(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
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1 (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: Baron and Kenny (1986) a b c c a b supressor variable model 2, [email protected]. 1 Baron and Kenny (1986) Maassen & Bakker (2001) 1
2 (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
3 a b σ ab z = a b σ ab (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
4 Sobel s test Sobel s test 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) 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 Bootstrapping method 1 a b davidpm/ripl/mediate.htm 5 davidpm/ripl/freqdist.pdf 4
5 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 (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) N N 1 8 SEM Figure 1 bootstrapping 9 SEM t z 5
6 Baron and Kenny (1986) SEM SEM (multiple mediator model) a b a b c MacKinnon et al. (2002) Baron and Kenny (1986) a b c Baron and Kenny (1986) a, b, c 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
7 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
8 (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
9 = Krull and MacKinnon (1999, 2001) (Kreft & De Leeuw, 1998) 2 1 Preacher X M Y 1 M Y Figure X M Y 1 2 9
10 3.3 HLM HLM ML-SEM 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 γ (3) 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
11 (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 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: Multilevel mediation model 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) group-mean centering (Enders & Tofighi, 2007) 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
12 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 ML-SEM straightforward 2 ML-SEM Mplus Preacher HP 14 2 between 2 1 between/within (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 preacher/syntax appendix pdf 15 Mplus model constraint Sobel s test 12
13 4 References Aroian, L. A. (1944). The probability function of the product of two normally distributed variables. Annals of Mathematical Statistics, 18, Baron, R. M. & Kenny, D. A. (1986). The moderator?mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, Bauer, D. J., Preacher, K. J., & Gil, K. M. (2006). Conceptualizing and testing random indirect effects and moderated mediation in multilevel models: New procedures and recommendations. Psychological Methods, 11, Edwards, J. R., & Lambert, L. S. (2007). Methods for integrating moderation and mediation: A general analytical framework using moderated path analysis. Psychological Methods, 12, Efron, B. & Tibshirani, R. (1993). An introduction to the bootstrap. Boca Raton, FL: Chapman & Hall/CRC. Judd, C. M., & Kenny, D. A. (1981). Process analysi: Estimating mediation in treatment evaluations. Evaluation Review, 5, Kenny, D. A., Korchmaros, J. D., & Bolger, N. (2003). Lower level mediation in multilevel models. Psychological Methods, 8, Kreft, I., & De Leeuw, J. (1998). Introducing multilevel modeling. London: Sage ( (2006). ) Krull, J. L., & MacKinnon, D. P. (1999). Multilevel mediation modeling in group-based intervention studies. Evaluation Review, 23, Krull, J. L., & MacKinnon, D. P. (2001). Multilevel modeling of individual and group level mediated effects. Multivariate Behavioral Research, 36, Ludtke, O., Marsh, H. W., Robitzsch, A., Trautwein, U., Asparouhov, T., & Muthen, B. (2008). The multilevel latent covariate model: A new, more reliable approach to group-level effects in contextual studies. Psychological Methods, 13, MacKinnon, D. P. (2008). Introduction to statistical mediation analysis. Mahwah, NJ: Erlbaum. MacKinnon, D. P., Fairchild, A. J., & Fritz, M. S. (2007). Mediation analysis. Annual Review of Psychology, 58, MacKinnon, D. P., Fritz, M. S., Williams, J., & Lockwood, C. M. (2007). Distribution of the product confidence limits for the indirect effect: Program PRODCLIN. Behavior Research Methods, 39, MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M., West, S. G., & Sheets, V. (2002). A comparison of methods to test the significance of the mediated effect. Psychological Methods, 7, MacKinnon, D. P., Lockwood, C. M., & Williams, J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, 39, MacKinnon, D. P., Warsi, G., & Dwyer, J. H. (1995). A simulation study of mediated effect measures. Multivariate Behavioral Research, 30,
14 Maassen, G. H., & Bakker, A. B. (2001). Suppressor variables in path models: Definitions and interpretations. Sociological Methods & Research, 30, Marsh, H. W., Ludtke, O., Robitzsch, A., Trautwein, U., Asparouhov, T., Muthen, B., & Nagensgast, B. (in press). Multilevel fully latent-variable models of contextual effects: Integrating multilevel and structural equation approaches with multiple indicators to control. Muller, D., Judd, C. M., & Yzerbyt, V. Y. (2005). When moderation is mediated and mediation is moderated. Journal of Personality and Social Psychology, 89, Preacher, K. J., Zyphur, M. J., & Zhang, Z. (in press). A general multilevel SEM framework for assessing multilevel mediation. Psychological Methods. Sobel, M. E. (1982). Asymptotic confidence intervals for indirect effects in structural equation models. Sociological Methodology, 13, Springer, M. D. & Thompson, W. E. (1966). The distribution of products of independent random variables. SIAM Journal of Applied Mathematics, 14, Zhang, Z., Zyphur, M. J., & Preacher, K. J. (2009). Testing multilevel mediation using hierarchical linear models: Problems and solutions. Organizational Research Methods, 12,
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