ESTIMATION OF THE
MEAN AND VARIANCE COMPONENrS
IN A 'nlO STAGE NESTED DESIGN WITH COMPOSITED SAMPLES
by
Keith L. Kussmaul and R'. L. Anderson
Institute or statistics
M1meograph Series No. 473
April" 1966
iv
TABLE OF CONTENTS
Page
LIST OF TABLES
v
LIST OF FIGURES
1.
vi
INTRODUCTION
1
2 • REVIEW OF LITERATURE
3.
000000
o
••
0
•
3
•
OPTIMAL DESIGNS FOR WITHIN-A COMPOSITING
5
..
3·1- The Within-A Compositing Model
·
5
6
3·2. Variances of Weighted Means Estimators
• • • • • • • .
8
3·3· Optimal Estimation of Parameters
11
3·4. Numerical Comparison of Equal-Cost Designs • • • •
/II
•
•
/II
4. PRECISIONS OF THREE ESTIMATION PROCEDURES FOR WITHIN-A
COMPOSITING •
5·
..........····
4.14.2.
4·3.
Estimation of O"A' O"B and O"A + O"B
4.4.
--.2
"'2 "'2
Variances of O"A' "'2
O"B and O"A + O"B
···
····
····
······
Estimation Procedures to Be Studied
Estimation of m .
2
2
2
· ·2 · ·
··
····
21
23
24
29
NUMERICAL COMPARISON OF ESTIMATION PROCEDURES FOR WITHIN-A
.
COMPOSITING .
39
5 .1. "Decidedly Unbalanced" Designs 0 " " "
5.20 Optimal Choice of ri's and nij's for Procedure (3)
5.3. Choice of Designs for Numerical Comparisons
5.4. Results of Numerical Comparisons •
5.5. Loss of Efficiency in Unbalanced Designs . • • •
39
41
43
54
56
....
0
. ..···
·····
0
0
6.
21
••
SOME DESIGNS FOR ACROSS-A COMPOSITING • • •
·..
o
6.1. A Model for Across-A Compositing
6.2. Precision of the Estimators •
7· SUMMARY
AND CONCLUSIONS •••
00
••
00
••
0
67
67
. ..
0
59
68
71
8. LIST OF REFERENCES
I)
•
59
62
7.1. Outline of Research •••
7.2. Conclusions of Research
7.3. Suggested Future Research.
9. APPENDIX
•
0
000
••
73
000
75
v
LIST OF TABLES
Page
3.4.
3·5·
Efficiencies of 32 designs for which 2a + N + 8R = 240,
relative to the optimal design for a given estimator
and a given value of p ••• • • • • • • •••
....
12
2
Degrees of freedom in the optimal designs for (jA
• •
14
Efficiencies of 20 designs for which 2a + N + 4R = 50,
relative to the optimal design for a given estimator
and a given value of p • • • • • • • • • . • • • • ••
16
Efficiencies of 28 designs for which a + 2N + 16R = 400,
relative to the optimal design for a given estimator
and a given value of p • • • • • • • • • • • • • • • •
17
Values of a-l and R-a in the optimal designs of cost studies
(1), (2) and (3) • • • • • • • • • • • • • • • • • • •
19
4.1. Design conditions for equivalence of the weighting procedtlI'es .. . • .
.. •
0
•
•
•
0
0
0
22
•
4.2.
Values of fij's and gij'S for procedures (1), (2) and (3) ••
4.3.
Values of
lS.
25
5·1.
and K for procedures (1), (2) and (3)
2
Example of the choice of the ri's in procedure (3)
5·2.
Distribution of random variable X used to generate the nils • 45
5·3.
Unbalanced designs for comparison of procedures (1), (2)
and (3) . . . . . . .
0
lit
•
•
lit
..
•
..
•
..
..
•
•
•
29
43
lit
•
46
•
5.4. Relative efficiencies of procedures (1), (2) and (3) for
designs 1-25
5.5.
0
lit
0
II
0
1(11
0
•
..
..
•
•
0
0
•
•
•
•
48
•
Average ranks in relative efficiency of procedures (1), (2)
and (3)
0
5.6.
...
•
•
0
lit
0
•
..
0
0
1(11
0
..
•
•
•
..
•
..
..
•
•
•
lit
•
•
53
Efficiencies of designs 21-25 relative to designs 6, 10, 15,
22 and 26 from study (1), Chapter 3 • . • • • • • • • • • • 57
Appendix Tables
9.1.
Variances of estimators for 3 designs of Chapter 3
75
9.2.
Variances of estimators for designs 1-25 of Chapter 5 using
estimation procedure (3) • • • • • • • • • • • • • • • • •
76
vi
LIST OF FIGURES
Page
6.1.
Representation of an across-A compositing model. . • • • . .
60
1.
nrrrRODUC'rIOn
The design of experi.ments for estimating the variance conponents
for a random model has become an important area of s tatis~~ical research.
rrhis dissertation is concerned only with two'-stage nested designs ,{here
certain samples (at the second stage) are composited, or pooled, into a
single observation prior to measurement.
A design of this kind will be
economical only if the cost of measurement 13 a very expensive rart of
the experimentj If measurements were cheap and easy to obtain, each
sanwle wOlud be measured.
For convenience the two stages of nesting will be referred to as
A-classes and D-samples, respectively.
selected from each A-class.
Chapters
Hence, several B-sanWles are
3, 4, and 5 of this dissertation
are concerned with within-A compositing, where each observation is
composed of B-samples from the same A-class.
In Chapter 3 optimal de-
signs for estimating the variance components and the overall mean are
discussed, using the traditional analysis of variance estimation procedure.
In Chapters 4 and 5 the analysis of variance estimation procedure
is compared with two other estimation procedures, on the basis of computed variances of the respective estimators.
Chapter 6 contains a brief discussion of across-A compositing,
where all B-samples from several A-classes are composited into a single
observation.
The estimation procedure used here is due to Cameron
(19511
and is based on altering the number of B- samples I,d thin different Aclasses.
2
It is assumed in this dissertation thiJ.t
J[leaSnreFent~i
cere f"[lde iTith·-
out error.
If this if, not true then the statistical lcodels in Chapters
3 and 6
not the correct models) and if used ,rill lead to incorrect
al'C
conclusions.
Consider) for exacple) the nurwrical differences bet"l'1ecn
cOInposited observations '\fithin the sarr;e A-class.
If r::eaSUreli'.ent error
is present) these differences reflect variation anong D-san;ples and also
error of measurement.
Using the model of Chapter
3) hO>Tever) these
differences Ifill be attributed solely to variation amonc; B-samplesj
hence) the true variation among samples >lolLld be over-estimated.
The
correct model for this situation should include a third component of
variation to represent measurement error.
3
2.
REV;J:EW OF LITERATURE
The practice of compositing has long been used in many fields of
research.
Until recently, however, the estimation of the mean has been
the primary objective, with estimates of the variance components used
for the sole purpose of estimating the variance of the sample mean.
the field of soil sampling, Cline
In
(1944) indicated that for estimating
the mean a single composite sample is adequate, but that more than one
sample is required to estimate variation.
Reed and Rigney
(1947)
discussed in more detail the use of composite samples in soil sampling.
Newton et ale
(1951) used composite samples of Malayan rubber to main-
tain a continuous check on variation due to laboratory testing.
Tanner and Deming
(1949) discussed the sampling of bulk materials
for determining some average value efficiently and economically.
Knowledge of the variances between primary units and between secondary
units within the same primary unit is required; usually this information is lost when samples are composited.
Tanner and Lerner
Cameron
(1951) and also
(1951) discussed methods of compositing in which the
variance information is retained.
These articles and others are in-
cluded in an extensive bibliography of bulk sampling literature by
Bicking
(1964).
While the emphasis of these references and of this dissertation is
on statistical method, mention should be made of the difficulties inherent in the sampling process itself.
Considerable study, effort and
ingenuity is often required to correctly implement a mathematical model
in a real-world. situation.
Bicking
(1965) discussed the general problem
4
of setting up a sampling procedure; Tanner (1965) studied the error
introduced by d:isproportionate sampling of secondary units from primary
unit.s,
The techniques of variance component estir:e.tion used in this
di.ssertation parallel c1osc:ly those of Crump (1954) and Prairie (1962)
in earlier di.ssertations.
Crump considered optimal designs for estimat-
ine; the parameters of a tW'o-stage rwsced model, and Prairie extended
Crump's ,Tork t,o a three-stage nested model.
The variance component
model used is Eisenhart's (l)1j.7) I·lodel II; the variance components are
estimated by equating computed sums of squares to their expectations
and solving for the estimators.
Chapter 3 of this dissertation pre-
sents corresponding results for the present model.
Other estimation procedures for random models have been studied by
several authors.
Attention has focused, however, mostly on cross-
classification designs rather than nested designs, since in the traditional study of fixed-effects models several different estimation
procedures are used for cross-classification designs, depending upon
assumptions regarding interaction.
Bush and Anderson (1963) summarized
this development and considered three procedures for a two-way classification.
Anderson (1961) discussed briefly the problem of best estimation
procedure for nested designs.
lIe has presented four or five alterna-
tive estimation procedures in his classroom teaching, two of which
correspond to procedures
(1)
and (2) in Chapter
when extended to include compositing.
4 of this dissertation
Procedure (3) of this disserta-
tion arises due to compositing and would have no application in the
usual type of nested design.
5
3. OPTIMAL DESIGNS FOR WITIITN-A COMPOSITING
3.1.
The
Tn thin-A
Compositing Model
The model of principal interest in this dissertation permits
compositing within first-stage (A) classes only.
It is assumed an
observation can be described by
Y
ij
::::::
m + A. +
l
nij
1
L:
n .. k=l B.lJok
lJ
i
::::::
1,2, ... ,a
j
::::::
1,2, ... ,r.
l
where m is a constant, and the Ai and Bijk effects are independent
.22
normal in distribution with means zero and varlances ~A and. ~B' respecThe experiment contains a total of N B-samples and R observa-
tively.
tions, where
N=
L: n. : : : L: L: n .. and
i
l
i j
lJ
R :::
L: r. (~N).
i
l
In this chapter optimal designs for the individual and joint esti.
matlon
0f
t h e parameters m,
2
~A'
2 and
~B
2 + ~B
2 will be studied.
~A
These
designs will turn out to be either balanced or nearly balanced in the
n.
J.
IS,
meaning that from each A-class are selected roughly the same
number of B-samples.
In this case the choice of estimation procedure
is essentially immaterial, as is shown in Chapters
4 and 5.
Hence the
traditional analysis of variance, or weighted means, estimation procedure will be used exclusively throughout this chapter, with the understanding, where designs with decidedly unequal n.-values emerge as
l
optimal, that
op~imality
in these cases may be partly a function of the
choice of estimation procedure.
Knowledge in general of the dependence
of optimality on the choice of estimation procedure is nebulous.
6
3,2.
Variances of Weighted Means Estimators
The derivation of the weighted means esti.mators and their variances
is contained in the more general theoretical framework of Chapter
will not be repeated here.
In Chapter
4 and
4 the variances of the estimators
are calculated as follows:
2
Z 2
O"B
2 ( -n..
~) + O"A
N
"
Var (m)
l~
4
_.
2uB
R-a
_.
20"~
"2 +cr
"2 )
Var (0"
A B
-
4
22
20"A (KAA ) + 4O"AO"B (KAB ) +
where
;;::
K
AA
K
AB
22
(KAA ) + 4O"AO"B (KAB ) + 2 O"B4 (~l)
~f L', n.2 - 2N
~
Z
rr?
~
20"~
(KB2 ) ,
+ (Z n.2)2
1
(~ - Z n2 )2
i
N
;;::
if
L n.2
1
l~l
:=
rf(a-l)(R-l)
22
(R-a)( ~ - Z n.)
1
~2
--
rf(a-l)
N(a-l) ]
+ (l)[l R-a
(if - Z n~)2
~ - Z n~
1
1
2
(3.5 )
7
The coefficients in these expressions bear a strong resemblance to the
corresponding ones in Crump's study
(1954) of two-stage nested designs.
It is noteworthy that these variances, while functions of a, N, R
and the no's, are not functions of the individual r o and no. values.
1
1
lJ
That is, the variances of these estimators depend only on the total
number of observations (R) and not on how they are apportioned among
the A-classes.
Also, these variances depend only on the total number
of B-samples within the i-th A-class (n.), not on how these B-samples
1
are composited into the r. observations within the i-th A-class.
1
Hence
we need not examine all possible "different" designs having the same
total cost, since designs with equal a, N, Rand n, values are equiva1
lent from the standpoint of variance, for the four estimators considered.
If each n. is equal to N/a, equations
1
(3.2), (3.4) and (3.5)
may
be simplified as follows:
Val' (In)
"'2
Val' (a- )
A
"'2 )
Val' ( ~A+aB
;::
::
;::
a-2
a-2
B
A
- - +-a
N
(3.6)
4
2a-A
--+ 4a-2a-2 [N(:-l)] + 2a-4 [
a-l
B
A B
4
2a-A
22
- - + 4:a-AaB [N(:-l)] +
a-l
2a-~
2
a (R-l)
]
(3.7)
if(a-l)(R-a)
2
[a
+ (N-a)2] •
rf(R-a)
if( a-l)
(3.8)
8
3.3.
Optimal Estimation of Parameters
(1954) and Prairie (1962) studied optimal designs for esti-
Crump
mating individual parameters in the two-stage and three-stage nested
cases, respectively.
Vlhile in their research the total sample size was
fixed, total cost wj.ll be assumed fixed here, disproportionality of
cost being the primary reason for using composited designs.
The cost
of an experiment will be represented as
C
=
where C ' C and C are the unit costs of an A-class, a B-sample, and
A B
M
a measurement, respectively.
For fixed C, compositing is likely to be
economically advantageous when C is large in comparison to C and C '
M
A
B
as will be illustrated.
It is obvious from (3.3) that Var
is as large as possible.
R
=
From
(3.9)
(~) is minimized when a=l and R
this means that
N
where R, of course, must be an integer.
(1954) and
This solution essentially agrees with those of Crump
Prairie
(1962) for the last-stage variance component.
However, since
we switched from fixed sample size with proportional costing to fixed
total cost with disproportionate costing, the remaining solutions do
not so agree.
A
For example, consider Var (m ) as given in
(
3.2 ) •
Crump's
problem was to minimize ~ n~ for fixed N = Ln., which is achieved when
l
a=N and n.=l, i
l
= 1,2, ... ,a.
l
No price is paid in his problem by
9
permitting a to be as large as N; however in the present context, each
unit increase in a (while holding Nand R constant) increases the total
cost by CA.
~
For fixed N and fixed a,
n
2
i
is minimized by an equal (or as
equal as possible) choice of the nils, i.~., n
2
Var (~)
=
i
; N/a.
In this case
2
(JA
a
(JB
+- ;
N
(3·11)
it may be verified that (3.11) is minimized when
N
a
a result whose utility for design purposes depends upon a good guess of
the ratio
p
= ~/(J~.
R was set equal to a in the derivation of (3.12),
as required in a within-A compositing model; however, if m were the
only parameter to be estimated and if across-A compositing were permitted, then R=l would be a more economical choice.
The problems of integer-valued solutions and of dependence on p
become far more complicated when seeking optimal designs for (J~.
fixed a, Nand R the coefficients K , K and ~l in
AB
M
by Crump to be minimized if the n. 's are equal.
l
For
(3.4) were shown
Even if each n. is
l
assumed equal to N/a, however, no success whatever has been realized in
using the cost restraint
(3.9) in connection with equation (3.7) to
obtain explicit solutions for a, Nand R.
The problem becomes still more complicated when seeking optimal
designs for (J~ + (J~, since
KTI2
in
(3.5)
is minimized for fixed a, Nand
10
2
R '-Then L: n. :::: N(H-R+l), which usually requires a drastically unequal
i
J.
choice of the n. 's (e.~., if a=lO, N=60 and R=20, then Z n~ = 2460
J.
~
-
..L.
J.
which is best satisfied when nl =49, n =3, ~=n4=···=nlO::::l).
2
this, lIe Y'ecall that
+ ( l ) [1 _
R-a
N( a-I)
L: n~
N2 -
To prove
2
]
J.
Differentiating with respect to L: n~ gives
J.
After setting (3.14) equal to zero, we obtain the solution
L:
n.2
J.
::::
N(N-R+l) .
KB2'
To verify that this solution minimizes
it is easily shown that
the second derivative evaluated at L: n~ = N(N-R+l) is positive.
J.
a choice of the nils which minimizes
K .
AB
KB2
Hence
is a poor choice for K and
AA
From examination of (3.5) this means that the optimal choice of
the n. 's, for fixed a, Nand R, will depend in addition on the size of
J.
p,
a complication not present in the preceding paragraph.
Faced with such difficulties regarding general solutions, the
sensible alternative approach is numerical investigation.
~1odern
com-
puters facilitate the rapid comparison of many different designs, which
helps greatly in isolating efficient designs for particular estimation
purposes.
11
3.4.
Numeri.cal Comparison of Equal'-Cost Designs
An IBM 1620 FORTRAN program was written to evaluate the four
variances given by (3.2), (3.3), (3.4.) and (3.5) for six values of
p
= ~~/~~, namely 1/4, 1/2, 1, 2, 4 and 8. For an initial study 32
designs for the joint estimation of the four parameters were selected,
each satisfying the equation
=
2a + N + 8R
240,
where it is assumed that (C
C , CM) = (2, I, 8) and that 240 units of
N B
funds are available.
These designs are enumerated in Table 3.1.
In
all cases the n. 's were chosen as equally as possible; it vas noted in
1.
Section 3.3 that such a choice is non-optimal only for the coefficient
4
A2
A2
of ~B in Var (~A + ~B)·
The accompanying increase in variance due to
this fact usually is more than offset by the decrease in the other two
coefficients, for which the choice is optimal.
These numerical comparisons were conducted with two objectives in
mind:
(1) To find optimal designs to estimate ~~ and ~~ + ~~,
(2) To find designs which are reasonably efficient for estimating
some or all of the parameters jointly, I-There efficiency is
based on the reciprocal of variance.
To achieve these objectives the efficiencies of each design, with
respect to the optimal design for a given estinmtor and a given p, are
tabulated in Table 3.1.
(p
= 1/4,
For example, consider the entry ".50" in the
E )-colurnn for design 1.
m
This indicates that when p
= 1/4
design 1 is half as efficient as the optimal design (design 4) for
Table 3.1. Efficiencies of 32 desie1jns for which 2a + N + 8R = 240, relative to the optirnaJ.. design for
and a given value of p~
£!
1
e = 1/4
e = 1/2 I e = 1
.
e=2
P=4
Design a N R
ni
E:a Em EA EA+B Em EA EA+B Em EA EA+B ! Em EA EA+B Em EA
--1- 21 """22 22 2,1(20)
--:04 .50 .02 --:941---:65 --:c54 ~ .79 .11 1.00: .91 ---:2S' 1.00 1.00 .57
2 18 52 193(16),2(2) .041 .87 .11 .16\ .97 .25 .23 1.00 .54 .36:1.00 .84 .59 .99 1.00
3 18 28 22 2(10),1(8) .171 .57 .10 .90' .69 .22 .97 .78 .45 .991 .83 .69 .96 .87 .85
4 15 82 16 6(7),5(8)
.0411.00 .32 .10 1.00 .62 .15 .95 .91 .25, .90 .99 .44 .87 .96
5 15 66 18 5(6),4(9)
.13 i .91 .45 .33 .94 .75 .45 .91 .97 .63 \ .88 1.00 .80 .86 .95
6 15 42 213(12),2(3) .25 .71 .32 .74 .79 .56 .88 .82 .79 .97 .82 .88 .95 .82 .89
7 15 26 23 2(11),1(4) .33 .52 .17 ·99 .62 .33 1.00 .69 .54 .95 .73 .70 .87 .76 .77
8 12 112 13 10(4),9(8) .041i .97 .66 .09 .90 .91 .13 .82 .99 .21 .75 .90 .37 .71 .80
9 12 88 16 8(4),7(8)
.171 .90 .83 .35 .86 .99 .48 .79 1.00 .64 .74 .89 .74 .70 .79
10 12 72 18 6(12)
.251 .84 .74 .54 .82 .91 .68 .78 .95 .81 .73 .87 .82 .70 .78
11 12 48 21 4(12)
.38 .70 .51 .83 1 .73 .71 .94 .72 .82 .95 .70 .80 .85 .69 .75
12 12 24242(12)
.50 .46 .21 .98, .55 .37 .95 .60 .54 .87 .63 .64 .76 .65 .67
I
I
10 132 11 14(2),13(8) .04 .89 .87 .081 .79 .97 .12 .70 .91
10 108 14 11(8),10(2) .17 .85 1.00 .32! .77 1.00 .43 .69 ·91
10 8417 9(4),8(6)
.29 .78 .88 .561 .74 .92 .69, .67 .86
10 6020 6(10)
.42 .70 .69 .80 .69 .79 .90 .65 .79
10 44 22 5(4),4(6)
.50 .60 .51 .94 .•62 .65 .98 .61 .71
10 28243(8),2(2)
.58, .47 .30 1000! .53 .45 .95 .55 .57
8 152 9 19(8)
.041 .77 ·96 .08\ .66 .89 .11 .57 .76
8 112 14 14(8)
.251 .72 .97 .45 .64 .87 .56 .56 .74
8 96 16 12(8)
.33 .70 .91 .58 1 .63 .83 .69 .56 .73
8 64 20 8(8)
.50 .62 .71 .84: .59 .72 .89 .54 .67
8 32 24 4(8)
.67 .46 .38 .991 .49 .49 .92 .48 .54
5 182 6 37(2),36(3) .04 ~52 .76 .071 .43 .59 .10 .37 .46
5 150 10 30(5)
.21 .51 .75 .34! .43 .58 .41 .37 .46
5 110 15 22(5)
~42 .49 .69 .63! .42 .55 .64 .36 .45
5 70 20 14(5)
.63 .45 .58 .85 .40 .50 ·77 .35 .43
5 30 25 6(5)
.83 .35 .34 .90i .34 .37 .73 .32 .36
2 212 3 106(2)
.04 .22 .23 .071 .18 .16 .09 .15 .12
2 156 10 78(2)
.33 .22 .22 .37' .18 .16 .28 .15 .12
2 92 18 46(2)
.67 .21 .21 .53! .18 .15 .32 .15 .12
....lL. 2...@ 26 14(2)
1.00 .18 ~....:21:\ .16 -:12....42. .14 .11
= [Var (~) for optimal design among those considered] / [Var (m)
variance components.
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
!!;;;:
£/1(4) indicates 1,1,1,1; etc.
.20 .63 .77
.58 .63 .77
.78 .62 .75
.88 .61 .71
.89 .58 .67
.82 .55 .59
.18' .51 .62
.63 .51 .61
.72 .51 .. 60
.80 .50 .58
.76 .47 .52
.16 .32 .36
.43 .32 .36
.54 .32 .36
.57 .32 .35
.52 .30 .32
.10 .13 .09
.17 .13 .09
.18 .13 .09
.16 -:12....:.22
for tabulated
a given estimator
EA+B
e=8
Em
EA EA+B
'l":OO 1.00 """"Jf5' 1.00
.81 .94 1.00
.90 .84 .87
.65 .80 .88
.85 1 .80 .87
.87 1 .78 .84
·791 .73 .76
.541I .65 .71
•72 1 .65 .70
.74, .65 .70
.74( .64 .69
.68, .62 .65
.33 .60 .67 .471 .55 .59
.64 .59 .67 .61 1 .55 .58
.73 .59 .66 .63,1 .54 .58
.75 .59 .64 .63j .54 .57
.74\ .57 .62 .61\ .53 .56
.68 .55 .58 .58/ .52 .54
.3 0 ; .48 .53 .39 i .44 .46
.59: .48 .52 .501 .44 .46
.621' .48 .52 '51, .44 .46
.63 .47 .51 .5°\ .44 .45
.59! .46 .48 .48 .43 .44
.23 .30 .31 .25 .28 .26
.37 .30 .31 .301 .28 .26
.39 .30 .30 .30 i .27 .26
.39 .30 .30 .30 I' .27 .26
.37 .29 .29 .29 .27 .26
.091 .12 .08 .07' .11 .07
.11 1\ .12 .08 .08 .11 .07
.11 .12 .08 .08 .11 .07
.10 .12 -..:.21 .08 .11...E1.
design]; similarly for the
.90
.86
.75
.81
.80
.74
.61
.66
.67
.66
.63
.51
.55
.55
.55
.54
.52
.41
.44
.44
.43
.42
.24
.25
.25
.25
.25
.06
.06
.06
.06
~
f\)
13
estimating m.
When p =
4 or 8, design 1 is itself the optimal design
for m.
The optimal designs of Section 3.3 for estimating ~~ and m do not
appear in Table 3.1, since it is now required that all parameters be
estimable in each design.
Hence a=2 rather than a=l is the optimal
design for ~~, when both ~i and ~i + ~~ must be estimable.
must exceed a in the optimal design for m if
Also, R
~~, ~~ and ~~ + ~~ are to
If the optimal design of Section 3.3 (R=a) is used to
be estimable.
estimate m, the individual variance components cannot be estimated; how2 +
ever N~A
a~B
from Table
4.2 and equation (4.23) of Chapter 4.
2 is estimable from SSA.
This fact can be readily verified
In order to estimate ~~ efficiently, the primary objective must be
to have a large number of A-classes, with enough measurements within
A-classes to provide an estimate of
2
~B'
It is helpful here to think in
terms of degrees of freedom; the model of this chapter provides a-I and
R-a degrees of freedom for the estimation of
2
~A
and
2
~B'
respectively.
Hence a-I should be large and consequently (from the cost restraint)
R-a fairly small.
to
2
~B
As p increases, ~~ becomes more important relative
as a source of variation,
SO
that intuitively a-I should become
larger relative to R-a.
Table
2
~A'
3.2 lists the degrees of freedom in the optimal designs for
It is clear that a-I is always large, the maximum possible value
being 20 in this example, and that as p increases a-I increases at the
expense of R-a.
14
Table 3,2,
Degrees of freedom in the optimal designs for
Design
a-I
-
p
4, 8
17
1
5
2
14
3
9
1
11
4
9
4
1/4, 1/2
A
R-a
2
14
2
'J
Suppose it is desired to estimate 'J~ + 'J~ efficiently.
From
Table 3.1 it is clear that design 1 is the optimal design if p
very close to optimal if p < 1.
2:
1, and
However, when p < 1 design 1 is a very
poor design for both 'J~ and a-~ when considered individually.
It should
be observed here that the addition of one A-class to design 1 (while
holding Nand R constant) would make a
have high efficiency for
= N = R;
such a design would
a-~ + a-~, but would fail completely to distin-
guish class variation from sample variation.
If p «1, variation between samples would be the major component
of
'J~ + a-~; hence an optimal design for a-~ + a-~ would approach an opti-
mal design for a-i.
•
c~es
.p
o~
This fact is apparent when examining the efficien-
d
'
7 and 18 f or es t"~mat"~ng a-2 + a-2 and a-2 .
es~gns
A
B
B
Perhaps the main objective of experimentation is to estimate some
or all of the parameters jointly.
If this is the case, a sensible
criterion of optimality must be formulated to fit the objectives of
estimation.
Suppose, for example, that reasonably good precision is
desired for each of the four parameters, for p ranging from 1/4 to 8.
15
Inspection of Table 3.1 reveals that design 17 is the only design among
those considered which is at least 50 per cent efficient for all parameters over all values of p; while designs 16 and 22 are always at least
40 per cent efficient.
Hence these designs are good designs for the
stated objectives, and might be useful in an exploratory experiment in
a situation where little is known about the parameters.
Additional research on good criteria of optimality is needed,
particularly in the area of joint estimation.
One possibility for
future investigation is a criterion of equal coefficients of variability for the four estimators;
!.~.,
equal ratios of the estimated
standard deviation (of the estimator) to the estimator itself.
Hence
small variances would be required only for estimates of small parameters.
The coefficient of variability would be chosen as small as
experimental resources permitted.
Optimal designs, of course, would
still be functions of the value of p.
To gain more insight into the role of the cost assumptions, two
additional cost situations were examined.
The three studies of this
section will be referred to as follows:
(1)
(C , C , CM)
A B
=:
(2,1,8),
C = 240
(2)
(CA' CB' CM)
=:
(2, 1, 4),
C
(3)
(C '
A
cB'
CM) :::: (1,2,16), C
The results of studies (2) and
respectively.
50
=:
400.
(3) are tabulated in Tables 3.3 and 3.4,
Some of the results in study (2) were presented in an
earlier paper by Anderson (1965).
Table 3.3. Ef't'iciencies of 20 designs for which 2a +
and a given value of p~
Design a N R
n £!
i
--1- b 15 -::; 2(4),1(2)
2
3
4
5
6
7
8
9
10
11
5
5
5
4
4
4
4
3
3
3
3
3
2
2
2
2
2
2
16 6 4,3(4)
J2 7 3(2),2(3)
8 8 2(3),1(2)
22 56(2),5(2)
18 6 5(2),4(2)
14 7 4(2),3(2)
10 83(2),2(2)
28 410,9(2)
24 5 8(3)
20 67(2),6
J2
16 76)5(2)
13
J2 8 4(3)
14
34 3 17(2)
15
30 4 15(2)
16
26 5 13(2)
17
22 6 11(2)
18
18 7 9(2)
19
14 8 7(2)
~ ~ ~ ~ 5(2)
!!:IEm
= [Var
(~)
~
I Em
p = 1/4
EA
EA+B
I
N
+ 4R = 50, relative to the optimal design for a given estimator
e=
Em
e=1
1/2
EA
EAoIil
Em
EA
e•2
EA+B
Em
EA
e•
EA+B
Em
e=8
4
EA
EA+B
Em
EA
EA+B
--:I4 .75 ---:I4 .77 1--:&;, ·30 ·90 .95 .55 .99, I:OO ---:'Sl LOO,I:OO .99 I:OO I:OO 1.00 LOO
I
.
.14.96.112.112 LOO .71 .57 LOO .94 .75 .98 LOO .86 .94 LOO .90 .91 .92 .89
.29 .80 .37 .83 1 .87 .62 .95\ .91 .83 LOO .92 .91 .951 .89 .94 .90 .88 .88 .87
.43 .60 .22 .961 .70 .40 .98 .77 .60 .92 .&2 .74 .84 .&2 .&2 .81 .&2 .80 .80
I
I
!
0
.141.00
.29 .91
.43 .80
.57.66
.14 .91
.29 .87
.43 .81
.57, .74
.71' .65
.14 .70
.29 .69
.43 .66
.57 .64
.71 .60
.86 .55
LOO .48
.79 .32\ .961.00
.76 .62 .90 .94
.61 .87 .&2 .80
.41 1.00 I .71 .60
LOO .271 .81 .96
.96 .511.79 .93
.86 .721.75 .86
.72 .89' .71 .77
.55 .98 .66 .64
.76 .24 .59 .60
.74 .43 .58 .58
.69 .59 .57 .56
.64 .72 .56 .54
.57 .80 .54 .50
.48 .84 .51 .45
.36 .79 .47 .38
.44
.76
.97
LOO
.371
.63 i
.80!
.911
.92',
.30
.48 1
.59!
.651
.681
.68 'I
.63
for optimal design among those considered]
.89
.86
.81
.74
·72
.71
.69
.67
.64
.50
.50
.49
.49
.48
.46
.44
J.OO
.95
.86
.71
.80
.78
.75
.70
.63
.45
.44
.43
.112
.41
.38
.35
1 [Var (~)
.60
.85
.94
.90
.48
.67
.75
.78
.76
.34
.43
.47
.49
.49
.48
.44
.84
.&2
.79
.74
.89
.86
.81
.72
.65
.64
.65
.65
.64 .62
.63 .60
.61 .57
.45 .34
.45 .34
.44 .34
.44 .33
.44 .32
.43 .31
.1l2 .30
.7 : .78
.80! .77
.81: .75
.771.72
.52! .60
.60 I .60
.62i .59
.61; .58
.60 i .58
.31i .40
.331 .40
.341 .40
.34[.40
.34 .40
·33 .40
.32 .39
.81
.80
.77
.72
.57
.56
.55
.54
.53
.29
.29
.29
.29
.28
.28
.27
.71
.74
.73
.69
.50
·52
.52,
.52,
.51'
.27
.27
.27/
.271'
.27
.271
.261
.75
.74
.73
.71
.57
·57
.56
.56
.56
.38
.38
.38
.38
.38
.38
.38
.72
.71
.70
.67
.49
.49
.48
.48
.47
.25
.25
.25
.25
.25
.24
.24
.69
.69
.68
.66
.47
.48
.47
.47
.47
.24
.24
.24
.24
.24
.24
.24
for tabulated design]; similarly for the
variance components.
£/1 (4) indicates 1,1,1,1; etc.
I-'
0\
Table 3.4.
Efficiencies of 28 designs for which a + 2N + 16R .. 400, relative to the optimal design for a given estimator
and a given value of p!/
Design a
N
R
b/
ni -
--1- 2022 21 2(2),1(18)
2 18 39 193(3),2(15)
3 18 23 212(5),1(13)
4 16 56 174(8),3(8)
5 16 40193(8),2(8)
6 16 24212(8),1(8)
7 14 73 15 6(3),5(11)
8 14 49184(7),3(7)
9 14 25212(11),1(3)
10 12 9013 8(6),7(6)
11 12 66 16 6(6),5(6)
12 12 50 18 5(2),4(10)
13 12 26213(2),2(10)
14 10 107 11 11(7),10(3)
15 10 75 15 8(5),7(5)
16 10 51 18 6,5(9)
17 10 27213(7),2(3)
18
8 124 916(4),15(4)
19
8 92 13 12(4),11(4)
20
8 60178(4),7(4)
21
8 2821 4(4),3(4)
22
6 141 7 24(3),23(3)
23
6 10112 17(5),16
24
6 69 16 12(3),11(3)
25
6 2921 5(5),4
26
2 175 3 88,87
2 103 12 52,51
27
~
22 12,11
-=. ..::
!/Em = [Var (~)
,
p
= 1/4
~ I Em EA
.05i"""'34 .CY2
.051..• 79 .07
.151 .54 .07
I
.051' .94 .18
.15,.77.22
.251 .54 .12
.05 11.00 .34
I
P
EA+B
Em
.99 -:b9
.251 .92
.99 . . 66
.15,1 ·99
.53j .86
.991 .65
.1211.00
.201.82.~ .521 .87
.351 .54 .181.00 .64
.05i .99 .57 .1O! .94
.20:.88 .70 .43: .87
.30' .77 .56 .661 .80
.451 .53 .25 .99: .61
.051.92 .81 .091 .84
.251.82 .89 .471 .78
.401 .71 .66 .76! .71
.55i .50 .32 .991 .56
.051.80 .97 .091 .70
.25 .75 1.00 .431 .68
.45 .66 .77 . 75 1 .62
.65 1 .47 .37 .97: .5 0
I
.05 .65 .94 .081 .55
.30 .61 .90 .47 1 .53
.50 .56 .76 .741 .51
.75 .41 .40 .941 .42
.05 .24 .26 .07 .19
.50 .24 .24 .46 .19
1.00 .19 .16 .501 .17
'1
= 1/2
P
EA EA+B
.05
.18
.16
.39
.44
.26
.65
.70
.36
.88
.94
.80
.44
1.00
.99
.82
.49
.96
.94
.81
.50
.79
.76
.69
.46
.18
.18
.14
Em
1.00 --:B2
.33 .99
.99 .77
.211.00
.65 .91
.99,.73
. 16 .95
.65 .88
.991 .71
.14 .86
.54 .82
.78 .78
.97, .66
.13; .75
.58 .72
.84 .68
.94 .58
.121 .61
.53: .60
.81j .57
.89, .50
.12! .47
.54: .46
.751 .45
.801 .40
.09i .16
.31) .16
.291 .15
1'
I
=1
EA EA+B
p
Em
=2
EA
I
EA+B i Em
p
=4
EA EA+B I Em
.131.00i .92 .32 1.00i~ --:bl
.40 .49\1.00 .71 .7211.00 .94
.34 .97 .83 .59 .941 .87 .80
I ,
.70 .33i .96 .94 .541 .93 1.00
.72 .801 .90 .91 .911 .89 .96
.48 .951 .78 .69 .89 .80 .80
1
.92 .27/ .89 1.00 .461 .84 .95
.90 .801 .84 .95 .87i .82 .91
.57 .93: .74 .72 .85! .75 .78
1.00 .23j .78 .94 .401 .74 .84
.99 .69: .76 .92 .78, .72 .82
.90 .861 .74 .87 .84\ .71 .80
.61 .891 .67 .70 .79! .67 .71
.95 .21'1· .67 .82 .3 6 1 .62 ·71
.92 .69 .65 .80 .71 .61 .70
.82 .85 . .64 .75 .76 1 .60 .67
.60 .821 .58 .63 .701 .57 .61
.81 .20 .•54 .67 .32! .50 .56
.79 .611 .54 .65 .59i .50 .55
.73 .76 .52 .62 .64 1 .49 .54
.55 .73' .48 .53 .59', .47 .50
:
.62 .18; .41 .49 .27 .38 .41
.60 .551 .41 .48 .471 .38 .40
.57 .64: .40 .47 .491 .37 .40
.46 .611 .38 .42 .461 .36 .37
.13 .101 .14 .10 .09: .13 .08
.13 .18 ' .14 .10 .11i .13 .08
.11 .1711 .13 .09 .111 .12 .08
for optimal design among those considered] / [Var
(~)
I
p
=8
EA
1.0oI:OO --:Ni
.90' .981.00
.91; .88 .87
.76 .90 .96
.92 .87 .92
.84 .80 .81
.66 .81 .87
.85 .79 .84
.79 .74 .76
.58 .70 .75
.75 .69 .74
.77 .69 .73
.71 .66 .68
·50 .59 .62
.65 .58 .61
.65 .58 .60
.60 ·55 ·57
.41 . .47 .49
.52 .47 .48
.53: .47 .48
.49 .45 .45
'
.32 .35 .35
.39 .35 .35
.39 .35 .35
.37 .35 .33
.08 .12 .07
.08; .12 .07
.081 .12 .07
EA+B
1.00
.96
.88
.85
.88
.80
.75
.80
.74
.65
.70
.70
.66
.54
.59
.58
.55
.43
.46
.46
.44
.32
.34
.33
·32
.07
.07
.07
for tabulated design]; similarly for the
variance components.
£/1(4) indicates 1,1,1,1; etc.
I-'
~
18
For one design in each of' the three studies, the variances of the
estimators are tabul.ated in Appendix Table 9.1.
The choice of design
in each study maximizes the minimum of' the 19 efficiencies, among all
designs in each study.
Hence, by utilizing Tables 3.1, 3.3 and 3.4,
the variances of the estimators can be reproduced f'or any design, with
a minimum of rounding errors.
The question of' interest here :is whether or not the optimal designs
for the various parameters vary appreciably with the cost situation.
It was decided to examine closely the values (for the optimal designs)
of a-l and R-a, the degrees of freedom for estimating rr2 and rr2 , respecA
B
tively.
Table
3.5 lists the values of a-l and R-a for the optimal de-
signs of Tables 3.1, 3.3 and 3.4, with the maximum values of a-l and
R-a in each study included as reference points.
In addition, the values
of a-l and R-a are listed for those designs where the minimum of the
efficiencies Em' E ,
A
~
and E + is maximized.
AB
Such designs would be
reasonable choices for estimating all of the parameters jointly.
Cost appears to have little influence on the optimal designs for
rr2 and m.
B
of cost.
p.
2
For estimating rr , a-l
B
For estimating m, R-a
=1
= 1,
and R-a is maximized, regardless
while the choice of a-l depends on
2
If p is large a-I is maximized; however, as p decreases rr becomes
A
2
less important relative to rr as a source of variation.
B
Hence a-l
decreases and N, the total number of B-samples, increases.
p
= 1/4,
When
a-l is equal to approximately two-thirds of its maximum value
in all three cost situations.
19
Table 3.5.
Values of a-1 and R-a in the optimal designs of cost
studies (1), (2) and (3)
Study (2)
Study (3)
Designs with
Designs with
2a+N+8R==240
2a+N+4R"'50
Designs with
a -t2N+16R=400
study (1)
p
Maximums over
all designs
Optimum for
er~
Optimum for m
1/4
1/2
1
2
4
8
Optimum for
er~
1/4
1/2
1
2
4
8
Optimum for
eri +er~
1/4
1/2
1
2
4
8
Maximization
of the minimum
of Em' EA, ~,
and EA+B
1/4
1/2
1
2
4
8
a-1
R-a
a-1
R-a
a-1
R-a
20
24
5
'7
19
20
1
24
1
7
1
20
14
14
17
17
20
20
1
1
1
1
1
1
3
4
4
5
1
1
1
1
1.
13
13
15
17
17
19
1
1
1
1
1
1
9
9
11
14
17
17
4
4
4
3
1
1
7
9
11
13
15
17
5
1
1
1
1
1
9
14
20
20
20
20
14
13
19
19
19
19
19
7
1
1
1
1
1
7
9
9
9
9
9
12
12
14
14
14
14
5
5
9
9
9
9
10
10
11
11
11
11
8
1
1
1
1
5
5
2
3
3
4
4
5
3
1.
1
1
1.
1.
1
1
4
4
3
4
2
5
1.
5
5
1
1
2
2
2
3
5
3
3
4
4
5
4
4
20
The optimal designs for ~~ and ~~ + ~~ exhibit more variation with
cost.
As indicated in Section
3.3, it would be much more difficult to
predict optimal designs for estimating
a numerical study.
~~ and ~~ + ~~ in the absence of
However, the general comments regarding study (1)
still apply to studies (2) and
(3) as follows:
(1) As p decreases, the optimal choice for R-a increases at the
222
expense of a-l, for estimating both ~A and ~A + ~B'
In study (3) it
appears from Table 3.5 that drastic changes occur at p
Table 3.4 shows that design 18 (a-l
cient for estimating
2
~A'
= 7,
R-a
= 1)
= 1/4;
however,
is 97 per cent effi-
.
)
and that design 1 (a-l
= 19, R-a = 1 is
99 per cent efficient for estimating ~~ + ~~.
Hence the "drastic"
changes reflect only flat variance profiles at p
= 1/4.
(2) Designs may be very efficient for estirr~ting ~~ + ~~ while
poor for estimating ~~ and ~~ individually, particularly if p < 1This is illustrated by design 1 in each of the three cost studies.
In order to have reasonably good estimates of all four parameters,
it is obvious from Table 3.5 that more degrees of freedom are needed
for B than for A (around 50 per cent more, in general).
is rather uniform over all cost situations
0
This property
As would be expected, the
proportion of degrees of freedom for B relative to A decreases as p increases.
21
4.
PRECISIONS OF THREE ESTH1ATION PROCEDURES FOR WITEIN-A COI·jPOSITING
4,10
Estimation Procedures to Be Studied
Frequently one must estimate the variance components from a
decidedly ".mbalanced design"
While this rnay be due to poor planning,
it also may be due to difficulties inherent in the physical situation.
In either case, it is no longer obvious that weighted means estimation
should necessarilJ<- be used; hence, a comparison of weighted means estimation with other reasonable estimation procedures might be profitable,
In this chapter estimators for m,
2
~A'
2 and
~B
2 +
~A
2
.11 be
~B Wl
derived using three different unbiased estima.tion procedures.
variances of these estimators 'rill also be derived,
The
'rhe procedures to
be studied :JJe as follows:
(1) Method of Weighted l/ieans.
Each observation is weighted in
proportion to the number of B-samples composited into it; i.£., each
B-sample is weighted equally.
SSB
=
1.
~-(~n<.Y".)
. n.
. lJ lJ
J.
1
J
2
(4.1 )
(4.2 )
This procedure corresponds to weighted least squares estimation in
fixed effects models ,where each observation i.s weighted in inverse
proportion to its variance.
SSB will be used in all three estimation
procedures, since it is distributed as a multiple of a chi-square
random variable, and also since it seems reasonable to weight each
22
B-sample equally for estimating
J~. Hence, the estimation procedures
will differ only in the choice of SSA.
(2) Method of Unweighted Class Means.
Each A-class is weighted
equally.
z (-l:.-..
Z
no.
s~ ==
i
n .. Y .. )2
lJ lJ
J
l
(3) Method of Unweighted Measurements.
weighted equally
0
= '" -l:.-.. ( L
L.J
i
4.1
Table
Each observation is
ri
Yo" )2
j
J.J
(4.4)
1 (L: L: Yo.)2
R i j lJ
summarizes the equivalence conditions for the three weighting
procedures.
4.1.
Table
Design conditions for equivalence of the weighting procedures
Equivalence
Requirement
Example
Identical class structures
n .. i
lJ
r
,
o
l
S
equal for all i
I
s equal, and n .. s
lJ
equal for fixed i
(1)
r . 's equal, and n. 0'8
l
lJ
equal for all i
23
For balanced designs, the three procedures are eqQivalent, and the
resultant estimators can be shown to be maximum likelihood estimators
adjusted f'or bias"
Very little is known about the small sample proper-
ties of' w.aximum likelihood estimators for unbalanced designs.
4.2.
Estimation of m
The respective estimators of m follow from the correction terms in
the sums of' squares for A:
A
:=
~
A
-
m
2
=
1
L: L: nijY
ij
N
i j
1
a
~
L;-1-
(4.6)
L: n .. Y..
n.l j
i
lJ lJ
L: L: Y..
i j
lJ
If each of these estimators is expanded in terms of random A and B
effects, it is easy to verify their variances to be as follows:
V(~) -
1
V(~rJ
c:.
1.
A
V(~)
:=
~
a
=
2]
(5A . n2i + N(5B
l
[ 2 L:
(4.8)
2
2
l]
2 [a(fA + (5B ~ ~
1
2
R
l
l
[ (fA
2 L: r.2 + 2 L: L: -1-]
O"B .
n
.
l
l
l
j
ij
(4.10 )
24
2
2
2 + 2
Estimation of (JA' (JB and (JA
(JB
4·3·
2
2
2
2
The estimators of (JA' erB' and erA + erB are derived by equating the
sums of squares for A and B to their expectations.
For calculating
these expectations, a notation will be adopted which will prove equally
convenient for calculating the variances of the sums of squares in
Section 4.4.
The sum of squares for A may be written
\.,
SSA
L
(L f .. Y. )2
(1
Y .0
i~ 000
oj
i
j
lJ lJ
i
j
)2
lJ lJ
(4.11)
)
'There the f .. I sand g .. I S may be defined as in Table 4.2 to yield the
lJ
lJ
three estimation procedures under consideration.
To ,;ri te SSA, SSB and
the covariance matrix V of the observations in a more convenient nota-
= 1,2, ... ,a)
tion, the r. x 1 vectors J., N., F. and G. (i
l
l
l
l
l
as follows:
J~
:L
=
( 1
N~
=
(nil
l
F~
=
(f
G~
=
(gil
l
l
il
1
1
n
i2
n
ir.
l
i2
f. )
lr.
l
gi2
gil'. )
l
f
Also, the R x 1 vector G is defined as follows:
G'
= (Gi
)
G~ ... G~) •
are defined
25
,
Values of' f' .. ' s and gij s f'or procedures (1), (2) and (3)
~J
Table 4.2.
Procedure
f' ..
(1)
n ..
-2:JL
gij
~J
~
-r;;
-yN
n
ij
~
(2)
ni
niF
-1-
(3 )
-1-
-yR
~
Using these def'initions the sum of' squares matrices A and B, where
SSA
= y'AY
and SSB
= y'BY,
may be written as
A
=
diag (FiF~) - GG'
B
=
diag (Bih where Bi
(4.12 )
=
diag (n
ij
) -
NiN~
n.~
The covariance matrix V may be written as
(4.14 )
The expectations, variances and covariances of' the sums of' squares
now can be calculated as
26
E(SSA)
=
Var(SSA)
tr (VA),
=2
E(SSB)
2
Var(SSB)
tr(VA) ,
Cov(SSA, SSB)
=
= tr
(VB)
=2
(4.16)
tr (VB)2
2 tr (VAVB) ,
where l'tr" denotes the trace of a matrix.
(4.17)
This result, from \fuittle
(1953) and others, was applied in a similar context by Prairie (1962).
Frequent use will be made of the trace properties tr (~+M2)
+ tr (~) and tr (~M2)
= tr
= tr
(v~)
(~!v!1.), where the indicated matrix opera-
tions can be performed.
From
(4.12), (4.13) and (4.14) the matrices VA and VB can be
written as
VA
=
diag (V.F.F~) - VGG'
VB
=
diag (V.B.) ,
~
~
:L
(4.18 )
~
~
where
(4.20 )
I(ri ) denotes the identity matrix of rank rio
(4.20) may be simplified as follows:
J.J~N.N~
J. J ' diag (n .. ) ~
i
~J
~
~
~
n.
~
~
;::
JiN~
The coefficients in
27
n.N~
-2:-1:. )
I(r.) - (diag (-1-)
l
n ..
lJ
n.l
J.N~
=
I(r.)l
l
l
n.l
Substituting these results into (4.20) yields
J.N~
V.B.
l
l
-
2
O"B [r(r.) - .2:..l:.]
(4.21 )
n.
l
J.
The expected values of SSA and SSB can now be derived, using the
result given in (4.15), as follows:
E(SSB)
tr (VB)
=
I: tr (ViB )
i
i
=
O"B I: [r.l
i
=
O"B ~
2
2
J.
=
2
[r.
J.
O"B (R - a)
1
n.l
1]
tr (J.N~)]
l
l
(4.22 )
28
E(SSA)
(VA)
= tr
= i~
=
tr
crA2 i"
6
[ViF.F: - V.G.G:]
~
~
~
~
~
[F'•J• J.F.
I
1']
- G.J.J.G.
~
~
~
~
~
~
From (4.22) an unbiased estimator of
(4.23 )
~
l
0i is
given by
(4.24)
and from (4.22) and (4.23) it is easily verified that unbiased estimators of
cri and cri -+
cr~ are given by
K
~ [SSA - (R-~ ) SSE]
=
where
K
l
=
K
=
2
l
S-K
S
[SSA + (_ _2) SSE] ,
R-a
~[(~f ..
i
j
~J
v 6~ l
6
i j n ij
)
(4.26 )
2
( f 2..
lJ
2
- g..
~J
) .
Table 4.3 specifies Sand K for the three estimation procedures of
2
interest; these values may be derived from the f .. 's and g .. 's of 'lb.blel~.2.
lJ
~J
29
Table 4.3.
Values of
Ki
and K for procedures (1), (2) and (3)
2
Procedure
(1)
N
_ 1
\~
N i~
2
a - 1
n.
1.
a - 1
1
2
R - -R L: r.
i
1.
4.4.
"'2
"'2
~A' ~B
Variances of
and
To find the variances of the estimators
"'2
~A
"'2
+
"'?
~A' ~
"'2
~B
and
"'2
~A
+
"'2
~B
it is
necessary to derive the variances of SSA and SSB, and the covariance
between SSA and SSB, using (4.16) and (4.17).
From (4.21), we see that
(V.B. )2
1. 1.
=
2
~B (V. B.) .
1. 1.
Hence using (4.16) and (4.22),
Var (SSB)
=
4
2~B (R-a) •
(4.28 )
30
Consider now the variance of SSA.
From (4.16) and (4.18) it
follows that
Var (SSA) ::: 2 tr (VA)2
(4.29)
::: 2[~ tr(V.F.F~-V.G.G~)2 + ~ ~ tr(V.G.Gk'VkGkG:)].
ill l
l
l
l
ifk
l
l
l
In order to simplify (4.29) the following identities will prove useful:
(4.30 )
(4·31)
These identities follow from the properties of traces and from the
symmetry of the matrix V..
l
Using (4.30) and (4.31), (4.29) may be re-
written in the simplified form
(4.32 )
Each term on the right-hand side of (4.32) will now be examined in
detail:
2
f .. 2
2 ( f ) 2 + 2 " 2.sl]
[
"
:::L.O"A~··
O"BL.
i
j lJ
j n ij
31
2
I: (F~V.G.)2 = 2::[erA2F~J.J~G. + er F~(diag (-..L))G.]2
.
l 1 1
.
l 1 1 l
n. .
l
B 1
1
l
lJ
~2 , fijgij]2
[ 2(
=LerAI:f
.. )( I:g .. ) + VBL>
i
j lJ j 1J
j nij
4
2
= erA I:[ (I: f .. ) (2::
i j 1J
j
2
g .. ) ]
1J
4
f. ·gi· 2
+ erB L( I: lJ J )
i j nij
2 2
+ 2er er 2::[ (2:: f .. ) (I: g .. ) ( L.
A B . . 1J . lJ .
1
J
J
f. ·gi·
lJ
n. j
J
J )]
l
I
I
2 2:: G.J.J.G.
(2:: G~v.G.)2 = [ erA
+ er2B I:. G.lI( diag (-..L))G.]2
.
1 1 l
.
l l 1 1
n. .
l
l
l
=
l
lJ
g2.. 2
2
4 [ ("
2
4
erA L: w g. . )] + v~ [",~
W
w 2.J.]
·
1
J'
B
lJ
. n ..
.
l
J
1J
2
+
2er~er~ [~ (~ gij)2][~ ~ :~~]
1
J
l
J
1J
Clearly Var(SSA) may be written as a linear combination of the parameters
er~, er~
and
er~er~;
the coefficients of the parameters are
2
f .. 2
2I:(L: -2:.J.)
. . n ..
1 J
lJ
4L:(I:
..
l
J
f. jg .. 2
l lJ)
n..
lJ
2
g .. 2
+ 2[L: I: 2.J.]
. . n ..
l
J
1J
32
2
4L[ (2: f .. ) (L
i
j
].J
f~J
l. )] - 8L.:[ (L f .) (L: g .. ) (L
j nij
i j iJ j l.J j
+
4[~
f .. g ..
l.J l.J)]
nij
2
(j2:
l.
gij)2][~ ~ :~~]
l. J
•
l.J
Replacement of the f .. 's and g .. 's as indicated in Table
l.J
l.J
produce Var (SSA) for the three estimation procedures.
2
Var (SRA_)
--"1..
=
20-4
[L: n~
Ai].
L;
i
4.2
will
n~
l.
--:;=--- +
N
(4·34)
Var (SSA )
3
4
r?
2 L
i
l.
[L r. - ~~- +
2
= 20-A i l .
R
Last of all consider the covariance between SSA and SSB.
(4.17)
and using
(4.18), (4.19)
and
(4.20)
it follows that
From
33
COY
(SSA, SSB)
==
2 tr (VBVA)
==
2 L tr (V.B.V.F.F~ - V.B.V.G.G~)
i
~
~
~
~
~
~
= 2 iL F~V.B.V.F.
- 2
~ ~ ~ ~ ~
1
~
~
~
(4.36 )
~ G~V.B.V.G.
i
~ ~ ~ ~ ~
Using (4.14) and (4.21),
F~J.N~
= eT2 [Fi' B
V.F.
J,. ~
~ ~ ~
n
2
1
i
(1- Z f .) N~];
ni j i J
==
eT [F~
B ~
==
2
+ eT2 [diag ( l ) • F. 1
eTA [J.~ J~F.]
~ ~
B
n ..
~
~J
==
2
.. ) J.l + eT2 [diag ( l )
eTA [(Zf
B
nij
j ~J
~
F,l
~
Hence
F-I1 V-I Bi VJ.' FJ.'
. . ....
4
== VB
iT
[F.,,I
....
•
dl' ag (....L) .
~J
G.', a
If F ' is replaced w-lth
..
i
n, .
~
1<' ~ .
1
(}.;....
n.
f
<;
L.".
.
J. J
J.J
) 1\
liT r
;:I •
(....L).
• • ulag
1
n. .
J.J
F• ]
~
. '1 ar argument 1 eads t 0 th e res ult .
S~IDl
(4·38 )
34
If
(4.37) and (4.38) are sunnned over i and substituted into (4.36),
it is clear that
2
4
f ij
Cov (SSA, SSB) = 2~ [~~
- ~
B i j nij
i
1
2
--- (~f )
ni
j ij
Replacement of the fij's and gij'S as indicated in Table
Cov (S~, SSB) = 0
(4.40)
=0
(4.41)
Cov (S~, SSB)
=
4
2~B
{R
--- ~ [(--R
i
ri
The variances of the estimators
calculated.
From
4.2 shows that
From
2
1
ri
}
- l)(~ --- - -)] .
j nij
ni
A2
eTA'
.....2
~B
""2
and ~A +
"2
~B
may now be
(4.24) and (4.28) it is immediate that
(4.25) and (4.26) it follows that
35
"2
Var (O-A)
K~
1
=-
~
[Var (SSA) ... (_C_)
R-a
2
Var (SSB)
(4,44)
K,..,
(R-~ )
- 2
~
"2
Var (o-A ... o-B)
1
=
Kl-K
K1,
K , Var (SSA) and
2
(~-K2)
R-a
COY
COY
(SSA, SSB)] ,
(SSA, SSB) are functions of the estima-
tion procedure, as specified by Table
and equations
2
~ [Var (SSA) ... ( R_a2 ) . Var (SSB)
... 2
where
(SSA, SSB)]
COY
4.3, equations (4.33) - (4.35),
(4.40) - (4.42), Substitution of these quantities into
(4.44) and (4.45) leads to the following results:
Procedure (1)
2
:<:
L: n, - 2N L: n~ +
Var
•
1.
J.
,1.
1.
(4.46)
[~
N
2]
L: rl"
o
1.
l
Procedure (2)
(4.48)
+ 2CT4 [(a-2)
B a (_1)2
a
L:.1:..- + ( 1 L: .1:..-)2
i n.2
a(a-1) ~ n.1
1
~
37
Procedure (3)
...
2
R 2: (R-2r. )(2:.l:....) ... (2: r i )(2: 2: .l:....)]
~
j
n.
.
.
i ' n ..
i
4~2 ~2
~J
~
J ~J
A B[
(R2 _ 2: r 2 )2
i
i
(4.50 )
(4.51 )
39
5.
NUMERICAL COMPARISON OF ESTIMATION PROCEDURES
FOR WITHIN-A COMPOSITING
5 .1.
"Decidedly Unbalanced" Designs
If only balanced or nearly balanced designs are considered, the
choice of estimation procedure is not critical; hence, as indicated in
Section
3.1,
one would probably use the weighted means procedure for
reasons of tradition and covenience.
When forced to deal with decidedly
unbalanced designs, however, the choice of procedure becomes important.
The intent of this chapter is to define "decidedly unbalanced" designs
more precisely, and to indicate when each estimation procedure is the
most efficient procedure to use.
From its usage here and in Section
4.1,
it is apparent that a
decidedly unbalanced design is one which leads to marked differences
in the efficiencies of various estimation procedures.
Certain types
of unbalanced designs are not as severe in this respect as might be
supposed at first glance; hence, they could be called nearly balanced
for present purposes.
Some general comments can be made here; refer-
ence is made to the variances derived in Chapter
equations
(4.8) - (4.10), (4.43)
and
4, as presented in
(4.46) - (4.51).
The property of invariance with respect to the individual r
nij values, noted in Section
procedure (2).
i
and
3.2 for procedure (1), holds also for
Hence once a, N, R and the ni's are fixed, the differ-'
ence between procedures (1) and (2) is completely specified and cannot
be increased by unbalanced choices of the ri's and nij's.
the ni's are equal, the procedures are equivalent.
If all of
40
On the other hand, the efficiency of procedure (3) varies with the
choice of the r. 's and n .. 's. For fixed a, N, R, ni's and rJ.. IS, the
J.
J.J
differences between procedure (3) and procedures (1) and (2) are not
completely specified, since the efficiency of (3) tends to decrease as
the nij's for fixed i become more unequal.
the nature of these differences further.
Varying the ri's confuses
More will be said regarding
procedure (3) in Section 5.2.
The discussion of the preceding paragraphs may be sunnnarized and
extended as follows:
(1) For fixed a, Nand R, a nearly equal choice of the nils
dictates that procedures (1) and (2) will be almost equivalent.
(2 )
""'2
....2
....2
The variances of (J"A and (J"A + (J"B in procedure (3
nearly equal case in
)
are, for the
(1) as well as in general, extremely
difficult to minimize with respect to the r. 's and n .. 's; it
J.
J.J
appears that the optimal choices are equal n .. 's for fixed i
J.J
and slightly unequal r. IS, as will be explained in Section
J.
5·2.
(3)
If the nils, ri's and nij's are chosen as in (1) and (2),
then all three procedures are roughly equivalent; hence, the
question of a best procedure becomes rather academic, as in
Section 3.1.
Hence designs with nearly equal ni-values will be classified as nearly
balanced for purposes of comparing estimation procedures.
The nils
will be noticeably unequal in all of the designs considered in this
chapter.
41
.. 's for Procedure . (3)
Optimal Choice of ri's and n~J
If a, N, R and the n. 's are fixed, the variances of the estimators
~
in procedures (1) and (2) are completely specified.
Many possibilities
remain for procedure (3), however; hence, it would be helpful to specify
a rule for determining the r. 's and n .. 's which would yield an effi~
~J
cient design in terms of the results obtained from procedure (3).
The
rule should be a simple one, if procedure (3) is to be used in practice.
Suppose first that in addition the ri's are fixed, so that for
each i the nij's must be chosen subject to the restriction
nil + ni2 + •.• + nir .
~
= ni ·
O"~
The equations
2
+ C12 O"B
V(~)
=
Cll
v(q)
=
4
C
4 + C 2 2 + C 0"B
220"AO"B
21 O"A
23
"'2 "'2
V(O"A+O"B)
=
4
C
4 + C 22+ C 0"B
21 0"A
220"AO"B
24
must be examined for dependence on the nij's, where the coefficients
are defined from equations (4.10), (4.50) and (4·51).
Cll and C do
21
not involve the nij's, while C and C are easily shown to be mini12
22
mized by equal nij ' s, for each 1.
The i-th term of C ' for example,
22
is proportional to
,
which is clearly minimized if the nij's are chosen as equal as possible.
Though not proved, it is suspected that C and C are also minimized
24
23
42
by equal nij's; no contradictions have been found, and it appears that
they would be rare if existent.
Hence the nij's will always be chosen
as equal as possible; i.e., In.j-n.kl < 1 for all j and k, for fixed i.
--
J.
J.
-
Next, consider the choice of the r. 'so
If the r. 's are equal
J.
(r.
J.
R
= -)
a
J.
procedure (3) is essentially dominated by procedure (2).
More
precisely, if an equal choice of the n .. 's for fixed i is possible
J.J
(i.e., if n./r. is integer-valued for every i), then procedures (2) and
--
J.
J.
(3) are equivalent; this fact follows from the variances of the esti-
= ni/r i
mators given in Chapter 4 if the substitutions nij
are made.
and r
i
=~
If an equal choice of the n .. 's for fixed i is not possible,
J.J
then it has already been noted that the variances for procedure (3)
probably increase, while those of procedure (2) are unaffected.
Hence there is no point in having the r. 's equal, since procedure
J.
(2) is then always at least as good as procedure (3),
of procedures (2) and (3) in the case nij
ly justified as follows.
= ni/r i
The equivalence
can also be intuitive-
The equality of the r.). 's in (3) gives each A-
class equal weighting, while the equality of the n.
,IS
J.J
within an A-class each B-sample is weighted equally.
insures that
Such a weighting
procedure is precisely procedure (2).
Another possibility is to set r
i
proportional to n ,
i
!.~.,
With the n .. 's equal for fixed i, this amounts to weighting
J.J
each B-sample in the experiment equally, a weighting which is precisely
procedure (1).
Hence here also nothing would be gained by considering
procedure (3).
With the preceding intuitive support, it was decided to choose the
ri's as the average of those of the two proposed methods, thereby
43
placing procedure (3) midway between procedures (1) and (2) in this
sense.
As an example, let a=6, N=lOO, R=15, and let the n.]. 's be 50,
20, 12, 8, 6 and 4.
Table 5.1 shows how the r.]. 's would be chosen in
this case.
5.3.
Choice of Designs for Numerical Comparisons
It was decided to select 25 designs for the numerical comparison
of procedures (1), (2) and (3).
For 20 designs, a=20; N, the number of
B-samples, will be variable with mean value 100.
Five of these designs
chosen at random contain 30 measurements each, 5 contain 35, 5 contain
40, and 5 contain 45.
For each of the remaining 5 designs, the choice
of (a,N,R) equals that of one of the designs in study (1) of Chapter 3;
these designs will be used for an additional study in Section 5.5.
The critical question is the choice of the n.]. 's, since they must
be noticeably unequal in order to meaningfully distinguish between procedures.
How may the n.]. 's be so chosen?
Owing to the difficulties in
defining what is meant by a typically <unbalanced choice, a Monte Carlo
technique was used to generate each n.]. as a realization of a discrete
Table 5.1.
Example of the choice of the r.]. i s in procedure (3)
N=lOO,
a=6,
n
I
i
R=15
s = 50, 20, 12, 8, 6, 4
Method
r
r.]. I s equal
2·5
2·5
2·5
r i ex: ni
7·5
3·0
Average
5
3
l
r
2
r
r
r
5
r6
2·5
2·5
2·5
1.8
1.2
0.9
0.6
2
2
2
1
3
4
44
random variable X with expected value 5 and distribution skewed to the
right.
The random variable used is defined in Table 5.2.
While X has
the disadvantage of perhaps deviating from reality in many applied situations, nevertheless it produces designs of the general type desired and
renders explicit the method of producing them.
Hence if a=20, N = L: n. will have mean value 100.
i
~
For the 5
designs where N is specified by a design from Chapter 3, the n.'s
~
chosen from X are scaled appropriately so that they sum to N.
Table 5.3 lists the 25 designs to be studied.
variances of nine estimators, those of m,
For each design the
<ri and <ri + <r~ for the three
procedures, were computed on an IBM 1620 and evaluated for p
1, 2, 4 and 8.
= 1/4,
1/2,
Note that ~ adds nothing to the present study since it
is the same for all procedures.
Hence with 25 designs, 3 parameters and
6 values of p, a total of 450 numerical comparisons of the estimation
procedures were made.
For each comparison the efficiency of each proce-
dure relative to the most efficient procedure was calculated; these
efficiencies are tabulated in Table 5.4.
estimation of m in design 1 for p = 1.
Consider, for example, the
V(~) is equal to .0908 <r~,
.0739 <r~ and .0768 <r~ for procedures (1), (2) and (3) respectively; the
corresponding efficiencies are .81, 1.00 and .96.
Use of relative
efficiencies in this way greatly facilitates the comparison of estimation procedures.
Ranking of relative efficiencies provides an additional reduction
of the relevant data; in the above example procedure (2) may be given a
rank of 1, procedure (3) a rank of 2, and procedure (1) a rank of 3.
a tie in efficiencies occurs, the two procedures are each assigned the
If
Table 5.2.
Distribution of random variable X used to generate the n
X
p(X)
1
.14
2
.17
3
.17
4
.14
5
.08
6
.06
7
.05
8
.04
9
.03
10
.02
11
.01
12
.01
20
.01
average of the two adjacent ranks.
i
IS
The ranks thus obtained are
averaged, over the 5 designs in each llR_groupll, and tabulated in
Table 5.5.
Appendix Table 9.2 contains the variances of the estimators for
procedure (3), for designs 1-25.
The corresponding variances for pro-
cedures (1) and (2) may be calculated from the relative efficiencies of
Table 5.4.
46
Table 5.3.
Unbalanced designs for comparison of procedures (1), (2)
and (3)
n
Design
a
R
N
r
r
i
i
I
s
s
7
2
7755333
2222211
2 2 2 2 2 111 1 1
1 1 1 11 1 1 1 1 1
15 11 10
3 3 2
7 6 4 4 433
2 2 2 2 2 1 1
3 3 3 2 2 2 2 2 2 1
1 1 1 1 1 1 1 1 1 1
100
18 16
3 3
966 5 553
2222211
3 222 222 111
1 1 1 1 1 1 1 1 1 1
30
105
20 17 15 10 6 5 4 3 3 3
4 3 3 222 1 1 1 1
3 3 2 2 222 111
1 1 111 1 1 1 1 1
20
30
115
19 17 10
3 3 2
9 776 5 4 4
2 2 2 2 2 1 1
443 3 3 3 3 2 1 1
1 1 1 1 1 1 1 1 1 1
6
20
35
73
12
4
6
2
5
2
4444433
2 2 2 222 2
3 3 3 3 332 2 1 1
222 1 1 1 1 1 1 1
7
20
35
92
9
3
9
3
9
3
876 4 444
2 2 2 2 222
44333 3 3 2 2 1
2 2 111 1 1 1 1 1
8
20
35
93
17 11
4 3
7
2
6665543
2 2 2 2 2 2 2
3 3_,3 3 3 2 2 2 1 1
2 2 1 1 1 1 1 1 1 1
9
20
35
93
17 13
4 3
9
3
875 4 444
222 2 2 2 2
3 3 2 2 2 2 1 1 1 1
211 111 111 1
10
20
35
100
18 17 10
4 4 3
8754333
2222222
3 333 2 2 111
111 1 1 1 1 1 1 1
11
20
40
76
19
5
6
3
6
3
5 5 443 3 3
2 2 222 2 2
3 3 2 2 2 2 1 1 1 1
2222211 1 1 1
12
20
40
87
19 12
5 4
7
3
6 5 4 4 4 4 3
2 2 2 2 2 2 2
3 3 3 2 2 2 1 1 1 1
2 2 2 2 1 1 1 111
13
20
40
88
14 9
4 3
8
3
8754444
3 3 2 2 2 2 2
3 3 3 2 2 2 2 2 1 1
2 2 2 2 1 1 1 1 1 1
1
20
30
73
2
20
30
89
3
20
30
4
20
5
9
3
9
3
9
2
3
(Table continued on next page)
47
Table 5 -3 (continued)
,
,
ni s
Design
a
R
N
14
20
40
101
18 15 15 12 8 lj, 3 3 3 3 3 3 2 2 2 1 1 111
5 4 4 3 3 2 2 2 2 2 2 1 1 1 1 1 1 III
15
20
40
126
20 19 17 10 7 7 7 6 5 4 4 4 4 3 222 I I I
4 4 4 3 2 2 2 2 2 2 222 1 1 1 1 1 1 1
16
20
45
91
14 8 8 8 6 6 5 544 443 322 2 I I I
5 3 3 3 332 222 2 2 2 2 2 2 ~'l 1 1
17
20
45
104
17 10 9 8 766 6 5 4 4 4 432 222 2 1
5 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1
18
20
45
107
19 17 14 7665444 3 3 3 3 2 2 2 111
5 5 4 3 2 2 2 2 2 2 2 22 2 2 2 1 1 1 1
19
20
45
112
13 13 10 9 8 7 6 6 6 6 6 5 443 2 1 1 1 1
4 4 3 3 3 3 3 2 2 2 2 2 2 2 2 2 1 1 1 1
20
20
45
152
20 19 12 9 998 7 7 7 7 776 5 4 3 3 2 1
4 4 3 3 2 2 2 2 2 2 2 2 222 2 2 221
21
15
21
42
22
12
18
72
18 14 13 9 3 3 3 2 2 2 2 1
3 3 2 211 111 1 1 1
23
10
17
84
20 18 11 11 6 5 5 4 3 1
3 3 2 2 2 1 1 111
24
8
20
64
24 12 11 6 4 42 1
5 3 3 2 2 2 2 1
25
5
15
110
ri s
5 4 4 4443332 21111
2 2 2 2221111 11111
41 23 17 17 J2
5 3 3 2 2
48
Table 5.4.
p
Relative efficiencies of procedures (1), (2) and (3) for
designs 1-25
Procedure 1
Procedure 2
Procedure 3
2
2 2
2
2 2
2
2 2
m
m
m
(JA (JA+o'B
(JA (JA+o'B
(JA (JA+o'B
Design 1
1/4
1/2
1
2
4
8
1/4
1/2
1
2
4
8
.96 1.00
.89 1.00
.81 1.00
.88
·74
.69
·77
.67
·71
.88 1.00
.81 1.00
·90
·72
.67
.78
.64
·71
.63
.67
.89
·90
.89
.82
·75
·70
.87
.87
.83
.76
·70
.67
.44 LOa
.60
·98
.85
·99
LOa
.98 1.00
1.00 1.00 1.00
1.00 LOa 1.00
1.00
.67
·99
1.00
.82 LOa
.96 1.00 LOa
·91 1.00
·96
.87
·91
·93
.88 .88
.85
Design 2
·94
·59 1.00
1.00
·79 1.00
LOa
.94 1.00
1.00 1.00 LOa
1.00 1.00 LOa
LOa 1.00 LOa
1.00
.81
.96
·99
·92 1.00
.88
.96
.86
·90
.87
.85
·95
.96
.96
·93
.89
.87
1.00
.67
1.00
.90
LOa
·95
·90 LOa
.87
·92
.88
.85
.96
.99
·99
·95
·91
.88
LOa
LOa
l.00
.68
·98
·93
.89 1.00
.82
·94
.85
·79
.81
·77
·95
·97
.95
·90
.84
.81
·92
·52 1.00
1.00
·76 1.00
1.00
.94 1.00
1.00
1.00
1.00
1.00
1.00 1.00
1.00 1.00 1.00
1.00
·75
·99
·95
·92 1.00
.88
·95
.86
·90
.85
.87
·95
·97
·96
·93
.89
.87
.87
·96
1.00
Design 3
1/4
1/2
1
2
4
8
1/4
1/2
1
2
4
8
.84 LOa
·76 1.00
.68
.87
.61
·75
.65
·57
.60
·55
.81 1.00
·73 1.00
.61
.83
.68
·55
.58
·51
.49
·53
.82
.84
·79
·71
.64
.60
1.00
.80
.80
.74
.64
·57
·53
Design
.42
.89
1.00
.69
1.00
.90
1.00 LOa
1.00 1.00
1.00 1.00
.87
·96
LOa
LOa
LOa
.43 LOa
.66 1.00
.87 1.00
LOa l.00
1.00 1.00
1.00 1.00
4
1.00
1.00
1.00
1.00
Design 5
1/4
1/2
1
2
4
8
.87 1.00
.80 1.00
.89
·71
.66
·77
.69
.63
.61
.66
.86
.86
.82
·75
.69
.65
(Table continued on next page)
49
Table 5.4 (continued)
p
1/4
1/2
1
2
4
8
1/4
1/2
1
2
4
8
Procedure 1
2
2 2
m
(JA+(JB
(JA
.98 1.00
·90 1.00
.83
·99
.78
.89
.82
.76
.79
·74
·97 1.00
·93 1.00
.87
.98
.83
.90
.80
.84
.81
·79
·98
·97
·93
.87
.82
·79
·95
·96
·93
.88
.84
.81
Procedure 2
m
2
(JA
2+ 2
(JA (JB
Procedure 3
2
2 2
m
(JA
(JA+(JB
Design
.66
·98
1.00
.81
1.00
·97
1.00 1.00
1.00 1.00
1.00 1.00
6
l.00
1.00
1.00
1.00
1.00
1.00
1.00
.84
·92
·97
·93 1.00
·90
·95
.88
·91
.89
.87
·99
·99
·97
.94
·91
.89
Design
.67
.96
l.00
.83
1.00
·97
1.00 1.00
1.00 1.00
1.00 l.00
7
1.00
1.00
l.00
1.00
l.00
1.00
1.00
.89
.98
.96
l.00
·93
·90
·95
.88
·91
.89
.87
.98
·99
·97
.94
·91
.89
Design 8
1/4
1/2
1
2
4
8
1/4
1/2
1
2
4
8
1/4
1/2
1
2
4
8
·90 1.00
.83 1.00
·73 ·91
.68
.80
.65
·72
.68
.63
.86 1.00
·79 1.00
.87
·70
.78
.63
.60
.67
.62
.58
.86 1.00
·77 1.00
.66
.87
·72
·59
.63
·56
.54
·59
l.00
1.00
1.00
1.00
l.00
l.00
1.00
·79
·99
·95 1.00
·99
.98
·93 1.00
.89
.96
·93
.86
·90
·90
.88
.87
.85
.94
·93
.86
.78
·72
.68
.56
.94
1.00
·77
1.00
·94
1.00 l.00
1.00 1.00
1.00 1.00
·90
.88
.83
.74
.66
.62
Design
.86
.44
.66
.96
.86
1.00
l.00 l.00
l.00 1.00
l.00 l.00
.96
·95
.98
1.00
1.00
1.00
1.00
·73 l.00
1.00
·94 l.00
.95 1.00 l.00
.96
.89 1.00
.86
·91
·90
.84
.87
.87
·91
.89
.81
·70
.63
·59
Design 10
.48 1.00
·92
1.00
.74 1.00
1.00
·93 1.00
1.00 1.00 l.00
1.00 1.00 1.00
1.00 1.00 1.00
1.00
.74 1.00
·95 1.00
·97
1.00
.89
·97
.84
·90
·93
.81
.86
.85
.82
.82
·79
9
(Table continued on next page)
50
Table 5.4 (continued)
p
Procedure 1
2
2 2
m
(JA (JA+o'B
Procedure 2
2 2
2
m
(JA (JA+o'B
Procedure 3
2 2
2
m
(JA (JA+o'B
Design 11
1/4
1/2
1
2
4
8
.82 1.00
·73 1.00
.86
.64
.76
·57
.66
·54
.61
·52
·96
·91
.83
·73
.65
.61
.90
·52
·91
.74
·98
·92
1.00
.89
·97
1.00 1.00 1.00
1.00 1.00 1.00
1.00 1.00 1.00
1.00
·79 1.00
1.00
·97 1.00
1.00
1.00
·95
·90
·97
·99
.87
·92
·91
.88 .88
.85
Design 12
1/4
1/2
1
2
4
8
.83 1.00
.76 1.00
.66
.85
·59
·75
.65
·56
.60
.54
.94
·90
.83
·72
.64
.60
.88
.48 ·91
·98
·72
·92
.88
1.00
·97
1.00 1.00 1.00
1.00 1.00 1.00
1.00 1.00 1.00
1.00
·79 1.00
1.00
.98 1.00
·93 1.00 1.00
.87
.98
·95
.84
.89
·90
.82
.85
.85
Design 13
1/4
1/2
1
2
4
8
·92 1.00
.86 1.00
.78 ·91
.83
·72
.69
·75
.67
·71
·96
·94
.89
.81
.74
.71
.56
·96
·91
·76
·96
·99
1.00
·92
·99
1.00 1.00 1.00
1.00 1.00 1.00
1.00 1.00 1.00
1.00
.83 1.00
1.00
·97 1.00
·95 1.00 1.00
.98
.95
·90
.88
·91
·92
.88
.89
.87
Design 14
1/4
1/2
1
2
4
8
1/4
1/2
1
2
4
8
.83 1.00
·75 1.00
.64
.82
·70
·56
·58
·52
·50
·53
.83 1.00
·96
·77
.68
.82
.62
·73
.64
·58
.60
·57
·91
.87
·79
.66
.58
·53
·91
.87
.81
·71
.64
.60
.89
.85
·39
.66
.98
·90
1.00
.86
·97
1.00 1.00 1.00
1.00 1.00 1.00
1.00 1.00 1.00
1.00
·74 1.00
1.00
·99 1.00
1.00
1.00
·90
.83
.96
·92
.85
.85
·79
.80
.80
·76
Design
.85
.45
·70
·97
1.00
.88
1.00 1.00
1.00 1.00
1.00 1.00
.81 1.00
1.00
1.00 1.00 1.00
·93 1.00 1.00
.87
.94
·97
.88
.84
.89
.82
.85
.85
15
.94
·94
.98
1.00
1.00
1.00
(Table continued on next page)
51
Table 5.4 (continued)
p
Procedure 1
2
2+ 2
m
(JA
(JA (JB
Procedure 2
m
2
(JA
{+cr~
Procedure 3
2 2
m
{
(JA+crB
Design 16
1/4
1/2
1
2
4
8
1/4
1/2
1
2
4
8
·93 1.00
.88 1.00
.81
·93
.86
·75
.78
·72
.74
·70
·91 1.00
.85 1.00
·90
·77
.82
·72
·70
·75
.68
·72
·97
.95
·90
.84
·77
.74
.88
·90
·52
.96
.91
·72
1.00
.89
.95
1.00 1.00 1.00
1.00 1.00 1.00
1.00 1.00 1.00
·96
·94
.89
.80
Design
.61
·94
1.00
.83
1.00
·95
1.00 1.00
1.00 1.00
1.00 1.00
·75
·72
1.00
.81 1.00
1.00
·95 1.00
1.00
1.00
·97
1.00
·92
·98
·90
·94
·93
.88
·91
·90
17
.96
.96
1.00
1.00
1.00
1.00
.81 1.00
1.00
1.00
·97 1.00
1.00
1.00
·95
·92
·96
·97
·90
·93
·93
.89
·91
·91
Design 18
1/4
1/2
1
2
4
8
.83 1.00
.76
·97
.82
.66
·71
·59
.63
·56
.59
·54
.94
.89
.81
·70
.62
.58
.88
.49
.90
.74
·92
·99
1.00
.98
·90
1.00 1.00 1.00
1.00 1.00 1.00
1.00 1.00 1.00
.80 1.00
1.00
1.00 1.00 1.00
·92 1.00 1.00
.86
,96
.94
.88
.88
.83
.82
.85
.85
Design 19
1/4
1/2
1
2
4
8
·94 1.00
.89 1.00
.85
·93
.78 .87
.80
·75
·73
·75
.96
·94
·91
.87
·79
.75
.42
.87
.83
.64
.88
·93
1.00
.84
·93
1.00
.98 1.00
1.00 1.00 1.00
1.00 1.00 1.00
1.00
.78 1.00
1.00
.94 1.00
1.00
1.00
·99
1.00
1.00
·93
.94
.95
·90
.88
.91
·91
Design 20
1/4
1/2
1
2
4
8
.92 1.00
.86 1.00
.80
·90
.83
.76
.78
.74
.76
·73
.96
.94
·90
.83
.78
.76
.98
.63
·95
.86
1.00
.98
1.00
·96 1.00
1.00 1.00 1.00
1.00 1.00 1.00
1.00 1.00 1.00
1.00
.83 1.00
1.00
·99 1.00
.96 1.00 1.00
.98 ·97
.94
·93
·95
·95
.93
·92
·93
(Table continued on next page)
52
Table 5.4 (continued)
p
1/4
1/2
1
2
4
8
1/4
1/2
1
2
4
8
1/4
1/2
1
2
4
8
Procedure 1
2
2 2
m
O"A
1.00 1.00
·99 1.00
·95 1.00
.99
·90
.86
·91
.86
.84
.84 1.00
·77 1.00
.66
.84
.60
·71
.62
·57
.58
·55
.90 1.00
.84 1.00
.76
·90
.80
·71
.68
·73
.67
·70
O"A+o'B
Procedure 2
2 2
2
m
O"A
O"A+o'B
.96
·93
.89
.86
Design 21
.89
·57 1.00
.67
·97
·99
1.00
.82 1.00
1.00
.98 1.00
1.00 1.00 1.00
1.00 1.00 1.00
.81
.81
·77
.68
.61
·58
Design
.88
.45
·99
·71
1.00
·90
1.00 1.00
1.00 1.00
1.00 1.00
·93
·95
·90
·90
.86
·78
·73
·70
22
1.00
1.00
1.00
1.00
1.00
1.00
Design 23
·50 1.00
·92
1.00
·77 1.00
1.00
.95 1.00
1.00 1.00 1.00
1.00 LOa LOa
LOa
LOa
LOa
Procedure 3
2
2 2
m
O"A
O"A+O-B
.76
·97
·99
1.00
.83 1.00
.98
·93 1.00
.98
·94 1.00
.96
·92
·95
·91
·92
·93
1.00
.70
1.00
·95
·92 1.00
.86
·97
.83
.89
.81
.85
.85
1.00
·77
.98
.96
·91 1.00
.87
·94
.85
.89
.84
.87
.98
·99
·97
·93
.89
.86
.94
.97
·97
·93
.88
Design 24
1/4
1/2
1
2
4
8
1/4
1/2
1
2
4
8
.82 LOa
.96
·75
068
.82
.62
076
.68
·59
.64
.58
.87
.85
.84
.83
.83
.83
.96
·93
·91
·90
.89
.89
·94
.89
.82
074
.68
.64
.48 .88
.85
089
.96
·73
1.00
.89
095
1.00 1.00 1.00
1.00 1.00 1.00
1.00 l.00 1.00
·98
·96
·92
·90
.89
089
Design
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
l.00 1.00
25
l.00
1.00
1.00
1.00
1.00
1.00
1.00
.82 1.00
1.00 1.00 1.00
·95 1.00 1.00
.98
.99
·90
.87
.92
·93
.89
.85
.89
092
·90
.89
.89
.89
.88
.98
·95
·94
093
·92
.92
·99
·97
·95
·93
.93
·92
5,
Table 5.5.
Designs
R
Average ranks in relative efficiency of procedures (1), (2 )
and (,)
p
m
(1)
2 2 2
(J'A (J'A +erB
m
(2 )
2 2 2
(J'A (J'A+o'B
m
(, )
2 2 2
(J'A (J'A+o'B
LO
LO
2.6
,.0
,.0
,.0
,.0
,.0
,.0
,.0
,.0
,.0
2.2
L4
LO
1.0
LO
LO
,.0
,.0
2.,
L,
1.0
LO
LO
L2
L2
LO
LO
1.0
LO
1.6
2.0
2.0
2.0
2.0
2.0
2.0
1'.1
1'.7
2.0
2.0
2.0
1'.8
1'.8
2'.0
2.0
2.0
1/4
1/2
1
2
4
8
2.6 LO
,.0 1.0
,.0 2.4
,.0 ,.0
,.0 ,.0
,.0 ,.0
,.0
,.0
,.0
,.0
,.0
,.0
2.4
1.2
LO
LO
1.0
LO
,.0
,.0
2.6
L1
1.0
LO
L,
L4
L2
LO
LO
1.0
LO
_L8
2.0
2.0
2.0
2.0
2.0
2.0
1.0
1.9
2.0
2.0
1'.7
1'.6
1'.8
2'.0
2.0
2.0
11-15
1/4
1/2
40 21
4
8
2.8 LO
,.0 L2
,.0 ,.0
,.0 ,.0
,.0 ,.b
,.0 ,.0
2.,
,.0
,.0
,.0
,.0
,.0
2.2 ,.0
2.0 ,.0
LO 2.0
LO LO
LO LO
LO LO
2·7
2.0
2.0
LO
LO
LO
LO
LO
2.0
2.0
2.0
2.0
2.0
L8
1.0
2.0
2.0
2.0
LO
1.0
1.0
2.0
2.0
2-.0
16-20
1/4
1/2
1
2
4
8
2.6
,.0
,.0
,.0
,.0
,.0
1.0
1.2
2.6
,.0
,.0
,.0
2.,
2.6
,.0
,.0
3·0
,.0
2.4
L8
1.0
1.0
1.0
LO
,.0
,.0
2.4
L,
LO
1.0
2·7
2.4
1.8
Ll
1.0
l.0
1.0
1.2
2.0
2.0
21-25
1/4
1/2
1
2
4
8
2.6
2.8
,.0
,.0
,.0
,.0
L4
L6
2.6
2.8
,.0
,.0
2.8
2.9
,.0
,.0
,.0
,.0
2.0
1.8
1.0
1.0
1.0
1.0
2.6
2-;6
2.0
1.4
1.0
1.0
1.4
L5
1.,
l.0
LO
1.0
1.4
1.4
2.0
2.0
2.0
2.0
2.0
1.8
1.4
l.8
2.0
2.0
L8
1'.6
1.7
2.0
2.0
2.0
All
1/4
1/2
1
2
4
8
2·7
,.0
,.0
,.0
,.0
,.0
L1
l.2
2.6
,.0
,.0
,.0
2·7
2.9
,.0
,.0
,.0
,.0
2.2
L6
1.0
1.0
1.0
1.0
2·9
2.9
2.,
1.2
1.0
1.0
L8
1.7
1.5
1.0
1.0
1.0
L1
1.4
2.0
2.0
2.0
2.0
2.0
1.9
1.1
108
2.0
2.0
L5
1'.4
1'.5
2.0
2.0
2.0
1-5
6-10
1/4
1/2
,0 1
2
4
8
,5
45
2.8
,.0
,.0
,.0
,.0
,.0
2.0
1'.8
1.0
L7
2.b 2.0
2.0 2.0
1.0 LO
L2
L9
2.0
2.0
54
5.4. ResUlts of Numerical Comparisons
Some surprisingly uniform conclusions can be drawn from the evidence of Section 5.3, considering that a rather wide variety of
unbalanced designs was encompassed.
It will be assumed here that all
three parameters are to be estimated with reasonably good precision.
(1) It is clear from Table 5.5 that procedure (3) is never the
worst procedure.
Close examination reveals that in all 450 comparisons
not a single instance occurs of procedure (3) ranking third, nor is its
relative efficiency for any of the three parameters ever less than .67.
Neither of these properties holds for procedures (1) and (2); in fact,
each of these procedures ranks third at least once for each of the
three parameters.
Hence procedure (3) minimizes the maximum loss in
efficiency, and is recommended, particularly if nothing is known about
the value of p.
(2) The only instances where procedure (3) is consistently less
efficient than procedure (1) are for estimating (f~ when p is 1/4 or 1/2,
which amounts to 2 cases out of 18 possible cases.
Were it not for
these two estimation cases, procedure (1) would be eliminated from consideration.
(3) If P is greater than 1, procedure (2) is most efficient for all
of the parameters.
p > 1 indicates that
(f~ is the larger and presumably
more important of the two variance components, so that intuitively equal
weight should be given to each A-class.
(4) If P is less than 1, the choice of procedure is less obvious.
Procedure (1) has been noted optimal. for (f~; for m and(f~ + (f~ a choice
between procedures (2) and (3) is involved.
Here alone the value of R
55
seems to exert a noticeable effect, especially for estimating ~~ + ~~,
namely that as R increases procedure (3) becomes dominant over procedure
(2).
For estimating ~~ + ~~ this effect is noticeable when p equals
1/4, 1/2 and 1; for estimating m it is noticeable only when p equals
1/2.
(5) If p is less than 1 and good precision is desired for all parameters, procedures (1) and (3) are clearly preferable to procedure (2).
The smallest observed relative efficiencies are .73 for (1) and .67 for
(3), while that of (2) frequently falls below .50 for estimating ~~.
Choosing between procedures (1) and (3) is less obvious.
If p is less
than 1/4 procedure (1) would be the best choice, since the efficiency
of (3) for ~i would decrease sharply.
p
< 1/4 indicates ~~ to be the'
more important of the variance components, so that i,ntuitively equal
weight should be given to each B-sample.
(6) If design 25 is used, procedure (2) is most efficient in all
18 cases of comparison between the estimation procedures.
noteworthy is the estimation of ~i when p
= 1/4,
Particularly
since in this case the
efficiency of procedure (2) varies from .39 to .67 in the other 24
designs.
This peculiarity stems from the fact that design 25 has only
4 degrees of freedom between A-classes.
Hence, the coefficient of
~~
in Var (~~) (for any of the estimation procedures) is larger with
respect to the
4
~i~~
and
~~
coefficients than in the other designs.
The
"'2
~A-term in Var (~A) is therefore the predominant term for all but very
small values of p; also, it can be verified from equations (4.46),
(4.48) and (4.50) that the
~~-term is
minimized by procedure (2).
Thus, in design 25 procedure (2) is best for estimating ~~ for all but
very small « 1/4) values of p.
56
5.5.
Loss of Efficiency in Unbalanced Designs
Although the desirability of equal n.-values for design purposes
J.
was emphasized in Section 3.3, no mention has been made of the actual
loss in efficiency due to an unequal choice of the n.
J.
IS.
Designs 21-25
of Table 5.3 were included to help evaluate this loss in efficiency.
The (a,N,R)-choices for designs 21-25 correspond to those for designs
6, 10, 15, 22 and 26, respectively, of study (1) in Chapter 3.
The
n.-values,
of course, are decidedly unequal in designs 21-25, as
J.
compared to the equal choices of Chapter 3.
Table 5.6 lists the efficiencies of designs 21-25 relative to the
comparable designs of Chapter
3. Estimation procedure (3) was used in
connection with designs 21-25, since it is the recommended procedure if
nothing is known about p.
Mention was made in Section 3.3 of the difficulties in finding
optimal designs for estimating
CT2
A
+ CT2 .
B
If P
= 1 / 4,
it appears from
Table 5.6 that one is actually better off using the unequal n.-values
J.
of designs 21-25.
Had procedure (1), the estimation procedure of Chap-
ter 3, been applied to designs 21-25, the efficiencies for estimating
CT~ + CT~ when p
= 1/4 would
respectively.
Hence, if procedure (1) must be used, an equal choice
have been 1.00, .99, .98, .94 and .98,
of the nils is still preferable.
The efficiencies for estimating m and CT~ are more easily interpreted.
Designs 21-25 are least efficient, relative to those of
Chapter 3, for estimating CT~ when p
= 1/4.
However, this is partly a
reflection of the inefficiency of procedure (3) in this case, as
pared with procedure (1).
co~
A second point to be noted is that, among
57
Table 5.6.
Efficiencies of designs 21-25 relative to designs 6, 10, 15,
22 and 26 from study (1), Chapter 3
Designs compared
Chapter 3,
Chapter 5
study (1)
21
22
23
24
25
2
m
erA
1/4
1/2
1
2
4
8
·9°
·90
·90
·91
·91
·91
·73
.78
.79
.79
.79
.79
.79
.52
.64
10
1/4
1/2
1
2
4
8
15
1/4
1/2
1
2
4
8
.82
.82
.82
.83
.83
.83
22
1/4
1/2
1
2
4
8
.80
.82
.82
.83
.83
.83
26
1/4
1/2
1
2
4
8
·9°
.89
.89
.88
.88
6
p
.88
·78
.84
.89
·90
·91
.73
.78
.80
.81
.61
·73
·79
~82
.84
.84
.60
.72
·79
.83
.85
.86
·92
·92
·92
·92
·92
·92
2 2
erAofu'B
1.07
1.02
·97
·93
·92
·91
1.15
1.07
.96
.87
.83
.82
1.06
1.01
·93
.87
.85
.84
1.01
.94
.88
.86
.86
.86
·99
.97
.94
·93
·92
·92
58
designs 21-25, those designs with the most unequal ni-values are least
efficient when compared to their counterparts in Chapter 3Clearly, knowledge of the relationship between design and estimation procedure is fragmentary; many avenues of future research remain
in this general area.
59
6.
SOME DESIGNS FOR ACROSS-A COMPOSITING
6.1.
A Model for Across-A Compositing
The statistical model for Chapters 3, 4 and 5 has been the within-A
compositing model expressed in equation (3.1).
Frequently it may be
necessary or expedient to composite more than one A-class into a single
measurement, a practice which will be referred to as across-A compositing.
Cameron (1951) considered the effectiveness of this type of
compositing for estimating the variance components in a three-stage
nested model.
For a corresponding model in the two-stage nested case,
an observation might be described by
a
Yij =m+ b L: A.
a k=l iJk
+l
a
~
L: Bijk.e
L:
(6.1 )
ani k=l .£=1
j
i = 1, 2
:=
1, 2,
Q
0
0 ,
R.
As in (3.1) m is a constant, and the A
and B £ effects are indeijk
ijk
pendent normal in distribution with means zero and variances
cri and cr~,
The experiment contains 2aR A-classes, aR(~ + n )
2
B-samples and 2R observations; it is represented graphically in Figure
respectively.
6.1.
In such an experiment, where across-A compositing is used exclusively, a major problem is to separate variation due to A-classes from
variation due to B-samples within the same A-class.
this problem may be overcome by nesting
~
Cameron showed that
B-samples wi thin all A-
classes composited into R of the observations, and
~
B-samples within
all A-classes composited into the other R observations, where ~
r n2 •
60
a
1
a
1
j = 1,2, ... ,R
Figure
If
~
6.1. Representation of an across-A compositing model
= n2 = n,
sa:y, then the variance components cannot be individually
estimated, although the linear combination
2
n~A
+
2
~B
is estimable.
The observations are split into two groups of Reach (as indicated
above) for purposes of analysis.
~
1
= (R-l)
R
[~
_..2
Y1j -
j=l
If the mean squares
1
R(
R
are defined, it is easily verified that
~~
~
a
ani
E(O_) = - + "'1.
and that
2
~ Yij ) l,
j=l
i
= 1,2
61
Var (CL)
--:i
for i = 1,2.
=
2
(-)
R-1
2
0"A
(a
2
O"B 2
+ -)
ani
(6.4)
Since the mean squares are defined on different observa-
tions, their covariance is zero.
2
From these results Cameron derived unbiased estimators for O"A
and
2
O"B together with the variances of these estimators.
These results are
presented in equations (6.5) - (6.12), together with corresponding
2
2
results for m ahd O"A + O"B:
'"
m
(6.6)
~
"'2
A+O-B
'"
Var(m)
= (~~n2) [n2Q,)~ -1)
=
var(~~)=
- ~~ (n2-1)]
2+n2
[( ~ 2) 2 + 2]
+
O"A O"B
aR(~ +n )
~ n2
2
1
(6.8)
(6.9)
62
(6.11)
(6.12 )
In the estimation of m, equal weight is given to each B-effect, as in
Chapter 3.
Also, it is convenient to assume
~
> n2 , so that the expec-
tations of the numerators and denominators in (6.6) - (6.8) will be
positive.
6.2. Precision of the Estimators
Some general comments can be made concerning the variances given
in equations (6.9) - (6.12).
(1) Frequently the sole purpose of experimentation is the estimation of m and of Var (~).
For this purpose the individual variance
components need not be estimated; rather, only the linear combination
of
cri and cr~ which appears in Var (~)
is no longer required.
is ~
= n2 = n;
need be estimable.
Hence ~
:f
n2
From (6.9) it is clear that the optimal choice
thus, (6.9) may be written
63
Var (~)
=
1
2aRn
(ml + cr2 )
A
2
2
(jA
(jB
B
= +-2aR
2aRn '
where 2aR and 2aRn are the total numbers of A-classes and B-samples,
respectively.
It is readily verified that
=
A)2
- m
1: 1: ( Y
ij
=i....lojt--
--
2R-l
,.
(6.14)
A
Therefore, if low variance for m is the sole objective, there is no
need for more than one observation (i.e., 2R = 1); however, if in addiA
tion a good estimate of Var (m) is desired, it is clear from (6.15)
that 2R must be fairly large.
(2) Equations (6.10) - (6.12) are not functions of a, so that for
2 2
2
2
estimating (jA' (jB and (jA + (jB it is economically advisable to design
a=l, so as not to waste resources which might better be concentrated on
R, n and n • This of course represents a degenerate case of across-A
l
2
compositing, one which might not be desirable in certain situations due
to physical factors other than the cost of repeated A-classes.
R, on
the other hand, should obviously be as large as possible, though often
restricted by measurement cost.
(3) Suppose that the total numbers of A-classes, B-effects and
observations are fixed, the only remaining allocation problem being the
choice of
I1.
and n2
(I1. > n2 )
for fixed
I1.
+ n2
= C.
It is readily
64
shown that all of the coefficients in equations (6.10) - (6.12) are
minimized by
~
= C-l, n2 = 1. For example, the last coefficient on
the right-hand side of (6.12) may be written
,
and it is clearly minimized if
~
is maximized.
(6.16)
Since all of the
coefficients are minimized in this way, the three variances in question
are minimized for all values of p,
From these comments some conclusions can be drawn regarding acrossA compositing,
estimating m.
First, across-A compositing is highly advantageous for
If account is taken of the change in notation, it is
A
evident that Var (m) is the same in equation (6.13) as in equation
(3.6).
However, in the within-A compositing context of Chapter 3 at
least as many measurements as A-classes were required.
If across-A
compositing is used, the same precision can be obtained with a single
measurement, although usually a few extra measurements would be taken
to provide a numerical estimate of Var (~).
2
2
2 + 2 are the sole objecSecondly, if t h e paramet ers O"A' O"B and O"A
O"B
tives of estimation, it appears that a design of the type considered in
this chapter should never be used,
Suppose, for example, that suffi-
cient funds are available for 32 measurements; Le"
R=16,
If a=l and
n =1, the experiment will contain 32 A-classes and 16 (~+l) B-samples.
2
In this case it is easily shown from equations (6.l0) - (6.12) that
65
(6.18 )
Consider now a less expensive design of the within-A compositing type,
with 16 A-classes, 16(~+1) B-samples and 32 measurements.
From equa-
tions (3.3), (3.7) and (3.8) it follows that
= -.£..
(6.20)
15
(6.21)
AI">
Var ( CT""2+ere:)
A B
:=
2
15
222
[4 + 2CTACTB + (__
1 _)2 (1 + l5n..)
4]
.L
CTA
~+l
~+l
16 CTB
2
2
For all ~ > 1 these variances are uniformly (in CT and CT ) smaller
A
B
than the corresponding quantities on the right-hand sides of equations
(6.17) - (6.19).
Hence greater precision is obtained with half as
many A-classes.
It should be emphasized that only one across-A compositing model
has been considered; other models might do a better job of estimating
2
CT2 , CT2 and CT2 + CTBo
A B
A
Also, models providing for both across-A and
66
within-A compositing might be considered, since a small amount of
within-A compositing would provide a much improved estimate of CT~.
Frequently the primary interest of the experimenter is in estimating m, with knowledge of the variance components needed for the sole
purpose of determining the best design for estimating mo
It is well
known that if the unit costs of an A-class and a B-sample are C and
A
~
respectively, the optimal solution for n, the number of B-samples
from each A-class, is given by
(6.23)
Hence, a preliminary within-A compositing experiment could be designed
to estimate p
estimating m.
2/CT2
= CTA
B
' in order to determine the proper allocation for
This allocation would then be used in a more extensive
across-A compositing experiment, to obtain a precise estimate of m.
Some sort of running check should be maintained on p, as it may vary
over time.
More research should be concentrated on the formulation and
statistical evaluation of sequential programs of experimentation, for
estimation of the mean and other parameters.
67
7.
SUMMARY AND CONCLUSIONS
7.1.
Outline of Research
This dissertation has emphasized estimation of the mean and variance
components for a two-stage nested design, where certain second-stage
samples are composited prior to measurement.
In particular, research
was concentrated on three problems.
(1) Optimal designs for the individual and joint estimation of the
parameters m,
2
A'
(J
2
(JB
22.
and (JA + (JB were studied, where compositing is per-
mi tted within first-stage classes only.
If the traditional analysis of
variance estimation procedure is used, it is necessary to specify only
a, N, R (the total numbers of A-classes, B-samp1es and measurements,
respectively) and the nils, where n
in the i-th A-class.
i
is the number of B-samp1es taken
Given R measurements, the particular compositing
procedure is immaterial.
Designs having equal total cost were compared
in seeking optimal designs, where total cost is assumed to be an additive function of the unit A-class, B-samp1e and measurement costs;
!.~.,
(2) Two other unbiased estimation procedures were compared with
the traditional analysis of variance procedure, for the estimation
obj ectives stated in the preceding paragraph.
General conditions were
sought under which a given procedure is superior to one or both of the
other procedures.
(3) A particular model was considered where compositing is per-
formed across A-classes;
!.~.,
all B-samp1es from one or more A-classes
68
are composited into a single measurement.
Class variation is distin-
guished from sample variation by altering the number of B-samples within
each A-class.
The variances of the estimators of the four parameters
were compared to the variances of the corresponding estimators for the
within-A compositing model.
7.2.
Conclusions of Research
The first question which arises in considering problem
(1) is the
a
choice of the n ' s for fixed values of a and N, where N = r: n . Among
i
i=l i
those coefficients in the variances of the four estimators which depend
on the n.-values, all but one were found to be minimized by an equal
J.
(or as equal as possible) choice of the nils; !.~., ni ; N/a. The lone
""2 + O"B
""2) ; however, the inexception was the coefficient of O"Bh. in Var (erA
crease in the
O"~-coefficient due to an equal choice of the nils is
4
2 2
usually more than offset by the decrease in the O"A- and O"AO"B-coefficients, for which the equal choice is optimal.
always chosen equally.
Hence, the n.'s were
J.
A brief study of the efficiency of a decidedly
unequal choice of the nils relative to an equal choice was conducted.
2
For estimating m and O"A the efficiency of an unequal choice appears to
fall roughly between .70 and .90; for estimating O"i + O"~ it is usually
quite close to 1.00.
2
For estimating O"B the choice of the n
,
i
s is
immaterial.
Values of a, Nand R must still be specified.
and m, the optimal choices were determined as follows:
a=l,
N
=R
,.
2
For estimating O"B
m:
a=R,
N
- =
a
Clearly, the optimal estimation of m requires at least a guess of the
ratio p
2
= (JA2/(JB'
2 + (JB'
2 no solutions of the
For estimating (JA2 and (JA
type expressed in (7.3) were obtained, due to the increased complexity
"2 ) and Var ( "2
of the coefficients in Var ( (JA
(JA + "2
(JB ) •
To gain knowledge
of these solutions, and also to find good designs for the joint estimation of the four parameters, numerical studies were conducted using an
IBM 1620 for three different cost situations (C, C ' CB' CM)' Since
A
compositing is used when the cost of measurement is high, ~ was chosen
considerably larger than C and CB in all three cost situations.
A
Particular attention was directed to the values of a-I and R-a,
the degrees of freedom for estimating (J~ and (J~, respectively.
No
pronounced differences were observed between the three cost situations,
as long as the permissible ranges of values for a-I and R-a were taken
into account.
For all four parameters, considered individually and
jointly, a-I increases relative to R-a in the optimal designs as p
increases.
The ratio of a-I to R-a was almost always greater than I
222
in the optimal designs for (JA and (JA + (JB' while less than one for the
joint estimation of all four parameters.
More detailed conclusions
appear in Table 3.5.
The objective of problem (2) was the comparison of three unbiased
estimation procedures, as follows:
(1) Method of Weighted Means
(2) Method of Unweighted Class Means
(3) Method of Unweighted Measurements
70
Procedure (1) is the traditional analysis of variance estimation procedure discussed above"
The three estimation procedures yield essentially
identical results for designs with equal ni-values.
Therefore, 25
designs with decidedly unequal ni-values were generated to compare the
estimation procedures.
"2 is the same for all three procedures; hence,
eT
B
" eTA
"2 and
in this study attention was restricted to the variances of m,
"2
"2
eTA + eTB"
Cases were found in which each estimation procedure is the best
one to use.
2
If little or nothing is known about p, and if m, eTA and
all of roughly equal interest, procedure (3) was found
optimal.
If p is suspected to exceed 1, however, procedure (2) was
preferable for the same estimation goals.
If p is suspected to be less
than 1, procedure (1) was found optimal for estimating eT~, while for m
and eT~ + eT~ a less obvious choice between procedures (2) and (3) was
indicated.
In problem (3) a particular across-A compositing model due to
Cameron (1951) was considered.
For estimating m, designs based on
Cameron's model were found superior to within-A compositing designs,
chiefly because fewer measurements are required.
For estimating eT~, eT~
and eTi + eT~, the across-A compositing designs were found inferior to
their within-A counterparts, due to the difficulty of distinguishing
class variation from sample variation.
Only a restricted type of
across-A compositing was considered; other models might well yield
better estimates of the variance components.
Since the optimal allocation for estimating m is a function of p,
a sequential procedure was suggested for estimating m.
A preliminary
71
within-A compositing design would provide an estimate of p, thus determining the allocation for a more extensive across-A compositing design
for estimating m.
7.3. Suggested Future Research
(1) Include for each measurement in the model a third variance
component, to represent measurement error.
(2) Investigate cost models of a more general nature than simple
additive functions of the unit class, sample and measurement costs.
"2
(3) Determine the distributions of estimators such as ~A and
"2
~A
+ "2
~B' empirically if not analytically.
(4) Formulate estimation procedures where the variance components
are not estimated from differences of quantities known as sums of
squares in fixed-effects models, to eliminate the problem of negative
estimates.
(5) Investigate additional criterj.a of optimality for the joint
estimation of several parameters, such as equal coefficients of variabi1ity.
(6) Calculate maximum likelihood estimators and investigate their
properties for small but unbal,anced designs;> by means of' an empirical
sampling study.
(7) Conduct more theoretical research on the relationship between
optimal designs and best estimation procedures.
(8) Formulate sequential plans of experimentation for estimating
the mean, where an initial design is used to estimate the correct allocation for a larger and more expensive design.
~
(9) Explore more general models of the across-A compositing type,
with allowance for mixtures of within-A and across-A compositing.
(10) Extend the use of prior information to the compositing prob-
73
8 • LIST OF REFERENCES
Anderson, R. L. 1961. Designs for estimating variance components.
North Carolina State University at Raleigh. Inst. of Stat. Mimeo.
Series No. 310.
Anderson, R. L. 1965. Non-balanced experimental designs for estimating
variance components. Report on Seminar on Sampling of Bulk Materials, U. S.-Japan Cooperative Science Program, Nov. 15-18, 1965,
Tokyo, Japan. Inst. of Stat. Mimeo. Series No. 452.
Bicking, C. A. 1964. Bibliography on sampling of raw materials and
products in bulk. Tappi ll:147A-170A.
Bicking, C. A. 1965. A survey of the applications of statistics to
the sampling of bulk materials. Report on Seminar on Sampling of
Bulk Materials, U. S.-Japan Cooperative Science Program, Nov. 1518, 1965, Tokyo, Japan; edited by Dr. Kaoru Ishikawa, Faculty of
Engineering, University of Tokyo.
Bush, N. and R. L. Anderson. 1963. A comparison of three different
procedures for estimating variance components. Technometrics 2:
421-440.
Cameron, J. M. 1951. The use of components of variance in preparing
schedules for sampling of baled wooL Biometrics 1:83-96.
Cline, M. G.
288.
1944.
Principles of soil sampling.
Soil Science
~:275­
Crump, P. P. 1954. Optimal designs to estimate the parameters of a
variance component modeL Unpublished Ph.D. Thesis, Dept. of
Experimental Statistics, North Carolina State University at
Raleigh.
Eisenhart, C. 1947. The assumptions underlying the analysis of
variance. Biometrics 2:1-21.
Newton, R. G., M. W. Philpott;! H. F. Smith and W. G. Wren. 195L
Variability of Malayan rubber. Indus. and Engr. Chem• .!!2:329-334.
Prairie, R. R. 1962. Optimal designs to estimate variance components
and to reduce product variability for nested classifications.
Unpublished Ph.D. Thesis, Dept. of Experimental Statistics, North
Carolina State University at Raleigh. Inst. of Stat. Mimeo.
Series No. 313.
Reed, J. F. and J. A. Rigney. 1947. Soil sampling from fields of
uniform and nonuniform appearance and soil types. Jour. Amer. Soc.
Agron. 22:26-40.
74
Tanner, L. 1965. Errors of' non-proportional composite samples of' bulk
materials. Report on Seminar on Sampling of' Bulk Materials, u. S.Japan Cooperative Science Program, Nov. 15-18, 1965, Tokyo, Japan;
edited by Dr. Kaoru Ishikawa, Faculty of' Engineering, University
of' Tokyo.
Tanner, L. and W. E. Deming. 1949. Some problems in the sampling of'
bulk materials. Proceedings, Am. Soc. Testing Mats. ~:1181-1186.
Tanner, L. and M. Lerner. 1951. Economic accumulation of' variance
data in connection with bulk sampling. AS'IM Symposium on Bulk
Sampling, STP No. 114, 8-12.
Whittle, P. 1953. The analysis of' multiple stationary time series.
Jour. Roy. Stat. Soc. (B) 12:125-139.
75
9.
Appendix Table 9.1.
CT2
B
2
CT
A
APPENDIX
Variances of estimators for 3 designs of Chapter 3
V( "2
CT )
B
"
Vern)
"2 )
V(CT
A
"2 "2
V(CTA+a'B)
.0595
.127
·346
1.12
4.02
15·2
.150
.218
.437
1.21
4.11
15·3
.210
.492
1.43
4.82
17.6
67·3
·522
.804
1.'74
5·13
17·9
67·6
.0460
.119
·372
1.31
4.90
19·0
.209
.282
·535
1.47
5·07
19.1
Design 17, study (1)
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.167
.0480
.0733
.124
.225
.428
.833
Design 12, study (2)
1
1
1
1
1
1
1/4
1/2
1
2
4
8
·500
.146
.230
.398
.734
1.41
2·75
Design 20, study (3)
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.222
.0481
.0794
.142
.268
·519
1.02
76
Appendix Table 9.2.
2
O"B
2
O"A
Variances of estimators for designs 1-25 of
Chapter 5 using estimation procedure (3)
"'2
"2
"2 "2
"
Vern)
V( O"A)
V( O"B)
V(O"A+o'B)
Design 1
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.200
.0318
.0468
.0768
.137
.257
.497
.0733
.115
.245
.692
2.33
8.62
.122
.164
.293
.740
2·38
8.67
.0281
.0431
.0731
.133
.253
.493
.0482
.0856
.207
.638
2.25
8.47
.134
.171
.293
.724
2.33
8.55
.0281
.0431
.0731
.133
.253
.493
.0544
.0921
.214
.646
2.26
8.48
.130
.168
.290
·722
2.33
8.56
.0292
.0458
.0792
.146
.279
·546
.0539
.0933
.223
.689
2.44
9·22
.132
.171
·301
·767
2·52
9·30
.0258
.0408
.0708
.131
.251
.491
.0402
.0752
.192
.613
2.20
8.38
.142
.177
.294
·715
2.31
8.49
Design 2
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.200
Design 3
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.200
Design 4
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.200
Design 5
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.200
(Table continued on next page)
11
Appendix Table 9.2 (continued)
2
CT
B
~
..
V( ""2
CT )
B
""2
""2
V(m)
V(""2
CT )
A
V( CTA+dB)
.0303
.0448
00131
.132
.248
.419
.0503
.0898
0214
.643
2.23
8.29
.0991
.139
0263
.693
2.28
8.34
00263
.0401
00697
.128
.244
.475
.0355
.0108
.181
.601
2.16
8018
.103
0138
.254
.668
2.22
8.25
00280
00429
.0121
.132
.251
.490
.0425
00798
0201
.629
2.23
8.40
0102
0139
.260
.688
2.29
8.46
.0287
.0440
0074()
.136
.258
.503
.0483
.0867
.211
.651
2.29
8.63
2.34
8.68
.0291
.0452
.0774
.142
.211
·529
.0450
.0841
.212
.669
2·38
9·01
.102
.141
.269
.725
2.44
9.06
Design 6
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.133
Design 1
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.133
Design 8
1
1
1
1
1
1
1/4
1/2
1
2
4
8
0133
Design 9
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.133
.0982
.137
.?61
.701
Design 10
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.133
(Table continued on next page)
78
Appendix Table 9.2 (continued)
er2B
2
erA
....
V( "2
er )
B
V(m)
V(~)
v(erA+o'B)
.0313
.0463
.0763
.136
.256
.496
.0532
.0941
.222
.665
2.30
8.54
.0827
.123
.252
.694
2·32
8.57
.0297
.0453
.0766
.139
.264
.514
.0462
.0857
.213
.661
2·33
8.77
.0828
.122
.250
.698
2·37
8.81
.0279
.0426
.0720
.131
.248
.483
.0401
.0773
.198
.623
2.21
8.33
.0833
.121
.241
.666
2.2.5
8.37
.0299
.0468
.0805
.148
.283
·553
.0456
.0861
.220
.696
2.49
9.41
.0834
.124
.257
·733
2.52
9.45
.0254
.0410
.0723
.135
.260
·510
.0322
.0672
.186
.618
2.26
8.67
.0843
.119
.238
.670
2·31
8.72
"2
"2
Design 11
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.100
Design 12
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.100
Design 13
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.100
Design 14
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.100
Design 15
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.100
(Table continued on next page)
79
Appendix Table 9.2 (continued)
2
O'"B
2
O'"A
V( "'2
O'"B)
'"
Vern)
"'2
V(O'"A)
"'2 "'2
v( 0'"A+cr'B)
.0273
.0417
.0106
.128
.244
.475
.0379
.0743
.192
.608
2.16
8.16
.0724
.109
'.227
.643
2.20
8.19
.0259
.0401
.0685
.125
.239
.466
.0335
.0682
.182
·587
2.11
7.98
.0729
.108
.221
.626
2.15
8.02
.0275
.0432
.0746
.131
.263
.514
.0367
.0739
.197
.638
2·30
8·73
.0731
.110
.233
.674
2·33
8·77
.0250
.0395
.0684
.126
.242
.473
.0320
.0660
.179
·588
2.13
8.12
.0724
.106
.220
.628
2.11
8.16
.0220
.0357
.0631
.118
.228
.447
.0234
.0537
.157
·537
1.98
7·62
.0748
.105
.209
·588
2.03
7.67
Design 16
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.080
Design 17
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.080
Design 18
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.080
Design 19
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.080
Design 20
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.080
(Table continued on next page)
80
Appendix Table 9.2 (continued)
2
O"B
2
O"A
'"
Vern)
"2
V( O"B)
"'?
V( O"A)
"'?
"'2
V( O"A+a'B)
Design 21
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.0455
.0642
.102
.176
·328
.625
.132
.191
.368
·963
3·11
11.2
.179
.238
.416
1.01
3·15
11.3
.0445
.0707
.123
.228
.438
.858
.0781
.141
·351
1.10
3.95
15·0
.228
.291
·501
1.25
4.10
15·1
.0453
.0756
.1"36
.257
.499
.984
.0559
.123
.357
1.22
4.53
17·5
.237
.304
.538
1.40
4·71
17·7
.0583
.0958
.171
·321
.621
1.22
.0715
.160
.460
1.56
5·73
22.0
.167
.255
·555
1.65
5.83
22.1
.0660
.123
.236
.463
·916
1.82
.0480
.162
.594
2.27
8.89
35·2
.228
.342
·775
2.45
9·07
35·4
·333
Design 22
1
1
1
1
1
1
1/4
1/2
1
2
4
8
·333
Design 23
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.286
Design 24
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.167
Design 25
1
1
1
1
1
1
1/4
1/2
1
2
4
8
.200