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Incomplete block designs for parallel line assays
Journal of Statistical Planning and Inference 20 (1988) 121-128 North-Lolland
Department of Statistics, Haryana Agricuhml
121
University, Hisar, India
Received 26 December 1984; revised manuscript received 27 July 1987 Recommended by .I. Seberry
Abstract: A systematic method of constructing incomplete block designs for parallel line assays from semi-regular group divisible designs is proposed. Many new series of designs are obtained through this procedure. These designs estimate all important bio-assay contrasts with maximum efficiency in addition to having simple analysis. They need a smaller number of experimental units as compared to known designs in literature. AMS Subject Classification: 62KlO. Key words: Incomplete block designs; semi-regular group divisible designs; parallel line assays; partially efficient;, balanced designs.
I.
Incomplete block designs have been used quite effectively in biological assays (Ho-assays). In bio-assays contrasts of interest are other than elementary contrasts. Incomplete block designs for symmetrical imum efficiency important bio-assay c Kulkarni (1966), lshreshtha (1969) and Niga sion, reference may be made to Finney (19?$), ons under which preparation (L,), the co ) contrasts are estimated orthogonally. asymmetrical parallel line (AP tions. Their designs do not follow any pattern an treatment or block pseudo infor method of matrix inversion.
0378~3758/88/$3.50 0 1988, Elsevier Science Publishers R.
P. D. Puri, L. R. Gupta / Incomplete block designs
122
designs. In many cases these designs need a smaller number oafexperimental units as compared to the designs available in literature. The proposed designs are partially efficiency balanced (PEB) with almost three efficiency classes and have very sim analysis.
es
In a parallel line (PL) assay, with u =gt + qZ doses, where q1 and q2 are the number of doses for standard and test preparations respectively, the U-- 1 degrees of freedom attributed to treatment sum of squares can be split into single degree of freedom orthogonal contrasts Lp, Lh and EL; h = 1,2, . . . , where Lh and LArepresent respectively the sums and differences of h-th power regression. It is useful to subgroup further Lh and LL into odd and even numbered contrasts L2h+ 1gL2h+2 and -Gh+f* G/l+2 respectively. If Q = q2 the PL assay is called symmetrical (SPL); otherwise the assay is called asymmetrical (APL). The doses of standard and test preparations are assumed to be equispaced on a log scale. Consider an incomplete block design D(v, b, P, It) with v = q1 + q2 doses arranged into b blocks of sizes kr, 4, . . . , kb such that the i-th standard preparation is replicated ri (i- 1,2, . . . , ql) times and the i-th test preparation is replicated ri (i=ql+ l,q,+2,..., q1 + q2) times. Let r = (rlr r2, . . . orJT, and k = (kl, k2, . . . , kb)T. ere T stands for transpose. Further, let the number of units in thej-th block receiving the doses of standard and test preparations be ksjand k,, j = 1,2, . . . , b, respectively. Clearly kj = ksj+ ktj. For convenience, the symboi si (or Q) will be used to designate the i-th dose of standard (or test) preparation as well as its effect. Under usual intrablock model for equi-replicated block designs, Kyi Win and Dey (1980) have shown that the important bio-assay contrasts Lp, L1 and L; will be estimated free from block effects if the following conditions are satisfied: ksj/ktj=ql’/q2 Pf-
for all j= ‘1,2,. . ..b.
(2. la)
+k,(q, =+19,
(2.lb)
r=3ktj(q2+ ‘9,
(2.icj
i =
f? l
where Ej I and Cyi denote the sum tion for standar and test preparation dose indices, respectively, in the j-th MO ones (1959) has shown that if ent contrast ST is the n vector of dose totals, s is the right eigenvectcr of the matrix (2.2)
P.D. Puri, L.K. Gupta / Incomplete Mock designs
is a vector of dose effects. total number of elements Y and were termed basic
123
atrices having diagona
with the same efficiency factor (I -pi), where pi (i = 1,2, . . . $,O:,are the evgenvalues with mmultiplicityQig k, of
The parameters of a PE
design are written as
where the Li’s are mutually orthogonal idempotent matrices of rank Qi satisfying rr/n, I is an identity matrix and is the vector of unities. For c designs the pseudo variance-covariance matrix Q of the estimated treatment effects “. i is given by +
5 [pi/(1 -pi)]
i=l
A PEB design is called simple ) if j4i takes only p and 0 with multiplicities Q and u-e - 1, respectively.
tW0
distinct ValUes
3. II3 design with two associate classes is said to be grou L-1, --_~-5-t, ~~~L---__ea.I * -- *-fimln@ nf ci7e d nvmbol U= TM symbols clau uc ak rarlg;eu ii%0 fib &LtGye -
Consider a semi-regul
p.D. Puri, L.R. Gupta / Incomplete block designs
124
+a,(ql+ 1). Let uii (i= 1,2, l ,m;j= 1,2, , t) denote the j-th symbol in the i-th group. btain D* by replacing aI t symbols uii (i = 1,2, . . . , a1;j = 1,2, . , t) of the first a1 groups with alt sets of standard‘ preparation and a2t symbols UQ t) of the last a2 groups by a2t sets of (i=a,+1,a2+1 ,..., a,+a2=nl;j=1,2,..., test preparation so formed. Since the basic design is semi-regular, every block of it contains exactly c symbols standard and a2 doses of test preparations are from each group and aI doses therefore, kS.= c erlal and ktj = c a2a2 and (2. la) associated with each symbol of is satisfied for *. It can be easily seen that the conditions (2. Ib) and (2. lc) are also ce the final design D* will estimate all the three important consatisfied for D free from block effects. trasts Lp L1 a We shall now illustrate the procedure of construction through an example. Consider a semi-reguiar GD design with para.,meters l
l l
v/=6, r=2, k’=3,
b=4, m=3, t=2, &=O, A2=!
(3.2)
with groups (1,2), (3,4) and ($6) with block contents (1,3,5), (1,4,6), (2,3,6) and (2,4,5). Suppose we are interested in an incomplete block design for an APL assay with q1=4andq2=$.V1ecantakeal=~2=2,al=l,~2- 2 and the sets of standard and test preparation as (sr, sq), (s2, sj); (tl, t8), (t2, t,), (ts9 td), (t4, ts) respectively. Now by replacing the S?lmb~iSof design (3.2) by these sets we get the new design in block size 6 with block contents as follows: (% s4, 61,
tl,
s39 tl,
f3r f6, @,
61,
f49 ts,
(‘92, s3, f2, t3, t6, t7!-
td,
s4, t2, t4r t5, t7),
It may be remarked here that two designs were reported by yi Win and Dey J a&mW%nc. An nnt (1980) for Q = 4, q2 = 8 in block size 6 by trial arrd error. I%,:, a Llb11 urq=JAau U” ..“C fdhw any pattern and, therefore, for analysis the inversion of the treatment or block pseudo information matrix has to be obtained by the usual method of matrix inversion, which is time consuming. The design reported here is available in the same block size and requires thq 11~1 same number of experimental units, and it is an C 0’ having very simple analysis.
e shall now study the properties of the design of the asic design can be written in partitione
w$er
be the incidence matrix
P.D. Pub, L.R. Gupta / Incomplete block designs
at if we arrange
designs. It can be ea then the r matrix of
t
tions are written. Then it can be seen that the =
[(r - Al)frk] diag[
where it can be easily be verified that @* is c=ql+q2,
b, r, k=k’v/v’,
er=aI(t- 1), I= 1,2, r =(l/q)diag(
with parameters I.+= a& - Al)/rk, p3 =
e3=v-(m-l)t,
_4J2= (1 /a2)
5, diag
and ere stands for null matrix. The pseudo variance-covariance matrix of f for design
where for I = 1,2,, )/r
and
s,=(r-_~))/[rk-ar(r-ii.l)p.
) be the vector of adjusted treatment t c x I) are the vectors of adjusted dose totals for stan preparations respectively. Let denote the (i, j, k)-th ele iS
where
125
P.D. Puri, L.R. Gupta / Incomplete block designs
126
any new designs can be constructtid by considering different semi-regular deaI (I= 1,2). Some sf the signs a by proper choice of aI ssay designs of Migam a of Kyi in and Dey (198Q)and S are particular cases of our designs. any of the designs are Ssimple analysis. In addition of having simple analysis, in many cases our design reantnl quires less experimb,. W units as compared to the designs available in literature. If al = a, I= 1,2, then we get an S-PEB design with parameters v=q,+q2,
b,r;
k=vk’/u’,
II It\11 = (l/‘a)I,~i~~-\l,r,~IJv
p=a(r+j/rk,
e=(t-I),
R
where a = q/a/t, i = 1,2. For a = 2, we devide the standard and test preparations into a1t and a2t groups each of size 2. The groups for tl;~opreparations can be taken as (.sj,Sag_++I), j = 1,2, . . . , al t, and ($ t,++ 1), j = 1,2, . . . , a2t, respectively. For D* in this case it can be shown that preparation contrast L,n and all odd order contrasts LLfi+ 1 and L& + l are estimated with full efficiency. To save space we shall not give the values of idempotent matrices t;$ from now onward. If we take a1al = a2a2, then we get an SPL assay design with parameters v = 29,
6. r, k = 2qk’/v’,
er= a,(t- l),
pi = a,(r - A1)/rk,
1= 1,2, e3 = v-m(t-
p3 = 0,
l)- 1, L1,
or example consider semi-regular G &sign with parameters (3.1) and a1= 1, a2= 2, al = 4 and a2 = 2. Then we get an SPL assay design for with block contents
let us take
(s,,s2,S7,s~,fl,t3rf6,fs), (S3rs4d5&,
tl, t4, f5, ts),
The parameters of this design are v=l6,
b=4,
r=2, c 8 was also re
to 32 in our case.
q1 = q2 = 8
P.D. Puri, dt. 6
Gupta / Incomplete block designs
127
into t disjoint groups ~a”size 2 ES(Sj,Szl-j+ I), j = into t disjoint groups of size 4 as
(8 t j9
j+l9
t
4f-j9t41-j+!
)9
j=
11,395,...,
with indices sum 2t + 1 and 8t + 2 respectiv series will estimate preparation and all odd the designs of this serial k= 6~.
full efficiency. For
Let ql =2t: q2= 3t; t is odd. re we can di-dide qr doses of sta prep;Pi on into t disjoint groups of size as in series 3 and the test prepar into t disjoint groups of size 3 with indices sum equal to $(3t + 1); clearly t odd to have this quantity integer. If t=3, the groups can be taken BQ es
(tl9te9td9
U,J49tg)9
(t39t59t7)-
For designs of this series LP, Lr and L; contrasts will be estimated free from block effects. For this series k = 5~. If q1= q2= 2t, then we get a design of size 4~. ere the groups can be taken as explained in series 1 with aI= I for i= i, 2. For this series all bio-assay contrasts are basic contrasts. Preparation and odd order contrasts are estimated with fuli efficiency whereas the even order contrasts are estimated with efficiency factor the above series if &‘= 2, the resultant design will have bloc lect semi-regular design D with parameters
f we se-
then we get the designs reported by Nigam a u=4t,
l&r,
k=4,
p=*,
@=2(t-i),
as a particuiar case of our designs. io-assay series c
e lis-
128
P.D. Puri, L.R. Gupta / Incomplete block designs
Kulshrestha, A.C. (1969). Gr I ihe efficiency of modified BIB designs for bio-assays. Biometrics 25, 391-393. Kyi Win and A. Dey (1980). Incomplete block designs for parallel line assays. Biometrics 36, 487-492. Nigam, A.K. and G.M. Boopathy (1985). incomplete block designs for symmetrical parallel line assays. J. Statist. Plann. Inference Pearce, S.C., T. Calinski and T.F. l+J-,,;iall(1974). The basic contrasts of an experimental design with special reference to the analysis of data. Biometrika 61, 449-460. Puri, P.D. and A.K. Nigam (1977). Partially efficiency balanced designs. Commun. Statist. Theory Methods A 6, 753-771.
Department of Statistics, Haryana Agricuhml
121
University, Hisar, India
Received 26 December 1984; revised manuscript received 27 July 1987 Recommended by .I. Seberry
Abstract: A systematic method of constructing incomplete block designs for parallel line assays from semi-regular group divisible designs is proposed. Many new series of designs are obtained through this procedure. These designs estimate all important bio-assay contrasts with maximum efficiency in addition to having simple analysis. They need a smaller number of experimental units as compared to known designs in literature. AMS Subject Classification: 62KlO. Key words: Incomplete block designs; semi-regular group divisible designs; parallel line assays; partially efficient;, balanced designs.
I.
Incomplete block designs have been used quite effectively in biological assays (Ho-assays). In bio-assays contrasts of interest are other than elementary contrasts. Incomplete block designs for symmetrical imum efficiency important bio-assay c Kulkarni (1966), lshreshtha (1969) and Niga sion, reference may be made to Finney (19?$), ons under which preparation (L,), the co ) contrasts are estimated orthogonally. asymmetrical parallel line (AP tions. Their designs do not follow any pattern an treatment or block pseudo infor method of matrix inversion.
0378~3758/88/$3.50 0 1988, Elsevier Science Publishers R.
P. D. Puri, L. R. Gupta / Incomplete block designs
122
designs. In many cases these designs need a smaller number oafexperimental units as compared to the designs available in literature. The proposed designs are partially efficiency balanced (PEB) with almost three efficiency classes and have very sim analysis.
es
In a parallel line (PL) assay, with u =gt + qZ doses, where q1 and q2 are the number of doses for standard and test preparations respectively, the U-- 1 degrees of freedom attributed to treatment sum of squares can be split into single degree of freedom orthogonal contrasts Lp, Lh and EL; h = 1,2, . . . , where Lh and LArepresent respectively the sums and differences of h-th power regression. It is useful to subgroup further Lh and LL into odd and even numbered contrasts L2h+ 1gL2h+2 and -Gh+f* G/l+2 respectively. If Q = q2 the PL assay is called symmetrical (SPL); otherwise the assay is called asymmetrical (APL). The doses of standard and test preparations are assumed to be equispaced on a log scale. Consider an incomplete block design D(v, b, P, It) with v = q1 + q2 doses arranged into b blocks of sizes kr, 4, . . . , kb such that the i-th standard preparation is replicated ri (i- 1,2, . . . , ql) times and the i-th test preparation is replicated ri (i=ql+ l,q,+2,..., q1 + q2) times. Let r = (rlr r2, . . . orJT, and k = (kl, k2, . . . , kb)T. ere T stands for transpose. Further, let the number of units in thej-th block receiving the doses of standard and test preparations be ksjand k,, j = 1,2, . . . , b, respectively. Clearly kj = ksj+ ktj. For convenience, the symboi si (or Q) will be used to designate the i-th dose of standard (or test) preparation as well as its effect. Under usual intrablock model for equi-replicated block designs, Kyi Win and Dey (1980) have shown that the important bio-assay contrasts Lp, L1 and L; will be estimated free from block effects if the following conditions are satisfied: ksj/ktj=ql’/q2 Pf-
for all j= ‘1,2,. . ..b.
(2. la)
+k,(q, =+19,
(2.lb)
r=3ktj(q2+ ‘9,
(2.icj
i =
f? l
where Ej I and Cyi denote the sum tion for standar and test preparation dose indices, respectively, in the j-th MO ones (1959) has shown that if ent contrast ST is the n vector of dose totals, s is the right eigenvectcr of the matrix (2.2)
P.D. Puri, L.K. Gupta / Incomplete Mock designs
is a vector of dose effects. total number of elements Y and were termed basic
123
atrices having diagona
with the same efficiency factor (I -pi), where pi (i = 1,2, . . . $,O:,are the evgenvalues with mmultiplicityQig k, of
The parameters of a PE
design are written as
where the Li’s are mutually orthogonal idempotent matrices of rank Qi satisfying rr/n, I is an identity matrix and is the vector of unities. For c designs the pseudo variance-covariance matrix Q of the estimated treatment effects “. i is given by +
5 [pi/(1 -pi)]
i=l
A PEB design is called simple ) if j4i takes only p and 0 with multiplicities Q and u-e - 1, respectively.
tW0
distinct ValUes
3. II3 design with two associate classes is said to be grou L-1, --_~-5-t, ~~~L---__ea.I * -- *-fimln@ nf ci7e d nvmbol U= TM symbols clau uc ak rarlg;eu ii%0 fib &LtGye -
Consider a semi-regul
p.D. Puri, L.R. Gupta / Incomplete block designs
124
+a,(ql+ 1). Let uii (i= 1,2, l ,m;j= 1,2, , t) denote the j-th symbol in the i-th group. btain D* by replacing aI t symbols uii (i = 1,2, . . . , a1;j = 1,2, . , t) of the first a1 groups with alt sets of standard‘ preparation and a2t symbols UQ t) of the last a2 groups by a2t sets of (i=a,+1,a2+1 ,..., a,+a2=nl;j=1,2,..., test preparation so formed. Since the basic design is semi-regular, every block of it contains exactly c symbols standard and a2 doses of test preparations are from each group and aI doses therefore, kS.= c erlal and ktj = c a2a2 and (2. la) associated with each symbol of is satisfied for *. It can be easily seen that the conditions (2. Ib) and (2. lc) are also ce the final design D* will estimate all the three important consatisfied for D free from block effects. trasts Lp L1 a We shall now illustrate the procedure of construction through an example. Consider a semi-reguiar GD design with para.,meters l
l l
v/=6, r=2, k’=3,
b=4, m=3, t=2, &=O, A2=!
(3.2)
with groups (1,2), (3,4) and ($6) with block contents (1,3,5), (1,4,6), (2,3,6) and (2,4,5). Suppose we are interested in an incomplete block design for an APL assay with q1=4andq2=$.V1ecantakeal=~2=2,al=l,~2- 2 and the sets of standard and test preparation as (sr, sq), (s2, sj); (tl, t8), (t2, t,), (ts9 td), (t4, ts) respectively. Now by replacing the S?lmb~iSof design (3.2) by these sets we get the new design in block size 6 with block contents as follows: (% s4, 61,
tl,
s39 tl,
f3r f6, @,
61,
f49 ts,
(‘92, s3, f2, t3, t6, t7!-
td,
s4, t2, t4r t5, t7),
It may be remarked here that two designs were reported by yi Win and Dey J a&mW%nc. An nnt (1980) for Q = 4, q2 = 8 in block size 6 by trial arrd error. I%,:, a Llb11 urq=JAau U” ..“C fdhw any pattern and, therefore, for analysis the inversion of the treatment or block pseudo information matrix has to be obtained by the usual method of matrix inversion, which is time consuming. The design reported here is available in the same block size and requires thq 11~1 same number of experimental units, and it is an C 0’ having very simple analysis.
e shall now study the properties of the design of the asic design can be written in partitione
w$er
be the incidence matrix
P.D. Pub, L.R. Gupta / Incomplete block designs
at if we arrange
designs. It can be ea then the r matrix of
t
tions are written. Then it can be seen that the =
[(r - Al)frk] diag[
where it can be easily be verified that @* is c=ql+q2,
b, r, k=k’v/v’,
er=aI(t- 1), I= 1,2, r =(l/q)diag(
with parameters I.+= a& - Al)/rk, p3 =
e3=v-(m-l)t,
_4J2= (1 /a2)
5, diag
and ere stands for null matrix. The pseudo variance-covariance matrix of f for design
where for I = 1,2,, )/r
and
s,=(r-_~))/[rk-ar(r-ii.l)p.
) be the vector of adjusted treatment t c x I) are the vectors of adjusted dose totals for stan preparations respectively. Let denote the (i, j, k)-th ele iS
where
125
P.D. Puri, L.R. Gupta / Incomplete block designs
126
any new designs can be constructtid by considering different semi-regular deaI (I= 1,2). Some sf the signs a by proper choice of aI ssay designs of Migam a of Kyi in and Dey (198Q)and S are particular cases of our designs. any of the designs are Ssimple analysis. In addition of having simple analysis, in many cases our design reantnl quires less experimb,. W units as compared to the designs available in literature. If al = a, I= 1,2, then we get an S-PEB design with parameters v=q,+q2,
b,r;
k=vk’/u’,
II It\11 = (l/‘a)I,~i~~-\l,r,~IJv
p=a(r+j/rk,
e=(t-I),
R
where a = q/a/t, i = 1,2. For a = 2, we devide the standard and test preparations into a1t and a2t groups each of size 2. The groups for tl;~opreparations can be taken as (.sj,Sag_++I), j = 1,2, . . . , al t, and ($ t,++ 1), j = 1,2, . . . , a2t, respectively. For D* in this case it can be shown that preparation contrast L,n and all odd order contrasts LLfi+ 1 and L& + l are estimated with full efficiency. To save space we shall not give the values of idempotent matrices t;$ from now onward. If we take a1al = a2a2, then we get an SPL assay design with parameters v = 29,
6. r, k = 2qk’/v’,
er= a,(t- l),
pi = a,(r - A1)/rk,
1= 1,2, e3 = v-m(t-
p3 = 0,
l)- 1, L1,
or example consider semi-regular G &sign with parameters (3.1) and a1= 1, a2= 2, al = 4 and a2 = 2. Then we get an SPL assay design for with block contents
let us take
(s,,s2,S7,s~,fl,t3rf6,fs), (S3rs4d5&,
tl, t4, f5, ts),
The parameters of this design are v=l6,
b=4,
r=2, c 8 was also re
to 32 in our case.
q1 = q2 = 8
P.D. Puri, dt. 6
Gupta / Incomplete block designs
127
into t disjoint groups ~a”size 2 ES(Sj,Szl-j+ I), j = into t disjoint groups of size 4 as
(8 t j9
j+l9
t
4f-j9t41-j+!
)9
j=
11,395,...,
with indices sum 2t + 1 and 8t + 2 respectiv series will estimate preparation and all odd the designs of this serial k= 6~.
full efficiency. For
Let ql =2t: q2= 3t; t is odd. re we can di-dide qr doses of sta prep;Pi on into t disjoint groups of size as in series 3 and the test prepar into t disjoint groups of size 3 with indices sum equal to $(3t + 1); clearly t odd to have this quantity integer. If t=3, the groups can be taken BQ es
(tl9te9td9
U,J49tg)9
(t39t59t7)-
For designs of this series LP, Lr and L; contrasts will be estimated free from block effects. For this series k = 5~. If q1= q2= 2t, then we get a design of size 4~. ere the groups can be taken as explained in series 1 with aI= I for i= i, 2. For this series all bio-assay contrasts are basic contrasts. Preparation and odd order contrasts are estimated with fuli efficiency whereas the even order contrasts are estimated with efficiency factor the above series if &‘= 2, the resultant design will have bloc lect semi-regular design D with parameters
f we se-
then we get the designs reported by Nigam a u=4t,
l&r,
k=4,
p=*,
@=2(t-i),
as a particuiar case of our designs. io-assay series c
e lis-
128
P.D. Puri, L.R. Gupta / Incomplete block designs
Kulshrestha, A.C. (1969). Gr I ihe efficiency of modified BIB designs for bio-assays. Biometrics 25, 391-393. Kyi Win and A. Dey (1980). Incomplete block designs for parallel line assays. Biometrics 36, 487-492. Nigam, A.K. and G.M. Boopathy (1985). incomplete block designs for symmetrical parallel line assays. J. Statist. Plann. Inference Pearce, S.C., T. Calinski and T.F. l+J-,,;iall(1974). The basic contrasts of an experimental design with special reference to the analysis of data. Biometrika 61, 449-460. Puri, P.D. and A.K. Nigam (1977). Partially efficiency balanced designs. Commun. Statist. Theory Methods A 6, 753-771.