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Structural multilevel models for longitudinal mediation analysis: a definition variable approach

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Abstract

Mediation analysis is used to assess the direct effect of an exposure on an outcome, and the indirect effect transmitted by a third intermediate variable. Longitudinal data are the most suited to address mediation, since they allow mediational effects to manifest over time. There exist several approaches to deal with longitudinal mediation analysis, and one of the most widely spread, especially in social and behavioural sciences, consists of using multilevel models. However, when applied to mediational settings, these models present some limitations that can be overcome moving to a structural perspective. In this paper we propose a new formalisation of multilevel models within a structural framework combining the reticular action model notation and the definition variable approach. We reconsider two multilevel mediation designs very frequent in longitudinal settings from this structural perspective, discuss the advantages and limitations of such an approach and provide an empirical example.

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Data availability

The SAFE data used in this study are freely available in the NAHDAP Repository, at the following link: https://www.icpsr.umich.edu/web/NAHDAP/studies/34368, https://doi.org/10.3886/ICPSR34368.v1.

Notes

  1. In the original notation proposed by McArdle and McDonald (1984) \(\varvec{\Gamma }\) and \(\varvec{\Psi }\) are called \({\varvec{A}}\) and \({\varvec{S}}\), respectively. These are also the names they take in specialised software or packages, like OpenMx and sem packages in R.

  2. Starting from Eqs. (11), (12), it is easy to derive \( {{\mathbb {E}}}[ {\varvec{o}}_i] = {\varvec{\mu }}_i^o = {\varvec{F}}_i ( {\varvec{I}}_r - \varvec{\Gamma }_i)^{-1} \varvec{\mu }_i.\)

References

  • Asparouhov T, Muthén B (2019) Latent variable centering of predictors and mediators in multilevel and time-series models. Struct Equ Model 26(1):119–142

    MathSciNet  Google Scholar 

  • Baron RM, Kenny DA (1986) The moderator-mediator variable distinction in social psychological research: conceptual, strategic and statistical considerations. J Pers Soc Psychol 51(6):1173–1182

    Google Scholar 

  • Bauer DJ (2003) Estimating multilevel linear models as structural equation models. J Educ Behav Statist 28(2):135–167

    Google Scholar 

  • Bauer DJ, Preacher KJ, Gil KM (2006) Conceptualizing and testing random indirect effects and moderated mediation in multilevel models: new procedures and recommendations. Psychol Methods 11(2):142–163

    Google Scholar 

  • Bollen KA, Curran PJ (2006) Latent curve models. A structural equation perspective. Wiley, Hoboken

    MATH  Google Scholar 

  • Boonk L, Gijselaers HJ, Ritzen H, Brand-Gruwel S (2018) A review of the relationship between parental involvement indicators and academic achievement. Educ Res Rev 24:10–30

    Google Scholar 

  • Castro M, Expósito-Casas E, López-Martín E, Lizasoain L, Navarro-Asencio E, Gaviria JL (2015) Parental involvement on student academic achievement: a meta-analysis. Educ Res Rev 14:33–46

    Google Scholar 

  • Chou C, Bentler PM, Pentz MA (1998) Comparisons of two statistical approaches to study growth curves: the multilevel model and the latent curve analysis. Struct Equ Model 5(3):247–266

    Google Scholar 

  • Cole DA, Maxwell SE (2003) Testing mediational models with longitudinal data: questions and tips in the use of structural equation modeling. J Abnorm Psychol 112(4):558–577

    Google Scholar 

  • Curran PJ (2003) Have multilevel models been structural equation models all along? Multivar Behav Res 38(4):529–569

    MathSciNet  Google Scholar 

  • Domina T (2005) Leveling the home advantage: assessing the effectiveness of parental involvement in elementary school. Sociol Educ 78:233–249

    Google Scholar 

  • Duncan TE, Duncan SC (2004) An introduction to latent growth curve modeling. Behav Ther 35:333–363

    Google Scholar 

  • Fan X, Chen M (2001) Parental involvement and students’ academic achievement: a meta-analysis. Educ Psychol Rev 13:1–22

    Google Scholar 

  • Henderson CR (1973) Sire evaluation and genetic trends. J Anim Sci 1973:10–41

    Google Scholar 

  • Jöreskog KG (1973) A general method for estimating a linear structural equation system. In: Goldberger A, Duncan OD (eds) Structural equation models in the social sciences. Academic Press, New York, pp 85–112

    Google Scholar 

  • Jöreskog KG (1978) Structural analysis of covariance and correlation matrices. Psychometrika 43:443–477

    MathSciNet  MATH  Google Scholar 

  • Jöreskog KG, Sörbom D (2001) LISREL 8 user’s reference guide. Scientific Software International, Chicago

    Google Scholar 

  • Kenny DA, Korchmaros JD, Bolger N (2003) Lower level mediation in multilevel models. Psychol Methods 8(2):115–128

    Google Scholar 

  • Krull JL, MacKinnon DP (1999) Multilevel mediation modeling in group-based intervention studies. Eval Rev 23(4):418–444

    Google Scholar 

  • Krull JL, MacKinnon DP (2001) Multilevel modeling of individual and group level mediated effects. Multivar Behav Res 36(2):249–277

    Google Scholar 

  • Laird NM, Ware JH (1982) Random effects models for longitudinal data. Biometrics 38:963–974

    MATH  Google Scholar 

  • Lee JS, Bowen NK (2005) Parent involvement, cultural capital, and the achievement gap among elementary school children. Am Educ Res J 43(2):193–218

    Google Scholar 

  • Maxwell SE, Cole DA (2007) Bias in cross-sectional analyses of longitudinal mediation. Psychol Methods 12(1):23–44

    Google Scholar 

  • Maxwell SE, Cole DA, Mitchell MA (2011) Bias in cross-sectional analyses of longitudinal mediation: partial and complete mediation under an autoregressive model. Multivar Behav Res 46:816–841

    Google Scholar 

  • McArdle JJ (1978) A structural view of structural models. Paper presented at the winter workshop on latent structure models applied to developmental data. University of Denver

  • McArdle JJ (1979a) The development of general multivariate software. In: Proceedings of the association for the development of computer-based instructional systems. University of Akron Press, Akron, pp 824–862

  • McArdle JJ (1979b) Reticular analysis modeling (RAM) theory: the simplicity and generality of structural equations. Paper presented at the American Psychological Association Annual Meeting, New York

  • McArdle JJ (2005) The development of the RAM rules for latent variable structural equation modeling. In: Maydeu-Olivares A, McArdle JJ (eds) Contemporary Psychometrics. Lawrence Erlbaum Associates, Mahwah, New Jersey, pp 225–273

    Google Scholar 

  • McArdle JJ, McDonald RP (1984) Some algebraic properties of the Reticular Action Model for moment structures. Br J Math Stat Psychol 37:234–251

    MATH  Google Scholar 

  • McNeal RBJ (2014) Parent involvement, academic achievement and the role of student attitudes and behaviors as mediators. Univ J Educ Rev 2(8):564–576

    Google Scholar 

  • McNeish D, Matta T (2018) Differentiating between mixed-effects and latent-curve approaches to growth modeling. Behav Res Methods 50:1398–1414

    Google Scholar 

  • Mehta PD, Neale MC (2005) People are variables too: multilevel structural equations modeling. Psychol Methods 10(3):259–284

    Google Scholar 

  • Mehta PD, West SG (2000) Putting the individual back into individual growth curves. Psychol Methods 5(1):23–43

    Google Scholar 

  • Meredith W, Tisak J (1984) “Tuckerizing” growth curves. Paper presented at the annual meeting of the Psychometric Society, Santa Barbara

  • Meredith W, Tisak J (1990) Latent curve analysis. Psychometrika 55(1):107–122

    Google Scholar 

  • O’Laughlin KD, Martin MJ, Ferrer E (2018) Cross-sectional analysis of longitudinal mediation processes. Multivar Behav Res 53(3):375–402

    Google Scholar 

  • Otani M (2020) Parental involvement and academic achievement among elementary and middle school students. Asia Pac Educ Rev 21:1–25

    Google Scholar 

  • Preacher KJ (2011) Multilevel SEM strategies for evaluating mediation in three-level data. Multivar Behav Res 46(4):691–731

    Google Scholar 

  • Preacher KJ, Zyphur MJ, Zhang Z (2010) A general multilevel SEM framework for assessing multilevel mediation. Psychol Methods 15(3):209–233

    Google Scholar 

  • Preacher KJ, Zhang Z, Zyphur MJ (2011) Alternative methods for assessing mediation in multilevel data: the advantages of multilevel SEM. Struct Equ Model 18(2):161–182

    MathSciNet  Google Scholar 

  • Rappaport LM, Amstadter AB, Neale MC (2020) Level-specific evaluation of model fit in multilevel structural equation modeling. Struct Equ Model 27(2):318–329

    MathSciNet  Google Scholar 

  • Reynolds CR, Kamphaus RW (2004) Behavior assessment system for children. American Guidance Service, Circle Pines

    Google Scholar 

  • Rogers M, Theule J, Ryan B, Adams G, Keating L (2009) Parental involvement and children’s school achievement: evidence for mediating processes. Can J Sch Psychol 24:34–57

    Google Scholar 

  • Rovine MJ, Molenaar PCM (1998) A nonstandard method for estimating a linear growth model in LISREL. Int J Behav Dev 22(3):453–473

    Google Scholar 

  • Rovine MJ, Molenaar PCM (2000) A structural modeling approach to a multilevel random coefficients model. Multivar Behav Res 35(1):51–88

    Google Scholar 

  • Rovine MJ, Molenaar PCM (2001) A structural equations modeling approach to the general linear mixed model. In: Collins LM, Sayer AG (eds) New methods for the analysis of change. American Psychological Association, Washington, DC, pp 65–96

    Google Scholar 

  • Ryu E, West SG (2009) Level-specific evaluation of model fit in multilevel structural equation modeling. Struct Equ Model 16(4):583–601

    MathSciNet  Google Scholar 

  • Tofighi D (2010) 12. Multilevel mediation analysis: statistical assumptions and centering. PhD thesis, Arizona State University

  • Topor DR, Keane SP, Shelton TL, Calkins SD (2010) Parent involvement and student academic performance: a multiple mediational analysis. Univ J Educ Rev 38(3):183–197

    Google Scholar 

  • Wilder S (2014) Effects of parental involvement on academic achievement: a meta-synthesis. Educ Rev 66(3):377–397

    MathSciNet  Google Scholar 

  • Woodcock RW (1997) Woodcock diagnostic reading battery: examiner’s manual. Riverside Publishing, Itasca

    Google Scholar 

  • Yuan K, Bentler PM (2007) Multilevel covariance structure analysis by fitting multiple single-level models. Sociol Methodol 37(1):53–82

    Google Scholar 

  • Zigler CK, Ye F (2019) A comparison of multilevel mediation modeling methods: recommendations for applied researchers. Multivar Behav Res 54(3):338–359

    Google Scholar 

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  1. Department of Economics, Business and Statistics, University of Palermo, Viale delle Scienze, Building 13, 90128, Palermo, Italy

    Chiara Di Maria

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  1. Chiara Di Maria
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Di Maria, C. Structural multilevel models for longitudinal mediation analysis: a definition variable approach. Stat Papers 64, 2161–2182 (2023). https://doi.org/10.1007/s00362-022-01378-w

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