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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

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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)

ikikiii xfxfxfY )(...)()( 2211

Problem: How to choose the independent variables = design of the experiment

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Optimality of a design

iii xY 21

• 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:

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Optimality of a design

iii xY 21Example:

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.

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Lack of fit test arbitrary) ( gxgY iii

• Use the specific model as null-hypothesis in the general model:

kkk ffg ... with ,..., are There :H 1110

ikikii xfxfY )(...)( 11

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.

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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.

iii xgY

ikikii xfxfY )(...)( 11

General model:

Specific model:

These are the designs which distribute at least r*100% of the observations ”uniformly” on all possible x.

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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

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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.

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An experiment: the model

iiii xxY 32

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• 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].

321ˆ,ˆ,ˆ

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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:

.81,...,60,2.1

,59,...,47,200

19584099,46,...,36,6.0

,35,...,23,200

1958899,22,...,1,0

i

ii

i

ii

i

xi

31

90

19510

45

19520

90

19510

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An experiment: the result

Estimation of the regression curve:22.135.556.201 xxy

P-value of lackof fit test (hereF-test): 0.579

hypothesisof quadraticregression can notbe rejected

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C-optimal designs

Polynomial regression model of degree k-1Estimate the highest coefficient in an optimal wayUse only designs which are efficient for a lack of fit test

The optimal design can be derived algebraically for arbitrary k.

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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.

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Dose response relationship in clinical trials

Nonlinear models are used,for example

xxf21exp1

1)(

The D-optimal design for the estimation of 1

and 2 has half of the observations on each of two doses: 215434.1 x

(see for example Minkin, 1987, JASA, p.1098-1103)The D-optimal design depends on unknown parameters

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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

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Thank you