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

Institut für Medizinische Statistik 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 8 th international workshop on Model-Oriented Design and Analysis Almagro, Spain June 4-8, 2007 MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 1

Outline Institut für Medizinische Statistik • Introduction • Statistical model • Within-B-Swap Design • Equivalence Theorems • Results • Discussion MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 2

Design Institut für Medizinische Statistik 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 MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 3

Statistical model Institut für Medizinische Statistik Fixed effects gene-specific linear model for log-ratios (Landgrebe et al., 2006): Z = X θ + ε (1) Z = (Z 1 ,…,Z N ) T ~ N × 1 vector of log ratios observed on N arrays ε = ( ε 1 ,…, ε N ) T ~ ( 0 N , σ 2 I N ) residual vector θ = ( δ g , δ r , τ 11 ,…, τ KL ) T ~ (KL+2) × 1 parameter vector δ g , δ r : (fixed) dye effects of green, red dye τ kl : (fixed) combination effect of cell line k (k = 1,…,K) and treatment l (l = 1,...,L) [ ] ~ = × + X 1 | - 1 | X ~ N (KL 2) (concrete) design matrix N N where + ↔ ⎧ ⎫ 1, array t : cell line k treatment l ( green dye labelling) ⎪ ⎪ ~ = ↔ = = = x - 1, array t : cell line k treatment l ( red dye labelling) (t 1,..., N; k 1,..., K; l 1,..., L) ⎨ ⎬ tkl ⎪ ⎪ ↔ 0, array t : cell line k treatment l ( not used ) ⎩ ⎭ MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 4

Linear contrasts Institut für Medizinische Statistik in general: • C ~ (KL+2) × q ; Rg( C ) := r, r<(KL+2), r ≤ q - contrast matrix: - linear contrasts: C T θ , C T 1 KL+2 = 0 q T ⎡ ⎤ ~ = ⊗ T C ⎢ 0 0 1 A ⎥ Linear contrast for treatment effect: • ⎛ L ⎞ ⎛ L ⎞ K L ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎣ ⎦ 2 2 ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ L ~ : matrix specifying all pairwise comparisons of L treatments ⎜ ⎟ × A ~ L ⎜ ⎟ L 2 ⎝ ⎠ ⎡ ⎤ 1 - 1 0 0 ⎢ ⎥ 1 0 - 1 0 ⎢ ⎥ ⎡ 1 - 1 0 ⎤ ⎢ ⎥ ~ ⎢ ⎥ 1 0 0 - 1 ~ f.e. , = A 1 0 - 1 = A ⎢ ⎥ ⎢ ⎥ 3 4 0 1 - 1 0 ⎢ ⎥ ⎢ ⎥ 0 1 - 1 ⎣ ⎦ ⎢ ⎥ 0 1 0 - 1 ⎢ ⎥ ⎢ ⎥ 0 0 1 - 1 ⎣ ⎦ 1 ~ ~ T = ⋅ = − T (centering matrix) useful result: A A L J , where J I 1 1 • L L L L L L L L MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 5

Within B Swap (BS) Design Institut für Medizinische Statistik • often used in practical situations ⎛ ⎞ L • equally weighted support points (SP) = ⎜ ⎟ S 2K ⎜ ⎟ 2 ⎝ ⎠ • pairwise comparison of L treatments (factor A) within each cell line (factor B) • estimability → candidate design ξ * MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 6

Example Institut für Medizinische Statistik → L=3 (1,2,3), K=2 (a,b) 12 support points (arrays) design matrix: a2 a3 a1 a3 a1 a2 SP weight g r a1 a2 a3 b1 b2 b3 SP 1 SP 2 SP 3 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 a2 a1 a3 a1 a3 a2 4 1/12 1 -1 -1 1 0 0 0 0 SP 4 SP 5 SP 6 5 1/12 1 -1 -1 0 1 0 0 0 6 1/12 1 -1 0 -1 1 0 0 0 b2 b3 b1 b3 b1 b2 7 1/12 1 -1 0 0 0 1 -1 0 ~ A 8 1/12 1 -1 0 0 0 1 0 -1 SP 7 SP 8 SP 9 3 9 1/12 1 -1 0 0 0 0 1 -1 10 1/12 1 -1 0 0 0 -1 1 0 b2 b1 b3 b1 b3 b2 11 1/12 1 -1 0 0 0 -1 0 1 12 1/12 1 -1 0 0 0 0 -1 1 SP 10 SP 11 SP 12 ~ − A 3 MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 7

Notations (1) Institut für Medizinische Statistik ⎛ ⎞ KL • = ⋅ ⎜ ⎟ m 2 : number of discrete design points x 1 ,…, x m ⎜ ⎟ 2 ⎝ ⎠ = • Ξ { x ,..., x } : (discrete) design space 1 m ⎧ x ,..., x ⎫ m ∑ • Ω : class of discrete designs = 1 m ≤ ≤ ∀ = = ξ ; 0 p 1 i 1,..., m; p 1 ⎨ ⎬ i i p ,..., p ⎩ ⎭ = i 1 1 m m M ∑ • moment matrix: = T : p x x i i i = i 1 • (corresponding) generalized inverse: G := M - MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 8

Moment matrix – BS Design Institut für Medizinische Statistik ⎡ ⎤ − T 1 1 0 ⎢ ⎥ KL ⎢ ⎥ = − T M 1 1 0 KL ⎢ ⎥ ( ) ⎢ ⎥ ⊗ 0 0 c I L J ⎣ ⎦ K L KL KL 1 (~ (KL+2) × (KL+2) ) MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 9

Institut für Medizinische Statistik Generalized inverse – BS Design ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ − T 1 1 0 KL ⎢ ⎥ = − T G 1 1 0 ⎢ ⎥ KL ⎢ ⎥ ⎢ ⎥ ⎛ ⎞ 1 1 ⊗ ⎜ ⎟ 0 0 I J ⎢ ⎥ ⎜ ⎟ K L KL KL L c ⎝ ⎠ ⎣ ⎦ 1 (~ (KL+2) × (KL+2) ) MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 10

Notations (2) Institut für Medizinische Statistik • ξ * : candidate design; estimable with respect to C T θ (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 MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 11

D-optimality Institut für Medizinische Statistik Theorem 1 (Generalized Equivalence Theorem for D-optimality): The design ξ * ∈ Ω is said to be D-optimal for estimating the linear contrast C T θ if and only if there exists a generalized inverse G = M - of the moment matrix M that satisfies the normality inequality [ ] ( ) + + ≤ T T T T T T (N1) x GC ( C GC ) C G x tr C GC C GC for all discrete design points x ∈ Ξ , with strict equality for all support points of ξ *. (Pukelsheim, 1993, p. 180/205) MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 12

A-optimality Institut für Medizinische Statistik Theorem 2 (Generalized Equivalence Theorem for A-optimality): The design ξ * ∈ Ω is said to be A-optimal for estimating the linear contrast C T θ if and only if there exists a generalized inverse G = M - of the moment matrix M that satisfies the normality inequality [ ] (N2) + + ≤ + T T T 2 T T T T T 2 x GC ( C GC ) ( C GC ) ( C GC ) C G x tr ( C GC ) ( C GC ) for all discrete design points x ∈ Ξ , with strict equality for all support points of ξ *. (Pukelsheim, 1993, p. 180/205) MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 13

Results Institut für Medizinische Statistik Theorem 3 (D-optimality of the BS design): The Within-B-Swap (BS) design is D-optimal for estimating the linear contrast C T θ for the treatment effect in model (1). Theorem 4 (A-optimality of the BS design): The Within-B-Swap (BS) design is A-optimal for estimating the linear contrast C T θ for the treatment effect in model (1). Proof: - shown: D-optimality (sketch !!) - similar: A-optimality MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 14

Proof – D-optimality (1) Institut für Medizinische Statistik T ⎡ ⎤ ~ • linear contrast: C T θ , with contrast matrix = ⊗ T C 0 0 1 A ⎢ ⎥ ⎛ ⎞ ⎛ ⎞ K L L L ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎣ ⎦ 2 2 ⎝ ⎠ ⎝ ⎠ • With some algebra, we obtain from C , M and G (as before) : ⎡ ⎤ ⎛ ⎞ L ⎛ ⎞ 2 L 1 ~ K ~ ~ = ⎜ ⎟ ⋅ ⊗ T T T = ⎜ ⎟ ⋅ T T C G K ⎢ 0 0 1 A ⎥ , C GC A A , ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ ⎛ ⎞ K L L L L L 2 L 2 ⎢ ⎥ L ⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎣ ⎦ 2 2 ⎝ ⎠ ⎝ ⎠ ( ) 1 ~ ~ + = ⋅ T Τ C GC Α Α L L ⎛ ⎞ L ⎜ ⎟ 2 K L ⎜ ⎟ 2 ⎝ ⎠ [ ] = 1, − T • design points: with x 1, x ,..., x 11 KL + ↔ ⎧ ⎫ 1, cell line k treatment l (g) 1 L ⎪ ⎪ ∑ ⋅ = and x x = ↔ = = x kl - 1, cell line k treatment l (r) ; k 1,..., K; l 1,..., L ⎨ ⎬ k kl L ⎪ ⎪ = l 1 0, else ⎩ ⎭ MODA 8, Almagro, Spain, June 2007, Dipl.-Stat. Sven Stanzel 15