-e NON BALANCED EXPERIMENTAL DESIGNS FOR ESTIMATING VARIANCE COMPONENTS by R. L. Anderson Presented at Seminar on Sampling of Bulk Materials Tokyo, Japan, November 15-18, 1965, sponsored by the National Science Foundation and the Japan Society for the Promotion of Science. Institute of Statistics l{tmeograph Series 452
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-e
NON BALANCED EXPERIMENTAL DESIGNS FORESTIMATING VARIANCE COMPONENTS
by
R. L. Anderson
Presented at Seminar on Sampling of Bulk MaterialsTokyo, Japan, November 15-18, 1965, sponsored by theNational Science Foundation and the Japan Societyfor the Promotion of Science.
Institute of Statisticsl{tmeograph Series 452
NON BALANCED EXPERD1ENTAL DESIGNS FOR
ESTIMATING VARIANCE COMPONEln'S*R. L. Anderson
North Carolina state University at RaleighRaleigh, North carolina
1. Introduction
This paper is a summary of recent reeearch at North Carolina State University
on the development of experimental designs to estimate variance components. Much of
the research has been conducted by doctoral candidates in Experimental Statistics
under my direction. The two most recent projects are not completed, but developments
to date are summarized here. In our experience, a knowledge of the magnitude of the
variance components would be useful in the following situations:
(1) Population changes often can be described in terms of variance components, e.g.,
in quantitative genetics. Hence a knowledge of the actual magnitude of these
components is reqUired in assessing various control programs.
The proper allocation of resources to reduce product variability depends on a
knowledge of the relative magnitude of the variance components.
t~) A knowledge of the relative magnitude of the variance components is also needed
to determine the best allocation of funds in sampling to estimate population
means and totals and in planning experiments to compare treatments.
Unfortunately I have had very little experience with the problems faced in the
sampling of bulk materials. It is hoped that my appearance on this symposium will
have two benefits:
(a) Enable me to become familiar with your problems so that some of our
research can be oriented in the direction of helping to solve them.
* Presented at Seminar on Sampling of :Bulk Materials, Tokyo, Japan, November 15-18, 1965;sponsored by the National Science Foupdation and the Japan Society for the Promotionof Science.'e, ",>\1,#$
2
(b) Furnish you with a brief description of current devel~ents in designing
experiments to estimate variance components in the hope that you will be
able to adapt some of the results to bulk swmpling situations.
In this paper, it will be assumed that all sources of variation are essentially
random. Two experimental and operational procedures are considered: the nested or
hierarchical type and the classification type. A balanced nested design and a two
way classification operation with nested-type sub-sampling are illustrated in Table ~.
I have been using as a typical example of a nested-type operation the following:
A pilot study was considered to assess the various sources of variability in the
production and assay for streptomycin before conducting an experiment on the efficacy
of various molds. There were five stages in this process: an initial incubation in
a test tube; a primary inoculation period in a petrie dish; a secondary inoculation
period in another petrie dish; a fermentation period in a bath; and the final assay
of the amount of streptomycin produced. This is a five-stage nested operation.
Another example is a study by Newton, et ale (1951) of the variability of rubber,
considering the following sources of variation: producer's estates, days at a given
estate, bales on a given day, sheets from a given bale, and samples from each sheet.
Technicians then took measurements on the samples.
As an example of a classification-type operation, consider the follOWing: Samples
from each of s =10 sources of a material are to be analyzed by each of f =5 labora
tories. Let us assume that n ;:; 2 samples are sent to each laboratory from each
source. This is a two-way classification operation. The purpose of the investigation
is to est1ma~e the lab-to-lab variation, the source-to-source variation, a possible
lab x source interaction (the failure of source differences to be the same from lab
to lab~ and the swmple-to-sample variation. It is also possible to determine if the
sample-to-sample variation is constant from lab to lab or source to source.
Table 1. Examples of Balanced Nested and Classification-Type Operations
Nested-Type Operations
2a
Test Tube
Primary Inoculation
Secondary Inoculation
Fermentation
Assay
~' ...,
Nodel: Yijk~m = I..l + t i + Pj(i) + sk(1j) + fe(ijk) + B.m(ijkt)
*************Two-Way Classification Operation With Nested-Type Sampling
5(10)
1 2
Source
Batch
Lab. Tech.
1 1
2
2 1
2
1
2
1
2
2
2
2
2
2
1
2
2
2
2
2
2
2
1
2
2
2
2
2
2
2 . . .2
2 . 2
2 ••• 2
2 • • • 2
2 • • • 2
. . . . .2 • • • 2
2 . 2
2
2
2
2
2
2
for example, Pj(i) is the
.3
Additional complications can be added to the last example1 which involve com
binations of nested and classification-type operations. Suppose each laboratory
selects 2 technicians for the study and from each source 2 batches are selected. In
this case1 4 samples are sent to each laboratory from each batch, 2 for each tech-
nician.
Many classification-type operations will involve more than two classes. Some
discussion of the general multi-way classification operation will be included in
this paper.
The stochastic models for the observations obtained by use of these operations
are given at the bottom of each example in Table 1. In all cases ~ is the general
mean and the other parts of the models are assumed to be independently identically
distributed random variables. We will assume these random variables are normally
distributed; however1 this requirement is Bot always essential. The parameters of
e interest are the variances of the random variables. For the nested-type operation,
these are
For the two-way classification operation, they are
sources 10Labs 4Tech. in labs 5S x L 36S x T(L) 45B(S) x L 40B(S) x T(L) 50Analyses* 200
e * Again this can be subdivided for internal analysis.
which, in general, there is no closed-for.m solution.
5
In other words, ML estimates
Crump also
The problem.
02 ) and thec
must be obtained in a complicated iterative computing procedure. In addition, small
sample properties of ML estimators cannot be obtained except by use of empirical
samp1ing methods for specified sets of parameter values; these properties may not be
too good, i.e., the estimators will be biased and may be rather inefficient.
If one is estimating the five variance components and \.1 for the five-stage nested
operation, it is recognized that a plan which (for a given total number of assays)
produces the minimum variance estimator of a2 will furnish no estimator of the othera
components; all l6t assays would be made from the same fer.mentation bath. The best
estimator of the total variance, 0t2 + 02 + 0
2 + a2f
+ 02 would involve using l6t test
p s a
tubes and only one sample from each test tube, from each primary and secondary
inoculant, and from each fer.mentation bath.
_ In his analysis of the design problem for two-way classification data, Gaylor
(1960) showed that the lower bound to the variance of the estimator of any linear
combination, O~, of the variance components is 20;/(N-l), where N is the total
number of samples obtained. Gaylor showed that the variances of the estimators of
the functions given in Table 3 could reach this lower bound, where 02 is the samplings .
variance Within cells.
On the other hand, if the interaction variance component were zero, the optimal
design to estimate the row component above would use only one column (or the optimal
design to estimate the column component above would use only one row).
of the number of rows to best estimate 02 (or columns to best estimater
number of samples per row (per column) was considered by Crump (1954) .studied the use of various estimators.
Prairie (1962) considered various designs for a three-stage nested operation and
advanced a possible general criterion for a two-stage operation.
e ~able 3. Two-way classification designs which produce most efficient estimatorsof certain linear functions of the variance components for fixed totalsample. *
Function estimated (cr~) r c nijl.
21 1 Ncr
s2 2
+ cr; N 1 1cr + crs rc
2 + (J2
+ a~ 1 N 1crs rc
2 2 2 2 **cr + (j + a + a N N 1 or 0s rc r c
* 1 with l' th .th d .th 1 ~ Nr rows x c co umns nij samp es In e l. rowan J co umn, .~.nij = •l,J
** Each sample to be taken from a different row and column, e.g., row 1 and column 1row 2 and column 2, •••• This design is also most efficient to estimate ~; thevariance of
**************
Table 5. Efficiency ratios (E) for a, ~~, '(j;, 'P, N = 100 1:/
(6.4) p = Prob [Vo(p) ~ t3J'where t3 is a fixed number between min Vo and max VO'
(9 min Vo + max VO)/IO j
(3 min Vo + max VO)/ 4 .
9
o ~ [Da'~ ~ D. It is assumed one unit of funds will reduce either O'~ by 1% or 0';by 100 A%, A a known constant.
20' will be minimized if
where C2 =kl + k2 and CI C2 = Dkl + [n(k2/kl ). If da < 0, set Da = OJ if da < D,
set D = D.a
In order to use these results, one must obtain an estimator of p, p. Unfortunately
the use of an estimate of p results in less than the optimal reduction in total
variancej hence, one objective is to minimize the loss due to estimation. The true
reduced variance ratio is given by (6.2). In order to make (6.2) small, it is
desirable to obtain a p as close to p as possible.
Prairie considered a two-stage nested design of the Crump-type with N = ap + r
"and investigated how the true reduced variances (based on p) changed with a, the
number of classes. The estimator of p is given by (6.3). Smallness of (6.2) might
be a small expected variance ratio,
2'"~(dRlp).
Unfortunately it was not possible to obtain a closed-form expressionj Prairie has
recently investigated this numerically at Sandia Corporation.
In his thesis, Prairie investigated the effect of changing ~ (for fixed N) on
the maximization of P, equation (6.4). Two values of ~, as shown in (6.5), were used
in the study. The parameter values studied were:
p,lOO A = 1/10, 1/4, 1/2, 1, 2, 4 and 10; D = 25,100; for
N = 24, a = 2(2)8(4)20; for N = 72, a = 2,4,8,12,18,24(12-)60.
In general, it appears that if one uses a design that is moderately near the optimal,
he will do qUite well in achieving his objective of reducing total variance. The
study indicated that for most situations an intermediate value of ~, say N/4 :s a :s N/2
will give almost optimal results.
be a =N/3.
10
If one value of ! were to be recommended, it would
P was computed by use of Incomplete l3eta approximations. Profiles of VO(~) are
given in the two Prairie references.
4. Extensions to Multi-Stage Production Processes
In the five-stage nested experiment mentioned earlier, I had proposed a so-called
staggered design. This design was constructed to eqUalize (as far as possible) the
degrees of freedom for the various parts of the analysis of variance. Prairie (1962)
also considered my staggered (Dl ) design plus other non-balanced three-stage nested
designs.. with the modeli=l,2, ••• ,aj=l,2, ••• ,b
ik=l,2, ••• ,nij
Some of these are presented in Table 1, plus a balanced design, for N = 48. vlhen
he was a graduate student at Raleigh, L. D. Calvin became interested in these non-
balanced designs. He has subsequently used some of them; see Calvin and Miller
(1961). Prairie developed a specific procedure for constructing his three-stage
designs, called D2-designs:
1) N = aql + r l , 0 S r l < a.
Assign ql + 1 units to each of r l A-classes (designated as group Gl ) and
ql units to each of a-rl A-classes (designated as group G2 )·
2) b =a~ + r 2, 0 S r 2 < a.
To each A-class assign ~ B-classes and then one extra B-class to each of
r 2 A-classes. Make sure that b > a.
lOa
*Table 7. Some three-stage nested designs with N = 48 observations.
Balanced D.F. (15, 16, 16)
16
In
Calvin-Miller and
Prairie (D2)Balanced Staggered (Dl )
A
rS7-1 n!?/A
C/B/A12 8 8
Other Prairie designs, unbalanced D.F.
-±-. +(23, 16, 8) r I
8 16
(15, 24, 8)r
8 8
(7,52,8) 1-1, +t +18
Special staggered (11.) design for (15, 24, 8)
l-S I~ +- +4 4
++8
* The number below each basic plan is the number of replications used for the completeexperiment. The degrees of freedom (D.F.) are between A-classes, B in A classesand C in B in A classes, respectively.
lOb
e ,+,ab1e 7a. Some Non-Balanced Four and Five-stage Nested Designs
Bainbridge (Calvin-Miller) designs
Four stages Five stages
1;\1 = 4
nij = .3,1
nijk = 2,1,1~
I ISome Prairie deSignS:
ni = 5nij = .3,2nijk = 2,1,2
nijk!= 2,1,1;11,1,2II . f
Five stases
ni = 4
nij =nijk=
Four stages
*Anderson five-stage staggered design
n+t+¥+ tl
+ta2 ni =16,8,4,2,1 a,; a4 a5
nij = 8,4,2,1,.1
nijk = 4,2,1,1,1
nijkf = 2,1,1,1,1
* ai replications of each basic plan plus ~ replications of the full design given
at the top of Table 1, giving i~lai A-samples. This design is given in Anderson
and Bancroft (1952) with 13.1=2, a2=2, a,;=4, a4=8, a5=O.
,
11
3) 'Vlithin each A-class" assign the observations (nij ) to the B-classes as equall:y
as possible.
When b ::: N/2" the D2-design will usually satisfy the relations:
Ini - ntl = 0 or 1 ;
Inij - ntml =0 or 1.
Consider the four D2-designs given in Table 7. In all cases" the ni are equal
for a given design; the nij are either 2 or 1. ~ staggered (D1 ) design does not
have these features" e.g., ni =2 and 4" nij =1 and 2. An additional benefit to be
derived from a design such as the D2
(15,,16,,16) design is the facility to detect
variance heterogene1ty. If we indicate the three observations for a given A-class
as x11,x12 and x21, then a single degree of freedom contribution to SSC is
(x11 - x12'f/2. This can be computed separately for each of the 16 A-classes and
tested for heterogeneity; if the A-classes are sampled on successive time periods"
one could determine if there was a time trend in the variance estimates. The same
procedure can be used with SSB" where the single-degree-of-freedom contribution is
(xU + x12 - 2x21 )2/6• For my staggered (&1) design" six observations must be
secured at each step" giving two degrees of freedom for C and B at each step but
with only 8 steps instead of 16.
For each of 14 non-balanced designs, Prairie computed the efficiency ratios
Ea, ~, and Ec" where
E = var(~) for a given non·ba1anced design
'1("'2)var ~i for the balanced design
assuming the analysis of variance estimation procedure of Table 8. The design
parameters considered were Pl = 02
/02 and P2 = 0b2/02 =1/10,1/4,1/2,,1,2,4,10.a cc
Special analyses for my staggered (D1 ) design and the Calvin-Miller (D2) design
are presented in Table 9.
12
e !ale computation of var(o~) for the D2
-designs (of which the Calvin-Miller is one
example) is complicated by the fact that MBA and MSB are usually correlated; however,
if the design consists of replications of one basic plan, all mean squares are2 .
multiples of X -variates. The mean squares for rrr:y staggered design are uncorrelated;
however, MBA and MSB are determined by pooling several sums of squares which have
different expectations. In Table 9a, each mean square is independently distributed
as a multiple ofax2-variate; however, the pooled mean squares (Table 8) are distrib
uted as weighted sums of X2-variates, e.S. ,
MBA = (7~ + 7 MBA2 + MBA,)!15
( 22 ..22 22;= Xlol + x202 + X,o,) 15,
-~ 2 2where x;: and X2 have 7 d. f. each and X; has 1 d. f •
A D2-design which consists of replications of one basic plan ~UCh as for
e (15,16,16») 1s. easier to administe~ and is more amenable to sequential experimenta
tion than is my staggered design or any design with several parts in the basic plan.
This feature is emphasized by T. R. Bainbridge (1965).
Calvin and Miller (1961) have constructed a four-stage analogue of the
D2(15,16,16) design given by Prairie. Bainbridge (1965) presents the same four-stage
design and extends this principle to the construction of designs for five and six
stages. The Bainbridge four and five-stage designs, my five-stage staggered design,
and possible Prairie four and five-stage designs are presented in Table 7a. It
should be indicated that the Bainbridge four-stage design does not meet the criteria
set up by Prairie, because nij = 1 or';, similarly for all higher-stage Bainbridge
designs. However the benefits mentioned for the three..stage design become even more
important for higher stage designs, because it is possible to teat for variance
heterogeneity at each successive stage and sequential sampling becomes more important
e as the total number of samples increases.
Table 8. Analysis of variance for general three-stage nested design
l2a
Among C- classes in B in A N-b
Source of variation
.Among A-classes
Among B-classes in A
Mean.m:.. Square
a-l MSA
b-a MSB
MSC
Average value ofMean Square
2 I 2 2°c + K10b + K20a
2 20c + Klcrb
O~
A2oc
IS. = [N - ~; (~/ni>]/(b-a) ;
Ki = r~ ~ (nij/ni ) - Ei~ (n~j)/~/(a-l)
b J J
K2 =[N - t ni/N]/(a-l) •
MaB - MaC 1'2 K1MSA - KiMSB - (Kl - Ki) MSC. ° =Kl ~ a K1K2
**************Table 9a. Analysis of variance for the staggerea (Di) ~design in Table 7
Source of variation D.F. M.S. Average value of M. S.-Al (Group 1) 2 2 2 2
7 MBAl 0c + 2crb + 4cra = 0'1
A2 (Group 2) 7 MSA22 2 2 2
crc + crb + 20 ::.:: O2.a
A3
(Between Groups) 1 MSA3 O~ + 4/3 O"~ + 8/3
2 20"a ::.:: 0'3
Bl (Group 1) 8 MSB2 2
1 O'e + 20'b
B2 (Group 2) 8 MSB22 2
0c + °b
C (Group 1) 16 MaC 20"
C
Table 9b. Analysis of variance for the Calvin-Miller ( D2 ) design in Table 7
Source of variation ~ M.S. Average value of M.S.
A 15 MSA 2/2 20"c + 5 3 O"b + 3cra
16 2/ 2B MSB O'c+ 43 Ob
C 16 MSC2
O'c
13
e It is difficult to summarize Prairie's comparisons in a few words, mainly because
a reduction in variance for the estimate of one variance component will be offset by
an increase for some other component. This reemphasizes the need for a single
criterion.
Table 10 presents selected efficiency ratios for the six non-balanced designs
presented in Table 7. Ratios < 1 indicate superiority for the non-balanced design.
There is a loss of efficiency for a2 directly proportional to the reduction in degreescof freedom from 24 for the balanced plan to 8 or 16 for the non-balanced plans. Only
2; 2 2if P2 =ab ac is large does the non-balanced plans give better estimators of ab, even
though the non-balanced plans have more degrees of freedom for MSB than does the
balanced plan with 12 d.f. The non-balanced designs often achieve considerable~2
reduction in the variance of a •·a
2e The following summary seems in order for estimating aa:
(1) Since m;y staggered and the D2-designs have about the same efficiencies,
the D2-designs might be preferred on the basis of simplicity of con
ducting the experiments and analyzing the data.
(2) For Pl small" D2 with assignment (7,32,8) is very good; also for Pl = 1.
and P2 large.
(3) For P1. and P2 1.arge, D2 With assignment (1.5,,24,8) is quite good; also
for P1. =1 and P2 S 1..
(4) For P1. 1.arge and P2 S 1." D2 with assignment (23,,1.6,8) is very good.
However, we note that in some instances, there is considerable 1.oss in the efficiency
2of estimating ab, e.g. (4) above. One of the merits of my staggered design is that
it does not entaU such a 1.arge loss. Additional research needs to be centered on
obtaining a single criterion to be optimized and on extending Prairie's comparisons
e to non-balanced designs with more than three stages.
l3a
Table 10. Selected efficiency ratios for comparing the non-balanced designs with thebalanced desiGn in Table 8*
!/ Negative results indicate the standard procedure issuperior; procedures using no prior are unbiased, i.e.,mean square error equals variance. Additional samplingis being conducted.
If no prior is used, N1 = 40 is recommended, because at worst it is only 6%
less efficient than the one-step plan (p=l) and is almost as efficient as N1 = 20
for large or small p. Of the priors, C in two-steps is recommendedj in aeaeral
this shows considerable imporvement over the no-prior two-step procedure, especially
for p between 1/2 and 2. As indicated in .Tab1e 15, further sampling is being conducted.
REFERENCES
Anderson, R. L. 1961. Designs for estimating variance components. Proc. Seventh
Conf. Des. Expts. Army Res., Dev., Test., ARODR 62-2, 781-823. Inst. of Stat.
Mimeo Series No. 310.
Anderson, R. L. and T. A. Bancroft. 1952. Statistical Theo;g in Research. McGraw
Hill, New York.
Bainbridge, T. R. 1965. Staggered, nested designs for estimating variance components.
~dustrial Quality Control 22(1):12-20.Bush, N. 1962. Estimating variance components in a multi-way classification. UnpUb
lished Pil.D. Thesis, North Carolina State University, Raleigh. Inst. of stat.
M1meo Series No. 324.
Bush, N. and R. L. Anderson. 1963. A comparison of three different procedures for
estimating variance components. Technometrics 5:421-440.Calvin, L. D. and J. D. Miller. 1961. A sampling design with incomplete dichotomy.
!\gron. J. 53:325-328.Cameron, J. M. 1951. The use of components of variance in preparing schedules for
sampling of baled wool. Biometrics 7(1):83-96.Crump, P. P. 1954. Optimal designs to estimate the parameters of a variance compo
nent model. Unpublished Ph.D. Thesis, North Carolina state University, Raleigh.
Gaylor, D. W. 1960. The construction and evaluation of some designs for the esti
mation of parameters in random models. Unpublished Ph.D. Thesis, North Carolina
State University, Raleigh. Inst. of stat. Mimeo Series No. 256.
Henderson, C. R. 1953. Estimation of variance and covariance components. Biometrics
9:226-252.
Newton, R. G., M. vi. Philpott, H. F. Smith and W. G. Wren. 1951. Variability of
Malayan rubber. Indus. and gr. Chem. 43 :329.
Prairie, R. R. 1961. Some results concerning the reduction of product variability
through the use of variance components. Proc. Seventh Coni'. Des. Expts. Army
Res., Dev., Test., ARODR 62-2, 655-688.
Prairie, R. R. 1962. Optimal designs to estimate variance components and to reduce
product variability for nested classifica.tions. Unpublished Ph.D. Thesis,
North Carolina State University, Raleigh. Inst. of Stat. Mimeo Series No. 313.
Yates, F. 1934. The analysis of multiple classifications with unequal numbers in
the different classes. J. lmJ.. Stat. Ass. 29:51-66.