Incomplete block designs for symmetrical parallel line assays

Journal

of Statistical

Planning

and Inference

111

11 (1985) 11 l-l 17

North-Holland

INCOMPLETE BLOCK DESIGNS PARALLEL LINE ASSAYS A.K.

NIGAM

and G.M.

FOR SYMMETRICAL

BOOPATHY

Indian Agricultural Statistics Research Institute, New Delhi 110 012, India Received

August

Recommended

1982; revised manuscript

received

22 October

1983

by D. Raghavarao

Abstract: New series of incomplete From these designs In addition designs

important

block designs for symmetrical parallel lines are proposed. contrasts like L,, Lt and L’, are estimated free from block effects.

to these, other odd order contrasts

are shown

are also estimated

orthogonal

to block effects.

The

to have simple analysis.

AMS Subject Classification: 62KlO. Key words: Symmetrical

parallel

line assays;

Partially

efficiency-balanced

designs.

1. Introduction Incomplete block designs have been used in varietal trials, factorial experiments and biological assays. In bio assays, not all contrasts are of equal importance. Contrasts of preparation (L,), combined regression (L;) and parallelism (L,) are of major importance for parallel line assays. The difference between preparations is L, and the average linear regression of response on dose metameter is L, . The parallelism contrast L; is the difference between the two regression contrasts of preparations. L, and L, provide an estimate of relative potency while L’, is important When the number of doses for each preparation is the same, say m, the assay is called symmetrical. For a symmetrical parallel line (SPL) assay with 2m doses, the 2m - 1 degrees of freedom can be split into single degree of freedom orthogonal contrasts L,, L,,, and Li, h = 1, . . . , (WI- l), where L, and Li represent respectively the sums and differences of h-th power regression. For instance, when m = 3, the contrasts are preparations (L,), average regression (L,), difference of linear regressions (L’,), average quadratic regression or quadratic curvature (L2) and difference of quadratics (L;). As will be seen later, it is useful to further sub-group Lh and Lk into odd and even numbered contrasts L,, + I, L,, +2 and L;, + , , L;, + 2 respectively. Let I,, 12,,+1, l;.++, 12,,+2 and &+2 respectively denote the vectors of coefficients of the effects of the dose levels (using appropriate transformations, if necessary) in the contrasts L,, L2,, + , , L;, + , , LZn+2 and L;, + 2. 0378.3758/85/$3.30

0 1985, Elsevier Science Publishers

B.V. (North-Holland)

A.K. Nigam, G.M. Boopathy / Incomplete

112

block designs

Consider now an incomplete block design in u = 2m doses and b blocks with the i-th dose occuring in ri blocks. Suppose further that the j-th block has size kj such that the number of units receiving doses of standard and test preparations are respectively k, and k,. Thus k, + k, = kj. Let S; (ti), i = 1, . . . , m, denote the effect of the i-th dose of standard (test) preparation on log scale. Let us assume the model _Yij= u+di+bj+eij,

(1.1)

where Yij is the response to dose i in the j-th block, u is the general mean effect, 6i is the effect of the i-th dose (Si or ti as the case may be), bj is thej-th block effect and eij are the random error components assumed to be normally and independently distributed with zero mean and variance cr2. Kyi Win and Dey (1980) have derived the conditions under which L,, L, and L; are estimated free from block effects in a parallel line assay, symmetrical or otherwise. We restrict our discussion here to the case of SPL assays with o = 2m, ri = r for each i and k, = k, for all j. We further assume that kj = k for all j SO that k,=k,=+k. It follows from Kyi Win and Dey (1980) that L^r, L^, and L^; given as

lp =

fJ(S;-T)/mr,

1=I

2CZIr = f [i - (m + 1)/2](Si ,=I

(1.2)

+ T),

CL; = fJ [i-(m+1)/2](Sj-7;,), i=l respectively

estimate c’i=

L,, L1 and L’, free from block C”i=+k(m+l)

effects

if (1.3)

where Cl i and Cl’ i denote respectively summation for standard and test preparations dose indices in j-th Si is total observations the dose standard preparation C=mr(m2This is with construction analysis SPL laid in blocks. a discussion existing the is ferred Finney Das Kulkarni proposed block in L, L, be orthogonal block Kulshreshtha and Win Dey have designs orthogonal of L, L;. of are to m kz8, those Kyi and suffer the back consideramount guess is in construction. present some procedures constructing block for assays. of designs Kyi and are cases our The designs nice and be withany

A.K.

2. Construction

Nigam, G.M. Boopathy / Incomplete

and analysis

113

block designs

of designs when m is even

Three series of designs for SPL assays when m is even are given. All the designs have the property that the important contrasts L,, Lzn + 1 and L;,, + 1 are estimated free from block effects.

Series I Parametersofthedesignarem=2q, v=2m(m=4,6,...), b=q2,r=qandk=4. For construction of the designs, m doses of the standard preparation are arranged into q groups such that each group contains two doses and the sum of the subscripts or indices of these two doses is equal to 2q+ 1, which is necessary for estimating L, and L; free from block effects. Groups so formed are (s~,s~~), (.s~,s~~-~), . . . , (s,,s,+,). Similar groups are formed for test preparation, which are (ti, t2q), (t2,t2,-,),...,(t,,t~+~).

The incidence

matrix

of the design

where N, and N2 are the incidence and are given by

is then obtained

matrices

as

of the test and standard

I... :1*

where J,,. is a matrix of order m x n having all elements as unity, matrix of order q and e4 is a matrix of order q such that 01 o... 01

eq =

Throughout,

NNT =

matrix

matrix

NNT for the above

J 2q,2q rV2, + e2q) i .........................

1

Jwl

Z4 is an identity

061

we shall write the incidence

N= [;;I. The concurrence

preparations,

i r(Z2q+e2q)

N in the partitioned

design

form,

i.e.,

is

. 1 1

It can be easily seen that for the above design all the contrasts L,, L2, + 1 and L;,, + I are estimated orthogonal to block effects, and the contrasts L2n+2 and L&,+2 are each estimated with a loss of information +. The design is therefore a simple partially efficiency balanced (PEB) design (see Puri and Nigam, 1977) with only two

114

A.K. Nigam, GM. Boopathy / Incomplete block designs

distinct losses of information 0 and 5. Using the results of Puri and Nigam (1977) on PEB designs with m-efficiency classes, the covariance matrix 0 can be easily written as $J= (l&J) [ Some of the designs of designs.

(3rI-J)+re i 0 . . . . . . . . . . . . ..; .*. . . . . . . . . . . . : (3rl-J)+re 0

. I

of Kyi Win and Dey (1980) are particular

cases of this series

Series 2 The second series of designs for SPL assays are obtained by deleting the blocks of design of series 1 where dose indices for standard and test preparations are the same. The parameters of this series of designs are u = 2m, b = q(q - l), r = q - 1 and k= 4. For such designs NNT is of the form

NNT =

r(I+e) ! J-I-e ..................... J-I-e i r(I+e)

1 .

Here again, L,, L2n + , and L;, + , are estimated free from block effects and the conwith losses of information (r- 1)/2r and trasts L2,,+* and Lin+* are estimated (r + 1)/2r respectively. The design is PEB with three efficiency classes with three distinct losses of information as 0, (r - 1)/2r and (r + 1)/2r. Following Puri and Nigam (1977) the Q-matrix can be easily worked out as sZ= [l/r(r*-l)](SZij) where .52,, =Q,,=

[(3r*-1)/2]I+[(r2+1)/2]e-[2(r*+l)/u]J

and a,* = Q2r = -r(Z+ e) + (4r/o) J. Series 3 Here also doses are arranged into groups: m doses of the standard preparation are arranged into (z) groups, each consisting of four doses such that the sum of the subscripts or indices of these doses is equal to 2(2q+ 1). Similarly another (z) groups are formed by combining one group of standard doses with another group of test preparations such that dose indices for both preparations are the same. The first block is obtained by combining (t,, t2qr t,, t2q_ 1) with (s1,szq,s2, Sag_ t). Since there are (z) groups, the total number of blocks will be (24). The parameters of this series of designs are o=2m (m=6,8,...), b=(z), r=q-1 and k=8.

115

A.K. Nigam, GM. Boopathy / Incomplete block designs

In general, for blocks of size 4p (p = 1,2, . ..). m doses will be arranged into (4) groups, each group has 2q doses of the of these is equal to p(2q + 1). For designs with blocks of sizes 8, we have the incidence matrix N = [;;I, where Nt = N2, and N, is given by

such that N,, of order q x (z) is the incidence matrix of the balanced incomplete block design with parameters u = q, b = (z), r = (q - l), k = 2, A = 1 and m,, is the incidence matrix of the complementary balanced incomplete block design. We then have

NAT = (q-2)(~2q+e2q)+J2q,2q. The matrix NNT of the above design can, therefore,

be written as

NNT= J2,2@[(r-1)(1+e)+J]. It can be shown that the effects L,, L2,,+ 1, Li,,+, and L;n+2 are estimated orthogonal to block effects, and the contrast L2n+2 has a loss of information equal to (r-1)/2r. The design is therefore simple PEB with Q-matrix as

a=1

1

(5r+3)1+(r-1)e : (r- l)(l+e) . . ... .. ... .. .... ... ......... ........... .. .

4r(r + 1) i

J 2%29

: (%+3)1+(r-1)e

3. Designs for SPL assays when m is odd

When the number of doses of each preparation is odd, m(2q+ 1) doses are arranged into (q + 1) groups of two doses each such that the sum of the dose indices in a group is equal to 2(q+ 1). The q groups for the standard preparation are ) and the (q+ 1)-th group is (s~+,,.s~+~). For the test ),(~2~~2q),...,(~q,~q+2 (W2q+l preparation, groups are obtained by replacing ‘s’ by ‘t’. The parameters are o = 2m (m=5,7 ,... ), b=m, r=2 and k=4. For the above set of parameters, the series of designs are given. These designs permit the estimation of L,, L2n+ 1 and L;,,, 1 free from block effects. Series I

First (q + 1) blocks are obtained by combining the i-th (i = 1,2, . . . , q + 1) group of the standard with the i-th group of the test preparation and the next q blocks are obtained by combining the j-th group of the standard with the (q-j + 1)-th group of the test preparation, j = 1,2, . . . , q. In this case the N, and N2 matrices are given by

116

The

A.K.

matrix

Nigam, GM. Boopathy / Incomplete block designs

Ni can be written

as Ni = (I+ e). The order

of matrix

Ni is

(2q+l)x(2q+l).

The concurrence

matrix NNT for the above design can be written as

where A = 2(1+ e) and

B mxm =

I+e I+e

0

Z+e

0

Z+e

The orders of 1, e and 0 in A and B are q x q, q x q and q x 1 respectively. It can be easily shown that the estimates &,, &,,+, and L^i,+ 1 are all orthogonal to block effects. The design is again PEB with a-matrix easily determined. Series

2

The first q blocks are formed by combining the first group of standard preparations with the q-th group of test preparations the second group of standard preparations with the (q - I)-th group of test preparations and so on. In the (q + 1)-th block, s4 + 1 and tq+ , are repeated two times. The first q blocks are repeated so that the total number of replications for each dose is 2. Here, the N, and N2 matrices are

N,=[i

% I],

Nz=[;

The matrix NNT of the above design is NNT=~

Z+ei C . ..... ....... C

:I+e .

I

where e C mxm =

0’

i I

.

I

0 2

0’ .

0

e

1

4

:I.

A.K. Nigam, G.M. Boopathy / Incomplete

117

block designs

The matrices I, e and 0 have the orders q x q, q x q and q x 1 respectively. Here again the effects L,, L2n+l and L;,+, are estimated free from block effects.

Acknowledgement

The authors are grateful to the referees for suggesting improvements in the eariler drafts of the paper.

References Das, M.N. and G.A. Kulkarni Finney,

D.J.

Kulshreshtha,

(1966). Incomplete

block designs for bio-assays.

(1978). Statistical Methods in Biological Assay. Charles A.C.

(1969).

On the efficiency

of modified

BIB designs

Griffin,

Biometrics 22, 706-729. London.

for bio-assays.

Biometrics 25,

591-593. Kyi Win and A. Dey (1980). Incomplete Puri,

P.D.

and A.K.

Nigam

block designs

(1977). Partially

efficiency

for parallel balanced

line assays. designs.

Biometrics 36, 487-492.

Comm.

Statist. 753-771.