AGIUQJLTUPM. VARIATION AIJD SECOND-ORDER STATIOW\RY
ItAHOOII I1EASURSS
r 1. 1. 11-10rumIT
University of Western Australia
and
University of North Carolina at Chapel Hill
INSTITUTE OF STATISTICS HIllEO SERIES #1125
June 1977
i)EPARTI,mr·lT OF STATISTICS
Chapel Hill, North Carolina
AGlUQJLWRAL VARIATION AND SECOND-ORDER STATIONARY
RANIJCN MEASURES
M. L. THORNETT
UNIVERSI'lY OF WESTERN AUSTRALIA
and
UNlVBRSI1Y OF NOR'lH CAROLINA AT OfAPJiL HllJ.,
Experimental measurements associated with n-dimensional regions or
"plots" are regarded as observations on random variables indexed by the
bounded BOTel subsets of lIfU;'
these random variables having finite
second moments and satisfying an obvious finite additivity property.
Certain stationanty and continuity assunptions about the first two
moments
allow spectral representations to be derived which are analogous
to those already familiar from the theory of generalized stationary
random processes.
The theory is used to justify four models of two-
dimensional spatial variation
~hich
are fitted to some well-known uni-
formity trial data.
Some key words:
Spectral measure; Transfonnably bounded measure;
Isotropy.
May 20, 1977
1
1.
INTROOOCfION
The success of split-plot and incomplete block types of experimental
design in increasing the precision of estimates of various "treatment"
effects in agricultural
oxporimont:c c~n £roqUQntly bo
explained by the
observation tha.t 't.he yields of neighbouring plots are more highly
correlated than those further apart.
Fairly extensive data are available
substantiating this observation, and also the accompanying observation
that long narrow plots tend to be less variable than square plots of
the same area
(see Matc£m (1960)
§
and the references quoted
6.12,
therein).
The variation in the plot yields of an agricultural experiment
is an eXarI¥>le of two-dimensional spatial variation.
The aim of this
paper is to present some models of such variation, and to fit them to
the u.'lifonnity trial data of H. Fairfield Smith (1938).
To this end,
we draw on some theory concerni.ng second-order stationary random
measures, the basic ideas for which are discussed by Daley (1971)
in the point process
(1974) and·
~glam
context~
(1958).
2.
Let
each A
and more generally by Vere-Jones
FORriULATION
N denote a reasonable class of subsets of
~ N,
mn ,
and for
let weAl denote some real or complex measurement
associated with A.
2
In IR for example,
N might denote the class
of all rectangles, while w(A) could denote this season's wheat yield
fram the plot determined by A.
With this example in mind the
2
following two assumptions about the ftmction w seem plausible.
Assumption 2". w is finitely additive, Le., w(A u B) •.w(A) + w (B)
for any two-disjoint sets A and B.
Assumption 2''': weAl is small whenever A is .small.
more precisely, if
~
;:)
Az ;:) ...
Somewhat
is a non-increasing sequence of sets
in Nwhose Lebesgue measures approach zero, then W(Aj ) approaches
zero, in some sense,::.M j..
CI!
•
The first of these assumptions is clearly easier to live with if·
we require
N to be closed with respect to finite unions.
Without
.further ado then,. we introduce out principal assumptions.
N consists of the botmded Borel subsets of
Assumption 1.
n
. Ilt •
Assumption 2.
The collection {w(A) , A e:
a stoc,.ltastic process
(2a) B( IW(A) 12) <
{W(A), A
CD
E:
is a realization of
N} for which
for all A
if for any y e: lRn , A
(2b)
E:
N}
i~,
E:
N (second-order property);
\-Te
put A+y
= {x+Y,
X E:
A} ,.then
we have E(W(A+y)) = EO'I(A)) and E(W(A+y)wtB+Y'J) = E(W(A)W(B) )
for all y E: mn , and A, B E: N (stationarity property);
(2c") \'l(A u B)
:II
WeAl + WeB) whenever A and B are disjoint (addi-
tivi ty property); and
(2c ....)
if
~;:)
A ;:) ••• is a non-increasing sequence of sets in
2
2
N whose Lebesgue measures approach zero, then lim E CIW(A.)1 ) :II 0
j-+'OO
J
(contilUlity property).
Clearly, (2c")
and (2c ....)
labelled (2c), that if ~,
Az, ...
are equiValent to the single conditioIb
are disjoint members of N whose
3
mion is also in N,
then
00
00
\'Ie U A.)
j=l J
=
1:
l'J(A.).
J
j=l
As usual, the equalities expressed in (2C'°) and (2c) are between
equivalence classes of square integrable randon variables.
2.1 Definition.
A stochastic process
(2a) , (2b). and (2c)
{W(A) , A
€
N}
satisfying
will be referred to as a second-order
stationary random measure.
It would be Iilore in keeping \.,rith the terminology of Vere-Jones
(1974) to call
{II (A)
, A
€
N}
a second-order stationary random
complex measure, but we shall not do this.
In fact, from now on,
we shall simply use the tenn "random measure" to indicate a process
of the above type.
Similarly, the word "measure!l will embrace only
n
the complex Borel measures on I1t , i. e., complex measures on the
n
Borel subsets of m which take finite values on N.
Regarding the first assumption, the two dimensional version of
such an
N would appear to contain all the subsets which. could have
any agricultural interest, eg., rectangles, circles, other conyex sets
and their tmions.
The second assumption indicates that our study of
spatial variation will be a study of the first and second moments of
random measures, with the implication that, for inference purposes,
we would be making the traditional assumption of joint normality of
the random variables
W(A) ,
A
€
N.
The relevance of random measures to agricultural experimentation
may be seen as follows:
experimental plots
Let Yi"" 'Yk be yields from k (disjoint)
Ar, ... ,~.
Then Yl'··· ,yIc perhaps edited and
4
transformed~
may be regarded as observations on jointly normally
distributed random variables Y1'."'Yk of the fonn
Y.
f. (8 , ... ~ B)
J 1
P
_.
J
+
W(A. )
J
j=l, •.• ,k, where f 1 , ... ,.fIe are l'Jlown functions of unknown parameters 81 , ••• ,8 which are related to the various treatments applied
p
to the plots, and H'J(A), A € N} is a random measure. Without loss
of generality, it may be supposed that E 0'1 (A)) = 0 for all A
E
ilj.
The suggestion is that if such a model is confirmed to be appropriate
for a series of experiments of the same type, then this would allow
a better choice of design for the
interest~
Sp •.. ,8p '
3.
put
estj~tion
of the main parameters of
in the experiment of current interest.
THE FIRST TWO !v{)i.1EITS OF RANDOM HEASURES
Let {W(A) , A
€
<A> = EO'J(A))
and <A,B> = E(l1(A)W(B)).
3.1 Theorem
<A> =
c~
(A)
N}
be a random measure, and for A, BEN,
There exists a complex constant c such that
for all A
€
fJ,
\'{here An denotes
n-dimer~ional
Lebesgue measure.
The proof of this theorem is an obvious simplification of the
proof of the following theorem.
In fact, both results can be established
by arguments similar to those used by Yaglom (1958) in the proof of
the t:leorer.l on p. 91 for the case n = 1 (for
interpretation of n).
bO~l
our a."1.d Yaz;lom' s
Before stating this next theorem however, some
further notation is required.
5
For any A
[A]
EO
[A] denote its indicator function, and let
let
!II
denote the corresponding Fourier transform, Le.,
if x E: A,
athenJise,
= {;
[A](x)
and
(A]
J e iu . x
(u)=
dx
A
n
where, as usual, u.x
=L
Note that for any y
€
for all
x
measures
(i)
J
n
IR
J
x = (xl'
00
.,~).
.
[A+yJ(x) = [A] (x-y)
IRn • Finally, let R denote the set of all non-negative
EO
such that
v
f I (A]~12
dv
<
(ii)
o
M such that
lim A
j~ n
3. 2 TIleorem
for all A,B
0
0
then lim
j~
J
J
€
N.
and
is any non-increasing sequence of sets in
(A.) = 0
::
for all A
00
There exists v
<A,B>
Proof.
for u = lUl'''. ,unL
u. x.
j=l
E
~
R "vi th
JI[A.(1 2dv =
J
v( {O})
~
Ic 1
O.
2 I:. such. that
...
[A] (u) [B] (-u)d,,(u)
(3.1)
N•
€
The main step is establishing the identity
f f[A](x) [A](y)
for all A, B
<
EO
B+x+t, B+y> dxdy = H[B](x)[B](y) < A+x+t:lA+y > dx ely (3.2)
N and t E m.n . To that end, for each integer m ~ 1,
let C .:;j = 0,1,2, ... , denote a listing of the mutually exclusive and
mJ
n
exhaustive subsets of IR of the form
6
=
Cmj
_. 1 )
{( xp ... ,x ) ' 3-m (Pmjk!
n
< ~($
3- m (
1) k
Pmjk+ 2 " -
= l, ... ,n}
where the P Ok are suitably determined integers, and let x be t.-"'e
ffiJ
mJ
mid-point of Crnj . Using the fact that
o
[Cmj ] (x)
for all x
€
=
[Cmr] (x-x
mj +
mn it may be shO\m that
are finite unions of t.-"'e sets
[A](x)
=
~)
I: a
0
mJ
j
[C ](x)
mJ
i
Co,
mJ
and
0
Le. ,
[B] (x)
=
I:
k
b
~
nll<
[C_,J ex)
1IU\.
,
where the amj s,
bm1< s
only take the values 0 or 1, and only a
finite number of them are non-zero.
To extend the identity to all
N we use (2e' ') in conjunction with the fact that for each
A,B
€
A
N there exists an m such that
€
(3.2) holds when both A and B
[A](x)
where
*)
An (Am
=
I: a
j
.0
mJ
[C
0
rnJ
](x)
+
is arbitrarily small.
Next it may be shown that for each
positive-definite function of y, and
S01
A~
<A+y,A>
is a <;:ontinuous
by Bodmer's theorem, there
exists a finite non-negative measure v A such that
This together with (2b) and Fubini's theorem allows (3.2) to be
re-written as
The measure v we seek is then uniquely determined by the relationship
7
for all D
u
E
D.
E
tv, where A may be chosen so that [A]A(U)
~ 0 for
If we now suppose B to be a "square" centered at the
=0
origin, then putting t
in the right-hand side of (3.2) we
get the identity
2
B- IJ[BlCx)[BlCY) <
where
B
=
An (B) •
A+x~}\.+y
B~
Letting
<A)A>
> d.x .d'l
= 13- 2fl[A]AI 2
I[B]A I2 dv
yields
= f I (A]~17
dv
which, by the usual inner-product or variance-cova riance manipulations,
implies
for all A, B
E
ill.
and that \) ({ O})
That \)
;::
E
R £ollO\'1s directly from (2a) and (2c") ,
Ie 12 is a consequence of letting B ~
00
in the
inequality
B- 2<B,B>
~ 8- 2 1<B>1 2
Band S being as introduced above.
3.3 Definition
TIle measure
\)1
E
R constructed above will be called
the spectral measure of the random measure
4.
{1!l(A) , A E Iii}.
QfARACfERI ZING SPECTRAL I·IEASURES
The results of the previous section are of course completely
analogous to tile spectral representations in the ItS-Gel'fand theory
of generalized stationary random processes
& Vilenldn (1964), and Yaglom (1957)).
(see ItS (1953), Gel'fand
In this theory the class of
spectral measures cons ists of the so- called tempered measures, i. e. ,
8
those measures
for some k
satisfying
v
0, where
?
Iu 1
2
=
The author is not aware of any
u °u.
correspondingly simple necessary and sufficient characterization of the members
of R.
: IowGver ~ some sufficient cllaracterizations can be fomulated from the
follm·ling considerations. Let N depote the space of complex-valued f1.m.ctions
= L a. [A. ](x)
o£ the form f (x)
J
j=l
J
where A ,A , •.• , are painvise disjoint members of N, and a l ,a 2, ••• are
l 2
complex mnnbers, only a finite number of which are non-zero. The
Fourier transform of f
...
€
N \vi1l be denoted by f,
and for f,g
€
N,
f*g '* will stand for the function
f*g '* (x)
4.1
Definition
f
= ff(y)g(y-x)
(i)
f'*f * d~
A measure
~
dy.
~
is said to be positive-definite if
0
for all fEN.
A non-negative measure v is said to be transfonnable if there
(ii)
exists a measure
~
such that
IrA] '* [A] '.*d~
for all A
€
(iii)
(4.1)
N.
A non-negative measure v is said to be transfonnably botmded
if there exists a transformable measure v 1 such that
f
for all A
We use
E
2
I [Aj" 1
dv
::;;
f
I [A]" 1
2
dVl
N
Rt
to denote the set of non-negative measures \'lhich are
9
transformable ~ \'lhi1e
R
will denote those that are transfoTInab1y
tb
botmded.
It is easy to see that if
Converse1y~
positive-definite.
t.~at
3.2 can be invoked to sho\'1
"E
R
if
j.I
t
then the measure
j.I
in (4.1) is
is positive-definite then Theorem
"E
such that (4.1)
t
For present purposes hmJever, the most important observation
is true.
there exists
R
is that
Le.,
if a non-negative measure "is transfonnab1e, or failing that"
is transfonnab1y
botmded~
random measure.
The first of these inclusions is strict.
then" can be the spectral measure for some
To see this,
consider the measure" given by
,,(D)
= f h(u)
du
D
DEN,
where h is a bounded continuous non-negative ftmction which is not
the Fourier transfonn of any finite measure.
T.hen
\I
I. Rt
but is
transfonnab1y botmded by
"1
= sup h(u) An·
u
I t is an open question whether or not
4.2
Rtb
is a proper subset of Rt •
The following theorem characterizes R :
t
Theoren
A necessary and sufficient condition for the spectral
of a ril:ndOl.:H~~)nSurc:{W(A),A
€
exists a positive constant
~fA
TIL
_
.• _
N} to be transformable is that for every Ae:N thert
such that for any selection AI' AZ' ..• of
pairwise disjoint members of N whose union is A,
.;
10
Proof.
Necessity follows by observing that we can take
= f [A]*[A] * dllli
H
~
<
0.>
For sufficiency, let C . and x. be as in the proof of Theorem
mJ
for A
€
mJ
3.2~
N let
a.
mJ
if C.
rnJ
=
A,
c
otherwise
•
Consider the ftmction b A~m defined by
=
r
-it.xm·
J (C
amj e
mj
] "(u) •
j
Then (i)
lim hA,m (t,u)
=
~
(ii)
(iii)
"
[A] (u-t)
f I\~m(t,u) 12 d v(u) ~
HA ;
and
for any function f integrable with respect to
A
n
we have
Iff(t) {f 111" (t,u) ,2 dv (u) }dt IsM" suup If (u) I •
""",m
.t\,
It follov;rs then that
h (t)
A
=
f
2
IrA] "(u-t) 1 dv(u)
is a bounded continuous non-negative function satisfying
for any f integrable with respect to An'
Thus (see Eberlein (1955),
Theorem 2) there exists a bounded measure ].lA such that
hA(t)
= fe- ix . t d].lA(x)
and
11
The identity
may then be employed in the same manner as (3.2) to show that the
measure
constructed according to
II
for all D
€
satisfies (4.1) ..
N,
In many agricultural situations it would be reasonable to suppose
that <A,B>
~
0 for all A, B
€
N.
Under this assumption the condition
of the previous theorem is met with
so that
v
€~.
In addition, the positive-definite measure
corresponding to v
must itself be non-negative.
II
The role of such
measures - called p.p.d. measures - in the theory of second-order
stationary point processes has been pointed out by Vere-Jones (1974).
,
Note also that if
v
is botmded, then it is transformable.
It is
clear then that although R does not exhaust R, its twot
dimensional version is a source of many useful models of agricultural
variation.
TIle arguments used in proving Theorems 3.2 and 4. 2 are
tailored versions of arguments well-known in abstract harmonic analysis,
e.g., see the second chapter of Argabright and de Lamadrid (1974).
Let G denote the group of all orthogonal transformations
12
(rotations and reflections)
g A
=
{gx, x
€
in IRn ~ and for g
E
G, A
e:
N, let
A}.
A random measure {W(A) ,A
€
N}
if <gA.>'= <A> and <gA,gB>=<A,B> for all A,B
E
N, g
5.1 Definition
is said to be isotropic
E
G.
\,
Isotropy is an appealing assumption in modelling instances of
natural variation, though perhaps not so appealing in the case of
crops planted according to a rectangular grid pattern. Even here
it offers a simplification which should be considered before looking
at models which are not orientation-free.
Since the relationship
<gA.>
=
is always true for any
<A>
random measure, the condition of isotropy relates only to the spectral
measure of such. a process. The following characterization' '0£ isotropic
spectral measures, as well as its justification, is identical with
tile corresponding result for generalized stationary random processes
(See Yaglom (1957), Theorem 11, or Gel'fand
~
Vilenlcin (1964)) p. 294).
To that end let J r denote the Bessel function of the first kind of order
r, r denote the ganuna function, and define the ftmction S"ln: IR+IR by
(2/t) (n- 2) 12
=
5.2 Theoren (Yaglom)
r (n/2) J
(n-2)/2
(t)
A necessary and sufficient condition for a
spectral measure v to be isotropic is that v (D)
all g
E
=
v (gD)
for
G, D € N.
In this case we have
co
<A,B>
= J {!![A](x)[B](y)
o
for all A, B
e:
~(slx-yl)dx
dy} dn(s)
(5.1)
N where dnCs) denotes the v-measure of the spherical
13
shell s ~ lui
in IRn
s+ ds
<
6.
SOHE TWO-DI!1ENSIONAL
~nDELS
In the remaining sections our attention is confined solely to
IR:2 . To avoid subscripts we shall denote typical points in IR2 by
and u = (v,w) where y~ z, v, w, as well as r = (y2+ z2) 1/2
2
and s = (v + w2) 1/2 are all real. Since only variances and covariances
x
= (y,z)
are of
interest~
zero.
The vers ions of
\I
the quantity c
is transformable -
\I,
T1 ~
and
of Theorem 3.1 will be taken to be
II
in the modeIs considered ,- - each
are all absolutely continuous with respect
to Lebesgue measure of the appropriate dtmension, so to save
notation~
the same symbols will be used to represent the respective derivatives
with respect to Lebesgue measure.
A positive parameter (J2,
represent-
ing a scaling factor or variance component, appears in all models.
The model Ma
a
~
0
with
ll(y,Z)
=
o2 C2n )-laCa 2 + r 2)-3/2
co
The model ",Me
!'I(s)
= (J2(2n)-l
S8
J
0-
e-ett3(t~+s2)-3/2 dt
with
ll(y, z)
=
2 -1
(J n
8(13 + r)
-3
with
ll(y,z) = {oy Cy-l)/2} 2 Iyz ly-2
'
s
?;
0,
14
The model Ho
n(S)
= 02
r(2-o){wr(o)}-1(s/2)20-l
0
1
0 'S 1,
<
with
lJ (y, z)
= 0 2(1- 0) I (wr 20 )
For Ma. and H/3 the compatibility betv.Teen the expressions for
follow from (3.1), (401),(5.1) and the identities (1),
lJ
and
n
and
§8.l~
The same thing for My and Ho may be
(7), §8.2 of Bateman (1954).
established by using the identity (iii), Theorem 6 2 5; of Okikiolu (1971).
0
Given aib
-b/2 <
Z S
>
0
0 let R(a,b) stand for the rectangle -a/2
b/2, and vrrite V(a,b;-) for <R(a,b),R(a,b»
dash stands for one of the parameters
a. 1 8,y, or o.
<
y
a/2,
S
where the
In the method used
for fitting the above models it is V(a,b;-) which plays the basic
role, rather than the spectral measure
as
n
or
V(a,b;a.)
lJ.
"v,
or some related measure such
Rather lengthy integrations yield the following formulae:
= (20 2/w) [Va (a,b)
b 2) 1/2 - a2
+
V (b,a)-a. (a. 2
+
ab tan- l {(ab)/(a4+ a2 a 2
a
+
a2
+
+
a. 2 b 2) lIZ}]
(6.1)
where
V (a,b) = a a sirih- l {a/(a 2
a
V(a,b;s) = (40 2/w) [VS(a"b)
\.,rhere
+
b 2) 1/2}_ a. a sinh- 1 (ala)
+
VS(b,a) - (313/2) (a 2
+
(3/32/ 2) log {l
+
+
+
a(a 2
+
a 2) 1/2
b 2) 1/2
(a 2 + b 2) 1/2 /s }]
(6.2)
15
+ 3Sb/2 + I3b log[ {b + (a 2+b 2) I(Z_a}! {a+b _ (a 2+b 2) I!Z}]
-13{(3S!2) + b} log {I + (b!S)}
V(a,b;y)
.,
= aW(ab)y
(6.3)
and
V(a~b;o) = (a 2!n) [Vo(a,b) + Vo(b,a) + 2(a4- 20 + b 4- 2o )!(3_20)
+ {(a 2+b 2)2- o _a 4 - 2o _ b 4 - 2o }!(2 -0)]
(6.4)
where
and F
stands for the Gauss hypergeometric series (see p. 429 of
2I
Batffinan (1954)) •
Concerning the appropriateness of such models, an initial problem is
that of deciding \'1hether a given one of them offers more than the
simplest model of two-dimensional variation, namely
(6.5)
lIe observe that this model is a particular case of each of the models
considered J being realized \'1ith
respectively.
CL:=
0J S
= 0,
y
= 1,
and
Thus, in each case the estimated value of
0
CL,
=1
13, y,
or 5
for a given set of data provides a measure of departure from this basic
model.
Apart from allowing some measure of departure from the situation of
orthogonal plot yields as represented by (6.S), there is little to
justify the models MCL and HS.
Their inclusion here is due to their
being isotropic and the fact that closed expressions for V(a,b;-) can
be written down.
From (6.3)', the model By is seen to correspond to
16
the empirical lmv of agricultural variation devised by H. Fairfield Smith
(1938).
~~ittle
has
(1956)
of fixed (rectangular)
(ab)Y ~ 1 ~ Y < 2~
shape~
shov~n
that when plots are small or large and
<R(a?b)$R(a?b»
varies in the limit as
for a large class of models.
On
these grounds Hy
may be expected to fit observed data fairly 'Illell? particularly when these
data show that <R(a,b) ?R(a?b) > depends only on plot size and not on
plot shape.
For the purpose of comparison, He has been included as
the obvious isotropic equivalent of I!ly.
that
<R(a,b)? R(a?b»
shape, namely (ab)2- e
local maximum when a
It too has the property.
varies as a power of a b
for a given plot
Hm-lever, for ab fixed? V(a,b;e)
has a
= b; which suggests that l'·fe might be an
appropriate model in those cases where long narrow plots are observed
to be less variable
truL~
square plots of the same area.
7.
FI'ITING THE fvDDELS
Given positive integers a, b, r, c consider a rectangular
region of length ra and breadth cb
0~vided
into rc contiguous
rectanoau1ar plots Pjk each of length a and breadth b, and let Yjk be
some response measurement for the plot Pj k' where j = 1,
,r and
0
k ='l, ••. ,c.
••
The data from many uniformity trials (see Smith (1938))
are of this form.
To model such data we suppose Yjk to be an
observation on a random variable Yjk having the form
YO k
J'
where {W(A), A
A
E
E N}
= e+
N(Po k)
J'
is a random measure with <A> = 0 for all
N.
To infer something about <A,A> , A E rJ?
from such
data~
an
17
obvious statistic to look at is
s
2
=
This is the observed value of a random variable,
S2 say) lv-hose
expectation satisfies
= {rc V(a~b;-) - (rc) -lV(ra)cb;-)}/{ab(rc - l)}
Clearly, it is possible for the observations Y.1_ to be added
JK
together in several different ways corresponding to different
values of a, b~ rand c, with a corresponding s2 calculated for
each of these combinations.
Thus, suppose there are
Til
such combi-
nations a jJ b j > r j , cj ' j = 1, ..• , m of the variables a~ b, r,
and c for which tr.is corrected sum of squares is calculated, and
let the resulting values be denoted by s~, j
J
any
urUQl~nl
= l; .•. ,m.
parameters in the specification of V(a,b;-)
Then
could be
")
estimated by those values \'lhich, in some sense, make sj
E(S~) for
j
= l, ... ,m.
close to
The criterion adopted here is w~e nunimi-
zation of
The main objective in devising this technique was to get estirnates
without expending too much computer time, using a procedure that is
directly applicable to the results of many unifoITtuty trials as they
are reported in the literature, Le.) effectively as so many combi-
18
2
a. ~ b. ~ r. ~ c. ~ and s.
J
J
J
J
J
the fit of competing models may be compared.
Also, through
nations of the quantities
$ ~
Applied to the 45 data combinations given in Smith's paper this
0 730,
0
and. 0 by 0 50,
a, S. y,
technique estimates
0
1 0256, and
the corresponding variance components being estinated as
12,739,
14,353,
3,596,
and
6,014, with
97 6, 38 8, and 80 8 respectively.
0
0046~
0
0
$- values of
The estimate 1 256 for
0
be conpared with the value 1 251 obtained by Smith.
0
superiority of By
103 5,
0
y
may
The clear
over the three isotropic models, particularly Ho
calls for some comment.
In the first place, one might attribute
the apparent lack of isotropy to the usual row-ey-row l,vheat-p1anting
tedmique adopted.
A more important consideration perhaps, is the
scale of this particular uniformity trial.
With a
and b
expressed
in terms of length-units of six inches, the total area occupied by
the trial was 5 x 12
= 60
square yards.
It is doubtful if any model
of variation based on a trial of this size could be a reliable guide
to the variation encountered in the usual rtm of wheat experir.lents.
What kind of experimental data "muld allow reliable model-building
is, of course, a good question.
In Smith's paper the statistic
t
2
= s 2/ Cab)
?
'\vas used rather t~an s.... Table 1 sets out nine of the combinations
a~ b, r, c, t 2 , together with ECT 2) for each of the fitted models.
The main purpose of the table is to indicate how all three isotropic
models predict that long narrow plots \lill be less variable
measured by ECT 2) - than square plots of tIle same area.
as
19
Table 1.
2
Fi tted values of E(T ) under Ha., Me ~ My? and
r,1Q
"for Smith's
data.
~
a
1
3
1
2
6
9
18
2
12
b
6
2
36
18
6
4
2
72
12
r
30
10
30
15
5
3
1
15
2
c
12
36
2
4
12
18
36
1
6
2
n63
976
276
272
241
214
275,
63
96
Hex.
861
1051
153
209
249
243
208
52
72
Me
864
1040
154
210
254
247
208
52
75
My
939
939
242
242
242
242
239
83
82
Mo
373
995
161
265
256
212
53
92
t
p(r 2)
---
"----
215
20
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