Modelling Heterogeneous Dispersion in Marginal Models for Longitudinal Proportional Data

Summary Continuous proportional data is common in biomedical research, e.g., the pre-post therapy percent change in certain physiological and molecular variables such as glomerular filtration rate, certain gene expression level, or telomere length. As shown in (Song and Tan, 2000) such data requires methods beyond the common generalised linear models. However, the original marginal simplex model of (Song and Tan, 2000) for such longitudinal continuous proportional data assumes a constant dispersion parameter. This assumption of dispersion homogeneity is imposed mainly for mathematical convenience and may be violated in some situations. For example, the dispersion may vary in terms of drug treatment cohorts or follow-up times. This paper extends their original model so that the heterogeneity of the dispersion parameter can be assessed and accounted for in order to conduct a proper statistical inference for the model parameters. A simulation study is given to demonstrate that statistical inference can be seriously affected by mistakenly assuming a varying dispersion parameter to be constant in the application of the available GEEs method. In addition, residual analysis is developed for checking various assumptions made in the modelling process, e.g., assumptions on error distribution. The methods are illustrated with the same eye surgery data in (Song and Tan, 2000) for ease of comparison.

[1]  P. Diggle.  Analysis of Longitudinal Data , 1995 .

[2]  Suber S. Huang,et al.  VARIATION OF PERFLUOROPROPANE DISAPPEARANCE AFTER VITRECTOMY , 1992, Retina.

[3]  J. Robins,et al.  Analysis of semiparametric regression models for repeated outcomes in the presence of missing data , 1995 .

[4]  L. Zhao,et al.  Estimating equations for parameters in means and covariances of multivariate discrete and continuous responses. , 1991, Biometrics.

[5]  J. H. Schuenemeyer,et al.  Generalized Linear Models (2nd ed.) , 1992 .

[6]  Ananda Sen,et al.  The Theory of Dispersion Models , 1997, Technometrics.

[7]  P. Diggle.  Time Series: A Biostatistical Introduction , 1990 .

[8]  G. Molenberghs,et al.  Linear Mixed Models for Longitudinal Data , 2001 .

[9]  M. Crowder.  On linear and quadratic estimating functions , 1987 .

[10]  P. Diggle,et al.  Analysis of Longitudinal Data. , 1997 .

[11]  R. W. Wedderburn.  Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method , 1974 .

[12]  C. Kastner,et al.  The Generalised Estimating Equations: An Annotated Bibliography , 1998 .

[13]  Gordon K. Smyth,et al.  Generalized linear models with varying dispersion , 1989 .

[14]  P X Song,et al.  Marginal Models for Longitudinal Continuous Proportional Data , 2000, Biometrics.

[15]  M C Paik,et al.  Parametric variance function estimation for nonnormal repeated measurement data. , 1992, Biometrics.

[16]  Ole E. Barndorff-Nielsen,et al.  Some parametric models on the simplex , 1991 .

[17]  S. Zeger,et al.  Longitudinal data analysis using generalized linear models , 1986 .

[18]  Bent Jørgensen,et al.  Longitudinal Data Estimating Equations for Dispersion Models , 2000 .

[19]  Myunghee C. Paik,et al.  The generalized estimating equation approach when data are not missing completely at random , 1997 .