The Within-B-Swap (BS) Design is A- and D-optimal for estimating the - - PowerPoint PPT Presentation

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

The Within-B-Swap (BS) Design is A- and D-optimal for estimating the linear contrast for the treatment effect in 3-factorial cDNA microarray experiments

Dipl.-Stat. Sven Stanzel

• Prof. Dr. rer. nat. Ralf-Dieter Hilgers

Institute for Medical Statistics RWTH Aachen

8th international workshop on Model-Oriented Design and Analysis Almagro, Spain June 4-8, 2007

Institut für Medizinische Statistik

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Outline

• Introduction
• Statistical model
• Within-B-Swap Design
• Equivalence Theorems
• Results
• Discussion

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Design

3-factorial experimental setting:

• N arrays
• 2 colours (red/green) [ → repeated observations at each exp. unit]
• L treatments
• K cell lines
• conditions A and B to be compared on each array can be chosen from L⋅K

possible combinations of L treatments and K cell lines

• balanced incomplete block designs [BIBD]:
• first blocking factor:

condition

• second blocking factor: dye

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Statistical model

Fixed effects gene-specific linear model for log-ratios (Landgrebe et al., 2006):

δg , δr: (fixed) dye effects of green, red dye τkl: (fixed) combination effect of cell line k (k = 1,…,K) and treatment l (l = 1,...,L) (concrete) design matrix

where

Z = Xθ + ε

Z = (Z1,…,ZN)T ~ N×1 vector of log ratios observed on N arrays ε = (ε1,…,εN)T ~ (0N , σ2IN) residual vector θ = (δg,δr,τ11,…,τKL)T ~ (KL+2)×1 parameter vector

[ ]

2) (KL N ~ ~ | |

N N

+ × = X 1

• 1

X

L) 1,..., l K; 1,..., k N; 1,..., (t ) ( l treatment k line cell : array t 0, labelling) dye ( l treatment k line cell : array t 1,

• labelling)

dye ( l treatment k line cell : array t 1, x ~

tkl

= = = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ↔ ↔ ↔ + = used not red green

(1)

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Linear contrasts

• in general:
• contrast matrix:

C ~ (KL+2)×q ; Rg(C) := r, r<(KL+2), r≤q

• linear contrasts: CTθ , CT1KL+2 = 0q
• Linear contrast for treatment effect:

: matrix specifying all pairwise comparisons of L treatments f.e. ,

~

T L T K 2 L 2 L

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⊗ =

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛

A 1 C

L 2 L ~ ~

L

× ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ A

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 1

• 1

1

• 1

1

• 1

~

3

A

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 1

• 1

1

• 1

1

• 1

1

• 1

1

• 1

1

• 1

~

4

A

• useful result:

T L L L L L L T L

L 1 where , L ~ ~ 1 1 I J J A A − = ⋅ = (centering matrix)

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Within B Swap (BS) Design

• often used in practical situations
• equally weighted support points (SP)
• pairwise comparison of L treatments (factor A)

within each cell line (factor B)

• estimability

→ candidate design ξ*

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 2 L 2K S

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Example

a1 a2 a1 a3 a2 a3

a2 a1 a3 a1 a3 a2

design matrix:

SP weight g r a1 a2 a3 b1 b2 b3 1 1/12 1 -1 1 -1 0 0 0 0 2 1/12 1 -1 1 0 -1 0 0 0 3 1/12 1 -1 0 1 -1 0 0 0 4 1/12 1 -1 -1 1 0 0 0 0 5 1/12 1 -1 -1 0 1 0 0 0 6 1/12 1 -1 0 -1 1 0 0 0 7 1/12 1 -1 0 0 0 1 -1 0 8 1/12 1 -1 0 0 0 1 0 -1 9 1/12 1 -1 0 0 0 0 1 -1 10 1/12 1 -1 0 0 0 -1 1 0 11 1/12 1 -1 0 0 0 -1 0 1 12 1/12 1 -1 0 0 0 0 -1 1

3

A ~ −

3

A ~

b1 b2 b1 b3 b2 b3 b2 b1 b3 b1 b3 b2 SP 10 SP 11 SP 12 SP 7 SP 8 SP 9 SP 4 SP 5 SP 6 SP 1 SP 2 SP 3

L=3 (1,2,3), K=2 (a,b) → 12 support points (arrays)

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Notations (1)

• : number of discrete design points x1,…,xm
• : (discrete) design space
• Ω: class of discrete designs
• moment matrix:
• (corresponding) generalized inverse: G:=M-

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ = 2 KL 2 m

T i i m 1 i i

p : x x M ∑

=

=

1 p m; 1,..., i 1 p ; p ,..., p ,..., ξ

m 1 i i i m 1 m 1

= = ∀ ≤ ≤ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ =

∑

=

x x

} ,..., { Ξ

m 1

x x =

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Moment matrix – BS Design

( )

L c 1 1 1 1

L K 1 KL KL T KL T KL

⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⊗ − − = J I M

(~ (KL+2) × (KL+2) )

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Generalized inverse – BS Design

L 1 c 1 1 1 1 1

L K 1 KL KL T KL T KL

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⊗ − − = J I G

(~ (KL+2) × (KL+2) )

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Notations (2)

• ξ*: candidate design; estimable with respect to CTθ

(here: BS Design)

• proof of optimality: Equivalence Theorem for Matrix Means

(Pukelsheim, 1972, p.180)

• extension for singular case (Pukelsheim, 1972, p. 205)

p = -1: A-optimality p = 0: D-optimality

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

D-optimality

Theorem 1 (Generalized Equivalence Theorem for D-optimality):

The design ξ* ∈ Ω is said to be D-optimal for estimating the linear contrast CTθ if and only if there exists a generalized inverse G=M- of the moment matrix M that satisfies the normality inequality

(N1)

for all discrete design points x ∈ Ξ, with strict equality for all support points of ξ*.

( )

[ ]

tr ) (

T T T T T T

GC C GC C x G C GC C GC x

+ +

≤

(Pukelsheim, 1993, p. 180/205)

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

A-optimality

Theorem 2 (Generalized Equivalence Theorem for A-optimality):

The design ξ* ∈ Ω is said to be A-optimal for estimating the linear contrast CTθ if and only if there exists a generalized inverse G=M- of the moment matrix M that satisfies the normality inequality

(N2)

for all discrete design points x ∈ Ξ, with strict equality for all support points of ξ*. (Pukelsheim, 1993, p. 180/205)

[ ]

) ( ) ( tr ) ( ) ( ) (

2 T T T T 2 T T T

GC C GC C x G C GC C GC C GC C GC x

T + + +

≤

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Results

Theorem 3 (D-optimality of the BS design):

The Within-B-Swap (BS) design is D-optimal for estimating the linear contrast CTθ for the treatment effect in model (1). Proof:

• shown:

D-optimality (sketch !!)

• similar:

A-optimality

Theorem 4 (A-optimality of the BS design):

The Within-B-Swap (BS) design is A-optimal for estimating the linear contrast CTθ for the treatment effect in model (1).

Institut für Medizinische Statistik

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Proof – D-optimality (1)

• linear contrast: CTθ , with contrast matrix

T L L 2 T

~ ~ 2 L L K A A GC C ⋅ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

( )

Τ L L 2 T

~ ~ 2 L L K 1 Α Α GC C ⋅ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

+

• design points: with

[ ]

T KL 11

x ,..., x 1, 1,− = x

L 1,..., l K; 1,..., k ; else 0, (r) l treatment k line cell 1,

• (g)

l treatment k line cell 1, x kl = = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ↔ ↔ + =

and

∑

= ⋅ = L 1 l kl k

x L 1 x ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⊗ ⋅ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛

~ L 1 2 L K

L T K 2 L 2 L T T

A 1 G C ~

T L T K 2 L 2 L

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⊗ =

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛

A 1 C

• With some algebra, we obtain from C, M and G (as before) :

, ,

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Proof – D-optimality (2)

Using the properties of the matrices , it can be shown that In summary, the normality inequality (N1) of general equivalence theorem 1 can be transferred into the equivalent form:

L

~ A To proof (∆), 3 cases have to be distinguished:

. x L 1 x

L 1 l ql q

∑

= ⋅ =

( )

∑∑ ∑

= = = ⋅ ⎥

⎦ ⎤ ⎢ ⎣ ⎡ − ⋅ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

K 1 k L 1 l K 1 q q ql kl 2 2 T T T L L T

x x x 2 L K ~ ~ x G C A A GC x

, where

( )

2 x x x

K 1 k L 1 l K 1 q q ql kl

≤ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⋅

∑∑ ∑

= = = ⋅

(∆)

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Proof – D-optimality (3)

Case 2: design points specifying comparisons of treatments j, j‘ in two different cell lines i,i‘

■

( )

L 1 1 L 1 1 1) ( L 1 1 L 1 1 1 x x x

K 1 k L 1 l K 1 q q ql kl

= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⋅ − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⋅ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −

∑∑ ∑

= = = ⋅

( )

2 L 1 1 L 1 1) ( L 1 L 1 1 1 x x x

K 1 k L 1 l K 1 q q ql kl

= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⋅ − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⋅ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −

∑∑ ∑

= = = ⋅

Case 3: design points specifying comparisons of cell lines i,i‘ for the same treatment j Case 1: support points of BS (comparison of treatments j, j‘ in same cell line i)

( )

( )

2 1 1) ( 1 1 x x x

K 1 k L 1 l K 1 q q ql kl

= − ⋅ − + ⋅ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −

∑∑ ∑

= = = ⋅

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Discussion

• The Within-B-Swap (BS) Design is A- and D-optimal for estimating the

linear contrast for the treatment effect in the Landgrebe model.

• generalized equivalence theorems for matrix means
• solution of optimization problem: independent of N, K, L !!!
• robustness of BS Design
• Further results:
• φp-optimality for integer p ∈ (-∞;1]
• E-optimality
• interaction effect (solution conditioned on relation between K and L !!)
• combination of treatment effect and interaction effect

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

Further research

• φp-optimality for non-integer p
• ptimal design for estimating cell line effect
• ther contrast sets:

Helmert contrasts / polynomial contrasts (→ poster Hilgers)

• ther multi-factorial experimental setups (4-factorial,…)
• ther models:
• „overall“ models (gene effect, correlation structures between genes??)
• comparison of experimental conditions with fixed reference condition

[including „Common Reference Design“]

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MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel

References

Harville, D.A. (1997): Matrix algebra from a statistician‘s perspective. Springer, New York. Landgrebe, J., Bretz, F., Brunner, E. (2006): Efficient design and analysis of two colour factorial microarray experiments. Computational Statistics and Data Analysis 2006; 50: 499-517. Pukelsheim, F. (1993): Optimal Design of Experiments. Wiley, New York. Searle, S. R. (1971): Linear Models. Wiley, New York. Searle, S. R. (1982): Matrix Algebra useful for Statistics. Wiley, New York. Speed, T. (2003): Statistical analysis of gene expression microarray data. Chapman and Hall, Boca Raton. Stanzel, S., Hilgers, R.D. (2007): The Within-B-Swap (BS) Design is A- and D-optimal for estimating the Linear Contrast for the Treatment Effect in 3-Factorial cDNA Microarray

• Experiments. Proceedings of the 8th international workshop on Model-Oriented Design and
• Analysis. Almagro, Spain, June 4-8, 2007.