Estimation of the linear mixed integrated Ornstein–Uhlenbeck model

Rachael A Hughes,Michael G Kenward,Jonathan A C Sterne,Kate Tilling

doi:10.1080/00949655.2016.1277425

Abstract

ABSTRACTThe linear mixed model with an added integrated Ornstein–Uhlenbeck (IOU) process (linear mixed IOU model) allows for serial correlation and estimation of the degree of derivative tracking. It is rarely used, partly due to the lack of available software. We implemented the linear mixed IOU model in Stata and using simulations we assessed the feasibility of fitting the model by restricted maximum likelihood when applied to balanced and unbalanced data. We compared different (1) optimization algorithms, (2) parameterizations of the IOU process, (3) data structures and (4) random-effects structures. Fitting the model was practical and feasible when applied to large and moderately sized balanced datasets (20,000 and 500 observations), and large unbalanced datasets with (non-informative) dropout and intermittent missingness. Analysis of a real dataset showed that the linear mixed IOU model was a better fit to the data than the standard linear mixed model (i.e. independent within-subject errors with constant variance).

Highlights

Linear mixed models, proposed by Laird and Ware [1], are commonly used for the analysis of longitudinal clinical markers (‘biomarkers’) of disease; for example HIV research on the time course of CD4 counts [2] and time course of progesterone in a menstrual cycle [3]
We classified the estimation process as converged if the optimization algorithm converged to a solution within 100 iterations and the information matrix with respect to θ was positive definite
We have conducted the first evaluation of the feasibility and practicality of restricted maximum likelihood (REML) estimation of the linear mixed integrated Ornstein–Uhlenbeck (IOU) model, for both balanced and unbalanced data

Summary

Introduction

Linear mixed models, proposed by Laird and Ware [1], are commonly used for the analysis of longitudinal clinical markers (‘biomarkers’) of disease; for example HIV research on the time course of CD4 counts [2] and time course of progesterone in a menstrual cycle [3]. In such settings the data are typically unbalanced: the number of measurements differs among subjects and the time interval between consecutive measurements differs within and between subjects. Taylor et al [5], Zhang et al [6] and Stirrup et al [7] allowed for a nonstationary covariance structure by including in their model random effects other than the random intercept and a nonstationary stochastic process (the integrated Ornstein–Uhlenbeck (IOU)

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