Abstract
As in cross sectional studies, longitudinal studies involve non-Gaussian data such as binomial, Poisson, gamma, and inverse-Gaussian distributions, and multivariate exponential families. A number of statistical tools have thus been developed to deal with non-Gaussian longitudinal data, including analytic techniques to estimate parameters in both fixed and random effects models. However, as yet growth modeling with non-Gaussian data is somewhat limited when considering the transformed expectation of the response via a linear predictor as a functional form of explanatory variables. In this study, we introduce a fractional polynomial model (FPM) that can be applied to model non-linear growth with non-Gaussian longitudinal data and demonstrate its use by fitting two empirical binary and count data models. The results clearly show the efficiency and flexibility of the FPM for such applications.
Highlights
Just as in cross sectional studies, longitudinal studies frequently utilize non-Gaussian data that involve binomial, Poisson, Gamma, inverse-Gaussian distributions, and multivariate exponential families. Fitzmaurice and Molenberghs (2008) categorized models for non-Gaussian longitudinal data into three types based on the way they account for the correlation among the repeated measures and interpret the regression parameters: (1) marginal or population-average models, (2) generalized linear mixed models (GLMM), and (3) conditional and transition models
generalized likelihood ratio test (GLRT) indicates that the Generalized Fractional Polynomial Mixed Model (GFPMM) with power 2 and the gender variable performs significantly better than the GFPMM with power 2 alone
The efficiency and parsimoniousness of GFPMM have been investigated for non-Gaussian longitudinal data, in the context of binary and count responses
Summary
Just as in cross sectional studies, longitudinal studies frequently utilize non-Gaussian data that involve binomial, Poisson, Gamma, inverse-Gaussian distributions, and multivariate exponential families. Fitzmaurice and Molenberghs (2008) categorized models for non-Gaussian longitudinal data into three types based on the way they account for the correlation among the repeated measures and interpret the regression parameters: (1) marginal or population-average models, (2) generalized linear mixed models (GLMM), and (3) conditional and transition models. In spite of the many estimating methods that have been suggested, growth modeling with non-Gaussian data remains somewhat limited when modeling the transformed expectation of the response when applying a linear predictor as a functional form for the explanatory variables. We introduce a more flexible functional form for the linear predictor, which is known as a fractional polynomial regressor (Royston and Altman, 1994; Long and Ryoo, 2010). Zi is often constrained within the random intercept model, since non-Gaussian data provide relatively little information about individual heterogeneity beyond variability in the random intercept (Long et al, 2009). Compared with CPMs, FPMs have received relatively little attention in the context of non-Gaussian longitudinal data in the social and behavioral sciences, even though FPMs are more flexible than CPMs and provide broader classes for model selection. This is referred to as the generalized fractional polynomial mixed model (GFPMM)
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