by Mielke, T..

**Series:** 2009-11, Preprints

- MSC:
- 62K05 Optimal designs
- 62J05 Linear regression

**Abstract:**

In population pharmacokinetic studies the blood samples of individuals are evaluated together in one model, assuming that the same regression function can be used for all subjects, with slightly different parameters for the different individuals. These differences from the population mean are modeled by random variables.

The purpose of this article is to study the design for quadratic regression in mixed effect models, in the case of two allowed observations per individual. Taking many

blood samples of one individual is costly, unethical and in some cases even not possible. If less observations are being made than parameters are to be estimated, the

occuring D-optimal individual design, will lead to a singular information matrix. The use of population designs with different observation groups helps to construct

estimates for the population parameter vector.

Cheng[2] and Atkins and Cheng[1] provide D-optimal designs for quadratic regression with random intercept, considering two observations per subject. In this article we generalize these results to polynomial regression with random slope and

random curvature.

In section 2 we introduce the mixed effects model. Section 3 will introduce Doptimal designs for quadratic regression with random parameters, considering less observations per individual than parameters are to be estimated. We will show

results on the efficiency of D-optimal designs compared to a trivial three-group design.

**Keywords:**

optimal design, mixed effect models, quadratic regression, random slope, random curvature.