Binary Operations on Orthogonal Models, Application to Prime Basis Factorials and Fractional...

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Binary Operations on Orthogonal Models, Applicationto Prime Basis Factorials and Fractional ReplicatesVera de Jesus a , Mexia Tiago João a & Paulo Canas Rodrigues aa Faculty of Sciences and Technology-Mathematics Department , Nova University ofLisbon , Monte de Caparica 2829-516, Caparica, PortugalPublished online: 30 Nov 2011.

To cite this article: Vera de Jesus , Mexia Tiago João & Paulo Canas Rodrigues (2009) Binary Operations on OrthogonalModels, Application to Prime Basis Factorials and Fractional Replicates, Journal of Statistical Theory and Practice, 3:2,505-521, DOI: 10.1080/15598608.2009.10411941

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Journal of2

© Grace Scientific Publishing Statistical Theory and PracticeVolume 3, No.2, June 20094

Binary Operations on Orthogonal Models, Application toPrime Basis Factorials and Fractional Replicates6

Vera de Jesus, Faculty of Sciences and Technology-Mathematics Department,Nova University of Lisbon, Monte de Caparica 2829-516 Caparica, Portugal.8

Email: [email protected]

João Tiago Mexia, Faculty of Sciences and Technology-Mathematics Department,10Nova University of Lisbon, Monte de Caparica 2829-516 Caparica, Portugal.


Paulo Canas Rodrigues, Faculty of Sciences and Technology-Mathematics Department,Nova University of Lisbon, Monte de Caparica 2829-516 Caparica, Portugal.14


16 Received: December 11, 2007 Revised: September 3, 2008

Abstract18

Commutative Jordan algebras (CJA) are used in the study of orthogonal models either simple or de-rived through model crossing and model nesting. Once normality is assumed, UMVUE are obtained20for relevant parameters.

The general treatment is then applied to models obtained from prime basis factorials or their frac-22tional replicates. Besides model crossing and nesting, factor merging is considered. In this way wemay extend our results to factors with whatever number of levels instead of factors whose numbers of24levels are powers of the same prime.

AMS Subject Classification: 62J12; 62K15; 17C50.26

Key-words: Commutative Jordan algebras; Orthogonal models; Binary operations; Factorial models;Complete sufficient statistics; UMVUE; Confidence regions.28

1. Introduction

CJA enable the presentation of orthogonal models in a very convenient way. These30

algebras are linear spaces constituted by symmetric matrices that commute and that containthe squares of their matrices.32

In the next section we present the strong of CJA emphasizing the role played by orthog-∗1559-8608/09-2/$5 + $1pp – see inside front cover

© Grace Scientific Publishing, LLC

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506 Vera de Jesus, João Tiago Mexia & Paulo Canas Rodrigues

onal matrices associated to them. Seely (1971) showed that any CJA had one and only onebasis, the principal basis pb(A), whose matrices Q1, . . . ,Qw are pairwise orthogonal and2

orthogonal projection matrices. Now Q j = A′jA j, with the row vectors of A j constituting

an orthogonal basis for the range space R(Q j) of Q j, j = 1, . . . ,w. Then, if ∑wj=1 Q j = In, A4

will be complete and P = [A′1 . . . A′

w]′ will be an orthogonal matrix associated to A. If A isnot complete we can join Qw+1 = In−∑w

j=1 Q j to pb(A) thus obtaining the principal basis6

of A, the completed of A.We will consider binary operations on CJA which, see Fonseca, Mexia and Zmyslang8

(2006), may be used to derive complex models from simple ones. The basic techniques onthese developments are:10

• model crossing, then the treatments in the new model are all possible combinations of thetreatments of the initial models;12

• model nesting, now all the treatments of a model are nested inside each treatment ofanother treatment.14

Besides models obtained through crossing and nesting we will also derive models throughfactor merging. Sub algebras will be useful in connection with these models.16

A mixed model given by

Y =m

∑i=1

Xiβββ i +w

∑j=m+1

Xiβββ i + e, (1.1)18

with βββ 1, . . . ,βββ m and the βββ m+1, . . . , βββ w and e independent with null mean vectors and variance-covariance matrices σ2Icm+1 , . . . ,σ2Icw and σ2In, is associated to a CJA A if the matrices20

Mi = XiX′i, i = 1, . . . ,w, constitute a basis for A (Fonseca, Mexia and Zmyslang, 2006).

In this paper we are interested in models strictly associated to CJA22

Y =m

∑j=1

(A′

j⊗1r)

ηηη j +w

∑j=m+1

(A′

j⊗1r)

ηηη j + e, (1.2)

where ⊗ represents the kronecker matrix product and the components of 1r are equal to24

1. The vectors ηηη1, . . . ,ηηηm will be fixed, the ηηη1, . . . , ηηηm and e will be independent withmean vectors ηηηm+1, . . . ,ηηηw and 0, and variance-covariance matrices σ2

m+1Igm+1 , . . . ,σ 2wIgw26

and σ2In. When Q j = A′jA j, j = 1, . . . ,w, constitute the principal basis of a complete CJA

A of n0× n0 matrices, the model is strictly associated to A. We study these models in the28

third section. First we obtain general results and then we consider separately fixed effectsmodels, random effects models and mixed models.30

In the last section we applied the results on strictly associated models to prime basisfactorials and their fractional replicates. Besides crossing and nesting of models we will32

consider factor merging. This merging will enable us to overcome the requirement that, inthese models, the number of factors levels must be a power of a prime. The reason for this34

requirement was that, for example see Dey and Mukerjee (1999), Montgomery (2005) orMukerjee and Wu (2006), there was a 1-1 correspondence between the factor levels and the36

elements of a Galois field.

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Binary Operations on Orthogonal Models, Application 507

2. Commutative Jordan algebras

2.1. Orthogonal associated matrices2

Jordan algebras were introduced by Jordan, von Neumann and Wigner (1934) to providean algebraic foundation for Quantum Mechanics. Later on these structures were applied4

see for instance, Seely (1970), Seely (1971), Seely and Zyskind (1971), Seely (1977), Van-Leeuwen, Seely and Birkes (1998) and VanLeeuwen, Birkes and Seely (1999), to carry out6

linear statistical inference.Jordan algebras are linear spaces constituted by square matrices that contain the squares8

of their matrices. If the matrices commute, the Jordan algebra will be commutative. Weare interested in CJA. Moreover, Seely (1971) proved that each CJA has an unique basis10

{Q1, . . . ,Qw}, constituted by orthogonal projection matrices, all of them mutually orthogo-nal and with the same dimension. This basis is called the principal basis of the algebra.12

Let {Q1, . . . ,Qw} be the principal basis of a CJA A. If

Q1 =1n

1n1′n =

1n

Jn (2.1)14

where 1′n is a matrix n×1 with elements equal to 1, the CJA will be regular. Moreover, if

w

∑j=1

Q j = In (2.2)16

where In is the n× n identity matrix, the CJA will be complete. In what follows we onlyconsider regular and complete CJA.18

Let M be an orthogonal projection matrix belonging to A. We have

M =w

∑j=1

a jQ j. (2.3)20

with a j = 0 or a j = 1, j = 1, . . . ,w, and we write M = ∑ j∈C Q j with C = { j : a j 6= 0}. Thusorthogonal projection matrices in a CJA will be the sum of some or all the matrices in the22

principal basis.If R(Q j) =∇ j, j = 1, . . . ,w is the range spaces of the matrices Q j, j = 1, . . . ,w, with24

{Q1, . . . ,Qw} the principal basis of A we have, see Mexia (1995), Q j = A′jA j, j = 1, . . . ,w

where A j is a matrix whose row vectors constitute an orthogonal basis for ∇ j, j = 1, . . . ,w.26

Then,

P =[A′

1 . . . A′w]′ (2.4)28

will be an orthogonal matrix associated to the CJA A with principal basis N(A)= {Q1, . . . ,Qw}.

Moreover, if the CJA is complete and regular, we will have Q1 = 1n Jn =

(1√n 1n

)(1√n 1′n

)30

and so A1 = 1√n 1n. Then the elements of the first row of P are equal to 1√

n and P will be,see Mexia (1988), orthogonal standardized.32

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Lastly we observe that, see Fonseca, Mexia and Zmyslony (2006), if

M =w

∑j=1

a jQ j (2.5)2

is a matrix of a complete CJA, its eigenvalues will be the a1, . . . ,aw. Since M is symmetrictheir eigenvalues will be real with multiplicity g j = rank(Q j), j = 1, . . . ,w and so4

det (M) =w

∏j=1

ag jj . (2.6)

Moreover since the Q j, j = 1, . . . ,w are pairwise orthogonal if M is regular we will have6

M−1 =w

∑j=1

a−1j Q j. (2.7)

2.2. Binary operations8

We represent by ⊗ the Kronecker matrix product. This matrix operation is studied inSteeb (1991).10

Let us establish the

Proposition 2.1. With X j ={

Q j,1, . . . ,Q j,w j

}the principal basis of the CJA A j constituted12

by the n j×n j, j = 1,2, matrices,

A1⊗A2 =

{w1

∑i=1

w2

∑j=1

ai, jQ1,i⊗Q2, j

}14

will be the CJA with principal basis{

Q1,i⊗Q2, j, i = 1, . . .w1, j = 1, . . . ,w2}

. If A1 andA2 are completes [regular] , A1⊗A2 will be complete [regular].16

Proof. As the Kronecker product is associative and Q1,i⊗Q2, j, i = 1, . . .w1, j = 1, . . . ,w2are mutually orthogonal projection orthogonal matrices, A1⊗A2 will be the linear space18

constituted by symmetric matrices that commute and contain the square of these matrices.Then A1⊗A2 will be a CJA.20

As the Q1,i⊗Q2, j, i = 1, . . .w1, j = 1, . . . ,w2 are mutually orthogonal and they are linearlyindependent, they constitute the principal basis of A1⊗A2.22

Besides this, ifwd∑

i=1Qd,i = Ind , d = 1,2, we will have

w1∑

i=1

w2∑j=1

Q1,i⊗Q2, j = In1 ⊗ In2 = In1n2

and, if Qd,1 = 1nd

Jnd , d = 1,2, Q1,d ⊗Q2,d = 1n1n2

Jn1n2 , which completes the proof.24

We can show that, see Fonseca, Mexia and Zmyslony (2006), if A1,A2 and A3 are CJA,

A1⊗ (A2⊗A3) = (A1⊗A2)⊗A3. (2.8)26

Thus, the ⊗ product of CJA is associative.

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If Pl is an orthogonal matrix associated to Al , l = 1,2, P1 ⊗P2 will be an orthogonalmatrix associated to A1⊗A2.2

We also have the

Proposition 2.2. P(A1⊗A2) = P(A1)⊗P(A2) .4

Proof. As we saw, P(Al) = P = [A′l,1 : . . . : A′

l,wl]′ with Ql, j = A′

l, jAl, j, j = 1, . . . ,wl ,

l = 1,2. Then, Q1,i⊗Q2, j = (A′1,i⊗A′

2, j)(A1,i⊗A2, j), i = 1, . . . ,w1, j = 1, . . . ,w2 and we6

will have

P(A1⊗A2) = [A′1,1⊗A′

2,1 : . . . : A′1,w1

⊗A′2,w2

]′ = P(A1)⊗P(A2).8

Another operation with CJA in which we have the ⊗ product, will be represented10

by ~. Given{

Q j,1, . . . ,Q j,w j

}the principal basis constituted by n j × n j matrices

of the complete and regular CJA A j, j = 1,2, the principal basis of A1 ~ A2 is12 {Q1,1⊗ 1

n2Jn2 , . . . ,Q1,w1 ⊗ 1

n2Jn2

}∪{In1 ⊗Q2,2, . . . ,In1 ⊗Q2,w2} .

Then, if14

P j =

[ 1√n j1′n j

K j

]; j = 1,2 (2.9)

is an orthogonal matrix associated to A j, j = 1,2,16

P =

[P1⊗ 1

n2Jn2

In1 ⊗K2

](2.10)

will be an orthogonal matrix associated to A1 ~A2. Thus, if Q j,l = A′j,lA j,l , l = 1, . . . ,w j,18

with A j,1 = 1√n j1′n j

, j = 1,2, we have

Q1,l ⊗ 1n2

Jn2 =(

A1,l ⊗ 1√n2

1′n2

)′(A1,l ⊗ 1√

n21′n2

), l = 1, . . . ,w1

In1 ⊗Q2,l =(In⊗A2,l

)′ (In⊗A2,l), l = 1, . . . ,w2

(2.11)20

With A1,A2 and A3 complete and regular CJA we have, see Fonseca, Mexia and Zmys-lony (2006)22

A1 ~ (A2 ~A3) = (A1 ~A2)~A3 (2.12)

that is, the ~ product is also associative. We will say that, ~ is the restricted Kronecker24

product of complete and regular CJA.While the operation ⊗ will be used in models crossing, the operation ~ will be useful in26

the study of model nesting.Another interesting application of ~ will be when we consider r observations per treat-28

ment. Thus, if r = 1 and we use the CJA A, with r > 1, the CJA will be A~A(r), with

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{ 1r Jr,Jr

}where Jr = Ir − 1

r Jr, the principal basis of A(r). We have Jr = K′rKr, with Kr

obtained deleting the first row equal to 1√r 1′r of a r× r orthogonal matrix.

2

2.3. Sub-algebras

A◦ is a sub-CJA of the CJA A if it is an algebra and is contained in A. With {Q◦1, . . . ,Q

◦w◦}4

and {Q1, . . . ,Qw} the principal basis of A◦ and A respectively, we have,

Q◦i = ∑

j∈Ci

Q j, i = 1, . . . ,w◦ (2.13)6

where the sets Ci are pair-wise disjoint.If A◦ is sub-algebra of A, and8

m◦

∑i=1

Q◦i =

m

∑j=1

Q j, (2.14)

the pair (A◦,A) is said to be (m◦,m)− compatible. If the pairs(A◦

l ,Al)

are(m◦

l ,ml)−10

compatible, l = 1,2, we can order the matrices of the principal basis of A◦1 ⊗A◦

2 and ofA1⊗A2 so that the pair (A◦

1⊗A◦2,A1⊗A2) will be (m◦

1m◦2, m1m2)−compatible, since12

m◦1∑

i1=1

m◦2∑

i2=1Q◦

1,i1 ⊗Q◦2,i2 =

m1

∑j1=1

m2

∑j2=1

Q1, j1 ⊗Q2, j2 . (2.15)

We point out that every CJA is sub-algebra of itself.14

Suppose that the pair (A◦1,A1) is (m◦

1,m1)− compatible and A◦2 is sub-algebra of A2.

If these algebras are all complete and regular, A◦1 ~ A◦

2 will be sub-algebra complete and16

regular of A1 ~A2. The pair (A◦1 ~A◦

2,A1 ~A2) will be (m◦1,m1)− compatible for having,

m◦1∑i=1

Q◦1,i⊗

1n2

Jn2 =m1

∑j=1

Q1, j⊗ 1n2

Jn2 . (2.16)18

Furthermore, if the pair (A◦2,A2) is (m◦

2,m2)−compatible, the pair (A◦1 ~A◦

2,A1 ~A2)will be (w◦1 +m◦

2−1,w1 +m2−1)− compatible.20

When we are working with mixed models and their sub-models, m and◦m would be the

dimensions of A and A◦, where A◦ is associated to the fixed effects part and◦m will be the22

separation value.

3. Strictly associated models24

3.1. General results

Let {Q1, . . . ,Qw} be the principal basis of a complete and regular CJA A constituted by26◦n× ◦

n matrices.

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The principal basis of A~A(r) will be constituted by the matrices

Q j⊗ 1r

Jr =(

A j⊗ 1√r

1′r

)′ (A j⊗ 1√

r1′r

), j = 1, . . . ,w, (3.1)2

and Q⊥ = In−w∑j=1

Q j⊗ 1r Jr.

If the matrices of A are◦n× ◦

n those of A~A(r) will be n× n, with n =◦nr and, if A is4

complete,

w

∑j=1

Q j = I◦n

(3.2)6

as well as

Q⊥ = I◦n⊗Jr =

(A⊥

)′A⊥ (3.3)8

with A⊥ = I◦n⊗Kr.

We also have10{

g j = rank(Q j⊗ 1

r Jr)

= rank (Q j) = rank (A j) , j = 1, . . . ,w

g = rank(Q⊥)

= rank(A⊥)

=◦n(r−1)

. (3.4)

We put Z ∼ N (ξξξ ,W) when Z is normal with mean vector ξξξ and covariance variance12

matrix W.The model Y∼ N (µµµ ,V) will be strictly associated to the CJA A~A(r) if14

µµµ =m∑j=1

(A′

j⊗ 1√r 1r

)ηηη j

V =w∑j=1

γ j

(Q j⊗ 1√

r 1r

)+σ2Q⊥

, (3.5)

with γ1 = . . . = γm = σ2.16

With

ηηη j =(

A j⊗ 1√r 1′r

)µµµ, j = 1, . . . ,w

ηηη∗j =(

A j⊗ 1√r 1′r

)Y, j = 1, . . . ,w

, (3.6)18

we will have

ηηη∗j ∼ N(ηηη j,γ jIg j

), j = 1, . . . ,w (3.7)20

with ηηη j = 0g j , j = m+1, . . . ,w, and γ j = σ2j +σ2, j = 1, . . . ,w.

Then, with e∗ = Q⊥Y, the model22

Y =w

∑j=1

(A′

j⊗1√r

1r

)ηηη∗j + e∗, (3.8)

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will be strictly associated to the CJA A~A(r) .As the model is normal with mean vector and variance-covariance matrix given by (3.5),2

it has the density

n(Y) =e− 1

2

m

∑j=1

‖ηηη∗j−ηηη j‖2

σ2 +w∑

i=m+1

S jγ j

+ Sσ2

(2π)n2 .σ

(n−∑w

j=m+1 g j

).

w∏

j=m+1(γ j)

g j/2, (3.9)4

with the complete sufficient statistics

ηηη∗j ∼ N(ηηη j,γ jIg j

); j = 1, . . . ,w

S j = ‖ηηη∗j‖2 ∼ γ jχ2p−1, j = 1, . . . ,w

S = ‖e∗‖2 ∼ σ2χ2g

. (3.10)6

Given the models

Yl =wl

∑j=1

(A′

l, j⊗1√rl

1rl

)ηηη∗l, j + e∗l , l = 1,2, (3.11)8

strictly associated to the CJA Al ~ A(rl) , l = 1,2, if we cross them, we will obtain themodel,10

Y =w1

∑j1=1

w2

∑j2=1

(A′

1, j1 ⊗A′2, j2 ⊗

1√r

1r

)ηηη∗j1, j2 + e∗, (3.12)

strictly associated to the CJA (A1⊗A2)~A(r) .12

If the first m1 [m2] terms of the first [second] model correspond to the fixed effects part, inthe final model the fixed effects part is constituted by the terms with indexes jl = 1, . . .ml ,14

l = 1,2.

If we nest all treatments of the second initial model inside each treatment of the first initial16

model, we obtain the model,

Y =w1

∑j1=1

(A′

1, j1 ⊗ I◦n2⊗ 1√

r1r

)ηηη∗j118

+w2

∑j2=2

(I◦

n1⊗A′

2, j2 ⊗1√r

1r

)ηηη∗w1+ j2−1 + e∗, (3.13)

strictly associated to the CJA (A1 ~A2)~A(r) .20

As the random effects factors do not nest fixed effects factors, if the first model has ran-dom effects factors, the second cannot have fixed effects factors. Similarly, if the second22

model has fixed effects factors, the first can only have fixed effects factors. Thus, represent-ing for Fx, [Al, Mt] the fixed effects models, [random, mixed], we will have the following24

possible cases:26

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1st model 2nd modelAl AlMt AlFx Fx, Al, Mt

2nd model 1st modelAl Fx, Al, MtMt FxFx Fx

3.2. Fixed effects models2

Now we have m = w and γ1 = . . . = γw = σ2. Then, with{

ηηη = [ηηη ′1

: . . . : ηηη ′w]′

ηηη∗ = [ηηη∗′1

: . . . : ηηη∗′w ]′(3.14)4

we will have

ηηη∗ ∼ N(ηηη ,σ 2Ig

), g =

w

∑j=1

g j (3.15)6

independent of S = ‖e∗‖2 .

We will be interested in testing hypothesis on vectors Gηηη , with G = [G1 : . . . : Gw] a8

matrix whose row vectors are linearly independent. Taking

ΨΨΨ∗ = Gηηη∗ =w∑j=1

G jηηη∗j

ΨΨΨ = Gηηη =w∑j=1

G jηηη j

, (3.16)10

according to the Blackwell-Lehman-Scheffé theorem, we have the UMVUE

σ∗2 = Sg

ΨΨΨ∗ = Gηηη∗∼ N(ΨΨΨ,σ2GG′) (3.17)12

for σ2 and for ΨΨΨ.As the ηηη∗j , j = 1, . . . ,w are independents of S, ΨΨΨ∗ will be independent of S.14

We can use the statistic S∼ σ2χ2g to construct confidence intervals for σ2. The 1−q level

confidence intervals can be used to obtain q level tests for the null hypothesis16

H0 : σ2 = σ20 . (3.18)

The hypothesis being rejected when σ20 is not contained in the q level interval. Furthermore,18

see Mexia (1995),

U0 = (ΨΨΨ−ΨΨΨ0)′ (GG′)+ (ΨΨΨ−ΨΨΨ0) (3.19)20

where (GG′)+ is the Moore-Penrose inverse of GG′, will be the product by σ2 of a chi-square with c = rank(GG′) degrees of freedom and non-centrality parameter22

δ =1

σ2 (ΨΨΨ−ΨΨΨ0)′ (GG′)+ (ΨΨΨ−ΨΨΨ0) . (3.20)

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We put U0 ∼ σ2χ2c,δ independent of S. Thus to test

H0 : ΨΨΨ = ΨΨΨ0 (3.21)2

we will have the statistics

F =gc

U0

S(3.22)4

with the F distribution with c and g degrees of freedom and non-centrality parameter δ .Since δ = 0 when H0 holds the test will be unbiased.6

3.3. Random effects models

In these models we have m = 1 and8

S j =∥∥ηηη∗j

∥∥2 ∼ γ jχ2g j

, j = 2, . . . ,w (i) S = ‖e∗‖2 ∼ σ2χ2g ,

where (i) stands for independence.10

Besides the γ j, j = 1, . . . ,w, we may be interested in the σ2j = γ j−σ2, j = 1, . . . ,w. The

estimators12

γ∗j = S jg j

, j = 2, . . . ,w

σ∗2 = Sg

(3.23)

and hence14

σ∗2j = γ∗j −σ∗2, j = 2, . . . ,w. (3.24)

will be UMVUE for γ j, j = 2, . . . ,w, σ2, and σ2j , j = 2, . . . ,w, since they are unbiased16

estimators which are derived from sufficient complete statistics.We can reason as above to obtain confidence intervals for σ2 and through duality, test18

H0 : σ2 = σ20 .

Putting20

υ j =γ j

σ2 , i = 1, . . . ,w, (3.25)

the statistics22

F j =gg j

S j

S, j = 2, . . . ,w, (3.26)

will be the products by the υ j, j = 2, . . . ,w, of variables with the central F distributions with24

g j, j = 2, . . . ,w, and g degrees of freedom. If fp,r,s is the quantile for probability p of thecentral F distribution with r and s degrees of freedom we have, for υ j, j = 2, . . . ,w, the 1−q26

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level confidence intervals

[F j

f1− q2 ,g j ,g

;F j

f q2 ,g j ,g

], j = 2, . . . ,w

[F j

f1−q,g j ,g;+∞

], j = 2, . . . ,w

[0;

F j

fq,g j ,g

], j = 2, . . . ,w

(3.27)2

These intervals may be used to derive q level tests for the hypotheses

H0, j(υ j,0) : υ j = υ j,0, j = 2, . . . ,w. (3.28)4

We point out that the usual hypotheses

H0, j : σ2j = 0, j = 2, . . . ,w, (3.29)6

are equivalent to the H0, j(1), j = 2, . . . ,w.Moreover, if g ≥ 3, F j has the mean value g

g−2 υ j, j = 2, . . . ,w, so we will have the8

UMVUE

υ∗j =g−2

gF j, j = 2, . . . ,w. (3.30)10

Nextly, taking

λ j, j′ =γ j

γ j′, j 6= j′, (3.31)12

the statistics

F j, j′ =g j′

g j

S j

S j′; j 6= j′ (3.32)14

will be the product by the λ j, j′ , j 6= j′, of variables with central F distributions with g j andg j′ , j 6= j′, degrees of freedom. Thus we obtain the 1−q level confidence intervals for the16

λ j, j′ , j 6= j′, given by

[F j, j′

f1− q2 ,g j ,g j′

;F j, j′

f q2 ,g j ,g j′

], j 6= j′

[F j, j′

f1−q,g j ,g j′;+∞

], j 6= j′

[0;

F j

fq,g j ,g j′

], j 6= j′

(3.33)18

which can be used, though duality, to test the null hypothesis

H0, j, j′(λ j, j′,0) : λ j, j′ = λ j, j′,0, j 6= j′. (3.34)20

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Lastly, if g j′ ≥ 3, we have the UMVUE

λ ∗j, j′ =g j′ −2

g j′F j, j′ , j 6= j′. (3.35)2

3.4. Mixed models

In these models we have the fixed effects part ∑mj=1

(A′

j⊗ 1√r 1

)ηηη∗j and the random4

effects part ∑wj=m+1

(A′

j⊗ 1√r 1

)ηηη∗j + e∗. We carry out the inference for the fixed effects

part as in the subsection 3.2 rewriting matrix G as G = [G1 : · · · : Gm] and taking ΨΨΨ∗ =6

∑mj=1 G jηηη∗j and ΨΨΨ = ∑m

j=1 G jηηη j. For the random effects part we reason as in subsection 3.3restricting ourselves to the γ j and to the υ j with j = m+1, . . . ,w.8

4. Prime basis and related factorial designs

4.1. Foreword10

As we hope it is now clear that the key to apply the preceding results is having anorthogonal matrix associated to the relevant CJA. This is true whether we are working with12

isolated models or if we are using binary operations to derive complex models from simpleones. Thus in considering prime basis and related factorials we will concentrate on deriving14

the required orthogonal matrices.

4.2. Prime basis factorials16

We will now consider the pN factorial design with N (N > 1, integer) factors each havinga prime number p of levels. The levels will be numerated from 0 to p− 1 which are the18

elements of the Galois field GF(p) of order p. The pN treatments may be representedby vector x = [x1, . . . ,xN ]

′, with x j ∈ GF(p), j = 1, . . . ,N. These vectors will constitute a20

dimension N vector space GF(p)N over GF(p). The vectors in GF(p)N can be ordered bythe indexes22

l(x) = 1+N

∑j=1

x j p j−1. (4.1)

Given a ∈ GF(p)N we have the linear application of GF(p)N on GF(p) given by24

La(x) =

(N

∑j=1

a jx j

)

(p)

(4.2)

where (p) indicates the use of module p arithmetic. These applications constitute a linear26

space LN[p] over GF(p) isomorphic to GF(p)N . Thus the La1(x), . . . .,Lam(x) are linearly

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independent if and only if a1, . . . ,am are linearly independent.The linear applications whose first non null coefficient is 1 will be reduced linear appli-2

cations. Let LNr[p] be the family of such applications. There are

kN(p) =pN −1p−1

(4.3)4

reduced applications.With L1, . . . ,Lu linearly independent reduced applications, the set6

[L|b] = [L1, . . . ,Lu|b1, . . . ,bu] (4.4)

of treatments x such that (Li(x))(p) = bi, i = 1, . . . ,u, will be called a block. Since the system8

of equations defining a block enables us to express u components of x as linear combinationsof the remaining components, in every [L|b] there will be pN−u treatments and there will be10

pu blocks.We may order the reduced applications L∈LN

r[p] according to the increase of indexes l(a).12

Thus if l(a1) < l(a2) we will have k(a1) < k(a2), k(a) = 1, . . . ,kN(p).To each L ∈ LN

r[p] we can associate a p× pN matrix C(L) with elements14

ci, j(L) ={

0, L(x j) 6= i−11, L(x j) = i−1

(4.5)

i = 1, . . . , p and j = 1, . . . , pN . It is easy to see that16

C(L)C(L)′= pN−1Ip (4.6)

since L takes each of its values for pN−1 treatments. Moreover if L1 and L2 are linearly18

independent

C(L1)C(L2)′= pN−2Jp (4.7)20

since [L1,L2|b1,b2] contains pN−2 treatments whatever the pair (b1,b2).

With q = pN−1

2 we consider the matrices22

A(L) =1q

KpC(L), L ∈ LNr[p]. (4.8)

If L1,L2 ∈ LNr[p] are linearly independent, we have, according to (4.7)24

A(L1)A(L2)′= 0(p−1)×(p−1) (4.9)

We now prove the following,26

Proposition 4.1. The matrix

P(pN) =

[1√pN

1pN : A(L1)′: . . . : A(LkN(p))

′]′

(4.10)28

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is orthogonal and it is associated to the CJA A(pN) with principal basis

N(pN) ={

1pN JpN ,Q(L1), . . . ,Q(LkN(p))

}(4.11)2

where

Q(L j) = A(L j)′A(L j), j = 1, . . . ,kN(p). (4.12)4

Proof. First of all, we have

A(L j)A(L j)′=

1q2 KpC(L j)C(L j)

′K′p =

1pN−1 Kp

(pN−1Ip

)K′p = KpK

′p = Ip.

and for i 6= j,

A(Li)A(L j)′

=1q2 KpC(Li)C(L j)

′K′p =

1pN−1 Kp

(pN−21p1

′p

)K′p6

=1p

(Kp1p)(Kp1p)′= 0(p−1)×(p−1).

Thus the 1pN JpN ,Q(L1), . . . ,Q(Lw) are symmetric, idempotent and mutually orthogonal so8

they will constitute the principal basis of CJA A(pN).

The matrix10

P(pNr) =

[1√rpN

1pN r, A′1⊗

1√rpN

1rpN . . . A′kN(p)⊗

1√rpN

1rpN ;(

A⊥)′]′

(4.13)

with A⊥ = IpN ⊗Kr will be orthogonal and it is associated to the CJA A(pN)~A(r) .12Let L1 be linear space with basis L1 = {L1, . . . ,Lu}. In L1 there are pu linear applications

of which ku(p) are reduced. If we give to these reduced applications the indexes 1, . . . ,ku(p)14

we can take

A(Bl) =[A(L1)′ : · · · : A(Lku(p)(p))′

]′, (4.14)16

then

P(Bl) =

[1√pN

1pN : A(Bl)′ : A(Lku(p+1))′ : · · · : A(LkN(p))

′]′

, (4.15)18

will be an orthogonal matrix associated to a CJA A(pN ,Bl) which we consider when ana-lyzing designs in which the treatments are grouped in blocks. That grouping of treatments is20

know as confounding since the linear applications in L1 have fixed values for the treatmentsin any block. Thus the variations ascertainable to these applications is confounded with the22

variation between blocks. The applications in L1 are said to be confounded.

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4.3. Fractional replicates

If we take only the treatments x1, . . . ,xpN−u in one of the blocks [L1, . . . ,Lu|b1, . . . ,bu],2

we will have a fractional replicate 1pu × pN . We give to the chosen treatments the first pN−u

indexes. To study these models we introduce in L[p]N an equivalence relation ρL1 , putting4

lρL1 g if g = (cl + l∗)(p), (4.16)

with c ∈ {1, . . . p−1} and l∗ ∈ L1, where L1 is the linear subspace of L[p]N spanned by L.6

If l∗ takes a fixed value b for all chosen treatments, we will have

g(x j) = (cl(x j)+b)(p), j = 1, . . . , pN−u. (4.17)8

Therefore, if C∗(l) is the sub-matrix constituted by the first pN−u columns of C(l), we seethat the rows of C∗(g) are obtained reordering the rows of C∗(l). Similarly, with10

A∗(l) =1√

pN−u−1KpC∗(l), (4.18)

we see that the rows of A∗(g) are obtained reordering the rows of A∗(l).12

When r = 1, we have

S∗(l) = ‖A∗(l)y‖2 (4.19)14

and, when r > 1,

S∗(l) =∥∥∥(A∗(l)⊗ 1√

r1′r

)y∥∥∥

2(4.20)16

so in both cases we have,

S∗(l) = S∗(g). (4.21)18

Then, we chose in each ρL1 equivalence class, a reduced linear application to which isattributed the difference between the groups of treatments chosen, that correspond to the20

different values taken by the applications.As we saw, if we have the relation lρL1g, the two applications define the same group of22

treatments.The space L should be chosen in such a way that the effects of factorial levels and also24

low order factorial interactions are isolated in their ρL1 equivalence classes. For p = 2 andp = 3, this problem has been studied (for instance, see Cochran, 1992).26

To study confounding in the case of fractional replicates we complete L1 = {l1, . . . , lu}to obtain a basis {l1, . . . , lu, . . . , lv, . . . , lN} for L[p]N . Taking L2 = {lu+1, . . . , lv} and L3 =28

{lv+1, . . . , lN} as well as L j = L(L j), L j which is the subspace spanned by L j, j = 1,2,3.Furthermore we have the subspaces L1⊕L2 and L2⊕L3 given by the direct sum of L1 and30

L2 and of L2 and L3. With,

gh =N

∑j=1

ah, jl j, j = 1,2 (4.22)32

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we have g1ρL1g2 if and only if a2, j = (ca1, j)(p), j = u + 1, . . . ,N, with c ∈ {1, . . . p− 1}.Let us establish the2

Proposition 4.2. L1 is a ρL1 equivalence class. Moreover, there are kN−u(p) classes dis-tinct of L1, each containing pu(p− 1) applications. The ρL1 equivalence classes are ρ-4

saturated and (L1⊕L2)\L1 is the union of kv−u(p) such classes.

Proof. If g1 ∈ L1 we have g1ρL1 g2 if and only if g2 ∈ L1 and, since when g1,g2 ∈ L1,6

g1ρL1g2 we see that L1 is a ρL1 equivalence class. Moreover, if g1 = g1,1 + g1,2 and g2 =g2,1 + g2,2 with g1,1,g2,1 ∈ L1 and g1,2,g2,2 ∈ L2 we have, as we saw, g1ρL1g2 if and only8

if g1,2ρg2,2. So, the ρL1 equivalence classes distinct from L1 will contain one and only oneapplication from (L1⊕L2)r. Thus, there will be kN−u(p) ρL1 equivalence classes distinct10

from L1. Since #(L[p]\L1

)= pN− pu, we see that they will contain pu(p−1) applications.

Lastly, if g1 ∈ L1⊕L2, g1ρL1 g2 when and only when g2 ∈ L1⊕L2 will be ρL1-saturated.12

The rest of the proof is straightforward.

Let us give the indexes 1, . . . ,ku(p) to the reduced applications in L1,r, the indexes ku(p)+14

1, . . . ,ku(p)+ kv(p) to the reduced applications in L2,r, the indexes ku(p)+ kv(p)+ 1, . . .,ku+v(p) to the reduced applications in (L1⊕L2)\(L1,r∪L2,r), the indexes ku+v(p)+1, . . .,16

ku+v(p)+kN−v(p) to the reduced applications in L3,r and the indexes ku+v(p)+kN−v(p), . . .,kN(p) to the reduced applications in L[p]Nr \(L1⊕L2)r. Given l ∈ (L1⊕L2)r, let mo(l) be18

a minimum order application ρL1-equivalent to l. We will then have

S(mo(l)

)= S(l); l ∈ (L1⊕L2)r. (4.23)20

Reasoning as for the previous section, we show that

P(

1pu × pN

)=

[1√pN−u

1pN−u : A∗(l)′; l ∈ (L1⊕L2)r]′

(4.24)22

is an orthogonal matrix associated to the CJA A( 1

pu × pN)

with principal basis

N

(1pu × pN

)=

{1

pN−u JpN−u , Q∗(l)}

(4.25)24

where Q∗(l) = A∗(l)′A∗(l), l ∈ (L1⊕L2)r. For a more general discussion of this problemsee Mukerjee and Wu (2006).26

4.4. Binary operations and factor merging

Crossing and nesting prime basis factorials and/or their fractional replicates is quite28

straight forward. Only two points need to be considered in detail. Firstly, when we crossmodels for which there is confounding, the blocks will be the cartesian products of the30

original ones. This is, in each of the new blocks, we have all the combinations between thetreatments in a block of each of the original models. If confounding had only been used32

for one of the original models the treatments of the other are treated as constituting a block.Thus in each of the new blocks we will have all the combinations between the treatments in34

a block and all the treatments of the model without confounding.

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The second point also refers to model crossing. Let us assume the factors in the twooriginal models had p1 and p2 levels. Then we may merge one factor in each of the models2

to obtain a factor with p1 p2 levels. Let there be pNl−ull treatments, l = 1,2, in the two models

and Al(Ll) be the matrix associated to the effects of the two factors, l = 1,2. Thus, after the4

merging to the new factors will corresponding the matrix

A′ =

A′

1(L1)⊗ 1√pN2−u2

2

1p

N2−u22

:1√

pN1−u11

1p

N1−u11

⊗A′2(L2) : A′

1(L1)⊗A′2(L2)

. (4.26)6

When we replace the sub-matrices by A′ we replace A1⊗A2 by a CJA compatible withthis factor merging. Likewise we may merge factorial interactions taking Al(Ll), l = 1,2,8

to be the corresponding matrices in the two initial models.This merging of factors and factorial interactions enables us to overcame the limitation10

that the factors had to have a prime or the power of a prime number of levels.

References12

Cochran, W. G., 1992. Experimental designs. 2nd edition. Wiley Classics Library. New York.Dey, A., Mukerjee R., 1999. Fractional Factorial Plans. Wiley Series in Probability and Statistics.14Fonseca, M., Mexia, J.T., Zmyslony, R., 2006. Binary operations on Jordan algebras and orthogonal normal

models. Linear Algebra and its Applications, 417, 75–86.16Jordan, P., Von, Neumann J., Wigner, E.P., 1934. On an algebraic generalization of the quantum mechanical

formalism. Annals of Mathematics, Series II, 35, 29–64.18Mexia, J.T., 1988. Standardized orthogonal matrices and the decomposition of the sum of squares for

treatments. Trabalhos de Investigação N◦2, Departamento de Matemática, FCT-UNL.20Mexia, J.T., 1995. Introdução à Inferência Estatística Linear. Edições Universitárias Lusófonas.Montgomery, D., 1997. Design and Analysis of Experiments. John Wiley & Sons.22Mukerjee, R., Wu, C.F., 2006. A Modern Theory of Factorial Designs. Springer.Seely, J., 1970. Linear spaces and unbiased estimators. Application to a mixed linear model. The Annals of24

Mathematical Statistics, 41(5), 1735–1748.Seely, J., 1971. Quadratic subspaces and completeness. The Annals of Mathematical Statistics, 42(2), 710–26

721.Seely, J., Zyskind, G., 1971. Linear spaces and minimum variance estimation. The Annals of Mathematical28

Statistics, 42(2).Seely, J., 1977. Minimal sufficient statistics and completeness for multivariate normal families. Sankhya,30

39, 170–185.Seely, J., 1971. Quadratic subspaces and completeness. The annals of Mathematical Statistics, 42(2), 710–32

721.Steeb W. H., 1991. Kronecker Product and Applications. Manheim.34VanLeeuwen, D., Seely, J., Birkes D., 1998. Sufficient conditions for orthogonal designs in mixed linear

models. Journal of Statistical Planning and Inference, 73(1-2) 373–389.36VanLeeuwen, D., Birkes D., Seely, J., 1999. Balance and orthogonality in designs for mixed classification

models. The Annals of Statistics, 27(6) 1927–1947.38

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