21 12 23 (mediation analysis) Figure 1 X Y M (mediator) mediation model Baron and Kenny (1986) 1 1) mediated moderation ( moderated mediation) 2) (multilevel mediation model) a M b X c (c ) Y 1: 1 1.1 Baron and Kenny (1986) a b c c a b supressor variable model 2, e-mail: murakou@orion.ocn.ne.jp. 1 Baron and Kenny (1986) 2009 12 10000 2 Maassen & Bakker (2001) 1
(X) (Y ) Y = intercept + c X + e (1) e c c M = intercept + ax + e (2) a Y = intercept + cx + bm + e (3) (1) X c c c (3) b X M Y (indirect effect) a b 1 c c X Y M X Y c M c c c c = a b (4) c M c 1 X Y X Y M (complete mediation model) c (partial mediation model) Kenny Judd & Kenny (1981) Baron & Kenny (1986) 1.2 X Y, mediation effect MacKinnon, Lockwood, Hoffman, West, and Sheets (2002) MacKinnon (2008) 2
1.2.1 1 a b σ ab z = a b σ ab (5) 1.96 5% MacKinnon et al. (2002) Sobel s test Aroian (1944) a b (4) c c a b c c (Clogg et al., 1992 ) Sobel s test Sobel (1982) a b σ ab = a 2 σ 2 b + b2 σ 2 a (6) σ a, σ b a b (delta method) X f(x) V ar(f(x)) = ( f(x) X )2 (V ar(x)) (7) X f(x) V ar(f(x)) = d V d (8) V X f(x) X d Mplus model constraint f(a, b) = a b a b b a d = (b, a) X ( ) σa 2 0 V = 0 σb 2 (6) (8) 1 0 a b 0 1 (MacKinnon et al., 1995) Aroian (1944) s test 1 2 Aroian (1944) a b σ ab = (9) a 2 σ 2 b + b2 σ 2 a + σ 2 aσ 2 b (10) 3
Sobel s test Sobel s test 1.2.2 Distribution of a product method PRODCLIN 1.96 critical value (confidence interval) 3 95% 0 a b Distribution of a product method Springer & Thompson (1966) Table MacKinnon, Fritz, Williams, & Lockwood (2007) PROD- CLIN (distribution of the PRODuct Confidence Limits for INdirect effects) http://www.public.asu.edu/ davidpm/ripl/prodclin/ a, b, σ a, σ b a b 0 0 X M t 2 t critical value Table MacKinnon Website 4 MacKinnon et al. (2004) Empirical-M method 2 critical value Table MacKinnon Website 5 1.2.3 Bootstrapping method 1 a b 3 4 http://www.public.asu.edu/ davidpm/ripl/mediate.htm 5 http://www.public.asu.edu/ davidpm/ripl/freqdist.pdf 4
95% 6 0 nonparametric bootstrapping SAS R N N a b a b a b 7 parametric bootstrapping (Efron & Tibshirani, 1993) a b a b 95% bootstrap a b Efron & Tibshirani, 1993 bias-corrected bootstrapping method MacKinnon et al. (2004) bootstrapping method bias-corrected bootstrap distribution of a product method parametric bootstrapping AMOS, EQS, Mplus SEM 8 distribution of a product method bias-corrected bootstrapping PRODCLIN bootstrapping 2 (X M 1 M 2 Y bootstrapping 1.3 1.3.1 (SEM) (structural equation modeling; SEM) Baron and Kenny (1986) Figure 1 c bootstrapping SEM Baron and Kenny (1986) SEM 9 Figure 1 X Y SEM 1 (full information maximum likelihood method) 6 2.5 97.5 7 N N 1 8 SEM Figure 1 bootstrapping 9 SEM t z 5
Baron and Kenny (1986) SEM SEM (multiple mediator model) 1.3.2 a b a b c MacKinnon et al. (2002) Baron and Kenny (1986) a b c Baron and Kenny (1986) a, b, c 1.3.3 H 0 : a b = 0 vs. H 1 : a b 0 (11) H 0 : a = 0 OR b = 0 (12) a b a = 0 b = 0 2 Moderated mediation model 2.1 mediation moderation mediation X Y M moderation X Y Z (moderator, moderator variable) X Z (Baron and Kenny, 1986) Y = intercept + c XX + c ZZ + c XZXZ + e (13) 6
X Z 10 X Z XZ X Z (Cronbach, 1987) Y = intercept + (c X + c XZ Z)X + c ZZ + e (14) X Z 2.2 moderated mediation model 2 moderated mediation Figure 1 a, b, c Z Z a b Z a b moderated mediation Z Figure 1 SEM Z mediated moderation Muller, Judd, and Yzerbyt (2005) Edwards and Lambert (2007) moderated mediation model Figure 2 moderation effect Z mediated moderation c XZ Edwards and Lambert (2007) Figure 2 moderated path analysis a X + a XZ Z M b M + b MZ Z X c X + c XZ Z (c X + c XZ Z) Y 2: moderated mediation model M M = intercept + a X X + a Z Z + a XZ XZ + e (15) 10 Z X 7
(2) a XZ X M Z Y Y = intercept + c X X + c XZ XZ + b M M + b MZ MZ + d Z Z + e (16) (3) Z moderator +1SD -1SD moderator bootstrap Edwards and Lambert (2007) SPSS 2.3 mediated moderation? mediated moderation mediated moderation mediated moderation Z mediated moderation Muller et al. (2005) Edwards and Lambert (2007) mediated moderation moderated path analysis (Figure 2) moderation mediated moderation Figure?? a XZ Figure?? (moderated path model) moderated mediation mediated moderation 3 (multilevel mediation model, multilevel mediation analysis) (Hierarchical linear model; HLM) (multilevel structural equation modeling; ML-SEM) Krull and MacKinnon (1999, 2001) (e.g., Bauer, Preacher, & Gil, 2006; Kenny, Bolger, & Korchmaros, 2003) Preacher 11 HLM (Zhang, Zyphur, & Preacher, 2009) ML-SEM (Preacher, Zhphur, & Zhang, in press) 11 8
3.1 1 2 1 1-1-1 2 2-1-1 2 2 2 = 8 1-1-2 3.2 Krull and MacKinnon (1999, 2001) 2-1-1 2 1 1 2 1 (Kreft & De Leeuw, 1998) 2 1 Preacher 2 1 2 2 1 2 1 1 2 1 2 2 1 2-1-1 2 1 1 1-1 1 2 X M Y 1 M Y Figure 3 2-2-1 2 2 1 2 1-1-1 1 1 1 2 2 1-1-1 X M Y 1 2 9
3.3 HLM HLM ML-SEM 1 2 2-1-1 HLM X j Y ij X 2 1 i Level 1 : Y ij = β (1) 0j + r(1) ij (17) Level 2 : β (1) 0j = γ (1) 00 + γ(1) 01 X j + u (1) 0j (18) Level 1 : M ij = β (2) 0j + r(2) ij (19) Level 2 : β (2) 0j = γ (2) 00 + γ(2) 01 X j + u (2) 0j (20) Level 1 : Y ij = β (3) 0j + β(3) 1j M ij + r (3) ij (21) Level 2 : β (3) 0j = γ (3) 00 + γ(3) 01 X j + u (3) 0j (22) β (3) 1j = γ (3) 10 (21) M ij 1 1 2 γ (3) 10 1 2 1 2 M ij 2 aggregate M.j 1 2 grand-mean centering 2 group-mean centering 12 (21) 12 Enders and Tofighi (2007) Y ij X ij grand-mean centering 2 X.j grand-mean centering Level 1 : Y ij = β (a) 0j + β(a) 1j X ij + r (a) ij (23) Level 2 : β (a) 0j β (a) 1j = γ (a) 00 + γ(a) 01 X.j + u (a) 0j (24) = γ (a) 10 (25) X.j γ (a) 01 1 X ij X.j 2 X.j 1 X ij (over and above) contextual effect compositional effect 10
(22) Level 1 : Y ij = β (4) 0j + β(4) 1j (M ij M.j ) + r (4) ij (30) Level 2 : β (4) 0j = γ (4) 00 + γ(4) 01 X j + γ (4) 02 M.j + u (4) 0j (31) β (4) 1j = γ (4) 10 1 group-mean centering 2 M.j 1 2 aggregation 1 2 2-1-1 Figure 3 Preacher et al. (in press) レベル 2 (2) γ 01 M.j (4) γ 02 Xj (4) γ 01 (1) 01 ( γ ) Y M 個々の得点 X Y Mij (4) γ 10 レベル 1 Y 3: 2-1-1 Multilevel mediation model 2-2-1 1-1-1 1 group-mean centering group-mean centering Level 1 : Y ij = β (b) 0j + β(b) 1j (X ij X.j ) + r (b) ij (26) Level 2 : β (b) 0j β (b) 1j = γ(b) 00 + γ(b) 01 X.j + u (b) 0j (27) = γ(b) 10 (28) 2 1 2 group-mean centering (Enders & Tofighi, 2007) 2 1 2 γ (b) 01 γ(b) 10 contextual effect 2 contextual effect γ (a) 01 = γ(b) 01 γ(b) 10 (29) (Ludtke et al., 2009) 1 2 group-mean centering 11
Zhang et al. (2009) 3.4 ML-SEM ML-SEM 1 2 HLM 1 X ij X.j ML-SEM X ij = µ + u j + r ij (32) 1 u j 2 r ij 2 ML-SEM 1 2 13 ML-SEM straightforward 2 ML-SEM Mplus Preacher HP 14 2 between 2 1 between/within 1 2 3.5 (4) 2 2 Sobel test 15 Preacher et al. (in press) nonparametric bootstrapping parametric bootstrapping Mplus 2 moderated mediation model multilevel mediation model Bauer et al. (2006) HLM 13 2 HLM ML-SEM (Ludtke et al., 2009) socio-economic status (SES) aggregation Ludtke et al. (2009) Marsh et al. (in press) 14 http://www.people.ku.edu/ preacher/syntax appendix 081009.pdf 15 Mplus model constraint Sobel s test 12
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