Ł\”ƒ.dvi

Transcription

1 ,

2 ,

3 PLM CCM PLCM BTM 54

4 MCMC MCMC BTM PLM PLM GRM BTM CCM 129

5 3 535 CCM Bias

6 4 11 N J 9 21 CCM (1,1),(1,0),(0,1),(0,0) ( ) PLCM (1,1),(1,0),(0,1),(0,0) ( ) PLM Bias,RMSE,cor(θ, ˆθ) CCM Bias,RMSE,cor(θ, ˆθ) CCM Bias, RMSE, cor(θ, ˆθ) PLCM Bias,RMSE,cor(θ, ˆθ) PLCM Bias, RMSE, cor(θ, ˆθ) BTM Bias,RMSE,cor(a,â) PLM Bias,RMSE,cor(a,â) Bias, RMSE, cor(a,â) Bias/σ a j RMSE/σ a j Bias,RMSE,cor(a,â) Bias, RMSE, cor(a,â) Bias/σ a j RMSE/σ a j BTM Bias,RMSE,cor(b,ˆb) 75

7 5 39 2PLM Bias,RMSE,cor(b,ˆb) Bias, RMSE, cor(b,ˆb) I BTM Bias,RMSE PLM Bias,RMSE PLM Bias, RMSE PLM Bias/I, RMSE/I PLM Bias,RMSE,cor(θ, ˆθ) GRM Bias,RMSE,cor(θ, ˆθ) GRM Bias, RMSE, cor(θ, ˆθ) BTM Bias,RMSE,cor(θ, ˆθ) BTM Bias, RMSE, cor(θ, ˆθ) CCM Bias,RMSE,cor(θ, ˆθ) CCM Bias, RMSE, cor(θ, ˆθ) σγ 2 d(j) = BTM (1,1),(1,0),(0,1),(0,0) (100 ) b jk = 1714 CCM (1,1),(1,0),(0,1),(0,0) (100 ) σγ 2 d(j) = 0155 BTM (1,1),(1,0),(0,1),(0,0) (100 ) b jk = 0612 CCM (1,1),(1,0),(0,1),(0,0) (100 ) 140

8 6 21 CCM Bias CCM RMSE CCM cor(θ, ˆθ) CCM Bias CCM RMSE CCM cor(θ, ˆθ) PLCM Bias PLCM RMSE PLCM cor(θ, ˆθ) PLCM Bias PLCM RMSE PLCM cor(θ, ˆθ) Bias/σ a j RMSE/σ a j cor(a,â) Bias/σ a j RMSE/σ a j cor(a,â) Bias/σ a j RMSE/σ a j cor(a,â) Bias RMSE cor(b,ˆb) Bias 90

9 7 314 RMSE cor(b,ˆb) Bias RMSE cor(b,ˆb) Bias/I RMSE/I Bias/I RMSE/I Bias/I RMSE/I GRM Bias GRM RMSE GRM cor(θ, ˆθ) GRM Bias GRM RMSE GRM cor(θ, ˆθ) GRM Bias GRM RMSE GRM cor(θ, ˆθ) BTM Bias BTM RMSE BTM cor(θ, ˆθ) BTM Bias BTM RMSE 154

10 8 515 BTM cor(θ, ˆθ) BTM Bias BTM RMSE BTM cor(θ, ˆθ) CCM Bias CCM RMSE CCM cor(θ, ˆθ) CCM Bias CCM RMSE CCM cor(θ, ˆθ) 4 164

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

12 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)

13 (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, pp ), 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,

14 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)

15 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

16 ,, 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

17 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)

18 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,

19 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

20 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 ) = 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

21 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 ),,,,

22 1 20,,,,,,,,,,,,,,, 2,, 3,,, 4,, 5,,,,,, 6,

23 ,, *1,, Bradlow et al (1999), 2 BTM, 2PLM,,,,, 95% (mean 95% posterior interval width, M95%PIW),, M95%PIW,, *1, (2010),

24 2 22, Junker (1991),,, (2013),, 2PLM,,,,,,,,,,,,, Bradlow et al (1999), 1000, 60, (2013), 1000, 12,,,,,,,, ( ),,,,,,,,,,,

25 ,, 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)

26 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, , 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, , 21 22, CCM 1,2 (1,1),(0,0) (100 ) 84% (73%+11%), 2PLCM 88% (52%+36%)

27 CCM (1,1),(1,0),(0,1),(0,0) ( ) 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) ( ) 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

28 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 , Bias(ˆθ i ) = RMSE(ˆθ i ) = 100 r= cor(θ, ˆθ) = 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 )

29 2 27 ˆθ i, RMSE(ˆθ i ) ˆθ i, cor(θ, ˆθ) ˆθ θ ,, 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),

30 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

31 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)

32 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 λ, (215) σ 2, 4 (l = 1,,N +2J) λ l,,, λ l 25% 50% σ 2,, , 1 (burn-in ), 20000,,, 3000 burn-in, 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

33 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),, 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

34 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,,,,,

35 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%),,,

36 ,, 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),,,,,,,,,,,,, ,,,,,,, Bradlow et al (1999) 2, 2 BTM

37 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)

38 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 = N = 100,J = N = 100,J = N = 300,J = N = 300,J = N = 300,J = N = 500,J = N = 500,J = N = 500,J = N = 1000, J = N = 1000, J = N = 1000, J =

39 CCM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 100,J = N = 100,J = N = 100,J = N = 300,J = N = 300,J = N = 300,J = N = 500,J = N = 500,J = N = 500,J = N = 1000, J = N = 1000, J = N = 1000, J = CCM Bias, RMSE, cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 100, J = N = 100,J = N = 100,J = N = 300, J = N = 300,J = N = 300, J = N = 500, J = N = 500,J = N = 500, J = N = 1000, J = N = 1000, J = N = 1000, J =

40 PLCM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 100,J = N = 100,J = N = 100,J = N = 300,J = N = 300,J = N = 300,J = N = 500,J = N = 500,J = N = 500,J = N = 1000, J = N = 1000, J = N = 1000, J = PLCM Bias, RMSE, cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 100, J = N = 100, J = N = 100, J = N = 300, J = N = 300, J = N = 300, J = N = 500, J = N = 500, J = N = 500, J = N = 1000, J = N = 1000, J = N = 1000, J =

41 CCM Bias

42 CCM RMSE

43 CCM cor(θ, ˆθ)

44 CCM Bias

45 CCM RMSE

46 CCM cor(θ, ˆθ)

47 PLCM Bias

48 PLCM RMSE

49 PLCM cor(θ, ˆθ)

50 PLCM Bias

51 PLCM RMSE

52 PLCM cor(θ, ˆθ)

53 ,,, *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),

54 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

55 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)

56 , 31, (11) 2PLM ( ) P j (θ i ) = 1 1+exp[ 17a j (θ i b j )] (32) 2 BTM ( (31) ) 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

57 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 = π (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, ,, 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)

58 ( 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 R Ip (2010), 2 BTM 2PLM 8 7 R *3 R *4, R

59 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= 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 = R = 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

60 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 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

61 ,,,, 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)

62 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

63 ,, 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)

64 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,,,

65 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,

66 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,

67 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

68 3 66, (310), a j r bj, 2 BTM 2PLM, â j, 2PL M Bias, 2, 2PLM RMSE,, â j ,,,, 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),,,,

69 3 67,,,

70 BTM Bias,RMSE,cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

71 PLM Bias,RMSE,cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

72 Bias, RMSE, cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

73 Bias/σ a j RMSE/σ a j Bias/σ a j RMSE/σ a j N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

74 Bias,RMSE,cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

75 Bias, RMSE, cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

76 Bias/σ a j RMSE/σ a j Bias/σ a j RMSE/σ a j N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

77 BTM Bias,RMSE,cor(b,ˆb) Bias RMSE cor(b,ˆb) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

78 PLM Bias,RMSE,cor(b,ˆb) Bias RMSE cor(b,ˆb) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

79 Bias, RMSE, cor(b,ˆb) Bias RMSE cor(b,ˆb) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

80 Bias/σ a j

81 RMSE/σ a j

82 cor(a,â)

83 Bias/σ a j

84 RMSE/σ a j

85 cor(a,â)

86 Bias/σ a j 4

87 RMSE/σ a j 4

88 cor(a,â) 4

89 Bias

90 RMSE

91 cor(b,ˆb)

92 Bias

93 RMSE

94 cor(b,ˆb)

95 Bias 4

96 RMSE 4

97 cor(b,ˆb) 4

98 ,,,,, 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),,

99 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), ,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 42,,,,,,,,,,

100 , (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) )

101 ,,,,,,, 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,

102 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, ,, 3 5 (J d(j) = 5) 3 (J d(j) = 3) 2 (J d(j) = 2),, ( d(j) 4) 4 3 ( d(j) 3) 4 2 ( d(j) 2) 4 1 ( d(j) 1) 4 ( d(j) 0)

103 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) * 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 , 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

104 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

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

106 4 104 Bias,RMSE, I, Bias,RMSE, I, 431,, I, BTM, 2 BTM Bias,RMSE,, PLM, 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,,

107 ,, 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,,,,,,,

108 4 106,,, d(j) 2,1,0,,,,,,,,, ,,,,,,,,,, 4,, Ip (2010), (2001), Keller et al (2003), Wainer & Wang (2000),,,,,

109 I I N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

110 BTM Bias,RMSE Bias RMSE N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

111 PLM Bias,RMSE Bias RMSE N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

112 PLM Bias, RMSE Bias RMSE N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

113 PLM Bias/I, RMSE/I Bias/I RMSE/I N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

114 Bias/I

115 RMSE/I

116 Bias/I

117 RMSE/I

118 Bias/I 4

119 RMSE/I 4

120 ,,,,,,,, *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 ), Bradlow et al (1999), 2 BTM, 2PLM BTM, BTM, 2PLM, BTM, 2PLM, M95%PIW *1, (2012a),

121 5 119 BTM, 2PLM, M95%PIW BTM 2PLM,, DeMars (2006), b, a, b,,,,, (2001), 2000 GRM 2PLM, 0987, (2013), 2PLM GRM, , Wainer et al (2007, pp ), b, b,,,,, 15, a, b, c 3,, 3,,,,, 3 ( d ),,, 4,,,,,,,,,

122 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) )

123 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, ,, 3 5 (J d(j) = 5) 3 (J d(j) = 3) 2 (J d(j) = 2),, ( d(j) 4) 4 3 ( d(j) 3) 4 2 ( d(j) 2) 4 1 ( d(j) 1)

124 ( 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) * 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 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

125 , 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

126 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) = , 4000 b 5000

127 5 125 c 4000 d ,,, 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),, 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)

128 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,

129 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 BTM, BTM Bias,RMSE,cor(θ, ˆθ),, 54, BTM Bias, RMSE, cor(θ, ˆθ),, 55, BTM Bias, RMSE, cor(θ, ˆθ), 510, 511, 512,

130 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

131 ,, 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,

132 , 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

133 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) = 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) = BTM 510, b jk = 0612 CCM ,, 59, 510, 511, 2 BTM,, j,k, j,k CCM,, j,k,, j,k,,, 2 BTM CCM,,, CCM Bias

134 ,,, 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 ), b d,,, BTM,,, 537,,, 3,,,

135 5 133,,,,,,,, 4,,,,,,,

136 PLM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

137 GRM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

138 GRM Bias, RMSE, cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

139 BTM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

140 BTM Bias, RMSE, cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

141 CCM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) CCM Bias, RMSE, cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)

142 σγ 2 d(j) = BTM (1,1),(1,0),(0,1),(0,0) (100 ) U k = 1 U k = 0 U j = 1 38% 20% U j = 0 13% 29% 59 b jk = 1714 CCM (1,1),(1,0),(0,1),(0,0) (100 ) k = 1 k = 0 j = 1 46% 5% j = 0 6% 43% 510 σγ 2 d(j) = 0155 BTM (1,1),(1,0),(0,1),(0,0) (100 ) U k = 1 U k = 0 U j = 1 38% 23% U j = 0 15% 24% 511 b jk = 0612 CCM (1,1),(1,0),(0,1),(0,0) (100 ) k = 1 k = 0 j = 1 32% 8% j = 0 8% 52%

143 GRM Bias

144 GRM RMSE

145 GRM cor(θ, ˆθ)

146 GRM Bias

147 GRM RMSE

148 GRM cor(θ, ˆθ)

149 GRM Bias 4

150 GRM RMSE 4

151 GRM cor(θ, ˆθ) 4

152 BTM Bias

153 BTM RMSE

154 BTM cor(θ, ˆθ)

155 BTM Bias

156 BTM RMSE

157 BTM cor(θ, ˆθ)

158 BTM Bias 4

159 BTM RMSE 4

160 BTM cor(θ, ˆθ) 4

161 CCM Bias

162 CCM RMSE

163 CCM cor(θ, ˆθ)

164 CCM Bias 4

165 CCM RMSE

166 CCM cor(θ, ˆθ) 4

167 , 2,,,,,,,,,,,,,, 3,,,,,,,,,,,,,,

168 6 166,,,,,,,,,,, 4,,,,,,, 5,,,,,,,,,,,,, 62,,,,,,,,

169 6 167,,,,,

170 168 1 Anastasi, A (1961) Psychological testing (2nd ed) New York: Macmillian 2 (2005), 1, Bock, R D (1972) Estimating item parameters and latent ability when responses are scored in two or more latent categories Psychometrika, 37, Bradlow, E T, Wainer, H, & Wang, X H (1999) A Bayesian random effects model for testlets Psychometrika, 64, Braeken, J (2011) A boundary mixture approach to violation of conditional independence Psychometrika, 64, Braeken, J, Tuerlinckx, F, & De Boeck, P (2007) Copula functions for residual dependency Psychometrika, 72, Chen, C, & Wang, W (2007) Effect of ignoring item interaction on item parameter estimation and detection of interacting items Applied Psychological Measurement, 31, DeMars, C E (2006) Application of the bi-factor multidimensional item response theory model to testlet-based tests Journal of Educational Measurement, 43, Ferrara, S, Huynh, H, & Michaels, H (1999) Contextual explanation of local dependence in item clusters in a large scale hands-on science performance assessment Journal of Educational Measurement, 6, Frank, M J (1979) On the simultaneous associativity of F(x,y) and x + y F(x,y) Aequationes Mathematica, 19, Geman, S, & Geman, D (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images Institute of Electorical and Electorics Engineers, Transactions on Pattern Analysis and Machine Intelligence, 6,

171 Guilford, J P (1936) Psychometric methods New York: McGraw-Hill 13 (1984), 26, (2000) < tokyoacjp/ haebara/local ind/> ( ) 15 Hastings, W K (1970) Monte Carlo sampling methods using Markov chains and their applications Biometrika, 57, Hoskens, M, & De Boeck, P (1997) A parametric model for local dependence among test items Psychological Methods, 2, (1992) 18 Ip E H (2000) Adjusting for information inflation due to local dependency in moderately large item clusters Psychometrika, 65, Ip, E H (2002) Locally dependent latent trait model and the Dutch identity revisited Psychometrika, 67, Ip, E H (2010) Interpretation of the three parameter testlet response model and information function Applied Psychological Measurement, 34, Ip, E H, Smits, D J M, & De Boeck, P (2009) Locally dependent linear logistic test model with person covariates Applied Psychological Measurement, 33, (2001), 30, (2013) 2PLM, 9, Jannarone, R J (1986) Conjunctive item response theory kernels Psychometrika, 51, Jiao, H, Kamata, A, Wang, S, &Jin, Ying (2012) Amultileveltestlet model for dual local dependence Journal of Educational Measurement, Junker, B W (1991) Essential independence and likelihood-based ability estimation for polytomous items Psychometrika, 56, Kan, C C, van der Ven, A H G S, Breteler, M H M, & Zitman, F G (2001) Latent trait standardization of the Benzodiazepine dependence selfreport questionnaire using the Rasch scaling model Comprehensive Psychiatry, 42,

172 Keller, L A, Swaminathan, H, & Sireci, S G (2003) Evaluating scoring procedures for context-dependent item sets Applied Measurement in Education, 16, Kelly, T L (1924) Note on the reliability of a test: A reply to Dr Crumm s criticism Journal of Educational Psychology, 15, Kreiner, S, & Christensen, K B (2004) Analysis of local dependence and multidimensionality in graphical loglinear Rasch models Communications in Statistics: Theory and Methods, 33, Lee, G (2000) A comparison of methods of estimating conditional standard errors of measurement for testlet-based test scores using simulation techniques Journal of Educational Measurement, 36, Lee, G, Kolen, M J, Frisbie, D A, & Ankenmann, R D (2001) Comparison of dichotomous and polytomous item response models in equating scores form tests composed of testlets Applied Psychological Measurement, 25, Li, Y, Bolt, D M, & Fu, J (2005) A test characteristic curve linking method for the testlet model Applied Psychological Measurement, 29, Li, Y, Bolt, D M, & Fu, J (2006) A comparison of alternative models for testlets Applied Psychological Measurement, 30, Looney, M A, & Spray, J A (1992) Effects of violating local independence on IRT parameter estimation for the binomial trials model Research Quarterly for Exercise and Sport, 63, Lord, F M (1980) Applications of item response theory to practical testing problems Hillsdale, NJ: Lawrence Erlbaum Associate 37 Lord, F M, & Novick M R (1968) Statistical theories of mental test scores Reading, MA: Addison-Wesley 38 Muraki, E (1992) A generalized partial credit model: Application of an EM algorithm Applied Psychological Measurement, 16, Nandakumar, R (1990) Traditional dimensionality versus essential dimensionality Journal of Educational Measurement, 28, Neal, R (1997) Markov chain Monte Carlo methods based on slicing the density function Technical Report 9722, Department of Statistics, University of Toronto, Canada 41 Patz, R J, & Junker, B W (1999) A straightforward approach to Markov chain Monte Carlo methods for item response models Journal of Educational and Behavioral Statistics, 24,

173 Rasch, G (1960) Probabilistic models for some intelligence and achievement tests Copenhagen: Danish Institute for Educational Research 43 Reise, S P, Horan, W P, & Blanchard, J J (2011) The challenges of fitting an item response theory model to the social anhedonia scale Personality Assessment, 93, Journal of 44 Samejima, F (1969) Estimation of latent ability using a response pattern of graded scores Psychometrika Monograph Supplement, 34, Simms, L J, Goldberg, L R, Roberts, J E, Watson, D, Welte, J, & Rotterman, J H (2011) Computerized adaptive assessment of personality disorder: Introducing the CAT-PD project Journal of Personality Assessment, 93, Sireci, S G, Thissen, D, & Wainer, H (1991) On the reliability of testletbased tests Journal of Educational Measurement, 28, Spiegelhalter, D J, Thomas, A, & Best, N (2003) WinBUGS 14 [Computer Program] Cambridge, UK: MRC Biostatistics Unit, Institute of Public Health 48 Spiegelhalter, D J, Thomas, A, Best, N, & Lunn, D (2003) WinBUGS User Manual Version 14 Cambridge, UK: MRC Biostatistics Unit, Institute of Public Health 49 Tanner, M A, & Wong, W H (1987) The calculation of posterior distributions by data augmentation Journal of the American Statistical Association, 82, Thorndike, R L (1951) Reliability In E F Lindquist (Ed), Educational Measurement Washington, DC: American Council on Education 51 (2010), 6, (2012a), 8, (2012b), 39, (2002) [ ] 55 ( ) (2005) [ ] 56 ( ) (2008) 57 Tuerlinckx, F, & De Boeck, P (1999) Distinguishing constant and dimensiondependent interaction: A simulation study Applied Psychological Measure ment, 23,

174 Tuerlinckx, F, & De Boeck, P (2001) The effect of ignoring item interactions on the estimated discrimination parameters in item response theory Psychological Methods, 6, Wainer, H (1995) Precision and differential item functioning on a testletbased test: The 1991 Law School Admission Test as an example Applied Measurement in Education, 8, Wainer, H Bradlow, E T, & Du, Z (2000) Testlet response theory: An analog for the 3-PL useful in testlet-based adaptive testing In W J van der Linden, & C A W Glas (Eds), Computerized adaptive testing, theory and practice Boston, MA: Kluwer-Nijhoff pp Wainer, H, Bradlow E T, & Wang X H (2007) Testlet response theory and its applications USA: Cambridge University Press 62 Wainer, H, & Wang, X H (2001) Using a new statistical model for testlets to score TOEFL TOEFL Technical Report No 16 Princeton, NJ: Educational Testing Service 63 Wang, X, Bradlow, E T, & Wainer, H (2002) A general Bayesian model for testlets: Theory and applications Applied Psychological Measurement, 26, Wang, C W, Cheng, Y Y, & Wilson, M (2005) Local item dependence for items across tests connected by common stimuli Educational and Psychological Measurement, 65, Wang, C W, & Wilson, M (2005) Exploring local item dependence using a random-effect facet model Applied Psychological Measurement, 29, (2012) 54, Yang, W L, & Gao, R (2008) Invariance of score linkings across gender groups for forms of a testlet-based college-level examination program examination Applied Psychological Measurement, 32, (2013) 11, Yen, W M (1993) Scaling performance assessments: Strategies for managing local item dependence Journal of Educational Measurement, 30, Zhang, B (2010) Assessing the accuracy and consistency of language proficiency classification under competing measurement models Language Testing,

175 173 27,

176 PLM ((N = 1000,J = 50) ) # setseed(1010) # 100 t1 <- rnorm(n = 100) # bt1 <- runif(n = 30, min = -20, max = 20) # at1 <- runif(n = 30, min = 03, max = 15) # 900 t2 <- rnorm(n = 900) t <- c( t1, t2) # 20 bt2 <- runif(n = 20, min = -20, max = 20) bt <- c(bt1, bt2) at2 <- runif(n = 20, min = 03, max = 15) at <- c(at1, at2) # (2 ) tvec <- rep( t, 50) tmat <- matrix( tvec, nrow = 50, ncol = 1000, byrow = TRUE) ft <- tmat - bt et <- exp(-10 * at * ft) Pt <- 1 / (1 + et) Qt <- 1 - Pt #

177 175 m <- matrix(nrow = 1000, ncol = 100) am <- matrix(nrow = 50, ncol = 100) bm <- matrix(nrow = 50, ncol = 100) Xe <- matrix(nrow = 1000, ncol = 100) DPe <- matrix(nrow = 50, ncol = 100) PRe <- matrix(nrow = 50, ncol = 100) pc <- matrix(nrow = 3, ncol = 100) kc <- matrix(nrow = 3, ncol = 100) sc <- matrix(nrow = 3, ncol = 100) #MCMC 100 for(r in 1:100){ # <- runif(n = 1000 * 50, min = 0, max = 1) <- matrix(, nrow = 50, ncol = 1000) U <- ifelse(pt >=, 1, 0) # gc() gc() # # e <- matrix(nrow = 1000, ncol = 20000) be <- matrix(nrow = 50, ncol = 20000) ae <- matrix(nrow = 50, ncol = 20000) # X <- colsums(u) Xe[, r] <- X <- mean(x) <- mean((x - ) ^ 2) <- sqrt( ) e[, 1] <- (X - ) / PR <- rowmeans(u) PRe[, r] <- PR be[, 1] <- 1 - PR DP <- apply(u, 1, cor, X) DPe[, r] <- DP ae[, 1] <- DP for(i in 1:19999){ # (2 ) vec <- rep( e[, i], 50) mat <- matrix( vec, nrow = 50, ncol = 1000, byrow = TRUE)

178 176 f <- mat - be[, i] e <- exp(-10 * ae[, i] * f) P <- 1 / (1 + e) Q <- 1 - P # ( ) t <- rnorm(n = 1000, mean = 0, sd = 17) c <- e[, i] + t # c (2 ) cvec <- rep( c, 50) cmat <- matrix( cvec, nrow = 50, ncol = 1000, byrow = TRUE) fc <- cmat - be[, i] ec <- exp(-10 * ae[, i] * fc) Pc <- 1 / (1 + ec) Qc <- 1 - Pc # c P <- dnorm( e[, i], mean = 0, sd = 1) P c <- dnorm( c, mean = 0, sd = 1) L <- (P ^ U) * (Q ^ (1 - U)) L c <- (Pc ^ U) * (Qc ^ (1 - U)) LP <- log(p ) LP c <- log(p c) LL <- log(l ) LL c <- log(l c) LL <- colsums(ll ) LL c <- colsums(ll c) L <- (LL c + LP c) - (LL + LP ) c <- exp(l ) dim( c) <- c(1, 1000) <- apply( c, 2, min, 1) urn <- runif(n = 1000, min = 0, max = 1) logic <- >= urn e[which( logic), i+1] <- c[which( logic)] e[which(! logic), i+1] <- e[which(! logic), i] # (2 ) vec <- rep( e[, i+1], 50) mat <- matrix( vec, nrow = 50, ncol = 1000, byrow = TRUE) f <- mat - be[, i] e <- exp(-10 * ae[, i] * f) P <- 1 / (1 + e) Q <- 1 - P

179 177 # a ( ) ta <- rnorm(n = 50, mean = 0, sd = 025) ac <- ae[, i] + ta # ac (2 ) ec <- exp(-10 * ac * f) Pc <- 1 / (1 + ec) Qc <- 1 - Pc #ac Pa <- dnorm(ae[, i], mean = 1, sd = 05) Pac <- dnorm(ac, mean = 1, sd = 05) La <- (P ^ U) * (Q ^ (1 - U)) Lac <- (Pc ^ U) * (Qc ^ (1 - U)) LPa <- log(pa) LPac <- log(pac) LLa <- log(la) LLac <- log(lac) LLA <- rowsums(lla) LLAc <- rowsums(llac) L <- (LLAc + LPac) - (LLA + LPa) c <- exp(l ) dim( c) <- c(1, 50) <- apply( c, 2, min, 1) urn <- runif(n = 50, min = 0, max = 1) logic <- >= urn ae[which( logic), i+1] <- ac[which( logic)] ae[which(! logic), i+1] <- ae[which(! logic), i] # a (2 ) vec <- rep( e[, i+1], 50) mat <- matrix( vec, nrow = 50, ncol = 1000, byrow = TRUE) f <- mat - be[, i] e <- exp(-10 * ae[, i+1] * f) P <- 1 / (1 + e) Q <- 1 - P # b ( ) tb <- rnorm(n = 50, mean = 0, sd = 03) bc <- be[, i] + tb # bc (2 )

180 178 fc <- mat - bc ec <- exp(-10 * ae[, i+1] * fc) Pc <- 1 / (1 + ec) Qc <- 1 - Pc #bc Pb <- dnorm(be[, i], mean = 0, sd = 1) Pbc <- dnorm(bc, mean = 0, sd = 1) Lb <- (P ^ U) * (Q ^ (1 - U)) Lbc <- (Pc ^ U) * (Qc ^ (1 - U)) LPb <- log(pb) LPbc <- log(pbc) LLb <- log(lb) LLbc <- log(lbc) LLB <- rowsums(llb) LLBc <- rowsums(llbc) L <- (LLBc + LPbc) - (LLB + LPb) c <- exp(l ) dim( c) <- c(1, 50) <- apply( c, 2, min, 1) urn <- runif(n = 50, min = 0, max = 1) logic <- >= urn be[which( logic), i+1] <- bc[which( logic)] be[which(! logic), i+1] <- be[which(! logic), i] } # gc() gc() # a b e <- e[, -c(1:3000)] ae <- ae[, -c(1:3000)] be <- be[, -c(1:3000)] <- rowmeans( e) a <- rowmeans(ae) b <- rowmeans(be) m[, r] <- am[, r] <- a bm[, r] <- b pc[1, r] <- cor( t,, method = c("pearson")) pc[2, r] <- cor(at, a, method = c("pearson")) pc[3, r] <- cor(bt, b, method = c("pearson")) kc[1, r] <- cor( t,, method = c("kendall")) kc[2, r] <- cor(at, a, method = c("kendall")) kc[3, r] <- cor(bt, b, method = c("kendall"))

181 179 sc[1, r] <- cor( t,, method = c("spearman")) sc[2, r] <- cor(at, a, method = c("spearman")) sc[3, r] <- cor(bt, b, method = c("spearman")) # gc() gc() } # a b Bias RMSE Bias <- rowmeans( m) - t Biasa <- rowmeans(am) - at Biasb <- rowmeans(bm) - bt RMSE 1 <- ( m - t) ^ 2 RMSE 2 <- rowmeans(rmse 1) RMSE <- sqrt(rmse 2) RMSEa1 <- (am - at) ^ 2 RMSEa2 <- rowmeans(rmsea1) RMSEa <- sqrt(rmsea2) RMSEb1 <- (bm - bt) ^ 2 RMSEb2 <- rowmeans(rmseb1) RMSEb <- sqrt(rmseb2) #Bias RMSE B <- mean(bias ) Ba <- mean(biasa) Bb <- mean(biasb) R <- mean(rmse ) Ra <- mean(rmsea) Rb <- mean(rmseb) B <- sqrt(mean((bias - B ) ^ 2)) Ba <- sqrt(mean((biasa - Ba) ^ 2)) Bb <- sqrt(mean((biasb - Bb) ^ 2)) R <- sqrt(mean((rmse - R ) ^ 2)) Ra <- sqrt(mean((rmsea - Ra) ^ 2)) Rb <- sqrt(mean((rmseb - Rb) ^ 2)) # writetable( m, file = " (10) xls", sep = "\t") writetable(am, file = " a(10) xls", sep = "\t") writetable(bm, file = " b(10) xls", sep = "\t") writetable(xe, file = " X(10) xls", sep = "\t") writetable(dpe, file = " DP(10) xls", sep = "\t")

182 180 writetable(pre, file = " PR(10) xls", sep = "\t") writetable(pc, file = " pc(10) xls", sep = "\t") writetable(kc, file = " kc(10) xls", sep = "\t") writetable(sc, file = " sc(10) xls", sep = "\t") writetable(bias, file = " Bias (10) xls", sep = "\t") writetable(biasa, file = " Biasa(10) xls", sep = "\t") writetable(biasb, file = " Biasb(10) xls", sep = "\t") writetable(rmse, file = " RMSE (10) xls", sep = "\t") writetable(rmsea, file = " RMSEa(10) xls", sep = "\t") writetable(rmseb, file = " RMSEb(10) xls", sep = "\t") sv <- cbind( B, Ba, Bb, R, Ra, Rb, B, Ba, Bb, R, Ra, Rb) writetable(sv, file = " sv(10) xls", sep = "\t") CCM ((N = 500,J = 30) ) # setseed(1010) # 100 t1 <- rnorm(n = 100) # bt <- runif(n = 30, min = -20, max = 20) # at <- runif(n = 30, min = 03, max = 15) # 400 t2 <- rnorm(n = 400) t <- c( t1, t2) # (2 ) tvec <- rep( t, 30)

183 181 tmat <- matrix( tvec, nrow = 30, ncol = 500, byrow = TRUE) ft <- tmat - bt et <- exp(-10 * at * ft) Pt <- 1 / (1 + et) Qt <- 1 - Pt # (item1 ~ item30) b12 <- (-2) ec <- at * ft eec <- exp(ec) d1 <- eec[seq(1, 29, by = 2), ] d2 <- eec[seq(2, 30, by = 2), ] d3 <- exp(ec[seq(1, 29, by = 2), ] + ec[seq(2, 30, by = 2), ] - b12) d <- 1 + d1 + d2 + d3 P00 <- 1 / d P10 <- d1 / d P01 <- d2 / d P11 <- d3 / d 2PLM 2PLCM ((N = 100,J = 10) ) # setseed(1010) # 100 t <- rnorm(n = 100) # bt <- runif(n = 30, min = -20, max = 20) # at <- runif(n = 30, min = 03, max = 15) # 10 bt <- bt[c(1:10)] at <- at[c(1:10)] # (2 ) tvec <- rep( t, 10) tmat <- matrix( tvec, nrow = 10, ncol = 100, byrow = TRUE) ft <- tmat - bt et <- exp(-10 * at * ft)

184 182 Pt <- 1 / (1 + et) Qt <- 1 - Pt # (item1 ~ item10) <- 300 C1 <- - * Qt C2 <- 1 - exp(c1) C3 <- log(c2) C4 <- matrix(nrow = 5, ncol = 100) for(i in 1:5){ C4[i, ] <- colsums(c3[c((2 * i - 1), (2 * i)), ]) } C5 <- C4 - log(1 - exp(- )) C6 <- 1 - exp(c5) C7 <- log(c6) Cop <- (-1 / ) * C7 P00 <- Cop P10 <- Qt[seq(2, 10, by = 2), ] - Cop P01 <- Qt[seq(1, 9, by = 2), ] - Cop P11 <- 1 - Qt[seq(1, 9, by = 2), ] - Qt[seq(2, 10, by = 2), ] + Cop 2PLM 3, 4, 5 ((N = 1000,J d(j) = 5, d(j) 0) ) # setseed(1010) # ( :1186 :0394) tt <- rnorm(n = 1000, mean = 0, sd = 1) bt <- rnorm(n = 20, mean = 0, sd = 1) at <- runif(n = 20, min = 05, max = 15) te1t <- rnorm(n = 1000, mean = 0, sd = 0394) te2t <- rnorm(n = 1000, mean = 0, sd = 0394) te3t <- rnorm(n = 1000, mean = 0, sd = 0394) te4t <- rnorm(n = 1000, mean = 0, sd = 0394) tet <- cbind(te1t, te2t, te3t, te4t) # subtract <- function(x, y){x - y} # (5 ver)

185 183 c1 <- apply(asmatrix(tt), 1, subtract, bt) c2 <- numeric() for(l in 1:4){ zp <- t(c1)[, (5*l-4):(5*l)] - tet[, l] c2 <- cbind(c2, zp) } #l Pt <- 1/(1 + exp((-17)*t(t(c2)*at))) # (row & sum) U <- vector(mode = "list", length = 50) S <- vector(mode = "list", length = 50) for(i in 1:50){ u <- matrix(runif(n = 20000, min = 0, max = 1), nrow = 1000, ncol = 20) U[[i]] <- ifelse(pt >= u, 1, 0) z <- numeric() for(l in 1:4){ zp <- rowsums(u[[i]][, (5*l-4):(5*l)]) z <- cbind(z, zp) } #l S[[i]] <- z } #i # for(l in 1:50){ writetable(t(asvector(t(u[[l]]))), paste("u_", formatc(l, format = "d", width = 2, flag = "0"), "txt", sep = ""), sep = ",", rownames = FALSE, colnames = FALSE) writetable(t(asvector(t(s[[l]]))), paste("s_", formatc(l, format = "d", width = 2, flag = "0"), "txt", sep = ""), sep = ",", rownames = FALSE, colnames = FALSE) } #l WinBUGS data file ( GRM J d(j) = 5 ) #model file (GRM) #5item ver model{ for(i in 1:N){ for(j in 1:J){ Ps[i, j, 6] <- 0 for(k in 1:5){ Ps[i, j, k] <- 1/(1 + exp(-17*a[j]*(t[i]-b[j, k]))) P[i, j, k+1] <- Ps[i, j, k] - Ps[i, j, k+1] } #k P[i, j, 1] <- 1 - Ps[i, j, 1] U[i, j] <- S[i, j] + 1

186 184 U[i, j] ~ dcat(p[i, j, ]) } #j t[i] ~ dnorm(0, 1) } #i for(j in 1:J){ a[j] ~ dnorm(1, 4) b[j, 1] ~ dnorm(0, 1) for(k in 2:5){ b[j, k] ~ dnorm(0, 1)I(b[j, k-1], ) } #k } #j } WinBUGS model file ( BTM J d(j) = 3 ) #model file (TEM) #3item ver model{ for(i in 1:N){ for(l in 1:4){ for(j in (5*l-4):(5*l-2)){ P[i, j] <- 1/(1+exp(-17*a[j]*(t[i]-b[j]-g[i, l]))) U[i, j] ~ dbern(p[i, j]) } #j for(j in (5*l-1):(l*5)){ P[i, j] <- 1/(1+exp(-17*a[j]*(t[i]-b[j]))) U[i, j] ~ dbern(p[i, j]) } #j g[i, l] ~ dnorm(0, tau[l]) } #l t[i] ~ dnorm(0, 1) } #i for(l in 1:4){ tau[l] ~ dgamma(3, 1) } #l for(j in 1:J){ a[j] ~ dnorm(1, 4) b[j] ~ dnorm(0, 1) } #j } #model WinBUGS model file ( CCM J d(j) = 2 ) #model file (CCM) #2item ver

187 185 model{ for(i in 1:N){ for(j in 1:10){ z1[i, j] <- (17)*a[2*j-1]*(t[i]-b[2*j-1]) z2[i, j] <- (17)*a[2*j]*(t[i]-b[2*j]) dp[i, j] <- 1 + exp(z1[i, j]) + exp(z2[i, j]) + exp(z1[i, j]+z2[i, j]-g[j]) P[i, j, 1] <- 1/dP[i, j] P[i, j, 2] <- exp(z1[i, j])/dp[i, j] P[i, j, 3] <- exp(z2[i, j])/dp[i, j] P[i, j, 4] <- exp(z1[i, j]+z2[i, j]-g[j])/dp[i, j] } #j t[i] ~ dnorm(0, 1) } #i for(l in 1:L){ a[l] ~ dnorm(1, 4) b[l] ~ dnorm(0, 1) } #j for(j in 1:4){ g[j] ~ dnorm(-2, 1) } #j for(j in 5:10){ g[j] <- 0 } #j for(i in 1:N){ for(j in 1:10){ U[i, j] <- V[i, j] + 1 U[i, j] ~ dcat(p[i, j, ]) } #j } #i } #model #ok WinBUGS model file ( 2PLM ) model{ for(i in 1:N){ for(j in 1:J){ P[i, j] <- 1/(1+exp(-17*a[j]*(t[i]-b[j]))) U[i, j] ~ dbern(p[i, j]) } #j t[i] ~ dnorm(0, 1) } #i for(j in 1:J){ a[j] ~ dnorm(1, 4) b[j] ~ dnorm(0, 1) } #j

188 186 } #model, WinBUGS MCMC, data file inits file, WinBUGS User Manual Version 14 (Spiegelhalter et al, 2003)

189 187,,,,,