Journal of Statistical Planning and Inference 15 (1987) 265-278 North-Holland

265

O P T I M A L P A I R E D C O M P A R I S O N D E S I G N S F O R F A C T O R I A L A N D Q U A D R A T I C M O D E L S

E.E.M. van BERKUM Department of Mathematics and Computer Science, University of Technology, Eindhoven, The Netherlands

Received 15 January 1984; revised manuscript received 2 January 1986 Recommended by L.C.A. Corsten and Jaya Srivastava

Abstract: The construction of D-optimal designs in paired comparison experiments is considered. The procedures that are used in paired comparisons to estimate the parameters yield a covariance matrix that depends on the unknown parameters. The assumption of no treatment differences is made to construct designs. Methods similar to the ones used in the standard experimental situa- tion can also be used for the construction of paired comparison designs. Results are given for fac- torial and quadratic models.

AMS Subject Classification: Primary 62J15; Secondary 62K05.

Key words: Paired comparison; Bradley-Terry model; Optimal designs; Factorials; Incomplete block designs.

1. In~oduefion

The paired comparison experiment has t treatments or items, T l, . . . , Tt, with nij judgements or comparisons of T/and Tj, nij>_O, nii=O, nji=nij, i , j = l , . . . , t . Let ni. ij be the number of times T/has been preferred to Tj when Ti and Tj were com- pared. Bradley and Terry (1952) provided a paired comparison model, which postulates the existence of parameters, 7ti for T~, 7ti>0, such that the probability 7tij of selecting Ti when compared with ~ is

:rtij=Tti/(Yti-J- 7tj) ( i#: j ) . (1.1)

The parameters lti have to be estimated. Since (1.1) is scale-independent, a conve- nient scale-determining constraint

t in 7ri = 0, (1.2)

i=l

is used. Paired comparison designs can also be seen as incomplete block designs with

blocks of size two. As will be seen in Section 3, results concerning optimal designs

for paired comparisons arc also applicable in this case when the within-block dif-

ference of observations is chosen as response variable.

0378-3758/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)

266 E.E.M. van Berkum / Optimal paired comparisons designs

In the standard experimental situation, where instead of paired comparisons, we have direct observations, many results have been obtained in constructing optimal designs. However, fewer results are available in the case of paired comparison ex- periments and these are listed as follows:

Quenouille and John (1971) present 2n-factorial paired comparison designs, which can be constructed in order to reduce the number of pairs required by ignor- ing information on higher-order interactions. Their analysis assumes that the obser- vations have the same variance. Springall (1973) assumes that the/~i are functions of continuous independent variables, x l , . . . ,Xk measurable without error, and for- mulates a linear model

k

l n r t i= E x u f l j . (1.3) j = l

He gives so-called analogue designs which are paired comparison designs whose in- formation matrix is proportional to the information matrix of the design with the same points in the standard experimental situation. More generally Beaver (1977a, 1977b) considers the model

where

F(rt) = X f , (1.4)

P=(#l,---,#k)', n=(nl,...,nk)',

and X is a design matrix. However, the particular case F ( n )= (In nl, ...,In rtt)' is considered. EI-Helbawy

and Bradley (1978) give the covariance matrix in factorial paired comparison ex- periments and present optimal designs in some cases. Fienberg and Larntz (1976) give a log linear representation for paired and multiple Comparison models.

2. The information matrix

We consider the model (1.4), given by Springall. The inverse of the covadance matrix of the ML-estimators of Pi is asymptotically equal to the matrix M defined by

where

= E E nu xj,)(x, - xjq). i<j

(2.1)

tpij = 7tij ltji = i t i l t j / ( l t i + ltj) 2. (2.2)

The methods considered by Beaver (1977a, 1977b), EI-Helbawy and Bradley (1978) and Fienberg and Larntz (1976) yield the same matrix. This can be seen easily. EI- Helbawy and Bradley (1978) and Springall (1973) both use a maximum likelihood

E.E.M. van Berkum / Optimal paired comparisons designs 267

method, and Beaver (1977a: 1977b) uses a 'minimum- logit-z 2' procedure, which under weak conditions is asymptotically equivalent to the maximum likelihood method. Many criteria for constructing optimal designs are functions of the covariance matrix. However, the information matrix is a function of the unknown parameters lti and no estimates are available when the design is chosen. The covariance matrix can be simplified by making the assumption of no treatment dif- ferences

~i--1, i = 1, . . . , t . (2.3)

This is the assumption Springall (1973) and E1-Helbawy and Bradley (1978) make when they give applications of their results. With this condition the information matrix can be written as follows:

Mrq= ~ ~ ~ nij(Xir-X./r)(Xiq-Xjq ). (2.4) i<j

This matrix is proportional to the information matrix related to an ordinary least squares method in which the variances of the observations are assumed to be the same. In what follows, condition (2.3) is assumed to hold.

3. A method of construction of D-optimal paired comparison designs for response surfaces

We reformulate the model (1.3). Let x ~ X , the experimental region, X C [R",

f (x) = ( f l (x), f2(x), ... , fk(x)) ' ,

with fj : X ~ ~, continuous on the experimental region X. Then

In nx= f l ( x ) fll + "'" +j~(x) ilk,

or

In 1ix= ( f ( x ) ) ' f l .

A paired comparison design may be rewritten as a collection of variables

(Ul, Ol), (U2, 02), . . - , (U m, Om),

l id /~2~ """, ?/m~ NI

where

m

n i = N and ui, v i e X . i = I

(3.1)

(3.2)

the design should be interpreted as follows. For each pair (ui, oi), ni comparisons

268 E.E.M. van Berkum / Optimal paired comparisons designs

are made. So, a design is constructed by choosing both the pairs (ui, oi) and the numbers n i. This is a more general viewpoint, because in most cases the items and therefore the pairs (ui, oi) are given and only the ni may be chosen.

The design given above is denoted by ¢(N) or #. A normalized design e(N) or e is a design in which the n i are replaced by Pi, with Pi = ni/N. This design is called exact in distinction with discrete designs, which are designs where the Pi are real numbers satisfying

,m

Pi =1. i:=l

The above definition is analogous to the definition of a design in the standard ex- perimental situation. Moreover, when the difference between two items, or within- block difference, is chosen as respons variable y, themodeI can be reformulated in terms of the standard experimental situation as follows

where

y=(g(u , o))" fl, (3.3)

g(u, o ) = f ( u ) - f ( o ) and u, o ~ X .

Choosing design points in the case of model (3.3) is the same as choosing pairs for a paired comparison design. The information matrix can be expressed by

m

M(e)= Pi(f(ui)-f(oi))(f(ui)-f(oi))'. (3.4) i=1

Evaluating this expression shows that the matrix M(e) is proportional to the matrix given in (2.4). Fedorov (1972) gives the well-known definitions of criteria and pro- perties of optimal designs. We mention some of them. A design e* is called D- optimal when the following holds:

det(M(e*)) = max det(M(e)), (3.5)

in other words the generalized variance is minimized. A design e* is called G-optimal or minimax when the maximal value of the variance function is minimized. The variance function is a normalized measure of the estimated response. Therefore, us- ing model (3.3), the variance function d(u, o, e) can be expressed by

d(u, o, e ) = ( f ( u ) - ( f ( o ) ) ' M - l ( e ) ( f ( u ) - f ( o ) ) , (3.6)

a normalized measure of the variance of the estimated difference between the points u and o. So, a design e* is called G-optimal when

max d(u, v, e*)=min max d(u, o, e). u, o c X e u, o e X

Some well-known properties are given in Lemma 3.1.

(3.7)

E.E.M. van Berkum / Optimal paired comparisons designs 269

Lemma 3.1. (a) The following assertions are equivalent: (1) the design e* maximizes det(M(e)), (2) the design e* minimizes maxu, vexd(u, o, e), (3) maXu, v+xd(u, o, e*) = k.

The information matrices o f all designs satisfying (1)-(3) coincide. A linear com- bination o f designs that satisfy (1)-(3) satisfies (1)-(3).

(b) I f X is compact and the functions f (x ) are continuous, then a discrete D- optimal design can be found with the number of pairs m<_-~k(k + 1).

(c) The function d(u, u, e*) attains its maximal value k on the set o f pairs of an optimal design e*.

Now the well-known iterative procedure that uses the equivalence of D-optimality and G-optimality can be used to construct D-optimal designs. In Section 6 some results will be given for the case of a factorial model with main effects and first- order interactions. Some of the results of that section are known in literature. In Sections 4 and 5 the case of a quadratic response surface is discussed.

4. D-optimal designs in the ease of a quadratic response surface

The model considered is

In nx=fll Xl + "" + flnxn + fin x2 + ... + flnnx~,

-b ~12 Xl X2-]- ... "q- ~n_ lnXn_ l X n,

where

and

X = ( X I , . . . , X n ) ' , x E . X ,

(4.1)

with

E J M - 1 (e) = a I + ~3 ,

61

t~ = - 4 ~ ( 4 . 4 )

(4.3)

X={xelPn[-l<_xi<_l for all i}.

The number of parameters is 2n + (,~, i.e. k = ~n(n + 3) and according to Lemma 2t

3.1 a discrete D-optimal design can be found with m pairs, where

m <_+n(n + 1)(n + 2)(n + 3). (4.2)

It can be shown that the covariance matrix has the following structure:

270 E.E .M. van B e r k u m / Optimal paired comparisons designs

where y l is related to the main effects, a I + ~J is related to the quadratic effects, J1 is related to the interactions, I is the identity matrix, and J is a matrix with J0 = 1 for all i , j .

The designs constructed in this section have the structure given in (4.3) and (4.4). It will be proved that these designs are D-optimal by showing that the variance func- tion is bounded by k. Then it follows by using Lemma 3.1, that any D-optimal design has a covariance matrix with a structure given by (4.3) and (4.4). Let e be a design having a covariance matrix of the form (4.3) (not necessarily satisfying (4.4)). Now, using (3.6), one can express the variance function d(x, y, e) by

gi 17

a(x,y,e)= =r E (x,- yi)2 + E y )2 i= l i=l -

i= 1 i<j

It is easy to prove that this function satisfies

d((xl, . . . ,xi , . . . ,xn) , (Yl, "..,Yi, ." ,Yn), e)

- - d ( ( X l , - - ' , -- Xi , " " , X n ) , ( Y l , - - - , - -Y i , " " , Y n ) , e ) ,

and

(4.5)

(4.6)

d((xl, . . . ,xi , ...,Xy, ... ,xn), (Yl, "" ,Yi, .." ,Yj, ... ,Yn), e)

-- d ( (Xl , . . . , x j , . . . ,X i , . . . , X n ) , (Yl , "'" , Y j , "'" ,Yi, ... ,Yn), e), (4.7)

where 1 __ i_< n, 1 _<j < n. The condition (4.4) is equivalent to

d((xl, . . . , x i , . . . , x n ) , (Yl, ... ,Yi, ... ,Yn), e)

"--d((Xl, " " , Y i , " " , X n ) , (Yl , " " , X i , . . . ,Yn), e) (4.8)

for all x, y and 1 <_ i _ n.

When constructing D-optimal designs we have to search for pairs (ui, oi) where the function d(u, o, e) attains its maximal value. After having found such a pair, one can find more pairs by using (4.6), (4.7) and (4.8). Therefore, the following def'mi- tions are given.

Definition 4.1. The set S((x, y)) contains all 2 n pairs that can be found by multiply- ing pairs of coordinates (xi, Yi) of (x, y) by -1 . The set SP((x, y)) is the union of all sets S((p(x) , p(y))) , where p(x ) is a permutation of x. The information matrix of SP((x, y) is denoted by MP((x, y)).

Definition 4.2. The set SP(kl, k2, k3; o) is the set SP((x, y)), where the pair (x, y) is defined as follows:

E.E.M. van Berkum / Optimal paired comparisons designs 271

x = (1, . . . , 1)', y = (1, . . . , 1, - 1, . . . , - 1, o, . . . , 0) ' ,

and k I is the number o f l ' s in y , k 2 is the number o f - l 's in y,/(:3 is the n u m b e r

o f o's in y , - 1 < 0 < 1, kl+k2+k3=n. The set SPt (0, 0, n; o) is the set con ta in ing the pairs o f SP(0, 0, n; o) and all pairs

that can be found by replacing I pairs o f coordinates (x i, Yi) by (Yi, xi) as is done in (4.8). The i n fo rma t ion matr ices o f these sets are denoted by replacing the letter

S by the let ter M. I f k3 = 0, t hen we write SP(k 1, k2).

L e m m a 4.3. Let

(n - 1)! (4k2 + g k3)2 n, sl + tl (n - 1)! k3 gh 2", P l - kl ! ](:2! k3! kl! k2 .i ka!

h - 2 ( n - 2 ) [ (k23)gh2n, kl[ k2! k3[

2 ( n - 2 ) ! (4kl k2 + k2 k3 h + kl k3 g + ( k23 ) gh)2, ' z i - kl ! k2 ! ](:3 !

with g = (1 - w) 2, h -- (1 ÷ w) 2 . Then

and

Let

MP(kl'k2'k3;w)= I PlI SlI+tlJ Zl I ] if k3>O'

MP(kl,/(:2, k3; w)=-~ Pl I

Sll+tlJ Zl I ] if k3=O"

k#_l / j gh2",

z,=[(;)

272 E.E.M. van Berkum / Optimal paired comparisons designs

Then

and

MPt(0, 0, n; w) =

MPn/2(O, O, n; w)=

P2 1

s 2 I + t 2 J z21 ]

P21 s2I+t2J z21 ]"

if t.-~n,

Proof. These results can be proved by using the fact that SP(kl, k2, k3; o) is the union of n!/kl!k2! k3l sets S(p(x), p(y)), and that

MP((x, y) )=

E poI

sol+ toJ

ZoI where

n

Po=2n(n - 1)! ~ (xi-Yi) 2, i=1

n

So+to=2n(n" l) ! E (X/2-y2) 2, i=1

t o = 2 n + l ( n - 2 ) ! X X (x~i-y2)(x2-.~2), i<j

Zo=2n+l(n-2) ! ~ ~ (Xixj-YiYj) 2. i<j

For pairs (u, o) belonging to SP(kl, k2, k3; u) or SPI(0, 0, n; o) the function d(u, v, e) depends only on kl, k2, k3 and o, and is denoted by d(kl, k2, k3; v) or d(kl, k2) when k3=0. The function d(kl, k2, k3;o) can be expressed as follows:

d(k 1, k 2, k 3; v) = 4k 2 y + k 3 g y + 4lc 1 162 J + kl 163 g

+ k2 k3 h J + (k3) gh J + k3 gh a + k~ gh ~. (4.9)

This expression can be obtained by using (4.3) and (4.5). Now the idea is that one can find a D-optimal design by choosing a combination

of some sets defined above and suitable weights. Using the iterative procedure men- tioned in Section 3, one can find a combination that yields a D-optimal design as follows. Start with some combination of sets and compute weights, such that the determinant of the information matrix is maximized. Leave out, those sets for which the weights are not positive. Now, a design e0 has b ~ n constructed. Compute the

E.E.M. van Berkum / Optimal paired comparisons designs 273

aaaximal value of d(x, y, e0) and add those sets that contain pairs where d(x, y, to) attains its maximal value. Compute suitable weights as above and a new design e 1 aas been found. This procedure converges and the design found will appear to be D-optimal. The most difficult problem in this procedure is the determination of the aaaximal value of d(x, y, e). The following lemma reduces this problem to the com- 3utation of a countable set of values.

Lemma 4.4. Let e be a design with covariance matrix o f type (4.3) that satisfies the :onditions d = - 4~ and a - 2~ >_ O, and let v* be the value o f o where d(kl, k2, k3; v) lttains its maximal value f o r given k 1, k2 and k3. Then the following assertions are ,quivalent:

(i) (x*, y*) is a pair where d(x, y, e) attains its maximal value, (ii) there exists a combination 11, 12, 13 such that Ii + 12 + 13 = n, (x*, y*) belongs to

]P(ll, 12, 13; o*) or SPr(II, 12, 13; v*) f o r some r, and

d(ll, 12, 13; o*)>d(kl , k2, k3; w*)

¢or all k 1, k 2 and k 3 with kl + k2 + k3 = n.

3utfine of the proof. Let (x*, y* ) be a pair where d(x, y, e) attains its maximal value ~thx*=(x~ , . . . ,x*)" and y*=(y~ , . . . , y*) ' . It can be proved that Ix/*l = 1 or ly*l = 1 'or all 1 < i< n. Therefore, using the results given in (4.5), (4.6) and (4.7), we can tssume without loss of generality

x* = (1, . . . , 1)',

y * = ( 1 , . . . , l , - 1 , . . . , - 1 , y t * + t 2 + p . . . , y * ) ' for some 11 and /2 .

t then can be shown that y/* = yy* for all Ii +/2 + 1 _< i, j_< n, but the verification is • ather tedious and has been omitted here. Suppose y* = w for all 11 +/2 + 1 _< i_< n. ;ince d(x, y, e) attains its maximal value at (x* y*) , we have w = o, where o* is the ,alue of v where d(ll, 12, 13;o) attains its maximal value.

In Berkum (1985) a complete p roof can be found. In Section 5 some D-optimal designs will be given.

;. Some D-optimal designs fOr a quadratic model with interactions

The structure o f the D-optimal design found by the use of the procedure given n Section 4, depends on n. We have to distinguish the three cases:

(i) n_> 6, n even, (ii) n>_3, n odd,

(iii) n = 2 or n = 4 . ['hough case (i) is of little practical interest, it will be discussed more extensively as

274 E.E.M. van Berkum / Optimal paired comparisons designs

an example how the weights can be computed. This case is rather easy to show.

(i) n even , n >_ 6.

Consider the design, consisting of (a) the pairs of S ( l n - 1, ~n + 1) with weights Vl; the number of these pairs is

(n /~- 1) 2n- 1, a n d

d(-~n - 1, ~n + 1) = (2n + 4) y + (n - 2)(n + 2) ~;

(b) the pairs of S(-~n, 1 ~-n) with weights rE; the number of these pairs is n xgn-1 and

i,I/2 J'."

d(~n, -~n) = 2n y + n 2 ¢~;

(c) the pairs of SP(O, O, n;01) with weights/z and - 1 < 0 1 < 1; the number of these pairs is 2 n, and

d(O, O, n; ol)= n gl Y+(~)gl hi t~ + n gl hl a + n2 gl hl ~,

with gl = (1 - 01) 2, h 1 = (1 + ol) 2. (d) the pairs of SPn/2(O, O, n;01) with weights A; the number of these pairs is

( " )2"-1 n/2 , so the number of pairs of the design is

N = [ l + 3 n + 4 (½ n - l ) ] 2n"

The values of a, t~, ~, ol and the weights can be computed as follows. If the design is D-optimal, then d(x, y, e) attains its maximal value at the pairs of this design. This yield four equations:

(2n + 4) y + (n - 2)(n + 2) ~ =-~n(n + 3), (5.1)

4y + 2n t~ = n + 3, (5.2)

g l y + ~ ( n 1 ) g l h l J + g l h l o t + n g l h l - - ~ - T(n + 3), (5 .3)

fi= -4~. (5.4)

A fifth equation can be found by using the fact that the function d(0, 0, n;o) con- sidered as a function of o is maximal at o = ol; this yields

o ,=-~-+~- [ 1 - 2y 11/2 a+n~+~ (n -1 ) t~J " (5.5)

Solving the equations (5.1)-(5.5) we obtain

n + 3 ~, = t$ = 2n + 4 ' (5.6)

(1 - o l ) 3 + 2(n + 2) ol = 0; (5 .7)

the value of a can be computed by means of (5.3).

E.E.M. van Berkum / Optimal paired comparisons designs 275

Now M(e) is known and the weights vl, v2,/z and ). can be computed from the equation

M(e)= vl M P ( I n - 1, i n + 1)+ v2 MP(In , i n )

+/1MP(0, 0, n; 01) + ~. MP~n(0, 0, n; ol),

which yields four equations with four unknown variables. It has been verified that the weights are positive for n _ 20.

(ii) n odd, n_>3. Consider the design consisting of (a) the pairs of S( i (n - 1), i ( n + 1)) with weights Vl; the number of these pairs is

n ~2 n - l , a n d ( n - 1)/2 j

d(Itn - 1), i ( n + I)) = (2n + 2) y + (n - 1)(n + 1) ~,

(b) the pairs of SP(0, 0, n;o~) with weights/z and - 1 < o 1 < 1 ; the number of pairs and d(0, 0, n; ol) are as under (i) (c).

(c) the pairs of SPtn_ 1)/2(0, 0, n; ol) with weights ~; the number of these pairs is n ~2 n and the number of pairs of the design equals ( n - 1)/2 y '

3 n n

(iii) n=2, 4. Consider the design consisting of (a) the pairs of S(In, in) with weights •2,

(b) the pairs of SP(0, 0, n;vl) with weights/z, (c) the pairs of SPn/2(0, 0, n; Vl) with weights A;

results concerning these sets are given under (i) and (ii); (d) the pairs of SP(In - 1, ±n2 ,1 ; 02) with weights Q; the number of these pairs,

say N 4, equals

I8 if n=2, N4= 192 if n = 4 ,

and

d(In - 1, ~-n,1 1; 02) = 2n Y+g2Y + n(n - 2) t~ + (21- n - 1) g2 t~

+in h2~+ag2 h2+ ~g2 h2.

where g2 = (1 - 02) 2, h2 = (1 + o2) 2. Some results for 2_< n <_ 7 are given in Table 2 and for n = 2 in Table 1; the arrows

in the picture indicate the pairs. In Table 2, NP denotes the number of pairs of the corresponding subset of e. In the row marked with 010 the total weights of the subsets are given as percentages. ND denotes the number of pairs of the design e, and ND* equals ~n(n + 1)(n + 2)(n + 3)(n + 4), the number given in (4.2).

276 E.E .M. van Berkum / Optimal paired comparisons designs

Table 1

A D-optimal design in the case n = 2

SP( I , I)

v2 = 0 .02480

((1, i), (1 , -1) ) ((l, 1), ( - 1 , 1)) ((1, 1), ( - 1 , 1)) ((1, 1), (1,--1))

SP(0, 0, 2 ; v l )

v I = - 0 .15029

/~ =0 .04621

((1, 1), (Vl, Vl)) ( ( -1 , 1), ( - v l , vl)) ( ( l , - 1 ) , (ol, --Vl)) ((1, I), ( - v l , - v l ) )

SPI(0 , 0, 2 ; v l )

A = 0.02975

((I, vl), (vl, I)) ( ( - I, v l ) , ( - Vl, I))

( (1 , - v l ) , ( v b - I))

(( -- 1, -- Of), ( -- Vl, -- 1))

SP(0, I , I ; v2)

v2 = - O. 12002

= O. 07462

((1, l), ( - l, V2) ) (( -- l , -- 1), (1, -- 02))

((-- l, 1), (1, V2) ) (0 , - D , ( - 1, - v2))

(0, D, (v~, - l)) ( ( - 1, - I) , ( - v2, 1))

( ( i , - i ) , (v2, 1))

( ( - 1 , 1), ( - 0 2 , - 1 ) )

Table 1 shows that N D > N D * , except for n = 3, but there exist designs for which ND _< ND * (cf (4.2)).

The discrete optimal designs given in Table l can be used to construct an exact design with efficiency I - q for any small positive value of I?. For such a design, since the product of the weights and the number of pairs must be an integer, a large number of observations has to be chosen. However, one is interested in reducing the number of observations, so exact designs which are useful for practical applica- tions will not have a very high D-efficiency in general.

Some results concerning these problems are given in Berkum (1985).

E.E.M. van Berkum / Optimal paired comparisons designs 277

Table 2 Values of constants determining discrete D-optimal designs and the information matrices of these designs

n 2 3 4 5 6 7

t~ 1.933 2.279 2.589 3.015 3.331 3.760

0.756 0.618 0.599 0.578 0.563 0.559

y 0.494 0.507 0.553 0.510 0.563 0.510

- 0 . 1 8 9 -0 .155 -0 .150 -0 .145 -0 .141 -0 .140

ol - 0 . 1 5 0 -0 .118 -0 .107 -0 .080 -0 .078 -0.061

o2 0.120 0.018

det a 0.5541 0.2901 0.1484 0.2737 x 10-1 0.5771 × 10 -2 0.3522 x 10 -3

v I 0.425 X 10 -1 0.402 x 10 -2 0.146x 10 -3 0.321X 10 -3

NP 12 160 480 2240

070 51.0070 64.3070 7.0070 71.9070

v2 0.248x 10 - l 0.711 x 10 -2 0.952x 10 -3

NP 4 48 640

070 9.9070 34.1070 60.9070

/a 0.462x 10 - l 0.189x 10 - l 0.678x 10 -2 0.241X l0 -2

NP 4 8 16 32

070 18.5°7o 15.1°7o 10.807o 7.7070

A 0.298 x 10 -1 0.141x 10 - l 0.492X 10 -2 0.875 x 10 -3

NP 4 24 48 320

070 11.9% 33.8% 23.6% 28.0%

Q 0.746 X 10 "1 0.164X 10 -2

NP 8 192

o70 59.7°70 31.407o

0.106X 10 -2 0.516X 10 -6

64 128

6.8070 0.007070

0.396 x 10 -3 0.105 x 10 -3

640 4480

25.3% 28.1%

ND 20 44 304 512 1824 6848

ND* 15 45 105 210 378 630

a det = det(M- l (e)).

6. D-optimal designs for a factorial model with main effects and f'wst-order interactions

The model considered is

l n n x = f l l X l + "'" + fln Xn + ~12 x l x2 + "'" + B n - l n x n - l xn -

Here we have to consider separately the cases n odd, n even.

(6.1)

(i) n odd, n >--3. Now SP(~-(n-1) , ~ n + 1)) constitutes an exact D-optimal design. This design

contains all the pairs with ~(n + 1) factors at a different level and ~-(n- 1) factors at the same level.

278 E.E.M. van Berkum / Optimal paired comparisons designs

(ii) n even, n >__ 2. In this case one has to choose both SP(~-n, 1 ~-n) and SP(~-n- 1, ½n+ 1), where

each pair has the same weight. So this design is an exact D-optimal design. It has to be remarked that choosing either SP(~n, ~-n)l or SP(~-n-1 , ~-n+ 1) does not yield an optimal design.

The G-efficiency of SP(-~n - 1, ½n + 1) equals

(n + 1)(n + 2)(n - 2) n ( n 2 .-k n - 4) '

and the G-efficiency of SP(~n, 1 ~-n) equals

n2(n + 1) (n + 2)(n 2 - n + 2)

Some of the above designs are listed in Quenouille and John (1971).

References

Beaver, R.J. (1977a). Weighted least squares response surface fitting in factorial paired comparison. Comm. Statit. Theory Methods A6(13), 1275-1284.

Beaver, R.J. (1977b). Weighted least squares analysis of several univariate Bradley-Terry models. J. Amer. Statist. Assoc. 72, 629-634.

Berkum, E.E.M. van (1985). Optimal paired comparison designs for factorial experiments. Doctoral dissertation, Technological University of Eindhoven.

Bradley, R.A. and M.E. Terry (1952). The rank analysis of incomplete block designs. I. The method of paired comparisons. Biometrika 39, 324-345.

EI-Helbawy, A.T. and R.A. Bradley (1978). Treatment contrasts in paired comparisons: Large sample results, applications and some optimal designs, J. Amer. Statist. Assoc. 73, 831-839.

Fedorov, V.V. (1972). Theory of Optimal Experiments. Academic Press, New York and London. Fienberg, S.E. and K. Larntz (1976). Loglinear representation for paired and multiple comparison

models. Biometrika 63, 245-254. Quenouille, M.H. and J.A. John (1971). Paired comparison design for 2n-factorials. AppL Statist. 20,

16-24. Springall, A. (1973). Response surface fitting using a generalization of the Bradley-Terry paired com-

parison model. Appl. Statist. 22, 59-68.