PAIRWISE PROBABILITIES IN HtOBABILITY
NON-REPLACEMENT SAMPLING
by
James R. ChroD\V
Institute of Statistics
Mimeograph Series No. 926
Raleigh - May 1974
iv
TABLE OF CONTENTS
Page
STATEftEBT OF THE PROBLEM
1
1
The Use of PPS Ron-replacement Sampling •• O~~
Existing Criteria for Pairwise
probabilities .000.OOQQ.0.oo~ooaooooo0090
••
6
The Variance of a Finite Population
Parameter oov-oa~~09~VQO~OOVQG9~~09~0000900~11
2.1
The Basic Superpopulation Model •• 00
2.1.1
2.1.2
2.2
2.~
2.4
The Functional Relationship
between Y(i) and X(i) '0<>000"90<><'00 15
Assumptions about the Error Terms . 0 17
General Approach co09~~~~.ooo~o.oooo 25
Method ,: Constrained Minimizaclon • 27
Method 2~ Compensating Additive
Adjustments •• 0 . 0 . ° 0 0 . 0 . 0 0 0 . 0 ° 0 0 0 . 31
Method 3~ Sampling Unit
Re~Definition
2.6
3.
3.3
<>00
"CQ''i~''-'
""0'"
32
Some Comments on the Choice of Constraints o. 33
Summary of Chapter 2 LOOCC"'Co~
kO'~~~~~'"
35
STRATUK SAMPLBS OF SIZE THREE
3.1
3.2
15
The Superpopulatton variance of the
Finite population Variance 00~.~~~.~uoooooo 18
The Algebraic solution for Optimal 'alu~a OL
P(ij} ~~OGQ.ooo~qqq~~~~~~ ~~~~O~~Jo~-qo~OUQGQ 22
Adjustment LJrocedures 0oOOQ~O".O.o<>90oo.ooo~. 25
2.4 1
2.4.2
2.403
2.5
••••• 0.0.
31
o~uoqc .~~.
Beed and Approach oooooco~
oo~~-.oo~~
37
The superpopulatioD Variance of the Third
Moment of the Horvitz~Thompso4. Estimator
38
The Algeuralc Solution for thp P(ijk) O v o o o o . 41
The General Solution mvm~~~oaan~DOon 41
A Sometimes Optimal Solution
43
00
Adjustment Procedures
Summary of
DoaO~LCcooa~o
'.
q,
0
46
~7
v
page
TABLB OP CONTBNTS (Continaed)
401
General Approach for Algebraic solutions
for the P (ij) 0 .. " "q><HI-" 0
,~"
~,,~ ~- ~
Applications with Some Specific Error.Models
0
4.~
Expected Squared
to the Measure
Expected Squared
to the squared
A Generalization
4~3
4.4
".
0"
'" '"
'"
0"
?
A ftixed Error Model "'99099Q""~U"9""''''Q'''oo.oooo 56
General Approach for Algebraic
S(Hutions for the P (ijk) vo • •
58
Application of Adjustment Procedures
~ .~uo 59
-
--
'" 0
'.
'"
0...,.
0<'
40501
4 5.2
Adjustment of the P(ij)
Adjustment of the P(ijk)
o
59
0ooq""ooo"o
60
o"'c~.ooou • •
505
General Approach "'90Q090000~.90
O.O."O.oo ••
The Populations Con~1aered ••• oU • • 90.o.o.ou~.
The Designs Considered o.ooO'_~ooooo~ '000000"
sensitivity of the Criter10fi runction to
Design Selection ." ••
ooooo~uuoo9.00
Effect of Design Selection on the Variance
506
Comparison of Theorettcal,and Simulated
5E 1
5.~
503
50.
•••
OQQ09" ....
Estimator
Jesuits
60
48
49
Error Proportional
of Size 00,",'"
49
Brror Proportloua~
Measuce of Size 000 52
55
0
4.5
0
61
62
64
66
~C~Qq9~.4q~.o-90on~~qQ~~9~OOO~Q~GQ68
V~09~~OOC~ooooOO~O~OQO~ooo~'o~'--~ooq70
SUftftlBY lIfO SUGGBSTIOHS POR FURTHBR RESBlRCH
" 0 . > '" 0 0 0
14
6 1
60l
Summary of Research ."~,-,~~o~.. ,,o.o~~~~~.o .. o. 14
Recommendations for Farther Study ooo~~_ • • oqo 16
8.1
Pattern Matrix Inversion Ot;l"l'0"''''''·'O''O'''~ooo.()O() 81
The Third Moment of the HorvLtz~Thompson
Estimator ~ ._~
._Q •• qu.OvUU~g~~UOUQD 82
sample Designs Cons~u~~ea 1n Lne ~mp1r1cdL
Study o~~~
qO~~Dq~quo~covOOO~-o~~4caoqoc 85
Listing of Computer Prog~ams OO_~~DOOOUDaooo~ 96
0
8.2
8u3
8 4
0
10
STATEMENT OF THE PROBLEM
The theory of sampling with probabilities
to
size (PPS sampling)
(lg43to
proportional
was introduced by Hansen and Hurwitz
The technique permitted
the
use
of
an
unbiased
ratio estimator of a population total,
N
T
= S Y (i) ..
i=l
The Hansen-Hurwitz estimator for the population
using
their
probability
replacement
(pr)
total
design
when
may be
written as
n
ttstpr) = n-.. 1 1(+) S r[s(a) VI[s(u)]
u=l
where
N
][ (+)
= S I (i)
q
i=l
a
necessarily
distinct
frame of N elements
proportional
to
sample
elements
selected
Xli)
of
from
with
not
n
a
sampling
probabilities
and with replacement
~fter
each draw q
I (i)
=a
positive-.-valued variate associated with the i-th
element in the universe q and
Y(i)
= the
variate
investigation
universe.,
of
for
interest
the
in
i-th
the
element
sampling
in
the
2
The
sampling
population
frame
is
obtained
dividing
correspond
populationj
im~oo
as
closely
The population
sampled
as possible with the target
the population ibout which information
desired (Cochran o 1963 0 ppo
is
For the purposes of this
6=7)0
studyo further consideration is limited to the
the
the
sampled or universe of interest into N parts and
calling these parts sampling unitso
should
by
elements
in
sampling frameo
These elements are identified by their
labels, i=l,2,0 900 8.
The labels of elements in a particular
sample, s, of n sampling units drawn from the sampling frame
of N sampling units are denoted by s(u), u=10200990no
each
label
s(u)
corresponds
to
where
one of the sampling unit
labels 1 through N depending upon the particular sample,
To this extent the labeling of frame elements by
and
of
sample
employed
elements
interchangeably
by
s(u),
with
label i refers to the i-th sampling
frame
and
that
i=102,oo~,N
u=1,2 o ooo,n q
the
s.
will
be
understanding that the
unit
in
the label s(u) refers to the
the
sampling
u~th
sampling
unit in a particular sample, s, and also corresponds to
one
of the labels o im
Assuming the
elements
in
values
the
universe,
reasonably correlated with
sampling
can
procedure
effectively
comparison
of
XCi)
and
the
are
the
values
known
values
of
for
of
I(i)u
all
H
Y(i) are
the
PPS
and the associated estimation procedure
reduce
the
variance
of
estimates
in
with unbiased simple expansion estimates used in
3
conjunction with siaple random
use
of
PPS
sampling
with
In
sampling~
bias
associated
estillation applied to simple random sampling
The Hansen and Hurwitz PPS
sampling
with
ratio
designs~
scheme
is
often
to as PPS sampling with replacement or probability
replacemen~
sampling
the
the ratio estimation procedure
avoids the risk of estimation
referred
addition,
The
sampling~
special
case
stratified
of
with one element selected with PPS per stratum may
be considered as a PPS non-replacement sampling
disadvantage
of
the
scheme is that it
estimation.
one
does
scheme~
One
element per stratum PPS sampling
not
allow
for
unbiased
variance
When sampling error estimates are required with
such designs,
a
collapsed
stratum
approach
is
commonly
employed; the method involves treating two similar strata as
if they were one stratum
sampling
stratum o
variance
The
and
using
estimation
collapsed
the
formula
stratum
with
replacement
within the enlarged
approach
is
generally
assulledto overestimate the true varianceo
1 theory for PPS non-replacement
stratum
samples
of
Horvitz and Thompson
Horvitz
and
two
The unbiased estimator
Horvit~-Thompson
has
within
=S
u=l
calle
to
estimator and is of the form
n
t{slpne).
within
class two of linear estimators for
probability non-replacement (pne) designs
known as the
for
or more elements vas presented by
(1952)~
Thompson's
sampling
Y[s(u)]/P{s(u))
be
4
where
= probability
P(i)
i~th
that the
element is
included
in
the sample.,
In practice q the P(i) are usually determined so that
P (i)
This
choice
P (i)
of
the
in
non-replacement
the
appear
sample
PPS
IX ( ~)
makes
estimators
Horvitz-thompson
elements
= n X (H
is
PP~
alike;
hovever q
distinct
employed.,
Thompson presented a formula for the
under
Hansen-Hurvitz
alway!:
are
sampling
0
Horvitz
variance
of
and
the
when
and
t(slpnr)
sa.pling without replacement given )y
N
=S
V[ t (s I pnr) ]
12
(it [ l-P (i)
]/P (i)
i=l
N If
+ S S l(i)Y(j)[P(ij)-P(i)P(j) ]/[P(i)P(j)]"
i#j
where
P(ij) = the probability that the i-th and j-th
both
included
in
the
sample
probability for th£ i-th and
(or
~th
p(ij)~O(
the pairwise
this
variance
namely
n
v( t (s I pnr) ]
=S
YZ[ s (u) ]( 1-P[ s (u) ] I/P2{ s eu) ]
u=l
n n
-S S{ Y[ s (u) ]/P[ s (u) ]
H I[ s
(v)
l/P( s (v) ]}w[ s (u)
uti'll
where
w (ij~
::: [P (i) P (j)'-P (ij) lIP (ij}'
are
anits).
They also presented an unbiased estimator of
for designs with all
units
~
11
S
(v) ]
5
Ya tes
expression
and
Grundy
for
the
(1953)
an
presented
variance
of
the
alternate
Horvitz-Thompson
estimator o namely
N N
n?
V[ t (s I pnr) ] :: SSW (i j)
(i j)
i<j
where
and
D(ij) = Y(i)/P(i)-Y(j)/P(j).,
They also presented an alternate
variance
estimator
which
may be expressed as
n n
v[t(s,pnr)]
=S
S w[s(1I)oS(V)]D2[S{U)l's(v)]
u<v
where
w(ij)
Other
other
probability
unbiased
= W(ij)/P(ij)0
non-replacement
estimators
have
schemes
employing
e"~Q
also been developed o
Midzuno (1950)0 Lahiri (1951)0 Harain (1951)0
Raj
(1956b)
0
(1957)g and Rao e Hartleyo and Cochran (1962).,
Since
the objectives of this research are restricted to the
study
Murthy
of properties of the Horvitz_Thompson
methods will not be reviewed here.,
general
probability
sampling
estimator~
A
thorough
procedures
and
these other
review
the
of
lise of
different classes of estimators is given by Koop (1963).,
6
Many
sampling
schemes
probabilities"
set
of
selection
the
of
are obtained empirically due to
values
of
the
Horvitz
P(ij) ..
non-replacement
variances
Different
estimator
resulting
same
the
!?(i) ..
Horvitz-Thompson
probability
fo~
yield
which
probabilities"
different
exist
selection
pairwise
and Thompson (1952" po616)
recognized the problem of the choice of the P(ij) and
tvo
related
problems:
(1)
to
determine
the
pairwise
probabilities which minimize the variance and (2) to
selection
Several
probabilitieso
developed
which
schemes
in
the
achieve
other
the
desired
criteria
literature
and
have
sampling
posed
devise
pairwise
since
been
schemes which
satisfy these criteria have been proposedo
Raj
between
varianceo
(1956a)
the
Y(i)
assumed
and
a
straight
relationship
line
the XCi) and sought to minimize the
Under his assumptions" the variance was minimized
by minimizing
N N
S 5 P(ij)/[P{i)P(j) ]c
i~j
Raj employed a linear
programming
approach
optimal P(ij) subject to the constraints
Plij)
~
0"
and
N
S P ( i j l :: ( n-~ 1) P (i)
j;li
c
to
solve
for
7
=
He considered samples of size n
achieved
2
onlyo
The
solutions
by the linear programming procedure permitted some
P(ij) to be zero; as a result u an unbiased estimate
of
the
variance could not be obtained from the sample Jata o
Hanurav
optimum
(1967)
sampling
proposed
several
strategieso
Four
requirements
for
of these requirements
involved the pairwise probabilities:
(1)
P(ij»Oc
(2)
P(ij)$ P(i)P(j)
(3)
.[)
q
~
(ij) I[ P(i) P (j) ]
C
(4) the value of P(ij)
and C
is
>
00
and
computable
from
a
simple
compact formulao
Hanurav's
first
three
conditions
may
summarized
by
1 somewhat stronger q but simpler q constraint
on
be
reqcicing that all P(ij) satisfy
CP(i)P(j)
$
P(ij)
$
P(i)P(j)
where
o <
for all io
C
.s
(n-1) /[ n-P (i) ]
C may be stated as
o <
{n-l)/n.
C S
HanuravVg first requirement is necessary
variance
estimation,
The
for
unbiased
second requirement provides the
set of sufficient conditions for the non-negativeness of
Yates-Grundy
requirement
variance
form
of
protects
estimate
the
variance
against
itselfo
a
The
estimateo
large
variance
~he
The third
for
the
fourth teguirement is lor
8
convenience onlyo
Brewer (1963 q po7) obtained a
the
P(ij)
for
achieving these
of
selection
samples
~(ijJ
convenient
formula
for
of size two and suggested a way of
in practice through appropriate choice
probabilities at each drawo
Brewer's formula
may be written as
P{ij) == KP(i)P(j)f[l=P(i) ]-t+[l-P(j) ]-t}
where K is constant in i and j
~nd
may be expressed as
N
K
(2 + S P (t) /( l-P (t) ]
::=
I-i
0
k=l
Brewer
developed
his
formulation
in
to
trying
model
systematic sampling when units are arranged in random ordera
Durbin (1967) obtained the same formulation
for
P(ij)
in tr:ying to develop sampling schemes with these properties:
(1) strict PPS sampling q
(2) simple calculations for
P(ij~o
selectiond and (3) calculable
Sampford (1967) presented
samples
some
of size greater than twoo
methods
as
of
the
same
those obtained by Brewer and Durbin o
values
special
tabular
procedureso
his method would yield P(ij)
requirement
and
variance estimateso
would Q
of
Calculation
of the P(ij. for samples greater than two requires
or
selecting
For samples of size two Q
SampfordQs general procedure produces
P(ij)
sample
computer
Sampford also showed that
that satisfy
therefore Q
not
Hanurav~s
allow
second
negative
9
Jessen (1969)
feasible
samples
JessenUs
method
presented
methods
of
generating
for probability non-replacement sampling ..
three
His
probabilities.,
four
guarantees
method
four,
all
nonzero
which
pairwise
is applicable to
samples of size two onlYQ determined the P(ij) approximately
as
P(ij)
=:
P(i) P(j}-i
where
W
Jessen
~djusted
= l/f 1(1-1)
11 N
] S S W(ij) <>
il:j
these approximations numerically to
satisfy
the constraints
N
S
P (ij) = (n..... 1) P (i) ..
j"'i
The selection of P(ij) in method
based
on
the
four
by
Jessen
desirability of stabilizing the i(ij) in the
variance formula for the Horvitz-Thompson estimator..
showedC/
that
unless
possible for
~ll
criterion
minimizing
~f
was
W(ij)
all
the
P{i)
to
equal
W.
are
Jessen
eqaa10 it is not
Jessen
proposed
the
• If
S S W2 (ij!
il:j
as a measure of "goodness of fit" of the P(ij), but
not
attempt
to
obtain
an
exact
he
did
solution subject to the
constnaints on the P(ij)<>
Rao
and
Bayless
(1969)
examined
the
stability
of
10
population total .stimators and of variance estimators for a
number of sampling schemes including several
the
estimatoco
Horvitz~Thompson
included
empirical
different
finite
comparisons
The
based
populations
and
which
utilize
approaches
on
a
used
number
comparisons
of
based
on
expected ,variances under a superpopulation modele
Dodds and Fryer (1971) studied parametric solutions for
the
P(ij)
as
a
generalization
to
the
sampling
proposed by Brewer for samples of size twoo
three
general
parametric
For
scheme
of the
~ach
schemes u the algebraic solutions
for the P(ij) produced valid probabilities
over
a
limited
region of the parameter spaceo
Dodds and Fryer examined the
problem of choosing parameters
empirically
super population modelo
Using
nonzero intercept g Dodds
general
criterion
variance
as
parametric
had
~
and
of
Fryer
arrived
at
the
been
proposed
representation
the parameterso
spaceo
boundary
solutions
of
the
by
Raj
(1956a)0
same
The
P(ij) allowed Dodds and
by
varying
the
In testing their approach u they
tested
the
limitations
on
were
possible
were directly related to the relative
values of the minimum and the maximum size measures
population sampledo
the
of the acceptable region of the parameter
They also noted that
parametric
a
for minimizing the expected value of the
found that the solutions obtained in each case
on
through
superpopulation model with a
Pryer to approach the minimization problem
values
and
of
the
11
1 .. J Ihe_
Varian~e
Qf..L..lillite
~PoP1!latiQ1LPlllleter:
In the strictest sense o the only parameters of a finite
population
are
the
actual
values
characteristics of all of the N
The
frame
of inference normally
is limited to the
contrast
are
units
to many other areas of
based
on
assumed
in
~ssumed
population
~inite
of
the
the
populationo
in sampling theory
itself..
statistic~
distributions
observable
for
This
is
in
where inferences
hypothetically
infinite populations ..
The strict interpretation
makes
it
very
difficult
estimation schemes..
must
be
based
would
finite
sampling
theory
to compare alternate sampling or
The only valid
methods
of
comparison
on full knowledge of the entire population;
this is not useful for planning
sampling
of
be
or
design
purposes
since
unnecessary if the researcher possessed
full knowledge of the population parameters
in
sampling..
one
Furthermore g
knowledge
about
advance
of
particular
finite population .ay not be assumed to be characteristic or
typical of other finite populations..
Following this line of
reasoning leads one to ,;oncledef) as did Godambe P 955)
for
finite
populations
minimum variance
efficient,
schemes
and
assumptions
estimator~
if
not
always
estimation
or
that
there does not exist any uniformly
Practitioners u who are responsible for the
of
l!
guesses
optimal,
procedures o
about
the
deve10pment
sam pIp
must
make
properties
selection
several
of
the
12
population being sampled and then use these assumptions as a
basis
for
choosing
a
particular
sample
design
and
a
particular estimation procedurec
When =onsidering PPS sampling procedures
option q
it
is
useful
to
error termc
error
ele.en~s
possible
write the characteristics to be
observed as a function of some
known for all
a
~s
measure
of
size
which
of the population plus a deviation or
Further assumptions can then be made
terms
with
variances D and
respect
their
is
~bout
the
to their expected values q their
covariancesc
Such
models
for
the
characteristic values to be observed in a sample survey have
been called superpopulation models or hypothetical
infinite
population modelsc
Cochran (1939 u PPc
the
superpopulation
4g4~491)
model in considering ways of improving
saaple-based estimates of timber
estimates
for
all
volume
obtaining
relationship
and actual measurements c
Jessen (1942 u ppo
by
eye
plots in the universe studied and Bsing
the sample plots to estimate the
estiaates
developed an early form of
44-48)
1
Smith
between
(1938~
ppc
and Kabalanobis (1946,
pv
eye
4~9)D
368~
used model approaches to study optimum plot 3r sampling unit
sizes
for
studying
statisticso
The
crop
yields
and
other
concept of superpopulation
by Cochran (1946) in studying the properties
samples;
by
sample from
treating
i
the
entire
finite
agricultural
mod~1
of
was used
systematic
population as a
hypothetical infinite population e Cochran
was
13
able
to construct and interpret correlograms based upon the
finite population datao
Superpopulation models have since been used extensively
in
the
finite
population sampling literatureo
Several of
the studies discussed above were based, at least in part, on
consideration
of
group are Raj
(1956a),
(1969),
to
cases~
variance
the
superpopulation
expected
estimatoro
construction
all
(1967),
Rao
and
Bayless
models
have
been
study properties such as the expected value of the
variance or the
of
Hanurav
Included in this
Dodds and Fryer (1971)0
~nd
In most
used
superpopulation modelso
the
~f
value
Both
of
of
the
these
variance
cases
of
the
involve
the
finite population parameters as
units
in
a
function
the finite population and using the
superpopnlation model to obtain the expectet value
of
such
parameters over all hypothetical populations as generated by
the
model~
Consideration of these superpopulation
expected
values has not provided any direct optimization criteria for
the values of the
P{ij)
under
the
usual
superpopulation
model assumptlonso
The approach taken in this research has been to use the
super population
variance
Horvitz-Thompson
estimatoro
snperpopnlation
of
the
variance
Minimization
of
of
the
this
variance under appropriate assumptions does
lead to a direct solution for the P(ij)o
Some justification
for this approach has already oeen liven by Jessen (l969 g po
14
191) :
"The ten schemes selected from the literatuce and
the two introduced here vary widely in their
variances of estimated totals~ Y. when applied to
different populationso
It would be helpful to
have some studies describing the
nature
of
populations met or likely to be met in practiceo
until more is known about natural populations it
may be wise to i~~hose methods ~~ose aGaura~I
is_!~ast se!!2itive 1Q_~lation_characte£istic!!o"
(Underscoring added)c
1
Chapter
reasonable
2
superpopulation
Under
0
model,
this
superpopulation
expectation
Horwitz-Thompson
estimCltor
is
proposed
shown
is
it
the
of
is
model
variance
in
that
the
of
the
not a function of the P (ij)
[This result has previously been shown by Godambe (1955,
216) ]0
It
will
further be shown that the superpopulation
variance of the variance of the
is
a
function
of
the
HorVitz-Thompson
as
the
means
estimator
ftiniaization
P (ij) •
superpopulation variance by proper choice of
proposed
the
of
is
to protect against unusually large
of the particular realization
from the distribution of
this
P(ij)
variances of the Horvitz-Thompson estimator occurring
result
p..
all
~f
possible
represented by the superpopulation
.odel~
~
as
a
finite population
finite
populations
15
2..
THEORETICAL DEVELOPMENT
2 .. 1 :the Basic
Sup~rpopula1:ioD
,!ggel
2.. 1.. 1 the Punct.ional Rela1:ionshiJ,LBetweell , I1il
The model considered most relevant
Horvi1:z-Thompson
estimator
for
.~nd
Xli,,)
study
of
the
and probability non-replace.ent.
sampling is
Y (i) .- BI (i)+ e (i)
where
B
= an
unknown constant. D and
e(i)- the error term which measures the
deviation
from
the model ..
As pointed out by Horvitz and Thollpson
t(slpnr)
will
have
zero
proportional to Y(i) for
designed
the
to
pel)
t(slpnr:
variance
i..
all
if
The
P(i}
proposed
to
I(i)g
a
nonzero
intercept
saaplede
is
By aaking
variance
of
with
a
non-
Such a model may be written as
where 1 and B are both unknown
is
model
was more realistic for the population to be
Y {i} :: A + BZ (i)
Z(i)
exactly
occurs only if some of the e(i) are nonzero ..
Suppose one believed that a linear model
zero
is
take advantage of this possibility..
proportional
p.. 669)D
(1952 v
a
measure
of
size..
but
+. e (1)
nonzero
constants
and
Use of the Horvitz-Thompson
estimator and probability non-replacement sampling with P(i)
proportional to Z(i) would not provide the opportunity for a
16
zero variance of t(slpnr) even if the eli) were all zeroo
zero
variance
would occur with the eli) equal to zero only
if the P(i) were
Unless
one
exactly
proportional
to
[AlB
+
Z(i)]o
wishes to rely upon an extremely fortuitous set
of error ter.s e elite to rescue the
plans
A
from
producing
a
large
design
variance
and
of
estimation
t(Slpnr)e it
appears mare appropriate to use the no-intercept model
Y (i) = BI (i) + e (i)
where
Xli)
To define f(i)-e
individuallYe
it
but
is
only
= lIB.
not
Z(i)o
necessary
their
to
ratioo
know
Bven
1
such
B
a poor guess
should lead to a better deSign than one that sets
in
and
the
PCi)
a way that the principal advantage of PPS sampling
cannot be achievedo
In summarYe the premise put forth here is that
uses
the
Horvitz~Thompson
estimator
in
is
to
set
If
the
no... intercept
proportional
to
to
model does not present a tJood
approximation of the Y(i) to Xli) relationshipu
course
course
P(i) exactly proport:ional to some known
Xli) which is believed to be apprOXimately
Y(i) "
one
conjunction with
probability non...replaceaent sa.pling u the proper
follow
if
the
proper
to follow does not involve a magical choice of P(ij)
to compensate for this shortcomingo
selection
of
some
It
does
involve
the
other variate as a measure of sizeo
It
should be noted that Xli) could be defined as a function
of
11
several other known variables.,
Consideration of an intercept model can be defended
the
instance
of
multi-purpose sample surveys for modeling
the variance of estimates associated with
purposes
of
the
surveyo
some
less
fits
the
major
The choice of the P(i) and P(ij)
should probably still be based on some known variate u
which
in
I(i)u
no-intercept model for the particular Y(i)
that is associated with the major purpose of the survey.,
2., 102
~um2t:iqns
t:~!L B{£o£
about
ler.~
Suppose that the sampling statistician assumes the
no-intercept
available
model
to
and
him
utilizes as much information as is
to
construct
proportional to the unknown Y(i).,
about the error terms
appear
reasonable
(2) B[e(i)/I(i)]2
=1,2 u.,
= S2 (i)
<I"
approximately
The following assumptions
collection stage of the sample design
(1) Be (1) = 0 for i
Xli)
l!
in
the
pre-data
process~
Nu
(finite) for i
=
1£>211.,.,~uNu
and
(3) the error terms are independento
The assumption of independence is not
the
I(i)
unreasonable
are properly chosen and if stratification is used
to its fullest extent consistent with reqUirements
estimability
PPS
sampling
utilized.,
if
of
is
the variance.,
applied ll
deep
for
the
For most applications where
stratification
is
also
18
If one first assumes that the value of B is
that
the
error
terms
elements" then the
Prior
to
sum
to
zero over the
independence
sampling,
however,
assumption
N
is
and
?opulation
erroneous..
B is not known and cannot be
estimateda
In this • 2!iari sense,
terms
be
may
fixed
independence
of
error
assumed as an indication of complete lack of
further knowledge about the Y(i) given the I(i)o
2.,2 the ~u2erpo2ulat.ion.YAriango(~~
'ini~e
.PapIlatia! ,agiance
1 particular outoome of the super papulation
model
may
be represented by an N-dimensional vector of error terms and
designated as!c>
The transpose of
~
may be wrictten as
[e(l) ,e(2) "e(3) ,,';~o&>e(N) ]0
Por brevityu the Horvitz-Thompson estimator viII be
by
t(s). and its variance for a particular finite population
CWo, for a particular
The
~.
indicates
selected
from
indicates
a
a
the
In
the
~)
sample
finite
of
R
N
population
the
expectation of V[t(S) I
finite
population
~
s
elementso
sense,
So
the
variance
model
] over
variance
to
all
for
is
The finite population
variance is still a function of the error vector
under
The
sample of n elements selected from
computed over all possible samples
reasonable
]0
error terms
superpopulation~
hypothetical
particular
~
will be denoted by V[t(s)1
particular
the finite population of
The
denoted
consider
possible
~o
It
taking
outcomes
is
the
~o
the HorVitz-Thompson
19
estimator may be expressed in terms of
model
selectively
by
the
super population
substituting nl(i)/X(+) for P(i) and
1.1.
BI(i)+e(i) for Y(i) in the expression given in Section
lfter some simplification this yields
n 2 1- 2 (+) Vet (s) I 2 ]
N
=S
[B+e (i) /1 (i) ]2P (i) (
1-P (i)
]
i=l
!
!
+ 55 [B.e (i) /1 (i) ][ B+e (j) /X (j) ][ P (i j) '~P (i) P (j) ].
i#j
lfter
collecting terms in B2 and B" this expression becomes
N N
!
82 {
S P(i)[l-P(i)] + 5 5 [P(ij)-P(i)P(j)]}
i=l
i#j
N
+ B (2 S [e (i) /X (i) ]P (i) [ 1-P (i) ]
i=l
NR
.. S S [e(i)/ICi) ][P(ij)-P{i)P(j: ]
i#j
N N
+ S 5 [e (j )./1 (j) ][ P (i j) - P (i) P (j) ]}
i#j
R
• S
[e(i)/X(i)]2P(i)[l-P(1)]
1=1
N N
.. S 5 [e (1) /1 (i) ][ e (jl/l (j) ][ P (ij)-P (1) P (j) ) ..
i#j
lfter noting that
N
5 [P (ij) -P (i) P (jl ] :: -P (i) [l-P (i) ]"
j#1.
the expression simplifies further to
20
N
n 2 X- 2 (+) i[ t (s) 1
]
~
=S
[e (i) /X (i) ]2P (i)l l-P (i) ]
i=l
N N
+ S S [e (i) /1 (1) ][ e (j) /X (j) ][ P {ij)-P (i) P (j)] ..
i~j
Given the assumptions about the error terms, e(i),
in
Section
2 .. 102,
the
~s
stated
superpopulation expectation of the
variance is simply
EV[ t (s) I
=
]
~
N
n~2X2
(+)
5 P (i) [l-P (1)
]S2
(i) ..
i=l
Under the assumed model, the expectation of the variance
not
a
is
function of the pairwise probabilities and does not,
therefore,
p~ovide
any basis for selection of
the
pairwise
probabilities ..
Since the particular choice of
cannot
be
used
criterion must
statistician
large
to
be
reduce
variance ..
want
To
probabilities
the expected variance, another
developed ..
shDuld
pairwise
to
protect
Intuitivel"
the
sampling
pnotect against an extremely
against
a
large
variance
occurring most of the tiae, the sample design should produce
variances that have
about
their
expected
suggests minimizing
finite
a
the
tight
value..
This
population variance; this
the
results
variance, V2[t(S) I
obtained
~
~
distribution
intuitive
superpopulation
V{ V[ t (s), § . ]) = I( V2[ t (s) t
Osing
super population
varianc~
variance
the
of
the
may be written as
. ]} - B2{ V[ t (s),
above,
criterion
square
]0 may. be represented as
~
]} ..
of
the
21
N
n.I~.
(+) V2[
N
t
(5)
I
~
] = { 5 [e (i) /1 (1) ]2P (i) [l-P (i) HZ
i=1
If
+ (5 5 refit/XCi) ][e(j)/X(j)][P(ij)-P(i)P{j) l}Z
iflj
If
+ 2{
5 [e(i)/X(1) ]2P(i)[1-P(i)])
i=l
X
N N
( 5 5 [e (i) /1 (i) ][ e (j,/I (j) ][ P (i j)-P (i) P (j) ]}e
i~j
The third term u or the cross product term
have
a
zero
can
be
seen
to
expectation under the assumed superpopulation
modele
lfter expandinq the first two terms and notinq which
terms
in
the
products
superpopulation
BVZ[t(s),
~
BV 2 [t (s) t
have
expectation
nonzero
the
of
expectation,
squared
the
variance,
]0 may be represented as
~
] :: n.... ·X· (+)
[
•
5 E[ e (1) /1 (1) ].pz (i)[ 1-P (1) ]Z
i::1
If If
+ S 5
S2
(i)
S2
(jl P (i) P (j):[ 1-P (i) ][ 1-P (j)]
iflj
N N
+ 2 55
s2(i)S2(j)[P(ij)-P(i)P(j)]2}o
iflj
It most be further assumed that £[e(i)/X(i)]4 exists and
is
finiteo
The square of the superpopulation
variance q 8 2 V[t(s).
~
expectation
]0 may be written as
N
£2V[ t
(5)
I
~
of
] :: n-.X. (+) ( 5 s· (i) pz (1) [ l-P (i)
i=1
F
the
22
11 11
+ S S
S2
(i)
(j) P (i) P (j)[ 1-P (i) ]( 1-P (j»))e
S2
i~j
Combininq the above
B2V(t(S).
~]
results
EV 2[t(s)t
for
]
~
and
to obtain the superpopulation variance of the
finite population variance produces
I
n.X-. (v)VV[t(s) I
~
]= 5 (E[e(i)/J:(i) ]......s.(i) }P2(i)[ 1-P(i)]2
1=1
11 If
+ 2 5S s2(i)5 2 (j)[P(lj):"P(1)P(j)]2
0
i"j
To miniaize this variance by choice of
necessary
P(ij'
it
u
is
only
to consider those teras in the expanded fora that
are a function of
the
P(ij)
This
c
approach
reduces
to
ainiaizlnq
11 If
5 5
r. P (1)
P (j)
-
P(lj:
]29 2
(i) S2 (j),
l"j
or, in aore compact notation
If
H
5 5 W2 (ij)s2(i)s2(j'
0
i"j
If the
s2(i)
are
constant
over
all
elements,
i
o
this
criterion reduces to the one suggested by: Jessen (1969)0
~9I.yt-iQn .15!.L.Qpti.a!~Valu~s
203 IU-61gebJi:lic
The solution for the P(ij).
must
satisfy:
constraints
H
S
j~i
P(ijj :: (n-.1)P(i)
0
fi .P,Ii j)
the
set
of
23
In terms of the W(1jl" these constraints are
equivalent
to
requiring that
II
S i (ij)
= P (i) [1-P (i)]..
j"1
The aethod,of
obtain
the
Lagrange
constrained
subsequently for the P(ij) '"
multipliers
sO~(Jtion
may
for
The function
be
the
to
used
i(ij)
be
to
and
minimized
lIal be written as
11 I
P(;i(ijl"K(i):i#j} = 5 5'W 2 (ij)s2(1)s2(j)
1#j
II
-
Ii
4 5 I (ll{ 5 'I (ij),- ,p (i) [1-P (i) ]) ..
i=1
j#l
In·the abovei :the lei) are the Lagrange
multipliers~
Taking
derivatives with respect to the W(ij) and setting them equal
to zero as a necessary condition
for
obtaining
a
ainimum
produces 1(1-1)/2 independent equations of the form
Wtij)
= 5- 2 (i)s-2 (j)(K (i)
+ It (1)) ..
Taking derivatives with respect to the K(i) and setting them
equal to zero yields N additional equations of the form
H
5 W(i j), = P (1)11-P (i) ] ..
j#.i
The
entire
simultaneoQs
process
equations ..
yields
The
a
number
set
of
of
equations
solved initially can be reduced by applying the
of
H (N+ 1) /2
second
to be
set
equations to the first set to obtain 1 equations in K(i)
only:
24
P (i)
N
r l=P (i)]5 2 (i) = K (i)
N
.. S K (j) 5- 2 (j)
5 5- 2 (j)
j~:i
=
jtlii
I
K (i) [
N
S~2
5
(j)
~
2s- 2 (i) ] •
j=1
5 K (j1 5- 2 (j)
e
.;" 1
This set of equations
may
be
represented
in
matrix
notation as
="
£
!.,
The Nxl vector, £0 has elements c(i) defined as
c (i) = PCl) [1·-t- 'i)
The
Hxl
vector
5
moltipliers g K(i)o
is
the
]S2 (i)"
vecto£
of
unknown
La<jrange
The IxN matrix, K, may be represented
"=D
+
where D is a diagonal matrixe
a~
1 12'
The diagonal
ele.en~s
of
the
matcix D may be expressed as
= Sf
deli)
The term
5(1)
... '2s-~
t)
(i)
c
is defined as the folloWing soa:
N
Sf
I)
=5
S-2
(t) "
1t=1
The
vector! consists of I ones"
b (i) ==
If the two soms
5(
2)
:::
N
5
5«2)
and
5(3)
The vector
5(2) _
ele.en~s
8- 2 (i)"
are defined as
N
~
(kl d..,.l1 (kit) and S (3)> = 5 b (It) c (k) d-=,i (kt)"
It=l
and if
R has
It=l
-1,
then~
as is shown in Appendix
simultaneous equations in K(i) have
K(i) =
a- ll (ii)[::(i)-(l
so~utions
8.,9)
<jiven by
+ 5(2»)-rS>!31>1.,
the
N
25
Having obtained algebraic solutions for the
algebraic
so~utions
for
the
W(ij)
may
substituting the solved K(i) values in
be
the
I(i)o
obtained
first
the
by
1(1-1)/2
equations q namely as
= S-2 (i) S..,2 (j) {K (i)
W(ij;
+ K (j)
I.,
The P(ij) may be obtained as
P (ij)
To be valid
probabilitJ
= P (i) P (j)
solutions
sampling
in
-
W(ij)
the
0
sense
of
yielding
design, all of the PCij;
a
must satisfy
the conditions
o
S P(lj) S
minimum{P(i)iP(j~}.,
These may be called the "weakconstraints R .,
An eyen stronger set of constraints on
lo~q
considered in Section
the
P(ij)
was
namelJq
CP(i)P(j) S P(ij
S P(i)P(j}
where
o <
C S
{n... 1) /n.,
These may be called the -strong constmaints·o
The methodology for adjusting the solutions to
either of these constraints is given in the next
satisfy
section~
2,'J W!l§tmen! P£Qc!du!i!s
2. 'J. 1 fiuna!
.!IU!!:g~ch
The phenomenon
optimization
problems
sampling theorJm
has
of
adjusting
to
fit
algebraic
reality
Neyman allocation in
is
solutions
to
not foreign to
stratified
sampling
been known to allocate more sampling units to a stratum
26
than
exist
(Cochrane
stratu.~
to
in
The
stratumo
pol03)
1963 Q
is
the
corrective
procedure
to enumerate the overallocated
adjust the cost or variance
allocate
stratao
that
remainder
and
functions~
proceed
of the sample to the remaining
It may then again occur that the sample
allocation
exceeds the size of some strata and the procedure becomes an
iterative
which
process
prescribing
full
enumeration
to
a
after
terminates
ultimately
number
of strata and
allooating the remainder of the sample in an optimum wayo
Three alternative methods for adjusting the P(ij)
the
ahen
algebraic solutions fall outside of prescribed boundary
values are discussed
belov~
constrained
(1)
minimization Q
(2) compensating additive adjustments, and (3) sampling unit
re-~~finittono
among
these
The
three
choice
of
adjustment
particular
application Q
populationsize Q and other design options available to
the sampling stattsttciano
This topic 1s treated further in
Chapter 4 when specific applications are
discussed~
All three adjustment methods may be handled as
for
from
alternatives depends upon the nature of
the assumed variance model for the
the
procedure
adjusting
constnain~su
the
W(ij) rather than the P(ij)u
methods
The "weak
may be stated in terms of the R(ij) as
P(i)P(j)~lIinillulB{P(i)uP(j)}~
W(ij)
~
P(i)P(j) ..
The waak constraints g of course, do allow negative vdla0s of
1
! ~
" '., .J
21
o
where 0
S W(ij)
S
(l-q P{i)P(j)
< C S (n-1)/no
20~c2 ~~~hod l~_Cons!r~~d
Minimization
Method 1 is the most direct
approach
to
obtaining
a
near optimum solution when some of the initial solutions for
the W(ij) fall
prescribed
For
outside
the
intervals
determined
by
the
set
of
~onstraintso
discussion
purposes g
the
particular
constraints employed may be written as
L (ijl
S W(ij;
where L(ij) and O(ij) are the
S U (ij)
prescribed
bounds u respectivelyu for the W(ij)
lower
W(ij)
Suppose
r
bonndso
solutions
against
solutions
fall
their
below
upper
to
check
0
The first step of the Method 1 procedure
all
and
is
lower
their
bounds
respective
L(ij,o
lower
The corrective action prescribed by Method 1 is to
set the r associated values of these i(ij)
obtain
an
optimal
at
their
lower
bounds
and
solution for the remaining
W(ij)o
In terms of a constrained minimization u this amounts
to considering the additional constraints
W[ i( It) u j( k) ]
=
L{ i (k) • j (k) ] u k =1 u 2"
in the general constrained minimization
v
"
~ vr
problem
q
considered
earlier.,
The second step of the Method 1 procedure is
the method
~f
Lagrange multipliers to minimize
to
apply
28
N II
S S i
2
(ij)s2 (1) S2 (j)
1#j
sUbject to
N
S i (ij)
it' .L
i( i (k)
Q
=
P (i) f l-P (i) ]/1
j (Ie) ] = L[ i (k)
£
j (k) ]
This may be done by minimizing the function
N N
rf il(ij' ,K{i) ::i#j} = S 5 W2 (ij)s2(i)s2(j.
i#j
I
-
I
4 S K(i)r 5 W(ij) 1.=1
j"i
P{U[l-P(i)]}
["
-4 5 It (N+k)(
if i
(k)
/I
j (It) ] -
L[ i (k)
II
j (k) ]}.,
«=1
The expanded minimization problem now involves
multipliers,
algebEaic
condition
K{i),
solution
for
rather
given
obtaining
than
in
a
W(ij)
and K(i)
i(1j) and K(i)o
Section
2.,3.,
constrained
necessary
1
minimua
of
respect
the
to
be set to zero by appropriate choice of
This process
equations in the i(1j
Lagrange
the 1 used in thr initial
objective function is that the derivatives with
the
N~r
yields
1(1-1)/2
independent
of the form
r
W(ij) =S-2 (i) 5..,2 (j) (K (i) +1 (j) + S K (N+Ie) I[k I (ij) ]}
1e=1
where r[lq (ijJ] == 1 if [1{1t) "j(t)]
=
(ij) and IrkW (ij)
l::::
0
29
in
all
other
caseso
Setting
the derivatives taken with
respect to the K(i) at zero yields N equations of the for.
N
= P(i)[l-P(i)
S W(ij)e
jf!1
]
and r equations of the form
W[i(k)l1j(k)] = L[I(k)l1j(k)]o
The total set of 1(1-1)/2 + H
r
+
equations
can
be
reduced to H+r equations in the K(i)
S2
1
S
(I) P (i) [ 1-P (i) ] =K (i)
;-2
(i'
j~i
N
H
+ S K (jl S-2 (j)
r
S K(n+k) I[ k I (i j) ]
j#.i k=1
.. S
j~i
= I[i(k)]
s2[i(kl]S2[j(k) ]L[i(k),j(k)]
10
obtained
analytic
for
or
pattern
matDix
.. K(j(k)] to K(Ntok)
approach
has
been
solution of these simultaneous equationso
I+D is not exteemely large,
a
numerical
solution
can
If
be
obtained through use of standard matrix inversion procedures
employed with electronic computerso
limited
by
the
accuracy
The method is therefore
of computerized matrix inversion
procedures a
Given that solutions for the
solutions
for
W(ij)
may
be
substitution of the K(i) in
the
first
obtained aboveo
the
K(i)
are
obtained
obtained o
directly
8(1-1)/2
the
by
equations
30
The third step in the Method 1 procedure
the adjusted solutions for
U(ij}.
Suppose s
upper
bounds.
The
fall
prescribed
above
above their lower bounds.
their
adjustment
analagous to that prescribed above for
stay
to
check
against their upper bounds w
W(ij~
solutions
is
respective
procedure
adjusting
W(ij)
is
to
In terms of the constrained
ainiaization u s additional constraints of the form
W[ i (Ie) "j (k) ] ;: U( i (Ie),
for k ::
lo2~~~~os
The
fourth
minimization
inaluded~
for
],
are addedo
step
is
procedure
to
with
repeat
the
the
values
of
constrained
s additional constraints
As before o numerical solutions
I+r~s
the
1(t)
can
be
obtained
the Lagrange multipliers o K(i).
Thes9 may then be used to solve for the Weij) directly.
At first glance o it lay appear unnecessary to
the
lower
and
upper
bounds
separately.
consider
In most cases u
congtraints recognizing both upper and lower limits could be
added
to
the
minimization
problem
in
exception occurs when for some particular i
W(ij) in ascending order of size)
if
(ij)
<
one
stepo
(after
ordering
0
L (ij}
W(ij) > O(ij)
for j :: k+l, k+2 (/.
In this particular case, setting
,~No
if
(ij; :: L (ij)
The
31
= U(ij)
W(;.j)
for j= t+l,k+2 q c a • • Bo will usnally violate the constraint
N
S W(ij)
=
P (i) [ 1-P (i) ].
j~i
By handling
the
lower
and
upper
bounds
in
two
steps,
occurrence of this problem is avoided.
The entire process 3f checking
prescribed
solutions
against
the
bounds and addinq equations to the linear system
is repeated iteratively until all W(ij)
fall
within
their
upper and lower bounds.
2c4.;l t!ethog_~ CO.!Jl!!nsat-ing-!!!ditiv~.Adjl1stllents
Method 2 may be applied without accuracy limitations of
computer matrix inversion procedures.
W(ij) at a time
as
follows c
prescribed upper bound,
Suppose
W(ij)
9
S
N
in
the
W(ij)
constraints
W(ij)
j#i
the
=a
Corrective adjustments may be made
without violating the
exceeds
~oQ
W(ij) - U{ij)
and a>O.,
The method treats one
-
The reco••ended procedure is
P
(i)( l-P (i) ] .
.
0
(1) adjust W(ij) by "a,
(2) adjust W(ik) by +al {B-2) for k#i or jo
for kIt or jq and
(3)
adjust
W (jk)
by +a.1 (9-2)
(4)
adjust
W(klli)
by -2a/r (N<-2) (N-3) ]
m#i or j"
for
""i
me
j
and
32
If several W(ij) must be adjusted g they can be
individually in an iterative fashiono
for different V(ij)
the
iterative
Since the adjustments
lRay tend to work against each
process"
it
is
recollmended
iteration continue until sufficient
handled
that
numerical
other
in
the the
accuracy
is
obtained"
2,,'4.;4 Betbqg.-h_SaIRl!n.g.,jlnit
Under
sOlie
algebraic
COllllon
ae-Defin!~!2!!
superpopulation
models v
solutions for the VeijJ are associated with pairs
of sampling units g i and ju that have lIucb slIaller
of
size
negative
than
the
remainder
of
the populationo
measures
In lIany
applicatlonsl/ PPS non-replacement sallpling is applied at the
first
or
second
stages
of multi-stage samples"
applications, the primary sampling units
for.ed
generally
be
soaevhat arbitrarilyu and their definition is within
the contDol of the
that
may
In these
the
sampling
statistician"
problem of negative Veijl
This
suggests
be treated by adjusting
the stDuctur,e of the sampling frame by combining two or more
small
sampling
combining small
units
into
sampling
one larger sampling unit or by
units
with
~ny
other
adjacent
sampling unit"
Some side effects that might be expected
this
adjust.ent
~s
a result of
procedure are (1) to increase the expected
cost of data collection since the subsequent stages
sample
are
drawn
from
larger
first
average o and (2) to decrease the variance
stage
of
of
the
units on the
the
estimate
33
due to more dispersion of the ultimate sampling units across
the entire populationo
Method 3 is the simplest method of the three suggestedo
Its
general
effect
is
to
limit
application of PPS nOD-
replacement sampling to populations where the inequality
of
the .easures of si2e is not extremec
2c 5
~m~_~Q!lIeJ!ts~.Q1!.
gqp'~~ra1,nts
the Cho1,ce af
Is noted earlier, requiring that
P(ij)S P(i)P(j)
guarantees that
variance
~ll
will be
estimator
also
terms in the Yates-Grundy
non-negativ~
will
be
form
of
the
and the Yates-Grundy variance
non-negative
for
all
possible
sallpleso
The
initial
form
of
the
strong
constraints
was
discussed in Section 102 and was stated as
~
CP(i)P(j) S P(ij)
for.all P(ij) and some small
left-hand
j
P(UP(j)
C>O.
If
one
considers
the
part of the inequalities and sums both sides over
not equal i, the result is
CP (i) [n-p (i) ] S (n-1) P (i)
for all i
q
or after factoring out P(i} and dividing
by
[n~
P (i) ],
c
for all io
~
(n-. 1)1{ n-P (i) ]
If a single value
~f
; is to be used q it must be
less than the minimum value over all i
(n-.' 1) I[ n- P (i) ]0
~f
34
Setting C less than
or
equal
to
(n-1)/n
satisfies
this
requirement for an apper bound of Co
strong
If the
(n~l)/n,
some
applied with C set at
constraints are
immediate
effects
on the finite population
variance and variance estimates may be notedo
Stating the constraints in :erms of the W(ij)
o
The
nl (i)
right
x: (j)
S W(ij) S P(i)p(j}/n
term
hand
Tht
liZ (+) "
gives
may
be
also
variance
of
expressed
the
as
Horvitz-Thompson
estimator for designs employing this constraint will then be
bounded above by
I
[nIl (+)
..
51 (i) I (j)
i#j
]ZS
!)Z
(ij)
where
D (ij)
= Y (i) /P (i)
-
Y (j' IP (j)
or equi "alent 1 ,
::: [I (+l/n][ Y (i) I I (iJ -
D (ijl
~
(jJ
This apper bound is exactly equivalent to
the
total
estimate
for
probability replacement
variance
for
Stuart C1966 q po
terms
of
sam~
(j)
the
standard
1..
variance
of
Hansen~Hurwitz
This particular form of the
probability replacement sampling is developed
as an analogue to the
The
design~
the
f"{
non-~eplace.ent
form
by
Kendall
and
171).
constraint on the P(ij) may
be
expressed
in
coefficients Q w(ij}u of the YateS-Grundy variance
estimator by first noting that
35
1f
(ij) :: P (i) P (jl/l? (ij)
Then the effects of the constraints
L,
-
on
the
.1
(ij)
can
be
noted in the following steps:
(n... l)P (i) P (jl/n S P (ij)
S P (i) P (j)
Q
l/[P(i)P(j)] S l/P(ij) S n/[ (n-l)P(i)P(j)],
1 S P(i)P(j)/P(ij) S
n/(n~l)u
and
o
The
variance
S 1/(n-1)0
S v(ij)
estimators
for
designs
employing
this
constraint viII be bounded above for each sample by
n n
(n-1)-15 5D2(ij)o
i<j
This expression is exactly equal to
for
the
Hansen~Hurvitz
the
variance
estimate
probability replacement design wher
applied to samples of distinct elements (Kendall and Stuart u
1966/1 po
171)
0
It may then be concluded that application of the strong
constraints
C
with
equal
to
(n~l)/n
will produce designs
with variances and variance estillates that are never greater
than
those
that
would
be
obtained
under
probability
replacament designso
In this
proposed"
chapter/1
The
a
basic
superpopulation
is
relationship of the [(i) to the I(i) in the
stated model is one of direct proportionality plus
terti"
model
Assumptions
about
the
error term
~r&
ift
error
very Jeneral
36
requiring only that the errors have zero expectation and
independent.,
be
The expectation of the squared error terms is
left unspecified but must be finite.,
The
criterion
variance
of
of
minimizing
the
super population
the finite population variance is proposed and
the aethodology for obtaining algebraic as well as realistic
adjusted solutions is presented.,
The choice of criteria for
setting limits on the algebraic s01utions is
some
of
the
~re
estimator
discussed
implications on the variance and the variance
showno
It ihould be noted that the solutions for the
probabilities,
P(ijf,
a
sa.pie
of
pairwise
are obtainable for any sample size n
and are not limited to so1utions for samples
If
and
size
two
is
of
size
~he
desired,
two.,
pairwise
probabilities actually completely specify the sample designo
If
a
sample
of
three or greater is to be selected from a
stratu., a further challenge must be
identified
selection
by
scheme
probabilitieso
Horvitz
which
This
and
.et~
Thompson
achieves
general
three is addressed in Chapter 3.,
the
problem
This challenge as
is
to
desired
devise
the
pairwis~
for samples of 31ze
37
STRATUM SAMPLES OF SIZE THREE
3~
!~~g=!ng~£Gach
3.1
Although many survey designs which
non-replacement
sampling
employ
probability
also employ stratification to the
stratum~
extent of selecting two sampling units per
it
does
appear desirable to have available the option to select more
than two elements per stratum in some cases.
As an
example
if the total sample size allocated to a specific population,
sucb as a region g is an odd
nu.ber~
it is
not
possible
to
fora strata which permit the selection of two sampling units
stratum~
from each
allocation
of
If a
one
stratum
sampling
the
pairwise
established
unit q
the
~gain
nonestimability of variance must
of
is
old
be faced
with
an
problem
of
since
probabilities will then be zero.
one $tratum with an allocation of three sampling
tbe
some
Forming
units
and
remaining strata vith allocations of two sampling units
permits the use of a maximum
conjunction
with
degree
estimability
of
stratification
in
of the variance for all odd
sample sizes greater than or equal to three.,
In this chapter a method
is
three-vise probabilities II P(ijk)
values
for
discussed
the
in
P(ij)
Chapter
as
2.,
developed
function
determined
The
based
assu.ptious is minimized
on
SUbject
obtaining
which preserve the optimal
II
by
general
somewbat analagous to that used in
objective
for
Chapte£
the
to
the
procedures
approach taken is
2
in
that
superpopulatioD
certain
an
model
constraints.
38
The natural selection of the objective function turns out to
be the superpopulation variance
of
the
finite
population
third moment of the esttmatoro
lAg
3o~
?g~efe9!ula\~Vari,qc~of
t~eTbiE~~Homent
2!.!he Bo(vit!""'!aompsoD IstiJ!!!H
Proceeding
particular
finite
super population
vector v
as
in
Chapter
population
model
by
2
and
has
assuming
been
generation
that
generated
of
the
a
by the
error
term
an expression can first be obtained for the third
~,
moment of the Horvitz-Thompson estimator as a function of
As shown in Appendix 802 0 the finite
moment of the HorvitZ-Thompson
H [ t (s) I
~
I
estima~or
population
~o
third
can be expressed as
•
] = 5 [Y (1) /P (i) PP (i) [- 1-P (i) ][ 1-2P (i) ]
i=l
••
- 3 5'5 [Y(il/P(i) ]2[Y(j)/P(j)][ 1-2P(i) ]W(ij)
i~j
H• N
+ 5 5 5 (Y (i) /P (i) ][ Y (jllP (j)" ][ Y (k,/P (k) ]Q (i jk)
itlj,.k
where
Q (ijlG)
= P (ijlt) -P (i) P (jl P (It) +J] (i) W(jlt) +P (j) Ii (ik)
The tbird moment may
be
expressed
in
+J] (It) i
terms
(ij).
of
0
the
superpopulation model by selectively substituting nI(U/X(+)
for
pet)
~nd
BI(1).e(i) for Y(i)o
and cancellation of terms in 8 3
0
lfter some simplification
82
0
and Bo this yields
N
n:JX- 3 (+) II C t (s) a !it ]= 5
31
i=i
[3
Ii) "I (i - . PP (i)
>
l-P (i) J[ 1·· 2P {iJ ]
39
N N
~.
]S S( e (i) /x ti) ]2[ e (j) /X (j) )( 1-2P (i, ]W (ij)
ilj
N N
»
+ S S S( e (i) /x ti) ][ e (j) IX (j) ]{ e (k) IX (t) ]Q (ijt)
0
iil:j~k
Given the model assumptions stated
the
superpopulation
in
section
201~2v
expectation of the third moment of the
Horvitz-Thompson estimator is simply
EI![ t (s) I ~
]
=
$
N
=( X (+} /n]3 S E( e (i) /X (i) ]3P (1) [ l-P (i) ][ 1-2P (i) ]0
i=l
If it could be further assumed that
B[ e (i} /X (i) ]3 = 0 6
as is the case if the eli) are
dist~ibutionu
third moment of
In
zer.oo
then
the
the
generated
Horvitz-Thompson
normal
estimator
would
of
~f
the
P(ijk)o
the
the
P(ijt)~
third
best
moment
that
statistician can hope to do is to protect
moment
is
the
not
a
sampling
against
a
large
(in absolute value) by choosing the P(ijk) to
lIin!mize thf' superpopulation variance ·:>f the
third
moment o
This variance may be expressed 'is
VI! ( t (s) I !l ] = B{ M [t (5) t ~ ] ~
be
any case o the superpopulation expectation of the
Since the expectation of the
third
the
super population expectation of the
thicd moment is not a function
function
from
3
or in terms of the model paralleters as
BM [ t (s) i §
3
]
J2
N
[X
/n ]-6VM [ t (S) I
(+~
~
]=E{ S{ e {i)/1 {i) TIp (i)[ l-P (i) }[ 1-2P (1) ]
i=1
3
N
-
5 8{e(i)/I(i) ]3P(i)[ l-P(i) ][ 1-2P(i) ]
i=l
N li
-35 5[ e (i) /1 (i) ]2[ e (j)/X (j) ][ 1-2P (i) ]11 (ij)
i"'j
N N N
+5 5 S[e(i)/X(i) ][e(j'/I(j) ][e(k)/I(t) ]Q(ijk) }2 c
i#j#k
It must be assumed that the first siX moments
exist
meaningfullYG
sums
and
P(i1k).
with
to
order
in
consider
The terms associated
their
cross
eli) /X(i)
of
the
above
with
the
expression
first
three
products clearly do not involve any
Cross product terms of the terms in the fourth
terms
in
the
first
super population
expectation
assumption
that
e (jl/I (j)
operator
q
and
one
or e It) /x (t)
with
a
power
three
due
or
sums
to
clearly
the
sum
have zero
independence
more of the terms e(i)/I(i)o
will lppealJ after the expectec' value
of
1
and
will
thus
have
zero
expectationG
For the purposes of
third moment through
minimizing
~ppropriate
the
variance
of
the
choice of the P(ijk)v it is
sufficient to consider only the term
N N H
(S S S[ e (i) /X (i) ]( e (1) /1 (j) ][ e (t) /X (t) ]Q (ijk) }2
i .. j#k
whose expectation ander the assumed model is
41
N N N
S S ss2(i)S2(jrS 2 (kl0 2 (ijk'.
i#j#k
3.3.1 The
The
G~!§!al§olution
so~ution
for the P(ijk) must
satisfy
the
set
of
these constraints are equivalent
to
constraints
tI
S P (i jk) = (n"92) P (ij)
k#i or j
In terms of the Q(ijk)
0
requiring that
II
= R(ij)
S O(ijk)
It#i or j
wher.a
R (ij) = 2{ 1-P (i)-P (j) }i
The method of
obtain
the
Lagrange
multipliers
solution
constrained
subsequently for the P(ijk,.,
(ij~
for
.,
.ay
be
the
The function to
used
O(ijk)
be
to
and
minimized
may be written as
II tI If
F{Q(ijkloL(ij}:i#j#k)=S S S s2(i)S2(j}S2(k)QZ(ijk)
i#j#k
N N
~
In this problem u the L(ij)
Taking
N
6 S S L(ijl{ S Q(ijk)"'R(ij)].,
i#j
k#i or j
derivatives
with
are
the
Lagrange
multipliers.,
respect to the Q(ijk) and setting
tbem eCjual tc zero as a necessary conditior for)btaining
a
42
minimum
produces 8(N-1) (1-2)/6 independent equations
~f
the
form
Q (ijk)
=S-'2 (1) S~2 (j)S~2 (k)
{L (ij) +L (it) foL (jk) }"
Taking ierivatives vith respect to
the
and
L(ij)
setting
thea equal to zero produces 1(8-1)/2 additional equations of
the form
N
= R(ij).,
S Q(ijk)
klt1 or j
The
entire
simultaneous
process
yields
The
equations"
a
set
number
of
of
(1-1)1(1+1)/6
equations
solved initially can be reduced by applying the
of
equations
to oe
second
set
to the first set to obtain 1(1-1)/2 equations
in the L(ij) only:
B(ij~s2(i)s2(jt=L(ijJ
=L (lj) [S< 1
) .... 2s- 2
N
N
S S-2(t). S [L(ik)+L(jk) ls-2(t)
klti or j kit! or j
N
(1) -2s- 2 (j»)+ S
8
L (it) S-2 (It)
5 L (jlt)
..
S-2 (It) "
K'J
k,i
These equations can be expressed in aatrix notation and
a numeric solution is theoretically obtainable for the L(ij)
and therefore for the entire
viII
I(N-1)/2
numeric
.dtri~
ordinarily
be
usually
system
to
the
general
sampler~
solution
the
solution
super population .odel"
for
the
L(ij)
will
not
The problem that
remains to be solved is to obtain an analytic
of
Since
equations"
be excessively large for the use of
procedures o the
useful
of
under
c~presentation
the
general
43
an
Although
intractable,
an
analytic
~nalytic
constant over all i.
same
solution
general
appears
solution is obtainable if s2(i) is
This does not require the use
assumption in computing the
of
the
since the values of
P(ij)~
the P(ijk) are constrained to satisfy
the
regardless of how they were obtained.
The solution obtained
for the P(ijk} will be optimal in the
sense
the
superpopulation
variance
only if the s2(i) actually
usable
of
can
P(ij)
of
the third
be
solutions
minimizing
~oment
assumed
0f t(S)
constant,
bu.
solutions may be obtained even if the equal variance
It
assumption does not holdc
should
be
noted
that
the
optimal properties of the P(ij) are preservedo
If the s2(i) are constant, the
equations
N(H-l)j2
in
the L(ij) become
H
R(ij)=(N-4)L(ij)+
~
N
L(ik)+ S L(jk).
k#i
k#j
This system of equations may
le represented
in
matrix
form as
The N (N'~l) 12 x 1 vector. £u has elements B[ k (ijY ]
k
is
associated
with
the
pair
of
labels
~
here each
(i<j) by the
relationship
k(ij)
H
is an
defined
N (N~1) 12
itS
;.::
(i-~1)
square
if .-. i (i-1, /2 .. j
lllatrixu
The
~
i_
elements
of
Hare
44
h[k (ij) "k (ij) ]
=
N-2"
h[ It (i j) " k (i j I)] = 1"
=
h[k(ijr"k(iOj)]
10
and
h[k(ij)" k(i'j')]
= 0,
where i'#i or jq j'#i or j" and iO<jOo
vector
of
unknown
Lagrange
The vector y is
multipliers
also
the
indexed by
k (1j):0
The inyerse of
H
can
be
obtained
by
defining
the
elements of B-1 as
h--:[k(ij)fk(ij)]
= ao
h-[ k (ij) uk (ijO) ] = b,
h....[k(ij) uk(i'j! ]
= b,
and
h-[k(ij)lik(i'j')]
with i' and jO limited
IS
= c"
in·80
If the product of H and B-1 is then set
equal
to
the
identity matDix" three equations in,a" b o and c are obtained
as follows:
(N-2) a + 2(H-2)t = 1"
a + 2(N-2)b + (B-3}c
= 0,
and
4b + [(N-2) +2 (N-it) ]c = 00
These equations may be solved to obtain
a = (3{f2-18N+26)/[3 (N-2) (N-3) (1-4) ],
b ;: - (3N-10) 1[6(N-2) (N-3) (N-4) ]/1
45
and
c = 2/[ 3
(N~2)
(N-3) (N-4) ]0
It may be verified by computing 88- 1
does
produce the inverse
~f
=I
that this
approach
Solutions for the L(ij) may
80
then be obtained by computing
After some simplificatioD g the individual solutions for
the L(ij) may be written
~s
L (ij) = [3 (N-2) (N-3) (N-4) ]-1{ 3 {N-2) (N-3) B(ij)
N
... 3(N-2)j2[
S
R(ik)+
t#i
For this special
case u
I
N •
S
(jt) ] .. S S R(kl) }e
k#l
k#j
the
solution
for
Q(ijk)
may
be
wrixten simply as
Q(ijk)
=
L (ij) .. L (ik) .. L (jk)
0
Given the Q(ijk)u the solutions for the P{ijt) are
P(ijk)=Q(ijt)+P(i)P(jrP(k)~P(i)W(jk)-P{j)W(ik)-P(k)W(ij)o
These solutions are algebraic solutions
not
be
valid
probabilities
in
some
only
caseso
and
~ay
The minimal
constraints on these solutions should be that
o S P (ijk) S minimum{ P (1j) t>P (ik) liP (jt) }o
The methodology for adjusting algebraic solutions
do
not
sectiono
satisfy
these
constraints
is
given
that
in the next
46
Technicallyo
approaches
three
for
0
Method
P(ijk)o
considered
const.rained
feasible
only
for
saal1
general matrix solution approach
algebraic
discussed
the P(ij) in Section 2 u can be adapted to adjust
adjusting
t.he
the
solut.ions;
it
is
minimization
is
populations when the
applied
in
obtaining
requires numerical inversion of a
mat.mix with dimensions greater than 1(1-1)/20
Method 2 0 compensating
applied
iteratively~
a factor "a"
adjustments D
say
be
The steps required to adjust P(ijk) by
are~
(1)
adjust P (ijk) by a.
(2)
adjust
P (ij/lk)
all
P (te jk 0)
probabilities
of
form
the
P(ijk 8 )
,
and P(i'jk' by -a/(N-3);
fI
(3) adjust all
(4)
additive
0
probabilities
of
the
form
P(ij:'k ll ) "
and P (PjllJt) by +2a/[ (N-3) (1-4)]; and
P(i~jUk"
by
parenthesis
are
adjust all probabilities of the fora
-6a/[ (N-3) (N-ft) (N.-S) ]0
where sampling
unit
distinct u
iG#i or j or k o j'#i or j or t
to
and
labels
within
each
tests of this method indicate slow
algebraic
solutions
to
acceptable
initial algebraic solutions have
u
tG#i or j or
convergence
solutions
only
one
or
of
the
unless
the
two
values
outside the acceptable rangeo
If it makes sense to combine small samplinq
larger
units
in
the
particular
application~
uni~s
Methvd
into
3.
41
sampling unit redefinition g may be
strongly
recommended
as
a
means
P(ijk) since the other two methods
~pplied"
This method
is
of obtaining acceptable
are
either
numerically
difficult or slow to converge"
3" 5
The theory
Chapter
3
and
designs
for
~!..Y.!~.2f
developed
applied
samples
probabilities g
in
of
the
Chapter
2
is
extended
in
to the problem of obtaining sample
of
size
3
given
that
the
unit
P(i)g and the pairwise probabilities g P(ij).
have already been Jbtained"
suggested
Cha2teI ..1
A general numerical solution is
based on arbitrary assumptions about the variance
superpopulation
analytic
Adjustment
solution
is
procedures
solutions a.re discussed ..
model
error
obtained
for
terms"
under
handling
a
A
specific
special
case"
nonprobabilistic
48
4
401
APPLICATIONS
0
GeB!£a!.lHEQ~h.
-'2I. l!gebraiQ
.§2!Q~i~
f9I. P lij)
This section summarizes the steps that must be taken to
obtain
algebraic
solutions
statistician must first
believes
best
for
decide
represents
the
upon
the
The sampling
P(ij)~
the
error
population
(1)
he
sampled and the
variable to be observed; ioe<>o he must define
52
model
the
function
for each ele.ent in the population sampled,
As noted
in Section 2 1.2,
0
52
(i)
= E[ e (il/l (1)
Once having defined
52
in
(1)
= 1- 2 (i) Be 2
]2
some
(i) .,
manner
and
the
unit
selection probabilities, P(i)o as
= nl (1) II (+)
P (i)
algebraic
solutions
probabilities,
for
P(ij),
may
pairwise
the
be
11
obtained
by
selection
computing the
following (These terms have already been defined in
2 and ace presented bere for summary purposes only)
Cbapter
~
N
SC I)
=S
S-2
(1) ,
i::;\
If
=S
S(2}
s-2{i)/[S~')-2s-2(i)]0
1=1
H
5('3)
=5
P(i)[l-P{i) ]1[5 (1
)·2s- 2 O) 1.
i=1
K (i)
= (P (ir [1-P (1)
WfijI
and
:.:
]5 2 (i)-5 C J>/[
5- 2
1+S0:> H/[S'
(i) 5- 2 (j! [K (i) +K (jl ]0
1)'~2s-2 (i)
)v
49
=
P (ij)
P (i) P (j),-W (1j) .,
Some model specific solutions for the P(ij)
are
noted
in the following sections.,
4., 211u!lications !ith
~!Re9ted=sg~t~g.
4.,2.,1
M~
Error
Specific Error: Hadels
l!2R~!~i2n!l ~~
the Measure.of
ilig
If
the
sampling
superpopulation
mod~l
assumes
statistician
b~;
the
squared error terms have expectation
proportional to the measure of size u X(i)u
may
that
this
assumption
written as
Ee 2 (i! ::; tx (i)
where k is a positive constant.,
~his
utilizing
model has a particular appeal for sampling designs
.ulti~stage
sampling
formed primary sampling
units ..
and
somewhat
Suppose
arbitrarily
primary
sampling
units i and j are adjacent to each other and for some
may be
treated
more
conveniently
as
a
single
combined
primary sampling unit with a new size measure XCiV)
x: (i • )
::;
x: (1)
~eason
with
+ X (j) ..
Under this particular snperpopulation model r it makes
to
sense
combine primary sampling units and still assume that the
error model applies u since
Ee 2 (i II ) "'tx (1 II) ::;kX (1)
~kX
(j' =8e 2 (1) +£e 2 (j)
tor computational purposes k may be
since
it
appears
denominator of the
1S
assnmed
0
to
a factor in both the numerator.
final
solutions
tor
the
be
~nd
P(ij~u
1
the
The
50
intermediate
computational
values
computed
assuming this
model redqce to the following:
S2
-=
(i)
X-I (1) ,
=
5(1)
X(+),.
N
S (
> = S X (i) I[ X (+) - 2 X (i)
2
],.
i=1
=
5(3)
K(i)
=
N
S P(i)[
i=1
1~P(i)
]/[X(+)-2X(i)],
{n[ 1-P(i) ]/X(+)-S(3)/[1+S(Z)]}/[X(+)-2X(i)],
=
W(ij)
X(i)X(j){K(i)+K(j)},
and
P(ij) = X(i)X(j}{n 2 X-2(+)-K{i)-K{j)}
:; X(i)X(j){n 2 X-2(t)-n[1-P(i) ]X-l (+)[X(+)-2X(i) ]-1
-
+ X (i) X ( j) 5 (
For
3 )[
1 +S (
samples
pa~entbeses
co~resPQnds
1.2.
n( 1-P(j) ]X-l (+)(X(+)-2X(j) ]-1}
is
2 ) ]- 1{ [
of
X (+) - 2 X (i) ]- 1 +( X (+) - 2 X (j) ]- 1
size
equal
2
to
(n=2),
zero,
the
first
term
}.
in
and the formula for P(ij)
to that given by Brewer
and
cited
in
Section
This can be seen by noting that the following hold for
n=2;
N
5 ( 2)
~
(
112 ) S P (i) I[ 1- P (i) ] f
i=1
5<
3)
=
nX- t (+),
and
[X (+) -21 (i) ]
=
X (+)[ l-P (i) ].
The first term in parentheses in the formula for ·P(ij)
then
51
becomes
4 X'-:2 ( t) - 2 ;~ - 2 (-I) -:..:, ;{ - 2 (+)
Toc! second terF'
tb,'~
in
=
O.
for P(ij)
nYDu'ssicn
beco!fles
N
P (i) P (j) { [1-' P (i)
J-- 1
I
[1- F ( j) ] -
H 2..
1
S P (k) Ii 1- P (k) ]}1\.=1
1 •
particular solution not ')nly :'';dtisfies the constraints
This
[(iJ)
:> 0,
~(ij)
>
hut also satisfies
as can be shown
0
noting that
~y
N
W(i j )
-
N
{2 + S 1:' (k)
If
1- P (k) )}-
1
P (i) P {i) { 2 + S P (k)
k~1
- [ l-P (i) ]-1 -
The
t:!["t:,
all
the
in the first
terms
I[ 1- P (k) ]
~=1
r.
l-P (j) ]-lJ.
pal0.nth<:;~:i);
is
c}::~arly
positive
in th2 summation are rositive.
The
!: incp
tar~
in
Ll
2t
S P (k) [1- P
lui or j
(k)
1- 1 + P (i) ['
1- P (i) ]- 1 + P (j) [1 - P (i)
N
- [ '1- P (i) ]-
1- [
1 - l:' (j)
;;; S P ()c;)
r 1- P (I\.)
1- 1
1- 1
kli or i
and is therefore also seen to
3iz2
For samples of
:.:;olution~;
fur
by r e- \~ x d min i n <1
P(ij)
t.1H~
be
greater
a
sum
than
of
2,
all
the
positive
algpbraic
r;onti.nue to be posit.ivp a.s C;ln be
for f1'. U 1 a for P (i j).
'T h C'
fir s t
ter
SC~(~r,
11i
1. n
parentheses beccmes positive as can be seen by writing it as
52
nl- 2
(+)
~[I
rn-[ I
(+)
-nl (1)
][ X (+)-21:
(i) ]-t
J... l I ..
(+) -nX (:1) ][ X (+) "'21 (j)
[I (+) -nl (1) ][ 1 (+)-21 (1) ]-1 <1
and the entire term is bounded below by
nl- 2
which is positiveg
can
be
seen
to
(+~
(n.... 2
J
The second tera of the formula for P(ijl
be
positive
The
for
all
value
of
n because all its
n
is
of
course
ltmited in any case by the constraint that
<
P (1)
1
or equivalently tbat
n < X (+),/1(1)
for all i ..
This
[ In
scbemes ..
general
constraint
to
applies
practice, sampling units with
1(1)
all
PPS
exceeding
I(+)/n are generally sampled with probability one; in lIultistage
samples,
they
are
t~eated
as
self-representing
stl:ata.,]
~iVlflt..Ii£GE
111!lltl·OJ.~~iz!
4., 'l. 2 112ISt,!.
..neRGrt !2!l.!!.,.!!~ihe··s !I¥~ .-
another reasonable 1I0dei to assulle in
~ertain
cases
is
tbat
Be 2 (i)
= tl Z (1)
where t is a positive constant ..
An exaaple of
t:ppears
tn
apply
a
sampling
involves
problem
where
this
.odel
a recent study on the regional
53
destinations of import
port
terainalse
shipments
through
certain
The measure of size c ICi)c employed in the
study was the dollar value of
obtained
passing
~ach
shipment which
could
be
by applying standard conversion factors to tonnage
figures which were available in a
computerized
filee
The
data on the regional destination 3f the import shipments did
not appear on the computerized file, but could
be
obtained
by searching additional paper copy files created at the time
of deliveryo
Estimates of regional dollar values of imports
were
based
to
be
on
a sample of items selected from the
computerized file listingo
as
the
The observations q f(ri)
u
defined
dollar value of imports to each region u r, could be
mode.l ed !is
f (ri) = B Cr) X(i) + e (ri J
The values of f(ri)
values~
J(ri)
=
0
were assumed to take only
XCi)
possible
with probability B(r} and f(ri) = 0
with.probability [l-B(r)]o
8e 2 (ri)
tvo
=
Under these assumptions q
R(r)[ 1-B(r} ]1 2 (1)0
For a particular regional estimate, the model fits with k
=
B (r) [1-B (r) ]0
For computational purposes g k may again be
be
1
assumed
to
since it appears as a factor in the the numerator and
the denominator of the algebraic solution
for
P(ij)o
The
intermediate computational values assuming this model Eedoce
to the following:
54
5(:1)
=
K(i)
=
5(2).
= 1/(8-2),
(N-2)
-1
8
5 P(l) [l-P(i) ]"
i=1
11
5 P(i)[l-P(i)]}"
(1I-2)'"'"t(P(i)[1-P(i)1~(2(1I-1)]-1
i=1
= (1-2) - I ( P (1) r 1-P (i)
W(1j)
]+1' (j}
r l-I? (j)
)}
R
..
(N-1)..,1 (8-2)..,.1 51? (i)[ l..,P(i) )
i:;:l
and
P (ijt = P (i) P (j)
In practice, algebraic
model
are
-if (ij~
so~utions
C>
using this
particular
aore likely to fall outside the acceptable range
specified by either the weator the strong constraints
than
with the previously discussed aode10
As
noted
micialzed
when
in
Section
using
2.. ~"
this
model
the
function
criterion
is the one suggested by
Jessen, namely
N R
5 S W2
(ij) ..
1#j
Jessen applied his method 4 to the problem
saaple
of
2
measures 1,
compares
fcom
20
three
3"
in
most
and
selecting
a
a population of 4 elements with size
40
respectively ..
4o~o201
Table
of his approximate solutions with.the exact
algebraic solution"
were
a
of
Jessen~s
approximations
to
cases within rounding error of the
the
W(ij)
algeb~aic
55
solution and all came very close to
obtaining
the
minimum
value of the criterion function o
Table 402 .. 201 Comparison of algebraic solution with Jessen's
Bethod 4
____
It~ _I;.:er~:~~~rr{~id~~hltI:erati~~=d]i:~:~
'I (12)
Ii (13)
'I (24}
Ii (23)
110 0
o
00
C?.~
q- 4jl
001
001
002
.,10
.,07
007
001
.. Q6
oQ3
011
006
.. 07
001
.. 08
001
009
.08
oQ7
.,0666667
00666667
00266661
.,1066667
.. 0666661
00666667
o .,
.. 0300
.. 0300
.,0308
00298667
.e
lQ
'9' q'
0
C~C"~~'~
c)OOO-IQ~
W(24)
'1(34)
~
~ C)
0 0 ' If:
eo
G- 0
Q
N N
S SW~
(ij)
i<j
----_.
The two specific error models discussed
above
can
be
considered as special cases of the more general model
Be 2 (i) = kX 9 (i)
where k and 9 are constants o
considered
to
The value of
be between 1 and 2 as noted by
and Cochran (1962 u po
486) and
Bayless
96-104)
estimates
(1968,
of
g
ppo
9
for
ten
Cochran
obtained
large
natural
is
generally
Rao~
(1963 u
Hartley,
po
259) ..
aaximum likelihood
populations
and
concluded that values of 9 between 1 and 2 appear plausibleo
The
algebraic
computational
solutions
values
which
are
required
obtaining
optimal under this general
model do not simplify to any great extent.,
that (assuming k=l)
for
It may be
noted
56
in the general notation pt:esented in Section 11.,1.,
403 1 8ixed..nrOE. Hod!!
1 typical use of PPS sampling involves estimation of
population
attributeo
total
for
estimate
population
~f
the
time
or
Census
count,
a
particular
the
~ov
a county in the most recent decennial
of
the survey, the
~ctual
generall, not known or the priaary
special
with
Generally the size measure, Xli), is based on
previous
At
population
a
subpopulation q
!£g~.
a
a
total
censuso
population size is
interest
aay
be
in
a
all aarried feaales 15-35; the
population size at the time of the survey may be denoted
by
8(i)0
The special attribute is possessed by Y(i) of the
M(i)
population elements; if B(i) is the nuaber of feaales 15-35,
Y(i) aight be the number of females
15~35
that
have
had
a
live birth during the previous calandar ,eaco
1
Y(i)
superpopulation model could
nuaerically
in
two
[OSB(i)~l]o
The
the
stages as followso
stage, the superpopulation model
B(it
generate
~alues
generates
At
H(i)
8(i)
th~
[>0]
and
first
and
!(i) and a(i) are then used as
the paraaeters of the binomial distribution which
generates
the lei) at the second stageo
The 8uperpopulation model ma, be written as before as
Y(i) :: ax (i) .. e (i) "
The error tera can be partitioned as
e
(i) :;
e ('H) . ~ e (li)
51
where
e(11) .: l(i)[B(i)"(i)/I(i)-B]
and
e (2i) .: [Y (i) -B (i)
Under
this
model"
the
"(1) ]e>
squared
error
term
has
expectation
£e 2 (i) ::: £e 2 (li) + Ee 2 (2i)
with
Ee 2 (li) :::
12
(i)E[B(i)ft(i,/I(i)-B]21'
and
Be 2 (2i) ::: "(i) B (i)[I-B (i) ]0
It may be noted that if "(i,=I(i) for
i:::lv2"oo~uN,
then
Be 2 (li) ::: 1 2 (i)8[B(i)-B]2
and
Be 2 (2i) ::: 1
(i) I( B (i) [
1-B (i)
]
Io
This particular model specializes further for two cases:
(1)
B(i) ::: B for 1=1"217<><10,,N, and
(2) B(i) ::: 1 for BN of the sa.pling units and B(i) :::
0
for the remaining (1-B)8 sampling unitso
In the first caseq £e 2 (li)
squared
error
can
be
-
and
01'
expressed
as
the
total
expected
Xli) multiplied by a
constant"
Ee 2 (i, .: X (i) B ( 1-B)
In the second case" Ee 2 (2i)
= 0;
and the total squaced error
is a function of 1 2 (i'o
Ee z (i)
~
X~
(i) B (l-B) .,
58
Both of
these
special
cases
literature and are discussed in
As discussed in Chapter 3,
ainimizes
the
have
been
4~lcl
and 4.102 above.
a
aodeled
general
in
solution
the
which
super population variance of the third moment
of the Horvitz-Thompson estimator for any assumed values
Ee 2 (1)
is
solution
theoretical
obtainable
t.o
theoretically but requires a numeric
If(N-ll/2
general
of
simultaneous
approach
does not appear practical
is
even
equations.
The
not recommended since it
with
the
availability
of
present dayco.puters.
The recommendeo procedure for .ost applications is
one
discussed
in
Section
303o~0
required for this procedure,
given
the
The compotational values
the
P(i),
P(ij),
and
1(1j) have previously been determined are
&(ij)
L fij)
:: 2( 1-P(i) -'P(j) ]W(ij),
== [3 (1f-2) (If-3) (1-4) ]-1
(3 (1f-2} (N-3) B (1 j) ... (3/2) (8-2)
o (ijk)
r
If
x
•
M If
S R (ik) .. S R (jk) ]+5 SR (kl)},
k#i
k#j
k#l
= L (lj) + L (ilt) .. L (jk) ,
and
P (I jk) :: Q (ijk)
t'{)
til P (j) P ft) -P' (1) W(jk) -P (j) i (ik) ".p (It) W(i i) . ,
Iu the case uf
Ee2(i~~kX2(i),
in the sense discussed abov€c
this solution is
optimal
59
4 .. 5 .. 1 ldj!!§tment.. 9f..!lJ.~_P(!j:
The adjustment procedures for the P(ij)
in
detail
in
of
discussed
Sect.ion 204 and cannot be further summarized
without omitting essential aspects..
choice
are
adjustment
method
Some
for
comment.s
applications
appropriate..
Method 3, sampling unit. redefinition,
simplest
apply
to
and
also
1,
constrained
minimization
satisfy the prescribed
in
exp~rience
limitations
precludes
on
the
populations..
fOL
the
constraints
applying
this
accuracy
use
of
may
is
t.he
be
the
may have the added effect of
reducing the superpopulation expectation
Kethod
on
of
allows
exactly;
method
the
variance ..
the
user
some
indicates
to
initial
t.hat
the
of
matrix
inversion
this
method
except for very small
If the user sets the upper
and
procedures
lower
limits
P(ij) within the bounds reqUired for the solutions
to be valid in a probabilistic 3ense, these arbitrary bounds
can
then
be
satisfied
compensating additive
apprOXimately
adjustmentso
using
The
Method
accuracy
of
2~
the
apprOXimation may be controlled by the criteria used to stop
the iterative process a
by
Kethod
2
is
The accuracy of
solutions
obtained
not seriously affected by the size of the
population being sampled ..
After applying either Method
advisable
to
test
~hether
satisfy the constraints
the
1
or
adjuste~
Kethod
2~
it
is
probabilities still
60
I
S P(ij) = (n-l)P(i).
j#i
This
is
provides
particularly
use
in
using
8ethod
Method
to
1
1 possible coapromise to follow
satisfy
the
and
for
is
prescribed constraints
exactly and to shift to Bethod 2 whenever the
obtained
1
a warning that the aatrix inversion process is not
working satisfactorily.
to
important
solutions
so
the P(ij) do not satisfy the test of accuracy
stated a.bove ..
It aay be noted that ftethod 2 aay
initial array of numbers g T(ij)
N
S T(ij)
=
#
be
applied
to
any
that satisfy for each i
(n-l)P(i)
j"'i
and that tbe properly adjusted array may
spQcify
the pairwise
probabilities~
then
be
used
to
P(ij), of a probability
non-replace_ent designo
Adjustment procedures for the P(ijk) are
Section
3.~.
These
procedures
aust
be
discussed
in
used with care;
checks of the adjasted probabilities to determine compliance
with the constraints
R
S P(ijk) ;: (n-2)P(ij)
k#oi or j
sho~ld
be performed after either of the first two ad
procedures are appliado
at_eDt
61
SG
SOME EMPIRICAL COMPARISONS
5G 1 §eneral.· AP2roach
Two general qaestions cannot be
exaaination
chapterso
directly
of
first
the
question
is
concerned
superpopulation
with
the
population
to
be sampled.
superpopulation
variance
to
structure
The second guestion is
concerned with the effects )f the choice of
the
the
varianc~ ~riterioD
poor or wrong initial assumptions about the error
of
by
of the theoretical development in the preceding
The
sensitivity
answered
constraints
criterion
and
on
on
other
statistical properties such as the stability of the variance
estiaator o
An attempt was made initially
to
stUdy
these
issues
solely on a simulation basis by generating
,1
large nUllber of
saall
a
probabilistic
populations
through
super population
model.
values
Y{i)
of
distribution
the
was
the
Since
are
lse
in
aost
The
problem
applications
non~negativev
considered
most
lognormal distribution was utilized
1(I)e
0f
encountered
a
one-tailed
appropriate
to
With
generate
this
the
and
~he
realistic
approach
was
apparently that the higher order moments of the jistribution
could
not
be 3atisfactorily reproduced through the digital
co.puter siaulations methods usede
An a.lternate approach lias then
attemptedJhic~
utilized
the binoaial distribution and both theoretical and eRpirical
simulation results could be ottainedG
62
The
sections
remaining
descriptions
of
considered,
simulated
the
and
of
populations
discllssion
a
properties
this
of
the
chapter
studied
of
the
include
and the designs
theoretical
and
variance and of the variance
estimatoro
5.. 2
Tva
special
lli-f~l!!l!!:i211L~m!Sid~:r~!
superpopulation
models
with
squared errors proportional to X (1) and XZ (i)
can be simulated through the use
trials.
respectively,
Il
independent
Bernoulli
Both can be expressed in terms of the general model
Y(i}
In
of
expected
the
Population
first
the
1,
:: 8X(i)
+ e(i).
population
considered~
Y(i)
generated from the binomial
were
distribution with parameters Band lei).
designated
onder this
as
model,
the error term has second and fourth moments
leZ(U : l(i)B(1-B},
and
Be" ti)
respectivel,.
= X(1) B(1-B) { 1+3[ X(1) -2]8 (1-8)
These theoretical results
directly into the formulas for BV[ttS)'
as shown in section
In the
Population
si~gle
Y(i)
second
2.
could
~
}.
be
] and V,[t(s) I
population
each· Y{i)
assigned
~
]
2D2~
was
considered,
designated
as
generated on the basis of a
Bernoulli tDial with probability of success
were
applied
80
The
the values XCi) for a succeSSful trial
and a value of zero for an unsuccessful
t~ialu
th~
second
63
and fourth moments of the
Ee z (1)
erro~
terms under this model
~re
= 1 2 (1) B (1~B) ,
and
Ee· (i)
= X· (i) B (1""8) [
1-3B (1-B) ],
respective~y.
In a single simulation run,
~ener~ted
100
finite
e~ch s~perpopulation
populati9ns
of
size ten.
finite population, the variance, ,(t(s) I
An estimate of "[t(s) I
sample
pf
100
~
finite
superpopulation model.
], was evaluated.
~
generated
The entire Prooess
with a dif.erent initial random
estJ.a:ates
of Vf[ t (s) I !!
For each
] was obtained on hhe basis
populations
1 and
numb~r
~odel
co~ld
~f
by
the
the
be repeated
to obtain independent
a110'1 f()r
of
the~r
for
al~
ev~luation
pJ;eclsion.
T~e
size
measures,
populations
and
The value
B uas
~f
XCi),
weJ;e
were equal to the
arbitrarily
the
numbe~s
set
at
same
one through ten.
O.~
to
0btain
a
soaewhat 3kewed distribution of errors.
1 single simulation experiment
repl~cation
of
a
~inglr
may
b~
considered
population 100 tJ,mes, but may also
be considered as the simUlation of a single population
100
stData,
all
super popUlation model.
of
as
which
have
the
3ame
with
underlying
64
503 The Designs Cen2idere!
Twelve designs were considered in the
based
on
four
terms
and
for
study
sets of assumptions about the squared error
three
solutions
empirical
choices
of
the P(ij)o
restrictions
on
the
final
All the designs involved sallples
of size 20
The assa.ptions about the squared errOE terlls
in
obtaining
algebraic
solutions
were
that the expected
squared error terms were (1) constant q (2)
I(i)u
1 3 (i)
proportional
(3)
0
to
employed
proportional
to
1 2 (i). and (4) proportional to
,
The
constraints
applied
solutions lIay be stated
to
the
initial
'as~
(1)
oOlP (i) P (j' S P (ij) S lIiniauII(p(il.P(j) Jq
(2)
oOlP(i)P(j) S
P
(3)
oSOP (i)P (j) S
P (ij)
The choice of
constraints
is
C
algebraic
(ij' S P (ilP (j) u and
=
arbitrary
S P(i)P(j)o
001
in
but
the
first
intended
tvo
sets
of
to be a very weak
constraint which still preserves the Don-negativeness of the
P(ij)
aDdq therefore. the estimability of the variance..
general the choice of C bounds the coefficients.
the
Yates~Grundy
coefficients,
variance
form of the variance by
w(ij)q
estimator
of
by
the
(l-C)/C~
W(ij)q
(l-C~P(i}P(j)
Yates-Grundy
form
of
of
and
the
Therefore the uppel. bounds
for these coefficients under constraints (1) and (2) are
W(ij)
In
S o~9P(i)P{j)
..
65
and
w(ij)S 99.
The upper bounds for these coefficients under constrain"
(3)
are
W(ij) 5.S0P(i)P(ji,
and
w(ij) 5 1.
Choice of upper bounds
bounds
for
the
W(ij)
for
and
negative values of W(ij)
the
w(ij).
P(ij)
affects
lower
Under constraint (1),
and w(ij) are possible.
They
are,
however, bounded below as
W(ij)
~
- minimum{P(i)[l-P(j)], P(j)(l-P(i)]},
and
w(ij)
~
- minillulI{ l-P(i). l-P (j)
J.
Note that in no case will the values of W(ij) and
less
values of W(ij) and
by
Unde~
than negative one.
~(ij)
are bounded below by zero.
~mpirical ~tudy
two-digit
in
identifiers
Table
referenced hy these
identifiers
The
of
values
P(i),
catalogued in Appendix Section
Designs (01) and (02)
algebraic
be
constraints (1) and (2), the
The designs ased in the
actual
w(ij)
are
5.3.1
identified
and
for, discussion
P(ij),
W(ij)~
will
be
purposes.
and w(ijl are
8.~.
are identical since
the
initial
solutions satisfied both constraints tl) and (2).
The salle is true for designs (11) and (12).
66
Designs (Ot),(02)q (11)" (12), and (21)
directly
the
fro~
design
the algebraic 501ut10no
constraints
constnained
were
minimization
were
obtained
Adjustments to meet
by
achieved
Method
1,
(described in Section 2.4.2), for
designs (Oll, (13)" and (22); and by fIIethod 2,
coapensating
additive
2o·4o-~) ,
adjustments
(described
in
Section
for
designs (23), (31), (32) "and (33) ..
~able
Design' identifiers
5~3.1
Be a (1) =It ...'.....
(01)
(02)
(03)
Be 2 (1l=kl (1)
(11)
(12)
( 13)
Be 2 (1)=tX 2 (1)
(21)
(22)
(23)
lea (1) -=ltl' (1)
(31)
(32)
(33)
5 ....
~.D.i~iyitI
gf-llJl .. ~fiterj;stD_l!lDg'j.on
~.1.cti2!!
tg J!!sign
The theoretical values of the sQperpopulation
of
the
finite
evaluated
for
population
each
super population
5.4.1..
,,[test.
models..
It gay be
~
]
of
in
variance,
the
12
designs
and
~.
], were
the
tvo
The results are recorded in Table
verified
each
VV[t(st.
variance
coluan
that
the
coincides
lowest
value
of
with the design
assuaption that fits the true model of the population..
61
~able
5.q.l Theoretical
superpopulation
finite population variance
variances
of
the
Assumed mOde~.
--·c_~~-rt;;ints a lied to~ the__P (ij)
for design
.. Q1P(i)P(j)SP(ij) .OlP(i)P(j).S [.50P(i)P(j)S
£Q~pose~
.
~~!_~t!!~) ~_ULL.~.t~)_SP (i)P.Ul _~!!JL~p (i) P (j)
True model: !e 2 (i) ;: .161(i)
Be 2
Be 2
Be 2
Be 2
(i);:k
<>
288.,68
281.,95
292061
305058
~ 'H'
(1) =kl (i)
(i) ;:kX 2 (i)
(i) =kX 3 (i)
288.,68
281 095
289011
291020
288026
287.,96
288.,14
288028
True model: Be 2 (i) ;: .16X2(1)
Be 2
Be 2
Be 2
Be 2
9,686.,55
9,692061
9,664.,04
9,,128 40
(i) =k .,
(i}=kX (1)
0
. . .,
(i) =kX2 (i)
(i) =kX:J (i)
- - - - _.._--_ ...
0
9,,683012
9,,685 .. 03
---
As more restrictive
P(ij) by the design
(1) the
9,Olj!.80
9,,690036
9,686 .. 55
9,692 .. 61
9,671054
9,689.40
constraints
specification~
minimQm
achievable
are
imposed
on
the
two things happen:
value
of
~
V,(t(s) I
]
increases; and
(2) the difference in "(t(s) I
the
worst
design
(under
~
] between the best and
the
particular
set?f
constraints) decreases.
These phenomena may be observed in Table 5.4 .. 1 by
the
left
column
(least
comparing
restrictive constraints) with the
middle column (moderately restrictive) and the right
column
(most restrictive).
Probably, the most interesting aspect of Table 5.4.1 is
that
the expected increase in V'[t(S) I
~
) due to using the
poorest design of the twelve instead of the best
only
design
is
about 6 .. 1 percent under the first population model dnd
68
less than 061 percent under the second population modelo
Considering
proportionately
design
1 2 (i)
smaller
purposes
proportional
VV(t(s) I
to
loss
that
the
I(i)
when
than by assuming that
proportional
I(i)o
to
]
~
1 2 (i)
as
is
a
loss
when
squared
squared
by
assuming
that
is
error
is
in fact it is proportional to
If this is a general phenomenon, a
taken
error
fact it is proportional to
expected
~he
a
incurred by assuaing for
expected
in
function,
the
saaller
squared
error
risk
teras
is
are
proportional to the I(i).,
Based
on
Table
50401"
the
design
strategy
which
miniaizes
the aaximum possible loss is the one that imposes
the
restrictive
most
constraints
(right
column
in
the
table).
It aay be noted that if in the true model o the expected
squared error is proportional to 1 2 (i)
504~1),
squared
(bottom half of Table
the sample designs which assume a constant
error"
coae
closer to the optimum than those that
assume proportionality to the X(i)o
VV[t(s) I
expected
The
minimum
value
of
] is o however" achieved under the correct model
~
assuaptionso
505 Bfllc!
at
J;)esig!Li!!!eetion 2!!.. .!~e fariance
Forsa.ples of size two u the commonly used
~stima!:or
fates~Grundy
variance estimator is
v[t(s) I
where
~
]=w[s(l) us(2)]{Y[s(1) ]/P[s(l) ]-Y[s(2) 1,fP[s(2)]}2
69
v (i j) = [P (i) P (j) - P ( i j) l/P (i j) ..
The finite population variance of this estimator is
V{v[t(s) I
~
])
=
N N
S SW(ij)lf(ij)[Y(i)/P(i)-Y(j)/P(j)]4
i<j
- V2{ t (s) I
To
study
variance
the
effect
estimator~
in
]
expectation
of
the
. ] ..
design
selection
on
this
the super population expectation of this
variance vas examined..
EV2[ t (s) ,
of
~
on
Based
Section
finite
the
202~
derivation
the
population
of
superpopulation
variance
of
the
variance estimator may be expressed as
It It
BV(v[t(s)'
~
]} = S SW(ij)v(ij)(E[e(i)/P(i)]4
i<j
+ B( e (j)/P (j)]4 + 6E[ e (i) /P (i) ]2E[ e (j) /P (j) ]2)
N N
.. 2S SEre (i) /P (i) ]2E[ e (j)/P (j) ]2P (i, P (j) [ l-P (i) ][ l-P (j) ]
i<j
It N
- liS SE[e(i)/P(i) ]ZE[e(j)/P(j) ]ZHZ(ij)
i<j
N
- S E[ e (i) /P (i) ]4 PZ (i) [ l-P (i) ]2
i=l
This theoretical value of EV(v[t(s),
each
of
the
considered~
~
]} was
0
computed
for
12 designs and the tvo superpopulatiotl models
The results are shown in Table 5 .. 5 .. 1..
10
TabId 50$01 Theoretical superpopulation expectations of the
finite population variance of the Yates-Grundy
variance estimator
~::"::::::d;iI;1p~~f~~~p~m-~~~-~~:
purposes
= 0161(i)
True model: ReZ{i)
4,641
4,.014
35,252
393,893
le 2 (i} =It " 9 9 9
Be z (il :;:kl (i)
Be 2 (i) =k1 2 (i)
Ee a (i)=kI3 (i)
4,641
4,014
1,135
21, 701
True model:: Ee 2
Be 2 (i)
Be 2 (i)
Be z (i)
Be 2 (i)
e) 0-
Ol)
::k1 3 (i)
On the basis
squared
of
the
val~~
other
of
proportional
~
designs consideredQ
BV(v{t(s) I
restricttYeness
in
~
the
P(ij)
control
66,')85
62 .. 826
65,,031
66 .. 511
Ii
based
on
an
are clearly
I(i)
]} criterion )ver
with
constraintso
any
increased
Designs which
(ij) (designs (21) and
the values of v (ij) between 0 and
(33)]
(i)
With one exception, the
are clearl, inferior to the others.,
and
to
decreases
]}
permi t negatiiYe values of the
(23),
4 1 1110
4,~68
designs
5.,S.,1,
superior in terms of the EV(v[t{s) t
of
o161 2
3,919
3 1 944
12",003
62 17 815
81,,915
166,,0684
Table
error
=
(i)
12,.003
62 .. 815
170",715
1 q 51&9,)99
=k o
=kl (I.)
=k1 2 (i)
ass~~ed
lf1m;:-
u.B_f!inI'P(i) "P(j) } P(ij)SP(i) P(j) P(ij).SP(i) P{j)
(31)]
Designs which constrain
1
[designs
BV(v{tfS) I ~ ]}
(03),
(13)"
effectively. under
either superpopulation modelo
Five
generated
sets
under
or
replicates
each
of
the
of
100
populations
two supeLpopalatton
were
~odelsu
11
Bach
of
four
generatedo'
designs
Only
were
designs
applied
to
(11),
(01)6
each
population
(21)" and (31) were
used in the simulation experiment;' these designs utilize the
weakest
set
of constraints on the P(ij) and should exhibit
the largest iifferences between model
specifying
BY(v[t(s) f
the
design ..
~}
were
combination.,
The
The
estimated
values
for
individual
assumptions
of
each
used
yy[t(s) I
~
in
] and
replicate-design
esti.mates
are shown
in
BW[ v[ t ($)]1 !. ).,
Table
506~1
Simulated superpopulation variances
of
finite population variance, YV[t(s)I !. ]
assumed mOdel--~~
~~~p::;:~~
Simulated values
--L R~P_.]=_~r J
the
-
l.__R_;_P
R;P_~ ~I~__
True model: Be 2 (i) =ot6X{i)
80 2 (1)=k
Be 2 (i) =kl (i)
C)q,~~.C?'«;t
Be~
{i) =k1 2 (i)
Be 2
ti)
=lt13 (i)
0 ..
C>
..
2050)1
2050~7
204036
202048
2980ij8
307094
305016
319036
290070
292021
285084
291016
311068
311087
364<>~7
384 021
289011
290.,49
289 46
295034
0
True model: Ee 2 (i) -= ",161 2 (i)
Be z (1);1£ .. 'I 9 <> <l .,
Ee 2 (1) =kl (i) ., 'I
Ee 2 (1) =k][2 (1)
0
Ee Z (i) -=)(1 3 (1)
.,
9/1119
9,150
8,923
8,798
9,097
9,,134
9,234
9,645
9,,718
9 6 850
9",.12
9,956
8 0 365
8 0 422
8,465
8,532
1",296
1,,461
1,,228
7,055
12
Table 5d6c2 Simulated snperpopulation exp~ctations of the
finite
popalation variance of the variance
estimator Q BV( ve t (s) ]I ~ }
- - - - - - - r - - - - ---.----'--~--._-----.-----.
Assumed model
Simalated walues
for design
J
B;P
R:p._~I R~P__
porposes
I
_:;p_[ R;P
__
= 0161(1)
True model: le 2 (i)
le 2
Ee 2
Be 2
Be 2
4",163
4,,4_6
32,228
396",480
4':186
3,661
32,,410
141.358
(i) =Jt
CI~ 'rfl~
(i) =kI (i)
q
(i) =Jt12 (i) c
(1) :k1 3 (1)
0
0
0
4.511
4,..139
38,,381
430,837
Trae model: Be 2 (i)
5 .. 588
5;014
37.634
387.926
::
4 9 533
41'136
29,,922 .
325,830
.. 161 2 (i)
14,011
66,,124
76,,813
72,,198
12" 142
63,395
64,,189
59,229
67,605
64 .. 024
Be 2 (it~kxZ (i.) .. 172,899 181 .. 938 150,,664
178 .. 601 162,641
Be z (i)-kX3 (i) 0 1c$8xl0 6 1.. 62x10 6 '0]5:110 6 lc60x10 6 1.. "1x10·
Be 2 (i)=k o <IV"''''''
Be 2 (1):-kX (I) .. ""
Table 5 .. 6.3 Analysis of variance of the si.alated "alues of
Vy[tes) t g ] from Table 5.6 .. 1
..........
Degrees of Ee 2 (1) :: 0161 (1)
Be 2 (1) :: .. 161: 2 !!L
Sout'ce
freedom Mean sCJoareIP....rati.o "ean 3CJuare C!.:rat1o
--
Replicates
4
14.971068
652 .. 054
3,826 9 240
122 .. 821
3
67 .. 21
2 .. 926
9,460
0304
Brror ......
12
22097
Total
19
..
Designs
o
~
CI
31,153
lrom an exallinatton of the data in Table 5.. 6.. 1.. it
be
concluded
that
no
(1)
dramatic
di£ferences
criterion function occur due to choice of
while
the
in
the
and
(2)
theoretical results cannot be rejected; somewhat
different results
populations
desiqn g
may
may
be
obtained
for
any
set
ot
generated by the same superpopulation model,
100
!
13
two-vay analysis of variance (Table
each
true
5~6.3)
of the data
superpopulation
model
indicates
significant differences exist
among
replicates,
significant
that
from
large
and
that
differences among iesigns cannot be detected on
the basis of this experiment.
The simulated data on
the
expected
variance
of
the
variance estimator (Table 506c2) exhibit the 3ame pattern
differences shown
conclude
in
the
theoretical
resultso
of
may
that the properties of the variance estimator will
be strongly affected by the choice of desiqn
set
one
~f
in
any
given
populations generated by the superpopulation modelo
74
6a
SUMMARY AID SUGGESTIONS FOR FURTHER RESEARCH
60 1
~llarY.JU~urch
The problell of con.rolling the pairwise
P(ij),
in
probability
non-replace.en~
since the developllent of a
general
probabilities,
sampling has existed
theory
of
probability
non-replacement
sampling in conjunction with the use of the
Borvitz-Thollpson
~stillator(Horvitz
nllllber
of
A
researchers have treated the subject directly or
indirectly including aaj
(1963),
and Thollpson, 1952)0
Durbin
(1956a)#
Hanurav
(1967)
Brewer
IT
(1967J, sampford (1967)6 Jessen (1969), lao
~nd
and Bayless (1969),
Dodds and
Pryer
(1971)0
Salle
of
these apnroached the problem through a superpopulatioD model
and
exaained
such
expectation
estillator
Jessen
of
or
the
the
posed
criteria
an
as
variance
stability
intuitive
of
of
the
super population
the
Horvitz-Thompson
the
variance
criterion
which
closely to the approach followed in this
estimatoro
relates most
research,
but
he
did not solve for optillull designs directlyo
A general
defined
to
superpopulation
.odel
model
is
~escribed6
general
Under
employed
in
seeking
probabilities,P~ij).
optimum
designs
super population iariance Jf the finite population
1
measures
the superpopulation expectation of the variance
is not a function of the pairwise
criterion
size
potentially achieve the principal advantages of
probability non-replacellent sampling
this
with
solution for the
P(ij~
in
ter~s
is
The
~he
variancec
of the variances
15
of the error terms
in
the
superpopulation
Although the solution can be generalized to
n (within ,certain restrictions),
three-wise
to
1 lIethod for
the
is
~ny
problem
developed.,
sample size,
of
choosing
n-wise probabilities lIay still be difficult.,
obtaining
consistent
with
discussed.
Both a
three-wise
the
pairwise
general
and
probabilities,
probabilities,
a
specific
P(ijk),
P(ij)q
solution
is
are
presented.
The general methodologies developed for specifying both
the P (ij)" and the P (ijk) are based on
linear systems.,
These
nUllerical
results
properties
of
developed
for
solutions
adjustment
to
solutions
not
occasionally
or
within
may
with
functions.
constraining
procedures
may
consistent
probability
occur
~lgebraic so~utions
adjusting
be
stringent constraints on the acceptable
P(ij).; the
~pplication
in
the
to
are
algebraic
These
impose more
solutions
for
the
of these stringent constraints can be
used to control the range
appearing
the
bounds.,
used
yield
requisite
Procedures
specified
also
the
of
sums
of
for
values
of
the
coefficients
the variance and the variance
estillator"
Specific solutions
about
the
error
terlls
under
of
some
the
alternate
assumptions
superpopulation model are
discussed.,
The results of an empirical test
designs
of
twelve
different
applied to two different superpopulation Ikodeis are
76
reported..
The superpopulation variance of the
found
be
to
relatively
~
considered, although the
The
superpopulation
flat function
value
~iDimum
expectation
of
can
the
variance
~ver
the designs
be
identified.
variance
of the
variance estimator is also evaluated and found to vary
more
widel,
is
much
over designs and particularly as a function of
the constraints applied to the algebraic
solutions
of
th~
P (i j) c
Both the
adoption
theoretical
eapirical
results
support
of ,he proposed procedures for obtaining algebraic
solutions for che
super population
In
and
~d4ition~
P(ij)
as
a
means
of
controlling
the
variance of the finite population variance.
the superpopulation expectation of the variance
of tha variance; estimator.may be
con~rolled
the algebnaic solutions to satisfy
the
by adjustment of
strong
constraints
with the factor. ·C q set at (n-1)/n.
6 .. 2 ,bge••,IftgUi2qs . fOl:r"ther:StudI
The present research IIncovered a number of topics which
may warrant further 3tudy.
(1)
developmen~
!hese include:
of aore
general
methods
of
handling
Within-stratum samples of size three or greater:
(2) investigation of
including
alternate
methodology
for
adjustment
obtaining
procedures
more
rapid
convergence of iterative methods;
(3) investigation
superpopulation
of
the
possible
use
of
the
expectation of the variance of the
11
variance estimator
as
a
criterion
function
for
procedures
for
generating the P(ij);
(4) development of efficient
generating
populations
finite
superpopulation
computer
models
as
a
from
assumed
of
studying
means
complex sample design problems;
(5) development
of
"
procedures
and
to
application
of
hypothetical superpopulation models or to test
the
of
the
estimation
parameters
applicability
estimate
of
super population models to finite
populations; and
(6) study of other
size
measure
superpopulation
configurations
models
to
criterion function is as flat in
and
other
determine if the
the
region
near
the optimum as was shown in the empirical studyo
78
7.
LIST OF REFERENCES
Bayless, D. L.
1968. Variance Estimation in Sampling from
Finite
Populations.
Ph.d.
thesis,
Texas A&M
University, College station, Texas.
Brewer, R. K. W.
1963.
A model of systematic sampling with
unequal probabilities.
Australian J. Statis~. 5:5-13.
cochran, w.. G.
1939. The use of the analysis of variance
in enumeration by sampling.
J. Amer ..
Statist. Ass.
34:492-510.
Cochran, W. G.
1946.
Relative accuracies of systematic and
random samples for a certain class of
populations.
Ann.
Math.
Statist.
17,164-171.
strat~fied
Cochran, w..
G.
1963. Sampling Techniques.
Sons, lnc., New York.
Jobn Wiley and
Dodds, D. J., and Fryer,
J. G.
1971.
Bome families of
selection probabilities for sampling with Probabil~ty
proportional
to
size.
J. R.
Statist.
Soc.
33 (2) : 263-274.
Durbin, J.
1967.
Design of multi-stage surveys for the
estimation of sampling errors.
Applied Statistics
16 : 152-164 ..
(;odambe, V. P..
1955..
A unified theory
finite
populations.
J. R.
817:269-277.
of sampling
statist.
from
soc,
Graybill, Franklin A.
1969. Introduction to Matrices with
Applications
in
statistics.
Wadsworth PUblishing
Company, Belmont, California.
Hansen, M. H., and Hurwitz, W. N.
1943.
On the
theory of
sampling from finite populations.
Ann. Math.
Statist.
14 (2) : 333-362 ..
T. V.
1967.
Optimum utilization of auxiliary
information:
ps sampling of two units from a stratum.
J. R.
Statist.
Soc.
B29: 371-391.
H~nurav,
Ho~vitz,
D. G., and Thompson, D.. J.
1952.
A generalization
of sampling from a fin~te universe..
J. lmer.
Statist.
Ass.
47:663-685.
19
Jessen, Re J" 1942" statistical investigation of a sample
survey
for Obtaining farm factso
Iova Agr~ Exp"
Stao Res" Bull" 3040
Jessen, R" jo
19690
Some
non-replacement samplingo
64:175-1930
methods
J" Amero
of
probability
statisto
Asso
Kendall} Maurice Go, and Stuart, Alan" 1966" The Advanced
Theory of Statistics, Volume 3, Design and Analysis,
and Time 3erieso Charles Griffin and company Limited,
Londono
Koop, Jo:o
1963" On the ~xio.s of sample formation and
their bearing ~n the construction of linear.estimators
in sampling theo~y for finite universeso Parts I
and
II in Metrika 1(2) ~81-114 and Part III in Metrika
1 (3) : 165-204
0
Lablri, Dc Bo
19510
unbiased ratio
33:133-140"
A method of sample selection providing
)stimates"
Bull" Intc stat" Inst"
Mahalanobis, Po So
19460 On large
Phil" Trans o ROY" 'oc~
London
Mid2!JnO, H.
systemso
scale sample surveys"
Series B. 231:329-451"
An outline of the theory of aa.pling
(Japan) .: 149-156e
1nn4ls Inst" Math"
~950o
Murthy, Mo ~o
19570 Ordered and
sampling without replace.ent~
unordered estimates
Sankhya 54:596-612.
in
Haraln, Ro Do
1951" on sampling ~ithout replacement with
varying probabilitieso
Jo Indian SOCo 19ric" stat.
3 (2) : 169... 174"
Raj, Des"
1956a" A note on the determination of
probabilities in samplingvithout replacement.,
11:191-200"
optimum
Sankhya
Raj, Des"
.956b., .:>ome estimators in sampling with varying
probabilities without replacement., J, Amero statist o
Ass" 51:269-2840
Rao,
J" No K", and Bayless, 0" Lo
1969"
An empirical stUdy
of
che
stabilities
Jf
estimator~ and
variance
estimators in unequal probability sampling of tvo units
per;stratum~
J" Amero
Statist, Ass" 64:540~559~
so
Rao"
1962..
A
J .. N.. Ko o HartleYI1 H.. 000 and Cochran. W.. G..
simple
procedure
of unequal probability sampling
without
replacement..
J .. B..
statist ..
SOCo
824: QS2-Q I} 1 ..
sampford" M.. B.. 19670 On sampling without replacement with
unequal
probabilities
of
selection.
Biometrika
54:499-513"
Smith" Ho Fairfield.. 1938. An empirical law describing
heterogeneity in the yields of agricultural cropso The
Journal of Agricultural Science 28:1-23 ..
1953 ..
selection without
Yates" F.. o and Grundy" P.. M..
replacement from within strata and with probability
Statisto
Soc..
J .. ,B ..
proportional
to
size..
B15~253-261..
81
APPENDICES
80
8,,1
Pat!~l1at~i~.Ill!lI:siQ!l
Consider the set of N simultaneous equations
£
where k is
~n
If M caD be represented as
!! =
elements
a
!
Mxl vector for which a 30lntion is sought, and
M is an NxN matrixo
where D is
="
nonsingular
d{ii),
Dt 1-!!'
diagonal
matrix
with
1 is an Nxl vector of l's and h is an Nxl
vector with elements, b(i)
then if
Q
5(2)
is defined as
N
5(2)
:
b(i)d~l(ii),
5
1=1
and
5(2)
"-1
#
-1,
can be written as
The elements of the vectors y
u (i)
=
~nd
yare
lId (il) ,
and
veil
= b(i)/d(ii)~
respectivelyo
The proof is given by shovin<j that "-1ft
Note that
diagonal
= Ie>
82
= 1,
D Y
and
= y ..
D-t
!!
!! .I'
1
Therefore,
D Y y' +
[
1 h'
::;
(1+5(Z»
1 hi.
D..... I
,
and
ft- 1 fI
=I
+·1·!! II,
If the elements of
D.... 1 -
1 h•
D- I
=I
lesignated as eli)
£~re
k
and
as
K(i), the set >f salutions Ear the K(i) may be written as
.k
= [D- I
-
(
1 +5 ( 2 ») ~ 1 .!! .!. ] £,
or
K(i)
= d~l(ii)(
c(i) -
(1+S<Z»-lS<3)}
where
5(3)
1
= v'
C
N
: 5
b(k)c(k)d-1(kk)o
i=1
somewhat more general form
inv~rsion
of
this
is given by Graybill (1961, p.
pattern
matrix
110).
The third moment of the Horvitz.Thompson estimator
may
be written as
B{[t(s)-T]31 !! } ::; B[t 3
-
T3 -
lTV[ t
(s)
(S)
I .! ]
I !! ].
For economy of notation and convenience
derivations,
The cube
~f
the substitution
~(i)=Y(i)/P(i)
t(s) may be written as
in
subsequent
is used oe10w.
83
n
t
3
(S)
:.:: (
Z(S(U)])3
S
u=l
n
n n
= S Z( s (u) ] + P/2) 5 S Z[ s (u) ]Z[ s (V) ]( Z[ s (a) ]+Z[ s (v) ])
u=l
u~v
n n 11
+ S S ; Z(s(u) ]Z[s(v) ]Z[s(q) ]..
ui'v~q
The expectation of t
3 (S)
overall samples s of
size
n
is
tI
B[ t3 (S) f
] = S Z3 (i) P (i)
~
i=1
I
N I N
J
+(3/2)5 SZ(i)Z(j)[Z(i)+Z(j) ]P(ij)+S 5 SZ(i)Z(j)Z(k)P(ijk)o
ii'ji'k
i~j
The cube of the population total g T, for a given finite
popttlation,
~,
may be written as
T.J
={
If
S Z (i) P (i)
P
i=l
=
I
5 Z3 (i) p3 (i)
i=l
If If
+ (3/2) 5 SZ (i) Z (jl{Z (i) P (i) +Z (j)P (jl ]P (i) P (j)
ii'j
B N l\t
+ 5 S SZ(i)Z(j)Z(k)P(i)P(j),P(k) ..
i~j~k
The variance as given in Chapter 1 may be restated as
N
V[ t (S) f
~
] = 5 Z2 (1) P (i) [ 1-P (i) ]
i=l
84
If It
-S SZ(i) Z(j) W(ij)
iflj
where
= P(i)P(j)-P(ij)o
W(1j)
The pwoduct of the population total
and
the
variance
may be expanded as
if
TV[ t (s) I
~
=S
]
Z3 (1) p2 (1)[ 1-P (i) ]
i=1
if tf
• (1/2) S SZ (i) Z (j) (Z (j).[ 1-P (j). ]+Z (i) [ 1-P (i) ]}P (i) P (j)
i#j
• H
-
S SZ(i)Z(j)[Z(1)P(1)+Z(j),p(jrlW(ij)
i"j
11 I H
- (1/3) S 'S SZ (1) Z (j) Z (It) [P (i) W(jk) +P (j) W(ilt).P (It) Ii (1j) ]0
i"j#It
SUbstltutinq
the
above
results
into
the
initial
expression for the third moment yields
11
B{[ t (s) -T PI.! ]
=S
Z3 (i) P (1) [ 1-P (i) ][ 1-2P (i) ]
i:1
-
H 11
IS SZ2 (1) Z (j) [ 1-2P (i) ]W (1 j)
iflj
NNN
+S S SZ (1) Z (j);Z (t) Q (ijk)
iflj#1t
where
Q(ijlt) :;:; P(1jk) -P(i)P(j)P(It) +P{i) W(jk) +P(j) W(it) +P(It) W(ij)
0
85
The twelve designs studied in Chapter 5 are
in this appendixQ
documented
All aesigns were based an the same iet of
size measures, X(i), and selection probabilities,
shown
Table 8Q3Q1Q
in
coefficients
or
Tables
weighting
8o~o2
8~3al1.
through
factors,
W(ij)
and w(ij),
employed in the Yates-Grundy forms of the variance
the
as
The pairwise probabilities for each
of the designs are shown in
The
PCi),
and
the
variance estimator, respectively, also are displayed in
the tables.
probabilities dnd weighting factors shown in the tables
are
rounded
to
six
decimal
places.
empirical comparisons, all computations
coapllter
in
double
precision
In
were
conductinq the
performed
and
intermediate results were stored in double precision
by
all
arrays
within the computer.
Table 8.3.1 Size measures and selection probabilities
~~~=~===-_-i -=~~==-:J~~==--:-'---' 1-(IT--~=l-.m.-.__.-.. - ....-.~~-J-!C. _.. __-.
__.. -.~_
1
2
2
3
4
4
5
1
3
7
5
6
7
8
9
8
9
10
10
6
0,,036364
0.,012127
0,,109091
0 ... 145455
0., 181818
0.218182
0,,254545
0.,290909
0,,321213
0 ... 363636
86
Table 8,,3J2 Pairwise probabilities and weighting factors for
designs (01) and (02)
L --!. (iJJ __
W (i j ).._._:_.__
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
4
2
3
4
5
6
7
8
9
10
3
4
5
6
7
8
9
10
4
5
6
7
8
9
10
5
4
4
6
7
4
4
4
8
5
5
5
5
5
6
6
E,
6
7
9
10
6
7
8
9
10
7
8
9
10
8
7
1
8
8
9
10
9
10
10
0.,002145
0.,002982
0.,Q03660
00004181
0.,004544
0.,004754
00Q04814
00Q04138
0 .. 004546
00Q05681
0.,007062
0.,008186
0 .. 009057
00009687
00010097
0,,010331
00010480
00010181
0.,Q11978
0,,013484
0.,014725
0.,015751
0.,016658
0.,017651
00015539
0.,017800
0.,019834
0.,021730
0.,023660
0.,025989
0.,022009
0.,025021
0.,028045
0.,031352
0.,035508
0.,030347
0 0034775
0,,039832
00046334
0.,042101
0,,049330
00058748
0.,060293
0.,073303
0.,091079
0.,000500
0.,Q00985
0 .. Q01629
0.,:002431
0 .. 003390
0.;004503
0.,005164
0"Q01163
00008678
0 .. Q02253
0,,003516
0,,-Q05031
0.,Q06810
00'008825
0.,011060
0.,013470
0.,015966
0.,005687
0 .. 007857
0.,010317
0.,013043
0,,015985
0 .. 019044
0,,022018
0,,010908
0,,-Q13936
0,,017191
0 .. 020584
0.,023-943
0 .. 026904
0.,017660
0,,:021260
0,,024848
0 .. 028152
0.,030608
0,,025191
0.,028696
0.,031573
0 .. 033005
0,,031949
0.,033976
0.,033814
00034913
00032483
0.,027929
00232917
0.,]30352
0.,444987
0.,581410
0,,145904
0 .. 947170
1.,197225
1.,512023
1,,909042
0.,' ]96663
0,,497896
0.,615302
00751926
0.,911057
1.. 095315
1.,303807
1.,523459
0.,558556
0.,655937
0,,765155
00885788
l."Q 14859
1.,143257
1.,247396
0.,701980
00-182941
0.. 866750
00947277
10011935
1.,035222
0.,802402
0.;849703
0.,885995
0.,897958
0., 861985
00830094
0 0825189
0 .. 792659
0 712335
0., 15885l~
0
0.,688753
00515589
0.,579062
00443Ln
0.,306&52
81
Table 80301
~
~airwise probabilities and weighting factors for
design (03)
i==rJ :I~~--~~{!ji_
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
8
9
10
2
3
/I
2
5
2
2
6
2
2
7
2
8
9
2
3
10
3
3
5
3
7
8
3
3
3
4
4
4
4
4
4
5
5
5
5
5
6
6
/I
6
9
10
5
6
7
8
9
10
6
1
8
9
10
1
8
6
9
6
7
7
1
10
8
8
9
0000"1155
00002212
00002645
00003306
00003967
0.,004628
0 0005289
0.,005950
0.,006612
0.,005091
0.. 006182
00006801
0.,007934
0.,009256
0.,010579
0.,011901
00013223
00009923
00011568
00012858
0 .. 013884
0.,015868
0 .. 017851
0.,019835
00015853
00018187
00020278
00022139
0.,023802
0.,0264q6
00022587
0.,025615
0.,028633
0.,031802
00035588
0.,031091
00035363
00040153
Oo0460Q3
8
0., 0 i~2565
q
00049341
10
9
10
10
0., 05782'1
0.,059442
00071032
0., O~H031
J_
if (i
it-
0.,000890
00,001755
0.,002645
0.,003306
0.,;003967
00004628
00005289
0.,005950
0.;006612
0.,002843
00004397
0.,006416
0 .. 001934
0.,,009256
0.,010519
00011901
00013223
00005944
0.,008266
00'010944
0.,01388Q
0.015868
00011851
0.,019835
0.,010593
0.,'0135Q8
00016747
0 .. 020115
00023802
0.,0264Q6
0.011083
0.,·020606
0.,024260
0.,027102
0.,030528
0.,0244Q6
0.,028108
0.,031252
00033296
0 .. 031485
0.,033965
00034735
00035764
0.,034753
0,,031978
l_ _ !=iiil__._
0.,506949
0.,193384
1.,000000
1.,000000
1.,000000
1.,000000
l.,OCOtDO
1.,000000
1.,000000
00558305
00711309
0.,.942619
1.,000000
1.,000000
'0000000
1.,000000
'0000000
0.;599013
0,,714580
0.,851144
1.,000000
1.,000000
'0000000
1.,000000
0.,668230
00144915
0.,fJ25878
00'911305
1.,000000
1.,000000
00756329
0.,802545
0.,847270
0.,8110"18
0.,857831
0.,.786286
00794865
00778343
00J23139
00739695
0 .. 6883hJ
0,,600666
O.,60~65)
0",43)2
3
00 3 6];t~,.,~
'~
88
Table
~~--i
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
8~304
rJ -
-=r==-p-ii'jl-~~-J-'
2
3
4
5
6
1
8
9
10
3
4
5
6
7
8
9
2
3
10
3
5
1
3
6
1
3
8
9
3
10
4
5
6
4
4
,!
4
4
5
5
5
5
4
1
8
9
10
6
1
8
9
5
6
6
6
10
6
7
7
10
1
8
8
10
9
Pairwise probabilities and weighting factors for
designs (11) and (12)
'7
8
9
8
9
9
10
10
0~001186
0",001816
00002414
00003166
00003895
00004666
0,,005487
00006365
00007310
00003700
00005040
00Q06446
00007926
00009492
00Q11156
0.,Q12935
00014848
00007108
00009853
00012111
0,,014496
00017030
0,,019135
0"Q22642
0"Q13406
0,,016469
0"Q19103
0.;023135
0.,026195
00,030125
00021023
0,,()25139
0 .. 029501
0.,034150
0,,039134
00()30836
0.. 036166
00041840
00047911
00043173
00049915
00()57126
00058433
00066829
0.,011105
---WCij)--,'=
T~~==~- ~~(i j)~~.~=
00001459
00002151
00002815
0.,003446
0.,004039
0.. 004590
00-005092
0.;005536
0.,005913
0,,004234
0,,005539
0.006717
0 .. 007942
00009021
0,,010001
0.Q10867
0.011598
00008160
0.,009981
0.011691
0,,013272
0.Q14706
00'015967
0 .. 011027
0,,'013041
00015267
0,,011322
Oci019179
0,,020808
00022168
0,,018646
0.,021142
00'023391
0 .. Q25354
00026981
00024701
0 .. 021305
00029565
0.,031422
0,,030877
00033391
0.;035435
0 .. 036774
0.,038956
0.041903
'0230331
'0184885
1.. 137620
'0088426
10037182
0098.3759
0 J28011
00869786
00608915
'0144485
'0098934
1.. 051484
'0002016
0.,950397
0
00~96484
00840120
00181134
1.. 058636
'0012971
0..-965321
001)15553
00863523
00309073
00152031
0.,912784
00926996
0 .. 879126
0 .. Q29030
0.,176549
00 721507
00 ~86927
001341004
00192894
001112438
00689458
00801067
00154996
0.,706621
0.;655'165
0,,715202
00668961
0062029~
0062'1332
00582916
00543456
89
Table 8.l.S Pairwise probabilities and weighting factors for
design (13l
---.-=r=-----_.--...._--m
Ii (ij)
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
3
3
3
3
J
2
3
4
5
6
7
8
9
10
3
4
5
6
7
8
9
10
4
5
6
7
8
9
4
10
5
6
4
4
4
7
8
9
4
10
5
5
5
5
5
6
6
7
8
9
I~
10
7
6
6
8
6
10
7
7
7
8
9
8
8
9
9
10
9
10
10
0.001322
00001983
00002645
00:003306
00003967
0.;004628
00005294
00'006148
00007070
00003967
00005289
00006612
00Q07934
0 .. 009314
0 .. 010962
00012118
00014609
00()01934
00009917
00011977
00()14364
0 .. 016892
0 .. 019582
00022474
0.013271
0 .. 016359
00019607
00023042
00026694
00030615
0 .. 020946
00025089
00029466
00034113
00039098
00030864
00036225
00()41910
0.048000
00043305
0.,050069
0.,057304
00058648
0,,061016
00077189
00001322
00001983
00002645
00-003306
00003967
0 .. 004628
00005285
0 .. 005752
00006153
0 .. 003967
0 .. 005289
00-006612
0.Q01934
0.009198
0 .. 010195
0.,011084
00011837
0.001934
0.009917
0.011825
0 .. 013404
0 .. 014844
0.016120
0 .. 017195
00013175
00015377
0 .. 017418
0 .. 019272
0.020910
0 .. 022278
00018724
00Q21192
0 .. 023427
0 ..:025391
00021018
0.,()24613
00Q21246
0 .. 029495
0 .. 031339
0 0030144
0 .. 033231
00035258
0 ..036558
0..-038109
0 .. 041619
W (ij'
-
1.,000000
'0000000
1.,000000
10000000
'0000000
10000000
0 .. 998319
0 ... 935615
0 .. 870225
10()00000
1.,000000
1.. 000000
10000000
0 .. 981511
0 .. 930036
0 .. 811482
00810284
1.. 000000
10000000
00987305
00933119
0 .. 818156
0 .. 823227
00165095
0 .. 992734
0 .. 939992
00888379
00:836415
0.,783325
00727612
0,,893898
0 .. 844677
0 .. 195058
0 .. 744300
00~91019
0 .. 799392
00752149
0 .. 103'156
00652881
00109938
0 663tH 8
0
0.615212
O.,62:E45
0.. 5"1':' 00
o0·5 ~.~7' '~'34
_
90
Table
aQlo~
Pairwise probabilities and
design (21)
-~LJI-1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
1
3
...
j
3
4
0~006079
5
6
7
8
9
0.004343
0.002938
0.001864
00001120
0.000701
0.000624
0.008062
0.001319
0.006905
0.006823
0.001011
0.001649
00008558
0.009798
00008889
0.009198
0.,011038
0.0'2608
00014509
0.Q16140
0.019302
0.Q13021
0.Q15583
0.018416
0.021699
00025253
0.029137
OQQ20459
0.024614
0.029219
0.034095
00039302
0.031203
0.037011
0.043269
0.,049798
0.045253
0.. 052113
0.060624
0.062608
0.071181
0.083269
10
3
4
5
6
1
8
9
10
4
5
6
1
8
9
4
4
7
8
9
4
10
5
5
5
5
5
6
7
6
6
6
7
7
1
a
8
9
6
8
9
10
7
8
9
10
a
9
10
9
10
10
factors for
_---Je-._. w
W-.:.(_i:=.:j)~,
. _ - - - - I L -_ _
OQ010542
0.008145
10
5
6
P....:..(1.....:·j::..;..)
2
3
3
4
4
I!
__
~eighttnq
(tj) _ _
-0.001891
-0~149129
-0~004178
-0~000190
-0.512965
-0.129909
0~522199
0.002268
0.004995
0.001392
00009458
00011194
0.012599
15.831110
20.'76460
-0~000129
-0~015945
0~003260
0~006118
0.009045
0.011442
0.013508
0~015243
0~016648
1.700000
3.965511
8.q42623
0.445420
0."4894
1.)25106
1.618182
1.765906
1.181116
1.699156
0.185124
1.024361
1.156405
1.202411
00006919
0.010031
0,,012164
00015161
00011221
'~'87342
0.018962
1~132748
00020367
1.055185
0.0134251.;031030
0.016152
'o~36534
0~018549
1.003976
0~020615
0~950064
0.. 022351
0.023756
00019210
0 ..-021601
0.023673
0.885091
0.815311
0~Q25409
00026814
0.024334
0.026400
0.028136
00029541
0.028791
0.030533
0.0319]8
0.032599
0~034004
0.Q35139
0.~38959
0.,815698
0.810182
00745220
0.682243
0.779871
00112162
0 .. 650255
0.593214
00636364
00$18563
0.526810
00$20681
00473111
0.429202
91
Table 80307 Pairwise probabilities ind
design (22)
W (ij)
1
1
1
1
1
1
1
1
2
3
4
5
6
1
8
9
1
2
2
2
2
10
2
1
8
9
2
2
2
3
3
3
3
3
3
.3
'.l
.:J
4
3
4
5
6
10
4
5
6
1
8
9
10
5
6
7
8
4
4
5
5
5
9
10
6
1
8
5
5
6
10
6
6
6
9
10
1
8
1
8
8
9
10
.,
9
1
8
9
9
10
10
0,,002645
00003961
0 0005289
00006488
00005083
00004008
00003264
00002851
00002169
00001934
00008824
00001992
00001909
00{)08157
00008136
00009645
00{)10885
00-009863
00{)10353
00011593
0 0 013163
00015064
00011295
0.,:019851
00012139
00()15301
00018194
00021411
00()2Q971
00028855
00019158
00023913
00028519
0",033395
00038601
00030502
00036310
00042568
0",049097
00044552
00052012
00059924
00061901
00011081
00082568
~eighting
factors for
---.-I-··-·------(ij)
00000000
00000000
0 ... 000000
0.,000124
0.,002851
0.,005248
00007314
00009050
0.,010455
O.,{)OOOOO
00001154
00,005231
00-001958
O.,:() 10355
00012421
0.,014157
00015562
00006004
0 .. 009481
00012209
0.,014605
0.,016611
0 ... 018Q01
00019812
0.,Q13107
0.,016434
00018831
0.020891
0.,022633
00024031
0 .. 019911
00'022308
0..-024374
00026110
0.;021514
00:()25035
0 .. 021101
00028837
00030242
0.. 029498
00031233
0.,032638
00033300
00034105
00036440
._---
W
--~-
._
.•. ~._----
00000000
0.,000000
00000000
00019108
00$60976
10309278
20240506
30113913
3 0 176119
00000000
00198777
00,654532
'0006,84
'0269409
'0421855
1.,461780
1.,429689
0.;608755
Oo~15718
1.,053102
'0109551
1.,106701
'0064264
0.,'991101
'0075951
1.,074036
'0035010
0.,975718
0",906362
0.,:e33043
'0007139
0<,:930536
OoeS4670
0.,781848
00712188
00:~20169
00745158
00617426
00:615958
00,662101
00599810
00544668
00531891
0c:i48824J
00441334
92
Table 8 .. 30~ Pairwise probabilities and weighting factors for
design (23)
_
i = O__
.I___~=(!j~- ____.J_________~Jij; .. _==C=:·- w(ij)
1
1
1
1
1
1
2
3
4
5
6
7
00001322
00001983
00002824
0 .. 003781
0.,003967
0.,004628
1
1
8
0.,005289
9
10
3
4
5
6
7
0.,Q05950
0.,006612
0 .. 004995
0.,006905
00006612
0.. 001934
0 .. 009256
00010519
0.,011901
0.,013223
0.,010819
0.,011319
0.,012060
0.,013900
0.,015928
0.,011992
00020034
0 .. 014452
0.. 016516
0 .. 018193
0 .. 021561
0 .. 024850
00028613
0,,020922
0.,024521
0.. 028612
0.,033224
0 .. 038369
00030552
00035965
0.,041899
0 .. 048361
00043531
0005018"'
0" 0585.,.'
0.,06016"'
0.,069279
00080502
1
2
2
2
2
2
2
2
2
3
3
3
3
....':1
3
3
4
4
4
/1
4
4
5
5
5
5
5
6
6
6
6
7
7
8
9
10
4
5
6
7
8
9
10
5
6
1
8
9
10
6
7
8
9
10
7
8
9
10
9
9
1
8
10
9
8
9
10
-_.-..
10
_---~---_
..
"'-_.~._~.~-'_.~~-------
_ _ _ _ •• _ "
00001322
0.,001983
0.;002465
0.,002824
Ocio03961
0.,004628
0.,005289
00005950
0.,006612
0.,002938
0.,003673
0.;006612
0 .. 001934
0.009256
00010579
0 .. 011901
00'013223
0 .. 004989
0.,008516
0.,011742
0.,013868
0.,015808
00011110
0.,0'9635
0.,011994
0 .. 015219
00018232
0 .. 020153
0.,022153
0.,024219
0.,018147
0.,021160
0 .. 024281
00026280
00027741
0.,-024985
0 .. 021506
0.,029506
00030972
00030519
0.,032519
0.,033985
00035040
0.,036506
O.,Q38506
_ _" _ - " _ ' _ · _ _ . ' _ · " _ ' _ ' _ ' . ' _ . _ _ • •
' ~ _ " ' _ .
" ~ " _ '
_ _ ,J _ _ _
1.,000000
1.,000000
0.,812902
0.,145641
10000000
1.,000000
10000000
1.,000000
1.. 000000
0.;588225
00531922
1.,000000
1.,000000
1.. 000000
1.,000000
1.,000000
1.,000000
0.,458545
0.,152408
0.,973606
0.,997681
0.,992480
0 .. 984352
0.,980084
0.,829881
0 .. 921484
0.. 910168
0.,962545
00915594
00844657
0.,896020
0088731'1
00-848631
0.,791016
0.,723156
00811191
0.. 764813
00104216
00640365
00101094
O064029 i J
0.,580116
00-582318
0 .. 526t:)4 i4
A""_ _ _...._ - _ . _ _
0.,4'1 BJ 18
. , " _• • ,,_ •. _
.•• M_ • • __
.•. · •• _ _ ........
,_~"_,_
93
Table 8.3,,9 Pairwise probabilities andleightinq factors for
design (31)
-~j=r
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
4
5
6
7
8
9
to
3
4
5
6
1
8
9
2
3
10
3
5
3
3
3
3
6
7
3
4
4
4
4
4
4
5
5
5
5
5
6
6
6
6
7
4
8
9
10
5
6
7
8
9
10
6
1
8
9
10
7
a
9
10
B
7
P
(ii)---[----i(ij,.-----J--------.,cij)_-_
0,,012902
0.,Q11638
0 .. 011229
0.,Q00066
0.,000079
0.,000093
0., 000106
0",000119
0.,Q00132
0",021366
0.015211
0",009231
0.,Q04118
0.000595
0",Q00616
0",Q00867
0",)01816
0.,)11015
00()10034
00Q08946
0",008857
00009146
00()10210
00 )11880
0,,012206
00Q13856
00()16188
00018674
00021162
0.,325314
0.,')20950
00025314
0.,029685
0.034546
00039779
0, 033029
o 039126
0.Q45632
U•. 052444
0.Q48665
0.-,056738
7
8
10
0 .. ~65068
9
(1
10
10
00J67543
0.011348
00089855
9
-00010257
-0.. 001611
-00005940
00Q06545
0.,001855
0.,009164
00010413
0.,011782
0 .. Q13091
-0 .. 019432
-00004632
00Q03986
00011750
0 .. 011911
0.,020541
0 .. 022934
00024630
0,,004853
00009801
0 .. 014855
0.,018912
00022590
0"Q25493
0.,027789
0 .. 014240
00J17879
00)20837
00Q23640
0 0 )25841
00Q27519
00018719
00Q20961
0.,023201
0.,124958
00Q26336
00Q22508
00024345
00025713
00()26894
00025385
00026568
0 .. )21494
0 0027664
0",028431
0 .. )29153
-00195015
-0 .. 659128
-00528982
990000000
99.,000000
990000000
99.,000000
99.,000000
990000000
-00110079
-00304531
00431411
20853711
30.,111660
33.,340200
260438460
13.,·564210
0.,440583
0.,976818
1.,660520
20135295
2.,469924
2.,496887
20339068
10166657
1.,290315
'0287211
'0265940
'0181455
10089492
00893513
0.,828293
0.,181775
0.,.722442
00662063
0.,681468
00622204
0.;564190
0.,512811
00521621
00468253
0.,422550
0.,4095715
O",36?6_l?
O.. 32L!4t't
94
Table 80lo10 Pairwise probabilities
for design (32)
_N£J
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
~
4
Q
~
4
4
5
5
5
5
5
6
6
6
6
7
7
7
8
a
9
j
2
3
4
5
6
7
8
9
10
3
4
5
6
7
8
9
10
4
5
6
7
8
9
10
5
6
7
8
9
10
6
7
8
9
10
7
8
9
10
8
9
10
9
10
10
I--~_~'(iji
and
weighting
C wc'ij)'==C=
factors
W-(ijt _ _
0~002645
0~003967
-O~OOOOOO
-O~OOOOOO
-O~OOOOOO
-O~OOOOOO
0.005289
-0.000000
0~006612
O~OOOOOO
-0.000000
0.000000
0.007236
0.006000
0.000424
00001069
0.001122
00007934
OuOl0519
0.013223
0.011992
0.008648
0.007120
0.005684
0,004903
0.01586B
00013603
0<>Q11766
0.011855
0,·013334
0.000697
0.Q03257
0.010155
0~010832
0~010101
-O~OOOOOO
-O~OOOOOO
0.000000
0~003876
0~009864
0.014037
0~0181t8
0~021543
0.000000
0~006232
00012036
00015914
0.. 018402
0~014513
0~021129
0,016192
0.Q13020
0.013921
0.016431
0.020107
00023370
0.026870
0.017050
00021593
0.023477
0.013426
0.017815
0.. 020594
0.022207
Oo~27154
0.. 025739
00027313
0.. 028744
0 .. 026979
0.027626
0 .. 028878
0.030052
0.032191
0.037372
0 0 028558
Ou035845
0,042527
0.049281
0.Q45562
0.053811
0.062089
0.065805
0.075559
0.088242
0~024233
00026022
0.. 022619
0~024688
0~028488
0~029495
0.Q30413
0.029401
0 .. 030226
0.. 030766
0~096375
0.542795
23 0953990
10.135370
3.234820
-0.000000
-0.000000
0.000000
0.323194
1.140630
1.971592
3.187469
4~39]681
0.000000
00458104
1.022989
1.342413
1.. 380129
1.449846
1.'49930
1.031140
1.279736
'0253402
1.104489
1.036899
009684ij4
1.326585
1.141366
0.947818
0.848476
0.769126
00944129
0.770710
0 .. 679060
O~609131
0 .. 625246
0.548116
0.490800
0.446781
0.40003£
O~348660
95
Table 8 .. 1.. l1 Pairwise probabilities
for design (33)
__
t _ r - j - r =__-.!~i1!~
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
1
.3
J
3
3
2
3
4
5
6
7
8
9
10
3
4
5
6
7
8
9
10
4
5
6
7
8
9
10
4
5
4
6
4
7
8
9
4
4
5
5
5
10
6
7
8
5
9
5
6
6
6
6
7
7
7
8
8
10
7
8
9
9
10
a
9
10
9
10
10
0 .. 001322
0 .. 001983
00003214
0 .. 003398
00003967
0 .. Q04628
0 .. 005289
0 .. 005950
00006612
0 .. 003967
0 .. 005685
0 .. 008860
0 .. 007934
0 .. 009256
0 .. 010579
0 ..·011901
0 .. 013223
0.. 012287
0 .. 011515
00-011901
0 .. 013884
0 .. 015868
0 .. 017851
0.. Q19835
0 .. 016266
0 .. 016612
0 .. 018679
0 .. 021423
0.,024158
00021070
0.,021548
0.. 024101
0.,027497
0 .. 031787
0 .. 036846
0 .. Q30572
0 .. 035694
0 .. 041628
0 .. 048261
0.,043422
0 .. 050924
0.,059079
0 .. 060753
0.. 070385
0 .. 082320
if
and
(tj)
weighting
==r=--
0 .. 001322
0 ..·001983
0 ..'002016
0 .. 003214
0 ..'003967
0 .. 004628
0 .. 005289
0.,005950
0.006612
00003967
0 .. 004894
0 .. 004363
00007934
0 ... Q09256
0 .. 010579
00011901
00'013223
0 ..·003581
0 .. 008320
0 .. 011901
0 .. 013884
0.015868
0 .. 017851
0 .. 019835
0 ... 010180
0.;015063
0 ..:018345
0,,020891
0 .. 023445
0 .. 025823
00018122
0.022180
00025395
0 .. 027117
0.;029210
0 .. 024966
0 ..·027177
0 .. 029777
00031012
0 .. 030628
00032382
0 .. 033483
0.,;034453
0 .. 035401
0 .. 036688
factors
w (ij)._ _
10000000
10000000
0 .. 645848
0 .. 945797
1.,000000
1 .. 000000
'I .. 000000
1.. 000000
1.. 000000
1.. 000000
0 .. 860788
0.,492371
1.. 000000
1.. 000000
1.. 000000
1.. 000000
1.. 000000
0 .. 291421
0 .. 122571
1.. 000000
1.. 000000
1 0000000
l .. QOOOOO
1 .. 000000
0 .. 625818
00903493
00·982108
00975209
0.;970479
00:953932
0 .. 841008
00920304
00923554
00871974
0.,:794314
00816632
0 .. 778206
00715303
0 .. 643760
00705348
00635900
00566137
0 .. 561105
00502958
0.,
44567!·~
96
Since
exploratory"
the
nature
a
of
nu.ber
subprograms were required.
this
of
new
research
co.puter
A listing of these
was
.ainly
progra.s
programs
provided in this appendix.
C
C
C
C
C
CONTHaL AND INPUT ROUTINES
MAIN
DATA
SPECS
MAIN PROGRAM
INTEGER NOSPEC/O/
INTEGER*2 IIT(200}
RB1L*8 A(4555)
D=4555
1 CALL DATA(I"NOSPEC~l{1)~30)
IP(NOSPECoBOoOlGO TO 11
DO 10 I=l,NOSPBC
CALL SPBCS (It)
CALL UIITS(I,M,l(l)
IP(M.LTo2)GO TO 10
NI= •• (1-1) /2
M1=1+1
M2;;:M1+IN
ClLL P1J(A(l),N,l(ltl)"NI)
ClLL CHBCK(l(l),N,A(M'),NN)
10 COITIIOB
GO TO 1
11 RETURN
END
SOBROUTIIIB DATA(N,I10SPBC,X,L)
RBAL-a X (L)
RBAD(1".;BND~1000) .,NOSPle
READ, (1(1),I=l,N)
RETURN
1000 NOSPEC'::::O
RITURI
BND
SUBROUTINE SPBCS(M)
COMMON/BLK1/G,MODBL/BLK2/CORR/BLK3/ITBR"TOL1,TOL2/
$8LI<LOW/1LOW
COMMON/BLK4/ITBB2
REAL*8 G,CORR
READ.M,G o CORR"MODBL v ITBR.ITER2,TOL1"TOL2 v ILOS
PRINT,,'lS1MPLB SIZB ~
~oM,
and
is
91
*
SUPERPOPULATION PARAMETER =
',G
MODEL = ',MODEL,
*
• C-PACTOR POR UPPER BOUNDO! W(I,J) =' ,CORR
PRINT,'
TOL = " TOLl, TOL2,' ILOW = ',ILOW
PUICH,'lSA8PLE SIZE =
',M,
*
, SUPIRPOPULATIOB PARAMETER =
, ,G
PUNCH, • MODEL = ',80DIL,
*
I
C-P1CTOR POB UPPII BOUID or W(I,J) =1 ,CORR
PUNCH,'
TOL = I , TOLl, TOL2,' ILOW = ',ILOW
RETUBII
END
~
PRINT v
C
C
C
C
C
C
•
GEIBRAL PUNCTIOR5
58112
SUM
IBDEI
LOWBID
UPPERB
BIlL PUICTION 58IN2*8(1)
COKftOB/BLK1/G,ftODIL
B11L*8 1,G
IP(80DBL&~Q.2)GO TO 2
58112=1** (2-0)
IBTUBI
2 58I12=1/(G*I+(100-0»
1I1'UII
BID
BIlL rOICTION 5U8*8(I,I)
IB1L*8 l(I),B
B=O
DO 1 J= 1, I
1 8=8+1 (J)
SU!t=B
IETUIII
BIID
INTBGBR PUNCTIOII IIDII(I,3,N)
IITIGII 1,3,1
1'(I-J) 1,2,3
1 K=(I-1)*I-I*(I-l)/2+J-I
GO TO It
2 PIIIT,'I EQUAL J IN INDII PUBcrION',I
J.<=1
GO fO 4
3 K=(J-l)*B-J*(J-ll/2+I-J
4 IlIDBI=K
IETURI
BBD
BBAL FUNCTION LOWBND*8(Pl,P2)
98
COMMON/BLKLOW/I
REAL*8 Pl 1l P2
IP(IoEQo2)GO TO 2
LOWBJD=O
RETURN
2
LOWBND~Pl*P2-DMINl
(Pl 1l P2t
RETURN
END
RIAL FUNCTION UPFEBB*8(P1,P2)
COMMON/BLK2/COBR
REAL*e Pl v P2 v CORR
UPP&RB=(1-CORR)*Pl*P2
RITURI
BND
C SUBPROGRAMS TO COMPUTE P(I)
C DilTS
SUBROUTINE UNITS(NgMIlP)
RBAL*8 P(N)oSUMrXPLUSrl
XPLUS=SUM (P, Nt
DO 1 I=l.,N
X=P (I)
P(I)=M*P(I)/IPLUS
1 PRIHT,I,X,P(I)
RBTURI
BID
C CHECK ROUTINE TO DETERMINE IF PIJ SOLUTIONS SlTISFY BASIC
C
RESTRICfIOIS
SUBROUTINB CHBCK(P,N,W,NN)
RBAL*8 P(N)qW(NR),PI,RD
RBIG::;Oo
DO tOO I=l,N
PI:·Oo
DO 10 J=l,H
IP(JoBQ,I)GO TO 10
K=I NDBX (IJ H)
J
PI=PI+P(I)·P(J)-W(K)
10 CONTINUE
RD= {PI> f! (1» I.' (I)
PRINTvP(I)IlPJ1RD
RO=DABS (RD)
!~(RDoGToRBIG)RBIG=RD
100 CONTINUE
PIINT,' LARGEST REL1TIVB DIFFERENCE IS =
RBTURN
litO
',RBIG
99
C
C
C
C
C
C
C
C
C
C
SUBPROGRAMS FOR COMPUTATIOH OF P(IJ)
PIJ
lITEST
ADJUST
SPICIAL FUHCTIOHS
TK
LOWBND
UPPBRB
OTHBR GENBRAL FUHCTIONS RIQUIRBD
INDBI
SUBROUTINB PIJ(P,NoW,HHN)
RE1L*8 P(N) oW (NNH) 15(3) I1SMIN2,TK
S (1) =0
(2) :0
~
S (3)=0
DO 1 I=l,N
1 S(1)=S(1)+SftIH2(P(I»
DO 2 I=l,N
S(2)=S(2)+SMIH2(P(I»/(S(1)-2*SftIN2(P(I»)
2 S(3)=S(3)+P(I)*(1-P(I»/(S(1)-2*SMIN2{P(I»t
NLI551:;1-1
DO .3I=1,HLISS1
IPLUS1=I+l
DO 3 J=IPLUS1,N
K=I IDE I (I, J 0 H)
3 W(K) =StlIH2·(P (I)) *5MIN2 (P (J»* (TK (P (I) ,5) +'l'K (P (J) ,5))
CALL WTBS'l'(P,N,i,IHI)
DO 101 1=1,11.1551
IPLUS1=I+1
DO 101 J=IPLUSl o H
K=I NDBI (1" J, H)
P1IR=PtI)*P(J)~i(K)
WBST=i(K)/P1IB
POlICH, I,J" PAIR, W(K), WEST
101 PBIHT"I,J"P1IR,W(K),WEST
RETURN
110
SUBROUTINB WTBST(P,N"W"NNN)
COMftON/BLK3/IOITIB,TOL1,TOL2
R81L*8 W(NNH),ADJ,P(H),LOWBND,UPPIIB
ITB8=0
lILBSS i::~M~ 1
1
11'IR~"ITBR+l
HCORB=O
DO 10:1=1"NLIS51
IPLUS1=I+l
DOlO :J=IPLUS1,lf
K=I NDII (1 .. ·J 0 N)
IF(W(K)-LOWBND(P(I) ",P(J» )2,2,3
2 ID]~{LOWBND{P(I) qP(J»-W(K)*(1.0+TOt1)
GO TO 5
100
3 IF {W (K) -UPPERB (P (I) ~ P (J» ) 10, 10 (,4
(UPPERB (P (1) ~ P (J) )
-W (K» * (1 .. O. rOL2)
5 CALL ADJUST(W~H~NNN~lDJ~I,J)
IP«ITEReEQo 1) .. ORo (ITER.,EQeNOITBB)}PRINT,·N(·,I~J.
* 'BY'ill DJ
NCOBR=NCORR+'
10 CONTINUE
PRINToNCORR,'ADJUSTMENTS MADB IN ITERATION'tITER
4 lDJ==
IP(NCORR)12~12,11
11 IP(ITER-NOITER) 1,12,12
12 RETURN
END
SUBROUTINB ADJUST(W,NoNNN,IDJ,ISTIR,JST1R)
RE1L*8 W(NNN),lDJ
NLBSS1=N-l
DO 10 I=l"NLBSSl
IP1.US1=I+1
DO to J=IPLUS1,N
K=INDEX (I J ' N)
IPOIIT=-l
IP«I .. EQoISTIR)oORo(IoEQoJSTAR)} IPOINT=IPOINT+l
IP {(J.,~QoISTAR).,ORo(JoBQ.,JST1R» IPOINT=IPOINT+1
IP(IPOINT) 3,5,,4
W(K)=W(K)-ADJ/(H-2)
GO TO 10
W(K)=W(K)+2*ADJ/«N-2)*(N-3»
GO to 10
W(K)=W(K)+lDJ
:0IT1IUB
BBTURII
END
(I
5
3
4
10
REIL FUNCTIOli TK*S(P"S)
REAL*8 P,S(3),SMIN2
TK= lP* (l-P) /SM1&2 (P) -S (3) l(l+S (2»' I (S (1)-2*5MIN2 (P)'
RBTURN
BND
C
C
C
C
SUBPROGRAMS FOB COMPUTATION )P PiIJ} USING
MATRIX INVERSION,
INCORPURATBS ADJUSTMENT MBTHOD 1..
MAIl ROUTINE CARDS REQUIRBD TO CALL THBSE jUBROUTI&ES
R1D=8* (D-UN) + 1
NA=o2S*(SQRT(RAD)-1)
N1UG=IU-N
NAUG:::: -"lXOf-MAUG,-100}
"3=K2+Nl*IA
M4=M3+Nl*lfl
KS=N'1UG+ 1
CALL PAIRS(A(l) "N"I(M1) ~NN,A(M2) ,A(K3) "NA~A(H4} Ii
101
*
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
IRT(l),IHT(MS),IAUG)
PAIRS
SETUPA
INVERT
WCOftP
WTIS'll
WTIST2
IHIT
lUGftliT
SPECIAL FUNCTIONS
INDEX
TTK
OTHER FUaCTIONS
SMII2
LOWOID
UPPBRB
SUBROUTINB P1IRS(P,N,W,RIN,1,lI'V,IA,AUG,IAUG,JAUG,
*N1UG)
RBAL*S P(N),W(NHR)tl(NA,NA),lINV(NI,NI), lUG(NIUG),
*Sl uSfUN2
INTEGBR*2IIOG(NIOG),JIOG(IIUG)
Sl=O
DO 1 I=l,R
1 Sl:S1+SMIN2iP(I»
ClLLSR'lUPI (A,Hl,S1,.,P)
NCOR9=0
CALL CHVBRT(l,lINV,Nl,H)
CALL WCOMP(W,NNN,P,N,AOG,I1UG,JIUG,IAUG,IIRV,NA,
*NCORR)
4 LSIVB=NCORR
:ILL WTBST1(W,INN,NCORR,IUG uIAUG,JIUG,IAUG,P,N)
PRINT,NCORR
IF(NCORR-L51VB)6,6,5
S ClLL tNIT(A,N,NCORR,NA)
CALL 3BTUP1(A,NA,Sl,N,P)
CALL AUGMNT{A,NA,N,NCORR,
IAUG,J1UG,N1UG,P)
NAR1NK=H+HCORR
CALL INVERT(AulINV,Nl,IAR1NK)
CALL WCOMP(W,RHHuP,N,AUG,I1UG,J1UG,HAUG,AIIV,NA,NCORR)
6 BSAVB=HCORR
CALL IITBST2(W,NNN,NCORR ulUG,IAUG,JAUG,NIUG uP,N)
PRINT,NCORR
IP(NCOBR-HSAVB)S,S,7
7 CALL INIT(A,N,NCOBR,NA)
CALL SETUPA(1,HA,51,N,P)
CALL IUGMNT(A,NA,N,NCORR,
IAUG,J1UG,N1UG,P)
N1RIIIK:;;I+ NCORR
CALL INVERT(A,AINV,Nl,H1BAHK)
CALL WCOMP (II u HNN, P" N, AUG uIAUG, J1UG uHIOG"lI.V 1U.,NCORR)
8 IF(HSAVE-LSAVB)9,,9 v4
9 IF(NCORB-HSAVB) 10,10,6
1l
102
10 NLBSS1=N-l
DO 101 I=l,NLESS1
IPLUS1=I+l
DO 101 J=IPLUS1,N
K=INDEX (I/I J, N)
PIIR=P(I)*P(J)-W(K)
WEST=W (K) /PAIB
101 PBINT,IpJ/lPl.IR,W(K)/lWEST
BE1"UBN
EID
SUBROUTINE SETUPA(l,Nl,Sl,N,P)
RllL*8 1(ll g I1) ,Sl,SftIN2,P(I)
DO 10 -I=l o N
DO 10J=l/1N
1P(I-J) 2,1.,2
1 A(I/lI)=Sl-SftIN2(P(I»
GO 1"0 10
2 1 (I/lJ).=SlUN2(P (J»
10 CONTINUE
DE1"Ual
BID
1
2
3
4
5
508ROU1"IIE INYERT(l,lINV,DIII,N)
INTEGBR DIlloN
RB1L*81(DIII,DIII),IINV(D1II,Dlft),DIV,IIUL1"
DO 3 1=1,N
DO 3 J=l,N
1P(1-J)2 r ',2
IIIIV(1.,1)=1
GO TO 3
lINY (I, J) =0
CONTUIUB
DO 101=1,1
DIV=l (1 0 1)
DO 4 J=l r N
l(IDJ)=l(I,J)/DIV
lIHY(I,J)=AINV(I,J)/DIV
DO 6 J=ll'N
IP(I-J)5,6 g 5
MOLT=l(J, I)
009 K=l o N
l(JoK)~l(JoK)-MULT.l(I,K)
lINV(JoK)~lIIV(J,K)-"OLT*lII'(I,K)
9 COITIIUE
6 CONtINUE
10 ::OITINUI
RETOIU'
lUiD
SUBROUTINE
*NA,NCOBR)
WCOMP(W,NNN~P,N,lUG,IAOGoJAUG~IAUG~ArNVu
103
INTEGEB*2 IAUG(NAUG~.J1UG(11UGJ
REAL*8 1IIV(Nl oI1)
RE1L.a i(NINl*P(N) ulUG(N1UG) oSKIN2,TTK
NLESS1=N~
1
DO 4 I=1.HLESS1
IPLUSt=I+l
DO ~ J=IPLUSl o H
K=IHDEX U.JoN)
W(K)=SKIN2(P(I»*SKIN2 (P(J)}*
•
(TTK(AINV,NA,I1UG,JAUG,AUG,NAUG.HCORR"
*P"R.I)
•
+TTK(AINV,HA.IAUG,JAUG,lUG.NAUG.NCORR.
*P"HoJ»
4 CvRTINUE
IP(HCORRoEQoO) GO TO 10
DO 6 K=l I1 NCORR
II=I1UG (I)
JJ=JAUG (K)
J=INDBX(IIoJJoN)
6 W(J) =lUG (K)
10 RBTURB
BND
SUBROUTINE WTBST1(W"
DUB,NCORR~lOGI1IAUG.JAUG,DI!,
*P"lI)
INtBGBR DlfloDUB
INTBGEB*2 I1UG(DIK)pJ10G(DIK)
RBAL*8 \UG(DIH)~P(I ),W(DUB),LOWBRD
NLBSS1=If-1
DO 10 I=l.lILISSl
IPLUS1:;I+l
DO 10 J=IPLUSl o If
I=IIDBI (l.J. I)
IP(W(K)-AOWBID(P(I)I1 P (J»)2,10.10
2 NCORB=ICORR+l
lUG(HCOBR)=LOWBHD(P(I).P(J»
I10G(HCORB)=1
JAUG(NCOBR)=J
PRIIT. IAUG(NCORB) .J1UG(NCORB) .AUG(IfCOBR)
10 :OITlIUB
RETURN
UfO
SUBROUTINE WTBST2(W o
DUB,HCORR.1UG.I1UG.J10G~DIHo
*P l1 l1)
I~!rEGER DIH" DOB
XSTBGEB*2 I1UG(DI8)qJ1UG(DlfI)
BB1L*a AUG (DIK) .P(N ) .W(DUB).UPPBRB
NLBSS 1=8-1
DO 10 I=l"HLBSSl
IPLUS1:I+l
DO 10 J=IPLUS1"N
104
K::;:INDEX (I"J"N)
IF (iCK) -UPPERB(P(I) ,p(J»)e10,10,4
4 NCORR=»CORB+l
AUG(NCOBR)=UPPERB(P(I) ,P(J»
I1UG(ICORR)=I
J1UG (NCORR) =J
PRINT, I1UG(HCORR) ,J1UG(NCOBB) ,AUG (NCOBB)
10 CONTINUE
RETUBN
END
1
SUBROUTINE INIT(1,N,NCOBB,DI8)
INTEGER DIM"N,NCORR
RE1L*8 A(DI8,DI8'
8=R+NCORR
DO 1 1=1,8
DO 1 J=1,,"
A (IjJ) =0
BETURN
EID
SUBROUTINE AUGHNT(A"DOM~N,ICORR,
I1UG,JAUG,DIK,P)
INTEGBR DIK,DOK
INTEGER*2IAUG(DI8),J1UG(DI8)
REAL*8 A(ODK,DDK),
P(R),SKII2
DO 10 K=l"NCORR
KPLUS=K+N
1 (lAUG (K) , KPLUS)=SMIN2(P (JIGG (K) ) )
A(JIUG(K),KPLUS)=S"IR2{P(I1UG(K)})
l(KPLUS"IIUG(K»=1
l{KPLUS.JAUG(K»=l
10 A(KPLUS,KPLUS)=l
RETURN
END
RBIL FUNCTION TTK*8(AINV,NI,I1UG,JIUG,IUG,NIUG,NCORR,
*P,N"I)
RB1L*8 1IIV(Nl,ll) ,lUG(N1U~,P(I),SKII2
INTEGBR*2I1UG(N1UG)v J1UG (11UG)
TTK=O
DO 1 J=l" N
1 TTK=TTK+1INV(I"J)*P(J)*{l.0-P(J)./S8IN2(P(J»
IF (MCORrl) 10,,10,2
2 D03 J::;:1,NCORR
JJ=N{-J
3
TTr~=TTK+1INV
*311UG (J)
10 ·RETURN
END
C
(I" JJ) *lUG (J) / (S8IN2(P (lAUG (J) ) )e*SPJIN2 (P (
)))
SUBPROGRAMS FOR COMPUTATION OF ?(IJK)
105
C MAIN PROGRAM CARDS BEQUIREDTO CALL THESE SUBROUTIIES
IF(~ot~o3)GO TO 10,
911=1*(I-l)*(N-2)/6
H2=M1+NN
M3=M2+NNN
CALL PIJK(A(1)~NtA(Hl),NI,A("2~,HNR,A(ft3»
C
PIJK
C
TaITST
C
TRI1DJ
C SPECIAL FUNCTIONS
C
BIJ
C
TLIJ
C
TRIUPR
C OTBBh FUNCTIONS
C
INDEX
SUBROUTINE PIJK(P,R,W,IR,TBI,.NR,S4)
RE1L:4<8 P(I)QW(NNl,1'RI(N.1I),S4(1I~,1'T
RBAL*8 BIJ,TLIJ,SftII2,PSUft
TT=O
DO 9 1=1,1
S4(I)=0
DO 8 J=1,N
IF (I-J) 7,8,7
7 S4(I)=S4(I)+BIJ(P,I,i~II,I,J)
8 COITINUE
9 1'1'=1'1'+S4 (I)
TT=T.T/ (3* (1-4)* (N-3) * (1-2) )
L=O
NLES51=1-1
PSUft=O
NLESS2=N-2
DO 10I=1,ILES52
IPLUS1=I+1
DO 10 J=IPLUS1,NLBSS1
JPLUSt=J+1
IID1=INDBX(I,J,I)
DO 10 K=JPLUS1,N
IID2=INDBX(I,K,I)
IID3=INDBX(J,K,I)
L=L+1
TR1(L)=TLI.J(P Il I,W,II,S4 u TT,I,J).+TLIJ(P,.,W,NI,S4,TT,
*1 "K) +
*TLIJ(P.N,i,NI,S4"TT"J,K)+P(I)*P(J)*P(K)-P(I)*W(IND3)
*->P (J) *~ (IID2)
•
- P (K)
*W(I ND 1 )
10 COl:JfINUB
CiLL rRITST(P,N,i,II,TRI,IIN)
L=O
DO 11 I~lqNLESS2
IPLUS1=I+l
DO 11 J=IPLUS1,NLESS1
JPLust=J+l
106
DO 11 K=JPLUS1 QN
1.,;:1.-0-1
PSUM=PSUK+TBI (L)
PBINTuloJqKoTRI(L)oPSUK
11 CONTINUE
BETURN
END
SUBROUTINE TRITST(P,N,W,NN,TRI,NNN)
COKMON/BLK4/NOITER
R£&L*8 P(N),W(NN)QTRI(NNN),TRIUPR
REAL*8 ADJ
ITB8=0
&LESS1=N-l
NLBS52=1-2
1 ITBB=ITEB+l
NCOBR==O
1.==0
DO 10 I=l u NLES52
IPLUSt=I+1
DO 10 J=IPLUS1 q ILBSSl
JPLUS1:;J+l
DO 10 K=JPLUSl u H
1.=1.+1
IP(TBI (1.» 2,3,3
~
lDJ~-TBI(L).TRIUPB(P,N,W,.N,I,J,K)/(N-3)
GO TO 5
3 IP(T8I(L)-TBIUPB(P,N,W,IN,I,J,K»10,10,4
I,'; lDJ=TBIUPB (Po No WuRN, I, J, K). (1-4) / (1-3) -fBI (1.)
5 WRITB(10,*)I o J,K,IDJ
NCOBR=NCOBR+1
10 CONTINUE
IP(HCOBR .. BQcOlRBTUBN
RBWIND 10
DO 21 L=l o NCORR
RE1D(10 0 *)I q J,K o ADJ
I'«ITERoEQol).OR.{ITBR.EQ.. ROITER»PRIIT,'TBI(·,I oJ,K w
• ')BY'u 1DJ
21 CALL TBI1DJ(TRlqN,INN,ADJ,I,JqK)
REWIND 10
PRINT,HCORR,'ADJUSTMBNTS MADB IN ITERATION',ITER
11 IP(ITER-NOITBR) 1,12 0 12
12 RETURN
END
suaaOUTINB TRI1DJ(TRlqNqNHN,ADJqIST1B,JST1BuKSTARJ
1£11.*8 fRI(NHN)qADJ
NLI552::N-2
N1.B581;;1-, 1
1.=0
DO to 1=1"N1.:8552
IPLOS1=Ivl
107
DO 10J=IPLUSl o NLESSl
JPLUS1;..;J+l
DO 10 K=JPLUS1/l11
L=L+1
IPOIIT=l
IF{(I.~QDIST1R).OB.(J.EQoIST1R).01.(K.~Q.IST1R»
*IPOINT=IPOII'1'+l
IF ( (I .. ~Q. -1S'1'IR) ",OR (J .~Q.JS'l'II)o'OR. (K. BQ • .jST1R) )
0
*IPOIIT~IPOIIT"1
IF«I.~Q.KST1I)oOBo(J4EQ.KST1R).QR",(K.~Q.K5T1R»
*IPOINT=IPOINT.1
GO TO (l q 2 0 3 0 4) qIPOINT
1 TRI(L)=TBI(L)-6*lDJ/«N-3)*(1-4)*(I-S»
GO TO 10
2 TRI(L) ='1'81 (L) +2*IDJ/ «1-3) * (1-4).)
GO TO 10
3 TBI(L)=TRI(L)-ADJ/(1-3)
GO TO 10
4 TRI(L)=TBI(L).IDJ
10 CONTINOE
BBTORN
BND
REIL FUNCTION BIJ*8(P o NoWqNN.I oJ)
BEAL*8 P(N'oW(II)
K=INDEX (IoJol)
BIJ=W (K) * «1-2*P (I» .. (1~2*P (J) n
RI'l'OIN
BID
BElL PONCTIOI TLIJ*8{P~N,W.'1.54i'l''l',I.J)
BBIL*8 P(I).W(NN)qS4(1),T'l,RIJ
TLIJ=RIJ (P. 1 0 Wq NN 0 I, J) / (1-4)'- (54 (I) .54 (J) ) / (2* (1-4)
(1-3) +'l'T
RETURN
BND
*
RBALPUICTION TRIUPB*8(P~NoW,NN_I,J,K)
BRlt*S P (N) "i¥ (lUJ)
IND1=INDRX(I oJ,N)
IND2:INDEX(I o Kq N)
IND3=INDEX(J oKo N)
1=11 (I~!D 1)
IF{1~GT.W(!ND2»A=W(IND2)
IF{~.~ToW(IND3»
1=W(IID3)
!t'DiUpa"'l
aBTtJBN
END
C
C
5UBP~OGRAMS TO SIMUkATR POPUL£TIOHS 110 COMPUT~
THE SUPERPOPULATION ~lnIANCB OF THE VARIINCE
*
108
C
C
OF THE HORVITZ-THOKPSON ESTIMATOR
MAIN !ROGRAM CIRDS REQUIRED TO CALL THESE SUBROUTINES
NII={D~2*N-NN)/2
C
C
C
C
C
C
C
C
C
"3::M2+1
84="3+INI
CALL SIMLAT(A(1),N,A{Ml),NN,A(M2),A(K3),INN,M,A(M4»
SIMLAT
SORT
SPECIAL FUNCTIONS
'ARCOMP
YGBN
£21
OTHER FUNCTIONS REQUIRED
SUM
INDEX
C
SUBROUTIIB SIMLAT(P,N,W,NN,Y,FY1R,RNN,K,BFVAR)
BEAL*8 TEVYE
RE1L*8 P(N),W(NN),PVAR(INN),Y(I),B,I,YGEK,G,S2I,
*EPVAR(NNN),VCOMP
811L*8 B2I,VECOMP,MVAB
8£AL*8 YBIM,YBBRN,VVAR,II,IJ
INTBGBR IX (2)
COKftON/SIK2/S2I,G,MODEL
RBAD.IPOP
VO 1000 II=l,IPOP
REID,TOTX,TOTY,S2I,G,MODBL
kEAD.II,ITBR,IRBP
~RINT,' SUPBRPOPULATION PARAMETERS'
2'8INT, • '101.'1 = " TOTX, II TOTY = • ,'lOTY
PRINT,' S2I = ',S2I,IIG :: ',G,' BODEL= ',KODEL
PRINT,' IX :: ',II " ITER = ',ITER
rp(ITBRo~To'NN)ITBR=NII
B=TOTY/TOTI
18002=0
DO 299 IIII=l,IREP
DO 100I=1,IT,BR
DO 10J::1,N
I=P(J)*TOTX/M
INTI=I+.,5
IP(MOD£LoBQu3)GO TO 4
Y(J)=YGEN(B,I,IKOD2,IX)
GOTa 9
4 IP(DABS(G-2olotT.,Qol)GOTO 5
Y(J)~YBIN(B,I'TI,IMOD2,II)
GO
:to 9
5 Y(J)=YBBRN(B,X,IKOD2 v II)
9 IF(IoBQoloORotoEQoITBB)PRINT,X,INTX,Y(J),J
10 CONTINUE
PV1B(I)= VCOHP(Y,P,N~W•• N)
IP(MaGT.3)GO TO 100
ErVAR{I) =VECOMP flll P # N~ W,. NN I "'IR (I) )
109
100
CONTINUE
PRINT w e FINAL VALUES OF IX = ',IX
CALL ~ALLY(FVARoITER)
1F{"~Gro2)GO TO 250
CALL TALLY{EFVARg1TER)
250 DO 300 1=1,,1TB8
1F{1oLTollo0R.!oGT.ITER-l0~OR.IABS(I-ITBR/2).LT.2)
*PRINT"I"FVAR (I)
$EFVAB(I)
300 CO.T1IUB
299 CONTINUE
MVAR=Oo
IF{!ODBLoEQo3) GO TO 600
[;0 500 1= 1 "N
I=P (I) *'lOTI/M
500 MVAB=MVAR+P(I)*C'-P(I»*B2I(I)/I**2
KVAR={TOTX/M'**2*MVAR
GO TO 900
600 VV1R=0.
IF (Mo~Qo2)PRINT.e 'lHEORBTICAL EXPECTATION OF THE
*VARIINCE OF THE
S VIBIANCE ESTIMATOR =
" TEVVB{P,N,W"NN,TOTI,M)
IP(DIBS(G-2.)oLT.Oo')GO TO 700
DO 650 [=1,,1
I=P(I)tlTOTX/M
VV1B=VVAR+B*(1.-B)/X**3*('.+B*(1.-B)*2.*C X- 3 .)*(P(I)*
*(l.-P(I»)
***2
650 'VAR=MV1R+P(I)*('o-P(I»/X
HVIR=KYAR*B*(1-B)*(TOTX/Mt**2
NLBSS1=N-1
PBIJlTO' VVAR
DO 651 I=l,NLESSl
II=P{I)*TOTX/M
IPLUS1=I.'
DO 651 J=IPLUS1,N
IJ=P(J)*TOTX/"
K=ItfDEI (I"JIlH)
651
'YIB=VV1R+4.(B.(1-~)*·2/{XI*XJ)*W{K)*.2
PRINT" VVAR
VV1R=VV1B*(TOTX/M) **4
GO TO ag,t)
700 DO 750 1::::1"N
VV1R='VAR.(1.-4.·B.(1.~B»).(P{I).{lo-P(I»)
750
MV&B=KVAR+P(I)*(l.-P(I»
bilR=KVAR* (TOTX/M) **2*B*(1.-B)
NLESS 1=N·'·1
PBINT~ VVAB
DO 1~~ 1=1r.NLESSl
IPLUS1=I'"
DO 151 J=IPLUS1.N
K=<t NDE1 (I f J
v ~J}
•• 2
110
751
VV1R=VVAR~4*
8*(1-8) *W(K)**2
PBIN'1'u VVAR
VVAR=VVAR*(TOTX/M)**4*a*(lo-B)
899 PRINTvG THEORETICAL SUPERPOPULATION VARIANCE OF THE
*VARIANCE
•
*qVVAR
900 CONTINUE
PRINTu'THEORETICAL AVERAGE OF VARIANCE 'q~VAR
RELV=VVAR/(HV1R* MVAR)
PRINT q ' RELATIVE VIRIANCE = ',RBLV
1000 =ONTINUE
RETURN
END
SUBROUTINE TALLY(FVARuITBR)
R£IL*8 FVAR(ITBR).AVG.VARuSUM
AVG=SUM(FV1R uITBB)/ITB8
V18=0
DO 200 1=1 uITBR
200 V1R=VAR+(FV1R(I)-lVG)*(FV1R(I)-lVG)
V1R=Y1R/(ITER-1)
PRINT~AVG(JVAR,DSQRT(V1R)
CALL SORT (!'VARqI!BR)
RETURN
END
SUBROUTINE SORT(YuM)
RB1L*8Y(H) uTBHP
DO 10 1=2,H
IF(Y(1)-Y(1-l»6 q 10 p l0
6 fEMP=Y (I)
1M=1-1
DO 8 J=l,1M
L=I-J
IF(TEMP-Y(L» 7,9 q 9
7 Y(L+1)=Y(L)
8 CONTINUE
Y (1) ==TBMP
GO 'f0 10
9 Y(L+l)=TBHP
10 CONTINUE
HETURN
END
REAL FUNCTION
VCOMP*8(Y r P,N,W q NN)
RB1L*8 Y(N).P(N).W(NN)
VCOMP~O
NLESS 1:.:N--1
DO 1 1:=1"ILB551
IPLUS1;1+1
DO '1 lJ=:IPLtJS 1 ~
K"'"iUDEX (I (J.1,,; N)
N
111
1
vcoap=
RETURt"
END
VCOMP+W(K)*{Y(I)/P(I)-Y(J)/P(J)'**2
REAL FUNCTION YGEN*8(B,I,IMOD2,IX)
INTEGRR 11(2)
REAL*8 B,X,S
BE1L*8 E21
S=DLOG(1~Q+E2I(I)/{B*B*X*X»
U=DLOG(8*X)-S/2
SD=DSQRT(S)
CALL NORMAL(IX,IMOD2,SD,U,V)
YGEN=EXP(V)
",ErURN
END
REAL FUNCTION E2I*8(1)
COMMON/SIM2/S2I,G,MODEL
RE1L*8 S2I Q 1,G
IF(MODELo~Qo~)GO TO 3
IF(MODELoEQo2)GO TO 2
E2I=S2I*1**G
RETURN
2 E2I=S2I*(G*1**2+(1-G)*1)
RETURN
J IP(D1BS(G-2.).LToO.5)GO TO 4
E2I=S2I*1
RETURN
LJ E2I=S2I*A*1
RETURN
END
RBAL FUNCTION VECOMP*8(Y,P,N,i,NN,B)
RB1L*8 YeN) P(N),W(NN),B,VB,PIJ
VECOMP=O
NLBSS1-=N-l
DO 10 l=l Q NLESSl
IPLUS1=I+l
DO 10 J=IPLUS1I'N
K=INDEX (I 0 J £' N)
PIJ=P(I)*P(J)-W(K)
VE=W(K)/PIJ*(Y(I)/P(I)-Y(J)/P(J)t**2
10
VBCOMP~YECOMP+PIJ*(VB-E).*2
RETURU
ENr.
~UBROOTINE NORMAL(IX,IMOD2,SD,U,V)
REAL D(2) v Y(2)
INTEGER IX (2)
180D2=I80D2+1
r~QD2~MOD(!MOD2q2}
IP(IMOD2.EQ.O,GO
ra to
112
II{1)=rX(1)*16387
IF(IXP) aLToO' 11(1)=11(1) +2147483647+1
DO 3 J..;1,,2
II(2)=1X(2)*262147
IF(IX(2)oLToQtIX{2)=IX(2) +2147483647+1
Y(.1) =11 (.1)
3 Y(.1)=Y(J)/2141483647o
A=2*J.1415926536*Y(2)
B=-2., *ILOG (I (1»
D (1) =SQBT (B) *SII (A)
D (2)
=SQRT (B) *COS (A)
V""D {1} *SD-1-U
RETURN
10 V=D(2)*SD+U
RBTURN
END
RBAL FUNCTION YBIN*8(B q INTI,IMOD2 g II)
INTBGER IX (2)
BBAL*8 B
YBIN=O.,
DO 1 I=1,INTX
CILL RIR(IX g IKOD2 q V)
IF(VoLToB)YBIN=YBIN+lo
1 CONTINUE
RETURN
BID
REAL FUNCTION YBBRN*8(B g X,IKOD2 q IX)
INTEGBR IX (2)
REAL*8 B,X
YBBRN=O.,
CALL RIN{II"IMOD2"V)
IF(VoLT.,B)YBERN=X
RETURI
EID
SUBROUTINE RAN{IX q IKOD2 g V)
BElL Y (2)
INTEGBR IX (2)
IMOD2=IMOD2-o-1
IMOD2=MOD(IKOD2 q 2)
IF(IMOD2oBQoQ}GO TO 10
11(1)=11(1)*16387
IF(IX(1)6LToOtII(1)=IX{1)+2141483647+1
flO 3 .1=1".2
11(2)=11(2.*262147
IF (IX (2) ., LTo 0) IX (2) ""IX (2) +2141483647+1
y C.1} =IX (.1)
3 Y(J)=Y(J)/2147483647o
y:.:::y (1)
aBl'URN
113
10 V=YO(2)
RETURN
END
REAL FUNCTION TEVVB*8(P,N,W~NN,TOTX,M)
81AL*8 PIN) ,W(NN),XI,XJ,E2I,B4I
TEVVE:;::O"
DO 1 I=1,N
XI=P(I)*'l'OTX/M
1 TIVYB::::TBVVE-B4I (XI) / (P (I) *P(I»)* (lo-P (1» * p .. -P (1»
NLESS1=N-l
DO 2 I=l,NLESSl
XI=P(I)*'lOTX/M
IJ!I.US1=I+l
DO 2 J=IPLUS1,N
XJ=P(J)*TOTX/ll
K=INDEX(I,J 17 N)
2 TBYVE=TBYVE+W(K)*W(K)/(P(I)*P(J)-W(K»*(14I(XI)/P{I)**
$ 4
+B4I(XJ)/P(J) **4
*
+60*B2I(XI)*B2I(XJ)/CP(I)*P(J»**2)
* -20*B2I(XI)*12I(XJ)/(P(I)*P(J)
*)*(10-P(I»*(1.-P(J»)-4.*B2I(II)*B2I(XJ)*(W(K)/(P(I)*P
RETURN
END
REAL FUNCTION 141*8(1)
RE1L*8 S2I,117G
COMMON/SIM2/S2I,G,MODBL
IF (D1BS{G-2.lD~ToO.5) GO fO 4
E4I=I*S2I*('.t30*{A-2.)*S21)
RETURN
4 141=1**,4*52I*(10-3.*S21)
RETURN
END