A calibration function from change points using linear mixed models

Abstract

The change point problem is an interesting topic in both cross-sectional and longitudinal settings. In the cross-sectional scenario, the change point problem has been studied extensively. In longitudinal settings, authors usually suggest fitting linear mixed models but to find the change points in this scenario is not an easy task. In this way, identifying change points in linear mixed models (LMMs) is an open problem that has been studied by few authors. Recent contributions on this topic were done by \citeasnoun{lai2014}. However, to the best of our knowledge, there is neither proposal in which change points had been obtained for each subject nor about a calibration function fitted from these change points. The purpose of this proposal is to develop a solution to the change points problem under a linear mixed model when several covariates are considered into the model. If we obtain the change point for each subject under a longitudinal setting, this yields a change function instead of a single change point. We fit a calibration function that allows predicting the change point given some referenced values of time-independent variables or fixed effects. The solution is given by considering a linear mixed model (LMM) under the assumption that this model has a continuous change point for each subject, that is, a broken-stick model (profile) is associated with each subject in the data set. We considered both a parametric and a Bayesian approach, standard linear mixed models assumptions, and a first order autoregressive ($AR(1)$) covariance structure on the random errors. We found that there is not a close or analytical expression to obtain the change point for linear mixed models; this is why we suggest an adapted methodology to estimate subject-specific change points from linear mixed models. We show the results of both a parametric approach of the calibration function from change points, and some asymptotic properties of the calibration function parameters. %by executing a simulation study and formalize the results through a theorem. Additionally, we show the results of a Bayesian approach of the calibration function through a simulation study, by considering classical prior distributions of the parameters and random effects of the linear mixed model. Also, we illustrate this proposal in a practical situation with real data about dried Cypress wood slats \cite{botero1993}, and we compare the results obtained in the parametric case with the ones obtained by using the Bayesian approach. All algorithms and calculations were implemented by means of paralleling programmed routines in the statistical software R (team2014R) on advanced computational clusters, and high-performance computers. This proposal is useful because predicting a time in which the model changes is so important in productive processes, so that this prediction allows to avoid some additional drawbacks, and for example, it could help to decrease the storage expenses.