THE CONSTRUCTICN .A,ND EVALUATICN
OF SCME DESIGNS FOR THE
•
ESTIMATION OF PARAMETERS IN
RANDCM MODELS
r
by
DAVID W. GAYLOR
and
R.L. ANDERSON
-
•
INSTITUT,i OF
ST~TISTICS
MDmOGRAPH SERIES NO. 256
APRIL, 1960
:l;
Errata
p. 3, line 2 below (2):
change lin
p. 11, line 2 from botton:
p.
6.5, Table 10:
..
=~ n.1.".
change "the" to "an".
change "b(blocks)" to "b blocks (columns)".
2
2
for E(MS) of Columns, change "ncO"c" to "nrO"c".
p. 80, line below (143):
p. 87, line 2:
to "N
change "the" to "an".
p. 12, last line of Section 6.2:
p. 20, last line:
=~ ni "
change 11(172)11 to "(143)".
change "different" to "difficult" •
iv
T.ABIE OF CONTENTS
Page
1.0
LIST OF T4BIES •
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2.0
LIST OF FIGURES
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LITERATURE •
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3.0 INTRODUCTION
4.0 REVIEW OF
5.0
6.0
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LOWER BOUND FOR THE VA.RIANCE OF UNBIA.SED ESTIM.\TES OF
LINEAR COMBINATIONS OF COMPONENTS OF VA,RUNCE..
OPl'IMAL DESIGNS FOR ESTll1A.TING COMPONENTS OF V1\RIA,NCE
IN A TWO-WAY CROSSED CLASSIFICATION •
• •
6.1
Estimation of
~
•
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••
0
0
0
o
13
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• 14
o
o
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042
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59
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• 62
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6.6
Effect of Improper Choice of p in Determining
o
222
•
= 0
0
0
222
222
6.8 Estimation of crr/(cr
e +O'rc) or O'c/(O'e+O'rc).
6.9
Effect of Improper Choice of p in Determining
Optimal Design for ~ 0 0 •
0
••
0
6.10 Estimation of cr;/O': or O';/cr: when O';c
,
o
•
•
607 Estimation of O'r or O'c when O'rc
•
o
0
Estimation of O'r or O'c •
r
7.0
12
o
6.5
opt imal De sign f or ~2
222
2
SIMULTANEOUS ESTIMATION OF crr AND crr/O'
8.0 SIMULTANEOUS ESTIMATION OF 0'; AND 0';.
8.1
8.2
8.3
=0
0
o
2 2
OR O'c AND O'c/O'
•
•
0
11
o
0
Est imat ion of
2
o
o
o
6.4
2
7
•
0
Estimation of (0':+O';c+0';) or (0':+O';c+0';)
0
•
o
•
6.3
(ie +clrc+i+i)
r c.o .
1
011
o 12
••
6.2 Estimation of ~..
.vi
•
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L-shaped Design • 0
0
0
Disjoint Rectangles Design
•
Comparison of L-shaped and Disjoint Rectangles
Design.
0
o
47
63
• 66
v
TABLE OF CONTENTS (contirmed)
Page
9.0 PROCEDURES TO A,TTAJN SPECIFIED PRECISIONS OF
ESTIMA,TES OF COMPONENTS OF Vl\RI.t\NCE
9.1 One-way Nested Classification
9.2 Two-way Crossed Classification
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• 72
• 76
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10.0 SOME FURTHER RESULTS FOR .\ ONE-WU NESTED
CLASSIFIC~TION •
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11.0 SUMMARY l\ND CONCLUSIONS
•
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0
0
LIST OF REFERENCES
•
.
• 79
•
11.1 The Problem •
11.2 Optimal Allocation
11.3 Suggested Future Research
0
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• 72
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• 81
81
81
• 87
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o
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• 88
vi
1.0
LIST CF T4BLES
Table
1.
2.
A,nalysis of variance for a one-way nested
classification. • • • • • • • •
•
•
•
Analysis of variance for a two-way crossed
classification with nij = 0 or n
• •
•
•
•
•
• • 3
•
• 14
•
•
• 34
3.
2
V('cr ) for various vaJnes of t and
r
4.
Comparison of approximate and exact variances of a1'.2
r
•
•
• 40
5.
Effect on the approximate variance of ~ due to
r
using improper choices of p in determining
optimal designs
•••••••• •
•
•
•
• 43
Analysis of variance for two-way crossed
classifications with unequal numbers per cell
and no interaction
•• •
•
•
•
• 45
•
•
•
•
• 55
Effect on the approximate variance of
due to
using an improper cho ice of p while
determining the optimal design
• • •
•
•
•
•
• 58
Approximate values of Co for the optimal designs
for estimating a; and p (large N). • • • •
•
Analysis of variance for disjoint rectangles
with n observations per cell
••• •
•
•
•
•
• 65
Comparison of disjoint rectangles and L-shaped
"2
A2
2
2
designs with \l(a
r ) = V(ac ), where ar = a c • •
Analysis of variance ignoring last cl columns
•
•
•
• 68
•
•
•
• 69
Approximate total number of observations (Na )
required f or estimation of a~. • • • •
•
•
•
• 75
6.
7.
8.
9.
10.
11.
12.
13.
•
Page
p,
N;::25 and r=lO
Analysis of variance for optimal design shown
in Figure 3. • • • • • • • • •
p
•
• 60
3.0
INTRODUCTION
Considerable work has been done in the area of experimental
design to estimate treatment contrasts.
Estimates of components of
variance can be obtained as a by-product of these designs.
These
estimates often may be equally or more important than estimates of
the treatment contrasts.
However, designs used primarily for the
estimation of the latter may give very inefficient estimates of
components of variance.
Little work has been done in the general
area of designing experiments especially for estimating components
of variance.
Anderson and Bancroft [1952, p. 334] introduce a "stag-
gered ll design as a possibility for use with nested classifications.
Crump [1954J has considered optimal designs for estimating the com~
ponents of variance for a one-way nested classification.
In parti-
cular, Crump considered designs to estimate the component of variation
2 and the ratio p = O"a2/O"e,
2 where O"e2 ~s
. th e
among classes, O"a,
component of variance.
222
O"a/(O"a + O"e)
= p/(l
. th'~n c la ss
w~
The intra-class correlation coefficient is
+ p).
The purpose of this thesis is to investigate designs for estimating the components of variance fer a two-way crossed classification
· 'k
YlJ
where i
= 1,2, ••• r;
=~
II.
+ r.1 + c.J + (rc).~J. + e.~J'k
j = 1,2, ... c; k = 1,2, ••• nij;
the total number of observations, is fixed.
~;
the row effects, ri, are NID(O,
2
or);
(1)
and~ nij = N,
The general mean is
the column effects, Cj'
are NID(O, O"~); the interaction effects, (rc)ij' are NID(O,
2
O"~c);
the within cell effects, e ijk , are NID(O, O"e); and all effects are
uncorrelated.
2
The optimal design for a particular problem will depend upon
the method of estimation used.
This thesis will emphasize the
design of experiments for estimating components of variance rather
than the method of estimation.
In some cases, it will be possible
to couple the best design with the best estimator.
the best estimator may be unknown.
•
In other cases,
For these situations, a particu-
lar method of estimation will be selected and the allocation will
be considered for that estimator.
The simultaneous estimation of variance components and
sampling costs will also be investigated.
3
4.0
Hanunersley
[1949J
REVIEW OF LITERATURE
and Crump
[1954J
considered optimal designs
to estimate the parameters of the variance component model for a
one-way nested classification
(2)
= 1,2, ... a;
where i
tions,
N=~ ni,
j
= 1,2, ••• ~;
is fixed.
and the total number of obset'va-
The class effects,? ai' are
NID(O,O"~);
2
the within class effects, eij' are NID(O'O"e); and all effects are
uncorrelated.
Table 1.
The analysis of variance for (2) is given in Table 1.
Analysis of variance for a one-way nested classification a
Source of variation
DF
MS
E(MS)
Among cla s Be s
a-l
~
2
2
O"e + noO"a
Within classes
N-a
W
2
O"e
Total
N-l
a DF is the degrees of freedom, MS is the mean square,?
and E(MS) is the expected value of the mean square.
Hanunersley and Crump used the analysis of variance estimate of
2
O"a obtained by equating the mean
squares to their expected values in
Table 1 giving
(3)
4
Hammersley showed far a fixed
~
that the minimum variance of
~~ is obtained with equal number s of observations per class. That
is, N divided by ~ nmst give an integer, ni ... n ... N/a.
case, no ... n, and the value of
~
t~
which minimizes
In this
variance of
.-..2.
O'a ~s
a
2
l
... N(Np + 2 )/(Np + N + 1),
2
where p ... O'a/O'e, with N, a, and n all integers.
ni
=n
(4)
For this case wi. th
for all i, (J) is the maxinmm likelihood estimator and
Graybill and Wortham
[1956]
show that (J) falls in the class of
uniformly best (minimum variance) unbiased estimators.
Hammersley
offered no exact proofs of procedures to use for non-integers and
only suggested using the nearest integers.
Crump extended these remIts to the situation where N/a is not
necessarily an integer, but
N/a
:=
where N, a, p, and s are integers with
showed that the variance of
(5)
p + s/a
0
<s
~
a.
For fixed a, Crump
~~ is minimized by setting p + 1 observa-
tions i:p.! classes and p observations in the remaining a-s classes.
cr~
Crump also shows that the variance of
of classes is given by (4).
is minimized when the number
Crump hypothesized the folloWing pro-
cedure fer allocating samples to classes with fixed N:
(i)
(ii)
Find a
If a
l
l
fran (4).
If a
l
is an integer.9 set a ... a •
l
is not an integer.'l let
and below a
to (5).
l
~
be both the integer above
and allocate for each value of
Select the allocation which
~
variance of O'a.
~
accordirg
minimizes the
Hence, s classes would contain p + 1
5
observations and a-s classes p observations where p is the
1'2
largest integer less than N/a o
(Since the varia nce of Cia
differs very little for the two all ocations, a general
operational rule would be to choose a as that integer
closest to alo)
Crump did not give a proof that this procedure always gives the optimal design, but for numerous example s this procedure always
. . . d th e var1.ance
.
m1.nJ.lTl1.ze
Baines
0f
1'2
Ciao
[1944] used the F ratio, 4/W.9 in Table 1 to obtain an
estimator of p
= CI~/CI~o
(6 )
Baines restricted his investigation to the case of equal mmbers of
observations per class, ni = n = noS' and found the value of n which
'"
minimized the variance of po
of p,
p,
Crump considered the unbiased est:imator
which corrects the slight bias in
po
Crump extended the
results to the situation where N/a is a non-integer of
given by (5)0
'"P
For a fixed ~, Crump showed that the
the form
variance of
is minimized by putting p + 1 observations in s classes and p ob-
servations in a-s classes
0
The variance of
p is
minimized when the
number of cells is
a'
=1
+ (N - 5)(Np + 1)
2Np + N-3
0
It was hypothesized that tlE optimal design is obtained by setting
~ equal to the int eger below or above a
.
vanance
0
f /'0.
p.
i
which minimizes the
Then, s classes contain p + 1 observations and a-s
6
classes contain p observations where £. is the largest integer less
than N/a.
Crump shows that the guess or previous estimate of p required
in (4) and a' in (7) is not too critical. The design
l
may differ considerably from the optimal design without materially
to determine a
affecting the variances of
cri
po
or
Crump also ::hows that the optimal design for est:iJnating "" is
to set ni "" 1 and a "" N where the estimator is
~
A
"" ""..::.J y.~J)N
The optimal design for est:iJnating
0
(J~ is to set ni "" N and a ""
where the estimator is
"2
~
O:e ""..::.J
(y.
A
0
~J
-
",,)
2
/(N-l).
1
1
L<NlER BroND FOR THE VARIANCE OF UNBI.I\SED EST:J1IU\TES OF
500
LINE4R COMBIN4TIONS OF COMPONENrS <F V.A,RI4-JCE
Consider the general variance components model
Yi
where i
•
= 1,2 p
= J.L
(8 )
+ ei
eoN; J.L is the expected value of Yi; and the errors
ei are normally distributed with zero means and variance-covariance
matrix V(NxN).
Writing (8) in vector notation
1.. (Nxl) = IJ. (Nxl) + e (Nxl)
where E(~)
=Q
(Nxl) and E(~i)
(9)
= v.
Consider a quadratic est imator
(10)
of a linear combination of components of variance,
2
(J
,~
where
M :: M' (NxN).
Substituting from
(9) into (10) gives
Q,
=
(~' + ~,) M (i!
+ e )
or
Q, :: ~'~ + 2!i~ + ~iM~.
Since
E(~)
= Q,
the expected value of
Q is
E(Q,) :: ~t~ + E(~iM~).
From Whittle
[1953],
Lancaster
(11)
[1954],
(12)
and others the sth cunmlant
of e'Me is
K
or
s
= 2 s-l (s-l)J
tr(VM)
s
(13)
8
where A (NxN) is the diagonal matrix of latent roots of VM.
From
(13 )
giving from (12)
E(~)
= ~'~
(14)
+ tr(VM).
Squaring (11) gives
Q2
= (~,~)2
+
4(!,~)2
+
(!'M!)2 + 4(~'~)(!u~)
(15)
+ 2(~'~)(!'M!) + 4(!'~)(!'M!)o
Since the expected vaJnes of odd powers of e vanish for the normal
distribution
E(Q2)
= (~,~)2
+
~~
+
E(~'M!)2
+
2~'~
E(!iMe).
(16)
The var iance of Q is
From (12) and (16)
V(Q) =
~'~
+
E(~'M~)2
-
[E(~'M~)]2;
however,
E(!'M!)2 - [E(!'M!)]2
is the second cunm1ant of ! 'M!, which from (13) is 2 tr(VM)2 o Thus,
V(~) = ~'~
(17)
+ 2 tr(VM)2.
Restricting Q to unbiased estimators requires
E(Q)
= ~'~
+ tr(VM)
= a2
(18)
•
V and M do not contain I.L am M does not contain any est:imates from
the data of the variances or covariances of
J! 'M
= Q'.
the ei Us.
Thus,
That is, each row and column of M must sum to zero.
rank (M)
= q ~ N-lo
Thus,
9
Since ~'M
= Q',
(17) and (18) became
E(Q)
= tr(VM) = 0"2
V(Q)
=2
(19)
ani
2
tr(VM)
(20)
0
Since tr(VM)s = tr(}.)s, where }.(NxN) is the diagonal matrix of the
latent roots of VM with rank
= q,
=~}..J = 0"2
tr(A)
(21)
and
V(~)
V(~)
It is desired to minimize
E(Q)
= tr(}.) =
0"2.
=2
tr(}.)
2
~ 2
= 2~}.j.
(22 )
subject to the restriction that
Let
p
= ~}.~J -
where L is the LaGrange multiplier.
~}..J
L
-
i)
Setting the partial derivative
of P with respect to the j th non-zero latent root of VM equal to
zero gives
~p
d}.j
= 4 )..J -
L
= 0;
hence,
(23)
Summing (23) over j,
(24)
since
~}.. = tr(}.)
=
J
qL/4
2
0" ,
= 0"2
or
L
=4
2
0"
/q •
10
Substituting this result into (23),
Aj
= a2 /q
,
j
= 1,2, ••• q
,
and from (22) the minimm variance of Q is then
min V(Q)
=2
tr(A)2
= 2]A~ = 2a4/q.
(25)
It was shown that the maxlmm value that q can have is N-l for an
2
unbiased estimator of a.
Thus, the lower bound for the variance
of an unbiased estimator of a linear combination of components of
.
2.
var1ance, a , 1S
L.B.V(Q)
= 2 cr4/(N-l)
The lower bound uSlally will not be attainable.
•
(26)
However, i f a design
is found for which an unbiased estimator has a variance of
2 cr4/(N-l), then it is impossible to find an unbiased estimator and
design which give a lower variance.
11
6.0 OPI'IMAL DESIGNS FOR ESTIMJ\TING COMPONENTS OF VA.RIAACE
IN A, TWO-W'\Y CROSSED CL.\SSIFIC.'\TION
The fir st general type of problem considered is the optimal design for estimating a single function of components of variance in
the two-way crossed classification given by model (1).
The func-
tions considered are single components of variance, sums of components, and ratios of components.
In this section it will be
assumed that it is no more costly to sample different rows or
columns tha n to sample within a cell.
That is, the cost of' sampling
is directly proportional to N, the total number of' observations ..
Optimal designs for simultaneously estimating two components of'
variance are considered in later sections.
6.1
Estimation of
~
Consider the following unbiased estimator of the gereral mean
for the model given by (1)
N
~=~
i, j
The variance of
A
~
Yij/N.
is
,
where ni.
=~~,
j
J
and n.,
J
=~~
'.
i
J
The values of
~~
2
(27)
~
and ~ n.3 are minimized when nij = 0 or 1 and ni. = n. j = 1.
That is,
~ optimal
design is to select each observation from
a different row ani column, giving
2
~nij' ~ni.'
12
'Which is best since each
component of variance is divided by N, the
maxfmum possible number of observations.
6.2
Estimat ion of er;
2
Consider the following unbiased estimator of ere for the model
given by (1) with N observations in one cell
N
~ =]
k=l
(YiJ'k - y ..• )2 I(N-l) ,
lJ
N
where Yij.
=]
k=l
Yi "kiN, the mean of the ij cell.
J
The variance of
Since this attains the lower bound for the variance given by (26),
an/'
__ optimal design is to place all N observations in a single cello
6.3 Estimation of (er~ + er~c + er~) or (er~ + er~c + er~)
222
Consider the following unbiased estimator of (ere + errc + err)
with observations in
one column and N rows
N
~
.](Yij - Y.j)2 / (N_l)
l=l
N
=]
where y. j
Yij/N, the mean of the j th column and ".. means
i=l
"is estimated btl.
2
2
2 2
The variance of this estimator is
2 (ere + errc + err) I(N-l).
13
Since this variance attains the lower bound given by (26), the optimal design is to
select observations from only one colwnn and N
different rows in that column.
Similarly, the best design for estimating
(O"~
+ O";c +
O"~) is
obtained by sampling from only one row and N different columns in
that row.
The total variance of an observation, Yijb from (1) is
2
2
2
2
(O"e + O"rc + O"r + O"c). Consider the following unbiased estimator
of the total variance with each observation from a different row
and colwnn
2
2
2
2 ,.
( O"e + O"rc + O"r + O"c) =
N
~
(Yii - y•• )2 I(N-I)
,
i=l
N
where y .• =
]
i=l
Yii/N•
The variance
of this estimator is
Since this variance attains the lower bound for
the variance of
an estimator given by (26), the optimal design is to select each
observation from a different row ani column.
14
2
6.5
2
Esttmation of crr or crc
2
2
For estimating either crr or crc, we have not been able to develop
a general class of designs which can be proven to be opt:imal for all
situations.
Instead tte discussion ha s been limited to optimal de-
signs for the class of connected desigm with 0 or n observations
per cell, i.e. nij = 0 or n.
A connected design is one for which
the adjusted sum of squares for rows, as presented in Table 2, has
r-l degrees of freedom.
Similarly, when estimating cr;, the adjusted
sum of squares for columns has c-l degrees of' freedom.
The analysis
of variance, 'Which is obtained by the method of fitting constants,
is given in Table 2 for the model given by (1).
Table 2.
.Analysis of variance for a two-way crossed classification
with nij = 0 or n
Source of variation
E(MS)
DF
MS
Columns
c-l
C
Rows (adju sted for columns )
r-l
R*
N_r-c+l
r~r
cre + ncrrc
E
cre
Interaction (adjusted for rows
and columns)
n
Error
N-!!
n
Total
N-l
2
2
2
2
cre + ncrrc + c crr + r ocre
l
2
2
2
O"e + ncrrc + cocrr
2
2
2
2
An unbiased estimator of cr can be obtained by equating the mean
r
squares to their expected values in Table 2 giving
'(;-2
r
= (R*-I*)/c 0
,
(28)
15
where the value of coin general is given by Kempthorne [1952, p. 112]
c r
c
o
= (N - ~ n~ .In•. )/(r-l) ,
j i
~J
J
where
r
=~n .. •
i
~J
For the case being considered ha-e with n ..
~J
Co
=0
= (N-nc)/(r-l)
or n, Co is
(29)
•
Alternative analyses of variance would lead to alternative estima2
tors of a •
r
The investigation here will be limited to the estimator
given by (28).
It is realiZed tha t when the n .. are nat all equal,
~J
other estimators may be more efficient -than the one proposed ha-e.
It is hoped to study this problem in subsequent investigations.
However, f
QI'
this the sis, we will consider only (28) be cause th is
is one estimator in common usage and it has optimal pro perties
whe naIl n. . are equal.
~J
In Section
5.0,
a general quadratic estimator, Q, was con-
sidered for estimating a linear funct ion of components of
variance,
expressed in matrix notation.
By an orthogonal transfonnation, Q
can be expressed as
Q.
=~A.x~
~
(30)
~
2
where the X. are independent chi-square variates each with one degree
~
of freedom, and the Ai'S are the
Jatent roots of VM, where V is the
variance-covariance matrix of th e y' s.
Also, in Section
5.0,
it was
16
shown that the expected value of Q is
E(Q)
=~A.1.
and the variance of Q is
V(Q) = 2
~A~1.
(31)
In general, the row sum of squares adjusted for columns,
(r-l)R*, can be divided into its orthogonal individual sums of
squares
r-l
(r-l)R* = ~
j=l
(0";
+ ni + nc .i)x~
rc
J r J
,
(32)
2
where the X are irrlependent chi-square variates, each with one dej
gree of freedom.
Similarly,
N/n - r-c+l
~
(N/n - r-c+l)I =
k=l
From (28), Q =
•
(33)
~, which may be expressed in general as
r-l
,,2
O"r
222
(cre + ncrrc )X
k
N/n - r-c+l
2
= ~ A.X. +
j=l J J
~
where
A. =
J
~
k=l
~k'
(34)
222
O"e + nO"rc + nc.cr
J r
~--;:;-~--'""-....;;.
and
~
2
~
=-
N-nc
2
2
(r-l)(cre + nO"rc)
(N-nc)(N/n - r-c+l)
•
Since
hence,
r-l
~ c. = (N-nc)/n = c o (r-l)/n •
j=l
J
(5)
17
~2r is
From (31) and (34), th e var ian ce of
r-l
2~
nirc
2
(0" +
e
+ nc
.i)2
J r
j=l
2(r-l)
+
2
2
2
2 2
(0" + nO" )
e
rc
(36 )
•
(N-nc) (N/n - r-c+l)
=0
It will now be shown that a design with n ..
~J
or 1 can be
constructed which has a variance of ~2 smaller than or equal to the
r
variance of
~2r from a design
wi th n ..
~J
=0
or n.
two-way design with r rows, c columns, n ..
~J
Consider any
or n, and~ n ..
=0
~J
= N.
Expand the j th column into n identically filled columns with zero
observations in each of the n columns for the i th row if n
ij
=0
and one observation in each of the n columns for the i th row if
n ..
~J
j
= n.
Each column is expanded in this manner far all
= 1,2, ••• c.
In this way, a design with n~.
~J
structed which has r rows, c*
= nc
or 1 is con-
~ n~.
= N. It
~J
columns, ani
follows from (36) that the variance of
~ for this design is
*
-l (~2
v
+ 0"2 + C.O"2)2
= 2 ~ _e_ _r-:c~--=-J_r_
r
v*( ~2)
=0
(N_C*)2
j=l
r
2 2
2 2
+ 2(r-l) (O"e + O"rc)
(N-c* )2(N
where c * = no and o.* = nc..
Since
N-c*
= N-nc
J
J
N-o*-r+l
l
- r-c*+1)
~ (N-nc)/n
,
N/n - c-r+l
•
!I
(37)
(38)
18
Using the relationships c*
= nc
of "2
O"r given by (37) with n ..
~J
=0
*J = nc;
and (38), the variance
and c.
or 1 is always less than or equal to
~2r given by (36) where n ~J
.. = 0 or n. That is, from
the variance of
=0
any design with n ..
~J
or n, a design with n. j
~
=0
or 1 can be con-
structed which has an equivalent or smaller variance for ~2.
r
The problem row remains of finding the design with n ..
~J
A2
which minimizes the variance of O"r.
Choose any c'(2
=0
or 1
$N-r), the max:i.mu.rn. number of columns
SCI
Let c ~ be the coefficients in the orthogonal forms corres-
sampled.
J
ponding to (32) for n=l.
"A'j
,
~
2
= O"e
+irc
Similarly,
' 2
+ C.O"
J r
,
N-c'
2
2
(r-l)(O" + 0" )
e
rc
= (
)
N-c' )(N-r-c '+1
j
= 1,2, ••• ,
r-l;
,
,
k = 1,2, ••• , N-r-c +1.
Hence,
r-l
:z
1
The variance of
~;
,
c. = N-c
,
J
4
r-l
O"r
~
+ (N-c,)2~
j=l
222
where 0" = O"e + O"rc.
~ c'~
J
•
may be written in the followir:g form:
2 2
40" O"r
2 (r-l)0"4
N-c'
2
both c' and
,
= c 0 (r-l)
c'~
J
'
This expre ssion is obviously minimized 'When
are small.
(39 )
19
For a given N, r, and c', we wish to allocate the samples to the
given rows and columns so as to minimize ] c ~2, subject to the resJ
triction that ] c t = N-e I =
J
~
~Cl
2
j
CI
0
(r-l).
~
=~Cl
2
j
-
We note that
(N-ct)2
r-l
(N_C 1 )2
+ r-l
•
"2 becone s
Hence, V I (O"r)
V,(~2).
r
4
2(r-l)O"
4
2 2
0" O"r
(N-c')(N-r-c'+l) + N-c t
(40)
r-l
~
j=l
(c~
J
- c1)
20"4
r
2
+ ~
r-.L
0
•
Equation (40) is minimized, for a given N, rand c t , if the variance
of
is as small as possible.
c~
J
This is accanplished by making the
c t' s as nearly alike as possible.
J
N/r
= c,
If N is divisible by r, such that
an integer, then c' = c = c' is the minimum rn:unber of
o
columns and ]
(c ~ - C1)2 = O.
J
0
Therefore, for a given r, the
variance of ~; as given by (40) is minimized.
In this case, the
optimal design consists of c columna each with r rows in common.
If N/r is not an integer, we write
N
= r(c-l)
+ a
,
O<s~r
(41)
•
In this case, consider the design shown in Figure 1 which has
the property of having the smaJ.lest value of c' (c'
which
c~
J
2 (c J~
- C1)2 is zero.
0
=c
+ s - 1) for
For the design shown in Figure 1, the
(j = 1,2, ... , r-l) are identti'ied by (39) with the rn:unbers of
columns per row, c-l, in the first r-l rows, i.e., c~
J
= [r(c-l)
+ s - (c+s-l)J/(r-l)
= c-l.
= (N-c')/(r-l)
20
1
Row
number •
I I
r
1 •.•••.•.•• (c-1) •••• (c+s-1)
Column number
Figure 1.
Design for s! r, c'
= c+s-1
In Figure 1, i f any observati OIlS are taken from the first c-1
columns (subject to the condition that rand c' are rot decreased)
am added to the columns to the r
Figure 1,
~
(Cj -
c~)2 ~ 0
~ht
and c'
~
of the c-1 columns in
c+s-1.
Hence, the variance of
~ as seen by (40) would be greater than or equal to the variance of
~; for the configuration shown in Figure 1.
The only desi@1s remainirg which may be better than Figure 1 are
those obtained by taking some of the ob servations in the last s
co1wnns and
p1aci~
them in fewer columns.
As the number of columns
is decreased in this manner, N-c' is increased, which decreases the
variance of
A2
CJ.
r
~
But, ...::i (c ~ - c')
J
increases the variance.
sizes of
CJ;
am
i.
0
2
becomes greater than zero which
The solution for c' depends on the relative
This final question of allocating these s ob-
servations to minimize the varianoo of
~2r will now be investigated.
.4 result d:>tained by Shah [1959J will be used.
Shah considered
connected balanced incomplete designs with v treatments (rows) each
(~"--J
appearing r time sand b (b1oCkS)j\ of k plots each. Treatment i
21
appears n .. times in block j (n .. =
~J
~J
where Q = T - (l/k)NB,
A= r
° or 1).
1\
In this case At
I(vxv) - (l/k)NN', N
=
B represent vectors of treatment and block totals.
= Q,
(n .. ), and T and
~J
Shah shows that
and
A,
tA + a E(VXV)]-l = I(vxv) -
(l/v) E(vxv),
(42)
where a is any non-zero real rn.:unber and E(vxv) is a matrix of one's.
Apparently, these reSl.l1ts extend to any incomplete blocks situation
=~ n ..
regardless of th e values of n .• and n •. where n .•
n •.
J
=~i
~
n...
~J
j
~
J
~J
and
For our problem
A(rxr) R(rxl) = Q(rxl)
(43)
where R is a vector of NID(O,O";) variates, 4 is the adjusted sum of
squares and products matrix for rows, and Q, is the adjusted vector
of row totals (all adjusted for columns).
R=
In th e least square s sense
[4+aE(rxr)]-1Q,
;
since (A. + aE) is symmetrical,
SSR*
= Q'
SSR*
= R'AR
(4 + aE)-l Q,
•
Substituting from (43),
From (42) this becomes
- (l/v) R'EA,R
•
Since the sum of each column of the A rna trix is zero,
EA,=o.
Thus,
r-1
~
SSR* = R'AR = ~
. 1
~=
A.X.2
~ ~
,
22
2
where the X.]. are independent chi-square variates wi th one degree of
freedom each.
For the design being considered, the rows can be pernmted so
that the f:irst r-t rows have c-1 columns in cormnon ard the last
t rows have the same c-1 co1unms in common plus s observations distributed in columns, c, c+1, ••• ,c'-1, ct.
In this case the matrix
of sums of squares and cross products in the normal equations is
c1 •• .cc_1
c
1
···
·
c
c-1
Cc
·
·
··
·
cc'
r
1
•
·.
··
r r-t
rr-t+1
.
·•
··
r
r
cc·····cc'
r
1.
•
. ..
·•
··
·
0
.
0
..
r
1.
n· c
.
.
· ..·1
···
·
·
· . . ·1
l.
rr
r-t+1
· · . · · . . 1.
•
···
·•
··
. . 1·
1.
.
·
·
·
·
n c,r-t+1· . . n c,r
•
•
0
..
•
··
•
•
··
..
nc jr-t+1' . nc ' ,r
0
•
r
r 1 ••• •• rr_t
n· c '
.c-1
•
.
0
..
0
·•
c-1
n1 ·+c-1
··
.
0
·•
nt .+c-1
23
where the transpose of these elements appear below the diagonal.
A matrix of the adjusted row sum of squares and cross products is
A(rxr)
=[~l! ~r:t2 ~ ~r:t2 _1_ ~~ ~r:t2 ~ :
A{2 t x (r-t)
_]
~2 (txt)
:
•
The sub-matrices are
(c-l) E
~l = (c-l) I -
4.t2
=-
r
(C;l) E
~2 = (c-l) I - (C;l) E + B
where
0·
r-t+~
n•.
-~ {
i
n.
~,
~
•
•
B
=
(t
•
n.~,
r-t+l nil')n•.~
Adding (c-l)/r to every element in A gi ves
A + (c-l) E
= [_
~c:12 ~ __ :
0
r
~
(c-l) I + B
I
]
•
Shah shoW's that by using this technique, the latent roots of
[A
+ aE(rxr)] are also the latent roots of
root of 4 becomes ar.
the first r-t rows.
e.
4
except that the zero
The above matrix has r-t roots of c-l from
Let
D = (c-l) I
+
B
The
24
then
Since
I I=
B
In -
I I I·
(c-1) I = B
0,
In - (c-1) II=
0
•
Hence, at least one root fran til e last t rows is c-1.
(A + c-1 E) has at least r-t+1 roots of c-1.
r
One of the se roots
resulted from the technique of adding (c-1)/r.
of Shah, .\ has at least r-t latent roots of c-1.
c~
J
Therefore,
Using the results
That is,
= c-1 (j=1,2, ••• r-t).
41ternative1y, fro m
r-1
~ c~
j=l
J
= N - cf
= r(c-1)
+ s - (c+s-t)
= r ( c-1)
- c+t.
(44)
Similarly, fran the last trows
r-1
~ c t = t ( c-1) + s - (c+s-t )
r-t+1 J
= c(t-1)
Due to the balance from the first r-t rows
j = 1,2, •••• , r~t-1.
ct = c-1
J
The remaining root is
r-1
c'r-t
=~
j=l
r-t-1
c'j -
= r(c-1)
r-1
~ c' -
j=l
j
~
c~
r-t+1
J
- c + t - (r-t-1)(c-1) - c(t-1)
= c-1.
Hence, ther:'e are at least r-t values of c ~ equal to c-1.
J
Consider the desig n shown in Figure 2 which is constructed from
the design in Figure 1 by placing t-1
last t-1 columns into the c
th
(1~
column.
t S s) observations from the
Then, c t
=c
+ s - t.
1
r-t
Row
r-t+1
number
•
I I
r
1 •••••••• c-1
c •••••. (c+s-t=c')
Column number
Figure 2.
Design for 1 ~t ~ s
~
r,
= c+s-t
ct
It has been shown previously that this design
values of
Cj
equal to c-1.
Also, t-2 'values of
Cj
:ra s
at 1 east r-t
equal to care
obviously obtained from the t-1 rows r-t+1, r-t+2, .•• , r-2, r-l.
From (44), the remaining value of
OJ
is
r-1
0;_1
=
f
OJ -
(r-t)(c-1) - (t-2)c
=c
•
Hence,
c~
= c-1
j
c~
=c
j = r-t+1, •• , r-1
J
J
for the design shown in Figure 2.
= 1,2, •••• ,
r-t
For a fixed 0 1 , the variance of
26
~2 is minimized by minimizing ~. (c~ - c,)2. It will now be shown
r
J
0
that this is accompli med by the design shown in Figure 2.
corrected sum of squares among the
c~'s
J
is the same as the cOITected
sum of squares among the coded values c'! =
J
r-l
r-l
(t_l)2
r-l
~ (c~ - c,)2 =~
j=l
where 1
~
t
~
J
s
~
j=l
0
r.
J
r-l
~
j=l
- (c-l).
J
Then,
,
(45)
t),~(ct - c1)2
J
0
is
The sum of the c'."s is
J
r-l
c '.' =
J
~
j=l
Fran (39)
c t - (r-l)(c-l)
J
•
r-l
~
ct = N-c'
J
j=l
Then
c~
For any particular c' (or
minimized by minimizing ~ c ,.,2.
The
= r(c-l)
+ s - (c+s-t)
= r(c-l)
- c + t
(46)
•
~ c I.' becomes
J
r-l
~
j=l
c '.' = r(c-l) - c + t - (r-l)(c-l)
J
=t
- 1
(47)
,
am
ell = (t-l)/(r-l)
•
We note that
r-l
~
j=l
=
rs
amo~ first]
r-t, c'."s
J
+
[55
among last]
t-l, c'."s
+ rSS between first group]
L and second group
or
J
27
r-1
""'(
- I') 2
~
c'.'-c
j=l
J
r-t
=~
r-1
+
~
(C'J.'-C ,)2 +
1
j=l
(c 1.'-'0'2' )2
j=r-t+1
~r-t)cl'] 2
r-t
J
~t-1)C2' ]
+
(48)
2
t-1
where
r-t
'0"1
=~
c'.'/(r-t)
1
J
and
r-1
0"2
= ~
c '! /(t-1)
J
r-t+1
•
Due to the balanced porti ons in Figure 2 the latent roots are
easily identified gi villS
= 1,2, ••• , r-t
c I.' = 0
j
c','J = 1
j = r-t+1, ••• , r-1
J
Then, in (48)
r-t
"'" ( c I. 1
~
1
J
-
l'
c )2
=0
and
r-1
~
r-t+1
(C jl - -)2
c '2
=0
Uso
r-t
~
1
cl
.'
J
=0
and from (47)
r-1
~
r-t+1
c I.'
J
=t
- 1
•
•
•
28
Thus,
ss between first and]
~
= (t-1)
second group of c '." s
= (t-1)(r-t)
_(t_1)2
r-1
J
•
r-1
(49)
SUbstituting these results into (48), for the design shown in
Figure 2
r-1
(c'! _ ~'1)2
~
J
1
= (t-1)(r-t)
(50 )
•
(r-1)
SUppose th e t-1 observations are allotted to any of the s-t+1
columns beyond the first c-1 columns.
The rows can be permuted so
that at least the first r-t rows have c-1 columns in common.
Hence,
there are always at least r-t values of c ~ = c-1 or c'! = O. When
J
J
the t-1 observations are a110tt ed to any of the s-t+1 columns beyom
the first c-1 columns
r-t
f
(c j'
- ~1,)2 = 0
,
~ '2' ) 2 ~
,
r-1
~
r-t+1
(c'.' J
rSS between the two]
L
groups of c'." s
J
r-t
as in (50) since
~
1
=
0
(t-1)(r-t)
r-l
r-1
= a and
c '.'
J
fran (47)
~
r-t+1
c '.'
J
= t-l
as before
0
SUbstituting these results into (48) gives
r-1
~
1
( c ". _ -,,)2:>
c
J
(t-1) (r-t)
r- 1
,
when the t-1 observations are allotted to different columns.
(51 )
From
(50), the equality holds when the t-1 observations are all allotted
29
to the same column.
2 (c j'
quantity
Hence, for any particular value of c', the
- c11)2
for the allocation shown in Figure 2 is less
than or equal to th e quantity obtained by any other allocation of the
t-l observations to the last s-t+l columns.
~
In Figure 2, consider for any 1 S t
last s-t columns of the r
th
row.
s
~r
the observations in the
These observations are completely
confounded with columns and contribute no infonnation to the estima2
tion of err.
design.
Suppose these s-t observations are deleted from the
.As seen previously,
~ (c ~ - c') 2 will ranain unaffected by
J
0
the deletion since the c! are identified with the :n:uni::>ers of obserJ
vations per row in the first r-l rows.
Uso, i f the s-t observa-
=
tions are deleted, then s-t columns are deleted so that N-c'
[N - (s-t)] - [c' - (s-t)] is unaffected.
of
Therefore, the variance
~; as given by (40) is not changed by deleting the last s~t
columns in Figure 2.
Thus, the optimal design for fixed r is of the
form shown in Figure 3, consisting of c-l columns wi th r rows in
common and one column withu of the r rows, where
and r, c, and s are defined by (41).
u
=t
for 2 ~ t
OSu~s~r
(uFl)
That is,
ssS r
u = 0 for t = 1
since in the latter case, s observations are discarded.
From (37), the variance of ""2
err for the optimal design shown in
Figure 3 is
A2
V(er)
r
2
=-2
(N-c')
l
(r-t)
[
U2
J
+(c-l)ur2]2 + (t-l)(u2 + cUr)
2
+ (r-l) 2 er4/(N-r-c 1 +l)
(52 )
30
where t "" u for t
~
2 and t "" 1 for u "" 0, and the total sample size
actually used is N* "" N-s+u.
Similarly, it can be shown that placing the r-u empty cells in
more than one of the c columns increases] (c ~ - C1)2 and hence
J
0
1\2
increases the variance of CT •
r
1
r-u
Row
number
r-u+1
•
r
1 • • . • . c-1
c
Column number
Figure 3.
Optimal design, 0
~
u
~
s
$
r (u ., 1)
The question now remains of finding the best value of t for any
particular Nand r.
and placed in the c
1'2
variance of CT •
r
A.s observations are taken from the last columns
th
But,
column, N-c l increases, which decreases the
~
~ (Cj .. c~)
increases the variance.
2
becanes greater than zero which
For the form of the optimal design shown in
Figure 3,-1ro.bstituting the result from (50) into (40) gives far the
A2
variance of CT
r
r-1
[ (N-c'-r+l) +
2
2
tN-C')~r-l)
(r-t (t 1)
p
P +
+
e2 (N-c l ) J
r-l
(53)
31
where p
~
1 S t
22222
= crr/cr
, cr = cre + crrc '
s
~
r, and c'
:=
C
+
S -
in t and reaches a maximum at t
t
= (r+l)/2
(r odd).
= (r+l)/2
to t
N = r(c-l) + s, 1 ~ s ~ r,
t.
The term (r-t) (t-l) is quadratic
= r/2 = (r+2)/2
(r even) and
This term increases monotonically fran t
and. then decreases monotonically to t
= r.
=1
The last
term in (53) is constant fer all t and the f:irst two terms of (53)
are monotonically decreasing as t increases.
Thus, the variance of
&; is monotonically decreasing for t ~ (r+l)/2.
Since
N - ct
the best design when s
=r
= r(c-l)
,
+ t
is to set t = s
:=
r.
This clearly mini-
mizes the variance as given by (53), ani it uses all of the N
possible samples.
Wortham
[1956]
Also, in this balanced case, Graybill and.
show that the estimator of cr; used here (28) is a
uniformly best (minimum variance) unbiased estimator.
We need to consider t when s < r.
When p is large" the term
containing (r-t)( t-l) will dominate the change in the var iance as
t changes as shown by curve no. 1 in Figure
4.
Curve no. 2 would
result from a smaller value of p.
Curve no. 3 sh ows the case where a local minimum may occur.
The only term that contributes toward increasing
creases is (t-l)(r-t).
t
=1
to t
= 2.
as t in-
This term's greatest increase occurs from
If a local minimum occurs, the variance would have
to at least decrease from t
t > 2.
vee;)
=1
to t
=2
and then increase for some
This case would be fairly unlike1. y since the incremental
increases from (t-l)(r-t) become less as t increases.
also has the largest denominator in (53).
This term
32
When p is small, the first term of (53) dominates the variance.
This term is a monotonically decreasing function of t.
shown by curve no.
When p
= 0,
This case is
4.
the variance of
'8-;
is strictly a monotonically de-
creasing function of t as sh own by curve no. 5 in Figur e 4.
1
2.o.m ...•..••... s •......... r-l
r
Value of t
Figure 4.
~2)
.
V( O'r
versus t for varJ.ous
value s of P
The following procedure will lead to the value of t which minimizes the variance of
at t
= 1.
~;. Denote by VI (1) the derivative of V(d;)
33
(i)
(ii)
If s=r, choose t=r.
If s < r and VI (1) ~ 0, try t=l and t=s.
one whi c h
(iii)
. ..
mIDlID~zes
th3
.
var~ance
0
f A2
err.
If s <: r and VI (1) < 0, determine if V has a local
min:i.mum in the range 1 S t < (r+l)/2.
(iv)
Select the
If not, set t=s.
If s <r, VI (1) <. 0, and if V has a local min:i.mum in the
range 1 S t
.s (r+l)/2; denote by m the integer on the t
scale closest to this minimwn variance.
(a)
if sSm, set t=s.
(b)
If s >m, try t=m and t=s.
Select the one
which minimizes the variance of
6-;.
Case (iv)(b) is th e only case in which t may not be equal to 1 or s.
But, selecting t=s instead of t=m would probably have little affect
on the variance since both m and s would generally be fairly close
to each other when m is better.
A. good operational rule for determining t is:
try t=l (u=o)
and t=s (u=s) and select 1:he one which minimizes the variance of
'i-r
as given by (53).
It should be pointed out that if t=l gives the
minimum variance, only N-s observations are used and the design is
balanced with r rows and c-l columns.
Unless s=r, the use of t=s
resul ts in an unbalanced design with all N observations used.
The selection of t is generally not too critical as illustrated
by the following problem.
small N.
The effect due to t is greatest for a
The case was investigated where N=25 and r=lO for values
of t=l and 5 where p = 1/2, 1,2, and 10.
cases are summarized in Table 3.
The variances for these
34
Table 3.
2
V('cr ) for various values of t and p, N=25 and r=lO
r
p
t
v(a;)/(204)
0.5
0.5
1
5
.1389
.1059
1.0
1.0
1
5
.2778
.2381
2.0
2.0
1
5
.7222
.6761
10.0
10.0
1
5
12.28
12.51
To summarize: for the estimator given by (28) with fixed Nand
r, the design of the type n .. = 0 or n which minimizes the variance
lJ
of
~2r consists of c-l columns with r rows for each column and ore
column with u of the r rows, 0 ~u ~ s
~
r, with n = 1, i.e. one
observation in each of the occupied cells o
Up to this point, r has been kept fixed.
The question nCM
remains, having found the form of the optimal design for a given
r, of the type n .. = 0 or n, what is the best value to choose for
lJ
r?
This final question will now be investigated.
For the form of the optimal design shown in Figure 3, the
exact variance of
'dr2
as given by (52) is easy to find.
But, it is
still difficult to work with this function to determine the values
of rand u which minimize the variance since c and s are functions of r.
Thus, an approxilnate variance will be used which
depends on only one design parameter, c.
o
,.;
4 value, r, can be
....2
found which minimizes the approximate variance of oro
Then,
35
integers above and below
r will be substituted in the
formula for the
exact variance (52), using the optimal allocation for each, until the
1\2
value of r is found which minimiZes the exact variance of a •
r
This
scheme will now be developed.
Due to the near balance of the optimal design, the latent roots
=1
corresponding to the individual degrees of freedom for R* with n
do not differ by more than a;/(N-c).
variance of
(ie + i rc
'ir
is obtained by usirg the average latent root
+ c i)/(N-c) where c
0 r
0
= (N-c)/(r-1).
(a; + i rc
distributed approximately as
of freedom.
A good approximation for the
iJl
+ c0 r
Then (r-1)R* i,s
with r-1 degrees
Then, the variance of the estimator of
ir
given by
(28) is approximately
~2
v(ar )
~
2(i + a 2 + c i)2
e
rc
0 r
0 2 (r-1)
+
o
c
2
o
(N-r-c+1)
•
Let
a
222
=a +a
e
ro
and
Then the approximate variance may be written as
(_1_+_c_
op_)2_
[
r-1
+
-N--~---C-+-J
•
From (29)
(r-1)
=
(N-c)/c
o
where n
=1
Substituting this result into the above expression gives
2a4 (1 + 2c P + c 2p2 - 2p _ 0 p2)
0 0 0
V(ar ) ~
(c -l)(N-c)
o
~
2
•
(54)
36
From (41)
N = r(c-1) + s
where 1
~
~
s
r.
Substituting the limits of s into
r(c-1) + 1
~
this give
N ~ rc.
Substituting these limits into (29) gives
c-1
~
c
o
~
c
or
(55)
Choosing the mid-point of this interval" c = c
tuti~
o
+ 1/2, and substi-
this central value of c into (54) gi ves
4
2cr
~
V ( crr)~
22
2
(1 + 2c 0 p + C0 P - 2p - c 0 p )
(c -l)(N-c - 1/2 )
o
The approximate value
co
of c
0
which minimizes (54) is found by
derivative of (56) with respect to c
setting the
(56)
0
o
equal to zero
which gives
J
(N - ""
c - 1 j:2)(c
c - P2 )
'" - 1)(2p + 2p 2",
0 0 0
2""
[ - (1 + 2pc,..,; + p 2..,2
c 0 - 2p - p c 0 )(N - 2c 0 + 1/2)
o
_ 0
-.
This reduces to
c:
J
[p2 (N - 1/2) + 2p ] + Co [- 2p2 (N - 1/2) + 2 - 4p
+ [p2(N - 1/2) - (N - 1/2) + 2p - 1 ] ". 0
Solving for
C'o
•
give s
p2(N - 1/2)(N - 3/2)
2
P (N - 1/2) + 2p - 1 +
,v
c
o
=
+ 2p (N - 3/2) + 1
p2 (N _ 1/2) + 2p
•
(57 )
37
The negative radical is not a solution because this gives values
of c
o
<
1 which are impossible.
Since the result in (57) is only to be used as a first order
approximation, replacing (N - 1/2) under the radical by (N - 3/2)
still results in a good approximation for c •
Then, (57) becomes
o
C = p(N
o
- 1/2) + (N - 1/2) + 1
p (N - 1/2) + 2
I"\J
The limiting value of Co as N increases is 1 + lip.
(58)
For p small,
this limit may not be too useful.
From (29)
r = (N-c
+
co )/0 0
•
To obtain a fir st approximation for th e value of r which minimizes
the variance
or
'(;2, substitute the value of
r
c0
from (58) into the
above equation givirg
'If = (N
where
c is
value
or
-
c + Co )/'0
(59)
0
th e smalie st integer grea ter than or equal to
c.o
This
r will serve as a starting point from which to fim the
value of r which minimizes the exact variance
Now, to determine
variance
the value of r wh ich minimizes the exact
"'2
cr as given by (52)" co nsider as a fir st approximati on
or
r
for r, the value r given by (59) which minimizes tte approximate
.
,,2
varlance of cr • Let r be the largest integer less than r and let
r
l
r
2
be the smallest integer greater than or equal to r.
values of u
l
and u
2
Obtain the
....2
which minimize th e exact variance of cr for r
r
l
38
(i)
1\2
If the e.xa.ct variance of a
r
as given by (52) for r
l
is
less than the variance obtained wi th r , then choose r
2
3
~s the rext integer less than r • Find the value u
l
3
. ,,2
which minimizes the e.xa.ct variance of a for r • Conti me
r
3
in this manner until the variance increases.
Use the
-'2
value at: r which minimizes the variance of a •
r
(ii)
If the e.xa.ct variance of
~ as given by
(52) for r
less than the variance obtained with r p
as the rext integer larger than r •
2
which minimizes the exact variance at:
(iii)
0f
r
Wh
Find the value u
r
2
and u
2
3
3
~; for r 3 • Continue
•
••
the varlance
.
. ...J... mmJlTllZeS
lu.l
If the exact variances of
is
then choose r
in this manner until the variance :increases.
va1 ue
2
0
Use the
f ,..2
a •
'6; using either r l
r
am u
l
or
are equal, then choose the one which used
the smaller N*
=N -
s + u.
It has been suggested that one should evaluate these des:i.g ns
on the basis of maximum infonnation per observation used, where tl:e
number of ob servat ions use d is N~ This is equivalent to minirniz ing
the reciprocal of the information per observation used" i.e.
N* V(1i).
r
In this case, it appears that one would usually use a
balanced design with c-l or c columns and r rows, such that
(c-l)r = N-s or cr = N.
.A major exception to this procedure would
occur if the optimal r is such that c
= 2 and 2r > N.
In this
case, an unbalanced design would be recessary unless a sample size
larger than N is used.
It is hoped to e.xa.mine this rrethod of esti-
mating optimal designs in a future investigation.
39
To ootain the optimal design some previous knowledge of the
relative sizes of
ir
and
i
are required.
It will be shown in
Section 6.6 that the value actually uS3d for p can differ considerably from the true value wi. thm t appre ciably increasing the
.
f ,.2
varlance
0
(Jr.
Comparisons of the approximate variance given by (56) and the
exact variance given by (52) using t = u
for several cases.
= r/2
are shewn in Table
4
The percent differences were computed in the
following mamer
Percent
= (Exact variance) - (Approximate Variance)
100%
difference
Exact variance
x.
The percent differences in Table
4 were
computed using six decimal
places for the variances al thoogh the variances are srown with four
decimals.
The first three groups in Table
4 with one full ani one half-
full column of observati ons show fClr' each p that the percent differences do not increase appreciably as N decreases.
The fourth group
shows that adding one full column of observations and keepirg N=30
the same, the approximation becomes closer for each p.
The last two
grrops sh ow that changing the half column to 2/3 or 1/4 columns of
observations and keeping N=30 the same, the approximation is closer
for each p.
When N/r is an integer, a balanced design is obtained
and the approximate variance is then identically equal to the
exact variance.
Table
4 indicates that a design consisting of ore
full and one half-full column of ob servations apparently leads to
the worst approximations.
40
Comparison of approximate and exact variances of
Table 4.
Exact
N
r
N/r
p
var~nce
(x
)
Approxi rnate
var:;tnce
(x )
Percent
difference
75
75
75
75
75
50
50
50
50
50
1.5
1.5
1.5
1.5
1.5
0
1
2
4
8
.0559
.1561
.3472
1.0017
3.4006
.0559
.1515
.3288
.9282
3.1073
0
2.94
5.30
7.34
8.62
45
45
45
45
45
30
30
30
30
30
1.5
1.5
1.5
1.5
1.5
0
1
2
4
8
.0963
.2662
.5896
1.6972
5.7556
.0963
.2583
.5583
1.5719
5.2559
0
2.94
5.31
7.38
8068
30 20
30 20
30 20
30 20
30 20
1.5
1.5
1.5
1.5
1.5
0
1
2
4
8
.1508
.4110
.9059
2.5998
8.8039
.1508
.3989
.8576
2.4065
8.0336
0
2.94
5.33
7.1.14
8.75
30
30
30
30
30
12
12
12
12
12
2.5
2.5
2.5
2.5
2.5
0
1
2
4
8
.0509
.3884
1.1044
3.6723
13.3513
.0509
.3808
1.071.14
3.5525
12.8722
0
1.96
2.72
3.26
3.59
30
30
30
30
30
18
18
18
18
18
1.67 0
1.67 1
1.67 2
1.67 4
1.67 8
.1104
.3808
.9064
2.7228
9.4172
.1104
.3709
.8667
2.5642
8.7827
0
2.60
4.38
5.82
6.74
.l)
24
24
24
24
24
1.25
1.25
1.25
1.25
1.25
.3286
.5684
1.0020
2.4510
7.6754
.3286
.5583
.9618
2.2902
7.0326
0
1.78
4.01
6.56
8.37
30
30
30
30
0
1
2
4
8
rir
From (58), optimal designs with one full and one half-full column
of observations arise when p is approximately two.
For, N=30,
N/r = 1.5, and p = 2 the difference between the approximate and
exact variance is 5.33%.
It app3ars that the approximation for V(g.2)
will generally be close to the exact variance.
r
To surrunarize the results of this section:
(a) The estimator, ~2, considered in (28) was obtained by equating
r
the rean squares to their expected values giving a unique unbiased estimator where the mean squares were obtaired by the
method of fitting constants with the row mean squares adjusted
for columns and interaction adjusted for both rows and oolumns.
A.s indicated earlier, other estimators may exist which used in
connection with their opt:ilnal designs could give smaller variances under certain condi tions.
Even for the estimator selected,
it was not possible to make an exhaustive study of this problem.
In order to limit the size of this investigation to a reasonable
size, it was decided to consider only connected designs of tffi
type n .. = 0 or n.
l.J
(b) It was proved tha t the design of this type which minimizes the
variance of "'2
or for a fixed N and r is obtained wi th n .. = 0
l.J
or 1. The results of this study do not preclude the possibility
that a more general design would be better.
(c) N can be represented as follows
N
= r(c-l)
+ s
,
o<
s S r
The optimal design consists of one observation in each cell for
c-l columns and r rows and one observation in u(O~ u~
cells of one column.
Hence, the number of observations actuaJ.ly
used would be N* = r(c-l) + u.
determine u.
ss r, u:j=l)
A. procedure was developed to
In most cases u will be either 0 or s.
(d) The exact variance for
~; is relatively easy to find for the
design given by (c) due to the near baJ.ance.
42
(e) To find the value of r which minimizes the exact variance of
~;, it is easier
to
-v
obtain a first approximation, r, which
minimizes the approximate variance.
Then, values of r below and
values of r above '"
r are used to compute the exact varianc e.
For
each value of r tried, the value of u which minimizes the
variance of
. ..
m~nJ.lTl:LZes
~2r is determined. The combination of rand u mich
the exac t varJance
.
0f
"2.
. mama-.
CJ
~s f aund'
~n th ~s
r
(f) All of the results in this section apply to the est:iInation of
CJ~
and
by simply interchanging r ani c and by interchanging
2
r
CJ
2
•
c
CJ
6.6 Effect of Improper Choice of p in Determining
Optimal Design for
,,2
CJ
r
In order to determine the value of r wh ich minimizes tfu variance of
~2r accordirg to
t.h e procedure given in Section
knowledge of p is required.
Denote by pI, the incorrect value of p
used to determine the optimal design.
varianc e of
t rue va 1ue
CJ
r
Let V denote the approx:iIna te
~; given by (56) for the opt:iInal design based 0 n th e
0f
p.
. t e vanance
.
Le t V•. d enot e the approxJ.IlB.
the optimal design based on pi.
1\2
6.5, some
•
Table
5
IDeM'S
0f
1\2
...
CJ .Lor
r
how the variance of
changes when the value used for p to determine the optimal design
differs from the true value of p by a factor of two.
R.E. is the
relative efficiency of the design based on p' to the optimal design
based on p where
R.E.
= (V/V')
x 100%.
43
Table 5.
Effect on the approximate variance of
~2r due to
using improper choices of p in determining
optimal designs
....,
p
N
c
0
V/(J4
c'0
V' /cr4
R.E.
I"J
p'
(%)
30
30
.25
.25
4.04
4.04
.0849
.08Le
.125
.500
6.01
2.70
.0915
.0938
92.8
90.5
30
30
1.0
1.0
1.90
1.90
.36.31
.3631
.50
2.00
2.70 .3946
1.47 .3994
92.0
90.9
30
30
4.0
4.0
1.24
1.24
2.265
2.265
2.00
8.00
1.47 2.402
1.12 2.m
94.3
93.9
100
100
.25
.25
4.67
4.67
.0225
.0225
.125
.500
7.82 .0248
2.90 .0250
90.7
9000
100
100
1.0
1.0
1.97
1.97
.1026
.1026
.50
2.00
2.90 .1l25
1.49 .1129
91.2
90.9
100
100
4.0
4.0
1.25
1.25
.6514
.6514
2 000
8.00
1.49
1.12
.6914
.6963
94.2
9306
Far example, consider the first case in Table 5.
For N=30 and
p=.25 the optimal design according to the pro cedure developed in
Section 6.5 would have approximately
ly N/4 rows.
4 columns (0)
and approximateo
HCMever, if p is unlmown and a gre ss of p=.125 is
used, the procedure in Section 6.5 would give a design with approximately 6
columns (Co) and approximately N/6 rows.
Tte relative
efficiency of this design to the optimal design is 92.8%, where
the variances are computed from (56).
Table
5 indicates that the standard deviation
of
"2 from the
(Jr
optimal design based on p would vary from about 95 to 98% of the
44
standard deviation of
2p or p/2.
~2r obtained from the optimal design based on
Even though the procedure for finiirg an optimal design
requires some knowledge of p, the value use d for p can vary considerably from the true value without seriously affecting the variance
of the estimator of
cr~.
Due to the closeness of the approximate variance to the exact
variance, the relative efficiencie s based on exact varianc es will be
close to those mown in Table 5 which are based on approximate
variances.
For example, suppose p=leO ani N=30.
has the following pararreters:
1\2
4
0
variance of cr J.S 0.3571cr.
r
r=15, c=2 ani u=15e
The optimal design
The exact
Suppose a gue ss of p=2 e0 is used.
The optimal design based on p=2 eO consists of r=19, c=2 am u=ll.
4
1\2
The exact variance of cr is 0.3939cr.
r
The relative efficiency of
this design to the optimal design based on exact variances is 90 7%
0
as compared to 90.9% shown in Table 5 using approximations e
6.7
Estimation of cr; or cr; when cr;c = 0
When it is assumed that there are
!Xl
interaction effects
2
(cr
= 0), the model given by (1) for the two-way crossed classifirc
cation is
YoJ.JOk
=~
+ r
].
0
+
C
J
0
+ e Ok
J.J
0
(60)
Now, no restriction is placed on the number of 00 servations per cell
other than n
ij
~ 0 and ] nij = N.
The analysis of variance is gi van in Table 6 for a connected
design, using the method of fitt ing constants.
45
Table
6.
Analysis of variance for two-way crossed classifications
with unequal numbers per cell and no interaction
DF
MS
E(MS)
Columns
c-l
C
2
2
2
O"e + cIO"r + r o 0"c
.Rows (adjusted far columns)
r-l
R*
0"e2
Rows
r-l
R
2
2
O"e + rlO"o +
Columns (adjusted far rows)
c-l
C*
0"e2
E
O"e
Source of variation
N-r-o+l
Error
+ C*o 0"r2
C
0"2
o r
+ r * 0"2
o c
2
N-l
Total
If only one column were used ani all observations in the i th row
of the c-column design were put in the i th row of the l-column design,
E(MS~)
would be the same as E(MSR) above except that r
is, E(MSJL)
-~
= ie
+
C
0
i,r
where
0
0
possible situation for estimating
appear to occur when
mated and only
ir
ani
O"~ = O.
ie
= (N- ~ n~l ./N)/(r-l).
0";
l
= O.
That
The best
with the c-column design would
That is, only
0";
and
appear in the variance of
0";
6-r2 •
need be estiBut in this
most favorable case, MSR will have identically the same distribution
as
MS~.
Also, the variance of "'2
O"e will be greater with c columns
unless SSC* and SSE are pooled, in which case the variance of
be the same in the two cases.
tion of
~;
'Will
Since under the most favorable situa-
O"~ = 0, the c-column design
is no better than the l-column
46
design, it would appear that under the general condition of
()~ ) 0,
the l-colwnn design would be better.
! more formal approach would be -00 pro ve that under the
assumption that
()~ = 0
,
where
c*
o
=
[N -~~ ~]/(r-l)
n.
j i
j
•
Since
,
c.
*o
Hence, it is easy to show that the coefficients of
are never greater for V(MSR) than for V(M5R*).
is more complicated for the coefficients of
()~
and
The problem
()4rO In this case it is
required to prove tba t
~n~.
•
J.
-
2
-N
~
n~.
•
J.
J.
J.
+
~
[~n~.
-N-t:.
• J.
).
]2
2
co
is less than or equal to
2
n .• n ..
J.
J.J
n •.
J
•
The proof of this inequality bas not been attempted here,
since the
earlier argument that the I-column design is better than or equi valent
to the c-column design seemed sufficient.
47
2
A.ssuming we have established that, if arc
= 0,
the I-column
design is best, we rnxt proceed to find the best allocation in this
column.
This problem was solved by Crump
can be identified with his a classes.
situation was presented in Section
[1954J
where the
The optimal design for this
4.0.
The rea:r.lts of this secti on apply to the estimation of
= 0,
with a;c
by interchanging E. and
608 Estimation of
a;/(a:
The problem of estimating
E rows
ac,
2
'£0
+ a;c) or
a~/(a:
+ a;c)
a;/(a: + a;c) will now be studied.
Consider the analysis of variance given in Table 2 which is obtained
by the method of fittiI'lS constants for the model given by (1).
Alternative analyses of variance would lead to alterrative estima-
= i r I(ie
tors of p
+
i rc ).
The investigation here will be limited
to obtaining an estimator of p from this analysis of variance.
For a design with n.. = 0 or n, an est:linator of
J.J
il(i
r
e
+
nirc )
is readily obtained by considering tre ratio Fl = R*II" in Table 2
=0
Setting n ..
J..J
p
= i r'/(ie
+
0
or 1 leads to a simple estimator of
i rc ) based
on the ratio
general design with n .. = 0 or n the
J..J
= R*/I~
estimator cr:
Fl
For the more
p is far more
complicated.
*
R
can be divided into its r-l orthogonal. sum of squares giving
*
(r-l)R
where a
2
= ie
+
i rc
r-l
~
=~
i=l
and each
(a2
+ c.a2 )X.2
J..
r
J..
,
x~J.. has one degree of freedom. !lso,
(N-r-c+l) I
= a2X2
(61)
48
where
x2
has (N-r-c+l) degrees of freedom.
F'
(N-r-c+l)
= R*/I = ~--.:;-..,...-L.
(r-i)
r-l (~2
2
v
+
Then,
2) 2
Xi
C i (Jr
2X 2
i=l
(J
or
r-l
F'
= r-11
]
. 1
(1 + c.p) F.
~
~=
(62)
~
where F. is a F variate with 1 and N-r-c+l degrees of freedom.
~
The
expected value of F with ml and m degrees of freedom is
2
Therefore
E(F )
i
= (N-r-c+l)/(N-r-c-l)
•
Substituting this result into (62) gives
(63)
From (61) the expected value of R* is
*
E(R ) =
(J
E(R* )
(J
2
2
(Jr
+ ~
r-J.
r-l
~
~
. 1
~=
c.
~
(64)
and from Table 2
=
2 + C (J2
o r
•
(65)
Equating (64) and (65) gives
r-l
Co
=
r:l ']1
~=
ci
•
Substimting this realit into (63) gives
(66)
•
49
This result leads to the following unb:ia sed estimator of p based
on F'
A
1
_
C-o
p -
[(N-r-C-1)
(N-r-c+I)
(67)
where C is given by (29)
o
(68)
Co = (N-c)/(r-1) •
The variance of
p is
V(p)
From
(62),
(N-r-c-1)2
= (N-1'-c+1) 2
V(F')
c
2
•
(69)
o
the variance of F' is
V(F.) r-1
2
V(F') =
~ 2 ~ (1 + c.p)
(r-1) i=l
~
r-1
CoV(Fi'F j +i) ~
+
2 ...:::::l
(r-1)
i=l
The moments of F with
E(F ) =
~
and
(70)
degrees of freedom are
k - 1)1
(~72 - 1)1
Then, the variance of F is
V(F)
or
V(F)
...:::::l (1 +c. p)(l + c.p).
j =fi
~
J
(~/2 +
k
where k < m /2.
2
~
r-1
~
=
= E(F2 )
_ E(F) 2
2
2 m2 (~ + m2 - 2)
(m2 - 2)
2
(m2 - 4)~
(71)
'
In our case, Fi has m = 1 and m = N-r-c+1 degrees of freedom.
2
1
Thus,
V(F.)
~
=
2
2 m2 (m2 - 1)
(m2 - 2)
2
(m 2 -
4)
•
(72 )
50
Crump
[1954,
39]
p.
have orthogonal
x2
gives the covariance between two F variates which
with single degrees of freedom in the numerator
x2/m2,
and a cornman denominator of
2
Cov(F 0' F 0) ...
J.
m22
(m
J
2
•
- 2) (m - 4)
2
2
(73)
Substituting fran (72) and (73) into (70) gives
r-l
2
2 ~ (r-l)
(m -l)
2
-2
2
V(Ft) ...
~ (1 + c.p)2
o
1
J.=
r-l
+~
(m - 2) (m2 - 4)
2
i=l
J.
r-l
~
(1 + c.p)(l + c .p)
j=ll
J
J.
•
Substitut:i.IJg this result into (69) gives
,..
V(p)
for m >
2
2 (r_l)-2
= 2
--c o (m2 - 4)
4. It follows from
[(~_2)
~ (1 + ciP
)2J
2
+ l~ (1 + ciP)]
(66) that
~
(74)
(1 + cop) is a constant
J.
for any particular c
r-l
~
(1 + cop) = (r-l)(l + c p)
0
J.
i=l
•
(75)
Thus,
r-l
~1
o
(1 + cop)2
J.
=~(l +
c.p)2
J.
-[~(l +
J.=
+
[~(l + ~iP)]2
r-l
Cop)]
J.
r-l
2
51
or
r
~
~
. 1
~=
(1 + c.p)
2
~
2
= P2~
~ (c.-c)
0
~
+ (r-l)(l + c p)
2
(76 )
•
0
Substituting from (75) and (76) into (74) gives
2~
(m2 - 2) P ~ (c. - c )
2
0
~
+ (m2- 2)(r-l)(1 + cop)
2
2
+ (r-l) (1 + c p)
2
•
o
V(p) is minimized for fixed Nand r by making] (c. - c )2 small and
~
m2 and Co large.
0
Since
~
= N-r-c+l
Co
= (N-c)/(r-l)
and
these are both maxiInized men c is minimized.
,
Thus, the situation
here is analogous to that encCllntered in Section 6.5 for the esti2
mation of O'r.
vations.
In rome cases, it will be desirable to discard obser-
N can be represented as
N
= r(c-l)
+ s
,
0< s
~
r
•
As seen in Section 6.5, for the form of a variance function as sham
in (77), the variance of
p is
minimized by using c-l columns with r
rows in common and one colunm with u (0:5 u
rows.
:s s
~
r, u
f 1)
of the r
The value of u=l doe s not occur because in this case s obser-
vations are oomp1etely confourxied with columns and can be discarded
wi thou t changing the variance of
p.
The form of the optimal design
.52
for minimizing the variance of '"
p is shown in Figure 3.
.i\s before,
the rmmber of cib servations used is N* = N-s+u.
Up to this point, r has been kept fixed.
The question now re-
=0
mains, having fourrl the fonn of the design with n ..
lJ
minimizes the variance of
r?
p,
what is the best value to choose for
This final question will now be investigated.
balance of the optimal design, the
obtain.
or 1 which
variance of
Due to the near
p is
fairly easy to
But it is still difficult to work with this variance in
order to determine the value of rand u which minimizes the variance
of
p.
of
p will
Thus, as was done for estimating
i,r
an approximate variance
be used to determine an approximate value,
minimizes the exact variance of
p.
'r,
which
Then, integers above and below
,y
r are tried until the value of rand u is determined which mini-
mizes the exact variance of
"p.
F' /(1 + c o p) is distributed as F when the same columns are
sampled for each row.
Ft/(l + c p) is distributed approximately
o
as F when approximately the same columns are sampled for each row
as is the case here since it was shown that the variance of
p is
minimized when the number of columns sampled for rome rows is c-l
and c for the remaining rows.
Urrler these conditions the sum of
squares far rows is approximately distributed as
2
22
+ C 0" )x •
(0"
o r
This is the same approximation used in Section 6. .5 for obtaining an
.....2
approximate var iance of O"r.
Thus,
FI
When N is large,
-:::
(1 + c p) F
o
the variance of F from (71) is approximately
•
53
Hence,
Since
~
= r-l
and m2
= N-c-r+l,
2 (1 + c p)2 (N - c)
V(p) ~
2
(78)
0
co(r-l) [(N-C) - (r-l)]
From (29)
(r-l)
=
(N-C)/C
0
o
Substituting this result in (78) gives
(79)
o
From (55), Co S c S Co + 1.
interval, c
= co
Substitutirg the mid-point of this
+ 1/2, into (79) gives
V(p)
~ 2 (1 + c0
p)2/(c - l)(N - c - 1/2)
0 0
(80)
0
The value, '"
co' of Co which minimiZes (80) is obtained by setting the
derivative of (80) with respect to c
~
o
= 2p
o
equal to zero, which gives
(N - 1/2) + (N - 1/2) + 1
P (N - 1/2) + P + 2
(81)
0
From (29)
r =
(N -
c
o )/c 0
+ C
•
(82 )
~
To obtain a first approximation far r, substitute the value of c
o
from (81) into (82 ) giving
"oJ
r
=
N"';"';
(83)
(N - c + c o )/c 0
"
where I"J
c is the smallest integer greater than or equal to 'c.
o
This
~
value, r, will serve as a starting point from which to find the value
of r which minimizes the exact variance of
p.
To obtain the value of
54
r requires some previous knowledge of p since it appears as a nui-
p.
sance parameter in the variance of
It will be ::hown in SectLon
6.9 that the value actually used for p can differ considerably from
the true value of p without appreciably increasirg the variance of
The exact variance of
p will
p.
now be cbtained for the form of the
optimal design obtained previously in this section with n .. = 0 or 1.
~J
This design consists of c-1 columns with r rows and one column with
u rows.
The occupied cells contain one observation.
The form of the
design is illustrated in Figure 3 and th e analysis of variance showing the orthogonal parts of the row sum of squares is given in
Table
7. From Table 7
R*
* + (t-l)R2*
= (r-t-l)ELn_l-:;-
+____
R* ,
3
t=u for u 22
r-l
t=l for u = 0 •
Since ~ and
R; have the same expected values, they can be pooled
giving
R4=
[(r-t-l)~ +
R;]/
(r-t) •
Substi tuting these results into (67) gives
'"
(N-r-c-l)
p = (N-c)(N-r-c+l) [ (t-l)
Let R2
*/ I = F2 and R*/I =
4
Ft."
r~
+ (r-t)
r-1
rR4]- N:C
"
(84)
Then '"p becomes
'"
(N-r-c-l)
[
p = (N-c)(N-r-c+l)
(t-l)F2 + (r-t)Ft.
J-
r-l
N=C •
(85)
F /(1 + cp) is distributed as F with (t-l) and (N-r-c+l) degrees of
2
freedom and
Ft./[l
+ (c-1)p]
(N-r-c+l) degrees of freedom.
is distributed as F with (r-t) and
The variance of the unbiased esti-
55
p,
mator,
('"
V p)
=
based on the ratio
Table 7.
= R*jI
Analysis of variance for optimal design shown
in Figure 3
Columns
DE'
MS
E(MS)
c-l
C
2
2
2
r cr
cr + c l crr +o
c
R*
cr + c ocrr
r-t-l
~
cr + (c-l)crr
t-l
~
cr
1
R*
cr + (c-l)cr
r
I
cr
Rows (adjusted for columns) r-l
First r-t rows
Last trows
First r-t vs last trows
Error
N-r-c+l
Total
N-l
2
(86)
+ 2(t-l)(r-t)(1+cp)(1+cp-p)COv(F ,F )
2 4
Source of variation
where F
is
V(F )
2~(t-l) 2(l+cp) 2V(F2 )+(r-t) 2(l+cp-p) 2]
4
~
~
N-r-c-l
(N-c)(N-r-c+l)
Fl
3
2
2
2
2
2
2
2
+ ccr
r
2
2
is distributed as F with (t-l) and (N-r-c+l) degrees of
freedom and F
of freedom.
is distributed as F with (r-t) and (N-r-c+l) degrees
4
Formula (71) gives the variance of F. The munerator
mean squares of F2 and F are orthogonal but they have the same
4
denominator mean square, I. For this case, Crump [1954J gives the
covariance of these two F I s.
56
Now, to determine the value of r that minimizes the exact
variance of "p as given by (86), consider as a first approximation for
r, the value ~ given by (83).
Follow the same procedure of consider-
IV
ing integers around r and finding the best value of u for each as
given in Section
.
fA
varl.ance
0
p.
6.5
leads to the value of r which minimizes the
The value chosen for c is the largest integer greater
than or equal to N/r.
To summariZe the results of this section:
(a) Alternative estimators of p used in conjunction with their optimal designs may yield smaller variances under certain conditions.
(b) The connected design of the form n.. = 0 or 1 which minimizes
l.J
the exact variance was determined for fixed Nand r.
Designs consisting of n .. = 0 or 1 lead to a simple estimator
l.J
for
based on the ratio of mean squares. The optimal
cr;/i
design consisted of c-l columns wi th r rows each and one
column with u of the r rows where N = r(c-l) + sand 0 ~ u
u :J 1.
The total number of observations used is N*
~
s
~
r,
= N-s+u.
(c) The exact variance for '"p is relatively easy to find for the form
of the design designated by (c) due to the near balance.
(d) To find the
value of r that minimizes the exact variance of "p,
consider as a first approximation to r the value
which minimizes the approximate variance.
rV
r from
(83)
Consider integers
around r in the manner described in Section
6.5.
This leads
"
to the design which minimizes the exact variance of ,po
For
each value of r considered, the best value of u must also be
determined.
57
(e) All of the remIts of this section apply to the estimat ion of
O"~/i = O"~/(O":
+ O";c) by simply interchanging rand c.
6.9 Effect of Improper Choice of p in Determining
_ _ _ _,_--.,;O;.:.p_t_imal De si,:=,gn_f_o_r---a;.p
_
In order to determine the vallie of r which minimizes the variance of
p according
to the procedure given in Section 6.8, some kmw-
ledge of p is required since p is a nuisance pararreter in the formula
for the variance of
p as gi ven by
(86).
Denote by p' the incorTect
value of p used to determine the opt:imal design.
In order to
simplify the investigation of the effects d: us:ing pi instead of the
true value of p in obtaining the optimal design, the approximate
variance of
p for
p for
large N iodll be used.
The approximate variance of
the nearly optimal design based on p is obtained from (79)
,-v
f'J
2N
N
V = 2 (1 + c op) /(c 0 -l)(N-c)
(87)
-v
'V
where c is the smallest integer greater than or equal to c
o
IV
am c
0
is
obtained fr om ( 81 )
co -- ~-peN1/2)
+ (N - 1/2) +.1:,
- 1/2) + p +~
Similarly, the approxirra te variance of
p for
(88 )
•
the design based on pi
is
-
V'
_
=2
N
2""
N
(1 + c'p) /(c'-l)(N-c')
o
0
(89 )
N
c~
where c' is the smallest int eger greater than or equal to
~, _ ~(N - 1/2) + (N - 1/2) + 1
o p'(N - 1/2) + p' + ~
and
•
!'
(90)
Table 8 shows how the approx:i.ma te variance of p changes when the
value used for p to determine the opt :imal design differs from the
58
true value by a factor c£ four.
That is, p' = 4p and p' = p/4.
.....,;
R.E.
is the approx:ima te re1at ive efficiency of the design based on p' to
the design based on p where
""
R.E.
Table 8.
=
(V/V') x 100%.
p
Effect on the approximate variance of
due to using
an improper choice of p while determining the
opt:i1na1 design
-N
P
p'
V
R.E.
30
30
.25
1.00
1.00
.25
.1023
.5952
.1205
.7025
84.9
84.7
30
30
.50
2.00
2.00
.50
0.233
1.782
0.259
1.965
90.0
90.7
30
30
2.00
8.00
8.00
2.00
1.78
21.37
1.84
21.53
97.0
99.3
100
100
.25
1.00
1.00
.25
.0267
.1650
.0321
.2004
82.9
82.3
100
100
.50
2.00
2.00
.50
.0626
.4949
.0702
.5521
89.1
89.6
100
100
2.00
8.00
8.00
2.00
0.495
5.939
0.507
6.040
97.7
98.3
V'
For example, consider the first case in Table 8.
and p=.25 the optimal design is obtaired With c
o
~
For N=30
4.70.
However,
if P is unlmCMn and a gue ssp' =1. a is used to determine the opt:im a1
design then c I
o
~
2.75.
The approx:i1nate relative efficiency of this
design to the optimal design is 84.9%.
....
,..
Table 8 indicates the approximate standard deviation of p from
the optimal design based on p would vary from about 90 to almost
100% of the approximate standard devia tion of
optimal design based on 4p or
p/4.
p obtained
from the
Even though the procedure for
finding the optimal design for p requires some knowledge of p, th e
value actually used may vary considerably from the true value withoot
seriously affectirg the
variance of the
6.10 Estimation of cr;/cr; or
estimator of p = cr2/cr2 •
cr;/cr~
r
when cr;c = 0
Consider the special case of estimating cr;/cr: or cr;/cr: when
cr;c
= O.
Following the same argument used in Section 6.7, it
appears that a 1-co1umn design is always better or equivalent to
a c-co1umn design.
In this case, th3 best allocation to the rows
in this column is solved by Crump [19'4], where the ! rows can be
identified wi. th his a classes.
The optimal design for thi s
situati on was presented in Section 4.0.
60
From (58) the approximate value of c which min:imizes the vario
,,2
ance of O"r is
c (el) - peN - 1/2 ) + (N - 1/2) + 1
o r peN - 172) + 2
where p
of c
o
= 0"r210"2
and
0"2
=2
0"e +2
0"rc .
(91)
From (81) the approximate value
which minimizes the variance of
p is
(92)
When N is large, (91) be comes approximatel y
"V
2
co(O"r) ~ 1 + lip
(93)
ani (92) becomes approximately
~o (p) ~ 2 + lip •
(94)
For large N, the optimal design for estimating p consists of approximately one more column than t:re optimal design for estimating
Table 9 shews the approximate values of c
optimal designs for est:imating
Table 9.
0";
o
for large N for the
and p.
Approximate values of c for the optimal designs
2
0
for est:imating O"r and p (large N)
Approximate values of c
p
Estimating
ir
0
Est imating p
1/4
5
6
1/2
3
4
1
2
3
2
3/2
5/4
5/2
9/4
4
0";.
61
It was seen in Sections 6.6 am 6.9 that the design used could
vary considerably from the optimal design without 1 arge losses in
information.
From
Table 9, the designs for estimat:ing
not dif fer markedly vb en p
~
1.
ir
and p do
Tre se cases are well within the
limits of discrepancies studied in Sections 6.6 and 6.9.
Allocation for the estimation of
mates of p and vice versa when p
1.
If estimates of the two
a; and p, were equally important, one might consider a
values,
design with a c
when p
~
a; will also give good esti-
~
1.
o value about half-way between c 0 (i)
r and c 0 (p)
In this way, only a small amount of information would
be sacrificed on each estimate.
The remarks c£ this section apply to the simultaneous estimation of
a~ and a~/a2 by
:i..rJt, er chang ing
rand c in the discussion.
62
8.0 SIMtJLTANEOUS ESTIMA,TION OF (ir AND (ic
The simultaneous estimation of
O"~ and O"~ is complicated since
one must determine the type of design to use in addition to the
allocation far a given design.
efficient estimation of
columns.
0";
In general, it ha s been shown that
requ:ires a large number of rows and few
Similarly, efficient estination of
O"~ requires a large
number of columns and few rows.
It was not the original purpose of this the sis to investigate
this problem, but a few tentative observations will be presented.
We will consider two possible types of designs, which attempt
to sample both a large number of rows and columns.
8.1
L-shaped Design
0";
First, suppose we use N observations primarily to estimate
l
and N2 observations primarily to est:irnate O"~.
From section
6.S, the
N observations wc:uld consist of a large number of' rows ani a few
l
columns.
The N observations would consist of a large number of
2
columns and few rows.
A. ccmbination of these two configurations is
designated as an L-shaped design and is shown in lngure S.
total numbers of rews ani columns are r
=rl
+ r
2 ani c
= cl
Tre
+ c •
The N observations contain one oolumnwith r (0~r3<rl) empty
l
3
cells, ~ = rC - r y
The N2 observations contain one row with
2
(0 ~c3 <cl ) empty cells, N2 = cr 2 - c •
3
3
observations is N = Nl + N2 - r 2c 2 •
c
The total
number of
2
63
1
Rows
1
t
r2
I-
---;i 3-+:
c
I
I
:
I
-l_~--~I-------_!
I+- c2 ~i__--- cl
~I
-+:..
Columns
Figure
5.
L-shaped design
8.2
Disjoint Rectangles Design
2
The second design investigated fOt' simultaneously estimat:ing O"r
ani
O"~ is designated as a "Disjoint Rectangles" design and is shown
in Figure 6.
This design consists of g rectargle s, each containing
r rows and c columns.
The rectargles are disjoint in that each
rectangle samples a dif ferent set of r roW s and c columns.
In this
way several rows (gr) and columns (gc) are sampled.
The moo.el for the d isj oint rectangle s design shown in Figure 6
is
y.~J·kDv
= ....
+ g.~ + r ~J
.. + c ik + (rc) ~J
.. k + e.~J·kDv
(95)
64
Columns
1. . . . . . . c
1
•
t-1"""'------+--1
Rows.
• I---I~-----+--t
r
1
•
• Rows
..........
.... .
Columns
--1.......1
r
c •
. ,--,r-----...,...-,
11--1
+---1
Rows.
r~,--
1. . . . . .
Columns
Figure 6.
where i
..............
e
C
Disjoint rectangles design
= 1,2, •••g;
j
= 1,2, ••• r;
k
= 1,2, ••• c; t = 1,2, ••• n ijk ;
and the total number of cb servatic ns is:2 n"
= N. YiJ'kt is the
ijk lJ k
t th observation in the j th row and k th column of the i th rectangle;
2
~ is the general mean; the rectangle effects, g.,
are NID(0,a );
1
g
the row effects within rectangles, r. " are NID(O,i); the column
lJ
r
effects within rectangles, c .., are NID(O,i); the interaction
lJ
c
effects within rectangles, (rc). 'k' are NlD(O,i); the within
lJ
rc
2
cell effects, e
, are NID(O,a ); and all effects are uncorre1ated.
e
ijkt
The analysis of variance for the model given by (95) is shown in
Table 10 where n
ijk
=n
for all cells.
If
a~ = 0, G should be used
65
to improve the est:ilnates of
0'; and
This case will not be con-
sidered in this dissertation.
Table 10.
A,nalysis of variance for disjoint rectangles with
n observations per cell
E(MS)
Source of variation
DF
115
Rectangles
g-l
2
2
2
2
2
G O'e + nO'rc + nCO'r + nrcrc + nrccrg
Rows
g(r-l)
2
2
2
R cre + ncrrc + nccr
r
Columns
g(c-l)
/C2
2
2
C O'e + nO'rc + ntkY
c
Interact ion
g(r-l)(c-l)
I
2
2
O'e + nO'rc
grc(n-l)
E
2
O'e
Error
Total
N-l
For a balan ced case as shown in Table 10, Graybill and
Wortham [1956] show that the estirna tor of the oomponents of variance
obtained by equating the mean squares to their expected value s are
the best unbiased quadratic estimators.
"2
O'r
The estimator for
= (R-I)/nc
0'; is
(96 )
•
1\2
The variance of O'r is
1\2
V(O'r)
=
2r
N(r-l)nc
[
2 2 2]
2
2
2 2
(O'e + nO'rc)
(cre + nO'rc + nCO'r) +
c-l
•
(97)
, It will now be shown that (97) is minimized when n=l i f g is an
integral multiple of n.
Consider any disjoint rectangles design with
66
the parametersl g, r, c, and nijk = n.
From this design, a new
design can be constm cted with parana ters: g I =
and n ijk = 1.
The variance of
1(,,2) _
2r
V O"r - N(nr-l)c
[(U~
S;
gin,
r I = nr, c I = nc
for this design is from (97)
2
2 2
(O"e + O"rc)
2
2 2
+ 0"rc + nCO"r) + nc-l
].
(98)
The term in [brackets] in (98) is obviously less than or equal to
the similar term in (97).
Sim e
1
nr-l
<
2r
N(r-l)nc
2r
N(nr-i)c
Hence, VI (d;)
,
1
(r-l)n
•
~ V(~), where g is an integral multiple
Similarly, VI(O";)
~ V(O";). Then, the best disjoint rectangles de-
sign with n. 'k = 1 should be found.
lJ
been attempted.
of n.
A. more general proof has m t
It appears thcrt. in the case where n. 'k = n, setting
lJ
n=l is desirable.
The remaining discussion in this section will be limited to the
case where n=l.
8.3 Comparison of L-shaped and
~sjoint
Rectangles Designs
In this thesis, these two designs will be compared far the
special situation men
0";
=
O"~
and
V(~;)
=
V(~;).
Admittedly, this
may be somewhat artificial, but a comparison on this basis should
indicate the relative usefulness of these two designs in view of the
fact that it has already been shown that alloc ati on is not too
sensi tive to p.
It is contemplated that further research will be
67
devoted to the problem when the two components of' variame are
different.
Table 11 shows the results of some spe cjfic rnlIIler ical examples,
wh ere p
= O"r2/( O"e2
2)
+ O"rc
= O"c2/( O"e2
2) andn=.
I S '~nce O"r2
+ O"rc
2
= O"c'
4t
....
~
A2
seemed reasonable to consider designs which give V(O"r) • V(O"c).
The
value of the total number of observations, N*, was allowed to vary
slightly in order to obtain balance.
made on the basis of N*V(&;).
In this case, comparisons were
The reciprocal of this quantity is
the amount of information per obser-vation.
V
= V(~2r )/20"4 = v(ic )/20"4.
Figur es
In Table 11,
The design parameters are identified from
5 and 6. The estilna tor of a; for di sjoint rectangles is
given by (96) and the estimator for the L-shaped design is given by
The variames in the last column in Table 11, N*~2
V , are
(28).
based on an estimator,
Figure
5 are
ignored.
one given by (28).
~,
of
0";
in which the last c
l
columns in
Otherwise, the estimator is the same as the
The estimator,
&;,
could be improved slightly
by pooling the error sum of squares fran both parts of the des:ign.
Several tentative conclusions can be drawn from Table 11 0
The
best disjoint rectangles design is obtained when r ~ 2 + lip.
When p <2, the best disjoint rectangles design is better than the
best L-shaped design.
When p
~
2, there is little difference
between the best of each of the two designs.
When p ') 2, the best
L-shaped design appears better than the best disjoint rectangles.
As p becomes large, the est:imator of
discards tre la. st c
l
~2r
in the L-shaped design which
columns beccmes better than the estimator
using all of the observations.
This improvement is apparentJ.ya
68
Table 11.
Comparison of disjoint rectangles and L-shaped
designs with V(ci) = V(cl), where
r
c
r = c
i
i
L-shaped
Disjoint rectangles
p
N*
N*V
N~
36
33
36
4.24
3.29
3.24
5.00
4.12
3.24
0
0
0
36
33
36
10.89
9.46
9.84
10.00
10.08
9.84
2
2
3
8
0
0
36
36
-
33
25.6
26.0
30.2
2
2
12
0
36
-
79 .. 8
82.0
g
r=c
N*
N*V
r=c
r 2=c 2
.5
.5
.5
4
2
1
3
4,
6
36
32
36
3.38
3.ll
3.24
10
7
6
2
3
6
r =c
3 3
0
0
0
1
1
1
9
4
2
2
3
4
36
32
10.00
8.25
8.44
10
7
6
2
3
6
2
2
2
9
4
2
2
3
4
36
36
32
26.0
24.8
27.1
14
10
7
4
4
9
4
2
3
36
36
82.0
84.8
16
10
36
36
34.8
31.6
125
result found earlier in Section 6.. 5 where it was shown desirable to
keep the numbers of observations per row approx:imate1y equal.
was particularly true for large p.
This
.tUso the L-shaped design which
has partial rows and columns appears to be desirable when p > 1"
where c
2
= 2.
In this case a precise statement can be made concern-
ing the use of part ial columns (or rows).
ignoring the last c
1
The anal ysis of var iance
columns is given in Table 12!J where 02 = 2.
Let r4 = r - r3' the number of rows with two columns.
The est:imator of
cr;
used is
~r = (R*-E)jc 0
0
In this case
~2
crr =
•
(99)
69
Table 12.
Analysis of variance ignoring last c colwnns
l
Sour ce
of variat ion
DF
MS
1
C
r-l
R?~
Component 1
r -1
4
~
Component 2
r-r4
I?
2
2
a + 2ar
a2 + a r2
Error
r4-1
E
a
Total
Nl-l
Colwnns
Rows (adjusted for colwnns)
~
The variance of a
r
E(MS)
2
2
a + c o ar
2
is
2,...4 _
v(crA2 ) = -....;.v_
r
where p .
r
(r+r4-2)
2
2 2
= ar/a
•
To find the value of r4 which minimizes (100), set the derivative with respect to r
Nl
=r
+ r4'
equal to zero, subject to the restriction,
4
This leads to the following result. Use
r4
=1
r4
= Nl /2
+ (Nl - 2)/Pr
(2 ,
,
(101)
That is, when Pr > (2, one long colwnn and one shorter (partial)
colwnn is better than using two colwnns of equal ler.gth.
r4 must be at least two in order to provide an estimator.
Of
course,
70
It will now be shown that an L-shaped design with r
and r
= c,
where the last c
l
= N/4.
= c2 = 2
columns are neglected, is equivalent to
the disjoint rectangles design with r
rectangles, g
2
= c = 2.
For the disjoint
O'~
From (97), the variance of
is
(102)
For the L-shaped design w.i. th two complete columns, r4 = r.
r
=c
Since
and
N = 2r + 2c r
N
= 4'
+1
4 ,
•
Substituting these results into (100) gives
~2 = 2g4 [ (1
V(O'r)
+ 2Pr) 2 + 1
J
(103)
•
This is identical to (102) for disjoint rectangles.
borne out in Table li.
identical.
This result is
,,2
Similarly, the variances of O'c would be
i
No re strict ion that 0'2. =
has been imposed.
r
c
It ha s been shown when P > (2 that the variance can be reduced
r
for the L-shaped design by using a partial column and making the
first column longer.
better.
In some instances, the L-shaped would be
It appears, that a disjoint rectangles design with
r = c = 2 would not be used.
In Table l i it ,is indicated that the
best disjoint rectangles design has r
= c =2
for large p.
Since
the L-shaped design does better than 2x2 squares, this gives more
evidence to support the use of the L-shaped design for large p.
In general, as p increases, the L-shaped design is favored over
disjoint rectangles.
.Also, the analysis which discards the infor-
-e
71
mation in the other leg of the design in order to achieve balance
appears more useful as p increases.
When Pr >
If2 ,
the use of one
long column and a second shorter column is advantageous.
for pc.
Disjoint rectangles appear better for small p.
Similarly,
When p is
approximately two, it does not appear that one design is particularly
superior to the other.
The se observations must be taken with some
2 2
+ (Jc was not investigated; also the
rT
size of N-il- may have an infillence on where one design is preferable
reservation as the case for
to the other.
(J
72
9.0
PROCEDURES TO ATTAIN SPECIFIED PRECISIONS
OF ESTIMl\TES OF COMPONENTS OF VA,RIJ\NCE
9.1
One-way Nested Classification
Consider the problem of finding the mininmm value of N such that
the variances of the est:irnators of the components of variance in the
model given by (2) are less than or equal to
V(~~) S
pi
and
specified values
V(~:) S P2
(104)
or
(105)
and
(106)
Th e variance
0f
~2
O"a is a comp1ica t e d function, Crump [ 1954J
0
To
find the mininmm value of N which satisfies (105) would require a
trial and error procedure.
For a particular value of Nt the optimal
design is found by the procedure given by Crump.
of
Then the variance
~ would be determined for the optimal design based on Nt.
If
(105) was satisfied, then values of N less than Nt would be tried.
If (105) was not satisfied, then values of N greater than Nt would
be tried.
was found.
This would continue until the smallest N satisfying (105)
Such a procedure would be extremely tedious.
A. simple
scheme which gives the approximate mininmm value of N which satisfies (104) will now be developed based on an approximation for the
varia nce of
A,2
0"a
•
73
Consider the model given by (2) where the analysis of variance
is given in Table 1.
The estimator of
O'~ obtained by equating mean
squares to their expected values is
-"2
O'a
Hammersley
when n
[1949]
= n,
i
= (A-W)/no
•
(107)
showed that the minimum variance of
~~ is obtained
i.e., equal numbers of observations per class. However,
integer values can not always be obtained.
If N
= ak +
s where N,
a, k, and s are integers, Crump
[19.54]
consists of s classes with (k +
1) observations and a - s classes
with k observations each.
integer.
That is, n.
1
showed the optimal design
=k
or k + 1 where k is an
In order to s:implify further calculations, it will be
shown for this case that n ~N/a for large N.
o
n
o
1 [ N - ~n?]
=~
a-.L
rr-1
The value of n
o
is
(108)
•
The case that deviates most from equal numbers per class where
n.
1
=k
or k + 1 is when a/2 of the classes contain k observations
and a/2 of the classes contain k + 1 observations.
N
= ak
In this case
+ a/2
(109)
and (108) beccmes
n
o
1
='="""""""
a-.L
[ N-
ak
2
+ ak + a/2
- ]
N
N
=a -
a
4N(a-l)
•
Hence, as N increases
(110)
The var ianc e of
,..2
0'a is
V(~~) = [V(A)
+
v(w)J/n~
(111)
74
where
V(W)
= 20"4/(N-a)
e
•
(ll2)
Since the mmbers of observations per class are nearly equal,
V(4) can be approximated by
V(4)
where p
= ia /i.
e
';::!;
20"4 (1 + n p)2/(a-l)
e
0
(ll3)
SUbstituting from (ll2) and (113) into (lll) gives
(ll4)
From (llO),
(llS)
Using the same allocation
"2
4
V(O"e) ~ 20"e n o/N(no-l) •
From
(116)
(4), for large N and p not too small
no ~ (p + l)/p •
(ll7)
SUbstituting this result into (llS) gives for large N
,,2
NV ( O"a) ~
p
20"4
e 2 [ Np 2 + SNp + 4N - p - 1 ]
(p+1)
Np - P - 1
•
(118)
A.s N increases
(ll9)
for the approximate minimum obtainable value of NV(~~) for large N.
From (119) the approximate mininmm value for the square of the
coefficient of variation of ~~ is given by
2 ,.2
N·C.V. (O"a) ~ 2(p + 4)/p.
(120)
75
<: Pl w:h en
Then C.V. 2 (1\2)
CYa ,.,
2(p + 4)/Np S Pl
or when
(121)
Table 13 shows the approximate values of N required to satisfy a
specified C.V. (~) for various values of p.
good even for small values of N.
These results are quite
For example, C.V. is .53 for p=2,
N=24 and p=4, N=16.
Table 13.
Approximate total
for estimation of
~ber
a
of observations (Na ) required
Specified
p
c.v.(a-~)
.10
.20
.30
.50
1/4
3400
850
378
1)$
1/2
1800
450
200
72
1
1000
250
112
40
2
600
150
67
24
4
400
100
45
16
Substituting from (117) into (116) gives the approximate vari-
2
~2
ance of CY from the optimal design for estimating CYa for large N
e
1\2
4
NV(CY ) ~ 2CY (p + 1) •
(122)
e
e
The square of the coefficient of variation of ~; is
c.v. 2 (S;) ~
2(p + l)/N •
(123)
76
2(p + l)/N ~ P2'
or when
(124)
When N ~ N , then (105) and (106) are approximately satisfied for
a
e
large N by choosing N
= Na
and using the optimal design for estima-
2
'
t J.ng
eTa·
When N > Na , then additional observations are required so that
e
C.V. 2(~)
,s
C.V.2(~) ~
P2.
The approximate number of classes required for
PI is a
~
Nino.
already sampled, then each
to the estimate of
i.e
If observations are added to the classes
observation adds one degree of freedom
From (112), it follows that for a fixed a,
-
the allocation of the N samples to the .::: classes has no affect on
V(d;).
But, these extra observatims may as well be added so
that
the number of ob servations per class are almost equal in order to
minimize
V(~). The number of additional ooservations required to
satisfy (106) when N > Na is approximately N - Na for large N.
e
e
9.2 Two-way Crossed Classification
Consider a pro cedure such that the variances of the estimates
of the row and column components of variance of the model given by
(1) are less than or equal to specified values
A2) < ,
V( O"r
- PI
(125)
and
(126)
77
or
(1 2 1)
and
Consider the L-shaped design discussed in Section 8.1.
SUb-
stituting from (93) into (56) gives the approximate minimum value
~2
obtainable for V(O'r)' for large Nl'
V(g;) = 20'4Pr (Pr
+ 4)/Nl •
Similarly,
Or,
(12 9)
and
'V
2
~2
=2
(pc + 4)/N2 Pc •
Then (121) is approximately true "When
C.V. (O'c)
(130)
or
(131)
Formula (128) is approximately true when
or
(132)
Thus, the total number of observations required is
78
or from (131) and (132)
N ~ 2(Pr + 4)/Pr Pl + 2(p c + 4)/P c P2 -CPr + l)(p c + l)/PrP c •
(133)
Consider the disjoint rectangles design described in Section
8.2.
The variance of
variation squared of
~; is
-a-;
gi. ven
by (97).
The coefficient of
is less than PI when
(134)
Similarly,
(135)
A. procedure which leads to the minimum value of
(134) and (135) are satisfied is as follows.
cular values of r
~
2 and c
~
N
Slch that both
Compute for any parti-
2:
(136)
and
•
Choose g as
mum [gr' gc].
the smallest integer greater than or equal to the maxiThen, N
= grc.
Repeat this pro cedure for various
canbinations of r and c until the minimum value of N is found.
this way, both (134) and (135) are satisfied for a minimum value
of N.
(137)
In
79
SOME FURTHER RESULTS FOR A. ONE-WAY
10.0
NESTED CLASSJli'ICATION
Consider the one-way nested classification described by the
model given in (2).
variance, V(y.. )
J.J
Suppose it is desired to estimate the total
2
e
= era2
+ er , with a fixed cost.
Cons:ider the problem
of minimizing v(er; -; er;) with a fixed cost.
For a given value of the number of classes,
freedom for error is fixed, N-a.
since V(S2)
e
given
~
= 2ere4/(N-a).
~,
the degrees at'
Thus, the var iance of
The variance of
~a2
~; is fixed
is minimized for a
by making the numbers of observations per class as nearly
equal as possible, Crump [1954].
From the expected mean squares in Table 1 an estimator of
2
2
(era + ere) is obtained by
2'" 2
era + ere
= rLA.
The variance of this estimator at'
v(er~ :
er;)
]
+ (no-l)W /no •
er~
= [V(.I\)
(138)
+ er; is
+ (no -1)2v(w)J/n~
(139)
where
V(W)
= 2ere4/(N-a)
(140)
and as seen previously
since the numbers of observations per class are almost equal.
Sub-
stituting from (140) and (141) into (139) gives
2A 2
V(era + ere) ~
2er~
2
no
[(1 + n op)2
(no-1
a-I
+
N-a
)21
•
(142)
80
From (110), N ~n a.
o
Substituting this result into (142) gives for
large N
(143)
•
•
1'-13
Since an ~ N, the rmmerator in the [brackets] of (~) is approxio
mately for large N
2 2
2an p + an p + an •
000
Substituting this result into (14.3) gives
v(ia -:;: i)
~
e
2cr4 (2p + n p2 + l)/n (a-I) •
e
0
0
(1l.J4 )
Let
ca
c
= cost
= cost
of sampling a class
of obtaining an observation in a class.
e
The total fixed cost, C, is
(lLS)
Hence,
a ~C/(ca + no ce ) •
(146)
Differentiating (144) with respect to~, subject to the condition
given by (146), and setting this result equal to zero gives a
quadratic equation
a
r
in~.
Denote the solution of this equation by
As a first approximation to the value of
variance of
(cr~ ~ cr~),
~
which minimizes the
choose the integer closest to a
l
and N the
largest integer less than or equal to (C - aca)/c • 4llot the N
e
observations such that the numbers of observations per class are
as nearly equal as possible.
81
11.0 SUMMARY.AND CONCLUSIONS
11.1 The Problem
A sample of size N is to be seJe cted from a population which can
be classified in two ways, called rows and columns.
•
In selecting the
sample, it is assumed that r rows and c columns have been randomly
selected from the large IDJ.IIlber available and that for each row and
column cell (i, j) a certain IDJ.IIlber of measurements or observations
(n .. ) are obtained at randan. The measured value (y) of a g:bTen
J.J
· the J..th rowand J.th co1
sampl e t a ken f rom
umn·J.S assumed t 0 b e represented by the following linear random model:
y J.J
.. k
= ....
+ r.J. + c.J + (rc).J.J. + e.J. jk
'
where .... is the average value of the measurements in the population
and k represents the particular sample selected.
The random effects,
r., c., (rc) .. , and e ..k represent row, column, interaction, and
J.
J
J.J
J.J
sampling (Within a row-column cell) variation. It is assumed that
their effects are normally and independently distributed with zero
·
2 O"c'
2 O"rc'
2 a nd O"e'
2 respec t·J.ve 1 y.
means and
varJ.ances,
O"r'
The purpose of this investigation was to dev:i13e the best methcd
of selecting a sample of siZe N so as to obtain mininn.un variance
222
2
estimators of the parameters ...., O"r' O"c' O"rc and O"e.
11.2
Qptimal Allocation
If we use as the estimator of ....,
~ =~] yJ.J
. .k/N, where the
I'
summation is over the members of the sample, the variance of .... is
minimized if each observation is taken fran a different row and
82
= c = N.
column, so that r
n
ij
After a permutation of rows and columns,
= 1 for i=j am n ij = 0 for ifj (i,j
I:
1,2, ... , N).
If we consider any linear function of the components in Yo ok'
the variance of the linear function being designated as
•
(i,
~J
the
lower bound for the variance of an unbiased quadratic estimator of
(i
is 20'4/(N-l).
.Although it is not always possible to attain this
lower bound, estimators of the following can attain the lower bound:
(1)
0';, if all N observations are in one cell, r = c = 1 and
~l
(2 )
= N;
222
O'e + O'rc + O'r' if only one column is selected with N rows,
with one observation per cell, r = N, c = 1 and nil = 1
(i
= 1,2, ••• ,
irc + i,c
(3)
0'2 +
(4)
th
e t0 tal
e
N);
if rows and columns are reversed in (2);
o
var~ance,
O'e2+2
0'
+ 0'2 + 0'2
, i f eachb
0 serrc
r
c
vat ion is selected from a different row and column, so
that r
=c
= N, n ..
~J
=1
for i=j ar:d no
0
~J
=0
for ifj.
2
For other linear forms, in particular for 0'2 and 0' , it is not
r
possible to attain the lower bound.
c
The variance of the est:ilnator
is materially affected by the sampling procedure.
41so, optimal
allocation will vary from one est:ilnator to another.
The estimator used for
ir
was obtained by equating mean squares
to their expe cted values in an analysis of variance based on the
method of fitting constants, where the row mean square is adjusted
for both row and column effects.
tions per cell, the variance of
With unequal numbers of observa-
~2r is a very complicated function.
83
This investigation was limited to designs of the type where there
might be some vacant cells and each occupied cell contained n observations, so that n .. = 0 or n.
lJ
These designs led to a simple
estimator of 0"; based on the difference of two mean squares.
proved f or this type of design that the variame of
when n .. = 0 or 1 (n=l).
It was
~2r is minimized
For a given number of rows, r, the optimaJ..
lJ
design with n .. = 0 or 1 consists of c-1 columns and aJ..1 r rows in
lJ
connnon and one column with u (Osu~s~r,
uf1) of the r rows,
where
N • r(c-1) + s •
1'2
J\n approximate variance of O"r was developed to obtain a first
approximation,
r,
for the number of rows which minimizes the
·
exac tvarlance
0f
",2
O"r.
The value of
"wi
r
in which
c is
= (N
r
is given by
,.;,.;
I
f'J
- c + co ) c 0 ,
the smallest integer greater than or equal to
"'"c o =
Pr(N - 1/2) + (N - 1/2) + 1
P (N r
172)
+
2
c
,0
and
,
222
~
M
where P = 0" /(0" + 0" ). Integers below r and integers above r
r
e
rc
r
are tried until the value of r is obtained which minimizes the
exact variance of
~;.
The exact variance of
'ci-;
for this form of
the opt:i1na1 design is
-(r_t) [12+(C-l)P] 2+ (t-l)(l+cp) 2J
l
+ (r-1) / (N-r-c+1)
,
84
cl
where
=
(cle + irc ),
t=u for u
~ 2 and t=l for u=O. For each value
of r tried, the value of u must first be determined to minimize the
.....2
variance of O"r. Generally, the best value of u is 0 or s; when
s=r, u=r.
The determination of the optimal design depends on the value of
•
Pro
It was shown by several numerical examples that if the value of
P actually used in constructing the optimal design differed from the
r
true value by a factor of two, the relative efficiency
of the design
is still generally greater than 90%.
When
ing
0";
0"2rc
= 0, it was shown that the optimal design for estimat-
consists of one column.
Then, the analysis of variance
reduces to a one-way nested classification for Which the optimal
allocation for estimating
0";
is given by Crump [1954].
From the same analysis of variance used to estimate
n
ij
0";
with
= 0 or 1, a simple estimator of P is obtained from the ratio
r
of the row to interaction mean squares.
procedures which lead to
The form of the design and
the design which minimizes the variance of
2
A
P are analogous to those obtained for estimating O"r except that the
r
first trial value for r is determined by using
2Pr (N - 1/2) + (N - 1/2) + 1
Co =
P (N - 112) + P + 2
r
r
"-J
r-
and the variance of P for the form of the optimal design is
r
A
V ( Pr
)
2 (r_l)-l
= --'""l2~---':----
(N-r-C-l)(t-l)(r-t)p;
2]
(N-c) (N-r-c-3) [
+ (N-c-2)(r-t + (N-c)p )
r
where t=u for u
~
2, t=l for u=O.
85
The determination of the optimal design for estimating P rer
quires some previous knowledge of the value of P ' which is the
r
quantity one is attempting to estimate.
Even if the value of P
r
•
actually used to
•
value by a factor of four, several numerical examples shOiled that
determine the optimal design differs from the true
the relative efficiency of the design is still generally greater
than 80%.
2
When arc = 0, the optimal design for estimating P then was
r
shown to consist of one column (j=l) with no restriction on the
nil's, i.e. n ..
lJ
~
O.
The analysis of variance with one column re-
duces to a one-way nested classification for which the optimal
allocation for estimating Pr is given by Crump
All of the results far estimating
mation of
a;
[1954]
0
and P apply to the estir
a~ and Pc simply by interchanging rand c.
A few tentative conclusions were obtained concerning the
simultaneous estimation of
to two types of designs:
0";
and
O"~. The investigation was limited
(1) an L-shaped design consisting of Nl
observations with several rows ani a few columns and N2 observations with several columns and few rows and (2) a disjoint
rectangles design consisting of several non-overlapping rectangles,
each with r rows and c columns.
The comparison of these two
designs was limited primarily to some rmrnerical examples.
In order
to simplify the comparison, the only cases considered were for
2
2
a r = o"c so that Pr = Pc
= p.
This may be somewhat artificial, but
it should lead to some general notions.
,,2
(1'2
structed so that V(O"r) = V O"c).
The designs were con-
86
As p increases, the usefulness of the L-shaped design increases.
It was shown that the N observations should contain two or more
l
and one long colmnn and one shorter, colmnn when
columns when p ~
f2
(2.
Also, as p increases, the variance of ~; is smaller when
•
p >
•
only the Nl observations are used. The gain from discarding observations is due to the near balance which is achieved in this manner.
The usefulness of the disjoint rectangles design increases as
.
p decreases, rand c increase for the optimal disjoint rectangles •
When p becomes large, the optimal values of r and c equal two.
In
this case, it was shown that an L-shaped design could be constructed
which is better than disjoint rectangles.
This also supports the
observation that L-shaped designs are better for large p.
The
optimal designs of each type appear to be about equivalent when p
is around the value of two.
A procedure was developed to find the approximate minimum
.
.
A2
A2
value of N requ~red ~n order that both V(O'a) and V(O'e) are less
than specified values in a one-way nested classification.
The
approximate minimum values of N required in order that both V('i-)
r
"2
and V(O'c) are less than specified values were found for the L-shaped
design and a procedure was developed for finding the exact minimum
value of N for a disjoint rectangles design.
4 procedure was developed for a one-way nested classification
Which minimizes the approximate variance of the estimator of the
2
2
total variance, O'a + O'e' subject to a fixed total cost, where the
cost of sampling a class is not necessarily equal to the cost of
sampling within a class.
87
11.3
(1)
Construction of more general designs without the restriction
, dt
that n ..
lJ
•
•
Suggested Future Research
=0
or n.
4pparently this will be very diff~
because of the complicated general variance function •
(2 )
.
222
2 2
Construction of designs to estlmate 0rc' 0rc/oe' 0;l0e' and
2 2
These require that some n .. ' s must exceed one.
c /0e .
lJ
°
(3)
Study of robustness of the designs to non-normality and to
inequality of some of the components.
For example, estimate
,.,2
when ,.,2
and ,.,2
v
v
v
vary f rom co1umn t 0 co1umn.
e
rc
r
(4)
Development of procedures to combine estimators when the
analysis of variance contains more mean squares than components of variance to be estimated.
(5)
Consideration of estimators other than those based on the
usual analysis of variance.
Some simple examples indicate
that this should receive serious consideration.
(6)
Evaluation of designs on the basis of information per
observation rather than on the variance of the est:imator for
fixed N, or on the basis of other cost cons:iderations.
(7)
Development of a sequential estimation procedure, with at least
two stages, the first stage design to
be used to estimate
values of p to be used for designing the second stage.
•
.
(8)
Extension of the results to multi-stage nested and crossed
class ificat ions, with both fixed am ramom components.
(9)
Study of the effect of the n .. being randan, due to natural
lJ
or other causes.
(10) Consider other designs and estimating procedures when several
variance components must be estimated.
88
LIST OF REFERENCES
Anderson, R. L. and Bancroft, T. 4., 1952. Statistical Theory in
Research. McGraw-Hill Book Co., Inc. New York.
,
Baines, A. H. J., 1944. On the economical design of statistical
experiments. Ministry of Supply Advisory Service on Statistical Methods and Quality Control. Tech. 1lep. No. ~.C./R/15.
Crump, P. P., 1954. Optimal designs to estimate the parameters
of a variance component model. Unpublished Ph.D. Thesis,
North Carolina State College, Raleigh.
Graybill, F. A. and Wortham, A. W., 1956. A note on uniformly
best unbiased estimators for variance components •
J. Amer. Stat. Assoc. 51, 266-268.
.
Hammersley, J. M., 1949. The unbiased estimate and standard
error of the interclass variance. Metron 15, 189-205.
Kempthorne, 0., 1952. The Design and .Analysis of Experiments.
John Wiley and Sons, Inc. New York.
Lancaster, H. 0., 1954. Traces and cumulants of quadratic foms
in nonnal variables. J. Roy. Stat. Soc. (B) 16, 247-254.
Shah,
~. V., 1959. 4 generalization of partially balanced incomplete block designs. 4.rmals Math. Stat. 30, 1041-1050.
Whittle, P., 1953. The analysis of multiple stationary time
series. J. Roy. Stat. Soc. (B) 15, 125-139 •
•
,
..
ABSTRACT
GAYLOR, DAVID WILLIAM.
The Construction and Evaluation of Some
Designs for the Estimation of Parameters in Random Models.
(Under
the direction of RICHARD LOREE ANDERSON).
,
•
This dissertation considers methods of sampling, with fixed
sample size N, which would lead to good estimates of components of
variance in a two-way crossed classification model with n .. obserJ..J
vat ions in the (i,j) cell:
..
yJ..'J'k
= IJ,
+ r.J.. + c.J + (rc)..
J..J + e.J..J'k '
where the effects r i , c j ' (rc)ij' and e ijk are normally and independently distributed random variables with zero means and
.
2 O'c'
2 O'rc'
2
2 respec t·J..ve1 y.
vanames
O'r'
and O'e'
It was shown that the lower bound f or the variance of an unbiased quadratic estimator of a linear function of components of
variance with expected value, 0'2:; is 20'4/(N-l) where N is the total
IUlIIlber of observations.
were determined
2
2
and (0'e + 0'rc +
Procedures which achieved the lower bounds
for estimating 0';, (0'; + O';c + 0';), (0'; + O';c +
2
2
0'r + 0').
In estimating other functions of the
C
variance components, optimal allocation depends upon the estimators
used.
The estimator for
ir
was
obtained by equating mean squares
to their expected values in an analysis of variance based on the
method of fitting constants, where the row mean squares is adjusted
•
•
••
for column effects and interaction is adjusted for both row and
column effects •
A procedure was developed to
1\2
minimize the variance of O'r when
n ij = 0 or n; for this case, it was Shown that n
= 1.
w·
From the same analysis of variance used to estimate
estimator of
ir /(ae2 + irc ) can be
row to interaction mean squares.
a~, an
obtained from the ratio of the
The form of the design of the
type n .. = 0 or 1 which minimized the variance of this estimator
lJ
was determined.
Allocation for estimating
ic
and
i/(i
+ i ) were
c
e
rc
treated
similarly.
a;
..
1\ few tentative conclusions on the simultaneous estimation of
and
a~ were obtained. Two types of designs were compared •
Procedures were developed for both the one-way nested and twoway crossed classifications to find the approximate minimum value of
N such that the variance of the estimators of components of variance
were less than specified values.
A procedure
was developed for the one-way nested classification
which minimizes the approximate variance of an estimator of the
total variance subject to a fixed total cost where the cost of
sampling classes is not necessarily equal to the cost of sampling
within classes•
•
,
.•
a
..