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


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.


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



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)


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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

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