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Some Curiosities in Optimal Designs for Random Slopes

Thomas Schmelter12, Norbert Benda3, Rainer Schwabe1

1Otto-von-Guericke-Universität, Magdeburg 2Bayer Schering Pharma AG, Berlin 3Novartis Pharma AG, Basel

mODa 8

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Motivation

5 10 15 20 0.0 0.5 1.0 1.5 2.0 Zeit Konzentration

Original motivation: Population PK studies

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Motivation

5 10 15 20 0.0 0.5 1.0 1.5 2.0 Zeit Konzentration

Original motivation: Population PK studies Linear Random Coefficient Regression Model

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Motivation

5 10 15 20 0.0 0.5 1.0 1.5 2.0 Zeit Konzentration

Original motivation: Population PK studies Linear Random Coefficient Regression Model Random Slope Model

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Linear Random Coefficient Regression Model

jth observation of individual i given by Yij = f(xij)⊤bi + εij, i = 1, . . . , n, j = 1, . . . , mi

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Linear Random Coefficient Regression Model

jth observation of individual i given by Yij = f(xij)⊤bi + εij, i = 1, . . . , n, j = 1, . . . , mi εij ∼ iid N(0, σ2)

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Linear Random Coefficient Regression Model

jth observation of individual i given by Yij = f(xij)⊤bi + εij, i = 1, . . . , n, j = 1, . . . , mi εij ∼ iid N(0, σ2) bi ∼ iid N(β, σ2D)

• independent
• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Linear Random Coefficient Regression Model

jth observation of individual i given by Yij = f(xij)⊤bi + εij, i = 1, . . . , n, j = 1, . . . , mi εij ∼ iid N(0, σ2) bi ∼ iid N(β, σ2D)

• independent

Individual observation vector Yi = Fiβ + Fi(bi − β) + εi Cov(Yi) = σ2Vi with Vi = Imi + FiDF⊤

i

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Estimation of Population Parameters

Weighted least squares: ˆ β = n

• i=1

F⊤

i ViFi

−1

n

• i=1

F⊤

i ViYi

Covariance matrix Cov(ˆ β) = σ2 n

• i=1

F⊤

i ViFi

−1

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Definition of Designs

Elementary design

ξi = xi1 . . . xiki mi1 . . . miki

• xij exp. settings

mij replications

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Definition of Designs

Elementary design

ξi = xi1 . . . xiki mi1 . . . miki

• xij exp. settings

mij replications

Population design

ζ = ξ1 . . . ξl g1 . . . gl

• ξr elementary design

gr proportion of individuals

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Uniform Designs

All individuals are observed uniformly using the same setting

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Uniform Designs

All individuals are observed uniformly using the same setting

• mi = m,

xij = xj, Fi = F1, Vi = V1

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Uniform Designs

All individuals are observed uniformly using the same setting

• mi = m,

xij = xj, Fi = F1, Vi = V1

◮ Estimation of β does not require knowledge of D

(WLSE=OLSE).

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Uniform Designs

All individuals are observed uniformly using the same setting

• mi = m,

xij = xj, Fi = F1, Vi = V1

◮ Estimation of β does not require knowledge of D

(WLSE=OLSE).

◮

Cov(ˆ β) = σ2 n

• F⊤

i V−1 i

Fi −1 = σ2 n

• (F⊤

i Fi)−1 + D

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Uniform Designs

All individuals are observed uniformly using the same setting

• mi = m,

xij = xj, Fi = F1, Vi = V1

◮ Estimation of β does not require knowledge of D

(WLSE=OLSE).

◮

Cov(ˆ β) = σ2 n

• F⊤

i V−1 i

Fi −1 = σ2 n

• (F⊤

i Fi)−1 + D

• ◮ If non-integer replications are allowed for the observations,
• ne cannot improve by going away from the uniform

designs.

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Linear Criteria

◮ minimize

tr

• L((F⊤

1 F1)−1 + D)L⊤

= tr

• L((F⊤

1 F1)−1)L⊤

+ tr

• LDL⊤
• constant
• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Linear Criteria

◮ minimize

tr

• L((F⊤

1 F1)−1 + D)L⊤

= tr

• L((F⊤

1 F1)−1)L⊤

+ tr

• LDL⊤
• constant
• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Linear Criteria

◮ minimize

tr

• L((F⊤

1 F1)−1 + D)L⊤

= tr

• L((F⊤

1 F1)−1)L⊤

+ tr

• LDL⊤
• constant

(x1, . . . , xm) optimal in reduced model ⇒ (x1, . . . , xm) optimal in RCR model

Luoma (2000), Liski et al. (2002)

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Random Slope Model

Yij = µ + bixij + εij,

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Random Slope Model

Yij = µ + bixij + εij, D = d

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Random Slope Model

Yij = µ + bixij + εij, D = d

• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x

◮ standard interval

0 ≤ xij ≤ 1

◮ σ2 = 1

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

◮ Optimal designs are supported only on {0, 1}

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

◮ Optimal designs are supported only on {0, 1} ◮ Candidates for optimal designs can be characterized by

number of observations m1 to be taken at x = 1

• r by w = m1

m

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

◮ Optimal designs are supported only on {0, 1} ◮ Candidates for optimal designs can be characterized by

number of observations m1 to be taken at x = 1

• r by w = m1

m

allow wm to be non-integer

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

◮ Optimal designs are supported only on {0, 1} ◮ Candidates for optimal designs can be characterized by

number of observations m1 to be taken at x = 1

• r by w = m1

m

allow wm to be non-integer

◮ Then

Cov ˆ µ ˆ β

• =

1 nm 1 w(1 − w) w −w −w 1 + mdw(1 − w)

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

D- vs. G-optimality

The D-optimal proportion at x = 1 is w∗

D = (1 +

√ md + 1)−1

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

D- vs. G-optimality

The D-optimal proportion at x = 1 is w∗

D = (1 +

√ md + 1)−1 The G-optimal proportion at x = 1 is w∗

G =

1

2(1 − 2(md)−1 +

• 1 + 4(md)−2),

d > 0

1 2,

d = 0

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

D- vs. G-optimality

Optimal proportions at x = 1 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 m*d w IMSE-optimal proportion

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

D- vs. G-optimality

Optimal proportions at x = 1 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 m*d w IMSE-optimal proportion D-optimal proportion

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

D- vs. G-optimality

Optimal proportions at x = 1 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 m*d w IMSE-optimal proportion D-optimal proportion G-optimal proportion

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Efficiency of w = 1

2 20 40 60 80 100 0.70 0.80 0.90 1.00 m*d eff

D-efficiency of w = 1

2

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Efficiency of w = 1

2 20 40 60 80 100 0.70 0.80 0.90 1.00 m*d eff

D-efficiency of w = 1

2

G-efficiency of w = 1

2

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Discretization

Uniform designs:

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Discretization

Uniform designs:

m = 2:

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Discretization

Uniform designs:

m = 2: m∗

1 = 1 (number of obs. at x = 1) D- and G-optimal

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Discretization

Uniform designs:

m = 2: m∗

1 = 1 (number of obs. at x = 1) D- and G-optimal

m = 4:

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Discretization

Uniform designs:

m = 2: m∗

1 = 1 (number of obs. at x = 1) D- and G-optimal

m = 4: m∗

1 = 2 D-optimal for d ≤ 1 2

m∗

1 = 1 D-optimal for d ≥ 1 2

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Discretization

D-optimal non-uniform designs:

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Discretization

D-optimal non-uniform designs:

m1 = ⌊w∗m⌋ + 1 for a proportion α

• f the individuals

m1 = ⌊w∗m⌋ for a proportion 1 − α

• f the individuals
• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Discretization

D-optimal non-uniform designs:

m1 = ⌊w∗m⌋ + 1 for a proportion α

• f the individuals

m1 = ⌊w∗m⌋ for a proportion 1 − α

• f the individuals

m = 4: d < 1: (1 − d)n individuals with m1 = 2 dn individuals with m1 = 1 d ≥ 1: uniform design with m1 = 1

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Discussion

◮ D-optimal and G-optimal designs go into opposite

directions.

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Discussion

◮ D-optimal and G-optimal designs go into opposite

directions.

◮ Strange limiting behavior of G-efficiency.

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes

Discussion

◮ D-optimal and G-optimal designs go into opposite

directions.

◮ Strange limiting behavior of G-efficiency. ◮ What happens in more complex models?

• T. Schmelter, N. Benda, and R. Schwabe

Optimal Designs for Random Slopes