1. 1 Optimal design which are efficient for lack of fit tests Frank Miller, AstraZeneca, Södertälje, Sweden Joint work with Wolfgang Bischoff, Catholic University of Eichstätt-Ingolstadt, Germany DSBS/FMS workshop 2006-04-26, Copenhagen Statistical Issues in Drug Development

2. 2 Optimal design for regression models Y i observations (i=1,…,n) x i independent variable f j : known regression functions (j=1,…,k)  j unknown parameters (j=1,…k)  j iid error (E(  j )=0, V(  j )=  2 unknown) Problem: How to choose the independent variables = design of the experiment

3. 3 Optimality of a design We consider the LS-estimators of  1,  2. If it’s important to estimate the slope  2 : The variance of the estimator of  2 should be minimal If it’s important to estimate  1 and  2 : The covariance matrix of the estimators of  1,  2 should be “minimal” Important criterion: Minimisation of the determinant of the covariance matrix (D-optimality) Example:

4. 4 Optimality of a design Example: Consider the design: half of observations at lowest possible x i, half of observations at highest possible x i. This design is both, optimal for estimation of  2 (c-optimal) and D-optimal for estimation of  1 and  2. But we get no information if the above straight line regression is the true relationship between independent factor and observed variable. We want to be able to perform a lack of fit test.

5. 5 Lack of fit test Use the specific model as null-hypothesis in the general model: General model: Specific model: Different lack of fit tests possible (F-test, non- parametric tests) Power of lack of fit test should be optimised for functions in the alternative with a certain ”distance” from H 0.

6. 6 Optimal designs efficient for lack of fit tests We consider all designs which have an efficiency ≥ r (r between 0 and 1) for the lack of fit test. In this set of designs, we determine the optimal design (c-, D-optimal, …) for the specific model. General model: Specific model: These are the designs which distribute at least r*100% of the observations ”uniformly” on all possible x.

7. 7 An experiment Aim: to study the (toxicological) impact of fertilizer for flowers on the growth of cress Region of interest: a proportion of 0 - 1.2% concentration of fertilizer in the water N=81 plant plates with 10 seeds each

8. 8 An experiment Plate i is treated with a concentration x i of fertilizer, x i  [0, 1.2] After 5.5 days, the yield Y i (in mg) of cress in plate i is recorded.

9. 9 An experiment: the model In the focus: we want to estimate the parameters  1,  2,  3 as good as possible Here: The determinant of the covariance matrix of should be as small as possible (D-optimality). Moreover, at least 1/3 of the observations should be used to check if the above model is valid. We search for the D-optimal design within the set of designs having at least 1/3 of its mass uniformly distributed on the experimental region [0, 1.2].

10. 10 An experiment: the optimal design Solution (“asymptotic” design): 33.3% of observations uniformly on [0, 1.2], 26.6% of observations for x i = 0, 13.4% of observations for x i = 0.6, 26.6% of observations for x i = 1.2. Approximation with:

11. 11 An experiment: the result Estimation of the regression curve: P-value of lack of fit test (here F-test): 0.579  hypothesis of quadratic regression can not be rejected

12. 12 C-optimal designs Polynomial regression model of degree k-1 Estimate the highest coefficient in an optimal way Use only designs which are efficient for a lack of fit test The optimal design can be derived algebraically for arbitrary k.

13. 13 References Biedermann S, Dette H (2001): Optimal designs for testing the functional form of a regression via nonparametric estimation techniques. Statist. Probab. Lett. 52, 215-224. Bischoff, W, Miller, F (2006): Optimal designs which are efficient for lack of fit tests. Annals of Statistics. To appear. Bischoff, W, Miller, F (2006): For lack of fit tests highly efficient c-optimal designs. Journal of Statistical Planning and Inference. To appear. Dette, H (1993): Bayesian D-optimal and model robust designs in linear regression models. Statistics 25, 27-46. Wiens, DP (1991): Designs for approximately linear regression: Two optimality properties of uniform design. Statist. Probab. Lett. 12, 217-221.

14. 14 Dose response relationship in clinical trials Nonlinear models are used, for example The D-optimal design for the estimation of  1 and  2 has half of the observations on each of two doses: (see for example Minkin, 1987, JASA, p.1098-1103) The D-optimal design depends on unknown parameters

15. 15 Dose response relationship in clinical trials One possibility is to divide the trial into two stages. Use some prior knowledge about the unknown parameters  1 and  2 to compute two doses for stage 1. Perform an interim analysis and update knowledge about the parameters. Compute a new D-optimal design for stage 2. It might be desirable already in the first stage of the trial to have the possibility for a lack of fit test

16. 16 Thank you