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A joint marginalized multilevel model for longitudinal outcomes

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  • Samuel Iddi
  • Geert Molenberghs

Abstract

The shared-parameter model and its so-called hierarchical or random-effects extension are widely used joint modeling approaches for a combination of longitudinal continuous, binary, count, missing, and survival outcomes that naturally occurs in many clinical and other studies. A random effect is introduced and shared or allowed to differ between two or more repeated measures or longitudinal outcomes, thereby acting as a vehicle to capture association between the outcomes in these joint models. It is generally known that parameter estimates in a linear mixed model (LMM) for continuous repeated measures or longitudinal outcomes allow for a marginal interpretation, even though a hierarchical formulation is employed. This is not the case for the generalized linear mixed model (GLMM), that is, for non-Gaussian outcomes. The aforementioned joint models formulated for continuous and binary or two longitudinal binomial outcomes, using the LMM and GLMM, will naturally have marginal interpretation for parameters associated with the continuous outcome but a subject-specific interpretation for the fixed effects parameters relating covariates to binary outcomes. To derive marginally meaningful parameters for the binary models in a joint model, we adopt the marginal multilevel model (MMM) due to Heagerty [13] and Heagerty and Zeger [14] and formulate a joint MMM for two longitudinal responses. This enables to (1) capture association between the two responses and (2) obtain parameter estimates that have a population-averaged interpretation for both outcomes. The model is applied to two sets of data. The results are compared with those obtained from the existing approaches such as generalized estimating equations, GLMM, and the model of Heagerty [13]. Estimates were found to be very close to those from single analysis per outcome but the joint model yields higher precision and allows for quantifying the association between outcomes. Parameters were estimated using maximum likelihood. The model is easy to fit using available tools such as the SAS NLMIXED procedure.

Suggested Citation

  • Samuel Iddi & Geert Molenberghs, 2012. "A joint marginalized multilevel model for longitudinal outcomes," Journal of Applied Statistics, Taylor & Francis Journals, vol. 39(11), pages 2413-2430, July.
    • Handle: RePEc:taf:japsta:v:39:y:2012:i:11:p:2413-2430
    DOI: 10.1080/02664763.2012.711302
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    References listed on IDEAS

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    1. C. Y. Wang & Naisyin Wang & Suojin Wang, 2000. "Regression Analysis When Covariates Are Regression Parameters of a Random Effects Model for Observed Longitudinal Measurements," Biometrics, The International Biometric Society, vol. 56(2), pages 487-495, June.
    2. Erning Li & Naisyin Wang & Nae-Yuh Wang, 2007. "Joint Models for a Primary Endpoint and Multiple Longitudinal Covariate Processes," Biometrics, The International Biometric Society, vol. 63(4), pages 1068-1078, December.
    3. Geert Verbeke & Geert Molenberghs, 2003. "The Use of Score Tests for Inference on Variance Components," Biometrics, The International Biometric Society, vol. 59(2), pages 254-262, June.
    4. Patrick J. Heagerty, 1999. "Marginally Specified Logistic-Normal Models for Longitudinal Binary Data," Biometrics, The International Biometric Society, vol. 55(3), pages 688-698, September.
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    Cited by:

    1. Øystein Sørensen & Anders M. Fjell & Kristine B. Walhovd, 2023. "Longitudinal Modeling of Age-Dependent Latent Traits with Generalized Additive Latent and Mixed Models," Psychometrika, Springer;The Psychometric Society, vol. 88(2), pages 456-486, June.
    2. Gul Inan & Ozlem Ilk, 2019. "A marginalized multilevel model for bivariate longitudinal binary data," Statistical Papers, Springer, vol. 60(3), pages 601-628, June.
    3. Anna Ivanova & Geert Molenberghs & Geert Verbeke, 2017. "Mechanism for missing data incorporated in joint modelling of ordinal responses," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 66(5), pages 1049-1064, November.

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