Optimal design which are efficient for lack of fit tests

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. Optimal design for regression models • Yi observations (i=1,…,n) • xi independent variable • fj: 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. Optimality of a design Example: • 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)

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

5. Lack of fit test General model: • Use the specific model as null-hypothesis in the 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 H0.

6. Optimal designs efficient for lack of fit tests General model: • 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. Specific model: These are the designs which distribute at least r*100% of the observations ”uniformly” on all possible x.

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

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

11. An experiment: the result • P-value of lackof fit test (hereF-test): 0.579 • hypothesisof quadraticregression can notbe rejected Estimation of the regression curve:

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. 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. Statistics25, 27-46. • Wiens, DP (1991): Designs for approximately linear regression: Two optimality properties of uniform design. Statist. Probab. Lett.12, 217-221.

14. Dose response relationship in clinical trials Nonlinear models are used,for example The D-optimal design for the estimation of 1and 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. Dose response relationship in clinical trials One possibility is to divide the trial into two stages. Use some prior knowledge about the unknownparameters 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 designfor 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. Thank you