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heterogeneity (multivariate analysis) (homogeneous) ( ) ( ) (group) (multiple group analysis) (class) (latent class analysis LCA) Bartholomew Knott(1999) ( 1) (common factor analysis FA) (LCA) (latent profile analysis LPA) (latent trait analysis/item response theory LTA/IRT) Bartholomew & Knott, 1999; Heinen, 1996; Langenheine & Rost, 1998 (Common Factor Analysis) (Latent Profile Analysis) (Latent Trait Analysis) (Latent Class Analysis) (eta) (quantitative) (qualitative) ( Hicks, Krueger, Iacono, McGue, & Patrick, 2004; Pickles & Angold, 2003) 2

(Lubke, & Muthen, 2007; Lubke, Muthen, Moilanen, McgGough, Loo, Swanson, Yang, Taanila, Hurtig, Jarvelin & Smalley, 2007; Lubke & Neale, 2006; Muthen, 2006; Lubke & Muthen, 2005) (Lubke & Neale, 2006; Muthen, 2006) (Curran, 2004; Meredith & Horn, 2001) (Molenaar & van Eye, 1994) (local independence) (factor mixture model FMM) (Yung, 1997; Muthen & Shedden, 1999) (structural equation) (within class between group) 3

FMM (categorical factor analysis) Y i i p q Y = α + α η + ε ( i= 1, 2,, p) (1) i i0 ij j i j= 1 η j αi0 αij ε i αi0 η j ε i 0 σ σ 2 q p 2 j i η1 η q ( ) ( ) η1 η q Y1 Y p ε1 ε p 0 Yi 1 0 (1) Yi P r( Y i = 1 η) ( π i η) ( πi η) η i η η (link function) logit (1) (2) Bartholomew, Steele, Moustaki & Galbraith, 2002 q π ( η) log itπ ( η) = log = α + α η 1 ( ) ( i= 1, 2,, p) (2) i i e i0 ij j πi η j= 1 (2) π ( η) (3) i q exp αi0 + αijηj j= 1 πi ( η) = q 1+ exp αi0 + αijηj (3) j= 1 (2) (3) α i0 α ij α i0 α (categorical factor analysis) 4 ij

Bartholomew Knott(1999) FMM (Latent Class Analysis) (logistic regression) LCA Y i (4) v e 1 β (4) 1+ e i ( i = X i, ) = vi P Y vi = β 0 + β1x i Y i X C i i k = 1,2,, K Yi = 1 (unconditional probability) (5) P K ( Y 1) = P( C = k) P( Y = 1 C k) = i i i i = k = 1 (5) ( ) P Y = 1 C k i k i i = Y 1 i k P( Yi = 1 Ci = k) (4) P β e 1 (6) 1+ e k ( Yi = Ci = k) = βk ( ) β k P C i = k i C i K k = 1,2,, (multinomial logistic regression) (7) γ k e P( Ci = k) = K e k ' = 1 γ k ' (7) γ k ' (Factor Mixture Model) 5

(dimensional) (categorical) (cluster analysis) (model based) (factor mixture model) 1 C η Y 1 Y2 Yp 1 1 Y 1 Y p η η Y 1 Y p η C η Y 1 Y p C η 1 η C 1 η C Y 1 Y p 6

(Clark,2010) * 1if yij τ y = (8) ij * 0if yij < τ * i j yij τ yij 1 i p * Yi k = 1,2,, K (9) Y * = α + Λ η + ε (9) ik k k ik ik (9) i k α k p k ( p 1) Λ k k p q ( p q ) η ik i k q ( q 1) ε ik i k p ( p 1) p logit probit (9) k η = a + ζ (10) ik k ik (10) i k a ( k q 1) k ζ ik ζ ( ik q 1) Ψ ( k q q) (11) ζ ~N 0, Ψ (11) ik ( ) k (8) (11) (8) (11) 2 ( η C ) 7

η C Y 1 Y2 Yp 2 2 C Y α ( ) k C η Y ( ) Λ ( ) k C η k η 2 ik η ζ ( ik Ψ ) k τ k Λ k ζ ik k 2 1 k = 12,,,K 叁 (1997) (Ross L. Mooney) 1.97.75,.85,.84,.82,.81,.85.88 LCA, IRT FMM 18 1705 753 44.3% 950 55.7% 577 (33.8%), 517 (30.3%) 611 (35.8%) 8

28 35 5 1703 39.4% 38.9% 23.19% 34.5% 31.4% 1703 436 25.6% 5 87 5.1% 472 27.7% 2 3 4 19.7% 13.1% 8.8% 2 32 5 436 25.6% 4 133 7.81% 5 127 7.46% 5 87 5.11% 1 79 4.64% 3 77 4.52% 2 (Categorical Exploratory Factor Analysis, CaEFA) Mplus5.0(Muthen & Muthen, 1998-2007) 3 3 74.7 3.1 CFI.999.952 TLI.986.903 RMSEA.090.035 SRMR.065.013 1 0.83 0.51 0.73 0.57 0.47.05 ;.05 ( 0.60,0.36 0.96) ( 0.49 0.77) 0.39 1 CFI TLI RMSEA SRMR 9

(categorical confirmatory factor analysis CaCFA) 2 1 00000 436 25.60 2 00010 133 7.81 3 00001 127 7.46 4 11111 87 5.11 5 10000 79 4.64 6 00100 77 4.52 7 10111 73 4.29 8 10100 68 3.99 9 00011 66 3.88 10 01000 56 3.29 11 10011 55 3.23 12 10010 53 3.11 13 10110 49 2.88 14 11110 37 2.17 15 10001 35 2.06 16 11100 27 1.59 17 11000 24 1.41 18 01011 23 1.35 19 01001 23 1.35 20 00110 21 1.23 21 01010 20 1.17 22 11101 18 1.06 23 10101 18 1.06 24 01100 18 1.06 25 11001 17 1.00 26 11011 13 0.76 27 00111 12 0.70 28 11010 9 0.53 29 01111 9 0.53 30 01110 9 0.53 10

31 00101 7 0.41 32 01101 4 0.23 5 1 0 1 5 1 0 BIC ABIC entropy( ) BIC ABIC Entropy 0.90 3 BIC ABIC entropy LCA3( 3 ) (BIC 10028 ABIC 9941 entropy 0.771) LMR(LO-MENDELL-RUBIN adjusted LRT test) BLR( bootstrapped likelihood ratio test) 4 3 LCA4 MPlus LCA4 LCA3 3 3 3 EFA1 EFA2 LCA1 LCA2 LCA3 LCA4 FMM21 FMM22 FMM23 /CFI.952.999-5356 -4988-4951 -4930-4961 -4952-4941 BIC/TLI.903.986 10749 10059 10028 10032 10005 10008 10009 ABIC/RMSEA.090.035 10733 10024 9947 9959 9970 9963 9955 /SRMR.065.013-0.708 0.771 0.603 -.713 0.604 LMR - 2>1 3>2 4>3-2=1 3>2 BLR - 2>1 3>2 4>3-2>1 3=2 5 9 5 11 17 23 11 14 17 :LMR LO-MENDELL-RUBIN ADJUSTED LRT TEST BLR BOOTSTRAPPED LIKELIHOOD RATIO TEST > = (FMM) 1 2 3 FMM21 FMM22 FMM23 BIC ABIC entropy FMM21 BIC 10005 ABIC FMM23 9955 entropy FMM22 0.713 LMR 2 1 3 2 BLR FMM22 FMM21( 2 1 ) FMM23 FMM22( 3 2 ) FMM23 LCA4 FMM23 11

FMM22 FMM22 CaCFA2( ) LCA3( ) 1. (CFA FMM ) CaCFA FMM 4 4CaCFA.857.748.518.743.570 FMM 4.796.649.380.608.363 FMM 5 4 CaCFA FMM CFA FMM F1 F2 F1 F2 1..857(.031).796(.015) 2..518(.038).380(.104) 3..748(.032).649(.109) 4..743(.045).608(.085) 5..570(.041).363(.090) CaCFA FMM CaCFA 1.840 1.228-0.612 0.050.583 1.525 0.988-0.537 0.033 0.477 0.672 FMM 12

FMM 5 MPlus 0 5.080 4.587 FMM CFA 5.681 1.931 1.285 0.388 3 CaCFA 3 25.6% 5 32 1703 32 3 CaCFA 4 FMM CFA 13

3 436 25.6% FMM CaCFA Pearson.988.963.999.998 5 6 5 6 FMM 8 ( 8 + ) FMM 5 5 4 FMM 14

5 CFA FMM CFA FMM Factor1 Factor2 Factor1 Factor2 00000-0.612-0.537-1.599-0.820 00001-0.401-0.108-1.289-0.111 00010-0.309 0.099-0.995 0.569 00011-0.170 0.435-0.611 1.407 00100 0.003-0.273 0.297-0.474 00101 0.118 0.087 0.532 0.227 00110 0.178 0.276 0.808 0.983 00111 0.279 0.605 1.304 2.132 01000-0.225-0.368-0.606-0.639 01001-0.081 0.013-0.344 0.060 01010-0.012 0.207-0.072 0.764 01011 0.100 0.537 0.343 1.725 01100 0.229-0.180 1.036-0.338 01101 0.332 0.165 1.267 0.380 01110 0.387 0.354 1.586 1.222 01111 0.483 0.684 2.214 2.595 10000 0.184-0.199 1.122-0.321 10001 0.294 0.152 1.354 0.399 10010 0.355 0.342 1.682 1.256 10011 0.464 0.677 2.330 2.658 10100 0.588-0.037 2.585-0.027 10101 0.706 0.301 2.932 0.831 10110 0.779 0.501 3.595 2.096 10111 0.917 0.858 4.672 3.930 11000 0.407-0.109 1.817-0.188 11001 0.517 0.233 2.078 0.575 11010 0.580 0.426 2.515 1.583 11011 0.696 0.769 3.361 3.230 11100 0.835 0.058 3.376 0.162 11101 0.967 0.395 3.899 1.195 11110 1.055 0.606 4.889 2.769 11111 1.228 0.988 6.075 4.574 5 1 0 1 5 1 LCA FMM 15

5 FMM 6 FMM 2. (LCA FMM ) 16

6 6 1082 ( 63.5%) 211 ( 12.4%) 411 ( 24.1%) 27% 26% 2.3% 5 73.5% 0 211 7 7 6 LCA LCA 1 2 3 1..175.483.867 2..134.215.478 3..023.945.735 4..270.168.796 5..264.000.699 1082 211 410.635.124.241 17

18 7 LCA CFA LCA 8 8 LCA + LCA 8 LCA 8 LCA LCA LCA + LCA + 7 00110 21 LCA 2 k k-1 LCA 00100 00110 01100 10100 11100 2 00101 00111 3 8 LCA

LCA FMM 8 LCA 4 1 3 3 5 IRT 2 4 LCA 1 3 FMM 10111 11011 11111 FMM 19

7 CFA LCA FMM LCA FMM 1 2 3 1 2 00000 436 0 0 0 436 00001 127 0 0 0 127 00010 133 0 0 0 133 00011 66 0 0 0 66 00100 0 77 0 0 77 00101 7 0 0 0 7 00110 0 21 0 0 21 00111 0 0 12 0 12 01000 56 0 0 0 56 01001 23 0 0 0 23 01010 20 0 0 0 20 01011 23 0 0 0 23 01100 0 18 0 0 18 01101 0 0 4 0 4 01110 0 0 9 0 9 01111 0 0 9 0 9 10000 79 0 0 0 79 10001 35 0 0 0 35 10010 53 0 0 0 53 10011 0 0 55 0 55 10100 0 68 0 0 68 10101 0 0 18 0 18 10110 0 0 49 0 49 10111 0 0 73 73 0 11000 24 0 0 0 24 11001 0 0 17 0 17 11010 0 0 9 0 9 11011 0 0 13 13 0 11100 0 27 0 0 27 11101 0 0 18 0 18 11110 0 0 37 0 37 11111 0 0 87 87 0 5 1 0 1 5 1 LCA FMM 20

8 FMM FMM 4 5 4 2 FMM LCA LCA FMM LCA FMM LCA 8 LCA FMM LCA FMM 8 LCA FMM 0 1 2 3 4 5 LCA FMM 1 2 3 1 2 436 436 395 77 472 228 107 335 23 27 173 233 150 86 84 87 87 0 21

1703 5 (1) (2) (3) (4) (5) IRT LCA 3 2 5 32 3 3 5 LCA FMM CFA LCA FMM CFA 0.100 0.074-0.014.05 LCA FMM FMM FMM LCA 56% 60% 55% 22

LCA LCA FMM CFA LCA FMM FMM LCA LCA CFA FMM LCA AA BB( CC ) CFA FMM 2 5 32 32 (empty cell) LCA LCA 32 1703 32 4 LCA 6 64 LCA (local maximized solution) CFA LCA FMM 1703 32 23

FMM FMM ; (McLachlan & Peel, 2000) FMM (Bauer and Curran, 2003) CFA FMM LCA CFA LCA 2 5 00000 25.6% Kim Muthen(2009) Kim Muthen(2009) (two-part factor mixture modeling) 24

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