**Designing a dose-response study analysed by nonlinear mixed effects models **

Thu Thuy Nguyen, Caroline Bazzoli, France Mentré

UMR738 INSERM and University Paris Diderot, Paris, France

**Objectives: **Nonlinear mixed effects models (NLMEM) can be used to analyse dose-response trials with several doses in each patient. The choice of design beforehand has important impact on study results as on the precision of parameter estimates and on the power of tests. Design in NLMEM can be evaluated/optimised using the population Fisher information matrix, with first order approximation of the model [1,2]. This approach was implemented in the R function PFIM [3,4] as well as in other software. We aim to design a dose-response trial using PFIM 3.2.

**Methods**: To design this dose-response study, we use an Emax model inspired from a previous example [5]. We assume an exponential random effect model and an additive residual error model. In PFIM 3.2, we can include a covariate effect on the parameter D50 (dose to reach 50% of maximal effect). To study the design influence on the criterion and the precision of D50 estimation, we consider various designs with 100 or 200 subjects receiving 7, 4 or 2 doses as given in the example or optimised with Fedorov-Wynn algorithm [6]. To examine the covariate influence on optimisation, we compare the optimal designs obtained for a model without and with covariate effect decreasing D50 by 50% in half of the subjects. Using the standard error (SE) of covariate effect, we compute the power of the Wald test for D50 comparison and the number of subjects needed (NSN) for a power of 90%.

**Results**: As expected, the richer is the design, the larger is the criterion and the more precise is the estimation of D50. For the sparse design with 2 doses, the relative SE of D50 is 37% with the optimal design vs. 50% with the given design. The optimal design with 4 doses has 2 groups for the model without covariate but has 1 group for the model with covariate. With 2 times less samples than the given design with 4 doses, the optimal design with 2 doses provides an expected power remaining above 80% to detect a covariate effect decreasing D50 by 50%. In term of power, there is not much difference between the design with 100 subjects, 7 doses and the one with 4 doses; but there is one between the design with 2 doses, 200 subjects and the one with 100 subjects. It emphasises the impact of the number of subjects in power of tests.

**Conclusions:** We have illustrated different steps in designing a dose-response study using PFIM 3.2. It is a useful tool for design allowing users to take into account discrete covariates and to compute power and NSN.

**References:**

[1] Mentré, F., Mallet, A., Baccar, D. (1997) Optimal design in random effect regression models, *Biometrika*, 84(2), 429-442.

[2] Bazzoli, C., Retout, S., Mentré, F. (2009) Fisher information matrix for nonlinear mixed effects multiple response models: Evaluation of the appropriateness of the first order linearization using a pharmacokinetic/pharmacodynamic model, S*tatistics in Medicine*, 28, 1940-1956.

[3] Bazzoli, C., Retout, S., Mentré, F. (2010) Design evaluation and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0, *Computer Methods and Programs in Biomedicine**, 98**, *55-65.

[4] http://www.pfim.biostat.fr/.

[5] Plan, E.L., Maloney, A., Mentré, F., Karlsson, M.O., Bertrand, J. (2010) Nonlinear mixed effects estimation algorithms: a performance comparison for continuous pharmacodynamic population Models, *19th Meeting of the* *Population Approach Group in Europe (PAGE) Conference*, Berlin, Germany, Abstract 1880 [www.page-meeting.org/?abstract=1880].

[6] Retout, S., Comets, E., Samson, A., Mentré, F. (2007) Design in nonlinear mixed effects models: optimization using the Fedorov-Wynn algorithm and power of the Wald test for binary covariates, *Statistics in Medicine*, 26: 5162-5179.