journal of statistical planning

Journal of Statistical Planning and and inference ELSEVIER Inference 64 (1997) 353-367

Optimality of neighbour balanced designs when neighbour effects are neglected

Jean-Marc Azais*, Pierre Druilhet Laboratoire de Statistique & Probabilitds, UMR CNRS C5530, UniversitO Paul Sabatier, 118,

route de Narbonne, 31062 Toulouse Cedex. France

Received 9 February 1995; revised 21 October 1996

Abstract

The purpose of this paper is to study optimality properties of neighbour balanced designs when neighbour effects are not taken into account in the analysis model. Two types of designs for t treatments are considered: designs with t - 1 blocks of length t and designs with t blocks of length t - 1. When the designs are not randomized, new criteria on the bias matrix are introduced. When they are randomized, bias and variance are used in the comparison. In both cases neighbour balanced designs are shown to be optimal. @ 1997 Elsevier Science B.V.

A M S classification: 62K05; 62K10; 05B20

Keywords: Universal optimality; Neighbour-balanced designs; Bias optimality; Block designs; Circular designs; Randomization

1. Introduction

In many experiments (especially in agriculture), the response on a given plot may

be affected by treatments on neighbouring plots as well as by the treatment applied

to that plot. Azais et al. (1993) give many examples o f such situations, and they

give a catalog of circular neighbour balanced designs to be used. The situation differs

depending whether the neighbour effects are incorporated into the analysis model or

not. I f they are, it has been shown by Druilhet (1996) that neighbour balanced designs

are universally optimal in Kiefer 's sense (1975) among a wide class of competing designs (to establish this optimality, the definition o f neighbour balance used is stronger

than the one used in this paper). Often neighbour effects are actual but rather small;

* Corresponding author.

0378-3758/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved P H S 0 3 7 8 - 3 7 5 8 ( 9 6 ) 0 0 2 1 3 - 3

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incorporating them into the analysis model increases considerably the parametric di- mension of the models. That causes a diminution of residual degrees of freedom and an augmentation of variance of the treatment estimator. So, neglecting them by classical analysis of variance may be preferable. In that case, neighbour effects induce a bias on treatment estimates and, for randomized designs, an additional variance that we may wish to minimize by choosing carefully the design. In the present paper, we show that neighbour balanced designs are optimal for this purpose.

All designs are assumed to be in linear blocks, with neighbour effects only in the direction of the blocks (say left neighbour and right neighbour effects). Because the effect of having no treatment as a neighbour differs from the neighbour effect of any treatment, we consider only designs with border plots: in each block, two plots (the border plots) which receive treatments are added at both ends, but are not used for measuring the response variables (see Langton, 1980 for the interest of such designs). We call inner plots, those plots which are not the border plots. Moreover, we restrict our attention to circular designs in the sense of Definition 1.2 below.

A design d is defined as a function from the set of plots to the set of treatments. We denote by d( i , j ) the treatment assigned by d on the jth plot of the ith block; d(i,O) and d ( j , k + 1) are the treatments allocated to the border plots. We denote by 12(t,b,k) the set of all circular designs with t treatments and b blocks of size k, where size refers to the number of inner plots.

Let Yij be the response obtained on the plot ( i , j ) . Then all the observations are assumed to be uncorrelated with common variance; we propose three models for the expectation, depending on whether the orientation is taken into account:

(~ ' 1 ) F_(yij) = I~+fli+~d(i,j)+ 2d(i,j_l).

(J//2) ~-(Yij) = I~ + fli + Za(i,j)+ 2d(i,j--I) + Pd(i,j+l).

(~ '3 ) E(Yij) = I ~ + fli + za~i,j)+va(i,j-l)+Vd~i,j+l).

for l<~i<~b, l<<.j<~k.

Model (o//gl) corresponds to only one neighbour (say left) effect. It is particularly adapted to temporal problems with carry-over effects. Model ( J / 2 ) corresponds to specific additive influence from each neighbour. Model (~/3) corresponds to the same additive influence from each neighbour. These models are called true models.

We denote by Y, resp. B, Td, Ld, Rd the vector of observations, the incidence matrix of blocks, treatments, left-neighbours, right neighbours.

The unknown parameters have the following meanings: /2 is the general mean, ]3i is the ith block effect, zi is the effect of treatment i, 2i is the left-neighbour effect of treatment i, p~ is the right-neighbour effect of treatment i and vi is the neighbour effect (left and right) of treatment i.

We add the classical identification constraints:

t b t t t

~ z , = ~]~i = ~ 2 i = ~ P i = ~'~vi = 0. (1.1) i = l i - -1 i --1 i --1 i = l

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Remark. If only model ( J / l ) is used, right border plots are not necessary.

Definition 1.1 (Kiefer, 1975). A design d is said to be a Binary Block Design (BBD) if for all i,j: ndij E {0, 1} where ndij is the number of occurrences of treatment i on inner plots of block j.

Definition 1.2. A design d is said to be a Circular Block Design (CBD) if the treatment on each border plot is the same as the treatment on the inner plot at the other end of the same block.

Remark. In such designs, we have d(i,0) = d( i , k ) and d ( i , k + 1) = d(i, 1).

Definition 1.3. A design d is said to be a circular binary block design (CBBD) if it

is a binary block design and a circular design.

Definition 1.4. A design d is said to be a circular neighbour balanced block design (CNBD) if it is a CBBD and if, for each ordered pair of distinct treatments there exists exactly one inner plot which receives the first chosen treatment and which has the second chosen treatment as (say) right neighbour.

In this paper, we restrict our attention to the optimality of CNBD for two types of designs: (t - 1 ) blocks of size t, and t blocks of size (t - 1 ). Indeed, neighbour balance implies that the number of replications is divisible by t - 1 and an experimenter has rarely the resources for 2 ( t - 1 ) or more replications. Moreover, CBBD of the first type are complete block designs and equireplicated CBBD of the second type are balanced incomplete block designs (BIBD). Besides, if the block size is less than t - 1, neighbour balanced designs are, in general, not balanced incomplete block designs, and valid randomization does not exist for a neighbour-balanced design that is not a balanced incomplete block design (see Monod and Bailey, 1993). Note that some examples of constructions of neighbour balanced designs that are BIBD with small block size are given in Martin and Eccleston (1991), but they have relatively a great number of blocks; for certain values of the number of treatments and neglecting orientation (model (Jr'3)), neighbour balanced designs that are BIBDs with block size ½(t+ 1) can be constructed using the l-terraces defined by Morgan (1988).

In Section 2 we first define a global criterion of optimality for the bias introduced by neighbour effects, and then we show that CNBD's are optimal for this criterion. In Section 3, we consider restricted randomization of designs; we compare the usual randomization of a block design with the randomization of a CNBD introduced by Aza'is (1987). It is shown that in both situations, the bias is the same, but in the second case, the matrix of variances of the treatment estimators is lower in the Loewner sense.

356 J.-M. Aza?s, P. DruilhetlJournal of Statistical Planning and Inference 64 (1997) 353-367

1.1. Es t imat ion o f t reatments

We denote by ~at the set of labels of blocks containing at least once treatment l in design d, that is the set {i E { 1 . . . . . b} such as d ( i , j ) = l for some j E { 1 . . . . . k} ).

Ignoring the presence of neighbour effects, the intra-block least-square estimators of treatment effects are (see, e.g., Guttman, 1982):

t = (T~ pr±(B)Ta)+T~ pr±(B)Y, (1.1)

where pr±(A) is the orthogonal projection onto the subspace Im(A) x, and A + is the Moore-Penrose inverse: the choice of this generalized inverse ensure that the constraint (1.1) is satisfied.

For the two types of considered designs, estimators are given by the following more tractable formulas (Cochran and Cox, 1950):

(a) if the design is a complete block design ( t - 1 blocks, t plots per block):

1 t 1 1 [ l - - ~ Yil t( t - - 1) ~ Yi j . (1.2)

t - 1 i=1 l<~i<~t 1 <~j<~t--1

(b) if the design is an incomplete block design (t blocks, t - 1 plots per block)

t , )] [I -- t( t - 2~) i Yil t -- 1 Yij • (1.3)

1.2. No ta t ions

We denote by ~ the vector with entry i equal to [g and by ~ the vector of ones and by I the identity matrix, by J the matrix of ones with suitable dimensions.

If v is a random vector, we denote by Var(v) its variance matrix (also called variance--covariance matrix). The usual variance and covariance of a random scalar variable are denoted by var and cov.

2. Bias optimality of neighbour balanced designs for non-randomized designs

In this section, it is shown that the vector of biases is minimum in a global sense defined below.

2.1. Bias calculation

The vector of bias introduced by the neighbour effects is denoted by b. It depends linearly on the vectors of neighbour effects 2, p or v. So we can define two (t x t)

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matrices Md and ~/d such that we have:

for model (~#1): b =Ma2,

for model (J#2): b = M a 2 + ~4dp ,

for model (J#3): b = (Ma +-~ld)v.

Matrices Ma and A~td are given by the following formulae:

Md = (T~pr-L(B)Td)+T~ pr±(B)Ld, (2.1)

/~¢d = M~. (2.2)

Remark. Because of the constraint ~I=1 ,)~i = 0 (resp. E l= , Pi = 0), the matrices Md (resp. -~/a) are non unique: we can freely add a scalar multiple of the matrix J to them. So, in order to get unicity, we impose the following condition:

Man = 0. (2.3)

Note that the choice of the Moore-Penrose inverse in formula (2.1) ensures the condition above holds when the design is equireplicated. Note also that the formula (2.2) holds because of the circularity of the design and because of choice of the Moore-Penrose inverse.

We now define the (t × t) matrix of neighbourhood Nd as follows: Nao is the number of occurrences of treatment l in the inner plots preceded by treatment j (possibly in the border plots) (1 ~< l ~< t; 1 ~<j ~< t). Note that, because of the circularity of the design, we have the following: Najt is the number of occurrences of treatment 1 in the inner plots followed by treatment j (possibly in the border plots).

For the two types of designs considered here, we can state again formula (2.1) by using the neighbourhood matrix:

(a) for complete block designs:

i N. 1 j . (2 .4 ) = d - t

(b) for incomplete block designs with t block of size t - 1:

Md =/17/~ - t - 1 1 1 - - J . (2 .5 ) t(t - -~Nd t(t -- 2) I t

Example . For a CNBD design, we have

Nd = (J - I ) .

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2.2. No uniform optimality

Consider now the following design dl with its neighbourhood matrix Nal:

B.P. InnerPlots B.P. ( 0 0 l 1 i /

I l(i3 4 ]il 001 dl = 3 4 2 and Nd, = 2 2 0 0 . 3 1 4 1 1 2 0 3 4 1 1 1 0 2

If one chooses vector (+ 1, - 1,0, 0, 0) for the values of 2, p and v, then the bias vanishes in any of the three models. That means that we cannot expect a CNBD to be uniformly bias-optimal, where uniformly means that the norm of the bias is minimal whatever the vectors, 2, p or v are. So we are led to introduce another criterion, following the same idea as in Kiefer (1975).

2.3. Universal bias optimality criteria

We first define a property of optimality for the bias.

Definition 2.1. A design d* is said to be universally bias-optimal over a class ~ of competing designs if the matrix Ma* defined by (2.1) minimizes ~(Ma) for every function ~ from the set of all (t x t) matrices with zero row sums to (-o~, +c~] satisfying:

(a) ~b is invariant under any permutation applied simultaneously to rows and columns. (b) ~(~I4)~<~(M) if I~1 < 1. (c) • is convex.

Remark. Condition (a) means that criteria do not depend on the way treatments are labeled. Condition (b) means that criteria are non-decreasing.

We refer to Shah and Sinha (1989) and to Pukelsheim (1993) for discussions about similar optimality criteria in the case of information matrices.

As in Kiefer (1975), we have the following proposition:

Proposition 2.3.1. Consider a class ~ of designs with the same size. I f there exists a design d* in ~ such that • Ma. is completely symmetric (c.s).

• ItrMa • I = minae~ ItrMa[. then d* is universally bias-optimal over the class ~ of competing designs.

Proof. The proof is similar to that in Kiefer (1975): We denote by St the symmetric group of permutations of { 1 . . . . . t}, and by PC the permutation matrix corresponding

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to the permutation a on St. Using (a) then (c) in Definition 2.1, we have

V d E ~ ~ ( M a ) = l Y'~" ~(P~MaP~)~qb( 1 y'~ ) t! ~cs, fi" ~s, P~ MdP~ "

But ll)d = 1 ~ r ~ t M , p , 2-~a~s~ ra a ~ is completely symmetric with zero row and column sums thus it is proportional to Ma.. Moreover, it has the same trace as Md so tr(Md. )AT/d = tr(Ma)Md*. Applying (b) of Definition 2.1 gives the result. []

We now give two examples of classes of functions ~ under the models (~ '1)-( , / / /3) : The first class of functions is given by setting:

for model (~ '1 ) ~ ( M ) = I lgl l ,

for model (Jg2) 4~(M) = II(MIM')[I ,

for model (Jg3) ~ ( M ) = IIM + M'II,

where 1[ [I is any norm onto the space containing the bias matrices that is invariant by coordinate permutations.

The second class is defined by using the vector of positive singular values a(M) of the matrix of bias, defined such that al(M)~>az(M)~> . . . >~at(M)= 0. Let g be a non-decreasing convex function, we define:

t--1

for models (~/1) ~g(M) = ~ g(ai(M)), i=1

t--1 t - - I

for models ( ~ 2 ) ~g(M) = ~ g(ai(M)) + g(ai(M')) = 2 ~_~ g(ai(M)), i=1 i--1

t--1

for model (Jg3) ~o(M) = ~_, g(ai(M + M')). i : 1

For p ~> 1, and choosing g(x) = x p, we have the well known criteria (given here only for model (~ '1)) :

(19p(M) ( t l l t_l )l/p = ~ aP(M) i=1

Remark. Other examples of criteria can be constructed from the book by Marshall and Olkin (1979), using other non-decreasing Shur-convex functions applied to a(M).

We now give (only for model ( J / l ) ) some particular cases of these criteria which have concrete interpretations; we denote by I1" II the euclidean norm and by S* the set {xE~t/~xi-=O and [Ixll= 1}:

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(a) We first define the average bias AB(M) as the average norm of the bias summed over S*:

AB(M) = ~ ( ~ 2 ( m ) ) 2 = I ]lM21l dp*(2), JS

where/~* is the uniform measure onto S* and ~ is a real constant. (b) The maximal bias MB(M) is defined as the norm of the bias for the "worst" 2,

which leads to a minimax criteria:

MB(M) = q ~ ( M ) = al (M) = max IIMxll. xCS*

2.4. Optimality of CNBD

We now can state the main result of this section.

Proposition 2.4.1. Under model (Jr/ l) , resp. ( J / 2 ) and (J#3), a CNBD is universally bias optimal over the class of CBBDs with the same size.

Proof. It has been seen that for a CNBD, the neighbourhood matrix is completely

symmetric, so the matrix of bias is c.s. too. In addition because no treatment can be preceded by itself, the trace of the neighbourhood matrix of any CBBD is zero. Conse- quently, the matrices of bias of all CBBDs have the same trace. So by Proposition 2.3.1, a CNBD is universally bias-optimal. []

Remarks. The results of the proposition above are still true for a wider class of criteria: analysing the proof of Proposition 2.3.1, we can note that in fact the condition (b)

in Definition 2.1 is not necessary, since all matrices have the same trace. However this condition is natural and is necessary if we want to consider the optimality over a broader class of competing designs.

Under model ( ~ 3 ) , the neighbour balance condition can be weakened, using the fact that orientation is not taken into account. Neighbour balance is then defined by "each treatment (on an inner plot) has each other treatment equally often as neighbour

(right or left)".

2.5. Extension of the class of competin9 designs

We briefly discuss here the non bias-optimality of a CNBD over the whole class of equireplicated designs.

For a block design which is equireplicated but not binary, the matrix of bias has a more complicated form. Formulae (2.4) and (2.5) no longer hold and we have to use (2.1) which is less tractable.

We give below two designs which are equireplicated, but which have a bias matrix with a trace lower in absolute value than that of a corresponding CNBD. So, because

Z-M. Aza~s, P. DruilhetlJournal of Statistical Planning and Inference 64 (1997) 353-367 361

the trace is a ~-criterion of Definition 2.1, a CNBD is not universally bias optimal

over the class of equireplicated designs. Optimality of CNBD could be investigated

over the class of equireplicated designs with no treatment preceded by itself, or for the ~p criteria (for every p/> 1 ) only. However it is very technical and so it is not be developed in this paper.

Designs whose bias matrices have lower trace (in absolute value) than a CNBD:

B.P. Inner plots B.P. B.P. Inner plots B.P.

[i 3 4 Ii 3 3 1 [i] 3 4 2 1 1 4 4 " 4 5 3 1 1 2 5

d2 d3

The diagonal terms of the matrix of bias for a CNBD of the same size are ( - 0 . 2 , - 0 . 2 , - 0 . 2 , - 0 . 2 , - 0 . 2 ) ; the absolute value of their sum is equal to +1. For d2, the diag-

onal terms of the bias matrix are ( - 0 . 1 0 , - 0 . 0 5 , - 0 . 0 3 , - 0 . 1 0 , - 0 . 0 7 ) ; the absolute value of their sum is equal to 0.34. For d3, the diagonal terms of the bias matrix are (-0.03, +0 .00 , -0 .06 , -0 .09 , +0.17); the absolute value of their sum is equal to 0.02. These two examples show that two cases can occur: first, each diagonal term can be "better" than the CNBD's one, and second, there is a compensation between positive

and negative terms.

3. Optimality of restricted randomized CNBD

In this section, two randomizations are compared: it is shown that the biases intro- duced by neighbour effects are the same under classical and restricted randomization, but that under the classical one, the variance of the treatment estimators is greater in the Loewner sense than with the restricted one. Moreover, the strata decompositions are the same in both cases.

3.1. The unrandomized model

We give here the "conceptual model", which does not correspond to an actual ex-

periment: we assume only additivity of plot effects and treatment effects. We suppose that the yield z on a plot j of block i with treatment l is:

z(l, i , j ) - -Zl + fli + Uij + "neighbour effects".

where Uij is the constant effect of the plot (i,j). Moreover, we add the constraints:

k Vi= 1 . . . . ,b: ~ Uij=O. (3.1)

j= i

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3.2. Description o f the randomizations

Considering circular designs, randomization needs two steps: first we randomize the designs considering only the inner plots as described below, and then treatments are allocated to the border plots to make the design circular.

It is necessary here to introduce the notion of plot-names and actual plots (Bailey, 1981): we suppose given a virtual design d, where d(i , j ) is a treatment-label assigned to plot-name (i,j). The randomization can be described using types I and II among the three types defined by Preece et ah (1978):

Type I: In this type of randomization, the correspondence between plot-names and actual plots is randomly determined: we denote by q5 the random permutation of the blocks and by Oi the random permutation of the inner plots in block-name i.

Type II: In this type, the correspondence between the treatment-label and the actual treatment is randomly determined: we denote by ff the random permutation of treatment label.

Type III: In this type, theplan is selected randomly from a set of designs. It corre- sponds to a generalization of types I and II.

We denote by Sn the group of permutations on {1 . . . . . n}. (a) Classical randomization of a CBBD. Only type I is used here. The randomization consists of: (i) randomizing completely the blocks: q5 is drawn uniformly in Sb,

(ii) randomizing completely each block (only the inner plots) independently: for i in {1 . . . . . b}, 0 i is drawn uniformly in Sk.

(b) Restricted randomization of a CNBD. Here type I and type II are used to ensure strong validity of the randomization in

the absence of neighbour effects in the model (see Aza'is, 1987). The randomization consists of:

(i) randomizing completely the blocks, (ii) randomizing each block independently by a circular intra-block permutation of

the inner plots: for i in { 1 . . . . . b}, Oi is drawn uniformly in the subgroup of Sk formed by all the circular permutations.

(iii) randomizing completely the treatment labels. ~, is drawn uniformly in St. Steps (i)-(iii) may be carried out in any order.

3.3. The randomized models

We now give the general form of the randomized models: particular forms is given for each case. We denote by d C [~2 the set of plot-names. We suppose that plot-name (i, l) receives treatment l, and corresponds to the block-name i:

For designs with t complete blocks, d can be choosen as:

~¢={1 . . . . . t -- 1} × {1 . . . . . t},

J.-M. Azai's, P. DruilhetlJournal of Statistical Planning and Inference 64 (1997) 353-367 363

For designs of t blocks of size t - 1, ~¢ can be choosen as:

~ = { ( i , l ) / i ~ l , l<~i<~t, l<~l<~t}.

flil is the effect of the actual plot assigned to the plot-name (i, l), /~i/ the effect of the block assigned to the plot-name (i, l), ,~.i/ the left-neighbour effect assigned to the plot-name (i, l), Pit the right-neighbour effect assigned to the plot-name (i, l), and ~it the sum of the right and the left-neighbour effect assigned to the plot-name (i, l). Note that fli/, ,~it, fii/, vi/ are random variables the distributions of which depend on the type of randomization.

Denoting by yi/ the value observed on the plot-name (i, l), we now can write the general form of the randomized model:

(J//1 ) yi/='c/÷flil÷~i/÷fli/÷eil, (J//2) Yi/ = "C/ ÷ fli/ ÷ Si/ ÷ g~ ÷ flit q- ei/,

( J /3 ) Yi/ = "~1 ÷ fli/ ÷ V i l ÷ fli/ q t- ei/, for (i , j) E ~¢,

where ei/ are measurement independant errors with common variance O'e and indepen- dent of the randomization.

3.4. Designs with t - 1 blocks of size t

We compare here the bias and the variance of the treatment estimator under each randomization. Using the estimator (1.3) and because of the constraints (1.1), we have for all IE{1 . . . . . t}:

(X/l) tl = 27/ -~ ~ [EI~-~ Si/ ÷ ~I~--~ flil] ÷ "measurement errors".

t - I ~ t - t (~'2) t/ = ~/ -~- t--~ [ E i = I (~i/ ÷ Pil) ÷ E i = I fli/] ÷ "measurement errors".

(.///3) / ' /= z / + ~ [ ~--]~I-~ vi/+ Y]I-~ flu] + "measurement errors".

("measurement errors" means here a certain linear function of ei/'s).

3.4.1. Bias and variance under classical block randomization It is well known that for complete block designs, block randomization is not neces-

sary, so the classical block randomization assigns to the plot-name (i, l) the actual plots (i, Oi(l)). Since Oi is drawn uniformly in Sk, Oi(l) is drawn uniformly in {1 . . . . . t}. 2i/, Pil, ~:il are the effect of the neighbours of treatment ! in block i.

Proposition 3.4.1. The bias vector b of treatment estimators is

- 2 / ( t - 1) under model (Jgl) , = - (2 + p)/(t - 1) under model (Jg2),

- 2 v / ( t - 1 ) under model (J/¢'3).

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P r o o f . It suffices to establish the following relations:

- 2 t . - P t . ~-vil- - 2v t ~-~lil : O; []=~il = t -- 1' [Ffiil = t~- l ' t - 1"

Consider for example ,~il, properties of the randomization imply that it is chosen at

random among {2t,; l ' ~ l } , thus using constraints (1.1), we have E~il : - ~ l / ( t - l). E/~it and ~-Yil can be obtained in the same manner. Efia = 0 comes from the fact that the plot o f block i assigned to treatment l is chosen at random among all the plots o f

block i. []

P r o p o s i t i o n 3.4.2. The (t × t) variance matrix Var(~) of treatment estimators satisfies:

Var( / ) = Ve + 1I, + Vu.

where Ve is a variance matrix resulting f rom measurement errors only. It is not modified by the randomization. Vn is the variance matrix resulting f rom neighbour

effects only. Its expression is very complicated, especially when there are various neighbour effects (models s/¢2 and J43). V~ is the variance matrix resulting f rom plot effects only: we denote by ~ the vector with component ( ~ I - ~ 5it)t=l,...,t, then

V~ = V a r ( e ) = au 2 _ _ _ ~ Ui~. ~ - 1 I - where a u ( t - l ) 2 i,j

P r o o f . The fact that measurement errors are uncorrelated with neighbour or plot effect

comes from the fact that they are independent from the randomization. The calculation of V~ is classical. Thus it remains to show that neighbour effects are uncorrelated with

plot effects: - I f i ¢ i' then for all l and I', cov(~.a, fii, t , )= 0 because plots are randomized inde-

pendently between blocks. - I f i = i t, we can split up each permutation Oi into two permutations: a unique circular

permutation 7i and a unique permutation ai such that ~i(1 ) : 1 and 0 i = O~ i 0 ])i. Thus:

COV(~il,~lil,) : ~(~(~il~lil,/Oli)) -- 1 t t ( t - - 1) ~ 2s ~ Uij=O" s~l j : l

The second equality comes from the fact that 2ij depends only on 0~ i and because of the constraints (3.1). By the same way, we have: Vi, i', l, l t, cov(~il, fiit, ) = 0 and

COV(Yil, ~liU ) = O. []

3.4.2. Bias and variance under restricted randomization P r o p o s i t i o n 3.4.3. The bias vector o f treatment estimators for the restricted random- ization is the same as the one for the classical randomization.

P r o o f . We remark that the probability for a given plot to receive a given treatment depends neither on the treatment label nor on the plot label (so H:fia = 0). In addition,

J.-M. Azai's, P. DruilhetlJournal of Statistical Planning and Inference 64 (1997) 353-367 365

all possible outcomes of the randomization are CNBD, ~ i - ~ ~.it (resp. with /~it and vit) is a non random variable equal to -21 (resp. - P t and -2vl). []

Proposition 3.4.4. The (t × t) variance matr ix o f treatment estimators is

Var(i)----- V~ + lie,

where Vu and Ve have the same expression as in Proposition 3.4.2.

Proof. The absence of Vn comes from the fact that the sum of neighbour effects in i is non random. Thus it remains to check that the expression of Vu is the same as in Proposition 3.4.2: following Bailey and Rowley (1987), we just have to note that the probability for any two ordered given treatments (l, l ~) to be affected to any two distinct ordered actual plots (~,//) is equal for both randomization. []

3.5. Designs with t blocks o f size t - 1

We obtain here results similar to those of Section 3.4. We give first the expressions of it for all l C { l . . . . . t}:

t--1 (J¢'l) it = zt + ~ [ ~ i¢ t Sit - ~ + ~ i ¢ l Uil] + "measurement errors".

t - 1 21 +Pt ( . g 2 ) it = rt + ~ [ ~ i 4 t (Sit + ~it) - 7 - 1 + ~ i 4 t ~.]

+ "measurement errors".

t--1 [ ~ i ~ t ~it -- t~l -~ ~iT~t Uii] -r "measurement errors". ( ~ 3 ) it = st +

3.5.1. Bias and variance under classical block randomization

Unlike for complete block designs, it is necessary here to randomize the blocks to ensure that the randomization is valid. So the randomization assigns to the plot-name (i, l)i4l the actual plot (~b(i), Oi(l))i~l where Oi(l) and ~b(i)) are drawn uniformly resp. in { 1 . . . . . t - 1 } and in { 1 . . . . . t}). Sit, Pit, vit are the effect of the neighbours of treatment l in the block qS(i).

Proposition 3.5.1. The bias vector b o f treatment estimators is

-2 / ( t - 2) under model (Jgl) , = - ( 2 + p) / ( t - 2) under model (~'2),

- 2 v / ( t - 2) under model (~/¢3).

Proof. The proof is similar to that of Proposition 3.4.1. []

Proposition 3.5.2. The (t x t) variance matr ix o f treatment estimators is

Var(i)----- 17u + ~ + l?e,

where Ye is the variance matr ix due to measurement errors, which does not depend

on the outcome o f the randomization. Yn is the variance matr ix due only to neighbour

366 J.-M. Azai's, P. DruilhetlJournal of Statistical Plannino and Inference 64 (1997) 353-367

effects. Its expression is very complicated especially when there are various neigh- bour effects (models (,///2) and (,///3)). IT"u is the variance matrix due to plot effects: we denote by ~ the vector with component (~"]~i-~ fia)l=l,...,t, then

~ u = V a r ( a ) = ( t - 1 ) 6 2 ( J ) .2 1 1 where o u - ~ Ui~.

t - 2 t(t - 1) i,j

Proof. As in Proposition 3.4.2, we only show that neighbour effects and plot effects are uncorrelated. - If i ¢ i I then for all j and f , cov(2a,~i,e)= 0 because plots are randomized inde-

pendently between blocks. - If i = i t, as in Proposition 3.4.2, we can split each permutation Oi into two per-

mutations: a unique circular permutation ~i and a unique permutation ~i such that

0~i (1 ) = 1 and Oi =~i o ])i. Thus:

l t t - I

COV('~it, fiu )= ~-(E(~-(~il~lil'/O~i, ~)) /Cp))- t(t 1) ~ ~ ~ "~.i' ~ Uij = O. - - i=1 i ' l l j = l

By the same way, we have: Vi, i t, l, l', cov(~a, fiil, ) = 0 and cov(?it, flit' ) = 0. []

3.5.2. Bias and variance under restricted randomization Proposition 3.5.3. The bias vector of treatment estimators for the restricted random- ization is the same as the one for the classical randomization.

Proposition 3.5.4. The (t x t) variance matrix of treatment estimators is

Vat(i) = V~ + Ve,

where Vu and Ve have the same expression as in Proposition 3.5.2.

Proof. The proof is similar to that of Proposition 3.4.2. []

4. Discussion

We have seen that for non randomized designs, neighbour effects cause treatment estimators to be biased. Neighbour balanced designs (in the sense of the paper) give optimal reductions of this bias.

For a randomized design, a certain part of the preceding bias becomes random and the remaining bias is the same as for neighbour balanced design, as if "in average" a randomized design was neighbour balanced. Nevertheless, the non neighbour balanced designs inflate the variance, so balanced designs are optimal for variance reduction.

Considering the results of Dmilhet (1996) concerning analysis with neighbour effects, Aza'is (1994), Kunert (1985) concerning analysis with AR models and Kiefer and Wynn (1981) concerning analysis with lag-one correlation, we can see that neighbour-balanced

J.-M. Azais, P. Druilhet/Journal o f Statistical Planning and Inference 64 (1997) 353-367 367

designs (as a vague notion to be specified in each case) have optimality properties for most of the analysis models.

References

Azais, J.-M. (1987). Design of experiments for studying intergenotypic competition..L Roy. Statist. Soc. B 49, 334-345.

Azais, J.-M. (1994). Design of experiments and neighbour methods, In: T. Califiski and R. Kala, Ed., Proc. Internat. Conf. on Linear Statistical Inference L INSTAT 93. Kluwer Academic, Dordrecht, 211-221.

Azais, J.-M., R.A. Bailey and H. Monod (1993). A catalogue of efficient neighbour-designs with border plots. Biometrics 49, 1252 1261.

Bailey, R.A. (1981). A unified approach to design of experiments. J. Roy. Statist. Soc. A 144, 214-223. Bailey, R.A. and C.A. Rowley (1987). Valid randomization. Proc. Roy. Soc. Lond. A 410, 105 124. Cochran, W.G. and G.M. Cox (1950). Experimental Designs. Wiley, New York. Druilhet, P. (1996). Optimality of neighbour-balanced designs. Unpublished. Guttman, I. (1982). Linear Models: an Introduction, Wiley, New York. Kiefer, J. (1975). Construction and optimality of generalized Youden designs. In: A Survey of Statistical

Design and Linear Models. North-Holland, Amsterdam, 333-353. Kiefer, J. and H.P. Wynn (1981). Optimum balanced block and latin square designs for correlated

observations. Ann. Statist. 9(4), 737-757. Kunert, J. (1985). Optimal repeated measurements designs for correlated observations and analysis by

weighted least squares. Biometrika 72(2), 375-389. Langton, S. (1990). Avoiding edge effects in agroforestry experiments; the use of neighbour-balanced designs

and guard areas. Agroforestry Systems 12, 173-185. Marshall, A.W. and I. Olkin (1979). Inequalities: Theory of Majorization and its Applications. Academic

Press, New York. Martin, R.J. and J.A. Eccleston (1991). Optimal incomplete block designs for general dependence structures.

Z Statist. Plann. Inference 28, 67-81. Monod, H. and R.A. Bailey (1993). Valid restricted randomization for unbalanced designs. J. Roy. Statist.

Soc. B 55, 237~51. Morgan, J.P. (1988). Balanced polycross designs. J. Roy. Statist. Soc. B 50(1), 93-104. Preece, D.A., R.A. Bailey and H.D. Patterson (1978). A randomization problem in forming designs with

superimposed treatments. Austral. J. Statist. 20, 111-125. Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York. Shah, K.R. and B.K. Sinha (1989). Theory of Optimal Designs. Springer, Berlin.