Autoregressive Linear Mixed Effects Models
In the previous chapter, longitudinal data analysis using linear mixed effects models was discussed. This chapter discusses autoregressive linear mixed effects models in which the current response is regressed on the previous response, fixed effects, and
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Autoregressive Linear Mixed Effects Models
Abstract In the previous chapter, longitudinal data analysis using linear mixed effects models was discussed. This chapter discusses autoregressive linear mixed effects models in which the current response is regressed on the previous response, fixed effects, and random effects. These are an extension of linear mixed effects models and autoregressive models. Autoregressive models regressed on the response variable itself have two remarkable properties: approaching asymptotes and statedependence. Asymptotes can be modeled by fixed effects and random effects. The current response depends on current covariates and past covariate history. Three vector representations of autoregressive linear mixed effects models are provided: an autoregressive form, response changes with asymptotes, and a marginal form which is unconditional on previous responses. The marginal interpretation is the same with subject specific interpretation as well as linear mixed effects models. Variance covariance structures corresponding to AR(1) errors, measurement errors, and random effects in the baseline and asymptote are presented. Likelihood of marginal and autoregressive forms for maximum likelihood estimation are also provided. The marginal form can be used even if there are intermittent missing values. We discuss the difference between autoregressive models of the response itself which focused in this book and models with autoregressive error terms. Keywords Asymptote · Autoregressive Autoregressive linear mixed effects model · Longitudinal · State-dependence
2.1 Autoregressive Models of Response Itself 2.1.1 Introduction There are three major approaches for modeling longitudinal data: mixed effects models, marginal models, and transition models (Diggle et al. 2002; Fitzmaurice et al. 2011). Linear mixed effects models and marginal models with linear mean structures are discussed in the framework of linear mixed effects models through © The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2018 I. Funatogawa and T. Funatogawa, Longitudinal Data Analysis, JSS Research Series in Statistics, https://doi.org/10.1007/978-981-10-0077-5_2
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2 Autoregressive Linear Mixed Effects Models
Chap. 1. We discuss nonlinear mixed effects models in Chap. 5. Mixed effects models include both fixed effects and random effects, and random effects take account for variability across subjects. Marginal models directly model the marginal distribution of the response without random effects. The model with the unstructured variance covariance in Sect. 1.3.3 is an example of marginal models. In the case of a linear model, interpretation of the fixed effects parameter is the same between two models. Although linear mixed effects models have good properties, they are not always satisfactory to express nonlinear time trends. On the other hand, nonlinear mixed effects models with nonlinear random effects parameters are complicated and there is discrepancy between marginal and subject specific inte
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