A CLASS OF TWO REPLICATE INCOMPLE'rE BLOCK DESIGNS
by
J. Roy
University of North Carolina
•
This research was supported by theUn:i.ted
States Air Force through the Air Force
Office of Scientific Research of the Air
Research and Development Command. Reproduction in whole or in part is permitte.d
for any purpose of the United States
government.
Institute of Statistics
Mimeograph Series No. 201
June, 19$8
, .
I
A ClASS OF TWO REPLICATE INCOMPLETE BWCK DESIGNS
l
By J. Roy
Department of Statistics
University of North Carolina
o.
tptroduction and Summary.
Two replicate incomplete block designs for comparative trials are useful
when experimental units are costly and/or when experimental error is small.
many are known
£1_7 £3_7 £6_7 1:7_7 £8_7.
replicate designs
CE: 1 led
Not
In this paper a new class of two
Simple Partially Linked Block designs is introduced.
It is shown that with any of these designs, the variance of the estimate of the
difference in effects of two treatments can be at most of seven different types.
The general procedure of intra and inter- block analysis is developed and illustrated with a numerical example.
A list of these designs involving ten or fewer
plots per block is given together with the values of parameters required in the
analysis and the values of the efficiency-factor.
It turns out that most of these
designs are highly efficient with an efficiency-factor of the order of 75
0/0.
It
is indicated how other two replicate designs can be derived from these designs.
1.
Two general methods of analysis of experiments in randomized incomplete blocks.
Consider an experiment in randomized incomplete blocks in which v treatments
are tried in b blocks, each of k plots, each plot getting just one treatment and
each treatment being applied on at most one plot in a block and altogether on r
plots.
Let i\.
JU
denote the number of blocks in which the j-th and the u-th treat-
ments occur together and fJ. t the number of treatments common between the i- th and
i
the t-th blocks. (j,u
= 1,
2, ••• ,
Vj
i,t
= 1,
2, ••• , b).
Let the "effect" of the j-th treatment be denoted by O. and that of the i ... th
J
block by 13 • The usual assumption is that Yji the "yield" of the plot in the i-th
i
1. This research was supported by the United States Air Force through the Air
Force Office of Scientific R~search of the Air Research and Development Command.
Reproduction in whole or in part 1s permitted for any purpose of the United States
government.
- 2 -
'e
block getting the j- th treatment is given bY'
Y'ji
= ~i
+ OJ + e ji
where eji's are mutually uncorrelated random variables" each with expectation zero
and variance
cl.
For intra-block estimation ~i 's are regarded as constants whereas
for combined inter- and intra-block estimation 13 's are regarded as random variables
i
mutually uncorrelated and also uncorrelated with the eji's, each with the same
2 2 2
expectation ~ and variance r:J1. We shall write 6 = r:J /r:J l'
Let B denote the total "yield" of all plots in the i- th block and Tj that
i
for all plots getting the j- th treatment. We shall use the symbol { B j to denote
J
the total for all blocks in which the j-thtreatment occurs and similarly the sym.
J
bol {T i to denote the total for all treatments which occur in the 1-th block.
Thus
where n
ji
= 1 if the j-th treatment occurs in the i-th block and n ji
=0
otherwise.
Let
Qj = T j -
Qj
=
~{ B
J
j
Tj • kk {B 3j
Pi=Bi-itT}i
It is well known
£9_7
that the equations for intra- block estimation of treatment
effects are
(1.1)
j
=1,
2, ••• , v
- 3-
t>
r(l -
The design is said to be "connected" if the matrix C
u
if
j =
if
J ~ u
= «c.JU»
is of rank (v-l).
whatever follows, we shall be concerned with connected designs only.
In
We shall de-
note by (t , t , ••• , tv> any particular solution of these equations (1.1).
l
2
For combined inter- and intra- block estimation, the equations are
(1.2)
(J
=1,
2, ••• , v)
where
S
c jU
r(l - kia>
=u
if
j
if
J ~ u
=)
l _~
We shall denote a solution of these equations (1.2) by (tl'
t 2,
••• , tv).
We note that the equations (1.2) for combined estimation are obtained easily
from the equations (1.1) for intra- block estimation merely by replacing k by k+&
both in the expressions for CjU's and QJ's.
suggested in
£9_7.
This is somewhat simpler than what is
We shall call this the Q-method of estimation.
An alternative method
£1o_7
is to estimate the treatment-effects from the
equations
(1.;)
where
~
j denotes the total of the effects of all the blocks in which the J-th
treatment occurs:
.. 4 ..
'e
For intra-block estimation the block-effects are estimated from the equations
(1.4)
= l~
(i
2, ••• , b)
where
)
k(l-
~
t -fJ
it
r
=t
if
i
if
i ~ t
We shall denote a solution of these equations (1.4) by (bl~ b2 , ••• ~ bb).
For
combined inter- and intra-block estimation, the block.. effects are estimated from the
equations:
=1,
(i
2, ••• , b)
where
i
if
i ~ t
J.L
--it
r
=t
if
We note that the equations (1.4) and (1.5) are identical except that the diagonal
terms d
ii
and.
d ii differ by 6. A solution of the equations (1.4) will be denoted
by (bl~ b 2 , ••• , bb)·
The method here described is a somewhat simplified version
of what is suggested in
£10_7.
We shall call this the P-method of estimation.
To estimate the parameters a
2
and 8, the analysis of variance can be carried
out as follows:.
ANALYSIS OF VARIANCE
Variation due to
Sum of squares
Degrees of Sum of squares
freedom
Variation due to
Blocks (Unadjusted)
S~
b.l
SB
Blocks(Adjusted)
Treatments(Adjusted)
Srr
v-l
S*T
Error
SE
n-b.v+l
SE
Treatments(UnadJusted)
Error
Total
T
n-l
T
Total
- 5Here, n
= bk =vr
total of the
denotes the total number of plots, G is the total and G2 the
sq~res
of all the "yield's" and
S*
T
·2
Then unbiassed estimates of u
and u
2
l
=.!
v
E
r j=l
T~
G2
- J
n
are obtained from the fa.ct that the expecta-
tions of SE and SB are given by:
(~) = (n (SB)
b - v + 1)0'2
= (b-l)O'2
.
2
+ (n-v)ul
As an estimate of 8 one can take
(b-l)SE
(1.6)
d
='T(-n--:;b--v-+l='"')~S;-B--"-"(-n-- v--)~s='-E
in the sense that the ratio of the expectations of the numerator and the denominator
of d is equal to 5.
Generally d is a consistent estimate of 8.
Consider the variance of the intra-block estimate of OJ - 0u the difference
between the effects of the j-th and the u-th treatments,
The average variance of all such differences is given by
- 6_
V
1
:=
eV)
2
v
v
E
l:
j=l u=j+l
2
v jU
CT
whereas in a randomized block experiment any such difference would be estimated
with a variance
2 2
-r cr
if the error variance were the same.
E
(1.7)
This bears to the former the ratio
=V(V-l>/ ~
r
~
v
j=l u=j+l
jU
which is called the "efficiency-factor" of the design.
factors are generally preferred.
It has been shown
o<E <
1 - 11k
- 1 - llv
2.
Designs with high efficiency
£12_7
that
•
Dual1zation of designs.
New designs have sometimes been
£8_7 £11._7 £1"5_7
constructed by dualiza-
tion, that is by interchanging the role of the blocks and treatments of a given de-
* plots,
such that each treatment occurs at most once on each plot and altogether on r * plots.
It is easy to see that the dual design D will involve v =b * treatments in b = v*
blocks each of k = r * plots and each treatment will occur in r = k* plots. It is
also obvious that if the design D* is easily analyzable by the Q-method,the dual
design D can be readily analyzed by the P-method. It has also been shown £12_7
sign.
*
Consider a design D involving v
* treatments
i"n b
* blocks,
that the efficiency factor E of the design D is given by
(2.1)
E
=
* *
* *
*
(b -l)E
(b*-v)E + (v -1)
*
where E* is the efficiency-factor of the design D.
Consequently
each of k
- 7according as
b
* ~> v *
•
Therefore, if we start with a design D* with a reasonably high efficiency-factor in
which b* > v*, by dualizing it, we always get a design whose efficiency-factor is
still higher.
}.
Partially balanced association
scheme6~
Given ~ + ~ + ... + ~ + 1 objects, a rela.tion- satisfying the followins- -. conditions is sa1d-to-bea partially balanced assOciation scheme With
associate classes:
(i)
(ii)
m
L2_7, £4_7
Any two objects are either l-st, or 2-00, ... , or m-th associates.
The relation of association is symmetrical, that is, if the object a
is the i-th associate of the object
of a (i
=1,
~,
then
~
is the i-th associate
2, ••• , m)
(iii) Each object has nl first associates, n2 second associates, ••• , n
m
m-th associates.
(iv)
If any two objects a and ~ are i-th associates, then the number of ob.
jects which are the j-th associates of a and the k-th associates of
~
is Pijk, i~dependent of the pair of i-th associates (1,j,k = 1,2, ••• ,m)
i
The parameters pjk are not all independent. They satisfy, for instance, the following relations:
i
Pjk
1
i
=Pkj
j
ni Pjk = nj Pik
k
= ~P1j
if 1
i
P
i
1
jl + Pj2 + ••• + Pjm =
=j
•
- 8Partially balanced association schemes with two associate classes are classified in
£4_7
known cases.
and listed in 15_7.
Though not exhaustive, these cover all the
The five types discussed in £4_7 are (1) Group Divisible (GD) I (2)
Triangular (T), (3) latin square type (IS), (4) Cyclic (C) and (5) Simple (51).
In a Group Divisible type of association scheme, there are mn objects which
fall into m groups of n objects each.
Any two objects in the same group are first
associates and two objects from different groups are second associates.
n
Thus
2 = n(m - 1)
o
o
n(m-l)
n-l
n-l
n(m-2)
In a Triangular type of association scheme I there are p(p-l) /2 objects arranged in a square array of p rows and p columns as follows:
(1)
The positions in
the principal diagonal (running from the upper left- hand to the lower right- hand
corner) are left blank.
(ii)
The p(p.l)/2 positions above the principal diagonal
are filled by the numbers 1, 2, "., p(p-l)/2 corresponding to the objects.
(iii)
The p(P.l)/2 positions below the principal diagonal are filled so that the ar~yls
symmetrical about the principal dia.gonal.
Two objects a.re first associates if they
lie in the same row (or column) in this array, otherwise they are second associates.
For this scheme
p-2
4
2p- ,8
2p-8
(p-4)(p-5)/2
- 9 ..
In the Latin square type of association scheme with t constraints, there are
s2 objects arranged in a square scheme.
For the case t
=2,
two objects are first
associates if they occur in the same row or in the same column;
second associates.
otherwise they are
For the general case, a set of (t-2) mutually orthogonal Latin
squares are taken and two objects are first associates if they occur in the same
row or column or correspond to the same letter of one of the Latin squares.
In
this scheme
~ = (a-l)(s.. t+l)
nl • t(a-l),
t
2
..
3t + s
(t-l) (s.. t+l)
(t-l) (s- t+l)
(s- t) (s- t+l)
t(t-l)
t( s-t)
t( s-t)
(s-t)
2
+ t - 2
4. Simple partially linked block designs.
An
allocation of v treatments in b blocks each"f k plots will be called a
simple partially linked block (SPtB) design if the following conditions are satis..
fied:
(1)
Each treatment occurs at most on one plot in a block and altogether
on two plots.
(H)
Any
two blocks have at most one treatment in common.
(i1i) Two blocks are first (second) associates if they have one (no) treat..
ment in common and this association scheme is partially balanced with
parameters nl ,
~
i
and Pjk.
- 10 -
Thus,
and, of course,
r
= 2.
SPLB designs can be easily constructed as follows.
.
Given any partially
*
~
balanced association scheme V with two associate classes, a design D
with
+ n2 + 1 treatments and .k* = 2 plots per block can be easily constructed by
considering the objects as treatments and forming one block with each pair of treat-
v*
= nl
ments that are first associates.
b*
1
='2nl(nl+n:::+l)
In this design D*, obviously there are
.
blocks, each treatment occurs on r*
= n plots
and anypBirs··of
l
treatments occur together on one or no block accorcing as they are first or second
associates.
The dual D of this design D* is obviously a SPLB design.
By inter-
changing the nomenclature of "first" and "second" associates, a second SPLB design
can similarly be constructed from the same association scheme
'.
..;(
. 'u.
A SPLB design cen easily be analyzed by the P-method described in section 1.
Let us use the notation 8 to denote summation over first associates.
1
(1.4) for intra-block estimation reduce to
2
The equations
If Pll ~ 0, the general solution of these equations,except for an arbitrary constant,
is given by
(4.1)
where
f
= a/A
fl = l/A
- 11 -
and
The intra- block estimates 'of treatIi'len'c effects are then g:lvep. by
( 4.2)
where
t
b
j
j
=.!rT
- 1Cb 1Jj-7
2!. j
denotes the sum of the bils for the two blocks in which the j-th treat-
ment occurs.
Take two treatments:
say, the j-th and the u-th.
Two cases may arise: (X)
the two treatments occur together in a block or (y) they do not.
In case (X) there are three blocks in which at least one of the two treatments occurs.
In one of these blocks both the treatments occur.
other two blocks.
Consider the
We sh3.ll say that the j-th and the u-th treatments form a pair
of the type Xl if these two blocks are first associates and of type X2 if these
two blocks are second associates.
In case (y) there are four blocks, in two of which the j-th treatment occurs
and in the other two the u-th treatment occurs.
With these four blocks, it is
possible to form four differrmt pairs of blocks such that in each pair there is one
block containing the j-th treatment and one block containing the u-th treatment.
If v is the number of first associate pairs amongst these four pairs of blocks, we
sha.ll say that the j-th and the u-th treatments form a pair of the type Y (v
v
0, 1, 2, 3, 4).
=
vie have thus classi,fied all possible pairs of treatments into seven distinct
types:
Xl' ~ and YO' Yl , Y2 ' Y ' Y4 •
3
Consider nm., the variance of the intra-block estimate of (;). J
ference between the effects of the j-th and the u-th treatments.
computation, it is seen t!1at
Var(t.-t )
J
u
= v.JU0-2
Q ,
U
the dif-
After a little
- 12 where the value of vju depends on the type of pair formed by the j-th and the u-th
treatments and is tabulated below:
Type of pair
Value of v.
JU
of treatments
II
1 + , 1 + ,
1 +
21
+
211
1 + 2' + "1
1 + 2'
1 +
21 -
(1
1 + 2( - 2"1
We thus see that in all seven different precisions are possible.
T·,. compute the efficiency-factor E of the
SF!Jl design, we observe that the
efficiencj-factor E* of the dual design is given by
and therefore from (2.1),
E
=
( v-1)A
For combined inter- and intra-block estimation, we have from (1.5)
2P.~
- 1) -
fhe general solution of this (except for an
arbitrar~
constant) is given by
(4.4)
wnere
f
= a/ A
~l ,.. 1/ A
and
-a
= a + 28
1 = A + 28(a+~)
+
48
2
•
The combined estimates of the treatment effects are tnen given oy
J
where { b j denotes the sum of the hi's for the two blocks in which the
j-th treatment occurs.
The analysis of variance can be carries out as in section 1.
components are computed in the following order:
The various
first the total sum of squares
T, then the unadjusted block sum of squares S; and the unadjusted treatment sum
* next the adjusted block sum
of squares ST'
01'
the adjusted sum of squares ST = SB + S; -
s;
squares SB = .Eb b. P. and finally
i=l 1. 1.
and the error sum of squares
SE • T - S*
B - ST = T - SB - s*
T .
,.
A list of simple partially linked blOCk design with ten or ,fewer plots
per block.
A list of SPLB
designs with k
~
association schemes 1s presented here.
10 derivable from known partially balanced
fhe list is arranged in increasing order of
v and under the same v, in increasing order of k •. fhe values of the parameters
-14v,b,k = nl,a,~,E and the type of the association scheme are shown. The word "interchange" indicates that in the definition of the association schemes given in section
3, the nomenclature of first and second associates have to be interchanged. Of the
designs listed, the lattice designs are, of course, well known and a few others
with a GD type of association scheme are given in ~8_7. The other designs are new
List of SPLB
Serial
Number
v
k=n
1
b
a
designs with k
A
- 10
<
Association scheme
E
4
2
4
4
8
0.600
aD m = 2, n = 2 interohange
9
12
15
3
4.,
6
oJ
6
6
10
6
4
18
24
10
0.661
0.150
0.565
aD m = 2, n = 3 interchange
T p =4
T p .. S interchange
16
18
24
25
21
4
4
6
5
6
8
9
8
10
9
8
5
8
10
9
32
18
48
50
.54
0.114
0.680
0.807
0.150
0.796
30
36
39
40
45
6
6
6
8
6
10
12
13
10
1.5
7
12
1
10
8
40
72
39
80
4.5
0.182
0.718
0.760
0.841
0.7.5.5
48
48
49
54
.57
6
8
1
9
6
16
12
14
12
19
6
12
14
12
7
32
96
98
108
38
0.740
0.829
0.800
0.848
0.738
60
60
64
68
72
8
10
8
8
9
1.5
12
16
11
16
8
12
16
60
120
128
68
96
0.812
0.863
0.818
0.807
0.833
7.5
81
100
100
105
10
9
8
10
10
1.5
16
25
20
21
150
162
50
200
84
0.854
0.833
0.784
0.836
T
m = 2, n = 9 interchange
s = .5, i = 2
m = 2, n == 10
P =7
105
130
135
180
10
10
10
10
21
26
27
36
126
104
135
12
0.841
0.832
0.83.5
0.813
T
P :: 7 interchange
*Lattice
9
11
1.5
18
7
20
9
13
II
14
8
0.8J.~6
GD m = 2, n
=4
interchange··
18 s = 3, i .... 2
GD m = 4, n = 2 interohange
GD m = 2, n .. 5
GD m .... 3, n = 3 interchange
T P = .5
GD m = 2, n
c
GD
T
=6
interchange
m = 5, n = 2 interchange
p = 6 interchange
LS s = 4,
GD n = 4,
GD m"" 2,
GD n = :;,
i =2
m = :; interchange
n =7
m = 4 interchange
31.
P =6
m .. 6, n = 2 interchange
GD m = 2, n = 8 interchange
T
GD
C
L3 s = 4, i :: 3
m = :;, n = 5 interchange
GD
(ill
LS
GD
3L
Sl.
IS
s = 6, i
=2
- 15 -
6. Numerical illustration.
To illustrate the numerical procedure, let us consider the folloWing artificial data (Table 6.l) giving the plan and the yields of a randomized experiment
with a SPLB design involving 15 treatments in 10 blocks each of 3 plots. The
figures in brackettes indicate the serial numbers for the treatments and the figures
below them are the corresponding yields.
TABLE 6.1 FIELD PLAN AND YIELDS
Blocks
Blocks
(13 )
(6)
(3)
4.5
5.8
4.2
(10)
(13 )
(7)
2
7
9.9
5.3
6.7
(2)
(3 )
(l)
3
lj.
5
(14)
(ll)
( 4)
4.7
5.1
3.8
(10)
(11)
(12)
6.3
5·7
5.8
(15)
(8)
(5)
4.9
8.0
7.5
(12)
(15)
(l)
7.3
4.2
2.4
(2)
(9)
(14)
8.6
3.0
5.4
6
1
8
lj..6
4.4
2.3
(7)
(8)
(9)
9
3.9
7.1
0.8
( 6)
(4)
(5)
10
2.2
3.5
5.3
This was obtained by taking the
Triangular association scheme with p=5 represented by the follOWing square array:
x
1
2
3
4
1
x
5
6
7
2
5
x
8
9
3
6
8
x
10
4
7
9
10
x
- 16 in which each number stands for an object and two objects are first associates if
they do not occur together in the
row nor in the same column.
s~e
Thus
1
«P2» ['2 :]
iJ
=
oJ
Forming blocks With each pair of first associates, we get the design D*
Blocks
4
1
7
6
5
.8
10
9
11
12 13
14 15
Treatments 1,8 1,9 1,10 2,6 2,7 2,10 3,5 3,7 3,9 4,5 4,6 4,8 5,10 6,9 7,8
with parameters v*= 10, b*= 15, k*= 2, r*= 3. Dualizing we get the SPLB design:
Blocks
4
1
5
6
7
8
9
10
Treatments 1,2,3 4,5,6 7,8,9 10,11,12 7,10,13 4,11,14 5,8,15 1,12,15 2,9,14 3,6,13
with parameters v
= 15, b = 10,
k = 3, and of course, r
= 2.
For this design, we have
a
=~
+ pi1- pi1 = 3 + 1 - 0
2
A = bP11
f
=4
= 10.1 = 10
= a/A = 0.4
i1 = l/A
= 0.1
and
::I
(15-1) • 10
(15 - 10). + 2.3 t 4 • (10-1)- 3
140
140
6.(33) = 24S
= 50 +
J
= 0.565
In the actual lay-out the blocks and the plots within a block have been re-arr~ged
at raldom.
-17TABLE 6.2 DETAILS OF Cm1?UTATION
...i-. .
Blocks
i
1
2
3
4
5
6
7
8
9
10
Total
Treatments
Bi
i Tli
2Pi
14.;
21.9
11.3
11.8
1100
13.6
17.8
20.4
13.9
17.0
26.4
36.6
26.5
29.;
28.1
28.2
40.1
37.0
26.9
21.1
2.6
7.2
-3.9
-.5.9
-6.1
-1.0
-4.;
3.8
0.9
6.9
153.2 306.4
Tj
1
2
3
4
.5
6
7
8
9
10
11
12
13
14
15
4.7
13.2
8.6
7.3
12.8
8.0
10.6
15.1
3.8
16.2
10.8
13.1
9.8
10.1
9.1
Total
153.2
-*
**
Check:
Check:
First associates of i
2,
1,
1,
2,
1,
;,
2,
4,
3,
3,
3,
4,
9,
8,
6,
7,
6,
;,
7,
4,
7
10
10
8
10
9
9
8
6
3,
3,
1,
;,
5,
1,
2,
4,
4~
2,
6,
7,
1,
6,
8,
b
-2.8
-7.8
10.4
17.9
;.4
-3.7
7.1
-11.1
-4.6
-10.8
0.76
2.10
-0•.52
-0•.57
-1.90
-0.77
-1.09
0.41
-0.10
1.68
0*
0*
Blocks in which
treatment j occurs
sum is zero.
sum is G/2
5
Sl( 2Pi)
9
10
3
6
8
;
4
8
10
7
7
9
2
10
9
i. b}j
-0.62
1.16
0.24
-2.67
--1.49
-1.14
1.53
-0.16
1.11
1.01
-1.86
-1.19
2.86
0.91
0.31
0*
b
i
i
0•.5032
1.3916
-0.4848
-0.6391
-1.2317
-0.4017
-0.7128
0.4303
0.0160
1.1888
-0.0002*
0*
t.
S-l
l b
Jj
-t
2.66
6.02
4.18
4.98
7.14
4.57
4.54
7.63
1.34
7.60
6.33
7.14
3.41
4.60
4.40
-0.4688
0.7040
0.0184
-1.6334
-0.8014
-0.7285
0.7525
-0.2088
0.5491
0.6188
-1.1745
-0.7568
1.8948
0.7811
0.4463
2.584
6.248
4.291
4.467
6.801
4.364
4.924
1.654
1.625
7.791
5.987
6.928
3.953
4.656
4.327
J
76.60** -0.0006*
j
76.600**
- 18 To carry out the analysis of variance, we compute:
G = 153.2
G2In
n = 30
=
782.341
= G2- G2/n ..
118.859
2
ZB.=
2477.96
1
*= -1 EB2- G2/n =
SB
3
i
43.646
ET~= 1737.78
* "21 ZT 2- G2/n
ST=
j
86.549
2= 901.20
T
G
TABIE 6.3
=
ANALYSIS OF VARIANCE
Variation due to Sum of squares Degrees of freedom Sum of squares Variation due
to
Blocks
(Unadjusted)
43.646
9
26.406
Blocks
(Adjusted)
Treatments
(Adjusted)
69.309
14
86.549
Treatments
(Unadjusted)
5.904
6
5.904
118. !3~9
29
118.859
Error
Total
Error
Total
To test if treatment differences are significant, we compute the variance-ratio
F
which with
14
= 69.309L14
= ;).
c O~l
~ .90476oJ
and 6 degrees of freedom is significant at the .5
%
level.
To test any particular treatment difference, say that between treatments
1 and 2, we proceed as follows:
is
The best intra-block estimate of the difference
·.19 Now, treatments 1 and 2 occur together in block 3 and the other blocks in which they
occur are:
block 9 (in which treatment 1 occurs) and block 10 (in which treatment
2 occurs).
But the pair of blocks 9 and 10 are second associates because they do
not have a treatment in common.
Hence, the treatments 1 and 2 form a pair of type
X2• The variance of (tl - t 2 ) is thus
(1
+
t)
2
2
a = 1.4 a
and this is estimated by
1.4 x 5.904/6
=
1.3176
and the standard error is
~1.3776 = 1.17371.
We then have the Student's ratio
t =
e
-3.36
r.rr:fl'I
=
2 86
which with 6 degrees of freedom is significant at the 5
1
%
3
••
%
level but not at the
level against both-sided alternatives.
For combined estimation, we have:
_
9x
d - 6 x 26.406
a=a
+
A= A +
5~904
.-:t) x
=
6
= 0.7
I
043
2d = 5.52086
2d(a + n1 ) + 4d2
= 10 + 1.52086(4+3)
.,
5a~04
aI A=
.-11 = l./l"=
+
4 x 0.57825 = 22.95902
o. 2L.047
0.043;;6 •
The best combined estimate of the difference between treatments 1 and 2 is then
found to be
- 20 -
7. Construotion of other two replioate designs:
In like manner, two replicate designs can be constructed from any partially
balanced assooiation scheme with m > 2 classes, by first constructing a partially
balanoed incomplete blook design with k-2 and A - A2= .•• = Ap• 1, Ap+l= ••• =Am=O
l
and then dualizing it. By replacing each objeot by a group of t(t ! 2) objeots
in a partially balanced association soheme with m assooiate olasses, one gets
again a partially balanced assooiate scheme with (m+l) assooiate olasses.
result can be used in oonstructing other two replioate designs.
This
Another way would
-
be to replace eaoh treatment in a SPLB design by a group of t(t > 2) treatments.
This, however, Will not be pursued in this paper.
8.
Aoknowledgment.
I wish to thank Professor R. C. Bose for kindly reading the manuscript and
oalling my attention to ~3_7.
- 21 REFERENCES
7
r l R. C. Bose, npartially balanced incomplete block designs with two
associate classes involving only two replications.~ Cal. Stat. AsAcn. Bull.,
Vol. 3 (1951) pp. 120-125.
["2_7 R. Co:, Bose and K. R. Nair, "Partially balanoed incomplete block
designs. U Sankhya" Vol~ 4 (1939), pp. 337-372.
...
.
r3J R. C. Bose and K. R. Nair, "Resolvable incomplete block designs with
two rep!ications. 1t Institute of Statistics, University of North Carolina, Mimeograph Series No. 69, (1953).
r4_7 R. C. Bose and T. Shimamoto, 'Classification and analysis of partially
balanced incomplete block designs with two associate classes. n Jour. Amer. Stat.
Asscn., Vol. 47 (1952), pp. 151-184.
r5
7 R. C. Bose, ti. H. Clatworthy and S. S. Shrikhande, /lTables of partially
balancea designs with two associate classes. It North Carolina ~ricultural Experiment Station Technical Bulletin no.107, (1954).
~6_7 K. R. Nair, ItPartially balanced incomplete block designs involving
only two replications." Cal. Stat. Assgn. B~, Vol. 3 (1950), pp. 83-86.
e
["7 7 K. R. Nair, nSoma two replicate partially balanced designs II.
Stat. Aasen. Bull., Vol. 3 (1951), pp. 174-176.
Cal.
-
£8 7 c. s~ Rarnakrishnan, "'rhe o~al of a PBIB design and a new class of
designs With two replications." Sankhya, Vol. 17 (1956), pp. 13319 7 c. R. Rao, "Genera. methods of analysis for incomplete block designs. II
Jour. Amer: Stat. Asscn., Vol. 42 (1947), pp.541-56l.
[)07 c. B. Rao, "On the recovery of interblock information in variebal
trials." ~ankhya, Vol. 17 (1956), pp. 105rl17 J. Roy and R~ G, Laba, t~lassification and analysis of linked block
designs.It-Sankhya, Vol. 17 (1956), pp. 115-
7
~12
J. Roy, I~n the efficiency-factor of block designs." Sankhya,
Vol. l!, (1957), pp.
L13 7w. G. Youden, "Linked blocks: A new class of incomplete block
designs. Ii -(Abstract) ,Biometrics, Vol. 1 (1951), pp. 124-
---- .