Advertisement

Statistics in Biosciences

, Volume 10, Issue 1, pp 139–159 | Cite as

A Multivariate Generalized Linear Model Approach to Mediation Analysis and Application of Confidence Ellipses

  • Brandie D. WagnerEmail author
  • Miranda Kroehl
  • Ryan Gan
  • Susan K. Mikulich-Gilbertson
  • Scott D. Sagel
  • Paula D. Riggs
  • Talia Brown
  • Janet Snell-Bergeon
  • Gary O. Zerbe
Article
  • 137 Downloads

Abstract

Mediation analysis evaluates the significance of an intermediate variable on the causal pathway between an exposure and an outcome. One commonly utilized test for mediation involves evaluation of counterfactual effects, estimated from separate regression models, corresponding to a composite null hypothesis. However, the “compositeness” of this null hypothesis is not commonly acknowledged and accounted for in mediation analyses. We describe a generalized multivariate approach in which these separate regression models are fit simultaneously in a single parsimonious model. This multivariate modeling approach can reproduce standard mediation analysis and has notable advantages over separate regression models, including the ability to combine distributions in the exponential family with any link functions and perform likelihood-based tests of some relevant hypotheses using existing software. We propose the use of a novel visual representation of confidence intervals of the two estimates for the indirect path with the use of a confidence ellipse. The calculation of the confidence ellipse is facilitated by the multivariate approach, can test the components of the composite null hypothesis under a single experiment-wise type I error rate, and does not require estimation of the standard error of the product of coefficients from two separate regressions. This method is illustrated using three examples. The first compares results between the multivariate method and separate regression models. The second example illustrates the proposed methods in the presence of an exposure–mediator interaction, missing data and confounding, and the third example utilizes these proposed methods for an outcome and mediator with negative binomial distributions.

Keywords

Composite null hypothesis Counterfactual effects Exponential family of distributions Exposure–mediator interaction Likelihood ratio test for mediation Scheffe’ simultaneous confidence limits 

Notes

Acknowledgements

This work was supported by the National Institutes of Health (Grants P50 MH086383, K23 RR018611, 2T32AR007534-27, R01 HL113029, R01 HL61753, R01 HL079611, R01 AR051394, R01 DA034604 and R01 DA022284) and the Cystic Fibrosis Foundation (WAGNER15A0).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

12561_2017_9191_MOESM1_ESM.docx (248 kb)
Supplementary material 1 (docx 247 KB)

References

  1. 1.
    Albert JM, Nelson S (2011) Generalized causal mediation analysis. Biometrics 67:1028–1038MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baron RM, Kenny DA (1986) The moderator-mediator variable distinction in social psychological research: conceptual, strategic, and statistical considerations. J Pers Soc Psychol 51:1173–1182CrossRefGoogle Scholar
  3. 3.
    Blood EA, Cabral H, Heeren T, Cheng DM (2010) Performance of mixed effects models in the analysis of mediated longitudinal data. BMC Med Res Methodol 10:16CrossRefGoogle Scholar
  4. 4.
    Brown SA, Gleghorn A, Schuckit MA, Myers MG, Mott MA (1996) Conduct disorder among adolescent alcohol and drug abusers. J Stud Alcohol 57:314–324CrossRefGoogle Scholar
  5. 5.
    Casella G, Berger RL (2002) Statistical inference. Thomson Learning, Pacific Grove, CAzbMATHGoogle Scholar
  6. 6.
    Coffman DL, Zhong W (2012) Assessing mediation using marginal structural models in the presence of confounding and moderation. Psychol Methods 17:642–664CrossRefGoogle Scholar
  7. 7.
    Crowley TJ, Riggs PD (1995) Adolescent substance use disorder with conduct disorder and comorbid conditions. NIDA Res Monogr 156:49–111Google Scholar
  8. 8.
    Dabelea D, Kinney G, Snell-Bergeon JK, Hokanson JE, Eckel RH, Ehrlich J, Garg S, Hamman RF, Rewers M (2003) Effect of type 1 diabetes on the gender difference in coronary artery calcification: a role for insulin resistance? The coronary artery calcification in type 1 diabetes (CACTI) study. Diabetes 52:2833–2839CrossRefGoogle Scholar
  9. 9.
    Deboer EM, Swiercz W, Heltshe SL, Anthony MM, Szefler P, Klein R, Strain J, Brody AS, Sagel SD (2014) Automated CT scan scores of bronchiectasis and air trapping in cystic fibrosis. Chest 145:593–603CrossRefGoogle Scholar
  10. 10.
    Hayes AF (2009) Beyond Baron and Kenny: statistical mediation analysis in the new millennium. Commun Monogr 76:408–420CrossRefGoogle Scholar
  11. 11.
    Hayes AF, Scharkow M (2013) The relative trustworthiness of inferential tests of the indirect effect in statistical mediation analysis: does method really matter? Psychol Sci 24:1918–1927CrossRefGoogle Scholar
  12. 12.
    Imai K, Keele L, Tingley D (2010) A general approach to causal mediation analysis. Psychol Methods 15:309–334CrossRefGoogle Scholar
  13. 13.
    Koopman J, Howe M, Hollenbeck JR, Sin HP (2015) Small sample mediation testing: misplaced confidence in bootstrapped confidence intervals. J Appl Psychol 100:194–202CrossRefGoogle Scholar
  14. 14.
    Lange T, Vansteelandt S, Bekaert M (2012) A simple unified approach for estimating natural direct and indirect effects. Am J Epidemiol 176:190–195CrossRefzbMATHGoogle Scholar
  15. 15.
    Mackinnon DP, Fairchild AJ (2009) Current directions in mediation analysis. Curr Dir Psychol Sci 18:16CrossRefGoogle Scholar
  16. 16.
    Mackinnon DP, Fairchild AJ, Fritz MS (2007) Mediation analysis. Annu Rev Psychol 58:593–614CrossRefGoogle Scholar
  17. 17.
    Mackinnon DP, Fritz MS, Williams J, Lockwood CM (2007) Distribution of the product confidence limits for the indirect effect: program PRODCLIN. Behav Res Methods 39:384–389CrossRefGoogle Scholar
  18. 18.
    Marshall G, De La Cruz-Mesia R, Baron AE, Rutledge JH, Zerbe GO (2006) Non-linear random effects model for multivariate responses with missing data. Stat Med 25:2817–2830MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mikulich SK, Zerbe GO, Jones RH, Crowley TJ (2003) Comparing linear and nonlinear mixed model approaches to cosinor analysis. Stat Med 22:3195–3211CrossRefGoogle Scholar
  20. 20.
    Pearl J (2001) Direct and indirect effects. In: Proceedings of the seventeenth conference on uncertainty in artificial intelligence. Morgan Kaufmann Publishers Inc., Seattle, Washington, pp. 411–420Google Scholar
  21. 21.
    Pearl J (2010) An introduction to causal inference. Int J Biostat. doi: 10.2202/1557-4679.1203
  22. 22.
    Riggs PD, Winhusen T, Davies RD, Leimberger JD, Mikulich-Gilbertson S, Klein C, Macdonald M, Lohman M, Bailey GL, Haynes L, Jaffee WB, Haminton N, Hodgkins C, Whitmore E, Trello-Rishel K, Tamm L, Acosta MC, Royer-Malvestuto C, Subramaniam G, Fishman M, Holmes BW, Kaye ME, Vargo MA, Woody GE, Nunes EV, Liu D (2011) Randomized controlled trial of osmotic-release methylphenidate with cognitive-behavioral therapy in adolescents with attention-deficit/hyperactivity disorder and substance use disorders. J Am Acad Child Adolesc Psychiatry 50:903–914CrossRefGoogle Scholar
  23. 23.
    Robins JM, Greenland S (1992) Identifiability and exchangeability for direct and indirect effects. Epidemiology 3:143–155CrossRefGoogle Scholar
  24. 24.
    Sagel SD, Wagner BD, Anthony MM, Emmett P, Zemanick ET (2012) Sputum biomarkers of inflammation and lung function decline in children with cystic fibrosis. Am J Respir Crit Care Med 186:857–865CrossRefGoogle Scholar
  25. 25.
    Scheffé H (1959) The analysis of variance. Wiley, New YorkzbMATHGoogle Scholar
  26. 26.
    Sobel ME (1982) Asymptotic confidence intervals for indirect effects in structural equation models. In: Leinhart. S (ed) Sociological methodology. Jossey-Bass, San FranciscoGoogle Scholar
  27. 27.
    Taylor AB, Mackinnon DP (2012) Four applications of permutation methods to testing a single-mediator model. Behav Res Methods 44:806–844CrossRefGoogle Scholar
  28. 28.
    Thompson LL, Riggs PD, Mikulich SK, Crowley TJ (1996) Contribution of ADHD symptoms to substance problems and delinquency in conduct-disordered adolescents. J Abnorm Child Psychol 24:325–347CrossRefGoogle Scholar
  29. 29.
    Tofighi D, Mackinnon D (2011) RMediation: an R package for mediation analysis confidence intervals. Behav Res Methods 43:692–700CrossRefGoogle Scholar
  30. 30.
    Tukey JW, Brillinger DR, Cox DR, Braun HI (1984) The collected works of John W. Tukey. Wadsworth Advanced Books & Software, BelmontGoogle Scholar
  31. 31.
    Valeri L, Vanderweele TJ (2013) Mediation analysis allowing for exposure-mediator interactions and causal interpretation: theoretical assumptions and implementation with SAS and SPSS macros. Psychol Methods 18:137–150CrossRefGoogle Scholar
  32. 32.
    Vanderweele TJ (2014) A unification of mediation and interaction: a 4-way decomposition. Epidemiology 25:749–761CrossRefGoogle Scholar
  33. 33.
    Vanderweele TJ, Vansteelandt S (2009) Conceptual issues concerning mediation, interventions and composition. Stat Interface 2:457–468MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Vanderweele TJ, Vansteelandt S (2010) Odds ratios for mediation analysis for a dichotomous outcome. Am J Epidemiol 172:1339–1348CrossRefGoogle Scholar
  35. 35.
    Young DA, Zerbe GO, Hay WW Jr (1997) Fieller’s theorem, Scheffe simultaneous confidence intervals, and ratios of parameters of linear and nonlinear mixed-effects models. Biometrics 53:838–847CrossRefzbMATHGoogle Scholar
  36. 36.
    Zerbe GO, Jones RH (1980) On application of growth curve techniques to time series data. J Am Stat Assoc 75:507–509CrossRefzbMATHGoogle Scholar

Copyright information

© International Chinese Statistical Association 2017

Authors and Affiliations

  • Brandie D. Wagner
    • 1
    • 2
    Email author
  • Miranda Kroehl
    • 1
  • Ryan Gan
    • 3
  • Susan K. Mikulich-Gilbertson
    • 1
    • 4
  • Scott D. Sagel
    • 2
  • Paula D. Riggs
    • 4
  • Talia Brown
    • 3
  • Janet Snell-Bergeon
    • 2
  • Gary O. Zerbe
    • 1
  1. 1.Department of Biostatistics and Informatics, Colorado School of Public HealthUniversity of Colorado DenverAuroraUSA
  2. 2.Department of Pediatrics, Children’s Hospital ColoradoUniversity of Colorado School of MedicineAuroraUSA
  3. 3.Department of Epidemiology, Colorado School of Public HealthUniversity of ColoradoAuroraUSA
  4. 4.Department of PsychiatryUniversity of Colorado School of MedicineAuroraUSA

Personalised recommendations