 0145 0088 N = 1000,J d(j) = 2, d(j) 1 0162 0091 N = 1000,J d(j) = 2, d(j) 0 0187 0100

3 72 35 Bias,RMSE,cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) 4-0135 0216 0642 N = 300,J d(j) = 5, d(j) 3-0091 0199 0657 N = 300,J d(j) = 5, d(j) 2-0045 0175 0729 N = 300,J d(j) = 5, d(j) 1-0002 0163 0767 N = 300,J d(j) = 5, d(j) 0 0041 0154 0833 N = 300,J d(j) = 3, d(j) 4-0196 0251 0741 N = 300,J d(j) = 3, d(j) 3-0132 0219 0713 N = 300,J d(j) = 3, d(j) 2-0073 0188 0753 N = 300,J d(j) = 3, d(j) 1-0021 0176 0761 N = 300,J d(j) = 3, d(j) 0 0026 0155 0872 N = 300,J d(j) = 2, d(j) 4-0291 0328 0628 N = 300,J d(j) = 2, d(j) 3-0213 0273 0616 N = 300,J d(j) = 2, d(j) 2-0123 0218 0642 N = 300,J d(j) = 2, d(j) 1-0064 0206 0616 N = 300,J d(j) = 2, d(j) 0 0003 0166 0821 N = 1000,J d(j) = 5, d(j) 4-0207 0230 0797 N = 1000,J d(j) = 5, d(j) 3-0166 0198 0825 N = 1000,J d(j) = 5, d(j) 2-0117 0159 0856 N = 1000,J d(j) = 5, d(j) 1-0065 0120 0907 N = 1000,J d(j) = 5, d(j) 0-0017 0087 0945 N = 1000,J d(j) = 3, d(j) 4-0274 0287 0873 N = 1000,J d(j) = 3, d(j) 3-0203 0234 0735 N = 1000,J d(j) = 3, d(j) 2-0140 0179 0786 N = 1000,J d(j) = 3, d(j) 1-0075 0129 0840 N = 1000,J d(j) = 3, d(j) 0-0026 0097 0954 N = 1000,J d(j) = 2, d(j) 4-0389 0396 0825 N = 1000,J d(j) = 2, d(j) 3-0291 0317 0636 N = 1000,J d(j) = 2, d(j) 2-0195 0231 0679 N = 1000,J d(j) = 2, d(j) 1-0121 0170 0713 N = 1000,J d(j) = 2, d(j) 0-0051 0116 0934

3 73 36 Bias, RMSE, cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) 4-0229 0098-0073 N = 300,J d(j) = 5, d(j) 3-0164 0082-0220 N = 300,J d(j) = 5, d(j) 2-0105 0057-0163 N = 300,J d(j) = 5, d(j) 1-0050 0045-0105 N = 300,J d(j) = 5, d(j) 0 0014 0035-0015 N = 300,J d(j) = 3, d(j) 4-0308 0117-0046 N = 300,J d(j) = 3, d(j) 3-0217 0084-0177 N = 300,J d(j) = 3, d(j) 2-0140 0057-0148 N = 300,J d(j) = 3, d(j) 1-0071 0047-0123 N = 300,J d(j) = 3, d(j) 0 0002 0033-0007 N = 300,J d(j) = 2, d(j) 4-0432 0166 0002 N = 300,J d(j) = 2, d(j) 3-0310 0117-0256 N = 300,J d(j) = 2, d(j) 2-0197 0069-0240 N = 300,J d(j) = 2, d(j) 1-0112 0069-0237 N = 300,J d(j) = 2, d(j) 0-0012 0041-0016 N = 1000,J d(j) = 5, d(j) 4-0240 0170-0030 N = 1000,J d(j) = 5, d(j) 3-0187 0142-0138 N = 1000,J d(j) = 5, d(j) 2-0134 0097-0108 N = 1000,J d(j) = 5, d(j) 1-0082 0051-0046 N = 1000,J d(j) = 5, d(j) 0-0024 0016-0003 N = 1000,J d(j) = 3, d(j) 4-0308 0227-0001 N = 1000,J d(j) = 3, d(j) 3-0226 0169-0230 N = 1000,J d(j) = 3, d(j) 2-0159 0110-0182 N = 1000,J d(j) = 3, d(j) 1-0094 0052-0118 N = 1000,J d(j) = 3, d(j) 0-0033 0022-0005 N = 1000,J d(j) = 2, d(j) 4-0428 0331 0020 N = 1000,J d(j) = 2, d(j) 3-0315 0246-0326 N = 1000,J d(j) = 2, d(j) 2-0215 0152-0283 N = 1000,J d(j) = 2, d(j) 1-0140 0085-0232 N = 1000,J d(j) = 2, d(j) 0-0053 0033-0006

3 74 37 Bias/σ a j RMSE/σ a j Bias/σ a j RMSE/σ a j N = 300,J d(j) = 5, d(j) 4-0794 0339 N = 300,J d(j) = 5, d(j) 3-0567 0283 N = 300,J d(j) = 5, d(j) 2-0362 0197 N = 300,J d(j) = 5, d(j) 1-0173 0155 N = 300,J d(j) = 5, d(j) 0 0049 0120 N = 300,J d(j) = 3, d(j) 4-1066 0404 N = 300,J d(j) = 3, d(j) 3-0751 0292 N = 300,J d(j) = 3, d(j) 2-0484 0199 N = 300,J d(j) = 3, d(j) 1-0244 0164 N = 300,J d(j) = 3, d(j) 0 0006 0113 N = 300,J d(j) = 2, d(j) 4-1498 0575 N = 300,J d(j) = 2, d(j) 3-1072 0407 N = 300,J d(j) = 2, d(j) 2-0684 0238 N = 300,J d(j) = 2, d(j) 1-0387 0238 N = 300,J d(j) = 2, d(j) 0-0042 0141 N = 1000,J d(j) = 5, d(j) 4-0830 0588 N = 1000,J d(j) = 5, d(j) 3-0646 0492 N = 1000,J d(j) = 5, d(j) 2-0465 0336 N = 1000,J d(j) = 5, d(j) 1-0282 0178 N = 1000,J d(j) = 5, d(j) 0-0083 0056 N = 1000,J d(j) = 3, d(j) 4-1066 0786 N = 1000,J d(j) = 3, d(j) 3-0782 0584 N = 1000,J d(j) = 3, d(j) 2-0549 0381 N = 1000,J d(j) = 3, d(j) 1-0327 0181 N = 1000,J d(j) = 3, d(j) 0-0115 0076 N = 1000,J d(j) = 2, d(j) 4-1484 1147 N = 1000,J d(j) = 2, d(j) 3-1093 0852 N = 1000,J d(j) = 2, d(j) 2-0746 0528 N = 1000,J d(j) = 2, d(j) 1-0484 0294 N = 1000,J d(j) = 2, d(j) 0-0185 0115

3 75 38 2 BTM Bias,RMSE,cor(b,ˆb) Bias RMSE cor(b,ˆb) N = 300,J d(j) = 5, d(j) 4-0022 0173 0986 N = 300,J d(j) = 5, d(j) 3-0004 0148 0989 N = 300,J d(j) = 5, d(j) 2 0002 0146 0990 N = 300,J d(j) = 5, d(j) 1 0009 0134 0991 N = 300,J d(j) = 5, d(j) 0-0003 0127 0992 N = 300,J d(j) = 3, d(j) 4-0018 0166 0989 N = 300,J d(j) = 3, d(j) 3-0008 0152 0991 N = 300,J d(j) = 3, d(j) 2-0002 0148 0991 N = 300,J d(j) = 3, d(j) 1 0006 0141 0992 N = 300,J d(j) = 3, d(j) 0-0004 0134 0993 N = 300,J d(j) = 2, d(j) 4-0028 0142 0994 N = 300,J d(j) = 2, d(j) 3-0046 0142 0993 N = 300,J d(j) = 2, d(j) 2-0038 0138 0993 N = 300,J d(j) = 2, d(j) 1-0012 0132 0994 N = 300,J d(j) = 2, d(j) 0-0014 0126 0995 N = 1000,J d(j) = 5, d(j) 4-0005 0090 0996 N = 1000,J d(j) = 5, d(j) 3 0006 0078 0997 N = 1000,J d(j) = 5, d(j) 2 0006 0074 0997 N = 1000,J d(j) = 5, d(j) 1 0013 0073 0997 N = 1000,J d(j) = 5, d(j) 0 0013 0073 0998 N = 1000,J d(j) = 3, d(j) 4 0002 0089 0997 N = 1000,J d(j) = 3, d(j) 3 0005 0084 0997 N = 1000,J d(j) = 3, d(j) 2 0004 0078 0998 N = 1000,J d(j) = 3, d(j) 1 0009 0078 0998 N = 1000,J d(j) = 3, d(j) 0 0010 0076 0998 N = 1000,J d(j) = 2, d(j) 4 0009 0086 0997 N = 1000,J d(j) = 2, d(j) 3 0002 0086 0997 N = 1000,J d(j) = 2, d(j) 2 0002 0082 0998 N = 1000,J d(j) = 2, d(j) 1 0004 0081 0998 N = 1000,J d(j) = 2, d(j) 0 0011 0076 0998

3 76 39 2PLM Bias,RMSE,cor(b,ˆb) Bias RMSE cor(b,ˆb) N = 300,J d(j) = 5, d(j) 4-0055 0227 0987 N = 300,J d(j) = 5, d(j) 3-0039 0193 0984 N = 300,J d(j) = 5, d(j) 2-0022 0188 0984 N = 300,J d(j) = 5, d(j) 1-0005 0158 0989 N = 300,J d(j) = 5, d(j) 0-0019 0140 0992 N = 300,J d(j) = 3, d(j) 4-0023 0205 0990 N = 300,J d(j) = 3, d(j) 3-0024 0175 0990 N = 300,J d(j) = 3, d(j) 2-0016 0168 0991 N = 300,J d(j) = 3, d(j) 1-0003 0150 0992 N = 300,J d(j) = 3, d(j) 0-0013 0141 0993 N = 300,J d(j) = 2, d(j) 4-0020 0178 0993 N = 300,J d(j) = 2, d(j) 3-0040 0156 0993 N = 300,J d(j) = 2, d(j) 2-0033 0148 0994 N = 300,J d(j) = 2, d(j) 1-0011 0135 0994 N = 300,J d(j) = 2, d(j) 0-0018 0127 0995 N = 1000,J d(j) = 5, d(j) 4-0045 0144 0996 N = 1000,J d(j) = 5, d(j) 3-0030 0131 0991 N = 1000,J d(j) = 5, d(j) 2-0012 0125 0992 N = 1000,J d(j) = 5, d(j) 1 0006 0101 0994 N = 1000,J d(j) = 5, d(j) 0-0003 0076 0998 N = 1000,J d(j) = 3, d(j) 4-0014 0124 0997 N = 1000,J d(j) = 3, d(j) 3-0018 0101 0997 N = 1000,J d(j) = 3, d(j) 2-0012 0096 0997 N = 1000,J d(j) = 3, d(j) 1-0002 0089 0997 N = 1000,J d(j) = 3, d(j) 0-0001 0079 0998 N = 1000,J d(j) = 2, d(j) 4 0007 0110 0997 N = 1000,J d(j) = 2, d(j) 3-0012 0094 0997 N = 1000,J d(j) = 2, d(j) 2-0008 0089 0998 N = 1000,J d(j) = 2, d(j) 1-0002 0085 0998 N = 1000,J d(j) = 2, d(j) 0 0003 0075 0998

3 77 310 Bias, RMSE, cor(b,ˆb) Bias RMSE cor(b,ˆb) N = 300,J d(j) = 5, d(j) 4-0033 0054 0001 N = 300,J d(j) = 5, d(j) 3-0035 0045-0005 N = 300,J d(j) = 5, d(j) 2-0024 0043-0005 N = 300,J d(j) = 5, d(j) 1-0014 0023-0002 N = 300,J d(j) = 5, d(j) 0-0016 0013 0000 N = 300,J d(j) = 3, d(j) 4-0004 0039 0001 N = 300,J d(j) = 3, d(j) 3-0016 0023 0000 N = 300,J d(j) = 3, d(j) 2-0014 0020 0000 N = 300,J d(j) = 3, d(j) 1-0009 0010 0000 N = 300,J d(j) = 3, d(j) 0-0008 0007 0000 N = 300,J d(j) = 2, d(j) 4 0008 0035-0001 N = 300,J d(j) = 2, d(j) 3 0005 0013 0000 N = 300,J d(j) = 2, d(j) 2 0005 0010 0000 N = 300,J d(j) = 2, d(j) 1 0002 0004 0000 N = 300,J d(j) = 2, d(j) 0-0004 0001 0000 N = 1000,J d(j) = 5, d(j) 4-0040 0054 0000 N = 1000,J d(j) = 5, d(j) 3-0037 0053-0006 N = 1000,J d(j) = 5, d(j) 2-0018 0051-0006 N = 1000,J d(j) = 5, d(j) 1-0007 0027-0003 N = 1000,J d(j) = 5, d(j) 0-0016 0003 0000 N = 1000,J d(j) = 3, d(j) 4-0016 0035 0000 N = 1000,J d(j) = 3, d(j) 3-0024 0017 0000 N = 1000,J d(j) = 3, d(j) 2-0017 0018-0001 N = 1000,J d(j) = 3, d(j) 1-0011 0011 0000 N = 1000,J d(j) = 3, d(j) 0-0011 0003 0000 N = 1000,J d(j) = 2, d(j) 4-0002 0024 0000 N = 1000,J d(j) = 2, d(j) 3-0015 0008 0000 N = 1000,J d(j) = 2, d(j) 2-0010 0006 0000 N = 1000,J d(j) = 2, d(j) 1-0005 0004 0000 N = 1000,J d(j) = 2, d(j) 0-0008 -0001 0000

3 78 31 Bias/σ a j

3 79 32 RMSE/σ a j

3 80 33 cor(a,â)

3 81 34 Bias/σ a j

3 82 35 RMSE/σ a j

3 83 36 cor(a,â)

3 84 37 Bias/σ a j 4

3 85 38 RMSE/σ a j 4

3 86 39 cor(a,â) 4

3 87 310 Bias

3 88 311 RMSE

3 89 312 cor(b,ˆb)

3 90 313 Bias

3 91 314 RMSE

3 92 315 cor(b,ˆb)

3 93 316 Bias 4

3 94 317 RMSE 4

3 95 318 cor(b,ˆb) 4

96 4 41,,,,, Ip (2010), National Assessment of Educational Progress (NAEP),,, (2001), 2000 GRM 2PLM, 2PLM GRM, Keller, Swaminathan, & Sireci (2003),,, Wainer & Wang (2000), TOEFL,,,, Ip (2000),,, Lee (2000),,,, Wainer et al (2007, p 182),,

4 97,,, Anastasi (1961, p 121) Thorndike (1951, p 585), Guilford (1936, p417), Kelly (1924),,, Wainer (1995), Low School Admission Test (LSAT),, Wang & Wilson (2005), 17 154 8,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 42,,,,,,,,,,

4 98 421, (114) 2 BTM ( ) P j d(j) (θ i ) = 1 1+exp [ 17a j d(j) (θ i b j d(j) γ id(j) ] (41) 15, 2 BTM b a c,, a d(j),,, c, 3,,, 2 (Braeken et al, 2007; Hoskens & De Boeck, 1997; Ip, 2002; Ip et al, 2009),,, b 2 BTM 422,, 2,, (11) 2PLM ( ) P j (θ i ) = 1 1+exp[ 17a j (θ i b j )] (42),, 2 BTM ( (41) )

4 99 423 42,,,,,,, U i u i ( i ) i θ i,, θ i I(θ i ), I(θ i ) ((43) ) [ ( ) 2 ] I(θ i ) = E logl(u i θ i ) θ i θ i, ˆθ i,,, θ i ˆθ i σ 2ˆθi I(θ θ i ) (44) i (43) σ 2ˆθi θ i = 1 I(θ i ) (44),, I(θ i ) ˆθ i,,,, 2PLM 2PLM, (45) ( Lord & Novick (1968) (2002) ) J I(θ i ) = (17) 2 a 2 j P j(θ i )Q j (θ i ) (45) j=1, (45) P j (θ i ) 2PLM ((42) ), Q j (θ i ) 1 P j (θ i ), 2PLM (45),, 2 BTM 2 BTM,

4 100 (46) ( Wainer, Bradlow, & Du (2000) Ip (2010) ) I(θ i ) = J j=1 [ ] (17a j d(j) ) 2 exp(17a j d(j) (θ i b j d(j) γ id(j) )) (1+exp(17a j d(j) (θ i b j d(j) γ id(j) ))) 2 dγ id(j) (46), 2 BTM (46), 424,, 4, 5, 41,,,,,,, 4, 300, 1000 2,, 3 5 (J d(j) = 5) 3 (J d(j) = 3) 2 (J d(j) = 2),, 4 5 4 ( d(j) 4) 4 3 ( d(j) 3) 4 2 ( d(j) 2) 4 1 ( d(j) 1) 4 ( d(j) 0)

4 101,, 3, *1 1 θ i (i = 1,2,,N) γ id(j) (a = 1,2,,N; d(j) = 1,2,3,4) a j d(j) b j d(j) (j = 1,2,,20) N(0,1), N(0,σγ 2 d(j) ), U(05,15), N(0,1) *2 2 1 2 BTM ((41) ), N 20 A 3 U(0,1) N 20 N 20 B 4 A,B U, a ij b ij u ij = 1, a ij < b ij u ij = 0 5 U 422 2 6 423, 5 7 3 6 R 8 7 R, Bias(Î(θ i)) = 1 R Î r (θ i ) I(θ i ) (47) R r=1 RMSE(Î(θ i)) = 1 R ) 2 (Îr (θ i ) I(θ i ) (48) R, (47), (48) Î(θ i ) I(θ i ), Îr(θ i ) I(θ i ) r r=1, 7 R,, 1000 R = 50, 300 R = 100 *1, R *2, R

4 102 425 MCMC, 5 MCMC MCMC (Neal, 1997), WinBUGS 14 (Spiegelhalter et al, 2003) MCMC,,, 2PLM θ i N(0,1) a j N(1,025) b j N(0,1) 2 BTM θ i N(0,1) a j d(j) N(1,025) b j d(j) N(0,1) γ id(j) N(0,σγ 2 d(j) ) σγ 2 d(j) Γ(3, 1),,, 2PLM θ i 0 a j 1 b j 0 2 BTM θ i 0 a j d(j) 1 b j d(j) 0 γ id(j) 0 σγ 2 d(j) 3,, λ,, burn-in

4 103 2PLM 1000 burn-in 2 BTM 1000 burn-in,, WinBUGS User Manual (Spiegelhalter et al, 2003), 5%, 2PLM 4000 2 BTM 5000 43,,, 2PLM 2 BTM Bias,RMSE,, Bias, θ i = 300, 275,,275,300 25 Bias(Î(θ i)),, 425,, θ i N(0,1), N(0,1) θ i 0125 (-300 300, -2875 2875 ), Bias,, RMSE, Bias, θ i = 300, 275,,275,300 RMSE(Î(θ i)),,,,, 2PLM 2 BTM Bias RMSE, ( ),,, θ i = 300, 275,,275,300

4 104 Bias,RMSE, I, Bias,RMSE, I, 431,, I, 41 432 2 BTM, 2 BTM Bias,RMSE,, 42 433 2PLM, 2PLM Bias,RMSE,, 43, 2PLM, 2 BTM Bias,RMSE,, 44, Bias, RMSE, 2PLM Bias,RMSE 2 BTM, 2PLM, Bias RMSE I,, 45, Bias/I RMSE/I, Bias RMSE I, 2PLM Bias/I, RMSE/I, 41, 42, 2PLM Bias/I, RMSE/I, 43, 44, 2PLM Bias/I, RMSE/I 4, 45, 46, 2PLM,,

4 105 45,, 2PLM, 2 BTM, Bias,, d(j) 2,1,0,,, Bias I 01, 43,, Bias/I, 45,,, Bias/I, Bias,, 41,, Bias/I,,, Bias 45,, 2PLM, 2 BTM, RMSE,, d(j) 2,1,0,,, RMSE I 01, 44,, RMSE/I, 46,,, RMSE/I, RMSE,, 42,, RMSE/I,, RMSE 423,,,,,,,

4 106,,, d(j) 2,1,0,,,,,,,,, 434 41,,,,,,,,,, 4,, Ip (2010), (2001), Keller et al (2003), Wainer & Wang (2000),,,,,

4 107 41 I I N = 300,J d(j) = 5, d(j) 4 6645 N = 300,J d(j) = 5, d(j) 3 6322 N = 300,J d(j) = 5, d(j) 2 5752 N = 300,J d(j) = 5, d(j) 1 5140 N = 300,J d(j) = 5, d(j) 0 4519 N = 300,J d(j) = 3, d(j) 4 8035 N = 300,J d(j) = 3, d(j) 3 7836 N = 300,J d(j) = 3, d(j) 2 7519 N = 300,J d(j) = 3, d(j) 1 7155 N = 300,J d(j) = 3, d(j) 0 6834 N = 300,J d(j) = 2, d(j) 4 8833 N = 300,J d(j) = 2, d(j) 3 8658 N = 300,J d(j) = 2, d(j) 2 8389 N = 300,J d(j) = 2, d(j) 1 8217 N = 300,J d(j) = 2, d(j) 0 8010 N = 1000,J d(j) = 5, d(j) 4 6645 N = 1000,J d(j) = 5, d(j) 3 6322 N = 1000,J d(j) = 5, d(j) 2 5752 N = 1000,J d(j) = 5, d(j) 1 5140 N = 1000,J d(j) = 5, d(j) 0 4519 N = 1000,J d(j) = 3, d(j) 4 8035 N = 1000,J d(j) = 3, d(j) 3 7836 N = 1000,J d(j) = 3, d(j) 2 7519 N = 1000,J d(j) = 3, d(j) 1 7155 N = 1000,J d(j) = 3, d(j) 0 6834 N = 1000,J d(j) = 2, d(j) 4 8833 N = 1000,J d(j) = 2, d(j) 3 8658 N = 1000,J d(j) = 2, d(j) 2 8389 N = 1000,J d(j) = 2, d(j) 1 8217 N = 1000,J d(j) = 2, d(j) 0 8010

4 108 42 2 BTM Bias,RMSE Bias RMSE N = 300,J d(j) = 5, d(j) 4 1252 1291 N = 300,J d(j) = 5, d(j) 3 1066 1047 N = 300,J d(j) = 5, d(j) 2 0852 0889 N = 300,J d(j) = 5, d(j) 1 0546 0592 N = 300,J d(j) = 5, d(j) 0 0311 0366 N = 300,J d(j) = 3, d(j) 4 0977 1053 N = 300,J d(j) = 3, d(j) 3 0855 0925 N = 300,J d(j) = 3, d(j) 2 0670 0741 N = 300,J d(j) = 3, d(j) 1 0525 0599 N = 300,J d(j) = 3, d(j) 0 0416 0492 N = 300,J d(j) = 2, d(j) 4 0712 0830 N = 300,J d(j) = 2, d(j) 3 0691 0803 N = 300,J d(j) = 2, d(j) 2 0533 0648 N = 300,J d(j) = 2, d(j) 1 0427 0552 N = 300,J d(j) = 2, d(j) 0 0436 0530 N = 1000,J d(j) = 5, d(j) 4 0458 0501 N = 1000,J d(j) = 5, d(j) 3 0267 0325 N = 1000,J d(j) = 5, d(j) 2 0143 0215 N = 1000,J d(j) = 5, d(j) 1 0216 0274 N = 1000,J d(j) = 5, d(j) 0 0005 0160 N = 1000,J d(j) = 3, d(j) 4 0424 0487 N = 1000,J d(j) = 3, d(j) 3 0382 0437 N = 1000,J d(j) = 3, d(j) 2 0295 0360 N = 1000,J d(j) = 3, d(j) 1 0374 0428 N = 1000,J d(j) = 3, d(j) 0 0273 0329 N = 1000,J d(j) = 2, d(j) 4 0312 0394 N = 1000,J d(j) = 2, d(j) 3 0310 0380 N = 1000,J d(j) = 2, d(j) 2 0244 0327 N = 1000,J d(j) = 2, d(j) 1 0248 0325 N = 1000,J d(j) = 2, d(j) 0 0175 0258

4 109 43 2PLM Bias,RMSE Bias RMSE N = 300,J d(j) = 5, d(j) 4 3140 3148 N = 300,J d(j) = 5, d(j) 3 3562 3571 N = 300,J d(j) = 5, d(j) 2 4406 4414 N = 300,J d(j) = 5, d(j) 1 5021 5028 N = 300,J d(j) = 5, d(j) 0 5767 5773 N = 300,J d(j) = 3, d(j) 4 1833 1850 N = 300,J d(j) = 3, d(j) 3 2043 2060 N = 300,J d(j) = 3, d(j) 2 2507 2523 N = 300,J d(j) = 3, d(j) 1 2907 2921 N = 300,J d(j) = 3, d(j) 0 3347 3359 N = 300,J d(j) = 2, d(j) 4 1168 1195 N = 300,J d(j) = 2, d(j) 3 1364 1389 N = 300,J d(j) = 2, d(j) 2 1764 1767 N = 300,J d(j) = 2, d(j) 1 1955 1977 N = 300,J d(j) = 2, d(j) 0 2322 2342 N = 1000,J d(j) = 5, d(j) 4 2425 2428 N = 1000,J d(j) = 5, d(j) 3 2844 2847 N = 1000,J d(j) = 5, d(j) 2 3716 3718 N = 1000,J d(j) = 5, d(j) 1 4629 4632 N = 1000,J d(j) = 5, d(j) 0 5488 5490 N = 1000,J d(j) = 3, d(j) 4 1158 1165 N = 1000,J d(j) = 3, d(j) 3 1464 1471 N = 1000,J d(j) = 3, d(j) 2 2071 2076 N = 1000,J d(j) = 3, d(j) 1 2699 2703 N = 1000,J d(j) = 3, d(j) 0 3236 3240 N = 1000,J d(j) = 2, d(j) 4 0490 0515 N = 1000,J d(j) = 2, d(j) 3 0781 0799 N = 1000,J d(j) = 2, d(j) 2 1323 1331 N = 1000,J d(j) = 2, d(j) 1 1665 1673 N = 1000,J d(j) = 2, d(j) 0 2073 2080

4 110 44 2PLM Bias, RMSE Bias RMSE N = 300,J d(j) = 5, d(j) 4 1888 1858 N = 300,J d(j) = 5, d(j) 3 2556 2524 N = 300,J d(j) = 5, d(j) 2 3553 3525 N = 300,J d(j) = 5, d(j) 1 4475 4436 N = 300,J d(j) = 5, d(j) 0 5456 5407 N = 300,J d(j) = 3, d(j) 4 0856 0798 N = 300,J d(j) = 3, d(j) 3 1888 1135 N = 300,J d(j) = 3, d(j) 2 1837 1781 N = 300,J d(j) = 3, d(j) 1 2383 2321 N = 300,J d(j) = 3, d(j) 0 2932 2867 N = 300,J d(j) = 2, d(j) 4 0456 0365 N = 300,J d(j) = 2, d(j) 3 0673 0586 N = 300,J d(j) = 2, d(j) 2 1212 1119 N = 300,J d(j) = 2, d(j) 1 1528 1425 N = 300,J d(j) = 2, d(j) 0 1866 1812 N = 1000,J d(j) = 5, d(j) 4 1967 1927 N = 1000,J d(j) = 5, d(j) 3 2577 2522 N = 1000,J d(j) = 5, d(j) 2 3572 3503 N = 1000,J d(j) = 5, d(j) 1 4413 4358 N = 1000,J d(j) = 5, d(j) 0 5483 5330 N = 1000,J d(j) = 3, d(j) 4 0733 0679 N = 1000,J d(j) = 3, d(j) 3 1082 1034 N = 1000,J d(j) = 3, d(j) 2 1776 1715 N = 1000,J d(j) = 3, d(j) 1 2326 2275 N = 1000,J d(j) = 3, d(j) 0 2932 2911 N = 1000,J d(j) = 2, d(j) 4 0177 0121 N = 1000,J d(j) = 2, d(j) 3 0470 0419 N = 1000,J d(j) = 2, d(j) 2 1079 1004 N = 1000,J d(j) = 2, d(j) 1 1417 1348 N = 1000,J d(j) = 2, d(j) 0 1899 1822

4 111 45 2PLM Bias/I, RMSE/I Bias/I RMSE/I N = 300,J d(j) = 5, d(j) 4 0284 0280 N = 300,J d(j) = 5, d(j) 3 0404 0399 N = 300,J d(j) = 5, d(j) 2 0618 0613 N = 300,J d(j) = 5, d(j) 1 0870 0863 N = 300,J d(j) = 5, d(j) 0 1207 1196 N = 300,J d(j) = 3, d(j) 4 0106 0099 N = 300,J d(j) = 3, d(j) 3 0152 0145 N = 300,J d(j) = 3, d(j) 2 0244 0237 N = 300,J d(j) = 3, d(j) 1 0333 0324 N = 300,J d(j) = 3, d(j) 0 0429 0420 N = 300,J d(j) = 2, d(j) 4 0052 0041 N = 300,J d(j) = 2, d(j) 3 0078 0068 N = 300,J d(j) = 2, d(j) 2 0145 0133 N = 300,J d(j) = 2, d(j) 1 0186 0173 N = 300,J d(j) = 2, d(j) 0 0235 0226 N = 1000,J d(j) = 5, d(j) 4 0296 0290 N = 1000,J d(j) = 5, d(j) 3 0408 0399 N = 1000,J d(j) = 5, d(j) 2 0621 0609 N = 1000,J d(j) = 5, d(j) 1 0859 0848 N = 1000,J d(j) = 5, d(j) 0 1213 1179 N = 1000,J d(j) = 3, d(j) 4 0091 0084 N = 1000,J d(j) = 3, d(j) 3 0138 0132 N = 1000,J d(j) = 3, d(j) 2 0236 0228 N = 1000,J d(j) = 3, d(j) 1 0325 0318 N = 1000,J d(j) = 3, d(j) 0 0434 0426 N = 1000,J d(j) = 2, d(j) 4 0020 0014 N = 1000,J d(j) = 2, d(j) 3 0054 0048 N = 1000,J d(j) = 2, d(j) 2 0129 0120 N = 1000,J d(j) = 2, d(j) 1 0172 0164 N = 1000,J d(j) = 2, d(j) 0 0237 0227

4 112 41 Bias/I

4 113 42 RMSE/I

4 114 43 Bias/I

4 115 44 RMSE/I

4 116 45 Bias/I 4

4 117 46 RMSE/I 4

118 5 51,,,,,,,, *1,, Keller et al (2003), Lee (2000), Lee, Kolen, Frisbie, & Ankenmann (2001), Reise, Horan, & blanchard (2011), Tuerlinckx & De Boeck (1999), Wainer & Wang (2000), Wang, Cheng, & Wilson (2005), Zhang (2010),,, Braedlow et al (1999), DeMars (2006), (2001), (2013), Wainer et al (2007, pp 137-140), Bradlow et al (1999), 2 BTM, 2PLM BTM, BTM, 2PLM, BTM, 2PLM, M95%PIW *1, (2012a),

5 119 BTM, 2PLM, M95%PIW BTM 2PLM,, DeMars (2006), b, a, b,,,,, (2001), 2000 GRM 2PLM, 0987, (2013), 2PLM GRM, 0973 0995, Wainer et al (2007, pp 137-140), b, b,,,,, 15, a, b, c 3,, 3,,,,, 3 ( d ),,, 4,,,,,,,,,

5 120 ( a, b, c) ( d),,,,, 52,, a d,, ˆθ ir ˆθ i Bias(ˆθ i ), RMSE(ˆθ i ) cor(θ, ˆθ), 521, 421, (114) 2 BTM ( ) P j d(j) (θ i ) = 1 1+exp [ 17a j d(j) (θ i b j d(j) γ id(j) ) ] (51) 522, a d, a a, (112) GRM ( ) P d(j) (r θ i ) = 1 ] 1+exp [ 17a d(j) (θ i b r ) 1 ] (52) 1+exp [ 17a d(j) (θ i b r+1 ) b b, 2 BTM ( (51) )

5 121 c c, (116) CCM ( ) P(U j,u k θ i ) = exp[u j Z j +U k Z k U j U k b jk ] 1+exp[Z j ]+exp[z k ]+exp[z j +Z k b jk ] (53) Z j = 17a j d(j) (θ i b j d(j) ) (54) Z k = 17a k d(j) (θ i b k d(j) ) (55) d d, (11) 2PLM ( ) P j (θ i ) = 1 1+exp[ 17a j (θ i b j )] (56) 523,, 4, 5, 51,,,,, 4, 300, 1000 2,, 3 5 (J d(j) = 5) 3 (J d(j) = 3) 2 (J d(j) = 2),, 4 5 4 ( d(j) 4) 4 3 ( d(j) 3) 4 2 ( d(j) 2) 4 1 ( d(j) 1)

5 122 4 ( d(j) 0),, 3 *2 1 θ i (i = 1,2,,N) γ id(j) (i = 1,2,,N; d(j) = 1,2,3,4) a j d(j), b j d(j) (j = 1,2,,20) N(0,1), N(0,σγ 2 d(j) ), U(05,15), N(0,1) *3 2 53 2 BTM ((51) ), N 20 A 3 U(0,1) N 20, N 20 B 4 A,B U, a ij b ij u ij = 1, a ij < b ij u ij = 0 5 U 522 4 6 3 5 R 7 6 R ˆθ ir,, Bias(ˆθ i ),RMSE(ˆθ i ),cor(θ, ˆθ) ( ) Bias(ˆθ i ) = 1 R ˆθ ir θ i (57) R r=1 RMSE(ˆθ i ) = 1 R ) 2 (ˆθir θ i (58) R cor(θ, ˆθ) = 1 R r=1 R cor(θ, ˆθ r ) (59) r=1, 6 R, ˆθ ir, 1000 R = 50, 300 R = 100, c, 421, 2, c,, J d(j) = 2, *2, R *3, R

5 123 524, 5 MCMC MCMC (Neal, 1997), WinBUGS 14 (Spiegelhalter et al, 2003) MCMC,,, a θ i N(0,1) a d(j) N(1,025) b r N(0,1) b θ i N(0,1) a j d(j) N(1,025) b j d(j) N(0,1) γ id(j) N(0,σγ 2 d(j) ) σγ 2 d(j) Γ(3, 1) c θ i N(0,1) a j d(j) N(1,025) b j d(j) N(0,1) b jk N( 2,1) d θ i N(0,1) a j N(1,025) b j N(0,1),,, a θ i 0 a d(j) 1 b r r = 1,2,3,4,5, 0, 01, 02, 03, 04 b

5 124 θ i 0 a j d(j) 1 b j d(j) 0 γ id(j) 0 σγ 2 d(j) 3 c θ i 0 a j d(j) 1 b j d(j) 0 b jk -2 d θ i 0 a j 1 b j 0,, λ,, burn-in a 1000 burn-in b 1000 burn-in c 1000 burn-in d 1000 burn-in,, WinBUGS User Manual (Spiegelhalter et al, 2003), 5%, a J d(j) = 5 5000, 4000 b 5000

5 125 c 4000 d 4000 53,,, 2PLM GRM, BTM, CCM Bias,RMSE,cor(θ, ˆθ), 2PLM GRM, BTM, CCM Bias RMSE, θ i 523,, θ i N(0,1), θ i 1, Bias RMSE,, θ i 531 2PLM, 2PLM Bias,RMSE,cor(θ, ˆθ),, 51 532 GRM, GRM Bias,RMSE,cor(θ, ˆθ),, 52, GRM Bias, RMSE, cor(θ, ˆθ),, 53, Bias, RMSE, cor(θ, ˆθ), GRM Bias,RMSE,cor(θ, ˆθ) 2PLM Bias,RMSE,cor(θ, ˆθ), GRM Bias, RMSE, cor(θ, ˆθ), 51, 52, 53, GRM Bias, RMSE, cor(θ, ˆθ)

5 126, 54, 55, 56, GRM Bias, RMSE, cor(θ, ˆθ) 4, 57, 58, 59, GRM,,, 57,, 2PLM,, 2PLM GRM,, 51,, Bias, 2PLM,, GRM,, 57,, 2PLM,, 2PLM GRM,, 58, 2 3, RMSE, 5,,, RMSE 2 3, 2PLM RMSE, GRM, RMSE, 5, 57, 4 2PLM, GRM,, 2PLM GRM 57,, 2PLM,, 2PLM GRM,

5 127, 59, 2 3,, cor(θ, ˆθ), cor(θ, ˆθ), GRM 2PLM, 59, 5, 4 4, cor(θ, ˆθ), 4,, 2PLM, GRM, 4,,, 5,,,,, a,,,,, a 533 2 BTM, BTM Bias,RMSE,cor(θ, ˆθ),, 54, BTM Bias, RMSE, cor(θ, ˆθ),, 55, BTM Bias, RMSE, cor(θ, ˆθ), 510, 511, 512,

5 128 BTM Bias, RMSE, cor(θ, ˆθ), 513, 514, 515, BTM Bias, RMSE, cor(θ, ˆθ) 4, 516, 517, 518, BTM,,, 59, N = 1000, 2PLM 2 BTM,, 2PLM 2 BTM,, 510,, Bias,, 2PLM,, 2 BTM 59,, 2PLM 2 BTM,, 2PLM 2 BTM,, 514,, RMSE, RMSE, 2 BTM 2PLM,, 517, 5, 4, 4 RMSE, 5, 59, 4 2PLM 2 BTM, 2PLM BTM

5 129 59,, 2PLM 2 BTM,, 2PLM 2 BTM,, 518, 5, 4 4, cor(θ, ˆθ), 4,, 2PLM, BTM, 4,,, 5,,,, b,,,,,,, b, 534 CCM, CCM Bias,RMSE,cor(θ, ˆθ),, 56, CCM Bias, RMSE, cor(θ, ˆθ),, 57, CCM Bias, RMSE, cor(θ, ˆθ), 519, 520, 521, CCM Bias, RMSE, cor(θ, ˆθ) 4, 522, 523,

5 130 524, CCM,,, 511,, 2PLM CCM,, 2PLM CCM,, 519, Bias,,,, 2PLM Bias, CCM, 522,, Bias,, CCM Bias, 2PLM, 511,, 2PLM CCM,, 2PLM CCM,, 520,, RMSE, CCM 2PLM, RMSE 511,, 2PLM CCM,, 2PLM CCM,,,,, c

5 131,,,,,,, c,,,, CCM,, 535, 535 CCM Bias, CCM b jk, -1714, b jk -0612, 2 BTM CCM, 2 j,k 100, σγ 2 d(j) = 14075 2 BTM j,k (1,1),(1,0),(0,1),(0,0) (100 ) 58, b jk = 1714 CCM 59, 2 BTM CCM, 2 j,k (N = 100), σγ 2 d(j) = 0155 2 BTM 510, b jk = 0612 CCM 511 58,, 59, 510, 511, 2 BTM,, j,k, j,k CCM,, j,k,, j,k,,, 2 BTM CCM,,, CCM Bias

5 132 536 51,,, 3, GRM, BTM, CCM, 2PLM,, BTM 2PLM, Bradlow et al (1999), BTM,, BTM, 2PLM, M95%PIW BTM 2PLM,, DeMars (2006), a, b, d,,,,,,, Wainer et al (2007, pp 137-140), b d,,, BTM,,, 537,,, 3,,,

5 133,,,,,,,, 4,,,,,,,

5 134 51 2PLM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 300,J d(j) = 5, d(j) 4 0023 0530 0822 N = 300,J d(j) = 5, d(j) 3 0023 0501 0848 N = 300,J d(j) = 5, d(j) 2 0021 0459 0880 N = 300,J d(j) = 5, d(j) 1 0020 0429 0899 N = 300,J d(j) = 5, d(j) 0 0021 0385 0923 N = 300,J d(j) = 3, d(j) 4 0021 0423 0902 N = 300,J d(j) = 3, d(j) 3 0021 0409 0910 N = 300,J d(j) = 3, d(j) 2 0021 0393 0920 N = 300,J d(j) = 3, d(j) 1 0021 0381 0925 N = 300,J d(j) = 3, d(j) 0 0021 0366 0932 N = 300,J d(j) = 2, d(j) 4 0021 0398 0916 N = 300,J d(j) = 2, d(j) 3 0022 0388 0922 N = 300,J d(j) = 2, d(j) 2 0022 0377 0928 N = 300,J d(j) = 2, d(j) 1 0021 0371 0930 N = 300,J d(j) = 2, d(j) 0 0021 0361 0934 N = 1000,J d(j) = 5, d(j) 4 0016 0546 0804 N = 1000,J d(j) = 5, d(j) 3 0016 0517 0831 N = 1000,J d(j) = 5, d(j) 2 0015 0471 0867 N = 1000,J d(j) = 5, d(j) 1 0015 0435 0891 N = 1000,J d(j) = 5, d(j) 0 0015 0385 0919 N = 1000,J d(j) = 3, d(j) 4 0015 0424 0897 N = 1000,J d(j) = 3, d(j) 3 0016 0409 0907 N = 1000,J d(j) = 3, d(j) 2 0015 0391 0916 N = 1000,J d(j) = 3, d(j) 1 0015 0377 0923 N = 1000,J d(j) = 3, d(j) 0 0015 0363 0929 N = 1000,J d(j) = 2, d(j) 4 0016 0397 0913 N = 1000,J d(j) = 2, d(j) 3 0015 0386 0919 N = 1000,J d(j) = 2, d(j) 2 0015 0374 0925 N = 1000,J d(j) = 2, d(j) 1 0015 0367 0928 N = 1000,J d(j) = 2, d(j) 0 0015 0358 0932

5 135 52 GRM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 300,J d(j) = 5, d(j) 4 0021 0533 0817 N = 300,J d(j) = 5, d(j) 3 0020 0499 0858 N = 300,J d(j) = 5, d(j) 2 0020 0454 0888 N = 300,J d(j) = 5, d(j) 1 0020 0426 0904 N = 300,J d(j) = 5, d(j) 0 0020 0397 0918 N = 300,J d(j) = 3, d(j) 4 0021 0422 0906 N = 300,J d(j) = 3, d(j) 3 0021 0409 0913 N = 300,J d(j) = 3, d(j) 2 0021 0396 0920 N = 300,J d(j) = 3, d(j) 1 0021 0386 0924 N = 300,J d(j) = 3, d(j) 0 0022 0375 0929 N = 300,J d(j) = 2, d(j) 4 0022 0399 0918 N = 300,J d(j) = 2, d(j) 3 0022 0389 0923 N = 300,J d(j) = 2, d(j) 2 0021 0378 0928 N = 300,J d(j) = 2, d(j) 1 0022 0373 0930 N = 300,J d(j) = 2, d(j) 0 0022 0365 0933 N = 1000,J d(j) = 5, d(j) 4 0019 0541 0800 N = 1000,J d(j) = 5, d(j) 3 0019 0506 0848 N = 1000,J d(j) = 5, d(j) 2 0018 0457 0881 N = 1000,J d(j) = 5, d(j) 1 0018 0426 0899 N = 1000,J d(j) = 5, d(j) 0 0008 0397 0914 N = 1000,J d(j) = 3, d(j) 4 0018 0416 0902 N = 1000,J d(j) = 3, d(j) 3 0019 0403 0910 N = 1000,J d(j) = 3, d(j) 2 0018 0387 0917 N = 1000,J d(j) = 3, d(j) 1 0018 0374 0922 N = 1000,J d(j) = 3, d(j) 0 0019 0363 0926 N = 1000,J d(j) = 2, d(j) 4 0018 0396 0914 N = 1000,J d(j) = 2, d(j) 3 0019 0386 0920 N = 1000,J d(j) = 2, d(j) 2 0018 0375 0925 N = 1000,J d(j) = 2, d(j) 1 0019 0368 0927 N = 1000,J d(j) = 2, d(j) 0 0019 0361 0930