On block designs for symmetrical parallel line assays with even number of doses

Journal

of Statistical

Planning

and Inference

21 (1989) 383-389

383

North-Holland

ON

BLOCK

DESIGNS

LINE

ASSAYS

Sudhir

GUPTA

WITH

FOR EVEN

SYMMETRICAL NUMBER

PARALLEL

OF DOSES

Division of Statistics, Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, U.S.A. Received

10 November

Recommended

1986; revised manuscript

Abstract: A new approach parallel

is developed

line assays with even number

the existing

received

16 February

1988

by D. Raghavarao

designs

is obtained

for the construction

of block designs

of doses. A wide class of designs

using the approach

for symmetrical

which includes

many of

presented.

AMS Subject Classification: 62KlO. Key words and phrases: Balanced block

design;

factorial

design; contrast;

eigenvalue;

eigenvector;

incomplete

bioassay.

1. Introduction Consider a symmetric parallel line (SPL) bioassay experiment involving two preparations, standard and test, each at m equispaced doses. It is then desired to conduct the experiment using an incomplete block design; see for example Finney (1978, p. 177), and Das and Kulkarni (1966). The purpose of the present paper is to give general procedures to obtain suitable designs for such experiments when the number of doses m is even. In parallel line assays certain contrasts among treatment parameters are usually considered to be of major importance. These contrasts are: the preparation contrast L,, the combined regression contrast L1, and the parallelism contrast Lf . The L, and L, provide an estimate of the relative potency, and Li is important for testing deviation from parallelism of the linear regression lines for the standard and the test preparations. Therefore, it is usually desired to estimate these three contrasts with high efficiencies. Das and Kulkarni (1966) gave some designs in which L, and L, are estimated with full efficiency. Kulshreshtha (1969) and Kyi Win and Dey (1980) gave further designs which estimate all the above three contrasts with full efficiency. Designs of Kulshreshtha are for even m and block size at least 8, while those of Kyi Win and Dey involve considerable amount of guesswork. Recently, Nigam and Boopathy 0378-3758/89/$3.50 0 1989, Elsevier Science Publishers

B.V. (North-Holland)

S. Cupfa / Block designs for bioassays

384

(1985) gave some useful designs which also permit estimation of the above three contrasts with full efficiency. However, the designs of Nigam and Boopathy are limited in the sense that block size has to be a multiple of four. Finney (1979) indicated with the help of some examples how suitable designs can be obtained using factorial designs. We note that the choice of block size or some other considerations may dictate the use of a design in which all the three contrasts L,, L, and Li may not be estimated with full efficiency (see also Finney, 1979). Therefore, a more general approach to the construction of designs is considered in this paper where the preparation contrast may not necessarily be estimated with full efficiency. However, the L, and L; are still necessarily estimated without any loss of information.

2. Notations

and preliminaries

Let s,
$=(l

lit = (e: ei)‘,

-1

li2=(e:

-el)‘,

where 1, denotes a column vector normalized contrasts, clearly

L, = where ajj = G lowing type:

(

ei2

(2.1)

,..., m-l,

of l’s of size q. Then,

It will be helpful

Ceil

i-l,2

-IL)‘,

restricting

attention

to

Li= (/(I +/ail, L! = (1:2T)/aj2, i= 1,2, . . . . m - 1,

(l;r)@z,

(e;r er2 ... eiU -e,

ei =

. . . _l)‘=(l:,

...

e,

e,

to observe

-ej,U_l ei,u-l

Since ei’l,=O, clearly e!‘lU=e?‘lU=O, (i+j), wegetelej=2(e,!‘ej)=O(i#j=2,4,6 lemma.

that the structure

... -ei,)‘,

... ei,)‘=(e,”

ef’)‘,

of e, is of the fol-

i=l,3,5

,..., m-l,

i=2,4,6

,..., m-2.

(2.2)

i=2,4,6 ,..., m-2. Also, since ejej=O ,..., m - 2). We thus have the following

S. Gupta / Block designs for bioassays

The ef, i=2,4,6 ,... ,m - 2, constitute orthogonal contrast vectors of dimension u. Lemma

2.1.

385

a complete set of mutually

When a bioassay experiment is to be conducted using an incomplete block design, it will be desirable to use a design in which the covariances among estimates of the contrasts of interest are zero; a discussion on the importance of orthogonal estimation of parameters in constructing confidence intervals can be found in Box and Draper (1987). Another importance of such designs is realized while obtaining combined intra-block and inter-block estimate of a contrast of interest; under orthogonal estimation the combined estimate is BLUE (cf. Sprott, 1956 and Martin and Zyskind, 1966). Most of the bioassay designs considered by the previous authors also possess this orthogonality property. Therefore, we will restrict attention to such orthogonal designs only. Since L,, Li, Li, i = 1,2, . . . , m - 1, provide a complete set of mutually orthogonal contrasts, we have the following well known result, e.g. Pearce, Calinski and Marshall (1974). Lemma 2.2. A necessary and sufficient condition for obtaining mutually orthogonal estimates of L,, Lj, L,‘, i = 1,2, . . . , m - 1, is that the corresponding contrast vectors be eigenvectors of the C-matrix of the design.

3. Class G of designs for SPL assays (m even) A wide class of designs for SPL assays will be defined in this section. However, first it will be helpful to consider the following which can be proved easily. Theorem 3.1. A sufficient condition for I,, i= 1,3,5, . . . , m - 1; j= 1,2, to be eigen-

vectors of the C-matrix of the design D corresponding to the eigenvalue r is that the two treatments f + (j - l)m and mj t 1-f do not fall in different blocks of the design, j=l,2; f=1,2 ,..., u. Clearly, the contrast II;5 is estimated with full efficiency, i-1,3,5 ,..., m-l; j-1,2. In the class of designs where lr and lij are eigenvectors of the C-matrix, i = 2,4,6, . . . , m - 2; j= 1,2, we will restrict attention to designs which satisfy the requirements of Theorem 3.1. Let G denote the class of such designs. Clearly, using Lemma 2.2, in the designs belonging to G all contrasts of interest are estimated uncorrelated with each other. Also, L;, L!, i = 1,3,5, . . . , m - 1, are estimated with full efficiency. For obtaining designs belonging to G, we now define a class G, of designs having parameters u, = m = 224,bl, rl, kl. Let Nr and C1 denote the incidence matrix and the C-matrix respectively of a design D, E G,. Let e, ej and e: be eigenvectors of C, with eigenvalues 19~ , 19,and 07 respectively, i = 2,4,6, . . . , m - 2, where ej is

S. Gupta / Block designs for bioassays

386

given by (2.2),

e= (1 1 . . . 1 -1 -1 ... -I)‘= ej+=(ej’ -ef’)‘, Now, let the treatment

f

(1; -I’,)‘, (3.1)

i=2,4,6 ,..., m-2. in design

(f; rn -f+ 1) and the treatment m Let D be the design thus obtained. of design D is given by

D,

be replaced

by the pair

of treatments

-f + 1 by the pair (m +f, 2m -f + l), f = 1,2, . . . , u. It can then be verified

that the incidence

matrix

(3.2)

where I,, contains unity in (i, u f 1 - i) positions (i = 1,2, . . . , u), and zero elsewhere, I, is the identity matrix and 0 denotes a matrix of zeros, all of order ux u. It should be noted that for design D, r= r, and k = 2k,. Theorem 3.2. (i) The design D belongs to G. (ii) The L,, Li and L! are estimated with variances 02/6,, respectively, i = 2,4,6, . . . , m - 2.

a2/0i and 02/8T

Proof. To prove the theorem we need to show that (i) ID, f,i and lr2 are eigenvectors of C-matrix of design D with respective eigenvalues or, 6i and 97, i=2,4,6 ,..., m-2, and (ii) the conditions of Theorem 3.1 are satisfied. Note that e, ei and e? are eigenvectors of N, Nr’ with respective eigenvalues k,(r, - O,), k, (r, - 0,) and k,(r, - OF), i = 2,4,6, . . . , m - 2. Also, from (2.1) and (2.2), for i=2,4,6 ,..., m-2, we can write

(3.3)

Using (3.2), (3.3), k=2k,, r=rl and the fact that Ni N;ei = kl(r, - B;)e, it can be shown that NN’lir = k(r - 0;) lil, i = 2,4, . . . , m - 2. Therefore, Cl,, = (rl-~NN~)li, Similarly,

it can also be proved Cli2 = Si*li2, Clp = L9,1p.

= e;l;l,

i=2,4,...,m-2.

(3.4)

that

i=2,4 ,..., m-2,

(3.5) (3.6)

S. Gupta / Block designs for bioassays

387

Thus, part (i) of the theorem follows from (3.4), (3.5) and (3.6). Finally, observe that design D is obtained by replacing treatment (-l)j-if+ (j- l)(m + 1) in design D, by the pair of treatments {f+ (j- l)m, mj+ 1 -I}, f=l,2,... , u; j = 1,2. Therefore, clearly the conditions of Theorem 3.1 are satisfied. Hence the theorem. Example r,=k,=3:

3.1.

124 578

Consider

the

following

238 167

3 2

57 46

design

D,

having

parameters

ui = b, = 8,

468 135

The e, e, and e,? are eigenvectors of the C-matrix of this design with respective eigenvalues 2.0, 2.667 and 2.0 where i = 2,4,6. The design D is then obtained from D, by replacing in D, the treatments 1,2,3,4 by the pairs of treatments (1, S), (2,7), (3,6), (4,5) respectively, and the treatments 8,7,6,5 by (9,16), (10,15), (11,14) and (12, 13) respectively. 124578 4 5 9111416 2 7 11 12 13 14

2 3 6 7 916 9 10 12 13 15 16 1 3 6 8 12 13

3 1

4 5 6 10 15 810111415

It follows from Theorem 3.2 that Lj and Lf are estimated with full efficiency for i= 1,3,5,7, while L,, Lj and L,! are estimated with respective variances a2/2.0, a2/2.667 and a*/2.0, i= 2,4,6. In this paper bioassay designs belonging to G are obtained using class G, of designs as indicated above. The designs belonging to G1 can be derived through balanced factorial designs of Shah (1958,196O) which is shown in the next section.

4. Construction

of designs belonging

to G1

Consider a 2 x u factorial experiment where the two factors are denoted by F, and F2 respectively. The ui = 2~ treatments will be considered in the lexicographical order. Let D, be a connected balanced factorial (BF) design for the 2 x u experiment in bl blocks of size ki and r, treatment replications. In BF designs all the normalized contrasts belonging to a particular effect are estimated with the same variance. Clearly e = (1’ -1’)’ which represents the main effect of F, is an eigenvector of the C-matrix of design D2 with corresponding eigenvalue say 8,. Let be linear, quadratic, cubic, . . . contrast column vectors of order u WI, w2, . . . . J+*_l as given by orthogonal polynomials. Then, hi = (w: w;)’ (i = 1,2, . . . , u - 1) represent the u - 1 main effect contrasts of F2. Similarly, hi* = (WI - w,!)’ (i= 1,2, . . . , u - 1) are then u - 1 interaction effect contrast vectors. Since D, is a BF design, each hi has an eigenvalue f3, where all normalized main effect F, contrasts are assumed to be estimated with a constant variance a2/&. Similarly, we will denote by e4 the

S. Gupta / Block designs for bioassays

388

constant

eigenvalue

corresponding

to each of the interaction

contrast

vectors

hT,

i=l,2 )..., U-l. As a consequence of the fact that h,, hz, . . . , h, 1 all have the same eigenvalue &, it can be verified that each of the contrasts hk, hi,. . . , h!+2 also have the same eigenvalue 19,, where h! = (e,f’ e,“)’ (i = 2,4,6, . . . , m - 2). Here ei, et, . . . , efnP2 are a complete set of u - 1 mutually orthogonal contrast vectors of dimension u as shown earlier in Lemma 2.1. Similarly, each of hi*, hl”, . . . , hF_2 also have the same eigenvalue e4 where h!*=(ei -ef’)‘(i=2,4,6 ,..., m-2). Let in design D2, treatments u + i and 2u + 1- i be interchanged (U + i - 2u + 1- i), i.e. treatment u + i be replaced by 2u + 1 - i, and the treatment 2~ + 1 - i be the treatment u+i, i-l,2 ,..., uO, where u. is the integer portion of +u. Let D, be the design thus obtained. It can be verified that the incidence matrix of design D, is given by N, =

(4.1)

N2

where N, is the incidence

matrix

of design

D2.

Theorem 4.1. (i) The design D, belongs to G,. (ii) The eigenvalues corresponding to e, ei and e: are 19~)1512 and 8, respectively, i = 2,4,6, . . . , m - 2. Proof. To prove the theorem we need to show that e, ei and e:, i = 2,4,6, . . . , m - 2, as defined by (2.2) and (3.1) are eigenvectors of the C-matrix of D1 with respective eigenvalues el, e2 and 0,. Observe that for i = 2,4,6, . . . , m - 2 we have

h,! and The theorem

then follows

using

eT =

(3.1) and (4.2).

Example 4.1. Consider the BF design D, for a 2 x 4 experiment dence matrix as follows:

N2=

1000 1100 0110 1011 0101 0010 0001 0000

Then, using (4. l), the incidence ple 3.1 is obtained.

(4.2)

0101 0010 0001 0000 1000 1100 0110 1011 matrix

given by its inci-

.

of the design Di considered

earlier in Exam-

S. Gupta / Block designs for bioassays

389

BF designs are available in plenty in the literature, e.g. Muller (1966), Suen and Chakravarty (1986), group divisible designs tabulated by Clatworthy (1973). The method of construction proposed is thus widely applicable.

Acknowledgement Research for this paper was partially supported by Northern Illinois University Summer Research Grant. The author is thankful to D. Raghavarao, Associate Editor, for a careful reading of the paper and helpful suggestions.

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