XLII Mathematical Statistics Optimality of neighbor designs under mixed interference models Katarzyna Filipiak and Augustyn Markiewicz Pozna´ n University of Technology and Pozna´ n University of Life Sciences XLII Mathematical Statistics – p. 1/2
Rees (1967), Hwang (1973) XLII Mathematical Statistics – p. 2/2
Single block of circular design XLII Mathematical Statistics – p. 3/2
Single block of circular design D A B C D A ���� � �� � ���� border plot inner plots border plot XLII Mathematical Statistics – p. 3/2
Example A B C D d = n = 12 , b = 3 , t = 4 , k = 4 B C A D A D C B XLII Mathematical Statistics – p. 4/2
Example A B C D d = n = 12 , b = 3 , t = 4 , k = 4 B C A D A D C B 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 T d = L d = R d = 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 XLII Mathematical Statistics – p. 4/2
Design matrix of neighbor effects L d = ( I b ⊗ H k ) T d of left-neighbor effects: XLII Mathematical Statistics – p. 5/2
Design matrix of neighbor effects L d = ( I b ⊗ H k ) T d of left-neighbor effects: R d = ( I b ⊗ H ′ k ) T d of right-neighbor effects: XLII Mathematical Statistics – p. 5/2
Design matrix of neighbor effects L d = ( I b ⊗ H k ) T d of left-neighbor effects: R d = ( I b ⊗ H ′ k ) T d of right-neighbor effects: H k - the circular incidence matrix XLII Mathematical Statistics – p. 5/2
Design matrix of neighbor effects L d = ( I b ⊗ H k ) T d of left-neighbor effects: R d = ( I b ⊗ H ′ k ) T d of right-neighbor effects: H k - the circular incidence matrix 0 0 0 · · · 0 1 1 0 0 · · · 0 0 0 ′ 1 k − 1 = H k = 0 1 0 · · · 0 0 . . . . . ... I k − 1 0 k − 1 . . . . . . . . . . 0 0 0 · · · 1 0 XLII Mathematical Statistics – p. 5/2
Neighbor designs Neighbor designs Rees (1967), Hwang (1973) XLII Mathematical Statistics – p. 6/2
Neighbor designs Neighbor designs Rees (1967), Hwang (1973) There are t treatments to be arranged on b blocks containing k experimental units. Each treatment appears r times (but not necessarily on r different blocks) and is a neighbor of every other treatment exactly λ times. XLII Mathematical Statistics – p. 6/2
Neighbor designs Neighbor designs Rees (1967), Hwang (1973) There are t treatments to be arranged on b blocks containing k experimental units. Each treatment appears r times (but not necessarily on r different blocks) and is a neighbor of every other treatment exactly λ times. A design d is said to be a neighbor design if a matrix S d + S ′ d is completely symmetric with diagonal elements equal to zero, where S d = T ′ d L d is its left-neighboring matrix and d L d ) ′ = T ′ S ′ d = ( T ′ d R d is its right-neighboring matrix. XLII Mathematical Statistics – p. 6/2
Neighbor designs Neighbor designs Rees (1967), Hwang (1973) There are t treatments to be arranged on b blocks containing k experimental units. Each treatment appears r times (but not necessarily on r different blocks) and is a neighbor of every other treatment exactly λ times. A design d is said to be a neighbor design if a matrix S d + S ′ d is completely symmetric with diagonal elements equal to zero, where S d = T ′ d L d is its left-neighboring matrix and d L d ) ′ = T ′ S ′ d = ( T ′ d R d is its right-neighboring matrix. If blocks should also be a factor in the design, then the requirement that d is a BIB design is added. (Hwang, 1973) XLII Mathematical Statistics – p. 6/2
CNBD Circular neighbor balanced design (Druilhet, 1999) XLII Mathematical Statistics – p. 7/2
CNBD Circular neighbor balanced design (Druilhet, 1999) A circular BIB design in D t,b,k such that for each ordered pair of distinct treatments there exist exactly l inner plots which receive the first chosen treatment and which have the second one as right neighbor, is called a circular neighbor balanced design (CNBD). XLII Mathematical Statistics – p. 7/2
CNBD Circular neighbor balanced design (Druilhet, 1999) A circular BIB design in D t,b,k such that for each ordered pair of distinct treatments there exist exactly l inner plots which receive the first chosen treatment and which have the second one as right neighbor, is called a circular neighbor balanced design (CNBD). A circular (BIB) design d is said to be a CNBD if its matrix S d is completely symmetric with diagonal elements equal to zero. XLII Mathematical Statistics – p. 7/2
Complete block design - CWNBD Circular weakly neighbor balanced design (Filipiak and Markiewicz, 2012) XLII Mathematical Statistics – p. 8/2
Complete block design - CWNBD Circular weakly neighbor balanced design (Filipiak and Markiewicz, 2012) A circular BB design d ∈ D t,b,t , such that s d,ij ∈ { p − 1 , p } , i � = j , and S d S ′ d is completely symmet- ric is called a circular weakly neighbor balanced design (CWNBD). XLII Mathematical Statistics – p. 8/2
Complete block design - CWNBD Circular weakly neighbor balanced design (Filipiak and Markiewicz, 2012) A circular BB design d ∈ D t,b,t , such that s d,ij ∈ { p − 1 , p } , i � = j , and S d S ′ d is completely symmet- ric is called a circular weakly neighbor balanced design (CWNBD). If d ∈ D t,b,t , b = ( t − 1) / 2 , is CWNBD, then S d + S ′ d is completely symmetric; i.e. d is a neighbor design. XLII Mathematical Statistics – p. 8/2
CNBD2 Circular neighbor balanced design at distances 1 and 2 (Druilhet, 1999) XLII Mathematical Statistics – p. 9/2
CNBD2 Circular neighbor balanced design at distances 1 and 2 (Druilhet, 1999) A CNBD in D t,b,k such that for each ordered pair of distinct treatments there exist exactly l inner plots which have the first chosen treatment as left neighbor and the second one as right neighbor, is called a circular neighbor balanced design at distances 1 and 2 (CNBD2). XLII Mathematical Statistics – p. 9/2
CNBD2 Circular neighbor balanced design at distances 1 and 2 (Druilhet, 1999) A CNBD in D t,b,k such that for each ordered pair of distinct treatments there exist exactly l inner plots which have the first chosen treatment as left neighbor and the second one as right neighbor, is called a circular neighbor balanced design at distances 1 and 2 (CNBD2). A circular (BIB) design d is said to be a CNBD2 if its matrices S d and U d are completely symmetric with diagonal elements equal to zero, where U d = L ′ d R d is its left-neighboring matrix at distance 2. XLII Mathematical Statistics – p. 9/2
Fixed models M 1 a : y = T d τ + L d λ + ε XLII Mathematical Statistics – p. 10/2
Fixed models M 1 a : y = T d τ + L d λ + ε M 1 b : y = T d τ + L d λ + B β + ε XLII Mathematical Statistics – p. 10/2
Fixed models M 1 a : y = T d τ + L d λ + ε M 1 b : y = T d τ + L d λ + B β + ε M 2 a : y = T d τ + ( L d + R d ) η + ε XLII Mathematical Statistics – p. 10/2
Fixed models M 1 a : y = T d τ + L d λ + ε M 1 b : y = T d τ + L d λ + B β + ε M 2 a : y = T d τ + ( L d + R d ) η + ε M 2 b : y = T d τ + ( L d + R d ) η + B β + ε XLII Mathematical Statistics – p. 10/2
Fixed models M 1 a : y = T d τ + L d λ + ε M 1 b : y = T d τ + L d λ + B β + ε M 2 a : y = T d τ + ( L d + R d ) η + ε M 2 b : y = T d τ + ( L d + R d ) η + B β + ε M 3 a : y = T d τ + L d λ + R d ρ + ε XLII Mathematical Statistics – p. 10/2
Fixed models M 1 a : y = T d τ + L d λ + ε M 1 b : y = T d τ + L d λ + B β + ε M 2 a : y = T d τ + ( L d + R d ) η + ε M 2 b : y = T d τ + ( L d + R d ) η + B β + ε M 3 a : y = T d τ + L d λ + R d ρ + ε M 3 b : y = T d τ + L d λ + R d ρ + B β + ε ε ∼ N ( 0 n , σ 2 I n ) XLII Mathematical Statistics – p. 10/2
Mixed models 1 LL ′ + σ 2 I n M 1 a V : λ ∼ N ( 0 t , σ 2 1 I t ) , Cov ( y ) = V = σ 2 XLII Mathematical Statistics – p. 11/2
Mixed models 1 LL ′ + σ 2 I n M 1 a V : λ ∼ N ( 0 t , σ 2 1 I t ) , Cov ( y ) = V = σ 2 1 LL ′ + σ 2 I n M 1 b V : λ ∼ N ( 0 t , σ 2 1 I t ) , Cov ( y ) = V = σ 2 XLII Mathematical Statistics – p. 11/2
Mixed models 1 LL ′ + σ 2 I n M 1 a V : λ ∼ N ( 0 t , σ 2 1 I t ) , Cov ( y ) = V = σ 2 1 LL ′ + σ 2 I n M 1 b V : λ ∼ N ( 0 t , σ 2 1 I t ) , Cov ( y ) = V = σ 2 M 2 a V : η ∼ N ( 0 t , σ 2 1 I t ) , 1 ( L + R )( L + R ) ′ + σ 2 I n Cov ( y ) = V = σ 2 XLII Mathematical Statistics – p. 11/2
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