The Equivalence Theorem in Optimal Design

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1 he Equivalece heorem i Optimal Desig Raier Schwabe & homas Schmelter, Otto vo Guericke Uiversity agdeburg Bayer Scherig Pharma, Berli rschwabe@ovgu.de PODE 007 ay 4, 007

2 Outlie Prologue: Simple eamples. A shortcut to liear models. Desigs ad desig criteria 3. Approimate desigs 4. he Equivalece heorem 5. RCR models Epilogue: Ope problems

3 Prologue: Simple eamples Compariso of two groups Y i μ + ε i i radom error observatio,..., i group i, group mea ε i i.i.d. Var ε σ i δ μ μ?

4 Compariso of two groups estimated differece δˆ y y Var δ ˆ + σ + fi optimal choice eve: / odd: ± /

5 Compariso of two groups optimal choice / 00 % 80 % deficiecy: 60 % 40 % icrease i sample size 0 % 0 % 33 % 5 % 50 % proportio: /

6 Liear regressio Y observatio,..., i Y α + β + itercept slope ε eplaatory variable radom error ε i.i.d. Var ε σ

7 Liear regressio estimated slope ˆ β y y Var β ˆ σ desig regio: 0 optimal choice eve: 0 or, each / times

8 . A shortcut to liear models geeral liear model Y f β + ε radom error observatio,..., eplaatory variable ε i.i.d. Var ε σ regressio fuctios f f,..., f p β,..., β p β parameter

9 ε Fβ Y + Y Y Vector otatio f f F desig matri p I ε Y σ Cov Cov

10 Estimatio Gauss arkov heorem βˆ F F F Y least squares Is best liear ubiased esrimator for β covariace matri Cov βˆ F σ F iformatio matri F F Fisher iformatio f f

11 Y ε β β + + Liear regressio f f, regressio fuctios dimesio p F F iformatio matri

12 . Desigs ad desig criteria eact desig,..., iformatio matri,..., f f aim: choose,, m from desig regio X Covβˆ resp. to. miimise v f f predictio variace

13 Desig criteria miimise equivaletly l det D: A: det - trace - ISE: f - f d G: ma f - f

14 3. Approimate desigs... m... m desig poits weights iformatio matri f f D-optimal miimises det etc.

15 ......,..., iformatio matri f f Embeddig of eact desigs stadardised

16 Coveity set Ξ of approimate desigs is cove α + α is a desig stadard criteria Φ : Ξ, ] are cove Φ α + α α Φ + α Φ D-criterio: Φ l det

17 4. he Equivalece heorem directioal derivative F ; η lim [ Φ Φ α α 0 α + α η at i the directio of η Φ ] ote: F Φ ; 0 regularity coditio: { f ; X} is compact

18 he Geeral Equivalece heorem miimises Φ if ad oly if Φ-opt. F Φ ; η 0 for all η maimises mi F η Φ ; η

19 Φ diffferetiable Sesitivity FΦ ; η η FΦ ; δ δ ϕ sesitivity fuctio ; F ; δ Φ miimises Φ ϕ ; 0 for all miimises maϕ ;

20 Sesitivity ϕ sesitivity fuctio ; F ; δ Φ miimises Φ ϕ ; 0 for all miimises maϕ ; ote: ϕ ; 0 for > 0

21 D-criterio l det Φ tr ; F - p η η Φ p - ; f f ϕ p - ; v f f D-optimal for all Kiefer, Wolfowitz 960 G-optimal

22 Liear criteria tr - A Φ tr tr ; F η η A A Φ tr ; ϕ A f A f tr - - f f A-optimal for all Fedorov 97: AI c-optimal c c c f - -

23 Pros ad cos usually ot costructive eceptio:polyomial regressio efficiecy bouds algorithms

24 Polyomial regressio Y, β + β +,, + β + ε p p 0 ϕ ; polyomial of degree p- miimal support 0 0 D-optimal: p / p p

25 Efficiecy bouds ; ma ϕ X + Φ Φ coveity ; F η η Φ Φ Φ D-criterio ; mav det det eff p p - - e D A-criterio tr ma eff A - - f f Φ Φ

26 Algorithms X arg maϕ ; steepest descet Fedorov Wy + α α α δ add oe poit α arg mi Φ α + αδ X

27 σ ukow iformatio matri for, σ β θ σ σ θ σ σ θ costat same story as before

28 5. Radom Coefficiet Regressio idividual curves are give by a commo liear model Y f b i i i + ε i idividual i,..., replicatio,...,m i idividual parameters: eplaatory variable populatio parameter b i ε ~ i error N 0, σ ~ N β, σ D idepedet

29 Desig idividual desig μ μ i i i i i populatio desig ν ν ζ ζ ζ i i i i problem: set of iformatio matrices is ot cove m i m

30 Sigle group desigs ζ iformatio for populatio parameter β + D m fied model iformatio

31 miimise D tr A + m Liear criteria m A tr AD tr + costat! A, ISE, c result Luoma 000, Liski et al. 00 optimal i reduced model optimal i RCR model

32 A Equivalece heorem coveity Φ Φ f cove cove i D-optimal for β - tr β locally optimal at D β β + D m - - f for all Fedorov, Hackl 997

33 Variace compoets coveity fails withi idividuals eve for sigle parameters e.g. d equivalece theorems which? give oly ecessary coditios

34 A ultivariate Equivalece heorem geeralised multivariate desigs ~,..., w~ m L L l,..., w~ l lm geeralised multivariate iformatio ~ ~ if ad oly if ~ sup m ~ l i w i tr V Fi Im + F ~ ~ det maimises,..., i ~ ~ F DF i ~ F i F settigs weights D-opt. p Fedorov 97

35 Epilogue: Ope Problems ubalaced desigs e.g. idividual desigs sigular sigle group desigs ot applicable for treatmet comparisos o-liear models liear approimatio: appropriate? Need for a more geeral approach!!!

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