:.'~'5W'~.,~.~r~I~~I~l!Ii~".~.:~i~.·
_
•••
•
i':,.",.
,":,'.'V"i·'''l
. . _~\W~~,;""
..·"'-'-i("'~.;"":'~.'~."\.;'c.X<l·",~
~~~if.~~::*:~}tJ~¥1?""'~'-,Ii;.·:<,~J",\'.p;-\·,.:f:,:",
""'"-'~'"
-'..-
~
"-
'~'->""--"
._"
OPTIMAL DESIGNS TO ESTIMATE VARIANCE
COMPONENTS AND TO REDUCE PRODUCT VARIABIL
FOR NESTED CLASSIFICATIONS.
Richard R. Prairie
Institute of Statistics Mimeo #313
ABSTHACI'
PAAIHIE} HICHARD HOLAHD.
Optimal Designs to Estimate Variance
Components and to Heduce Product Variability for Nested Classifications.
(Under the direction of HICHARD LOREE ANDERSON.)
Two separate problems were considered in this thesis} both
involvine the design of experiments for estimating parameters for
nested classifications.
The first problem was concerned with the importance of experimental
design for the estimation of functions of variance components for a two
stage nested classification when there is a specified objective.
The
objective considered was that of efficiently reducing product} or total}
variability through the expenditure of a fixed amount of funds.
A model was proposed that related the reduction in total variance
to the amount of funds expended.
'I
f
With the model considered it was
found that the optimal allocation of funds is a function of p. =
2jc 2
CT A CT B}
where CT~ and CT~ are the variance components associated with the two
,
assignable sources of variation.
,I
I!
It was assumed that an estimate of p is obtained from an experiment based on the model
,
I'
x .. =m+A. +B.,}
~J
~
~J
(1)
j = 1}2} ••• }n ;
i
En.= N}
~
where m is constant and A. and B . are normally and· independently
iJ
l
distributed (NID) with means zero and respective variances
r
:e
CT~ and CT~.
A method of design construction was given so that different designs are
--'
obtained by varying the nwnber of classes a.
Two estimators of p based
on the usual analysis of variance were considered.
A criterion was developed for judging the influence of design and
estimator.
Some nwnerical results were presented which indicated that
for most situations an intermediate value of!:) say N/4
give results that are
~uite
<a < N/2)
will
close to the optimal.
The second aspect of this
thesi~
dealt with the design of experi-
ments for estimating parameters from a three stage nested classification)
for which C
(NID with mean zero and variance O"~) is added to (1) with
ijk
j = 1)2) ... )b.; k = 1)2) .. • )n .. j l: l: n .. = N.
l
lJ
i j
lJ
The estimators of the
variance components used were those based on the usual analysis of variance.
Variances of the estimators of
studied for various designs.
of m)
O"~
and
O"~
+0"
~
~J O"~ and O"~ were derived and
Designs were presented for the estimation
+ 0" ~ that minimized the variances of these esti-
Designs were also given for the estimation of 0" ~J O"~ and
.mators.
2
2
2
P = O"A/(O"C +O"B) that appeared to yield estimates whose variances were
near minimal.
Attention was also devoted to the simultaneous estimation of 0" ~J
0"2
2
B and 0" C·
A method of design was suggested and the variances of the
estimates from this design were compared numerically with other types
of design.
.,
;
,
I
,
,.f
e
0PrH1AL DESIGNS TO ESTD1ATE V.(\.RIANCE COMPONENTS AND
,
TO REDUCE PRODUCT VARIABILITY FOR NESTED CLASSIFICATIONS
by
RICHARD ROLAlill PRAIRIE
-e
A thesis submitted to the Graduate Faculty
of North Carolina State College
in partial fUlfillment of the
requirements for the Degree of
Doctor of Philosophy
I'
I,
I
11
DEPARTI<JIDlT OF EXPERJMENTAL STATISTICS
RALEIGH
il
I
'II
II
11
!'
APPROVED BY:
Chairman of Advisory Committee
,~
ii
BIOGRAPHY
Born:
July 26} 1933, Minneapolis, Minnesota
Parents:
Roland H. and Margaret Prairie
Married:
November 29} 1958, to Elaine M. Caroon
Child=en:
Michael Roland} born June 2O} 1960
Jennifer Yvonne} born July 17} 1961
Undergraduate work:
University of Minnesota, 1951-1955
Graduate work:
North Carolina State College, 1955-1958 and
Employment:
Graduate Assistant} North Carolina State College}
1955-1958 and 1960-1962
Operations Analyst} Cape Canaveral, Florida,
Summer 1957
Statistician, Sandia Corporation} Albuquerque}
New Mexico} 1958-1960
iii
ACKlIOWLEDGEMENTS
To R. L. Anderson I express my sincere thanks
during the course of this investigation.
~or
his cooperation
He suggested the problem and
,
provided the assistance that made the completion of this thesis possible.
I am grateful to W. J. Hall for valuable suggestions) and to
J. C. Koop) H. L. Lucas and R. G. D. Steel
~or
their carefUl review of
the final draft.
I am indebted to Miss Dianne Diano for her conscientious efforts
in preparing and running the programs that Were required to obtain the
numerical results contained herein.
I thank the Institute of Statistics for providing the financial
assistance that made my stay at Raleigh possible.
To W. S. Mallios I express my thanks for acting as a sounding
board for many of the concepts that are) and are not) contained in this
thesis.
I thank my wife for putting up with the whole thing.
~
!
II
iv
TABLE OF COllTElrfS
Page
LIST OF TABLES
LIST OF FIGURES •
vi
........
• .viii
....
1.0
INTRODUCTION.
2.0
REVIEW OF LITERATURE .
3.0
REDUCTION OF PRODUCT VARIABILITY THROUGH THE USE OF VARIANCE
COMPONENT ANALYSIS • '.' • • • •
• • •• •
1
4
3.1 Introduction
• • • • •
3.2 Notation
• • • • • • • •
3.3 A Cost Model
• • • •
3.4 Description of the Experiment for Estimating p • • • •
3.5 Some Mathematical Development for the Estimators of p.
3.6 Criteria for Assessing the Influence of the Design of
Experiment on the Reduction of Total Variance . •
3.7
Some Numerical Results for P
Prob [( O"Rldl) <
=
. 3.7.1 General Remarks • • • . • • • • • • •
3.7.2 Values of p) 100A) D) N) a and f3 Used
Computing P • • • • •. • • • • • • •
3.7.3 ·OUtline of Computational Procedure
3.7.4 Discussion of Results • • . • • • • •
4.0
4.3
5.0
....2
2
~
4.4
Variances of A) O"B
and ,C • • • • • •
4.5
"'2 "'2
Covariances among O"AJ
O"B and
~
c .
. • •
in
•
36
36
46
!
. • • • •
./
• •
. . . . .
0Pl'JJ.1AL DESIGNS FOR ESTll'lATING PAIWtlETERS FOR A THREE STAGE
NESTED CLASSIFICATION.
• • • • •
Introduction. • ••
••••••••
• • • •
Estimatipn of Parameters for a Three Stage Nested
Classification. • • • •
• • • • • •
5.2.1 Estimation of m • • • • • • •
5.2.2
~ ..
Estimation of O"~
26
36
Introduction. • • • •
• • • •
Derivation of Variances of Ql) Q2 and Q3' • • • • • ••
Derivation of Covariances among Ql) Q and Q~ •
~
11
12
15
19
22
f3] .
VARIANCES OF ESTIMATED VARIANCE COMPONE~rrS FOR A THREE STAGE
NESTED CLASSIFICATION.
• • •
• • • • • • • •
4.1
4.2
11
47
52
52
55
65
67
67
69
69
71
71
72
v
TABLE OF COIITErITS (continued)
Page
2
5·2·3
Estimation of
B
• • • •
72
5.2.. 4
Estimation OfO-~
•••••
72
5·2·5
5.2.6
0-
222
Estimation of 0- C + o-B + 0-A
222
Estimation of p = 0-A/(o- C + 0- B)
.
6.0
• • •
74
.
SOME ASPECTS OF THE Sn,1ULTANEOUS ESTIMATION OF VARIANCE
COMPONEHTS FOR A THREE STAGE NESTED CLASSIFICATION
6.1
6.2
Introduction. • . • • • • • • • • • • • e _ • •
Designs for Estimating Variance Components for a Three
.Stage Nested Classif~cation• • • • •
SUMMARY AN]) CONCLUSIONS
~
73
. . . . . . .. . . . . . .
75
75
77
. . .. . . . 103
• • •
• • • • • • • • 103
7·1 The Problems. • • •
The
Influence
of
Experimental
Design
in
Reducing
Product
7·2
7·3
7·4
Variability for a Two Stage Nested Classification • • 103
Designs for Estimating Parameters for a Three Stage
• 105
Nested Classification • •
108
Suggested Future Research •
LIST OF REFERENCES.
........ .. ......
. . . . . • 109
vi
LIST OF TABLES
Page
1.J.
An analysis of variance for a two stage nested classification. . . . . . . .
2
. . . . . . • . .
2.1.
An analysis of :variance for a two-way classification.
3.1.
An analysis of variance for a two stage nested classification
8
with p+l units in each of s classes and p units in each of
a-s classes • . • • . . • • . • . • . • . . . . • . . • . .
:s f3
2
of P = Frob [ ( CTR 1dl ) :s f3]
of P = FrOb[(CTR2Idl) :s f3J
2
of P = Prob [ (CTR !dl ) :s f3
Values of P = Frob-[ (CT 2 IAd )
R
]
1
for 100A
= 1/10
-
and D = 100.
38
and D = 100 .
39
40
for 100A
= 1/4
for 100A
=
1/2 and D = 100 .
for 100A
=
1 and D
= 100
41
for 100A
= 2 and D = 100
42
for 100A
=
for 100A
= 10 and D = 100 ••
for 100A
=1
3.3·
Values
3.4.
Values
3·5·
Values
3.6.
Values of P
3·7·
Values of
3.8.
Values of
3·9.
Values of
4.1.
An analysis of variance for a three stage nested classifica-
A
.
Frob [( CT~21 dl ) ::;
13]
P = prob[(CT~2Idl) :s f3]
P = FrOb[(CTR21~) ::; f3]
P = Frob [( CT~21 dl ) :s f3]
=
tion. . . . . . . . .
6.1.
]
20
....
4 and D = 100
and D = 25 • . •
. . . . . . . . ..
52
The usual analysis of variance for the Anderson design.
79
6.2 •. The usual analysis of variance for the Calvin-Miller design.
81
6.3.
Values of VA' VB and V for the balanced design with degrees
c
of freedom:
6.4.
Values of EN
of freedom:
-.
6.5.
6.6.
11, 12, 24 .
• • • • • • • • • • • • • ••
87
and E for the Dl and D designs with degrees
2
C
88
7, 24, 16.
•
~
·
Values of ~,
of freedom:
E:s
Values of E ,
A
of freedom:
E:s
··········· ···
and EC for the Dl and D2 designs with degrees
15, 16, 16
89
•
·
········· ·····
and E for the Dl and D2 designs with degrees
C
90
23, 8, 16.
·..······ ·········
vii
LIST OF TABLES (continued)
Page
6.7.
Values of E , ~ and E for the D and D designs with
1
A
2
c
. degrees qf freedom: 7, 32, 8
.
6.8 .. Values of EA,
6.9.
91
~
and E for the D and D designs with
1
C
2
degrees of freedom: 15, 24, 8. • • . • . • • •
92
Values of E , ~ and E for the D and D designs with
1
2
A
C
degree s a f freedom: 23, 16, 8. . . . . . . . .
93
6.10. Values of EA,
~
and E for the D and D designs with
1
C
2
degrees of freedom: 31, 8, 8
. . .
6.11. &'1alysis of variance for tobacco data •
-
6.12. Values of
freedom:
.
94
99
;
"2·
~2
var(~A)/var(~A) for the D -design with degrees of
1
15, 16, 16 . .
.............
102
~I
f
[
I
f
I
_c
viii
LIST OF FIGURES
Page
Profile of Vo when P -< PI •
Profile of Vo when P >
P
- 2
...
....
32
....
32
32
I
I
~
!
l.0
HrI'RODUCTION
In many situations in which a result from an experimental unit is
subject to several sources of variation the problem is to identify the
sources and to estirnate the variation associated with each.
Gener~lly
estimation and identification are accomplished through the use of experimental procedures.
While considerable effort has been devoted to
the efficient design of experiments for the estimation of treatment
contrasts and regression parameters) little has been done on designing
experiments for the estimation of variance components.
Crump
(1954)
proposed designs for the two stage nested classification that provide
II
i '/1
optimal estimates of the between class variance component) the within
.---
class variance component) or.the ratio of the two components.
Gaylor
(1960) presented optimal designs for the estimation of components of
variance and certain functions of the components for the two-way crossed
classification.
,
Anderson and Bancroft
(1952) and Anderson (1960) intro-
-
duced a staggered design for the estimation of parameters from a three
or more stage nested classification.
No known work has been done on
the problem of designing experiments for the estimation of variance
components with the objective of using the estimates for the efficient
planning of future experiments or courses of action.
Part of this thesis is concerned with the design of experiments
for the efficient estimation of functions of the components of vari~ce
with the objective of reducing product, or process, variability.
Only the case in Ylbich aresult is subject to two sources of variation
will be considered.
It will be assumed that a single observation can'
\-1
\
;
~"
i
I'
I:~
f
r,
I,
,,
'
i'
2
be represented by
x ..
m
).J
+ A.). + B).J
..
i
1,2, . , .a
j
1,2, ... n.).
L
n.).
=
(1.1)
N
wher'e m is a constant, Ai and B.. are both normally independently disJ.J
'
t'
1 y.
tributed (NrD) with means zero an d varJ.ances,
u2 ' and u2 ' respec~J.ve
B
A
Under this model the product variance, i.e., the variance of an observation, may be represented by
..,
\.
An analysis of variance for ~ta obeying
(1.1)
is given in Table
1.1.
Table 1.1.
An analysis of variance for
a tw
0
stage nested classifi-
cation
Source of
variation
-a/
Degrees of
freedom
Classes
a - 1
Within classes
N- a
Total
N- 1
Mean
square
Expectation Of~/
mean square
2
K
(N -
L n.
---:-::-NJ._ )/ ( a -1 )
The other part of this thesis is devoted to the study of experimental designs for the purpose of estimating parameters from a
stage nested classification.
can be represented py
l
three
It will be assumed that each observation
3
i
1;2;
a
j
1;2)
b
k- 1;2;
i
n ..
n ..
n.; I: L: n
L:
j
where
ill
==
lJ
lJ
1
i
j
ij
==
N
is a constant; A.; B.. and C. 'k are all NID with m~ans zero and
1
variances;
222
~A; ~B
and
~C;
lJ
lJ
respectively
0
In this work the emphasis will
be placed on design rather than method of estimation
0
The esti.mators
used throughout are the usual analysis of variance estimators obtained
by equating mean squares to their expectations.
Optimal designs for
the estimation of single functions of components of variance will be
·
given.
The analogy between these designs and those proposed by Crump
for the two stage classification will also be shown.
Some effort will also be devoted to the study of the influence of
design on the simultaneous estimation of the three components of variance from a three stage nested classification.
4
2 .0
REVTE'tl OF LITERA'I.' URE
As was mentioned in Chapter 1) no known work has been done on
determinine the influence of the experimental design when estimates of
variance cc,mponents are to be used for planning future
courses of action.
experi~ents
or
One of the most common uses of variance components
is in the planning of an experiment in which it is desired to minimize
the cost of obtaining a sample estimate when the precision is fixed) or
conversely) to maximize the precision of an estimate obtained for a
, i
given cost.
An excellent discussion of this use of variance components
is given by Marcure (1949) '. To point out the lack of concern', for the design of the experiment from which the estimates of the variance components are to be obtained) we quote Marcuse:
"This dilemma may be evaded by first carrying out a preliminary
experiment in nested sampling using a set of arbitrarily chosen
class frequencies. 1I (Page 192)
It certainly seems
th~t
one could do better than to arbitrarily choose
the design as Marcuse suggests.
Some work has been done on designing experiments for the specific
purpose of estimating variance components.
published by Hammersley
The first such work was
(1949). He considered the two stage nested
classification in which an observation may be represented by (1.1).
~
..
~
An analysis of variance for such observations is given in Table
2
,,2
Hammersley used as an estimator of ~A) ~A
the variance of
= (MA-MB)/K~
1.1.
He derived
~~) var (~)) and proposed a design to miniinize var (~) •
,,2
For N and ~ fixed he showed that var (~A) is minimized by choosi~ a
design that has an equal number of units in each class j i.e.) requiring
D. H. Hili LIBRARY
North Carolina $t4lte College
L,
I
5
that N/a
be an integer.
For N/a an integer he proved that the number
of classes that minimizes var (~) is
:=:
a
N
l
-[
No 2 + 20-2
A
B
N(o-~
+
o-~)
+
and therefore the optimum number of urtits per class is
" II
For either a
l
or b
l
not an integer Hammersley did not prove an optimum
procedure for determining a design, he only suggested trying the
closest integers.
Crump
(1954)
considered the problem of efficiently designing
experiments for the two stage nested classification but did not restrict himself to integer values of N/a.
~,
where N/a
:=:
He proved for fixed Nand
p + s/a(O ~ s < a), that the optimal design for the
estimation of o-~ consists of s classes with p+l units and a-s classes
with p units.
P
:=:
Crump proposed a scheme, which depended upon the ratio
2 2
o-A/o-B' for determining a design which is optimum.
He was unable to
prove that his scheme always leads to an optimal design, but for numerous examples his procedure always minimized the variance of
A2
0-A'
Crump's
results are elaborated on in Chapter 5 of this thesis.
Baines
(1944)
considered the ~iased estimator of p given by
p
:=:
(M A
~)/~
•
He determined the design that minimizes var
(p);
howeve~he considered
only designs with equal allocation to the classes.
l
6
Using an unbiased estimator of p, Crwnp extended the work of
Baines to situations in which N/a is not an integer.
~ fixed, and with N/a ~ p + s/a(O ~ s
Crwnp proved for
< a), that the optimal design
;
wi~h
consists of a classes with p+l observations and a-s classes
observations.
p
He also suggested a procedure, again depending upon the
parameter p, for determining a design which would minimize the variance
of his estimator.
Through the use of numerical methods Cr-wnp indicated that the
guess or previous estimate of p required for his procedure is not very
critical and that it may differ from the true value by a considerable
amount without greatly
affectip~
,
2
the efficiency of the estimator of rr
A
or po
Gaylor (1960) extended Crump's work to the two-way crossed classification in which a result may be represented by
X~J'k ~ m + R.
....
~
+ C.J + (RC) ~J
.. + E"~J k
i
1,2, .. or
j
1,2, ... c
k
1,2, ... n ..
~J
2: 2: n ..
~J
~
N
where m is a constant, R~, C., (RC) .. and E. 'k are NID with mea~s zero
....
and variances,
J
~J
,
-,
~J
_rr~, rr~, rr~c and rri, respectively.
Gaylor found designs
2222222
which yield estimates of rr , rr + rr
+ rr , rr + rr
+ rrCJ and.
E
RC
E
RC
R
E
. .
rr2 + rr2 + rr2 + rr2 ' ..,4th mnlmum
poss ibl e var i ance.
RC
R
C
E
n..L
I
fI
Ie
If ~ is the linear function of these variance co~onents which is
to be estimated, the minimum possible variance is 2rr~ /( N-l)
.
~
0
This
i';
r-
l'
".-,
I~
I
minimum can be attained only if a design can be constructed. so that all
!i
!
L
j.
of the information is used to estimate
TIl
and ~~.
J.
The designs he pre-
sented for the estimation of the above mentioned functions of the variance components aye as follows;
\ (1)
~i,
select all N observations from one cell, r = c
~l
= N;
2 +
o-E
(2)
2
~RC
1 and
2
+ o-R'
use only one column with N rows and one obser-
vation per cell, r = N, c = 1 and nil = 1 (i = 1,2, •.. N);
2 + o-RC
2 + ~C'
2 same as J.n
. (2 ) WJ.. th rows an d co 1 umns J.n
. t er-
(3 )
~E
-,,-
changed;
2+2
+~C
2+2
~RC
~R' select each observation from a different
(4)
~E
row and column, so that r = c = N, n
n .. = 0 for i
J.J
f
ij
= 1 for i
j and
j.
For other functions of the variance components, in particular for
~~
=
~~ or o-~J it is necessary to estimate other variance components as
well as
4
2
0-.,
J.
2~i/(N-:-l).
hence, the minimum attainable variance is greater than
Uni'ortunately, it was not possible to determine this minimal
attainable variance.
It was found that the optimum design depends on
the estimator used and upon the values of the parameters themselves.
As an estimator of ~~ he used an analysis of variance estimator
based on the arithmetic of the method of fitting constants which assumes
only E
to be random.
ijk
Because the variance of his estimator was ex-
tremely complex for designs with unequal n
with n
ij
equal to 0 or n.
ij
, he considered only designs
He proved that for n
ij
~
0 or n the variance
of the estimator is minimized by taking nij equal to 0 or 1.
An anal-
ysis of variance for such a situation is given in Table 2.1.
'.
I
~.
8
Table 2.1-
An analysis of variance for a two-way classification
Degrees of
freedom
Mean
square
Columns
c-l
C
CT
Rows (adjusted for columns)
r-l
R*
CT
Interaction (adjusted for
rows and columns)
N-r':'c+l
I*
CT
TOTAL
N-l
Source of
variation
Expectation of
mean square
\·2 + 2 +
2
2
c1CTR + rOCTC
E CTRC
2 +
E
2
2
CT
RC
+ c
2
0-
o R
2
E + o-RC
The estimator·of o-~ Gaylor considered was
"'2
CT
= (R* - I * )/c
R
.
.
0
"
where Co = (N-c)/(r-l).
!i
For a given number of rows, r, Gaylor shows
"I. li
:, II
that if N
=
rc, where c is an integer, the best design is the balanced
L _ \ {JL".J'",./'\.~," 'S tt,i=..( ~
r x c design with one unit per cell.
If N
= r(c-l)
f) It..·•. JI,
't.:!~ ,-,:·v1~l. 5
" I·
+ s where 1 < s < r,
.···11
F ~I
Ii:!
the best design is either the balanced r x (c-l) design which uses only
N-s observations, or the unbalanced design with r
plus one column with s of the r rows.
.
rows by c-l columns
;I:.'
II
,1
II
;I;:
~ ;. 11
II
!,
i; t' !!
On the basis of this result it
was proposed that future research might use as a criterion of effi-
I
,'I
jI
.J
1,
~
.!t
,.j
. I'
, ~. i.
ciency the maxi.mum information per observation.
To determine the number of rows that minimizes the exact variance
,
11
;
L
, 1·
I
of
~
~R'
~
r,
to r.
The value of
r
he obtained was
I:
in which
e,
c is
=
(N -
C + Co )/c0 ,
the smallest integer greater than or equal to Co and
.
1
i' {'
.
I.
,
r
'
I
. I'
Gaylor lLsed an approximate variance of the estimator o-R to
obtain a first approximation
'
11I
··t'!.
I
9
-co
where PR
r
"'2
==
==
(N - 1/2) PR + (N - 1/2) + 1
(N - 1/2) P + 1
R
2/( O"E2 + eJ"RC·
2 )
O"R
He suggested trying integers below and above
unti.L the value of r is found that minimizes the exact variance of
~R.
Since this exact variance changes very little for small changes in
r near
r,
it is generally recommended that one use as r the closest
integer to
r.
The estimator Gaylor suggested for P is
R
PR
==
[N-r-c-l
N-r-c+l
/c o .
Again he considered only designs for which n
==
ij
0 or 1.
His procedure
for determining the optimal design for the estimation of P is similar
R
2
to that given for the estimation of O"R except that the first trial
value for r is determined from
..
c
2P R(N - 1/2) + (N - 1/2) + 1
o
PR(N - 172) + PR + 2
2
The above results apply to the estimation of O"Cand
Pc by interchanging
r and c.
Gaylor concluded with a brief study of designs for the simultaneous
estimation of
O"~
and
O"~.
He considered two special designs, called the
L-design and Balanced Disjoint Rectangles.design, and compared them for
2
the case O"R
==
I
I
!
i
2
O"C·
I
In accordance with the results of Crump, Gaylor found that the
;
guess or previous estimate of P required for determining a design may
R
vary considerably from the true value, and yet the estimates will be of
high efficiency.
;
10
For the two stage nested classification) the two-way crossed
classification and the three stage nested classification) Searls
(1956»)
(1958) and (1961), respectively) presented a method of obtaining vari~ces
of estimated variance components by matrix procedures.
Anderson and Bancroft
(1952) and Anderson (1960) considered a
staggered design for the simultaneous estimation of variance components
from a three or more stage nested classification.
The purpose of this
design was to distribute more evenly the available degrees of freedom
to the sources of variation and thus to obtain more precise estimates
of some of the variance components than is possible with a balanced
design.
Through the use of a numerical example) it was shown that the
.,
t,
precision of certain of the estimates may be considerably improved by
"
use of a
stagge~ed
design.
",
,
~.,
..
'
,'"
ji'
11
3.0
REDUCTION OF PRODUCT VARIABILITY THROUGH
THE USE OF VARIANCE COMPONENT ANALYSIS
3.1
Introduction
In this chapter we consider those situations in which a result
from an experimental unit is subject to two independent sources of
random variation, and the problem is to reduce the total variance.
When a result is the sum of two independent parts, the variance of the
result, the total variance, may be expressed as
2
2
where O"A and O"B are the components of variance associated with sources
A and B, respectively.
Often a composite of many sources may be en-
compassed in a. single source.
For example, under a given sampling
scheme, the variation observed for a certain type of electron tube may
be attributable to variation among lots of tubes plus variation among
tubes within lots.
If A represents lots and B represents tubes in lots,
the variation associated with B may be due to both measurement error and
inherent variation among tubes.
2
If one is given the task of reducing O"T he could quite conceivably
proceed by affecting individually the sources that give rise to
2
O"B.
O"~ and
In the usual circumstance, only a limited amount of funds would be
available for reducing
0";
and one woUld attempt to devise a program
that would yield the greatest reduction in
of funds.
0";
for a given expend!ture
The crucial matter here is the proper apportionment of funds
toward reduction of the two sources of variation.
,I!
I
~!
J I·,
I
I
;
12
Two factors would determine the specific apportionment of fllilds.
They are:
(1) the relative rate at which
CT~ and, (2) the magnitude of p
==
CT;/CT~.
tions the relative costs of reducing
but p may not be.
CT;
can be reduced compared to
In many experimental situaand J~ may be adequately known
CT;
Often the estimate of p is obtained by performing an
experiment with a fixed sample size with units arranged in a specified
design.
The primary purpose of this chapter will be to investigate, llilder
a given cost model) the influence of the design of the experiment on
:1
'
,
the apporLionment of fllilds and therefore on the effectiveness of the
reduction of the total, variance.
The effect of using various esti-
mators of p will be briefly investigated.
It will be assumed that the
!i
11
i!
!i
cost of sampling for the experiment is directly proportional to the
sample size N.
In order to arrive at an explicit solution for the optimum apportionment of fllilds, it is necessary to adopt a model which relates funds
expended to reduction in variance.
Two such cost models will be dis-
cussed in Section 3.3.
/'
a
==
-3.2 Notation
total number of classes
lOOA
==
rate of percentage reduction in
2
CT
A
to that of
2
CT
B
for the same
expenditure of fllilds on each
A:t
refers to those classes with p+l units per class
~ refers to those classes with
e
~ ==
p units per class
2 A-,
a fixed number between min (CT~ Id ) and max (CT~21"~)
l
t
13
e
B1
= k1/c2
B2 =
k2/c2
Dk1 + In
k
~
+ k
1
- In k1
2
c2
= k
D
= total
d
A
=
amount of funds ex:Pended to reduce
CYA
~ =
amount of funds ex:Pended to reduce
CY
1
+ k
2
amount of funds ex:Pended to reduce the total variance
2
2
B
~
=
value of d that minimizes
A
~
=
estimate of
E
=
ex:Pectation or average value operator
~
2
R
cy
for fixed D [see equation (3.7)]
[see equation·(3.8)]
I
,
F(h,N-a)
gi
= Snedecor's
= scaling
mator
hi
= degrees
;.
i = 1,2 [see equations (3.13) and (3. 1 9)]
of freedom for the approximate distribution used with the
estimator Pi' i = 1,2 [see equations (3.14) and (3.20)J
k
1
= -In( .99)
-In(I-A)
"
K2
= [N(N-2P-l) + ap(p+l)] IN(a-l)
K2
=
(N-2p-l)/(a-2)
fi
=
(a-l)(Klp i + 1)/gl~' i = 1,2
-
J
II
F with h and N-a degrees of freedom
factor for the approximate distribution used with the esti-
~i'
I
,
-
.r
'i,
I
i
14
= lower
L
i
limit of integration on P >Then the
estirr~tor
A
Pi is used
[see equations (3.30) and (3.33)]
a
N
=
total number of units in experiment
n
E
i=l
n
i
= number of units in'ith class (E n.~ = N)
i
M , MAl' M~, M~,
A
1\
=
mean squares in an analysis of variance
(see Table 3.1)
p
= number of units in each of a-s classes (N = ap + s, 0 < s < a)
P
= Frob [ (~R2Idl) ~ ~] [see equation (3. 24 )J
P
= ~A/~B
2
PI = e
P2 = e
2
k
-C C
I 2
=
~
C (D-C ) .
l
2
= smallest
P*
u = largest
= number
s
-k D
1
e
k
=
~i = estimator
P~
l
~
,
k D
l e 2
~
of P, i = 1,2
[see equations (3.10) and (3. 11 )]
root of the equation V2(~)
A
root of the equation V2 (p)
=~
=~
of classes with p+l units per· class (see Section 3.4)
SA' S~, SA ' S~, SB
= sums
of squares in an analysis of variance
2
(see Table 3.1)
~~ = component of variance associated with source A
2
= component
~B
of variance associated with source B
222
+ ~B
~T = ~A
2
~~ = reduced
total variance =
2
~B( .99)
~~
.
d
+ ~i(l-A) A
15
~A
A
O"R I~=reduced variance ratio when ~ units have been expended on source
A and
D-~
on source B [see equation (3.9)J
2 0"11,0"12'
2
2
2 = expected values of the mean squares for sources A,
0"1'
0"13
A , A and
2
1
~
vs.
~,
respectively
A
of integration for P when the estimator Pi' i = 1,2
U = upper limit
i
is used [see equations (3.41) and (3. 44 )]
V
=0",21 d_
oR
-l
-k1D
V
2
= e
A
(Vo when P ~ PI)
+ P
VI = e
-kl(D-C1 )
A
k1/C2
p
+
-k2 C1
A-
J
k 2/ C2
n
pep
'I
J!,
.
V4
= max
W
1
=
W
2
e
(VI' V )
3
-k1(D-C )
1
=Pe
-k C
2 l
?Cij = observed value for jth unit in i
3.3
th
class = m + Ai + Bij
A Cost Model
There are several different models that could be used to express
reduction in variance as a function of funds expended.
simplest is the adelltive model.
One of the
Under this model the variance obtained
after expending d units is equal to the original variance minus an
amount proportional to d.
I
!J
I
~,.-
For the sitUfLtion at hand, the reduced total
16
i-
variance) symbolized by
source A and
CT~2) attained after expending dA units on
~
units on source B is given by
KB
are constants of proportionality.
I
I
i
'. f
where
I
II
RA
and
Such a model is quite unrealistic for depicting an actual real
world situation.
i
According to this model the amount that a variance
is reduced does not depend on the magnitude of the variance .
Generally)
more effort would be required to reduce a small variance a given amount
than to reduce a large variance by the same; amount.
Also the model
indicates that if enough funds are made available a variance can be
reduced to zero and even made negative) a circumstance which is quite
e·
absurd.
The basic model that is proposed here is of the power function
type.
Under this model it is assumed that the expenditure of one unit
2
'
of funds on source A reduces CT by 100 A' per cent and on source B reA
2
duces (TB by 100 B' per cent.
Hence) the
after expending d units on source A and
A
r~duced
~
total variance attained
units on source B is
i I
, I
II
In order to eliminate one parameter) the model proposed above will
! I.
Io
,
\
be slightly modified.
If it is assumed that one unit of funds expended
2
2
reduces er by one per cent and :r;educesCT by 100A
A
B
p~r
cent) the reduced
total variance may be written as
\..
i,..
I ••
;
17
Because it is more convenient to work with an exponential than with a
power f\mction"
k
l
= -In(.99)
CT~2 will be expressed in exponential form.
and k
2
=
By setting
-In(l-A) we have
,2 _ 2 -kl~
2 -k2 dA
- CT e
CT
+ CTA e
B
T
The model as given by (3.3) shall be the model used throughout the
remainder of this chapter.
When the total amount of funds is fixed at) say
D units)
CT~2
may
j
I
be written as
2 -kID
2
CTB e
+ CTA
d
A
'
=0
(~
= D)
2 -k (D-dA )
2 -k2 dA
l
CT e
+
CTA e
B
,
(3.4b)
d
A
=
D(~
= 0)
In determining the optimal allocation of funds to minimize
first find that value of d
A
which minimizes (3 .4b).
CT~2,
(3.4c)
f-
,
we
i
This value is
II
,
i
l)
If ~
r
<
0, we set d
A
=
OJ and if ~
>
0" we set dA
=
D.
Hence the
!
I
complete solution is
i
0
~ =
-
,
o~
--~
P ~ PI
Cl + C -,
-2
PI ~ P ~ P2
D,
P ~ P2,
(3.7)
18
k
where
= k
l e
-k D
1
2
and
For this work it is more convenient to use the ratio ~R.2
rather than ~~2.
(3.7),
Since d , as given by
l
.2/~B2
= ~T
also minimizes ~~2 and
since the statistical properties of both are the same, working with ~~2
is equivalent to working with
development.
~~2,
and
~~2
will be used in all future
The quantity ~~2 will be referred to as the reduced total
variance ratio.
As was mentioned previously, A is often known, but an estimate of
P is required before the allocation of funds can be completely speciIf an estimator of p, say
fied.
p,
is substituted into
(3.7)
for p, then
as an estimator of d , say '"
d , we have
l
l
I
!
I
U
f
~
L
/\
+~
C
I
=
I
::s P" ::s PI
"
PI ::s p ::s P2
o,
C
2
,
-
(3.8)
"
D,
f
I
!
I:
P ~ P2
where L, which may be negative, depends on the particular estimator of
p used.
H
l'
r
.'
"
Therefore the true reduced total variance ratio that exists after
1"
" on source B and "~ on source A is
expending D-~
,, ..
-k D
e
.2 "
~R 1 dl
e
I
"
,
+ P
L::S p
::s
PI
l,
-k1(D-CI ) "kl /C2
=
+ P e
!
P
-k2 CI ,,-k /C
P
-k D
1 + Pe
2
2
,
2
PI
,
"
::s "p ::s P2
P ~ P2
(3.9 )
19
It is important to realize that O"n21 aI' where
dl
is obtained by
experimental methods, may be considered as either a random variable or
as a conocant, according as ""~ is considered as a random variable or a
constant.
The reduced total variance ratio is a random variable in the
sense that if we were to perform a large number of experiments, they
would yield differing values of p"" and hence possibly differing values
""
of O"~2 I~.
Conversely, once the experiment has been performed, the
6l1ocation determined and the funds actually expended on the two
2 and O"A'
2 O"R,2/""d lS
. a constant associated with
sources contributing to O"B
l
the product or process.
I
i
I'
We shall be concerned with 0"~21dl as a random
variable.
In the next section an experiment for the estimation of p is
described and two different estimators of p are presented.
3.4 Description of the Experiment for
Estimating p
The situation shall be considered where there are N units available for an experiment and the units may be arranged in a two stage
nested classification with a classes.
The model that represents the
data collected according to such a scheme is taken to be of the form
given by (1.1).
As previously stated it will be assumed that the cost
of sampling the units is proportional to N regardless of the type of
design.
Further, it will be assumed that for any
allocated as equally as possible to the
N/a
=
~
classes.
~
the N units are
Specifically for
p + s/a (0 ~ s < a) there are p+l units assigned to each of s
classes (referred to as ~) and p units to each of a-s classes (referred
"
20
to as A ).
2
Crump
The motivation for this allocation scheme is given by
(1954). In Table 3.1 an analysis of variance is given for such
data.
Table 3.1.
Source of
variation
An analysis of variance for a two stage nested
classification with p+l units in each of s
classes and p units in each of a-s classes
Degrees of
freedom
Classes
a-I
s-l
A
2
Al vs. A2
2
CT
I
SA
M
A
a-s-l
SA
2
MA
2
1
S~
M~
1
Within
N-a
Total
N-l
SE
Expectation of~7
mean square
Mean
square
M
A
SA
Al
~/S =
Sum of
squares
1
~
CT~ [1
+ Sp]
2
CTll
CT~[l
2
CT12
= CT~[l
+ pp]
2
CT
13
2 [1
= CTB
+ ape pH) ]
N
P
+ (P+l)P]
2
CT
E
[N(N-2;-1) + ap(p+l)]/N(a-l)
It should be apparent that the partition of the class sum of
squares as given in Table 3.1 is relevant only for situations in which
° < s < a.
When s
=
0, a
N/p, ap(p+l)/N = p+l and the expectation of
=
the mean square for classes becomes E(M ) = CT2 (l+pp).
A
E
From the manner in which the units are assigned to the classes,
i.e., p+l units are assigned to each of s classes and p units to each
of a-s classes, it is realized that designating ~ completely specifies
I.
"
21
the design.
Therefore} varying the design would be accomplished by
varying a.
As was previously stated} the concern of this work is where p} and
hence d,} must be estimated.
.L
The estimator of P that we shall be pri-
marily concerned with is
This estimator will be considered because it is the one that is used
most extensively in practice.
One other estimator will also be con-
sidered}
(SA + SA )/(a-2) - M
B
1
2
where K
2
=
(N-2p-l)/(a-2).
A
A
The estimator P is the same as PI except that the one degree of
2
freedom due to Al vs. A is not used.
2
This estimator was tried to
)
learn if the exclusion of the one degree of freedom greatly detracted
from the worth of an estimator.
It was felt that for other than very
! :
, I
!, I.
,"I
Ii
I
}
small N and a the exclusion of the one degree of freedom would have
.
;
If'
little influence on the worth of an estimator.
A
The estimator P will be considered only for situations where the
2
experiment is not balanced} i.e.} N/a is noninteger.
As there is no
" does not exist
partition 'of the class sum of squares} the estimator P
2
for balanced designs.
A
A
In the next section some results will be presented for PI and P .
2
,I
. I
I
I.
I
" f
"i".
~
.
22
3.5 Some Mathematical Development for the
Estimators of P
The development for both estimators is very similar, therefore,
the details of the development
ar~
'"
given only for Pl.
From (3.10) it is noted that
~l = ~ [MA~MB] = ~ [ ~ 1],
and therefore a study of the properties of '"
~l becomes a study of MA/ME.
Now, ME is distributed as
~~
X(N_a)/(N-a), where X(v) is distrib-
uted as chi square with v degrees of freedom.
M is distributed as
A
ger F
= MA/M](l + ~
~~[l
+
~
.
P]x(a_l/(a-l).
I
If the design is balanced
Hence,
i~
N/a is an inte-
p) follows Snedecor's F distribution and the dis-
'" can be linearly transformed to that of F so that all
tribution of PI
the known properties of F can be utilized.
However, this work will not
be limited to considering only balanced designs.
integer, MA is not distributed as
-
When N/a is not an
~~[l + ~ P]X(a_l)/(a-l) and it is not
..
possible to make a simple transformation from the distribution of '"
PI to
that of F.
To circumvent this difficulty use was made of an approximate
distribution of a linear function of chi square variables [studied by
Satterthwaite (lJ46)].
The approximate distribution is set up such
that its mean and variance are equal to the mean and variance of the
linear function of chi square variables.
Consider the weighted sum of independent chi square variables
i~ _r
(
"
Let Q be represented by Z
gX(h) where g and h are chosen such that
23
E(Q)
~
E(Z) and var(Q)
~
var (Z).
It is easily seen that E(Q)
.
2
2
E(Z) ~ gh) var(Q) = 2 Z ~.v., and var(Z) ~ 2g h.
~ ~
and var(Q)
=
Setting E(Q)
~
~ivi)
E
= E(Z)
var(Z) we obtain
g
=E
2
~.V./E ~iv.
~
~
~
2
h=(E~.v.) 2/E~.v.
~
~
~
~
d
:. ...:
Hence) Q = E Aix(V ) is approximately distributed as gx(h) where g and
i
h are given above.
Assuming that the allocation of units to classes is as described
in Section 3.4 and using information contained in Table 3.1 we note
that
"~l
may be written as
i
"
= .~ [
(SA + SA
1
+ SA)/(a-l)
2~;;
i;
]
!
I'
,II
_ 1
I
I
I.
,
;
!
i!
I ~
I 1"
.,! i'!
~ f·l
i,
.i
l ,:
2
2
It is observed that SAl = ~ll X(S_l)' S~
and
~
=
~~
x(N_a/(N-a), where
~il
=
~~[l
.2
2
= ~12 X(a-s-l») S~
+ (P+l)PJ)
~i2
=
22
= ~13 X{l)
~~[l
+
: i!
I,
i
PPJ
2 _ 2[1 + ap(p+l) ]
and ~13 - ~B
N
p.
Therefore, Q in (3.12) can be written
l
as a linear function of chi square variables
I
!
I
I
i
I, ,
24
,e
Using the result concerning the approximate distribution of a
linear function of chi square variables it is seen that
mately distributed as
1
I.
:=
(s-l) [1+(P+l)P]2 + (a-s-l) [1+pp]2 + [1 +
!\
is approxi-
cr~ gl X(h ) 'Where
.
g
~
:=
ap~p+l)
pJ2
1
(s-l) [1+(P+l)P] + (a-s-l) [1+PP] + [1 + aPT ) p]
I
I
I
Ie
,
I
(3.14)
!
i
I
The substitution of
cr~
gl X(h ) for
.
1
~
and
cr~
x(N_a/(N-a) for
.
~
i
in
(3.12) gives
i
i
i
I
where
leI
~
means approximately equal.
Recall that
follows Snedecor's F distribution with vI and v
2
degrees of freedom.
I
I:
1
r
I
I
25
Let
(3.16)
which is distributed as F with hI and N-a degrees of freedom.
It may be noted that when N/a = p = K is an integer) hI
l
(1 +
B
p»)
a
Hence)
a-I)
and
....
a fl
PI
=N
~l + N p) Fl - 1 ] .
a
....
Using similar techniques) the following results were obtained for P :
2
(SA
1
+ SA )/(a-2)
2
-
~
where
,
and F is distributed as Snedecor's F with h and N-a degrees of free2
2
dom.
.
26
I.
In the next section the preceding resu_lts will be used in the
.formulation of a criterion.
This criterion will be used to determine
the influence of the design of the experiment and the effect of the two
different estimators on the effectiveness of reducing product variability.
3.6 Criteria for Assessing the Influence of the
Design of Experiment on the Reduction of
Total Variance
The reduced total variance ratio,
'"
~~2 1d
l
, has a distribution whose
lower limit is the minimum attainable variance ratio, namely,
-k.
(D-d)
~l
1
e;
Therefore, it is reasonable that one would seek a design or estimator
'" that has the largest
which would generate a distribution for ~~2 I~
possible density near min (~~21~).
gested concerns the expectation "Of
The first criterion that is sug-
~~2Idl'
Ostensibly, the best design
or estimator would be one that yields the smallest expectation of
Therefore an expression will be developed for the expectation
The development will be given only for the estimator Pl'
2 '"
From consideration of ~R Id as given by
l
(3.9)
and from
(3.10)
r
'" is
it is apparent that the expectation of ~~2 jd
l
i
j'
f
j
j
j
"I
.j
I
.:i
"":P"i':
''''l!'"'
-t._,·"~
27
(3.22 )
Making the linear tranSf01TJmtion
the eXpectation
w~y
'i
be expressed as
,
i-
where
;
~-:
!:
28
.
,
and g(F ) is the density function for the F-statistic with hI and N-a
I
degrees of freedom.
Ideally) one would carry out the integration required in
and then specify numerical values for p) A) D and N.
(3.23)
The effect of the
design would be deterr:dned by substituting various values of a and
observing the nw::erical results of
E(0-~21~).
A major difficulty with
using this procedure was that we were unable to evaluate explicitly the
I
second int~gral on the RBS of
I
(3.23). Numerical integration could have
been usedj however, we preferred to use a simpler criterion which may
be even more useful than that involving average variance ratio.
I
/:
:' ~
This criterion concerns the quantity
I:
. .
.
2 ....
2 ....
where 13 is a fixed numb er between min (0-~ I ~) and max (0-R I'\ ).
By
careful selection of 13, such a criterion should do well to pinpoint the
effect of design and estimator.
Obviously, extreme values of 13 either
small or large, would yield P quite insensitive to variation in design
2, . .
or estimator regardless of their effects on the distribution of O"R
~.
An expression for P is now developed.
2 A
A
First recourse is made to investigating O"R I~l as a function of ~l.
Consider the function
29
Referring to (3.9) and (~.10) it is seen that V
o
VI
e
V
e
-k D
1
-k
2
V
(D-C )
I_I
:=
+pe
1 + Pe
k
-C C
e
Let P
o
1 2
(l~e
:=
-
JS. -
<p
.1 -
1
r-,k1 /C2
l'k C r-,-k /C
2 2
2 1
PI
, '"
3
r-,
---<p
PI
0
V
1
,
+ P
takes ori.three~i'orrns,
-k D
2
,
,
-k D
l
e
k_2
-k1 D
-
I
)/(I-e
-k2 D
I
). An
inspection of VI and V
3
shows
that
C3 .26)
if
In addition we see that
Also it is apparent that -V is continuous in [PI' P ]. - The con2
2
timiity of V pIus the equalities given by
2
(3.27) establish that V0 is
coritinuousover its entire range.
-e
Next consider the behavior of V
2
at its end points.
in [PI' P2 ]·
First consider V2
To do this the first dni vative of V 'rl_i th respect
2
30
.....
to PI is needed.
The derivative is
(3.28)
From (3.28) it is easily verified that
It is well kno,vn that a necessary but not sufficient condition for
a function f to have a relative extremum at a point x
o
is that fl(x )=0.
0
2
Using this result for (3.27) we note that V (Pl) is zero when
Since C
l
~l =
P •
Hence) V2 (Pl) has at most one extremum) namely) V (p).
2
The second derivative at
. VI
2
PI
=
P is
(p)
I,
,
\
i
1
j
;1
e
>0
The second. derivative will always be positive because k , k
l
2
and
L
;
\
hence, fl(x )
o
:=
0
is a sufficient as well as necessary condition and
the extremum will be a minimum.
Collectip~
tnese results we see that V
o
can be characterized by
three different profiles, depending on the value of P, as shown in
Figures ;.1, ;.2, and ;.;.
These three ways in which V behaves are
o
described below:
(1)
When P -< PI' V0 increases trom a minimum of V0
When P ~ P , Va decreases from a maximum of Va
2
minimum of V0
(;)
VI to a
:=
VI to a
Y;'
maximum of Va
(2)
:=
:=
V;'
When PI < P < P2' Va decreases from a value of V
o
minimum of V0
:=
:=
VI to a
V (p) and then increa,ses to a value of V0
2
:=
V;'
> V; if P > Po' VI <. V; if P < Po;
In Figure;.; it is noted that VI
i:
I;
! .
r
regardless of whichever VI or V; is the smaller, V reaches a minimum
2
value which is less than the smaller of VI and V;,
equals VI or V; only when P
=
PI or P
The minimum of V
2
"
. !
?
,. ,
= P2'
Therefore, if V were graphed' and then a horizontal line were
a
drawn at V
:=
.
I
13, min (VO(;l)] < 13 <'max [VO(P l )], the line would inter-
sect the graph of V at either one br two places.
o
.
In Figures
;.2 there would be exactly one point of intersection.
;.i
"
and
In Figure ;.;
j ,
the line would intersect the graph of Vo at 'exactly
two places if both'
.
I
.'
\
'-
t
32
v
13
~ -----------;;>r.c.-:..----+----
VI
Figure
3.1.
Profile of V when p
o
<
-
P
I
.,,
y
~
.
V
13
~---'------~,--------
,!'~
. \
-, i
I
1
I• !"
(
!
•
e
33
VI > P and V > p) and wouJd intersect at exactly oue place if VI > p
3
and V ~ P or VI ~ P and V > p. It should be observed that in Figure
3
3
3.3) if VI = P and V > p) the lower point of intersection is taken to
3
A
be PI = -l/~, and if VI >
P and V = ~ the upper point of intersection
3
is taken to be at positive infinity.
Also, in each of the Figures 3.1-3.3 the line V
the values of PI' P~ and P~' at which VO(Pl )
=~
~ ~
is shown and
are indicated.
2 "
Using the info::m.ation rega.rding the behavior of Vo(P ) = cr~
l
is found. tha.t P may be exp:cessed as
Pl
P
=
=
Idl
it
Prob(Pl ~ p~) if P ~ PI or,:if Pl < P < P2 and Vl ~ p, V > ~
3
P2 = Prob(Pl ~ p~) if P ~ P2 or if PI < P < P2 and Vl > ~, V ~ P
3
P
3
~
"
*).
Prob ( P*
L ~ Pl ~ Pu lf PI < P < P2 and Vl >~, V3 > ~
(3 .31)
. of the
where P * and P* are the smallest and largest roots, respectively,
u
L
equation
I
I.
P is expressed in integral form as
"
f
.. l
f'-
<Xl
P
~
P
2
P7.
:;
=!* f(P
*
PL
=1:
PL
U
I
)
f(~)
clPl
i ,;.
_!
~.
11
f
t
I
~
~.
aPl
I<
.
t
t
"1
e
To evaluate the integrals
given by
(3.17).
(3.33)-(3.35) ~e
is made of the relations
Then, using the density function of F the general
form for P can be approximated by
where R* is the appropriate domain of integration.
given by
(3.36)
is exact when N/a is an integer.
Note tl:lat P as
By making the trans-
i
·L
j"
formation of variable
f
: It \
I
t
-(3.36)
1;3
. :;, . 1.········-'·
:'.;
transformed into the incomplete beta form
f
,:_
~
,t
h /2 - 1
()/
x l
(l-x) N-a 2
1
-----:h~-.L------
dx
(3.37)
1
.B( 2 , N;a)
where R is the transformed domain of integration.
Each of the integrals
(3.33)-(3.35) can thus
be approximated by
incomplete beta functions and P can be expressed as
( ~2
P
N-a)
'
2
( ~2 '
~
P3
== I
U
1
(~
"2'
N-a)
2
N-a)_
2
(3.39 )
IL
1
( ~2 ' ~)
2
o~
e
.
-
35
~
where
Ul
a-I
(
g (N-a)
1
=
1 +
a-I
gl(N-a)
Ki
*
)
Pu + 1
(Ki
*~+ 1)
Pu
a-I
(
*
)
g (N-a)
PL + 1
1
(
*
)
1 + g a-I
(N-a)
PL + 1
1
Ki
L
l
=
Ki
x xm-l ( I-x )n-l
Bm,n
()
J
I x (m,n) = 0
.
d.x •
As was previously stated, the general development of this section
e
as given for
" applie s also to P" • The only differences in the results
2
~l
that occur for the different estimators are in the parameter h and the
limits of integration for (3.38)-(3.40).
by ~ as given by (3.20).
The parameter ~ is replaced
The limits of integration U and L are rel
l
placed by U and L , whichever is appropriate, where
2
2
K2 P*u + 1 )
a-2
(
* )
~(N-a) K P + 1
a-2
g (N-a)
2
U2
=
1 +
2
2
=
1
.
j
u
K2 PL* + 1 )
. a-2
(
* )
+ ~(N-a) K2 Pi + 1
a-2
~(N-a)
L
(
(
. After we have an estimator and a value for
I.
~
and have selected a
set of values for the system parameters D, A and p,the influence of the
36
design can then be ascertained by varying the number of classes a for
fixed N, over a specified range and computing for each value of a the
(3.38)-(3.40). The merit of
quantity P as given by use of the forms
the two estimators would be determined by recomputing a set of P's :tor
each estimator.
Hence,an experimenter could determine for himself which design
he should use for his particular situation, i.e., for a specific D, A,
p and sample of N.
To make the results of this chapter more useful it
was decided to present some numerical results.
3.7 Some Numerical Results for
P = Frob [( (J;21 ~) .:s t3]
3.7.1 General Remarks.
3.6,
Using the results derived in Section
a series of tables of values of P were prepared.
It was hoped that
consideration of some numerical results would aid in understanding the
effect of design and
estiw~tor
on the specific objective considered,
Ii
ii, .
viz., reducing product variability.
3.7.2
Values of p, 100A, D, N, a &'ld t3 Used in Computing P.
To
compute P it is necessary first to specify the system parameters p,
i1
100A, and D; the design parameters N and -a, and the fixed number t3.
•
,"
~
Uni'ortunately, one is not able to examine all the combinations of systern and design parameters that he would like.
A set of parameters were
chosen that should represent many situations encountered in actual
practice.
10.
The values of p selected were:
The values of 100A selected were:
Note, for example, that p
=
1/10, 1/4, 1/2, 1, 2, 4 and
1/10, 1/4, 1/2, 1, 2,
10 means that
(JA2 is
4,
and 10.
ten times larger than
e
37
O"~,
and 100A = 10 means that O"i is reduced at ten times the rate at
which
O"~ is reduced.
selected.
For D, the two values D = 25 and D = 100 were
The majority of the results are presented for D = 100.
Two s@nple sizes were considered:
a relatively small sample,
N = 24 j and a relative large sample, N = 72.
of a used were:
With N = 24 the values
2, 4, 6, 8, 12, 16 and 20j with N
= 72
they were:
2,
4, 8, 12, 18, 24, 36, 48, and 60.
Two values of f3 were used:
f3( .90)
= [3
f3(.75)
e
= [9 ~n (0"~21~) +
min
(0"~21~)
max
+ max
(0"~21~)]
/10j
; :,.
(0"~21~)]/4.
The value f3(.90) represents a reduction in total variance of 90 per cent
of the ma..ximum attainable amount, and f3(. 75) represents a reduction of
75 per ce:ct of the maximum attainable amount.
For the 49 combinations of p and 100A, with D = 100, and using
f3
= f3(.90),
the values of P are presented in Tables 3.2 - 3.8.
included in these tables for interpretative purposes are min
and max
(0"~21~1)'
Because the results obtained for D
= 25,
Also
(0"~21 ~)
in terms of
the ir!.f1uence of design and estimator, are very s~mi1ar to those obtained
for D
= 100
only a small portion of the results are presented for D
viJ;., those for 100A
= 1.
presented in Table 3.9.
The values of P for D
=
25 and 100A
=1
= 25,
are
2 I"d )
l
Of course the differences between min (O"~
and max (0"~21~) are considerably greater with D = 100 than with D = 25:
For three combinations of p and 100A, with D
are presented using f3(. 75) in addition to f3( .90).
= 100,
valuesofP
The three combinations
,
'-.
38
e
Table 3.2.
Values of p:::: PrOb[(o-~21~1):S f3J for 100A:::: 1/10 and D ::::100
estimato~/
-a
p
1/10
1/2
1/4
1
2
4
10
·95
.98
·98
.98
·97
.94
.94
.84
.84 ;
.85
.88
.90
.90
.88
.83
.83
.72
·72
·75
·73
·73
·72
·71
.66
.66
.39
·53
·59
.63
.67
.68
.68
.69
.69
N :::: 72
.96
.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
.98
.98
.87
.91
.96
.98
.98
·99
·98
.96
.96
.89
.89
N :::: 24:
(1)
(1)
(1)
(1)
(1)
(1)
(2 )
(1)
(2 )
2
4
6
8
12
16
16
20
20
e
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(2 )
(1)
(2 )
2 ,.
2
4
8
l2
18
24
36
"48
48
60
60
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
·99
·99
·95
·94
·99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
.99
1.00
1.00
1.00
·99
.98
.98
.90
.89
·99
·93
.93
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
'1.00
1.00
1.00
1.00
1.00
1.00
l.00
1.00
1.00
-99
.99
:
.
·58
.58
.76
.76
·79
.81
.83
.84
.83
·79
·79
·71
·71
.39
·53
.63
.68
.73
.76
.79
.78
·78
.76
.76
,
.'
::
.~
(
T
J
.)
j ..:
: .1
'
,
,,,,
.~
'--1.
t
~
~.,
.
0"
~.
1 -~'
,
I
:!
!
:
,
I
; 1
1eJ.)
.466
.616
.866
1.366
2.366
4·366
10.043
"
max (o-~2 I').)
1.090
1.226
1.452
1·905
2.809
4.619
10.366
" '.1':-1
·528
.677
f
1.420
2.410
4·390
10.07~
, 'j
min (o-~
f3(.90)
.925
I :
\
"
~/The symbois (1)
respectively.
and (2) refer to the estimators Pl and P2)
,~ , Il"
',""
. " '~ : II'
~
; -,
,
,
,
.:
\
I
I
39
Table
a
3.3. Values of P
=
2....
Frob [ (cr~
I~1) -~
t3
]
for
100A
=
1/4 and D = 100
P
estimato~/
1/10
1/4
1/2
1
2
4
= 24
' .82
.83
.83
.83
.80
.74
.74
.63
.63
·73
·71
.71
·70
.67
.63
.63
·55
·55
.40
N
(1)
(1)
(1)
(1)
(1)
(1)
2
4
6
8
12
16
16
20
20
1.00
1.00
1.00
1.00
·99
.95
.95
.84
.83
(2 )
(1)
(2 )
.97
·99
.99
·99
.98
.92
·92
.80
.80
.92
.95
.95
.95
.92
.86
.86
.74
.73
N
2
4
(1)
(1)
8
(1)
12
18
24
36
48
48
60
60
(1)
(1)
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
·97
·97
(1)
(1)
(1)
(2)
(1)
(2)
.99
1.00
1.00
1.00
1.00
1.00
1.00
.99
.99
·95
.95
.94
.98
.99
1.00
1.00
1.00
.99
.98
.98
.91
.91
·53
·59
.63
.66
.67
.67
.68
.68
·52
.73
.81
-.86
·90
·90
.89
.88
.88
= 72
.83
.87
·91
.93
.95
.95
·93
.89
.89
.79
.79
.15.75
.76
.78
.79
.80
.78
.74
.74
.66
.66
.40
·53
.63
.68
.73
.76
.78
.77
·77
.74
.74
. ·52
.73
.88
·93
.97
.98
.99
·99
·99
.9'1
,
·97
(cr~21~)
.466
.616
.866
1.366
2.349
4.092
8.786
max (cr~21~)
1.078
1.195
1.389
1·779
2·557
4.366
10.366
.527
.674
.918
1.407
2.3.70
4.119
8.944
min
t3( .90)
~/The
symbols
(1) and (2) are defined in Table 3.2.
1I
40
Table 3.4.
Values of
P ==
PrOb[«(J'~21~1) ~ t3J for 100A
==
1/2 and
D ==
100
p
a
estimato~/
(1)
(1)
(1)
(1)
(1)
(1)
2
4
6
8
12
16
16
20
20
(2 )
(1)
(2 )
1/10
1/4
1/2
.
·97
·98
·98
·97
.92
.85
.85
.72
.71
·91
·93
·93
·91
.87
.79
.82
.83
.82
~81
·79
.67
·77
·70
·70
.60
.66
.60
N ==
.74
.73
.73
.72
.68
.63
.63
.55
.55
N ==
2
(1)
.99
4
(1)
8
(1)
12
(1)
18
24
36
48
48
60
60
(1)
(1)
1.00
1.00
1.00
1.00
1.00
(1)
·99
(1)
.97
(2)
(1)
(2)
A
min «(J'~ I~ )
~
max
«(J'~ I~)
t3(.90)
·97
.89
.89
.466
·99
·99
.99
·99
.84
.88
.92
.93
.94
.93
·98
.94
•94
.83
.83
·90
.85
.85
.74
'.74
·94
.98
.616
2
1
.77
.77
.80
.81
.82
.82
.80
.75
.75
.66
.66
,.866 1.352
24
b/ .14
(.21~
( .36 )
( .43 )
(.46)
(.47)
(.41)
(.40)
(.32)
(.32)
.24
.28
.31
.31
.28
.27
.21
.20
72
(.22) .14
( . 40 ) .26
(.56) .38
(.65) .45
(.71) .50
( .74) .52
(.74) .52
(.66) .47
( .66 ) . 47
( .63) .36
( .63) .36
2.147
4
10
.45
.63
.70
.74
.78
.61
.84
.92
.95
.97
.97
·78
·77
.77
.76
·97
.96
.45
.63
.76
.83
.87
.61
·90
·92
.90
.90
.86
.86
,
,
.
}
~I
.r
ii
.. 85
.96
·99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
.!"
.
- L
'!' .:,
i,
oJ
,•
_
""','.
_l:l ',I:; ~~
'
!
~q.)J;·.I·.·
.,
~ 'I"t
'j
_ .~~
: j, '. ~~
3.423
.670.910 1.377 (2.202)2.169 3 ·517
,I'
·96
il'1;-i .~',:.~
i ' .•. 1:,:i,
j
1.061 1.151 1.303 1.606
.525
If
i
.
7.389
:
t '
. ;',
j
,
!!:./The symbols (1) and (2) are defined in Table 3.2.
E./The
values under this column that are in parentheses are for
t3( ~ 75) .
~~
,
I ;
i :
!
I
I
I
,
i
l.
,\
e
J
:~
41
Table 3.5.
Values of P
-a estimato~/
,"
Ij
~
= Prob ~(<T~2"
I~1) .:~:(13' ] for 100A = 1 and D = 100
,~ ,
r'
p
1/10
1/4
1
2
4
10
·55
·55
= 24
(.22 ~7 .14
(.36 ) .23
(.41 ) .27
(.43 ) .28
(.42 ) .28
(.34 ) .24
(.33 ) .23
(.26 ) .17
(.26 ) .17
.45
.62
.68
.72
·75
·75
.74
.73
·73
.52
·73
.81
.85
.88
.88
.87
.85
.85
.66
.89
.95
.98
·99
.99
.99
'098
.98
-.80
.82
.85
.86
.87
.86
.82
.76
.76
.66
.66
= 72
(.23 ) .15
(.41 ) .27
(.57 ) ·38,
(.64 ) .44
(.69 ) .48
(.76 ) .49
(.68 ) .47
(.59 ) . 41
( .58) .40
(.45 ) .30
(.44 ) .30
1/2
N
2
4
6
8
12
16
16
20
20
(1)
(1)
(1)
(1)
(1)
(1)
(2 )
(1)
(2 )
.92
·92
.90
.88
.82
.73
.73
.61
.61
.83
.83
.81
.80
.74
.67
.67
.56
.56
.79
·77
.76
.74
.70
.64
.64
F
N
e
2
4
8
12
18
24
36
48
48
60
60
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(2 )
(1)
'(2 )
.96
·99
·99
.99
.99
.98
.94
.88
.88
.76
.76
min (CT~21"
~)
"
max'(<T~2 I~)
13( .90)
.466
.86
·90
.93
·93
.93
.91
.87
.76
.80
.69
.69
.616
-
1.210
.856
.45
.63
·75
.81
.86
.88
.89
.87
.87
.82
.82
.52
.74
.88
·93
.96
.98
.98
.98
.98
·95
·95
1·711 2.464
... 66
·90
.98
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
4.660
,
"
{
t
, 1
I
;:
I
,'It
I
I
f:
:
,
~'
,..
~
:
~
,..
!;
.,",~
"
t
j
' ..
1-1 l~il
i;: ; d~';
~' i .\ "
'--
I "\1
'f
I.
Ii:'
..
;.,
.!
.1!
'~
I
1.037 1.092 1.183
1.366
2.366 4.366 10.366
l ';"
:
~
.523
.664
.888 (1.249) 1.226 1.777 2.654
-
!
I
~
'
:
~
5·231
-
~/The symbols (1) and (2 ) are defined in Table 3.2.
e
:£/The
values under this column that are in parentheses are for
13( .75) .
,
j
. ! "
j : ;, \
:
I
!' ['
!r.~":
. !; '~
I
",'
,.,t"i<'
e
42
Table 3.6.
a
Values of p:=:
estimato~/
(1)
(1)
2
4
6
(1)
8
(1)
12
16
16
20
20
(1)
(1)
1/10
.85
.82
(1)
(2 )
1/4
.82
.81
.80
.78
.72
1/2
,
~
1
N :::
b/
(.3 6 r .23
(.53)
( .58)
(.59)
( .56 )
(.46)
(.45)
. ( .33)
.79
·72
.65
.65
.66
.66
·55
·55
.56
.56 1(.33)
(1)
4
(1)
8
(1)
12
(1)
18
24
36
48
48
60
60
(1)
min
.91
.94
10
.51
.71
.78
.82
.84
.83
.83
.81
.80
.59
.80
.88
.69
.92
.97
.37
.40
.41
.38
.31
.31
.22
·91
.94
.93
.93
·91
.90
·99
·99
·99
·99
·99
.99
.21
.31
:=: 72
.28
.48
·52
.73
.59
.82
.69
.92
.95
.91
.95
·92
.93
·91
.91.90
.85
.85
(.78) ·59
(.83) .65
(.85) .67
(.85) .67
( . 81 ) '.62
.65
.72
·77
.78
.76
.78
.78
.68
.67
(.72)
(.72)
( .57)
( .56)
.69
.68
.53
.53
.94
.98
.99
1.00
1.00
1.00
1.00
.98
.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
(2)
.77
.77
.66
.66
.87
.92
.95
.96
.97
.95
.95
.91
.91
.98
1.00
«T~21 ~)
.466
.609
(1)
(1)
(2)
(1)
2 A
max «T~ I~)
t3(.90)
~The
(.39)
(.66)
4
.26
.45
(1)
.85
.88
2
24
.26
.42
.48
.51
.50
.43
.43
.32
N
2
D::: 100
p
.86
(2 )
PrOb[«T~21~) S t3J for 100A::: 2 and
.965 1.215 1.530
'1.013 1.033
.521
.52
.52
.38
.38
1.366 2.366
4.366 10.366
.651' (.~42) .797 1.005 1.330 1.814
symbols (1) and (2)
~e
2.326
3.130
defined in Table 3.2.
£/The values under this column that are in parentheses are for
t3( .75) .
J
II
'"
!
43
e
Table 3.7. Values of P
estimato~/
a
=
Prob [ (~~2;\
Idl ) ~ f) ] for 100A = 4 and D = 100
e
1/10
1/4
1
2
4
10
24
.42
.64
·71
.74
.73
.66
.66
.52
.51
.56
.78
.87
.90
.91
.89
.88
.79
.78
.66
.89
.95
·97
.98
.98
·97
.96
.96
·75
·95
·97
1.00
1.00
1.00
1.00
1.00
1.00
1/2
II;
I
I'
N, =
2
4
6'
8
12
16
16
20
20
(1)
(1)
(1)
(1)
(1)
(1)
(2 )
(1)
(2 )
.88
.86
.84
.81
.74
.67
.67
.56
.56
.34
.48
·50
·50
.45
.36
.36
.25
.25
.33
.50
.55
.56
·52
.44
.44
•32
.31
i•
II
J
I,"
Ii' -,
I
.68
.68
.40
.62
.73
.76
.76
.74
.68
.58
.58
•43
.43
.37
.60
·75
.80
.82
.82
.78
.69
.69
.53
.53
72
.44
.70
.86
·91
·93
.94
·93
.89
.89
.78
.78
min (~~21~)
.466
.558
.640
.734
.842
.965
],..157
(~~21a;.)
1.002
1.004
1.008
1.366
2.366
4.366
10.366
N =
e
2
4
8
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(2 )
(1)
(2)
12
28
24
36
48
48
60
60
max
f)(
.90)
.92
.96
·97
.96
.95
.93
.88
.80
.80
.519
•603
.677
.797
.58
.83
·95
.98
·99
.99
·99
.99
.99
.96
.96
.994
.67
·90
.98
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1·305
·75
.96
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
2.077
r
!
,
i
I
I
-
: d~',
·T;:
, ....
i: ','
Ii ,: d"
u
(q.
r,;, i
i> i i ~:_ .
i' ;. i ~-:
ti
.
r
f'
,I,:r<::.1
I Ii'""'
Ii]
;
,
I:.
t,'~~
.
~--~
"
."
;
"":
~/The symbols (1) and (2 ) are defined in Table 3.2.-"
J ;.
j I'
If'
, t:
44
Table 3.8.
a
Values of
estimato~/
P =
PrOb[(cr~21~1) ~
t3J
for 100A
1/10
1/4
2
(1)
.98
; .49
1/2
1
4
(1)
.98
6
(1)
8
12
(1)
16
4
.69
.76
.84
.90
.94
.95
.98
.96
.97
.99
1.00
.98
1.00
1.00
1.00
.99
.99
1.00
·99
.98
1.00
1.00
(1)
.98
.97
.94
.87
.67
.68
.65
.60
.76
.77
.76
.71
.86
.88
.89
.86
16
(2)
.87
.60
.70
.85
·96
.96
20
20
(1)
.73
.73
.48
.59
.58
·79
.78
·93
.92
.48
N
4
(1)
8
(1)
l2
(1)
.99
1.00
1.00
1.00
.56
.76
.84
.86
10
= 72
.57
.63
.70
.79
.86
.93
..89
.92
·95
.98
.99
1.00
.77
.96
l.00
l.00
.84
·99
1.00
1.00
.93
.99
1.00
1.00
(1)
.84
1.00
.80
.99
.98
1.00
1.00
l.00
(1)
.93
.91
1.00
1.00
1.00
1.00
(1)
.98
·75
.86
.97
1.00
1.00
1.00
(2)
.98
.86
.97
1.00
1.00
1.00
60
(1)
~90
.78
.92
.99
60
{2)
.90
·75
.67
.67
.78
.92
.99
1.00
l.00
1.00
1.00
.506
.537
.570
.606
.656
24
36
48
48
min (cr~21"
~)
max (cr~21~)
t3(.90)
I,
.
~
I
.85
(1) .
t
1.00
1.00
1.00
18
·I
= 24
.64
(1)
II
~.
100
2
.. 60
.81
2
D =
,.
.52
.71
(2)
10 and
P
N
(1)
=
1.000
1.000
1.000
1.366
2.366
4.366
10.366
.495
.528
.555
.620
.750
.982
1.627
~(The symbols (1) and (2) are defined in Table 3.2.
I
i
!,
I:
I
:!
;1- .
~:.'
,., ,
I,
e
"
45
Table
3.9.
Values of P
estimato~/
a
= Prob [( 0"~1~) ~ t3]
for
100A = 1
and
D = 25
P
1/10
1/4
1/2
1
2
4
10
.58
.80
.88
.91
.94
·93
·93
·91
·90
·72
.94
·98
·99
l.00
l.00
l.00
·58
.82
.94
.98
.99
1.00
1.00
1.00
1.00
.98
.98
·73
.94
l.00
l.00
1.00
l.00
1.00
1.00
l.00
l.00
l.00
N = 24
(1)
(1)
(1)
(1)
(1)
(1)
(2 )
(1)
(2 )
2
4
6
8
12
16
16
20
20
e
.96
·97
.96
·95
·90
.83
.83
.69
.69
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(2)
(1)
(2 )
2
4
8
12
18
24
36
48
48
60
60
.99
l.00
l.00
l.00
l.00
1.00
.99
.96
.96
.86
.86
....
min (O"~I~)
.....
~/The
,
e
.78
.76
·75
.73
.69
.63
.63
.54
.54
(.09)
( .11)
( .11)
(.11)
( .09)
( .09)
( .06)
( .06)
.06
.07
.07
.07
.06
.06
.04
.04
.44
.61
.67
·71
.74
·73
·73
.73
·72
·92
.96
·98
.98
.98
.98
.95
.90
.90
·79
.79
N = 72
.80 ( .06)
.80 (.11 )
.83 (.16)
.85 ( .18)
.85 (.20)
.84 (.21)
.81 ( .20)
·75 ( .17)
·75 (.16)
.65 ( .12)
.65 ( .12)
.04
.07
.10
.11
.13
.13
.12
.11
.10
.08
.08
.44
.61
.74
.80
.84
.86
.87
.85
.85
.81
.80
·99
.99
-
I'I' "" ,
,
i.·
~(
.(-
iii! {
"';
I;
i
If'
~
i
i'
r' !,
I --
I,.,
I
I.
' ~1· .
.878 l.028 1.278
l.078 l.194 1.389
l.764
2.556 4.111
1.778
8.778
2·778 4.778 10.778
.898 1.044 1.289 (1.767) 1.765 2·578 4.178
' ,i
, ' j ;,<.
8·978
-
r
(1)
and
(2)
are defined in Table
3.2.
.;:::: •
f'.~-.j - '-~
,
<.
~
t ,'t
I- ,;
!
.-t
,"i"
;
symbols
.
[ik'!
. -,'
'.
-
.'
j(
~
!,
:£/The
(
;
I
max (O"~I~)
~(.90 )
( .05)'E./ .03
.89
.90
.89
.88
.83
.75
·75
.63
.63
{
values und,er this column that are in parentheses are for
.75).
,.
-
46
e
are:
= 2,
P
100A
= 1/2; P = 1,
100A
= 1;
and P
= 1/2,
results are included in Tables 3.4, 3.5 and 3.6.
with D
= 25,
100A
For P
~(.75) is also used in addition to ~(.90).
,
.
are included in Table 3.9.
;3.7.3 Outline of Computatio:nal Procedure.
=1
These
and 100A
The computations reAll the
A brief outline of the
prqgram is as follows:
(1) For each of the 56 combinations of P, 100A, and D, numerical
values were computed fpr ~, k 2 , Cl , C2 , PI' P2 , VI' V2 (p),
.
(2)
V , V4, WI' W2 , Bl and B2 •
3
A test on P was made and ~(.90) and ~( .75) were computed in
one of three ways depending on the ol.Jtcome of the test.
If
I
t
;
I
.,,
"
P ~ PI' ~(.90)
~(.75)
P
2:
= (9 Vl
=
3
l
P2' ~(.90) == (VI + 9V )/10
3
~(.75)
= (VI + 3v3 )/4
PI < P < P2 , ~(.90) ==
~(.75)
Note that if P
= PI'
V2 (p) == V3 and V4
(3)
+. V )/10
3
(3V + V )/4
==
Vg(p)
[9V2 (P)
+ V4]/lO
[3 V2 (p)
+V
= VI
4]/4
and V4 == V ; if P
3
= P2'
= VI·
The Newton iterative method was used to compute the root or
roots of the equation
1:
~
I
f
=1
These results
quired to obtain the res'ults were carried out on an IBM 650.
progrannning was done in the FORTRAN language.
= 2.
!
The number of iterations carried for each root was such that
where
P
~
P~
is the root from the i
th
iteration.
For P
~
PI and
P one root was computed using as the first approximations
2
the values P and PI' respectively. For PI < P < P2 and
2
V ,V > ~ two roots were computed. As a first approximation
l 3
to the smaller root, P*, the value PI was used) for the larger
L
root, P*' the value P2 was used.
U
(4)
For N = 24:
"
a = 2, 4, 6, 8 and 12, and for N
a = 2, 4,
72:
=
8, 12, 18, 24 and 36, the limits of integration
~l
and U
l
were computed for each combination of P, 100A and D.
that all of these designs are balancedj hence,
gl
(5)
=1
+
~
pj
L =
l
(a-l)(~ P~
+ l)/(N-a)(l +
~
Note
K2 = N/a
and
p) and
U = (a-I) (~ P~ + l)/(N-a)(l + ~ p).
l
For N = 24; a = 16 and 20, and for N = 72: -a = 48 and 60,
the limits of integration, L and U , and the parameters, gi
i
i
and h.,
were computed.
J;
(6)
The values of P = Frob [( ()~2 1~)
~ ~]
ing the integrals in (3.38)-(3.40).
were obtained by evaluat-
I
i;
I
j
,, .
I
Ii1<
i·j!
t::
,\1
II
I
A subroutine developed
j
at VPI for the IBM 650 was used to evaluate these integrals.
Discussion of Results. study of the results indicates that
2 ;..
2 '"
the difference between max (()~ I~ ) and min (()~ I~) is very sensitive
3.7.4
to the parameters p and 100A.
With some combinations of P and IOOA there
I
1';
i
.,
.. ,
48
12 1'" )
.
CO"R
is little difference between max CO"R121'"~ ) and nun
~1 •
For these
situations the type of design or estimator is not very important because the worst possible allocation of funds results in a total variance which is only slightily large~ than that resulting from' the best
Particular combinations of 100A and p
possible allocation of funds.
where the allocation makes little difference are:
100A
= 1/4,
100A = 1/10, p = 10;
p = 4; 100A = 1/2, P = 2; and 100A = 1, P = 1.
It is also obvious that the magnitude of P depends on the specific
combination of p and 100A.
For situations where 100A
~d
P are both very large, or both very
small, the values of P are near one.
The reason for this may be under-
stood by considering a specific case.
and 100A = 1/10.
With 100A
=
Consider the combination p
1/10
1/10 we would want to allocate all our
funds to source B unless p were quite large.
ed in the value for Pl which is
This condition is reflect-
3.67, i.e., we would want to allocate
all our funds to source B unless p were greater than
Frob
=
[Co-~21~) ~ ~C.90)] = Prob
[C;
~
3.67.
Hence,
4.39)ICp = 1/10)] is very large
no matter what design or estimator is used.
For some combinations of p and 100A, viz.,
p
= 2, 100A = 1/2;
p = 1, 100A = 1 and p = 1/2) 100A = 2) the values of P using ~C. 90) are
relatively low.
To study the results in a somewhat higher range of p)
values of P were computed for the above mentioned combinations of p and
100A using ~C· 75).
From general consideration of the manner in which P varies as a
function of
~
it appears that if one uses a design that is moderately
•
,
"oi.
,~
. ;t
near the optimal he will do quite well in achieving his objective of
reducing total variance.
The results may be summarized according to sample size as follows:
(1)
For N
a
= 24
use a
=8
= 8 or 12 if 1/4
if 100A ~ 1/2; a
~ 100A ~ 1/2.
or a ~ 4 if 100A ~ 1/2.
For N
= 72
use a
= 24
100A = 1,2 and a
if 100A ~ 1/4; and
= 2 if 100A > 1
Never use a
Never use a > 20 if 100A > 1.
It
> 16.
is probably best not to use a
(2)
= 12
or 36 if 100A ~ 1/2, a
= 18
12 or 18 for 100A = 4,10.
for
Never use a < 4.
Also never use a = 8 for 100A ~ 1/4 or never use a > 48 for
100A > 1.
The results may also be summarized according to cost, i.e., according
to value of 100A, as follows:
(1)
(2 )
100A
= 1/10:
p
= 10.
p
= 10.
100A
= 24
= 72
use a
For N
= 1/4:
p > 4.
for N
for N
= 24
= 72
use a
For N
= 24
= 72
use a
For N
use a
= 24;
= 24-36;
use a
do not use a
= 8-12;
= 24-36;
= 2,
do not use a
= 2,
do not use a
4 for
4, 8 for
= 2,
4 for
do not use a == 2, 4 for
p > 4.
(3 )
(4)
100A
1/2:
p
~
2.
p
~
2.
100A
For N
= 1:
for N
= 24
use a
= 8-12;
= 24-36;
use a
= 8;
and do not use a ~ 16 for p ~ 1.
= 2,
do not use a
do not use a
do not use a
= 2,
=2
4 for
4 for
for p ~ 1
For N = 72 use a = 18-24;
do not use a = 2, 4 for p ~ 1 and do not use a > 48 for p ~ 1.
r
Ir
1
.,'
. ,--'
. F"'
,
.Y~'"
-,
(5)
100A
= 2:
for N
= 24
use a
= 8;
do not use a
and do not use a ~ 16 for p ~ 1~
For N
=
=2
72 use a
for p ~ 1/2
= 18-24;
do not use a = 2) 4 for p ~ 1/2 and do not use a > 48 for
.
;
p ~ l.
(6 )
100A = 4:
for N = 24 use a = 8; do not use a = 2 for p
and do not use a = 20 for p
N
= 72
use a
=
100A = 10;
2 or a> 16 for p
18-24; do not use a
not use a > 48 for p
(7)
~
~
=
For
4 for p ~ 1/4 and do
8; do not use a = 2 for p ~ 1/4
and do not use a = 20 for p ~ 1/2.
=2
1/4
1.
for N = 24 use a
do not use a
= 2)
~ 1.
~
For N
=
72 use a = 12-24;
for p > 1/4 and do not use a = 60 for p ~ 1/2.
The results seem to indicate that for most situations an intermediate value of ~) say N/4 ~ a ~ N/2) will give results that are quite
close to the optimal.
would be a
If one value of a were to be recommended it
N/3.
It is of interest to compare the results concerning the optimal
~
in the situation considered here with the results given by Crump (1954)
for the estimation of p.
As the value of a which minimizes var(p) he
gives the value
a
=
1
1 + (N-5) (Np + 1)
2Np + N - 3
Using N = 72) and substituting for p the values:
obtain for a
1
the values: 7.6) 24
1/10) 1) and 10) we
and 33) respectively.
As expected)
comparison of the results for optimal ~ using Crump's formula indicates
close agreement with the resluts given in Tables 3.5 and 3.9 with
r:
t .i 1 -
~ ,.,.\! ."
I ' ,f ,
~,
ft
.
t
100A = 1.
f.
~
I
. d:-
5l
Comparison of results from the estimator ~2 with those from
indicate very little difference between the two.
th~
PI
HenceJthe discarding
of the information from the one degree of freedom due to Al vs.
appears to have had little effect for
·t
~
situations considered.
It
should be noted that the one degree of freedom was discarded only for
designs in which a
~3 N and therefore only a relatively small per-
centage of the available degrees of freedom for classes
~as
lost.
For
situations in which a is small it is likely that the discarding of one
degree of freedom would have a greater effect.
I
{
1
'·I · · ~
.'
.~
.":1
~
I
,
\.
I
r
,.
52
4.0 VARIANCES OF ESTIMATED VARIANCE COMPONENTS
FOR A THREE STAGE NESTED CIASSIFICATION
4.1
Introduction
The variances of the estimated variance components for the three
stage nested classification given by model (1.2) will be derived in
this chapter.
An analysis of variance for data described by (1.2) is
given in Table 4.1 •
. Table 4.1.
An analysis of variance for a three stage
nested classification
Source of
variation
Expectation of~7
mean square
Mear::/
square
Degree$ of
freedom
A classes
a - 1
~/(a-l)
er2+K.i.2+~2
erB
erA
C
B in A classes
E b.-a
~/(E b i -a)
i
2
erC +1).2
er
B
~/(N-~ b i )
~
ere
i
Within B classes
~
Eb
N
1
i
2
.
N - 1
Total
2
E E E X, 'k
i j k ~J
-2
i = 1,2, ... ,a
E E n, ,xi' ,
i j
~J
J
-2
-2-
Cl = E E n ,xi' - E n,x, , x, = E E x, 'k/ni
iJ J
~
i j
ill
1
j k ~J
. -2
NX
(N
EE
i j
,
2
nij )/(E b,-a)
ni
i 1
n2 .
i
(N - E N)/(a-l)
i
,
n
K.i.
= (E E
i j
2
ij
n,
~
j
= 1,2, ..• ,bi
k
= 1,2, ... , n,
!
n ij
,
~J
= ni
E n, = N
i
1
53
In this work the usual analysis of variance estimators of the
variance components will be considered.
The estimators, obtained by
equating mean squares to their expectations, are given by
"'2
(J
Q.
= --::-::-_J~_
C
N - .E b.
.
l
l
( 4.1)
The variances of these estimators are
.
I
I
(4.2 )
'I
I
,
I
\
j
t 2K2(Ki-K2)cov(Ql~)/(a-l)(N - ~ b i )
l
Therefore, to obtain the variances as given by (4.2) it will be
~
necessary to procure the variances of, and covariances among,
~, ~
f
I
.1.
.'
I
~
l'
...A. ~
e
and ~.
The technique used to obtain the variances and covariances of
these quadratic forms depends on a well known result from statistical
theory [see Whittle
(1953)] .
For the situation where
x has a multinormal distribution with
covariance matrix y it can be shown that the expectation and variance
of the quadratic form y = x'Mx are
E(y) = tr(~)
var(y) =
2tr(~)2
Also) the covariance between~two quadratic forms YI = x'Mx and Y2 = x'Nx
is
Let J .. denote a column vector with n. elements.
~J
~
This vector has
n .. elements that are "ones" corresponding to the x's in the i th A-class
~J
and jth B-class) and the remaining n.-n .. elements are zeros.
~
~J
let 0 .. denote a null column vector with n.. elements and U
ij
~J
~J
. vector With n .. elements all
~J
"ones " )
the transpose
JI j
If we
a column
is
.
O!b )
~
A vector with n
i
~
elements all "ones" will be indicated as U ) and one
i
with N elements all "ones" will be simply UN.
Certain useful proper-
.. ties of these vectors are ltsted below;
J!~J.~
.U.
n.
~
j
N •
55
Some general properties of the trace of matrices are indicated
in
(4.6)
tr(AB) ~ tr(BA)
_....
--
~
provided the
nuw~ers
(4.6)
tr(AB)
---- + tr(CD)
_.... ,
of rows and coluwns of the matrices permit multi-
plication and addition.
The relations given by
(4.5)
and
(4.6)
will be
used extensively throughout the development of this chapter.
The covariance matrix
y,
required in
(4.3)
and
(4.4)
of the vector
of observations from a three stage nested classification is given by
y
~
diag(Y )
i
~
•
L
where
As usual I(n.) is the identity matrix.
-
1
4.2
Derivation of Variances of Ql' ~ and ~
To obtain the variances of
~, ~
and
~,
matrix products of the
form (~)2 will be needed for each of the three sums of squares, ~, ~
and
~.
First consider
~,
I
. i
the sum of squares for the A-classes.
.I
I.
I.
The sum of squares Q. may be w.ritten as
1
~:= xt~
( 4.8)
where
A
-
Letting r.
1
VA
1
1
n.
N
= (-- - - )
1
:=
. ( -1
')''- -N
1 UNU , •
dlag
N
n. U.U.
1 l. '
l.
the matrix product
y~
is represented by
:=
!N V-aUa U'1
r
r
At this point it is easy to obtain the expectation of Ql'
ring to
(4.3)
and
(4.9)
E( Ql)
,
a 1.a Ua Ua
I-
Refer-
'.
I
I-
i
f
I ,,'
we have
:=
I';
tr ( V_A_)
Substitution of ~, as given in
:=
I: tr (r .V. U. U~) •
.
1
(4.7),
1-l 1
into
1
(4.10)
( 4.10 )
gives
I, .
It'l
1-',
Ii r-
L:
': \-,
Ii- -
"
f
. l
Then by making use of
(4.7)
and
(4.$)
~
we find that E(Ql) is readily
reduced to
'~
;
;,
,-, l,
,
;
,
I
57
"'t.
(4.12)
2
n ..
2
n.
2
2
~~2.J) (JB + (N - E 2:) (JA
N
i j
N
2
n ..
2 + (~ ~ 2.J
== (a-l) (JC
i j ni
To obtain the variance of ~ we require
tr(r.V.u.u~)2 + ; ~ E tr(.V.u.u~)(vuUuU~).
tr(VA)2 == E
--
i
~-1 ~ ~
N'"
iff
_1 ~ X
X X ~
(4.13)
We shall proceed with the evaluation of tr(~~)2 by considering the two
terms on the right hand side (RRS) of (4.13) separately.
Using (4.5)
and (4.6) and substituting V. from (4.7) we have that
-~
E tr(r.v.u.u~)2
i
~-1
1 1
f
(4.14 )
[2
,+2
2
==Er.2 tr(JCu.U.
(JB~n.. J .. U., +(JAn.u.u.
•
.
~
~
~
1
.
J
lJ lJ 1
1 1
For fixed "i" the coefficients of the squares and cross products of
,i' ,
I-
I
the components of (4.14) are derived below,
'
I, '
P
~i
r~ tr[(~ n .. J .. U~ )(~ n.
~
~J;J ~
j
,
j
t i·
J.
_ i
u!)l
m ~m ~m ~ 'J
j.'
.~
,
"
! '
- 2i tr[2
, .. U.'+ j fro nlJ.. n.1mJ lJ.. U.J.
"J
n .. J .. U.J
U.
j
1 1m
- r
~
~J
~J
~
~J
~
~ ~
~
,
- 2[ En4.. + EEn2.. n.2 ]
-r.
1
j
~J
j
fro
~J
lm
222
r.1 [~. n ~J
..
J ;
J
•
i
!)JiilI.<..•.
,..
."-
58
2r.2 tr [
U. ,
U. E n .. J . . D.r
~
~ ~.
~J
~J ~
J
=
'J
2r.2 tr [ U. E. n2.. U.
~
~.
~J ~
J
:=;
2r~ tr(n.U.U~U.U~)
l
l
l
:=;
~
2r~~
:=;
~
tr[(L n. oJ.
.
J
~J
j
~J
4
~
~
;
2r~n?
~
~
oU~~ )(n.u.u~)J
1 1 1
~J
~J
Next) for fixed "i" and
(4.13).
2
r.n.
'J
2r.2 tr [ L n . . J . . n.2 U.
1
of
0
,
~~~~
~
2
2r.n.
E n2
iJ
~ ~.
J;
:=;
')
r.22
n. tr ( U. U.r U. U..
~~
J
=
~ ~
lit'
2 2 L n.2 2r.n;
~
f
0
•
~J
j
"
consider the second tenn on the RHS
Substituting for ~ and
Yf
as given in
(4.7)
gives
~. . 1
, '1
'I
i
:Ii
, ~
(4.15 )
The coefficients of the squares and cross products of the components of
(4.15) are obtained below)
[J.
"
xm]
tr L -Ln. . n(J J. . UIIJ II U;
m
lJ
xm
~J
X
~
2
:=;
2
E E ni.n(J ;
J m J xm
59
2 U.I + UII 6 n.2 ,Uf!'J
= tr [ U. 6 nIl
~
Xm l
x.J ~J x
m
=
n. 6
l
m
2
+
Xm
nf!
2
6 n..
X.
lJ
J
nIl
j
j
- ,i
Finally, to obtain the coefficients of the squares and :cross products
of the components in
termS
"t'
t:rf
varC~)
the results from the first and second
J
·f
f
;.
the RHS of C4.13) are combined and then summed over "ill and
as indicated.
The coefficients are given below,
CT~:; [:~:ir n~
crt -
[6
2Nn + n~) + L L
i l
N i
ir-f
= 12
n.n ll ]
l x
r
- :f
60
2
N-ni ]
[
1:[ -N. n.
~
==
2 2
1: n ..
+
~
1
~
(
n~~J. n~~m
• .).11'
~2
J m
[if?
~
1:
2
2 1:[N- niJ 2 u. E u .
i
Nn
~ j ~J
+
i
1 2 ( 2N
~
t
2
1:.L
i ni
- 2N 1: !.- (1:
j
i ni
~J
~
1:
j
n~.
.~N
E r[n.
iff
~
1:
- 4N
~J
i
L:
ill
n~.
1:
j
~J
n~ .)2 +
~J
(1: 1:
i j
~m
[2if?
1:
.
~
.L
n.
~
1:
j
n~.
~J
~J
;
~J
X j
+ 2 1: n. 1:
i
~ j
n~.
.Jl
~1
n~ . )2]
n~ + n~ 1: n~.]
+ 1: 1: [ n. 1: n~m + nIl 1: n~
iff ~ m x
X j
~J)
==
J
1
n~ .)2
(1:
j
in.
N··
==
m
[_2 1
2 2
1
2 2
2 2
1: ~ - 2 (1: n .. ) - . 2N (1: n .. ) + (1: n .. )
.
. ~J
;
n.
.
~J
. ~J
~
n. J
:.
~
J
J
+ 1: 1: 1: 1:
==
2 2
1: 1: 1: 1: n . . n/l
. ,),11 j
~J ~ m
~TX
~
~TX
1
11
J
. ~J
J.
~J
.i '
. !
.: r
I
~
2
,
2
- 4N 1: 1: n i . + 2N 1:. 1:. n ij ]
..
J
~J
~J
2[
1
2.. - 1:. 1: n2]
..
,== -N N 1:. - n. 1:. n J.J
.,
J.J
~
J
~
J
~
•
2
N-ni ]
1: -Ni [
ni
4
n.,+
J.
1
2 2
21:L:n.nll
~~ x
iff
2N 1: n3 + 1: n.4 + 1: 1:
i
i
i
~
iff
2
1: n.
i
~
2N 1:
i
n?~
2
+ (1: n~) ]
i
~
22]
n.n~
~
x
!
i
>-H '
I; Ii
: I
1 '
,
61
2
N-n.]:;
2
2
n. + L L [ntln . . + n-{n tl ]
2 L ~
f
L .L~ni
i
iff
~
..
,,~
x
2
2
n~. +
2N L n L
i j
i
+ L L
iff
= _~
[N2
N-
2
2
2
2
(ntl L n. " + n.; L ntl )
x j ~J
... ill Xm
L L
i
j
n~"
- 2N L n. L
i
~ j
~J
2
~ ill
xm
n~
L
~J
2
+ n. L ntl )
(ntl L n ..
X j
~J
~
i
]
n~".
~J.
+ L
i
n~
~
L L
f
j
n~. ]
•
xJ
After collecting the preceding results the variance of Q is given by
l
2
Va:r(Ql) =2tr(~~)
= 2 ((a-I)
~
o-~ +
; [
~
~~
~
12
ni
(~ n~j)2
_ 2N
J
~ ~. (~ n~j)2
~ ~
J
2 ] 2 2
LL n.. o-c o-B
i
+22
~
[N2
L L
i
j
n~j
~
- 2N Ln. L
i
~ j
n~. +
~J
L
i
j
n~
~
~J
62
Next consider the sum of squares for the B-c1asses. This sum of
j
, j-.
squares is
L
I
!
~ == x'Bx •
I.
The matrix
~
L
is
P
!
F
,"
where
I,
~i == l: -
1
,
~J
I
(4.17)
J .. J ..
j n ij
~J
t
!
!
From (4.7) and (4.17) one sees that
--
VB == diag(V. B. )
(4.18)
-~-~
and
V.B.
-~-~
cle [l:j ~
n
==
.l:.u.~ u!J
n
~
J .. J!. ~J
ij
~J
i
2[
, - 1 "'"
,
+ CJB l: J .. J .. - l: n .. J.ju.]
j
~J ~J
ni j
~J ~
~
2[
,
']
+CJAl:U.J .. - U.U.
. ~ ~J
~ ~
J
==- CJ2[ l: - 1
e
. n..
J
~J
, - - 1 U.U.']
Ji.J.,
J ~J
n. ~ ~
~
I
I
!
i
\
_1.;..
Again by making use of (4.5) and (4.6) it is seen that
tr(V.B.)
:--~-~
== CJ2
e
2
(b.-1) + CJ (n. - .l:.- l: n~.)
B ~
~
n j
~J
i
,-
,
-
63
,
Hence
t
E(~) == ~ tr(~J4)
1
(4.19 )
2
==
(I: b. -a) (J2 + (N - I: I:
i
C
1
i
j
,n~j)
i
!
(J2
B
.
It is noted that
tr(Y£)2
tr
==
diag(Yi~i)2
==
~ tr(Yi~i)2
•
I
,
(4.20 )
1
To obtain the coefficients of the squares and cross products of the
".11
components in (4.20) we first fix
1
and then take the trace of
,
. .,
2
(Yi~i).
The coefficients are derived below,
1
t
~L:-2...J.
J~.U.U~
n. . n.. 1j 1J 1 1
•
tr [ L: -2 J .. J .. J .. J ..
. .
1J 1J 1J 1J
J n ..
1J
+
~ UiU~Uiu~J
n.
== (b i
.
1
tr
I~ JijJ~li/~j
1 J
+ 1)
- 2
1J
==
b
i
- 1
- - 2 L: n .. J .J t.. J . . U.t
n. . 1J i J 1J lJ 1
J
1
1[ 2
f
.-. + L: L: n . n
+ -2 L: n. . J . j U.t J. ~ U.
J. . U.t J U.
i J i m 1J 1 i m 1
. 1J 1 1 lJ 1
J'~
ni J
r1ll
= I:
j
==
2
n ..
lJ
2
L: n ..
j
1J
23
.
422
(L: n . + L: I: n .. n. )
i
j
J
jim 1J 1m
1
- L: n .. +-2
n. . 1J
1
J
n
Jj
i
3 +~
2 2
L:n..
2 (I:n .. )
j
j
1J
n
1J
i
-,...1
64
~
J~
2tr([ E n. J.~m
m ~m
'1
:=
-
~m
~
J'jJ~.
n i u. U~J [E
.
J
~
,
--!..
n.
~
:=
II
-
~J
,1
l:.n.~J·Ji J.u~~ ].1
n E
j
i
,
,
:
l
J
E U.U!J . . J~. + 1 E n ' .UiU!J .. u~J
2 . i J
. ~ ~ lJ lJ
l ~J ~
J
n J
i
!
t
2[n. - ..!E n~.J
n j
lJ
~
The variance of
over
~
2tr [ E J . . J .. J .. J .. - E J .. J .. J .U.
n.. ~J lJ lJ lJ . n. . ~J ~J i J l
j
lJ
.
~
,j
i
~
suw~ng
is found by
t1
the coefficients just derived
,j
;I
i" and multiplying by two, this, gives
;
CT~
+ 2(N - E E
j
i
2
nij)CT~ CT~
:!
)
~,
ni
3
+[E E n~. - 2 E E
i
j
i
lJ
ni j
ni
j
1
2
2
+ E 2
(E n .. ) ]
i n . j ~J
CT
41
B
,( 4.21)
•
~
The sum of
where C
:=
s~uares
diag(Qi) and
~
:=
oJ..
Then ~Q
:=
ViC.
- -l
for the C-classes is
r(n.) - E~ J .. J~j
l
. n.. lJ l
-
(4.22 )
J . ~J
diag(YiQi) and from (4.7) and (4.22)
2
:=
CT c[I( n.) - 1:
-
l
.
J
~
n..
lJ
J .. J!
.J
lJ lJ
+ CTB [E J .. J~. - E J . . J~.J + CTA
j
~J lJ
j
lJ lJ
2
==
CT~ [r(n.)
l
2
E
j
~
n
ij
J..lJ J~.]J .
[U.l U~l
.J
E U.J~
j
l lJ
(4.23 )
i
!
;,,"':
.~.
r
The variance of
~
is
2
=
I:
i
tr ( y. . Q. )
J. J.
2
From (4.23) we obtain
(V.C.)
-J. -J.
4
O-C [ f(nJ.') - I:
-.!.... J .. J~ . J
j n ij J.J 1J
2
and
tr (V. C.)
-1-1
2
4
o-c (n. -b.) •
=
1
1
Hence
var(~)
= 2(N -
~ b i ) o-~
Derivation of Covariances among Ql'
4.3
To obtain the covariances "among
obtain matrix products of the form
~,
~
combinations among the quadratic forms;
between Q and
1
(4.24 )
•
1
~
and
~
~
and
~
it is necessary to
'!
for each of the three pairwise
First consider the covariance
~.
II
To derive cov(~~), as given by (4.4), we require
[I
til
2tr(VAVB) = 2
----
I: tr(r.V.U.U!V.~.)
i
1-1 1 1-1 1
•
I'
Substituting V. from (4.7) and B. from (4.17) into (4.25) gives
-1
-J.
tr(Y~Y.6)= i: r.tr f[o-2CU ' U!'+ o-2B I: n .. J .. U~
•
J.
J.'
J. J.
.
J
J.J
J.J
1
2
+ O-A n.u.u!]
J. J. 1
• [ er2 (i: 1
C
+
er~
-,.. - -1- U.U.')
. n.. Ji.J
J J.J
n i J. 1
J lJ
(i: J ..
j
J.J
J~.
- .!.J.J
n
i: n .. J ..
lJ lJ
i j
u~]l
.
J. )
(4.26 )
66
For fixed "i" the coefficients of the squares and crossproducts of the
components of (4.26) are derived below,
4
r.tr[~
U.J~.
- u.u~J
1
.
1 lJ
1 1
O"c:
0 ;
==
J
4
O"B :
·r.tr[(~
n'mJ'mU~)(L:
1
1
1
1
.
J
m
r. tr [
1
=
J ..
.
1m lJ 1m lJ
3
1
2
J
1
.
J
.
lJ lJ 1
.
J
n . . J . . U~)]
1
,
lJ lJ 1
'I
I]
n. n2.. J. U.
mJ
1m lJ 1m 1
j
n
i
~
j
n2.. U. U.t
lJ 1 1
+ L n . . J . . U~ - ~ n . . J . . U~]
.J
J
J
1
r.tr [ ~ n .. U.J ..
- -1
lJ 1 lJ
.
1
2
1
L ~
n.
r.1[L:
n..
- -n. (L:. nlJ
.. )
.
lJ
j
m
t
L: n. n .. J. J..
- -1
~
mJ
1
J~. )-(L:n.1m .T.1mU~1 )(l:..
~
n. .
lJ lJ
== 0
lJ lJ 1
r.tr[n.
L: U.J~.
- n.u.u~]
1
1.
1 lJ
1 1 1
0
==
J
r. tr
1
[
-
n.
~
1.
J
n .. U.J t.. - U. U.1
lJ 1 lJ
1 1
~
.
J
n2.. ]
o
lJ
Cov( ~~) is obtained by summing (4.26) over
II • "
1
and multiplying by
two, the result is
,
l
(4.27)
o and
-t:·:
i •...
V.
II
,:,:I i.,
.
I ~" -'
;
67
"'2
d .....2
a:-B an ~C
"'2
4. 4 Varl' ances of a-.A'
.....2
A2
.-..2
The variances of D"A' D"B and D"C are obtained by substituting the
expressions for var(Ql)' var(~), ~ar(~) and cov(Ql~)' as given by
(4.16), (4.21), (4.24)
and
(4.27), "respectively,
.....2.-..2
into
(4.2) .
.-..2
Covariances among D"A' D"B and D"C
Once the variances and covariances of Ql' ~ and ~ have been derived it is a simple matter to derive the covariances among the esti.....2
mated components of variance. Consider the covariance between D"C and
Cov ( "'2
~C "'2)
D"B may be expressed as
(4.28)
From
(4.1)
we have that
"'02
...2
-
var(D"C + D"B) - var
.
l- N-~
~b. +
•
J.
J.
•
1 (.~
K:
~ b.-a
1 . J.
~)-J
-, N-~ b.
•
J.
J.
var(~) +
var(~) ]
(~
i
"'2
"'2
Substitution of var(D"C + D"B) as given by
as given by
(4.2)
into
(4.28)
J.
(4.29),
.
2
b.-a)
•
J.
~
~
of var(D"C) and var(D"B)
yields
:=--
(N-~•
b.J.
fK.--J.
J.
Proceeding in a similar manner the other two covariances are obtained.
. i,
68
They are:
(IS. -K~)
- -
-(N---L:-='b~.)~2~K-K-~' var( ~)
i
-
-
l
~~
[ Ki.
f
(4·3l)
r~
var(;)
+
(Lb.-'-a)
i
~
(4.32 )
:
,
,
•. j
1
f
69
5.0
OPTIMAL DESIGNS FOR ESTIMATING PARAMETERS FOR
A THREE STAGE NESTED CLASSIFICATION
5.1
Introduction
Crump (1954) considered a two stage nested classification where
I
I
j.
the data can be represented by the model (l.1).
•
A form of an analysis
.i
of variance for data obeying (1.1) is given in Table 1.1.
He indicated
2
2
the optimum designs for estimating m, (fB' (fA and p
total sample of N, when the usual analysis of variance estimators are
used.
For estimating m the optimum design has a. = N with n. = 1, and
~
for estimating (f~ it has a
1
-I
. 1.
= 1 with n i = N.
:1 ,
2
The estimator of (fA is given as
,h
with variance
i
"
~(N-l)(a-l)
_2
2 2
(N-a)(~ - l: n )
i
For fixed a the variance of
When the n
i
~ is minimized when ni
are equal,
=
~ (i
=
~J
. ,I
-,
I
I
2 2
(a (fB)
+
N-a
].
:ji
1"
,•
;
;
•
!
!
i
.I
·i
i
':
70
....2
var,(o-A) is minimized by taking the number of classes to be
a
==
1
N(N P + 2)
Np+N+l
(5·4)
For the sitll8.tion where N/a is not an integer but
°< S < a )
N/a == p + s/a)
....2
Crump proved that var(o-A) is minimized by putting p+l units in each of
s classes and p units in each of a-s classes.
He hypothesized the
following procedure to determine a design
i)
1i)
find a
if a
l
l
from
(5.4)
is nonintegral) choose, as
~
the closest integer to a
,
l
and allocate as given above:[p+l units in each of s classes
and
p units in each of a-s Classes].
II
Because this procedure may not always give the minimum variance) he suggested checking both the closest integer above
~~d
below a .
l
As an estimator of p) Crump used the unbiased estimator
.... ==
p
--:::_N_ _ [(N-a-2)(a-l)
N-a
_2 .
2
N"" - E n
1
- (a-I)] .
•
I 1.'
!
The variance of this estimator is
J
'.
+
2N(N-3)
Jl- -
E ~2
1
p +
N2(N-3)(a-l)
(rf_ E n~)2
~
~.
"1
,
"
I:
~'
~
..." ,...,.1"',,,....
....
_.
<
"':"'."~"""-
~~
,,~,~
7l
....
Crump showed that var(p) is also minimized for fixed ~ by having the ni
equal.
When the n
i
are equal,
....
var(p) ;:; .,--_2...>..(N,..--n3-<-)----.:-(N-a-4)(a-l)
a +
-N
[
p] 2
,
and var(p) is minimized by taking the nwnber of classes to be
• 1
a
If a
2
2
;:; 1 +
i
(N-5)(N p+l)
2N p
+ N-3
is nonintegral he suggested having
~
classes, where
~
is the
closest integer to a , with p+l units in each of s classes and p units
2
in each of a-s classes.
In the following sections we shall consider analogous problems of
design for a three stage nested sampling plan, where an observation can
be represented by (1.2).
The estimators used will be the same type as
considered by Crump, 1. e., the usual analysis of variance estimators.
5.2
Estimation of Parameters for
5.2.1
Estimation of m.
"
a
Three Stage Nested Classli'ication
r: r: r: x.lJ'k
m ;:; i j k
..
;
N
It is obvious that var(.,m) is a minimum when n
j.
•
J
An unbiased estimator of m is given by
i
;:; n
ij
;:; 1 :for' all i and
Hence the optimum design has one unit in each of N A-classes.
.
.
r,J
.1,1
".'I
~~
I'
L
t
iI
72
Estimation of
5.2.2
(4.1)
2
~C.
An unbiased estimator of
~~ is given by
and the variance of the estimator is given in Section
"'2
obvious that var( ~c) is a minimlUll when 2:: b.
i
when a = b
i
= 1 and n iJ
N.
=
b
=
~.
;
4.4.
It is
is a minimum, 1. e.,
Hence the optimum design has all N units
in one B-class in one A-class.
~
2
Estimation of
(4.1)
~B.
and the variance of the-estimator in Section
and b it can be shown that
n ..
~J
An unbiased estimator of
= -bN for
all i and J.
4.4.
~~ is given by
For fixed a
var(~) attains a miDimlUll when ni
=
I:
~ and
1
r
When there is equal allocation to the A-
"'2
, iI'
t,
~
'
I
1
classes and to the B-classes, ~B = b(~ - ~C)/N and var(~B) is given
I
by
I
2 2
+
(b ~C)
N-b
]
Obviously the value of a that minimizes the above expression is a = 1.
"2
When a = 1, var(~B) has exactly the same form as that given by
for
"2
var(~A)
in the two stage
nes~ed
plan.
(5.3)
Hence the suggested proce-
dure for obtaining the design that minimizes var(~~) is to choose a = 1
and proceed as described in Section
2
~C
5.1 with ~~ substituted for ~~ and
,
: ,I
!
'i
,
2
for ~B·
5.2.4
Estimation of
~~.
An unbiased estimator of
~~ is given by
(4.1)
and the variance of the estimator in Section 4.4. We have not
"'2
been able to prove for fixed!!: and ~ that var( rr.A) is minimized when a
completely balanced design is used, i.e., when n.
~
and nij = N/b is an integer for all i and j.
= N/a
is an integer,
However, it does appear
that a balanced design is sufficiently close to optimal for the estimation
, ,'.l.
t,-
"
~
I;
-
I
...-/&
73
of ~~ to warrant its consideration.
Also, analytical treatment of the
"'2
expression for var(~A)
as given in Section
cult.
4.4 appears to be very diffi-
Because a balanced design appears to be near optimal and because
the expression for var(~) from a balanc~d design is in a form that can
be manipulated, we shall first consider designs for which n
n.. = N/b and both are integers for all i and j.
~J
sign
i
= N/a,
With a balanced de-
~ = N (MA - ~) and var(~) may be expressed as
It is obvious that var({) is a minimum when b is a maximum, that is
when b = N.
When b = N,
th~-expression for var(~) reduces to
"'2 _~rl- { [a (~~ +~~)
+ N
~~J2 [a (~~ + ~~)J2}
a - 1
+
N - a '_
'
var(~A) -
... ......
'
The expression for var(~) given above is in exactly the same form as
that given by
(5.3)
" "'2
for var(~A) in a two stage nested plan.
Hence, the
procedure suggested for obtaining a design is to set b = N and then
proceed as described in Section
Estimation of
"'2
~
5.1 with ~~ + ~~ substituted for ~~.
222
+
+
~C
=
~B
N
~
i=l
An unbiased estimator, ~, of
~A'
(xi11 -
x)
2
/N-1 ,
where
N
x
=
~
i=l
,
xiI1/N.
I,
f1
,
74
The variance of this estimator is
(5.16)
Since this varimlce attains the lower bOQnd as given by Gaylor (1960)
the optimal design has a
5.2.6
= b = N.
2 ~2
2
Estimation of p = ~A/(~C + ~B)o
For designs which are not
balanced the estimator of p would be exceedingly complicated.
the components ~~ and ~~ appear i.n p only as a simple
SlUll}
there is no need to provide separate estimates of them.
Because
~~ + ~~,
Hence, it
appears that the appropriate design will be similar to the type sug2
gested for the estimation of
~A
.;
insofar as O!1.l.y one unit should be
When b
assigned to each B-class.
\
= N)
p = ~_N--::- .[(N-a-s) ~
~--E
n:
~
].
.; I
I
an Qnbiased estimator of p is
- (a-I)] ;
(5.18)
the variance of this estimator is
'" = 2 4
var(p)
N-a-
{
f(L. N-a-2)'
p
+
We note that var(p) as given by (5.19) is in the same form as that
giveh by (5.7).
b
= N and
-
Hence the procedure for obtaining a design is to let
then proceed as described in Section 5.1 where the estima-
tion of p is discussed for a two stage nested classification.
__....L._-=I
>
It'
,:
-
75
6.0
SOME ASPECTS OF THE SJJ<IULTANEOUS ESTIMATION
OF VARIANCE COMPONENTS FOR A TlffiEE STAGE
NESTED CLASSIFICATION
6.1
Introduction
In this chapter we shall be concerned with the designs of experiments for the simultaneous estimation of variance components where a
result or observation may be represented bY~(1.2). An analysis of variance fox' data obeYing (1.2) is given in Table 4.1.
As before, only
estimators of the vari8-l1ce components based on the usual analysis of
variance will be considered.
The designs presented in Chapter 5 are for the estimation of
single functions of the variance components.
lInfortunately, a design
which is presented for the estimation of a single component may be very
poor for, or may actually preclude the estimation of, other components.
For exanple, the design that is suggested for the estimation of U~ (see
Section 5.2.4) does not allow the separate estimation of u~ and U~;
2
2
only the sum erC + u is estimable.
B
Presently it is the usual practice to design balanced experiments
for the joint estimation' of the three variance components.
There are
several reasons for designing balanced experiments, among them are:
(1)
The calculation of the sums of squares, and the coefficients
of the components in the' expected mean squares are somewhat
.11
easier for data collected from a balanced than from an unbalanced plan.
e
(2)
Evaluation of the precision of the estimates from an unbalanced plan is vastly more complicated than from a balanced
I
I
~r
.J
76
plan.
With a balanced design and under the assumptions of
Ie
L
T
(1.2), the sums of squares associated with the sources of
I
i
,,
variation are uncorrelated and each is distributed as chi
square.
"2
~
;"2
The estimators ~A anderB' given by (4.1) are linear
functions of uncorrelated chi square variables and thus the
variancesof these estimators are relatively easy to obtain.
For the general unbalanced design the sum of squares assodated with sources A and B are correlated and neither is
distributed as chi square.
Hence, the variances of ~ and
~ are very tedious to derive~ This may be appreciated by
consulting Chapter
(3)
4.
.
With an assignment of degrees of freedom for which a balanced
",I
design exists, the allocation of units to classes is uniquely
specified, viz., equal allocation to the classes.
For an
assignment for which a balanced design does not exist there
is a multitude of possible allocations.
Unfortunately, requiring that a plan be balanced greatly restricts
the manner in which degrees of freedom may be assigned to the sources
of variation.
Particularly bad is the circumstance that one may assign,
at most, one-half of the available degrees of freedom to sources A and
B, together, while to source C must be assigned at least one-half of
the available degrees of freedom.
Thus, if cr~ and cr~ are large and im-
portant compared to cr~, one is forced to estimate {
little precision.
ang. cr~ with too
I. ,
I
Therefore, it appears that in situations of the type
just described it may be desirable to sacrifice balance in order to
i
J. -
II
77
gain precision in the estimates of U~ and u~.
However, as soon as an
assignment of degrees of freedom is proposed for which a balanced design does not exist, one is faced with the dilemma of choosing a method
for allocating units to classes.
Contrary to a. two stage nested plan
where units are allocated as e~ually as possible to the classes [see
Crump (1 95 4)] there is no obvious method for allocating units with a
three stage nested classification.
l:
\
~
L
Designs have been suggested by Anderson and Bancroft (1952),
Anderson (1960) and by Calvin and Miller (1961) for the situation where
l
,
I
I
i
j
the degrees of freedom are assigned as n-l, n and n to the sources A,
B and C, respectively.
The purpose of this chapter will be to study methods of designing
experiments for the simultaneous estimation of variance components for
a three stage nested classification.
It is not intended that a final
solution to this problem of design will be presented here; hopefully,
this work will be a starting point for future research.
6.2
Designs for Estimating Variance Components
for a Three Stage Nested Classification
Anderson's so-called staggered design was proposed for a five
stage nested classification where the assigbment of degrees of freedom
to sources of variation was n-l to the main classes and n to each of
the subclasses.
In this chapter the staggered design will be considered
for a three stage nested classification.
The design would be composed of two basic configurations, as
illustrated below:
78
(1)
(2 )
Type (1) has two C-classes in each B-class and two B-classes in each
I
;
A-class.
Type (2) has one C-class in each B-class and two B-classes in
each A-class.
The design would consist of n/2 configurations of type
(1), and n/2 of type (2).
For this design, as with the usual unbalanced
design, the sums of squares due to A-classes and due to B-classes are
not distributed as chi square.
However, unlike the usual unbalanced
design, the sums of squares due to A-classes and due to B-classes are
uncorrelated.
For this particular sit;uation, the sum of squares due to
A-classes may be partitioned into three independent parts;
~ vs. ~ with respective degrees of freedom:
~, ~
and
(n-2)/2, (n-2)/2 and 1.
The sum of squares due to B-classE;!s may be partitioned into two independent parts, B in A , and B in~, each with n/2 degrees of freedom.
1
The usual analysis of variance for the Anderson design with the above
described partition is given in Table 6.1.
From' Table 6.1 two analysis of variance type estimators of cr~ are
suggested.
"2
The first is ~A as given by (4.1).
The second is the
1
"
i
t
r
II
:1
weighted sum
I
"'·2
+ W O"A
2
2
where
"
\
,
'I'
(6.1)
t
r
... -t
..,J"'
79
Table 6.1.
Source of
variation
The usual analysis of variance
for the Anderson design
Degrees of
freedom
Expectation of
mean square
2+grlO) 2+9n-10
2
uc .
n-l u B
3(n-l) uA
~/(n-l)
n-l
A classes
Mean
square
2 + 2 2
u +
B
c
(n-2)/2
2~1/(n-2)
U
~
(n-2)/2
2~2/(n-2 )
U
Al vs. ~
1
Q
U
B in A classes
B in ~ (B )
l
n/2
2 + 2
2
c uB + 2uA
2
c
13
~/n
n
2
40-A
+
2
2
4/3 uB + 8/3 uA
2
+ 3/2 uB
U~
2 + 2 2
u
c
B
2~1/n
U
·1 '
"
II
B in A (B )
2
2
n/2
Within B classes
n
Total
3n-l
~/n
and W and W are predetermined weights.
2
l
"
ji
2 + 2
c uB
2~2/n
II
U
I
fl
2
" - II
U
c
Because
~ is a linear func-
tion of uncorrelated chi square variables, it would be relatively easy
to derive its variance.
.i
The difficulties with this estimator are that
it does not use the information provided by the one degree of freedom
due to
i
I
•i
,e
i
~vs. ~,
Wland W •
2
and that there is no obvious way to choose the weights
Anderson (1961) has suggested use of i terateCl. linear esti-
mates (linear in the mean squares) for such situations where the weights
for the j+lst iteration are based on the estimates obtained from the jth
iteration.
I
•
<f.
i
~ ~.
i
t I,' ,
•
I
..r~'
/'-
80
As an alternative to be cOlupared with the staggered design a
second design will be considered.
For the assignment of degrees of
freedom n-l) n and n this design will consist of n configurations of the
type
,
That is) there are two C-classes in one B-class and one C-class in the
t,
other B-class and there are two B-classes in each A-class.
JI
l-I'
This is
.~
the design given by Calvin and Miller (1961) for the three stage nested
,
1t
classification.
.
I
The usual analysis of variance for such a design is given in Table
6.2.
~
The estimator of rr~ is
~.A =fl
- 5/4 ~n + ~
Q.;]/3 .
[n-l
(6.2 )
Unlike the circumstance for the Anderson design, the sums of squares
I,
I:
due to A-classes and due to B-classes ~e each distributed as c?i square.
ri
I
However) contrary to the result given by Calvin and r1iller, the sums of
II
squares due to the A-classes and due to the B-classes are correlated.
(4.27)
I
!
;
,
:
Referring to
the cov~iance between the two mean squares
~/(n-l) and ~/n is calculated to be
cov~~ , ~]
=
9: ,,~
(6.3 )
,
I, -
Therefore, the variance, as given by
(4.2),
of the estimator
such a design would be relatively easy to obtain.
~, ~
and
~
!
j
~ from
Using the fact that
are distributed as chi square, along with the result for
i,
..
.
,
81
Table 6.2.
The usual analysis of variance for
the Calvin-Miller design
Degrees of
freedom
Mean
square
A classes
n-l
Ql/(n-l)
, CTC
B in A classes
n
~/n
CT
Within B classes
n
~/n
CT
Source of
variation
_Total
Expectation of
mean square
2
2
2
+ 5/3 CTB + 3CTA
2
C + 4/3
CT~
2
C
3n-l
'J
the covariance of ~/(n-l) and ~/n as given by (6.3), the variance of
"2.
.
CT ~s glven
as
A
+ 25
lb
(1 + 4/3 P )2
2
+ l6n
1
---n---::;:-
2
2
P2 ]
9 n, ,
(6.4)
where Pl
2
2
== CTA(CT
C
and P2
2
2
C
== CTB/er
For assignments of degrees of freedom other than n-l, nand n,
where balance cannot be achieved, there is no known procedure for constructing the best design.
We shall consider a specific procedure for
constructing a design, called the D -design:
2
(1)- Write
and assign ql + 1 units to each of r
i
l
A-classes (this group of A-classes
will be designated by G ) and ql units to each of the r~maining a-r
l
,1
I
A-classes (designated by G ).
2
,-
I
82
(2)
j
Write
L
,
)
I
f·
To each A-class assign
each of r
(3)
2
A-classes.
~
B-classes and tfen one extra B-class to
!
I
> a.
Make sure that b
I
Within each A-class) assign the units to the B-classes as
equally as possible.
i'
i
It may be noted that when b ~ N/2 and a D -desi gn is constructed)
2
the design will usually satisfy the relations
In.~ - nfl
==
0 or 1
;
In~J
.. -nfmI
==
0 or 1
(6.5 )
.
For the assignment of degrees of freedom n-l) nand n this procedure
,
will yield a design which consists of n configurations of the type
)
and is the same as the Calvin-Miller design mentioned previously.
In order to gain insight as to how the D -design compares with
2
some other methods of design) including the Anderson design where the
comparison is feasible; it was decided to make use of some numerical
examples.
For these examples) a sample size of N
==
Eight assignments of degrees of freedom were chosen.
48
was selected.
They were:
.i
)
83
(4)
(2)
(1)
11
12
24
7
24
,
16
,
15
16
16
(6)
23
,
8
16
,
15
,
a reference point.
23
16
,
24
8
It is noted for the assignment (11, 12, 24) that
design may be constructed.
(8)
,
8
.80
completel.y1:cl.anced
This particular assignment was included as
The eight assignments listed above are intended to
encompass the range of assignments for which b
where b < N/2 were considered.
~
,.
N/2.
No assignments
There were two reasons for this;
I·
(1)
When b < N/2, there are many assignments for which a balanced
i
design could be constructed.
(2) . We are mainly concerned with the situation where cr;' and cr~ are
large and important compared to cr~; therefore, we presumably
are not interested in designs where more than half the available degrees of freedom are assigned to source C.
Two designs were considered for each,allocation of degrees of
freedom.
The first was constructed by dividing the A-classes into
groups and then assigning the B-classes and C-classes so that balance
would
~e
~esign
maintained within each group.
To simplify presentation, this
will be referred to as a D1-design.
For those assignments where
it was possible, a design of the type proposed by Anderson was constructed.
Care was taken so that a D1-design was not given that was
the same as a D -design.
2
The designs considered for each of the
assignments (2)-(8) are given below;
(A, B in A, and C in B in A with
the number of A-classes for each design indicated below the design.)
.
, .
.
84
(2 )
7, 24, 16
D1 :
rtrllln fill
nnTI
4
4
D2 :
j-,
8
i
i
l:
l.
(3)
15, 16, 16
D1 :
~
8
D2 :
~
;
Ii
n
I
t
,!'
,
)
I
--
8
n±
,'ii.
-
I
,
t,
!.
I
,I
1
16
(4)
i
II
23, 8, 16
D :
1
~.
8
D :.
2
A
16
±
16
n
l
I!
I
I
i
I!
t!
J
I
I
I
.i
I
I
I
I
1
,
D :
1
-lJ
f--"
+-t~
:t _ Ii',
~
4
D :
2
..
l .-
8
~
(5 ) 7, 32, 8
~-
:
I'
,1-
".
i
J
. - I':;..
4
I")
!lrtn
j
I
I
8
!
I· '
I
t-!--i
:J::
I
:~'!I
85
(6)
15, 24, 8
D
1
:
~
4
2:
D
h1
8
fill
4
n
,
8
ill
8
,
}'
(7) 23, 16, 8
1:
D
6n
A
4
2:
D
8
(8)
31, 8, 8
1:
D
~
4
2:
D
A
8
n
n
12
I
,
±
.;
8
16
n
n
4
8
1
24
1
16
The usual analysis of variance estimators given by (4.1), with one
e,.
exception, were the only estimators considered in this study.
Criteria
I
!
l
j .
I
I~
.L ':~
Ii
~
li.'~
i'
used to compare the designs were the quantities,
E
A
~
E
C
/[var(~~) for (11,12, 24
1
;=
[var\0"A
1,,2) for the assignment in qUestion]
;=
[var(~~ )
for the assignment in
qUe~tionJ I [var(~~)
[var(~)
for the assignment in
questionJ/[var(~~) for (11,12,24)] .
for (11,12, 24 ~
The results given in Section 4.4 were used for the calculation of
"2
"2
"2
var(0"A)' var(0"B) and var(0"C)'
the parameters, PI
;=
~
i
L
.
i
As the above criteria are functions of
2 2
2 2
0"A / 0"C and P2 = 0"B / 0"c' it was necessary to specify
numerical values for these parameters.
P2 each equal to 1110, 1/4, 1/2, I, 2,
The 49 combinations of PI and
lr
j
C
I.
!
,I
4 and 10 were selected.
; I.
i'
\"
The numerical results are presented in Tables 6.3-6.12.
Study of
the tables indicates, as expected, that the manner in which the degrees
of freedom are assigned to the sources of variation has a large effect
on the variances of the estimated variance components.
with PI
;=
10 and P = 2 the non-balanced assignment (31, 8, 8) yields
2
for the two designs D and D , respectively, E
2
l
A
~
;=
For example,
1.88 and 2.35 and E
C
;=
3.00.
;=
.59 and .53,
In this situation, the assignment
(31, 8, 8) naturally is much better than (II, 12, 24) for the estima2
2
tion of 0"A' much poore! for the estimation of (5B and very.much poorer
for the estimation of
(5~.
On the basis of the results some general observations can be made:
(1)
The result in Tables 6.4 and 6.5 can be used to compare the
Anderson principle of design and the D -design.
2
From compar-
ison of the results in Table 6.4 with those in Table 6.5 it is
noted that the D -design does better relative to the Anderson
2
design at the assignment (7,24,16) than at the assignment(15,16,J..6) .
•
!
i
87
Table 6.3.
Values of VA' VB and Vc for the balanced
design with degrees of freedom: 11, 12, 2~/
Values of VA for P2 equal
Pl
i/l0
1/4.
1/2
1
2
4
10
1/10
.0220
.0322
.0536
.113
.296
.924
4.90
1/4
.0350
.0472
.0720
.138
.335
.990
5.05
l/2
.0657
.0813
.112
.189
.409
loll
5.30
1
.161
.184
.225
.325
.590
1.38
5.84
2
.488
·525
.589
.734
1.09
2.06
7.08
2.64
3·97
4
1.69
1.75
10
9·65
9.80
.0404
1.86
10.0
2.10
10.6
11.6 .
14.1
10.1
..
23.4
j!
.0573
.531
.0937
1.70
I,:
9·20
il
d
1:
.~
Vc = .0417
iIIo--:
88
Table
6.4.
1/10
1/4
Values of E , ~ and E for the D and D
C
1
A
2
designs with degrees of freedom: 7, 24, 1~/
.4
10
1/10
1/4
1/2
1
2
.65
.57
.50
.42
.37
.33
.31
.66
.59
.52
.45
.40
.37
.34
.88
.78
.67
.54
.44
.38
.33
ji
.87
.78
.68
.56
.47
.40
.35
I
j
I:
I
:
t
I
1.13
1.02
.89
.12
.56
.44
.36
1.08
.99
.87
.72
.58
. 47
I
! ..
!! '.
.38
i
!.
1.37
1.28
1.16
.98
.76
.54
•41
.
1.28
1.20
1.10
.95
.76
.59
.44
1.53
1.48
1.39
1.25
1.04
.79
·53
1.41
1.37
1.30
1.18
.99
.78
.54
1.63
1.60
1.54
1.46
1.30
1.07
.72
1.49
1.47
1.42
1.35
1.22
1.02
·72
1.69
1.67
1.66
1.60
1.54
1.35
1.09
if!
1.53
1.53
1.52
1.41
1.29
.1.04
l'
D
1.30
1.10
.91
.75
.59
·57
D2
1.39
1.17
.96
·77
.59
.56
1/2
1
i
i
,
J
2
4
10
1.
~
;
i
90
Table 6.6.
Values of E ,
~
A
and E
e
for the D and D
designs with degrees of freedom:
1
2
23, 8, It}!:./
Values of E
D
1.94
1.89
1.85
1.81
1.81
1.81
1.81
D
6.00
5.31
4.63
3.95
3.45
3·13
2·90
D
l
1.54
1.56
1.60
1.64
1.70
1.74
1.78
D
4.06
3.88
3.64
3.37
3·13
2·97
2.83
D
1.20
1.25
1.31
~.42
1.54
1.64
1.73
D
2.48
2·53
2.59
2.66
2·71
2·73
2.74
D
.95
.98
1.05
1.15
1.31
1.48
1.65
D
1.37
1.47
1.61
1.82
2.08
2.34
2.55
,,
D
.82
.83
.87
.93
1.06
1.25
1.50
,!
I,
D
.82
.88
.97
1.14
1.42
1.79
2.23
II
D
.76
.77
.78
.80
.88
1.01
1.28
I!j:
D
.61
.63
.67
.76
·93
1.23
1.77
1
2
1
1/2
2
1
1
2
1
2
2
1
4
10
1/4
2
1/4
4
1/10
PI
1/10
A for P2 equal
2
1/2 '. 1
2
,t '
!'
I
Ii! .
'"
I'
D
·73
1
10
.74
.74
·73
D
l
.52
1.50
·52
1.50
.54
1.50
.56
1.50
D
5.30
4.50
3.67
2.84
D
2
2
.77
.62
.82
1.50
·1-.50
2.24
1.88
1.65
A
(var(~)
~ = [var(~~)
E
e
==
[var(~~)
for
for
.,
.'"":~
! I
ij
,,>
,
c
-.
f
\I
I
(23,8,16)J/[var(~~) for (11,12,24)J
/[var(~~)
;i
~~.
"'7
L
(23,8,16)J/[var(~) for (11,12,24)J
for (23,8,16)J
LI;
!:
e = 1.50
==
1.12
.74
1.50
E
!!:./E
1.02
for (11,12,24)]
.' }J
~
~:'
. i~:o---~.
Ij .- ,
.~'"
·:t
- ..1;
,';'t
\:
-l
-I
-J
{.
'1I
-
91
Table 6.7.
Values of EA'
E.s
and EC for the Dl and D
2
designs with degrees of freedom: 7, 32, 8!!:/
Values of E for P
A
2
P1
D1
D
2
Dl
.1/10
1/4
1/2
1
2
1/4
1/2
1
2
4
10
.68
·59
·51
.43
·37
·33
.31
.61
·52
.43
·35
.28
.24
.21
·90
·79
.66
.53
.43
.37
.32
.83
·72
.59
.46
.35
.28
.23
1.14
1.02
.87
.70
.54
.42
.35
D2
1.05
.94
.80
.62
.46
.34
.25
Dl
1.37
1.27
1.14
.94
.73
.54
.40
D2
D1
1.26
1.17
1.05
.86
.65
.46
.31
1.53
1.47
1.37
1.22
·99
.74
.49
D
1.41
1.35
1.26
1.12
.90
.66
.41
L
II
D
1.62
1.59
1.53
1.43
1.26
1.02
.67
I,
1f
,l1·
i!
1
,:
t1
,I
t I
D
1.49
1.46
1.40
1.31
1.15
·93
.59
D
1.69
1.67
1.65
1.59
1.52
1.35
1.04
r,
1.52
2.43
1.51
1.71
1.45
1.08
1.39
1.24
·70
·53
.94
.44
II
D
1.53
3·19
2·97
2.27
1.60
1.02
.67
·51
.43
2
1
D2
10
1/10
D2
D1
2
4
e~ual
1
D2
I
1
,
1
E.s
[Yar(~~)
for (7,32,8)J
/[Yar(~~)
for (11,12,24)J
= [Yar(~) for (7,3 2 ,8)J /[var(~~) for (11,12,24)J
EC = [Yar(~~) for (7,32,8)J /[Yar(~~) for (11,12,24)J
t
i:
U
4~
II
EC = }.OO
!!:/EA =
-
II'
II
.---tt
.I
-
tIl
,',
r
I
I
I
I
..
. J
• ...1
I
92
Table 6.8.
Values of E ,
and E for the Dl and D2
C
designs with degrees of freedom: 15, 24, ~/
A
~
Values of E for P
.89
.78
.67
.59
.54
·51
D
2
1.09
.97
.85
.72
.62
.56
·52
D
l
.96
.88
.79
.69
.61
·55
·51
D
1.00
.92
.83
.72
.64
·57
·52
.86
.80
.57
·52
.91
1
.91
,.87
.81
D
.88
.84
.81
D
.84
.82
D
.85
D
.53
·75
.67
.60
.54
i
.79
.74
.67
.61
·55
I'
.83
.81
.77
.72
.65
.56
i'
2
.79
.78
.77
.74
.70
.64
·57
I'
11
D
.83
.82
.81
.79
.75
·70
.61
2
1
1
.74
,
i
i
,'
; I
jI
'Il
;
.75
.74
·72
.68
.60
1I
D
.82
.82
.82
.80
.79
.76
.69
I
D
2
D
1
.75
3·22
.74
2.50
.74
1.81
.73
1.21
.73
.85
·71
.67
.66
D
3.54
2·72
1.95
E
C
=
1.27
.86
.65
3.00
for
~=
[Yar(~~)
for (12,24,8)J
E
C
[Yar(i~)
for (15 ,24,8)J /(Yar(~) for (11,12,24)]
--
.59
.56
iI
!
l.
I:i
l
Ii
,]
[Yar(~~)
=
I;
.76
2
A
l:
.76
1
=
-
D
2
~/E
; I:
.58
1
, 10
.63
I
.65
2
4
.72
\,
~
D
2
10
1.00
D
1
4
D
2
1/2
2
1/4
1
1/4
l'
1/10
PI
1/10
2 equal
A
1/2
(15,24,8)J/[Yar(~~) for (11,12,24)J
/[Yar(~)
.
for (11,12,24)J
.__ H
I
I
I
.1
1
I
H
11
II
11I'
I
93
Table 6.9.
Values of E , ~ and E for the D and D
l
A
C
2
designs with degrees of freedom: 23, 16, e}!:./
Values of E for P2 equal
A
.1/10
1/4
1/2
1
2
4
D
l
1.68
1.52
1.36
1.20
1.09
1.02
.96
D
2.56
2.25
1.94
1.65
1.44
1·30
1.20
D
l
1.35
1.28
1.20
1.12
1.04
.99
·95
D
1.89
1.77
1.64
1.48
1.35
1.25
1.19
D
l
1.06
1.05
1.03
1.01
·98
·95
.94
D
1.32
1.30
1.29
1.26
1.22
1.19
1.16
D
l
.84
.84
.85
.86
.88
.90
.91
D
.89
.91
.95
.99
1.04
1.08
1.11
·71
·71
.72
.74
·77
.82
.86
.66
.68
.71
.75
.83
·92
1.02
.64
.65
.65
.66
.61
.68
.72
·79
Pl
1/10
2
1/4
2
1/2
2
1
2
D
1
D2
2
4
10
D1
D2
.56
D1
.61
.61
.62
.62
.63
.64
.69
2
D
1
·51
3.24
·51
2.60
·52
2.01
.52
1.46
·55
1.13
.59
.96
·70
.88
D
2
4.98
3·89
2.84
1.89
1.29
·99
.84
D
.58
.57
c
A
=
~=
E
c
=
.66
.75
.89
!
·1 '
r:
j
,
E
~/E
10
[var(~i)
for (2 3 ,16,8)J
[var(~~)
for
[var(~)
=
3·00
!
I'
!1
I
j'
I
!
I
I:
; I.
'1'--/[var(~)
for (11,12,24)J
(23,16,8·»)/[var(~) for
for (23 ,16,8)J
I
/[var(~)
(11,12,24)J
for (11,12,24)J
I,
t ,
~ ~
,
L
1'
,
!
I
I
j,,
.
.'
94
Table 6.10.
Values of EA for P2 equal
,
1 . 2
1/10
1/4
1/2
Pl
1/10
2.94
2.74
2.55 . 2.44
2.36
2·31
D
6.00
5.31
4.59
3.88
3.34
3.00
2.76
D
2.27
2.25
2.22
2.22
2.23
2.24
2.30
D
4.06
3.83
3·58
3.29
3.04
2.84
2·71
D
1.52
1.59
1.67
1.80
1.94
2.07
2.19
D
2.44
2.48
2.53
2.57:
2.59
2·59
D
·99
1.04
1.14
.
1.30
2·59
1.54
1.79
2.04
D
2
1.32
1.40
1.53
1.73
1.98
2.21
2.41
,
D
1
·71
•74
.79
.90
1.10
1.40
1.79
'
I
D
2
·75
.81
2.09
I
D
1
·59
.60
D
·53
l
l
2
1
2
4
1
2
10
10
3.16
2
1/2
4
D
1
2
1/4
Values of EA, ~ and EC for the D1 and D2
designs with degrees of freedom: 31, 8, 8!!:./
.89
1.05
1.31
1.67
.62
.,
·55
.67
-
.67
.59
.78
1.00
1.44
.83
loll
1.63
,
j
I
II
\
I
I
·53
·53
.5 4
.55
·59
.67
.94
D
.43
.45
2.44
.47
2.11
.53
1.88
.64
1.76
.99
1.71
4.33
3.16
2.35
1.91
1.66
2
D
1
3·19
.44
2.81
D
6.83
5·58
2
.
I
D
1
.'
I
j
III:
I
ii
,
EC = 3·00
!!:./EA =[var(~~) for (31,8,8)J/Ivar(~) for (11,12,24)J
~ =[var(~) for (31,8,8)] /[var(~~) for (11,12,24)J
"2 for bl,8,8)J/[var(~~) for (11,12,24)]
EC = [var(~C)
i
\
;
f
I-, I
I'
t
95
For both PI} P > 1 there is not a large difference
2
between the two designs.
The D -design tends to be somewhat
2
better when P2 < PI and slightly poorer when P2 > Pl'
example} at PI == 10 and P2 == 2 the
D2-desig~
For
is about 9
per cent better for the assignment (7}24}16) andS per cent
better for the assignment (15}16}16) than the Anderson design.
At PI == 2 and P2 == 10 the Anderson design is about 3 per cent
better for the assignment (7)24)16) and 2 per cent better for
the assignment (15)16)16) than the D -design.
2
For both P }P
l 2
~
1 the Anderson design with the assignment
I
j
I.
(15}16}16) is generally better than the D2~design while with
the assignment (7}24}16) the two designs yield very similar
results.
For PI
~
1 and P > 1 the Anderson design is generally a
2
little better than the D -design.
2
The greatest difference is
10 per cent with the assignment (15)16}16) at PI == 1/10 and
i
I
II
1~
,
11;
4 !
I
P
2
1,
= 2.
For PI > 1 and P
2
i~
~
1 the D2 -desi gn is generally a little
better than the Anderson design.
The greatest difference is
II
11
.1
'i
I
!.
i!
10 per cent with the assignment (7}24}16) at PI ==10 and
P2 == 1/10.
The D -design appears better for the estimation of ~ if
2
i:
I!
;I
I!
"il1.
1<
P
2
(2)
> 2 and the Anderson design better if P2 < 2.
A comparison between the Dl-design and D -design for the
2
assignments (23)S}16)} (23}16}8) and (31}S}8) should show the
t
l
.,11
fl ..'
11\ I
, I
'I
·it
j!
'\
I!
11
I!
I
j
I
I
1
same general characteristics.
This is because the D1-designs
are very much alike for the three assignments, as are the
D -designs.
2
Consideration of Tables 6.6, 6.9 and 6.10 indi-
cates a distinct pattern in the comparison between the D 1
design and D -design.
2
The D -design tends to be better only
2
for large ~1 [PI ~ 4 with (23,8,16) and (31,8,8) and PI ~ 2
with (23,16,8)J and when P2 < Pl.
The difference between
the two desi.gns is the greatest with the assignment
(23,8,16).
It may be noted that the two designs have no
configurations in common with the assignment (23,8,16), but
.
<
do have some in common with the ~ssignments (23,16,8) and
(31,8,8).
For the estimation of O"~ the D -design is always poorer
2
(
i
II
with the single exception of P = 10 for the assignment
2
I!
(23,16,8) .
(3)
The D -design for the assignment (7,32,8) is similar in
2
its basic features to the D -design given for the assignments
2
(7,24,16) and (15,16,16).
However, the D -desi gn is quite
l
different from that given for the assignments (7,24,16) and
(15,16,16).
For the assignment (7,32,8) the D -design is
2
always better than the D -design for the estimation of both
1
2
2
O"A and O"B·
(4)
Both the D -desi gn and D -design for the assignment
1
2
(15,24,8) are quite similar in their basic configurations to
the designs given for the assignment (15,16,16).
As expected,
,I
II
. ,-1 .
I:
ill
III
i
It
:,·1
,I: I
II'
I:
97
a comparison between designs with the assignment (15,24,8)
yields results for the estimation of both
22
and ~B that are
~A
very similar to those obtained for the assignment (15,16,16).
The major difference is that the D -design is somewhat better
2
,
relative to the Dl-design for (15,24,8) than for (15,16,16).
(5)
Consideration also may be given to selecting assignments
which are best in general.
We will not consider situations
where Pi and P are both small, i.e., where both P , P < 1.
2
l
2
For both P , P > 1, it appears that the D -design with
l
2
2
the assignment (15,24,8) is quite good.
combinations of P
l
There are specific
and P for which another assignment is
2
better, but generally the assignment (15,24,8) does the best
job.
In comparison with the balanced assignment (11,12,24)
I,
it is noted that E ranges from .66 to .70.
A
For P
l
~
I'
\ I
1 and P > 1 the D -design with the assignment
2
2
(7,32,8) appears to be very good.
In comparison with the
balanced assignment (11,12,24) it is noted that E ranges
A
from .21 to .65.
For Pi > 1 and P2
~
1 the D2 -design with the
(23,16,8) appears to be very good.
as~ignment
In comparison with the
balanced assignment (11,12,24) it is noted that E ranges
A
from .51 to .75.
The assignments and designs mentioned above for the
estimation of ~i are also quite good for the estimation of ~~
I!
When P is small, P < 1, the assignment
2
2
2
suggested to estimate cr is rather poor for estimating
A
when P2 > 1-
cr~.
However, this may not be of particular concern as a good
estimate of
cr~ may not be necessary'when cr~ is small compared
2
2
to crA and cr C·
It is rather interesting to note that each of the three
suggested designs is a D -design in combination with a given
2
assignment of degrees of freedom.
1-
One important point that evolves from this brief study of designs
for the simultaneous estimation of variance ¢omponents is the need for
defining some meaningful criteria.
It is apparent with regard to assign-
ment of degrees of freedom that changes which increase the precision of
one component invariably reduce the precision of one or both the other
,
components.
types of designs considered.
stat~ng
I
This often also is true with regard to the two different
i
Hence, an important problem is one of
I
IiIt
the objectives for which the estimates are to be used and then
determining the experimental design that does the best job of accomplishing the stated objectives.
I
III:
1!
!.
1·
Iii
:'j I
i:, I
An actual experiment using a staggered design for the simultaneous
~I
jI
j I
estimation of variance components is given by Calvin and Miller (1961).
The experiment was performed on field grown, flue-cured tobacco with
five fertility treatments replicated four times.
was the concentration of nitrogen in plants.
each were randomly selected from each plot.
The response measured
Two groups of five plants
From the first group one
sample was drawn while from the second group duplicate samples were
i
!
I
l;
99
~
drawn.
One chemical determination was made on the sample from the
group and one was made on one sample from the second group.
sample from the second group was analyzed in duplicate.
fir~
The other
Hence) there
were twenty configurations of the type
The analysis of variance obtained from the actual data is given in
Table 6.11.
Table 6.11.
Source
of
variation
Analysis of variance for tobacco data
Degrees
of
freedom
Mean
sCluare
Expectation
of
mean. SCluare
I
ji
I;
: I
I·
I
!
.,
e
100
The estimates of the components of variance obtained were
.....2
cr =
C
.....2
0:
B
""2
.001767
= .03652
crA =
o
.05579
To fit these results into the frrunework of this chapter we shall
consider the results only from stages A} B and C.
With a three stage
nested classification the D -design which allocat~s the degree of free2
dom nearly eClually to the sources is n configurations of the type
, ,
.
,,
I
,,
'
: 1
Ii
i;
\
as suggested by Calvin and Miller.
To match nearly the experimental
situation described by Calvin and Miller} let n = 2O} and assign the
degrees of freedom as (19}20}20).
; 1
l:
If we take the estimates of the
components to be their true values then P = 19.8 and P = 2O.7} thus}
l
2
for purposes of computation} we shall consider P = P
l
2
= 20.
With the
"24
above information we may compute VA = var(cr )/2cr
as given by (6.4).
A
C
The calculated value of VA is VA = 57.6.
If the covariance between the
sums of SCluares due to A and due to B is neglected} as Calvin and Miller
do} the calculated value of VA is VA = 58.9.
For this case the exclu-
sion of the covariance term does not have much effect on VA'
Although
the covariance term had little effect in this particular case one should
I'
101
be very careful for other situations, particularly for higher stage
classifications where more than one covariance may not be zero.
For purposes of comparison, VA was computed for P
l
=
P
2
=
20 with
the assignment of degrees of freedom (14,15,30), an assignment for
which a balanced design is available; VA
= 76.6.
In addition to the work concerning design, brief consideration was
given to a second method of estimation which involves using a weighted
linear function of mean squares.
Anderson (1961) presents a weighted
least squares procedure and gives a numerical example.
He finds the
variance of the weighted estimator to be very close to that of the
usual analysis of variance estimator.
For the assignment of degrees
of freedom (15,16,16) used with the Anderson design, we shall consider
.
-
.
~2
.
the weighted estimator ~A as g~ven by (6.1).
1'2
1'2
The independent estimators ~~ and ~A2 as given by (6.1) have
i
variances
I
,
• I
~
; i
•
1'2
4
For large Pl and/or P2 var(~~)/2~c is approximately equal to
1'2
4
'.
1'2
1'2
var(~A )/2~C'
Therefore, as ~A_ and ~ are each based on the same
.
-~
2
A
2
-
number of degrees of freedom the estimator suggested for large P and/or
l
~2
~
.A
=
,
. I
102
~2
"'2
Values of var(rrA)/var(rr ) for these combinations of P and P were
1
A
2
given previously but are repeated in Table 6.12.
results indicates that the
wei~hted
Consideration of the
estimator is better than the usual
analysis of variance estimator when e~ther P or P is greater than
l
2
four.
\
However, when an advantage exists for the weighted estimator it
is slight,never exceeding four per cent.
As expected, the weighted
estimator is quite poor for small P and P .
l
2
that
UA
~~
One should keep in mind
"'2
is based on one less degree of freedom than is GA'
Although this study of a weighted estimator vas confined to one
simple example, the results tend to Support the use of the usual
analysis of variance estimator for situations where the degrees of
freedom are evenly distributed to the sources of variation .
Table 6.12.
.....2
~2
Values of var(rrA)/var(rr ) for the Dl-design
A
with degrees of freedom: 15, 16, 16
P2
P1
1/10
1/4
1/2
1
2
4
10
1/10
·76
.80
.84
·90
.96
1.00
1.04
1/4
.80
.83
.86
·91
.96
1.00
1.04
:
I
i
I
.\
~ I
, i
I
., ,
'
I
~
~
,..
I
~
I
I
e
r7
1/2
.86
.87
.89
·92
•9 t
1.00
1.04
1
·92
·92
·93
·95
·96
1.00
1.04
2
·97
·97
·97
.98
·99
1.00
1.04
4
1.00
1.00
1.00
1.00
1.00
1.01
1.03
10
1.02
1.02
1.02
1.02
1.02
1.02
1.03
'r '
103
7.0
SUMHARY AND COnCLUSIONS
7.1
The Problems
Two separate problems were considered in this dissertation:
(1)
The importance of experimental design for the estimation of
fUnctions of variance components for a two stage nested
classification when the objective is to economically reduce
product variability.
(2)
Comparison of a number of experimental designs for estimating
parameters for a three stage nested classification.
7.2 The Influence of Experimental Design in Reducing Product
Variability for a Two Stage Nested Classification
,.
When a result from an experimental unit is the sum of two independent random parts) the variance of the result) the total variance) may
be expressed as
where rr
2
2
and rr are the variance components associated with sources A
B
A
and B) respectively.
The objective considered was that of
redUcing~. It was assumed
that a fixed amount of funds) say D) are available to expend on sources
2
A and B in order to reduce rr .
T
The model was proposed that related
reduction in product variance to the amount of funds expended.
Under
the model it was assumed that the expenditure of one unit of funds on
2
2
source B reduces rr B by one per cent and on source A reduces rr A by 100A
per cent.
With the above model) it was determined that the optimal
.
2 2
allocation of fUnds is a function of p = rrA/rrB •
";'-
104
It was assumed that an experiment is to be performed to estimate
P in a two stage classification with N units assigned to a classes)
based on a model of the form
i=~1)2) ... )a
X •.
1.J
=
ill
+ A. + B ..
1.
1.J
j
=
'1) 2) • ' • ) n .
1.
En.= N
1.
where m is constant) A. and B.. are normally and independently dis1
tributed
(NID)
lJ
with means zero and respective variances
~~ and ~~.
The
type of experiment considered was such that) for N/a = p + s/a (0 < s < ~
there are p+l units assigned to each of s,classes and p units to each of
a-s classes.
Under such an experimental ~etup with N fixed) different
.
,
designs are obtained by varying the number of classes a.
~(o
estimators of p were considered) the most important of which
WaS
A
Pl = (l>). - ~) /Kl~
where M is the source A mean square) ~ 'is the source B mean square)
A
and K is the coefficient of
l
2
~A
in the expectation of M in the usual
A
analysis of variance.
i
The other estimator was introduced to determine the effect of
\
I
i
1
! '
not using the one degree of freedom that represents the contrast between
classes with p+l units per class and classes with p units per class.
As a criterion for judging the influence of design and estimator
on the reduction of
2
~T)
the quantity considered was
:,
i
I
i
I
105
2
is the reduced total variance, divided by
~B'
that is
A
attained after
~l
units of funds ha.ve been expended on source A and
A
D-d on source B, and f3 is a fixed number in the interval
l
max(~R2Idl) .
min(~R21~1)'
It was found that P could be expressed approxi,rnately in
the form of an incomplete beta fUnction whose parameters and limits of
integration are functions of A, p, D, Nand a.
It was then noted that
the influence of design and estimator could be ascertained by specif,ying A} p, D, and N, and evaluating the appropriate incomplete beta fUnction for various values of a.
To aid in appreciating the influence of design and estimator some
numerical results were presented.
The results indicated that there is
a significant decline in P only for designs deviating quite severly
; i
from the optimal.
say
N/4
~
For roost situations an intermediate value of
~,
I
;,'.
:
,
ii
a ~ N/2} will give results that are quite close to the
i
optimal.
I
Comparison of the results from the two different estimators
;
indicated the values of P from the two estimators to be very close.
,
i
I
7.3 Designs for Estimating Parameters for
I
,!
a Three Stage Nested Classification
The mathematical model considered
I
jI
i
j,l I
here was
I J
1
; ;
i = l,2} ..• ,a
I,
I
I
"" I
j = l,2, ••. ,b.
I; j
J.
k= 1}2, ..• }n ..
J.J
=n.,L:L:n .. =N
J.
•
J.
•
J
J.J
where m is constant} Ai} B } and Cijk are all NID with means zero and
ij
:i i
"
!i
-I
l;
I
1 'I ..
106
2
respective variances) u ' u
A
2
,
;
2
B and u c . Emphasis was placed on design
rather than on the method of estimation.
The estimators used were
those that are obtained from the usual analysis of variance by equating
meaD squares to their expectations.
"'2 ",2
"'2 were studied
Variances of the estimated componentsu , u ) and U
A B
c
for various designs.
"'2
It was found that the expression for var(~B)
was
"'2
complex and that for var(cr.A) was very complex.
Designs were presented for the estimation of m) U~) and
u-?
= U2c +
u
2
B
+ u
2
A
-
that minimized the variances of the estimates.
signs were also given for the estimation ,Of
u~, U~
and p
= U~/(U~
De+
U~)
that appeared to yield estimates whose variances were near minimal.
designs turned out to be analogous to those given by Crump
The
(1954) for
the estimation of components for a two stage nested classification.
, I,
Some attention was also devoted to the simultaneous estimation of
u
2
c'
2
2
B
A
u ' and u '
A method of design referred to as a D -design was
2
I,
,
suggested such that
,
: if
l'
il!
i
In..-nl/.
I=
lJ xm
I
I:
:11
0 or 1
, I
. I
II
I,
,
! ;
whenever the ntMber of B-classes) b, is greater than or equal to N/2,
i.e.) b> N/2.
For the assignment of degrees of freedom n-l) n, n this
design turned out to be the same as that presented by Calvin and Miller
(1961).
,.
To learn how the D -design compares with other designs some
2
,numerical examples were considered.
degrees of freedom with N =
i
,
\
i ,
I I
'I
For each of seven assignments of
'3I.
:!HI"
48 one additional design was considered
I:'.' I'
fi,
,
\
---------.
'. ,
,.
i,
;,
I
"
,
107
4IJ
along with the D -design.
2
Where it was possible this additional design
was the type of design suggested by Anderson and Bancroft (1952) and
Anderson (1960). For the balanced assignment (11}12}24), the quanti. . .2
-4
.....2
4
.....2
4·
ties var(J )/2u C' var(JB)/2u and var(~c)/2Jc were computed over a
c
A
2 2
2 2
range of values of Pl = J A/J C and P2 = J B/J C. For the seven unbalanced
assignments the quantities
EA
=
[
A2
A2
var(~A) for the assignment in qUestion] / [var( ~A) for (11,12,24)J
EB
=
[
A2
var(~B) for the assignment in qUestion] /[var(5) for (11,12,24)]
EC
=
[ var(J
A2 ) for the assignment in qUestion] /
C
[var(~)
for (11, 12) 24)J
were computed.
As expected, it was noted that assignment of degrees of freedom
has a large effect on the variances of the estimated variance components.
For those assignments of degrees of
freedo~
f.
,
where an Anderson design
was possible, it was observed that the relative merit of the D -design
2
depended upon the parameters P and P ·
l
2
For some combinations of P
l
\
I'
and P , the Anderson design was better than the D -design and for other
2
2
combinations the D -design was better.
2
.
,I
; If
t
I
I
The results wece summarized by suggesting assignments and designs
for the three situations:
.. i
!:
i
and
I
:
i
!
i
I
Ii
l I
\
108
7.4 Sugf,ested Future Research
(1)
Consideration of methods of estimation other than those based on
the usual analysis of variance.
In
particula~ weighted
lea?t
squares and maximum-likelihood esttmators might be compared with
the usual analysis of variance estimators.
(2)
Consideration of additional objectives for which the estimated
functions of the variance components are to be used.
(3)
Extending to the
ffi1~lti-stage
nested classification the concept of
reducing product variability throuGh the use of variance components.
(4)
Consideration of seQuential estimation procedures} using results
from the first stage to constructjdesigns for later stages.
(5)
Further study of methods of design for the simultaneous estimation
of variance components for mUlti-stage classifications} with special attention to the development of realistic criteria for
optimization.
I·
I
i
II
I
:i
'. i
11
.
ri t
, i
'
; i
I
109
LIST OF
REF&qf~CES
Anderson} R. L. 1960. Uses of variance component analysis in the
interpretation of biological experiments. Bulletin Int. Statist.
Instit. 37:3:71-90.
Anderson} R. L. 1961. Designs for estimating variance components.
North Carolina State College, Raleigh. Inst.,of Stat. Mimeo
Series No. 310.
Anderson, R. L. and Bancroft, T. A. 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. Rep. No. Q.C./R/15.
Calvin, L. D. and ~liller, J. D. 1961. A sampling design with incomplete dichotomy. Agron. J. 53:325-328.
Crump} P. P. 1954. Optimal designs to estimate the parameters of a
variance conJionent model. Unpublished Ph.D. Thesis, North Carolina
State College, Raleigh.
Gaylor, D. W. 1960. The construction and evaluation of some designs
for the estimation of parameters in random models .. Unpublished
Ph.D. Thesis} North Carolina State College} Raleigh. Inst. of
Stat. Mimeo Series No. 256. (University Microfilms, Ann Arbor)
Hammersley, J. M. 1949. The unbiased estimate and standard error of
the interclass variance. Metron. 15:189-205.
Marcuse, S. 1949. Optimum allocation and variance components in
nested sampling with an application to chemical analysis.
Biometrics ~:189-206.
Satterthwaite} F. E. 1946. An approximate distribution of estimates
of variance components. Biometrics Bulletin 2:110-114.
:1
II
:I
,I
Searle, S. R. 1956. Matrix methods in components of variance and
covariance analysis. Annals Math. Stat. 27:737-748.
!
II
Searle, S. R. 1958. Sampling variances of estimates of components of
variance. Annals Math. Stat. 29:167-178.
ill
Searle, S. R. 1961. Variance components in the unbalanced 2-way
nested classification. Annals Math. Stat. 32:1161-1166.
I, !
Whittle, P. 1953. The analysis of multiple stationary time series.
J. Roy. Stat. Soc. (B) 15:125-139.
1
·1
I
"
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, I
i i
·1,
,
I!
, !
, i
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,
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