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A note on A-efficiency of connected designs
ent of
ima University,
n
ctober 1986; revised manuscript receiv Recommended by
AbGr dc3.’ Khatri (1982) justified a statement “ mong the variance balanced designs, one should choose the design which has the minimum nu r of plots”, in terms of t2S of connected block designs are here presented to disprove hatri’s statement. Some general results are also provided to stress our statements. A
Subject Classi~cation: 62K10.
Key words and phrases: A-efficiency; BIB design; ‘VBdesign.
As the overall efficient esign, based on all the estimablep c functionsof a connecte ) design with parameters O,b, ri, kj9 e shows that t
where for N being the incidence matrix of t
*
0378-3758/88/$3.50
esign,
e
0 1988, lsevier Science Publishers
S. Kageyama / A-efficiency of connected designs
222
$y producing a new additional design of higher A-efficiency than the designs preri as follows, n being the total number of plots. design with u=8, b=28, r=7, k=2, d=l, and n=56, tr( 17.9.
design given by hatri (1982) with v=8, b=20, r= , kj=J for j= (ii) 1,2,3,4 and kj =2 otherwise, and n =48, tr( design given by Tyagi (1979) misprint; one block ($7) should be (3,8)], with v=8, b=16, ri=S for i=1,...,6 and r7=~=7, kj=6 for j= 1,2,3 and kj = 2 otherwise, and n = ith u=B, b=12, r=5, design given by Agarwal kj=4 for j=l,..., 8 and kj = 2 otherwi The same observation is again confirmed for block designs with v = 13,20 and 24, by Agarwal and mar (1985,1986). Now we produce some counterexamples (having large n and small tr( hatri’s statement. design with parameters v = 8, b = 14, r = 7, k= 4 v) onsider an existing and 11=3. Then we have for v= 8, n = 56,
tr(W-‘) = 4.726. design with parameters v = b = 8, r = k=7 and
(vi) Consider an existing il = 6. ‘Then n = 56,
tr(
-I) = 2.0594.
(vii) Consider a V design with parameters v = 8, b= 9, r==8, kj =7 cjr 8, having blocks (2,3,4,5,6,7,8),
(1,3,4,5,6,7,8),
(L&4,5,6,7,8),
(L2,3,5,6,7,8),
(1,2,3,4,6,7,8),
(1,2,3,4,5,7,8),
(1,2,3,4,%6,8),
(1,2,3,4,5,6,7),
(l&3,4,5,6,7,
S),
in which C= (55 b). Then n = 64 (being the largest among the above examples) and t (being the smallest). hese examples (v)-(vii) show that hatri’s statement is not valid in general. design e+ts for that v, it gives re of constructin
S. Kageyama / A-efficiency of connected designs
223
parameters v = b, r = k = v lt is easy to show t u=b,
rlk=v-1,
A=v-2
esig
rameters
g
(1.1) that
v2
1.
k2
The qtuntity F is now minimized on r and k un er r=k and u-- 2 k 2 2 for v. The pijrtiaf Arivative of F on k is then given by
given
k) + (v - l)2(r + 1) r(k- 1)2 < 0 for any parameters outwhenk=v-1. which implies that the minimum of F will be fo since rr k, it is easy to ove that the minimu ofF9 for k=v-1 (v is attained only when =k. Then we get v=b and r=k=v-1. r(k - l)/(v - 1) = v - 2 which completes the proof. Note that there always exists a design with parameters v = b, r = k = v A = v - 2. In fact, such a design with v = 8 p uces example (vi). the design of example (v s the only design mum (i.e., 2.05 17275). increases, then dentally, in general, for given v and r i fficiency factor tr ‘) decreases. This is because there is a re E (==h/(rk)) and tr( -*) as follows: tr(#-!
j
=
b-v 1 k +;+ '
(r+k)(v-1) rkE
’
224
S. Kageyama / A-efficiency of connected designs
author would like to thank the referee for his helpful suggestions. o due to Professor Y. Shintani for his comments.
G.G. Agarwal and S. Kumar (1984). On a class of variance balanced designs associated with GD designs.
Calcutta Statist. Assoc. Bull. 33, 18’7-190. G.G. Agarwal and S. Kumar (1985). A note on construction of variance balanced designs. .I. Ind. Sot. Agr. Statist. 31, M-183. G.G. Agarwal and S. Kumar (1986). On a class of variance balanced incomplete block designs. Commun. Stat&.-Theory Meth. 15, 15294533. C.G. Khatri (1982). A note on variance balanced designs. J. Statist. Plann. Inference 6, 173-W. S.C. Panandikar (1986). A note on A-efficiency of certain equiblocksized equireplicated connected designs. Calcutta Statist. Assoc. Bull. 35, 99401. B.N. Tyagi (1979). On a class of variance balanced block designs. J. Statist. Plann. Inference 3,333-336.
ima University,
n
ctober 1986; revised manuscript receiv Recommended by
AbGr dc3.’ Khatri (1982) justified a statement “ mong the variance balanced designs, one should choose the design which has the minimum nu r of plots”, in terms of t2S of connected block designs are here presented to disprove hatri’s statement. Some general results are also provided to stress our statements. A
Subject Classi~cation: 62K10.
Key words and phrases: A-efficiency; BIB design; ‘VBdesign.
As the overall efficient esign, based on all the estimablep c functionsof a connecte ) design with parameters O,b, ri, kj9 e shows that t
where for N being the incidence matrix of t
*
0378-3758/88/$3.50
esign,
e
0 1988, lsevier Science Publishers
S. Kageyama / A-efficiency of connected designs
222
$y producing a new additional design of higher A-efficiency than the designs preri as follows, n being the total number of plots. design with u=8, b=28, r=7, k=2, d=l, and n=56, tr( 17.9.
design given by hatri (1982) with v=8, b=20, r= , kj=J for j= (ii) 1,2,3,4 and kj =2 otherwise, and n =48, tr( design given by Tyagi (1979) misprint; one block ($7) should be (3,8)], with v=8, b=16, ri=S for i=1,...,6 and r7=~=7, kj=6 for j= 1,2,3 and kj = 2 otherwise, and n = ith u=B, b=12, r=5, design given by Agarwal kj=4 for j=l,..., 8 and kj = 2 otherwi The same observation is again confirmed for block designs with v = 13,20 and 24, by Agarwal and mar (1985,1986). Now we produce some counterexamples (having large n and small tr( hatri’s statement. design with parameters v = 8, b = 14, r = 7, k= 4 v) onsider an existing and 11=3. Then we have for v= 8, n = 56,
tr(W-‘) = 4.726. design with parameters v = b = 8, r = k=7 and
(vi) Consider an existing il = 6. ‘Then n = 56,
tr(
-I) = 2.0594.
(vii) Consider a V design with parameters v = 8, b= 9, r==8, kj =7 cjr 8, having blocks (2,3,4,5,6,7,8),
(1,3,4,5,6,7,8),
(L&4,5,6,7,8),
(L2,3,5,6,7,8),
(1,2,3,4,6,7,8),
(1,2,3,4,5,7,8),
(1,2,3,4,%6,8),
(1,2,3,4,5,6,7),
(l&3,4,5,6,7,
S),
in which C= (55 b). Then n = 64 (being the largest among the above examples) and t (being the smallest). hese examples (v)-(vii) show that hatri’s statement is not valid in general. design e+ts for that v, it gives re of constructin
S. Kageyama / A-efficiency of connected designs
223
parameters v = b, r = k = v lt is easy to show t u=b,
rlk=v-1,
A=v-2
esig
rameters
g
(1.1) that
v2
1.
k2
The qtuntity F is now minimized on r and k un er r=k and u-- 2 k 2 2 for v. The pijrtiaf Arivative of F on k is then given by
given
k) + (v - l)2(r + 1) r(k- 1)2 < 0 for any parameters outwhenk=v-1. which implies that the minimum of F will be fo since rr k, it is easy to ove that the minimu ofF9 for k=v-1 (v is attained only when =k. Then we get v=b and r=k=v-1. r(k - l)/(v - 1) = v - 2 which completes the proof. Note that there always exists a design with parameters v = b, r = k = v A = v - 2. In fact, such a design with v = 8 p uces example (vi). the design of example (v s the only design mum (i.e., 2.05 17275). increases, then dentally, in general, for given v and r i fficiency factor tr ‘) decreases. This is because there is a re E (==h/(rk)) and tr( -*) as follows: tr(#-!
j
=
b-v 1 k +;+ '
(r+k)(v-1) rkE
’
224
S. Kageyama / A-efficiency of connected designs
author would like to thank the referee for his helpful suggestions. o due to Professor Y. Shintani for his comments.
G.G. Agarwal and S. Kumar (1984). On a class of variance balanced designs associated with GD designs.
Calcutta Statist. Assoc. Bull. 33, 18’7-190. G.G. Agarwal and S. Kumar (1985). A note on construction of variance balanced designs. .I. Ind. Sot. Agr. Statist. 31, M-183. G.G. Agarwal and S. Kumar (1986). On a class of variance balanced incomplete block designs. Commun. Stat&.-Theory Meth. 15, 15294533. C.G. Khatri (1982). A note on variance balanced designs. J. Statist. Plann. Inference 6, 173-W. S.C. Panandikar (1986). A note on A-efficiency of certain equiblocksized equireplicated connected designs. Calcutta Statist. Assoc. Bull. 35, 99401. B.N. Tyagi (1979). On a class of variance balanced block designs. J. Statist. Plann. Inference 3,333-336.