NOTE ON THE ANALYSIS OF A RANDOMIZED BLOCK DESIGN
by
Junjiro Ogawa
University of North Carolina
This research was supported by the Office of
Naval Research under Contract No. Nonr-855(06)
for research in probability and statistics at
Chapel Hill. Reproduction in whole or in part
for any purpose of the United States Government
is permitted.
Institute of Statistics
Mimeograph Series No. 218
January, 1959
NOTE ON THE ANALYSIS OF A RANDOMIZED BLOCK DESIGN'
By Junjiro Ogawa
Institute of Statistics, University of North Carolina
Suppose that we are given b blocks each of which contains
or less homogeneous experimental units or plots and
varieties to be compared by experiment.
k more
k treatments or
We shall say that the design
is a randomized (complete) block design if we assign k treatments at
random (for instance take k cards with the numbers
l, ••• ,k on them,
shuffle them well and lay them out in a row to determine the position of
the first block.
Repetition of this process will produce the assignment
of the treatments in the second block and so forth.)
VA mathematically
rigorous treatment of this arrangement is at present not yet available.
An approximate test of varietal effects is possible by treating the arrangement as a two-way classification design ignoring the variation of
soil fertility within the rows." [1]
The purpose of the present note is to give a justification of the
usual analysis of the randomized block design.
Although the same argument
can easily be extended to general incomplete balanced block design, for
the sake of simplicity the simplest case, a randomized complete block,
is treated.
'This research was supported by the Office of Naval Research under
Contract No. Nonr-855(06) for research in probability and statistics at
Chapel Hill. Reproduction in whole or in part is permitted for any
purpose of the United States Government.
2
To begin with the explanation of some useful concepts of design, we
shall be concerned with general incomplete block design with v treatments, and
b blocks each of which contains
=kb
There are n
k plots.
experimental units or plots on the whole.
We
number them in some way but once for all; for instance, the i-th experi(j - I) + i .
mental unit in the j-th block bears the number
tion at i-th plot is denoted by xi
sented by an n-dimensional vector
called an
~servation
The observa-
and the whole observations are repre~
whose i-th component is
xi' and is
vector.
We shall define the incidence vectors of treatments as follows:
t:al
(1 )
~ =
t:a2
,
•
••
where
t:af =
1, if the plot f receives
the treatment a
0, otherwise
t:an
and the matrix
is called the incidence matrix of treatments.
is generated by
~1
•••
5v
The linear subspace which
is called treatment space.
Likewise the incidence vectors of blocks are defined by
I
al
1'J
1'J
(3)
]a
=
a2
•
••
,
where
1'J
af
=
1, if the plot f belongs
to the block a
0, otherwise
an
1'J
and the incidence matrix of blocks is defined by
and the linear subspace which is generated by ]1 •••
~b
is called the
-
block space.
~
The components of
projected into the block space with respect to
the basis vectors W is expressed as
N = (p'W
(5)
and is called the incidence matrix of the design.
(6)
1, if treatment a occurs
in block a
N = linaa II, where
0, otherwise
Now from the very definitions it is seen that
r,
~I¢
=
•
,
••
where r a stands for the replication of treatment a.
particular
Thus if in
= rv = r
then
(8)
T* - 1 qxpl
-r
'
*
B
1 HI
=k
are idempotents and they are the projection operators.
Evidently the treatment space and the block space has the intersection
which is generated by the vector
1 whose
components are all unity.
projection operator on this intersection is
where G is the
n x n matrix
whose elements are all unity
The
4
In the special case which we are now going to discuss
(10)
v=k,
r=b
and four matrices
(11)
I , G , B = kB* , T = bT*
are so-called lIrelationship matrices" of the design [2].
It is known that the linear closure of the matric set {I,G,B,T} is
a linear associative algebra and it is commutative.
Indeed, the multipli-
cation table is as follows:
I
B
T
G
B
kB
G
kG
T
G
bT
bG
G
kG
bG
nG
•
The decomposition of the unit element into orthogonal idempotents is
given by
I
1
(1
1
(1
1
1
1
1
=n
G + j(B'. - ~) + bT - ~) + (1 - i(B - bT + ~)
hence
(12 )
_ (1
1
I - ~ -
j(B -
1)
fiG
(1
1)
1
1
+ bT - ~ + (I - j(B - bT +
1
fiG)
•
~B - ~ is the projection operator of the
The meaning of (12) is:
1
1
contrast space into block space, bT - fiG is the projection operator into
treatment space and
I - ~B - ~T + ~
is the projection operator into
=,..1 'If\T{ x -
=-is -
error space, i.e.,
1 . 1)
(-kB
-.:.(3 x
n
-
I
K
-
1
I
-n--II x
-
I x,
-
where
n
B
=
L\'
f e block i
xf
and -x -- n
1
L
f=1
5
and
1T
1~)
1 ;Mo,'
1 I' 1
(b
- ~ ~ =b~~ ..! ~
n
( 14)
-_
t'fn"
':V~
-
1.
-
x ,
where
T,
•••
Tk
T =
,
E xf ,
Ta-rc
- 1
a=l, ••• ,v.
~af=l
The essential difference between the randomized block design and
two-way classification design is that the incidence vectors of treatment
are random in the former case whereas they are fixed vectors in the
latter case.
If the plot effect can be ignored, the underlying model for
a randomized block design is that the conditional distribution of the
residual
!
=~ - g! -
~.
1-
11' ~ 2
where g is the general mean and 11 = (t"
••• , t k ) and 2'= (b" ••• , bb)
are treatment effects and block effects respectively, given ~ is
There will be no loss of generality by assuming that
k
b
Eta = L a =
(16)
b
a=l
0
•
a=l
Obviously the probability of ~ is discrete and l/(kl)b •
If we denote the
n x k matrix whose elements are all unity by 'J, .
then
(17)
G
=Jq>1 =~JI
and since
~T -
tp = (~T - io) (~T - ~) = (f;¢ - *J)~'~(~~I
= b(~
- 1J) (~I
b
n b
and
- 1JI)
n
-~')
6
1
1
1 _ (
1
1
1)(
1
1
1
1- bT - i(B+ ~ - I - bT - i(B + ~ 1 - bT - i(B +. ~) ,
it follows that
(18)
(~I - ~JI)~ = (~,- ~JI )! + g (~I - ~J I )!. + (~I - ~J')q>. i + (~I - ~J' ) 'i'h
= {~,
- !J')e
+t
b
n
and
(19)
•
Thus
1
1
1
1
(1
1
.
~' (bT- ~)~ = !l(bT- ~)! +2i'b ~' - r/')! +
( 20)
and
1
1
1)· _
1
1
1
x' (1 - -bT - -kB+.:G
x - -e' (1 - -bT - -kB + .:G)e
..
n ....
n ...
(21 )
e·
Take an orthogonal matrix of the form
(22)
p
=
••
•
I
&
and let
e*
=P e
or
(2.3)
then we have
-e = P'e*-
•
bi'i
7
•
••
I
(J4> _ !J)
~ b
n
and
•
••
1
1
1)
P I( I --T
_n
b - -B
k +.:<3
n
and hence
1
o
• • • • • •
o
•
•
o
and since
Q
o
8
(~,
b
-1
J 1) (I -IT _lB +10) = (~, _lJ' ) (I - 1ftl )- (~I _lJ,)(~_lJ)q>'
n
b
k
n
b
n
k
b
n
b
n
_1
1. 1
1
-;41' - -J' - =4>'ff l + -J'ft'
b
n
n
_1
1
1.]*
- ~I - _JI - - 'l"
b
n
n
where J* stands for the
nk
1~
+ -.r'l"
n
-
1
-;4>'
b
(1
1)
n k q>1
-I -.':G
b k
1
+ _JI
n
_
- 0
'
k x b matrix whose elements are all unity
o
o
0
•
•
•••
o
R
o
Since
rank
1
1
o - .':G
n
~T
k -1
and
111
I - -T
- -B
+.':G
b
k
n
are orthogonal idempotents with
and (b-l){k-l) respectively, Q and R are orthogonal idem-
potents of rank
k - 2 and (b-l) (k-1) respectively.
Finally since
(26)
••
•
and
,
•
•
•
e*
n
if we consider the conditional joint distribution of ~/o~ and ~ /o~ ,
given q>, then they are mutually independent, i.e., T/02 obeys the
non-central chi-square distribution of d.f. k-l with non-centrality
parameter bt't/0 2
and
:t /0
2
obeys the chi-square distribution of d. f.
9
(b-1) (b-1).
Thus the conditional (given~) distribution of the statistic
(28)
is the non-central F distribution of d.f. (k-l~ (k-l)(b-l»
non-centrality parameter bl'!,/0 2
•
with
Consequently the absolute distribu-
tion of the above statistic is the same.
Thus this seems to offer a way of justification of the traditional
treatment of this problem provided that we can ignore the plot effect.
References
[lJ Mann, H.B., Analysis and design of experimen~. Dover Publications,
Inc., 1949, Chapt. VII, p. 76.
[2J
James, A.T., "The relationship algebra of an experimental design,"
Ann. of Math. stat., vol. 28, 1957.