On Optimal Design in Random Coefficient Regression Models and Alike

Transcription

1 PODE 26 May 5, 26 On Optmal Desgn n Ranom Coeffcent Regresson Moels an Alke & homas Schmelter,2 Otto von Guercke Unversty Mageburg 2 Scherng AG, Berln raner.schwabe@mathematk.un-mageburg.e

2 Outlne Prologue: Examples. RCR Moel 2. Desgn Issues 3. Ranom Intercept 4. Ranom Slope 5. Several reatments Eplogue: Messages

3 Prologue: Why Populaton Parameters? Example : neural scence socal etect the nfluence of eucaton on bran growth (apcal spnes / µm repeate measures octoon egus solate But: Each nvual has ts ownmean!

4 Why Populaton Parameters? Example 2: permetry etermne normal values for the vsual fel (vsual ablty n ecbel lumnescence mecal agnostcs retna But: Each nvual has ts owncurve! hll of vson

5 Example 2: permetry nvual hlls of vson: rght eye age: 3-4 fferental lumnance senstvty [Bs] horzontal eccentrcty

6 . RCR Moel Example: nvual regresson lnes = a + b x + ε nvual =,...,n replcaton j=,...,m explanatory varable ε ~ error N (, σ 2 nvual curves: = = 2 = 3 = 4

7 . RCR Moel Example: nvual regresson lnes = a + b x + ε nvual =,...,n replcaton j=,...,m explanatory varable ε ~ error N (, σ 2 nvual parameters: a b µ N, σ β 2 ~ populaton parameters µ µβ µβ β

8 Ranom Coeffcent Moel nvual curves are gven by a common lnear moel = a + f ( x nvual parameters: populaton parameters b + ε ε ~ N (, σ 2 ( a, b ~ N(( µ, β, σ D 2 nepenent

9 Invual Observaton Vector m b a ε β µ β µ + + = = M = ( ( m x f x f M M nvual esgn matrx θ

10 Invual Covarance Matrx 2 Cov( = σ V where V = I m + D example: ranom ntercept V = I m + µm D = m µ

11 Estmaton of Populaton Parameters ( = = = n n V V θ θ ˆ ˆ general least squares ( V V = θˆ nvual estmates nepenent of D ( = ( 2 = = n V σ θ ˆ Cov( covarance matrx epens on D, n general

12 2. Desgn Issues all nvuals uner the same regme: m m unform esgn x x j, V V estmaton: mean of nvual estmates ˆ θ = n = n oes not requre the knowlege of D (WLS=OLS ˆ θ

13 Covarance n Unform Desgns Cov( θ ˆ σ 2 = V n ( = 2 σ n ( + D covarance n the moel wthout ranom effects: = f ( x µ + β + ε (D =

14 Optmal Unform Desgns How to choose x,..., x m? qualty measure n terms of the stanarse covarance: ( + D Unform esgns are optmal! weghte generalse esgns Schmelter (24

15 mnmze trace Lnear Crtera ( L + ( D L constant! ( ( ( L L trace LDL = trace + A, IMSE, c result Luoma (2, Lsk et al. (22 ( x,..., x m ( x,..., xm optmal n reuce moel optmal n mxe moel

16 D- an G-crtera mnmze D-crteron et ( + D (( et( D et + resp. sup x ( ( f ( x + D f ( x sup x optmal esgns may ffer! ( ( ( f ( x f ( x + sup f ( x Df ( x x G-crteron

17 3. Ranom Intercept parallel nvual curves = a + f ( x β + ε D = example: parallel lnes = a +β x + ε

18 no nvual effect = f ( x hree Moels µ + β + ε ranom ntercept = a + f ( x β + fxe nvual ntercept = µ + f ( x β + no block effects ε ranom block effects ε fxe block effects Info (β Info (β m Info ( β = Info ( β + Info ( β + m + m Info (β

19 Unform Desgn Info ( β = Info ( β = Info ( β G- an D-optmal esgns o not epen on = µ resp. on the ntra nvual correlaton = + γ

20 4. Ranom Slope common ntercept = µ + b x + ε D = stanar nterval x

21 Unform Desgn G- an D-optmal esgns epen on the varance rato = β m = optmal frequency m n esgn pont D-optmal G-optmal m = m - m β

22 Unform Desgn G- an D-optmal esgns epen on the ntra nvual correlaton γ = + m = optmal frequency m n esgn pont D-optmal G-optmal m = m - m γ

23 5. Several reatments nvuals neste n 2 groups k =,2 k = a + b x + ε =,...,nk k k k k lnear regresson, common ntercept k = k = 2 a b k ~ µ N, σ β k k 2 µ µβ µβ β

24 Ranom Intercept lnear regresson, 2 groups, common ntercept k = a +β x + ε D-opt = D β -opt k k k k k = µ + β x + ε = µ + β k k k k k x + ε k k D-opt = D β -opt D-opt D β -opt

25 Unform Desgns are Optmal (n the settng of generalse esgns Schmelter (25 D-optmal esgns epen on = µ resp. γ m = optmal frequency m n esgn pont D-optmal m = m - m γ

26 Eplogue: Messages ranom ntercept Stanar esgns are sutable! ranom slope Don t o D- org-optmalty!