Modellering van niet-normale longitudinale data in continue tijd, gebaseerd op de likelihoodfunctie.

摘要

Building conditional models for non-normal longitudinal data turns out to be both an interesting and challenging problem to be considered. An extra motivating problem was to concentrate on likelihood based approaches to enable possibly non-nested models to be compared.;Two techniques were considered to attain this goal. The first one is based on empirical Bayesian arguments, and the second one is the generalized autoregression model (GARM). They can both deal with continuous and discrete observations in continuous time.;Consider the first family of models in the special case of series of overdispersed count data. The basic idea is to generalize the dynamic generalized linear model to model count data in continuous time. Basically, the intercept in a log-linear model, is given a gamma prior (conjugate) distribution. This prior is used to predict the state of the process at the next observation time, simply by considering a gamma predictive distribution with the same mode as the original prior, but with a Fisher information (at this point) decreasing with the time lag since the last observation. As soon as one observation is available, the distribution of the intercept, which is a kind of residual in a model assuming independence, is updated using Bayes theorem. The resulting (unconditional) likelihood is then the product over observation times of negative binomial densities.;The second family of models, or GARM, expresses some transform of a location parameter as a possibly non-linear function of the covariates plus a term involving the last cumulated residual. This last quantity is used to model serial association.;In both situations, the likelihood is maximized using a non-linear optimizer to obtain estimates for the parameters used to build the model. This allows one to consider non-linear models when these are found more useful to describe the data generating mechanism than polynomial or spline-based methods. Another interesting advantage of such an approach is that one can consider distributions outside the exponential family. This is particularly important in practice, because conclusions can be very sensitive to the distribution choice.;But this numerically demanding method is of course not feasible with large data sets. An important exception to this rule arises when one considers a distribution in the exponential family together with a linear model for the systematic part. Under these conditions, the GARM can be rewritten as a GLM for fixed values of the serial association parameters, making the use of the powerful IWLS algorithm possible.;Several examples, mainly in veterinary medicine, are considered to illustrate the flexibility of the above two approaches. Two data sets involving series of overdispersed counts are analysed using the gamma-Poisson model. In one of these, the (non-linear) generalized logistic growth curve is considered to describe the growth of colonies of Paramecium aurelium in a nutritive medium.;It is also shown how the GARM can be used in practice to model series of positive, binary, binomial, multinomial and count data. (Abstract shortened by UMI.)