Assessing Variability

Once a model has been chosen to represent the data, measures of variability for the estimates and confidence regions on the model’s parameters are usually needed for inferential purposes. Asymptotic theory can certainly be used at a preliminary stage. These results have only been proven for the linear mixed effect model (cf. chapter 3), but, at least as a first order-like approximation, could also be used for nonlinear mixed effects models.

More refined methods, such as likelihood profile traces and contours (Bates and Watts, 1988), can also be used to assess the variability in the estimates, but these will generally be computer intensive. A compromise between the asymptotic and profiling methodologies is to use a linear approximation to the loglikelihood, as in (5.1.2), to calculate the profile traces and contours. This constitutes a considerably less intensive computational problem than profiling the loglikelihood directly. Alternatively, if the fixed effects are the primary object of interest, we could profile the penalized nonlinear least squares problem corresponding to a fixed D =D.

Another (computationally intensive) alternative to assess the variability in the estimates is to use bootstrap methods (Efron and Tibshirani, 1993) to esti-mate standard errors and confidence regions.

More research is needed to determine which methods provide the most reli-able statistical results and to compare their relative computational performance.

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