Individual prediction regions for multivariate longitudinal data with small samples

Follow-up is more and more used in medicine/doping control to identify abnormal results in an individual. Currently, follow-ups are mostly carried out variable by variable using "reference intervals" that contain the values observable in 100(1-α)% of healthy/undoped individuals. Observations of the evolution of the variables over time in a sample of N healthy/undoped individuals, allows these reference intervals to be individualized by taking into account the possible effect of covariables and some previous observations of these variables obtained when the individual was healthy/undoped. For each variable these individualized intervals should contain 100(1-α)% of observable values compatible with previous observed values in this individual. General methods to build these intervals are available, but they allow only a variable by variable follow-up whatever the possible correlations over time between them. In this article, we propose a general method to jointly follow-up several correlated variables over time. This methodology relies on a multivariate linear mixed effects model. We first provide a method to estimate the model's parameters. In an asymptotic framework (N large enough), we then derive a (1-α) individualized prediction region. Sometimes, the sample size N is not large enough for the asymptotic framework to give a reasonable prediction region. It is for this reason, we propose and compare three different prediction regions that should behave better for small N. Finally, the whole methodology is illustrated by the follow-up of kidney insufficiency in cats.