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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

• border plot

A B C D

• inner plots

A

• border plot

XLII Mathematical Statistics – p. 3/2

Example

d =

 

A B C D B C A D A D C B

 

n = 12, b = 3, t = 4 , k = 4

XLII Mathematical Statistics – p. 4/2

Example

d =

 

A B C D B C A D A D C B

 

n = 12, b = 3, t = 4 , k = 4 Td =

                

1 1 1 1 1 1 1 1 1 1 1 1

                

Ld =

                

1 1 1 1 1 1 1 1 1 1 1 1

                

Rd =

                

1 1 1 1 1 1 1 1 1 1 1 1

                

XLII Mathematical Statistics – p. 4/2

Design matrix of neighbor effects

• f left-neighbor effects:

Ld = (Ib ⊗ Hk)Td

XLII Mathematical Statistics – p. 5/2

Design matrix of neighbor effects

• f left-neighbor effects:

Ld = (Ib ⊗ Hk)Td

• f right-neighbor effects:

Rd = (Ib ⊗ H′

k)Td

XLII Mathematical Statistics – p. 5/2

Design matrix of neighbor effects

• f left-neighbor effects:

Ld = (Ib ⊗ Hk)Td

• f right-neighbor effects:

Rd = (Ib ⊗ H′

k)Td

Hk - the circular incidence matrix

XLII Mathematical Statistics – p. 5/2

Design matrix of neighbor effects

• f left-neighbor effects:

Ld = (Ib ⊗ Hk)Td

• f right-neighbor effects:

Rd = (Ib ⊗ H′

k)Td

Hk - the circular incidence matrix Hk =   0′

k−1

1 Ik−1 0k−1   =         0 0 0 · · · 0 1 1 0 0 · · · 0 0 0 1 0 · · · 0 0 . . . . . . . . . ... . . . . . . 0 0 0 · · · 1 0        

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Neighbor designs

Neighbor designs Rees (1967), Hwang (1973)

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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 Sd +

S′

d is completely symmetric with diagonal elements equal to

zero, where Sd = T′

dLd is its left-neighboring matrix and

S′

d = (T′ dLd)′ = T′ dRd 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 Sd +

S′

d is completely symmetric with diagonal elements equal to

zero, where Sd = T′

dLd is its left-neighboring matrix and

S′

d = (T′ dLd)′ = T′ dRd 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)

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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 Dt,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 Dt,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

Sd 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 ∈ Dt,b,t, such that sd,ij

∈ {p − 1, p}, i = j, and SdS′

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 ∈ Dt,b,t, such that sd,ij

∈ {p − 1, p}, i = j, and SdS′

d is completely symmet-

ric is called a circular weakly neighbor balanced design (CWNBD). If d ∈ Dt,b,t, b = (t − 1)/2, is CWNBD, then Sd + 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 Dt,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 Dt,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 Sd and Ud are completely symmetric with diagonal elements equal to zero, where Ud = L′

dRd is

its left-neighboring matrix at distance 2.

XLII Mathematical Statistics – p. 9/2

Fixed models

M1a : y = Tdτ + Ldλ + ε

XLII Mathematical Statistics – p. 10/2

Fixed models

M1a : y = Tdτ + Ldλ + ε M1b : y = Tdτ + Ldλ + Bβ + ε

XLII Mathematical Statistics – p. 10/2

Fixed models

M1a : y = Tdτ + Ldλ + ε M1b : y = Tdτ + Ldλ + Bβ + ε M2a : y = Tdτ + (Ld + Rd)η + ε

XLII Mathematical Statistics – p. 10/2

Fixed models

M1a : y = Tdτ + Ldλ + ε M1b : y = Tdτ + Ldλ + Bβ + ε M2a : y = Tdτ + (Ld + Rd)η + ε M2b : y = Tdτ + (Ld + Rd)η + Bβ + ε

XLII Mathematical Statistics – p. 10/2

Fixed models

M1a : y = Tdτ + Ldλ + ε M1b : y = Tdτ + Ldλ + Bβ + ε M2a : y = Tdτ + (Ld + Rd)η + ε M2b : y = Tdτ + (Ld + Rd)η + Bβ + ε M3a : y = Tdτ + Ldλ + Rdρ + ε

XLII Mathematical Statistics – p. 10/2

Fixed models

M1a : y = Tdτ + Ldλ + ε M1b : y = Tdτ + Ldλ + Bβ + ε M2a : y = Tdτ + (Ld + Rd)η + ε M2b : y = Tdτ + (Ld + Rd)η + Bβ + ε M3a : y = Tdτ + Ldλ + Rdρ + ε M3b : y = Tdτ + Ldλ + Rdρ + Bβ + ε ε ∼ N(0n, σ2In)

XLII Mathematical Statistics – p. 10/2

Mixed models

M1aV : λ ∼ N(0t, σ2

1It), Cov(y) = V = σ2 1LL′ + σ2In

XLII Mathematical Statistics – p. 11/2

Mixed models

M1aV : λ ∼ N(0t, σ2

1It), Cov(y) = V = σ2 1LL′ + σ2In

M1bV : λ ∼ N(0t, σ2

1It), Cov(y) = V = σ2 1LL′ + σ2In

XLII Mathematical Statistics – p. 11/2

Mixed models

M1aV : λ ∼ N(0t, σ2

1It), Cov(y) = V = σ2 1LL′ + σ2In

M1bV : λ ∼ N(0t, σ2

1It), Cov(y) = V = σ2 1LL′ + σ2In

M2aV : η ∼ N(0t, σ2

1It),

Cov(y) = V = σ2

1(L + R)(L + R)′ + σ2In

XLII Mathematical Statistics – p. 11/2

Mixed models

M1aV : λ ∼ N(0t, σ2

1It), Cov(y) = V = σ2 1LL′ + σ2In

M1bV : λ ∼ N(0t, σ2

1It), Cov(y) = V = σ2 1LL′ + σ2In

M2aV : η ∼ N(0t, σ2

1It),

Cov(y) = V = σ2

1(L + R)(L + R)′ + σ2In

M2bV : η ∼ N(0t, σ2

1It),

Cov(y) = V = σ2

1(L + R)(L + R)′ + σ2In

XLII Mathematical Statistics – p. 11/2

Mixed models

M1aV : λ ∼ N(0t, σ2

1It), Cov(y) = V = σ2 1LL′ + σ2In

M1bV : λ ∼ N(0t, σ2

1It), Cov(y) = V = σ2 1LL′ + σ2In

M2aV : η ∼ N(0t, σ2

1It),

Cov(y) = V = σ2

1(L + R)(L + R)′ + σ2In

M2bV : η ∼ N(0t, σ2

1It),

Cov(y) = V = σ2

1(L + R)(L + R)′ + σ2In

M3aV : λ ∼ N(0t, σ2

1It), ρ ∼ N(0t, σ2 2It),

Cov(y) = V = σ2

1LL′ + σ2 1RR′ + σ2In

XLII Mathematical Statistics – p. 11/2

Mixed models

M1aV : λ ∼ N(0t, σ2

1It), Cov(y) = V = σ2 1LL′ + σ2In

M1bV : λ ∼ N(0t, σ2

1It), Cov(y) = V = σ2 1LL′ + σ2In

M2aV : η ∼ N(0t, σ2

1It),

Cov(y) = V = σ2

1(L + R)(L + R)′ + σ2In

M2bV : η ∼ N(0t, σ2

1It),

Cov(y) = V = σ2

1(L + R)(L + R)′ + σ2In

M3aV : λ ∼ N(0t, σ2

1It), ρ ∼ N(0t, σ2 2It),

Cov(y) = V = σ2

1LL′ + σ2 1RR′ + σ2In

M3bV : λ ∼ N(0t, σ2

1It), ρ ∼ N(0t, σ2 2It),

Cov(y) = V = σ2

1LL′ + σ2 1RR′ + σ2In

XLII Mathematical Statistics – p. 11/2

Information matrix - fixed model

The information matrix for estimation of treatment effects τ under the model

y = Tdτ + Zγ + ε: Cd = T′

dQZTd,

where QZ = In − PZ = In − Z(Z′Z)−Z′ and

XLII Mathematical Statistics – p. 12/2

Information matrix - fixed model

The information matrix for estimation of treatment effects τ under the model

y = Tdτ + Zγ + ε: Cd = T′

dQZTd,

where QZ = In − PZ = In − Z(Z′Z)−Z′ and

Z = Ld in M1a Z = (Ld : B) in M1b

XLII Mathematical Statistics – p. 12/2

Information matrix - fixed model

The information matrix for estimation of treatment effects τ under the model

y = Tdτ + Zγ + ε: Cd = T′

dQZTd,

where QZ = In − PZ = In − Z(Z′Z)−Z′ and

Z = Ld in M1a Z = (Ld : B) in M1b Z = Ld + Rd in M2a Z = (Ld + Rd : B) in M2b

XLII Mathematical Statistics – p. 12/2

Information matrix - fixed model

The information matrix for estimation of treatment effects τ under the model

y = Tdτ + Zγ + ε: Cd = T′

dQZTd,

where QZ = In − PZ = In − Z(Z′Z)−Z′ and

Z = Ld in M1a Z = (Ld : B) in M1b Z = Ld + Rd in M2a Z = (Ld + Rd : B) in M2b Z = (Ld : Rd) in M3a Z = (Ld : Rd : B) in M3b

XLII Mathematical Statistics – p. 12/2

Information matrix - mixed model

The information matrix for estimation of treatment effects τ under the model y = Tdτ + Zγ + Xδ + ε:

CdV = ˜ T

′ dQ ˜ Z ˜

Td,

where ˜

T = V−1/2T, ˜ Z = V−1/2Z and

XLII Mathematical Statistics – p. 13/2

Information matrix - mixed model

The information matrix for estimation of treatment effects τ under the model y = Tdτ + Zγ + Xδ + ε:

CdV = ˜ T

′ dQ ˜ Z ˜

Td,

where ˜

T = V−1/2T, ˜ Z = V−1/2Z and Z = 1n in M1aV Z = B in M1bV

XLII Mathematical Statistics – p. 13/2

Information matrix - mixed model

The information matrix for estimation of treatment effects τ under the model y = Tdτ + Zγ + Xδ + ε:

CdV = ˜ T

′ dQ ˜ Z ˜

Td,

where ˜

T = V−1/2T, ˜ Z = V−1/2Z and Z = 1n in M1aV Z = B in M1bV Z = 1n in M2aV Z = B in M2bV

XLII Mathematical Statistics – p. 13/2

Information matrix - mixed model

The information matrix for estimation of treatment effects τ under the model y = Tdτ + Zγ + Xδ + ε:

CdV = ˜ T

′ dQ ˜ Z ˜

Td,

where ˜

T = V−1/2T, ˜ Z = V−1/2Z and Z = 1n in M1aV Z = B in M1bV Z = 1n in M2aV Z = B in M2bV Z = 1n in M3aV Z = B in M3bV

XLII Mathematical Statistics – p. 13/2

Universal optimality

Cd1t = 0t

XLII Mathematical Statistics – p. 14/2

Universal optimality

Cd1t = 0t

• Proposition. (Kiefer, 1975).

Let Dt,b,k be the class of block designs. Assume that we have a design d∗ ∈ Dt,b,k such that Cd∗ is com- pletely symmetric and that trCd∗ is maximal over

Dt,b,k. Then the design d∗ is universally optimal in

the class Dt,b,k.

XLII Mathematical Statistics – p. 14/2

Universal optimality under M1a, M1aV

Theorem If there exists an equireplicated design d∗ ∈ Dt,b,k, such that Sd∗ is completely symmetric, sd∗,ii = 0, then d∗ is universally optimal under the models M1a as well

M1aV over all possible designs with no treatment pre-

ceded by itself in Dt,n. Example: every CNBD.

XLII Mathematical Statistics – p. 15/2

Examples

Every CNBD;

XLII Mathematical Statistics – p. 16/2

Examples

Every CNBD; "Half" of CNBD2 if t is prime and t = k:

d =

  

1 2 3 4 5 6 7 1 3 5 7 2 4 6 1 5 2 6 3 7 4

   ;

XLII Mathematical Statistics – p. 16/2

Examples

Every CNBD; "Half" of CNBD2 if t is prime and t = k:

d =

  

1 2 3 4 5 6 7 1 3 5 7 2 4 6 1 5 2 6 3 7 4

   ; Equireplicated design:

d =

  

1 2 3 4 1 3 7 1 5 7 4 5 2 6 2 4 6 3 5 6 7

   .

XLII Mathematical Statistics – p. 16/2

Universal optimality under M1b, M1bV

CNBD is universally optimal for the estmation of treat- ments effects in Dt,b,k under M1b ( Druilhet (1999)) as well as under M1bV ( Filipiak and Markiewicz (2007)).

XLII Mathematical Statistics – p. 17/2

Universal optimality under M1b, M1bV

CNBD is universally optimal for the estmation of treat- ments effects in Dt,b,k under M1b ( Druilhet (1999)) as well as under M1bV ( Filipiak and Markiewicz (2007)). Theorem 4. (Filipiak and Markiewicz, 2012, 2014) If there exists a BIB design d∗ ∈ Dt,b,t, such that sd,ij ∈

{p − 1, p}, i = j, and SdS′

d is completely symmetric

(CWNBD), then d∗ is universally optimal under the models

M1b as well as M1bV over the class Dt,b,t if b ≤ t − 1

• r b = p(t − 1), and over the class of equireplicated

designs with no treatment preceded by itself from Dt,b,t if

p(t − 1) < b < (p + 1)(t − 1), p ∈ N.

XLII Mathematical Statistics – p. 17/2

Examples

CNBD, CWNBD

XLII Mathematical Statistics – p. 18/2

Examples

CNBD, CWNBD Examples of CWNBD:

d∗ : t = 7, b = 3 d# : t = 7, b = 4

XLII Mathematical Statistics – p. 18/2

Examples

CNBD, CWNBD Examples of CWNBD:

d∗ : t = 7, b = 3 d# : t = 7, b = 4

Sd∗ =

         

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

         

Sd# =

         

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

         

XLII Mathematical Statistics – p. 18/2

Universal optimality under M2a, M2aV

Theorem If there exists an equireplicated design with no treatment preceded by itself at distances 1 and 2, d∗ ∈ Dt,n, such that Sd∗ + S′

d∗ and Ud∗ + U′ d∗ are completely symmetric,

then d∗ is universally optimal under the models M2a as well as M2aV over the class Dt,n with no treatment pre- ceded by itself at distances 1 and 2.

XLII Mathematical Statistics – p. 19/2

Universal optimality under M2a, M2aV

Theorem If there exists an equireplicated design with no treatment preceded by itself at distances 1 and 2, d∗ ∈ Dt,n, such that Sd∗ + S′

d∗ and Ud∗ + U′ d∗ are completely symmetric,

then d∗ is universally optimal under the models M2a as well as M2aV over the class Dt,n with no treatment pre- ceded by itself at distances 1 and 2. Examples: Every CNBD2;

XLII Mathematical Statistics – p. 19/2

Universal optimality under M2a, M2aV

Theorem If there exists an equireplicated design with no treatment preceded by itself at distances 1 and 2, d∗ ∈ Dt,n, such that Sd∗ + S′

d∗ and Ud∗ + U′ d∗ are completely symmetric,

then d∗ is universally optimal under the models M2a as well as M2aV over the class Dt,n with no treatment pre- ceded by itself at distances 1 and 2. Examples: Every CNBD2; "Half" of CNBD2 if n is prime and t = k:

d =

 

1 2 3 4 5 6 7 1 3 5 7 2 4 6 1 5 2 6 3 7 4

  .

XLII Mathematical Statistics – p. 19/2

Universal optimality under M2b, M2bV

Theorem A BIB design d∗ ∈ Dt,b,k, such that Sd∗ +S′

d∗ and Ud∗ +U′ d∗

are completely symmetric, is universally optimal under the models M2b (Filipiak, 2012) as well as M2bV over the class Dt,b,k with no treatment preceded by itself.

XLII Mathematical Statistics – p. 20/2

Universal optimality under M2b, M2bV

Theorem A BIB design d∗ ∈ Dt,b,k, such that Sd∗ +S′

d∗ and Ud∗ +U′ d∗

are completely symmetric, is universally optimal under the models M2b (Filipiak, 2012) as well as M2bV over the class Dt,b,k with no treatment preceded by itself. Examples: Every CNBD2;

XLII Mathematical Statistics – p. 20/2

Universal optimality under M2b, M2bV

Theorem A BIB design d∗ ∈ Dt,b,k, such that Sd∗ +S′

d∗ and Ud∗ +U′ d∗

are completely symmetric, is universally optimal under the models M2b (Filipiak, 2012) as well as M2bV over the class Dt,b,k with no treatment preceded by itself. Examples: Every CNBD2; "Half" of CNBD2 if n is prime and k = t:

d =

 

1 2 3 4 5 6 7 1 3 5 7 2 4 6 1 5 2 6 3 7 4

  .

XLII Mathematical Statistics – p. 20/2

Universal optimality under M3a, M3aV

Theorem If there exists an equireplicated design d∗ ∈ Dt,n, such that Sd∗ and Ud∗, sd∗,ii = 0 and ud∗,ii = 0, are com- pletely symmetric, then d∗ is universally optimal under the models M3a as well as M3aV over the class Dt,n with no treatment preceded by itself at distances 1 and 2. Example: every CNBD2.

XLII Mathematical Statistics – p. 21/2

Universal optimality under M3b, M3bV

CNBD2 is universally optimal for the estmation of treat- ments effects over the class Dt,b,k with no treatment pre- ceded by itself at distances 1 and 2 under the models

M3b (Druilhet (1999)) as well as M3bV ( Filipiak and

Markiewicz (2007)).

XLII Mathematical Statistics – p. 22/2

References

Bailey, R.A., Peter J. Cameron, P .J., Filipiak, K., Kunert, J., Markiewicz, A. (2016). On optimality and construction of circular repeated-measurements designs. Statistica Sinica, accepted. Druilhet, P . (1999). Optimality of circular neighbor balanced designs.

• J. Statist. Plann. Infer. 81, 141–152.

Filipiak, K., Markiewicz, A., (2007). Optimal designs for a mixed interference model. Metrika 65, 369–386. Filipiak, K., Markiewicz, A., (2012). On universal optimality of circu- lar weakly neighbor balanced designs under an interference model.

• Comm. Statist. - Theory Meth. 41, 2356–2366.

Filipiak, K. (2012). Universally optimal designs under an interference model with equal left- and right-neighbor effects. Statist. Probab.

• Lett. 82, 592–598.

Filipiak,

• K. and Markiewicz,
• A. (2014).

On the optimality of circular block designs under a mixed interference model. Comm.

• Statist. - Theory Meth. 43, 4534–4545.

XLII Mathematical Statistics – p. 23/2

References, cont.

Filipiak, K., Markiewicz, A., (2017). Universally optimal designs under interference models with and without block effects. Comm.

• Statist. - Theory Meth. 46, 1127–1143.

Hwang, F .K., (1973). Constructions for some classes of neighbor

• designs. Ann. Statist. 1, 786–790.

Kiefer, J. (1975). Construction and optimality of generalized Youden designs. J.N. Srivastava (Ed.), Survey of Statistical Design and Linear Models, North-Holland, Amsterdam, 333–353. Rees, D. H., (1967). Some designs of use in serology. Biometrics 23, 779–791.

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