·e
SOME NON-RESPONSE SAMPLING THEORY
FOR TWO STAGE DESIGNS
by
GEORGE THOMAS FORADORI
Institute of Statistics
Mimeograph Series No. 297
November, 1961
ERRATA SEEET
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TABLE OF CONTENTS
Page
LIST OF TABLES
LIST OF FIGURES •
1.0 THE PROBLEM
...
• • • • •
... . . . . . .
....
..... ...
1.1 Introduction.
1.2 Review of Literature •
1.3 Notation. • • • • • • •
vi
vii
1
1
...... ......
..........
16
2.0 THE THEORY OF SAMPLING WITH NON-RESPONSE AT THE SECOND STAGE.
21
Sampling with Unequal Probabilities and without Replacement at the First Stage • • • • • • •
2 .1.1 The Sampling Procedure
• • • •
2.1.2 An Estimator of the Total • • • •
.• • •
2.1.3 Variance of the Estimator, T • • • • • • • • .
l
2.1.4 An Unbiased Estimator of V(T ).
• • •
l
2.2 Sampling with Unequal Probabilities and with Replacement at the First Stage • • • • • •
• • • • • .
2.2.1 The Samp~ing Procedure • • • • •
• • • •
2.2.2 An Estimator of the Total •"",.
• • • •
2.2.3 Variance of the Estimator, T2 • • • •
"" ) • • • • •
2.2.4 An Unbiased Estimator of V(T
2
"'"
"'"
..
2.3 A Comparison of V(Tl ) and V(T2 ) Where the FSUs Are
Selected with Equal Probability for Each Case .• ••
5
2.1
3.0 OPI'IMUM SAMPLE ALLOCATION FOR NON-RESPONSE • • • •
22
22
23
25
28
32
32
33
34
36
40
....
A Cost Function • • • • • • • • • • •
• • • •
Sampling with Replacement at the First Stage • • • • •
3.2.1 Fixed Variance and Minimum Cost • •
3.2.2 Fixed Cost. and Minimum Variance • • • • • ••
3~3 Comments on the Optimum Solutions • • • • • •
• •
3.4 Sampling Wi thou~ Replacement at· the First Stage Wi th
Equal First stage InClusion Probabilities
3.5 Relative Efficiency • • • • • •
• • • • • • •
3.6 Application of Theory to Data • • • • • • • • • •
3.7· Extension to Stratified Case. • • • • • • • • • •
3.8 . Some Graphic Solutions for Optimum Recall Rates •••
42
44
44
49
50
53
55
58
66
69
v
TABLE OF CONTENTS (continued)
Page
4.0 TWO STAGE NON-RESPONSE THEORY EXTENDED TO MULTIPHASE .
~NG
•••••••••••••••••
~
•••
...
4.1 Sampling with Unequal Probabilities and without Re;.
placement at the First Stage • • • • • • • • • • • •
4.1.1 An Estimator of the Total • • • • • • • • • •
4.1.2 Variance of the Estimator, ~1E • • ••.• •.••
4.1.3 . Optimum Allocation • • • • • • • • • • • • • '.
4.2 Sampling.with Unequal Probabilities and with Replacement at the First Stage • • • ••
•••• • •
4.2.1 An Estimator of the Total •• • • • • • •
4.2.2 Variance of the Estimator, ~2E • • • • • • • •
4.2.3 Optimum Allocation. • • • • • • • • • • • • •
4.3 Comments on the Solutions • • • • • • • • • • • • • • •
4.4 Extension of the Theory to Stratified Sampling • •
........
5.1 Summar,y or Res.ults
•• -.
5.2 Summary of ConcJ-usions • • • • . . . . . . . . . .
5.3 Suggestions,for Further Research
......
LIST OF REFERENCES • • • • • • • • • • • • • • . . . . . .
5.0 SUMMARY AND CONCLUSIONS
·e
••••
77
77
78
81
85
85
86
86
88
90
94
94
95
95
97
vi
LIST OF TABLES
Page
17
Notation relevant to unstratified single phase sampling • •
2.
Notation relevant to unstratified multiphase sampling •
18
3 • Population data • • • • • • • • • • • • • • • • • . • •
60
4:
·e
..
1.
Population values of. reSPOndent SSUs. to ya.;rious mailings. for
th FSU • • • • • • • • • • • • • • • • • • • • • • • •
the i
76
vii
LIST OF FIGURES
P~e
1.
2.
3.
·e
Contours of equal f 1-1 for,. = 1 and R1
Contours of equal f 1-1 for,. = 1 and R1
Contours of equal f~1 for ,. = 1 and R1
= .25
= .50
= .75
··• .• ··
··· ..··
• · · . . . . · ·
71
72
73
1.0 THE PROBLEM
1.1
Introduction
The theory of statistical sampling is principally concerned with
the measurement of variation in an estimate based on the ultimate units
selected in the context of the frame used to identify these units and
the probability system constructed for the selection of these units.
The variation in an estimate arising purely from the operation of sampling is called sampling error and arises because not every sampling
unit in the population is enumerated.
However, through the use of prob-
ability theory, appropriate to the selection scheme, the variation can
be assessed and even controlled.
For those sample surveys where measurements of the ultimate sam-
··e
pling units depend on the response from human beings, difficulty may be
encountered in enumerating the selected units.
Thus, a sample drawn
properly, according to some known probability scheme may admit of errors
due to such non-response.
Certainly, the general class of non-sampling
errors includes much more than non-response errors.
Items such as
interviewer bias, coding errors, false or erroneous replies and plain
mistakes also fall into this class.
IIi essence, we shall include all
variations other than the random variations introduced by the selection
process itself in the class of non-sampling errors.
In this thesis, the
non-response errors will be singled out for critical analysis.
Unless
otherwise stated, the term bias used in this thesis will refer to that
arising from non-response.
2
A non-response situation will be said to exist when some of the
selected sample un1 ts fail to be enumerated.
It should be noted that
the problem of non-response is generally greater in surveys conducted
by mail than in those conducted by personal interview methods.
Notwith-
standing, the significantly lower cost of mail studies may make their
use desirable.
In the context of this thesis, the reasons for such incompleteness
are of no consequence.
Rather, we will consider the theoretical impli-
cations of non-response and the subsequent impact it has on sample
design.
The primary assmnption made is that by expenditure of suffi-
cient funds, all selected un1 ts can be contacted and enumerated.
This
is not absolutely true in practice but it is a suffiCiently close approx-
·e
imation to the real situation that departures can be ignored as having
insignificant effects on the results.
One simple, if questionable, means of avoiding modifications in the
development of sampling theories brought about by non-response is to
postulate complete homogeneity among both responding and non-responding
population units.
This being the case, one need merely substitute
another sampling unit (chosen according to some a priori probability
system) for any non-responding one.
The net effect is to attain the
predetermined sample size prescribed by the sampling plan.
However, to
ascribe homogeneity to both respondents and non-respondents without
strong evidence is not realistic, in most instances, especially in human
populations.
Such a replacement scheme will usually introduce some bias
in the estimates, except in cases where the measured characteristics
3
among responding and non-responding units in the population are, in fact,
homogeneous.
A more nearly acceptable means of correcting for non-response is to
expend some effort in the direction of the non-respondents themselves.
Thus, by subsampling the non-respondents one or more times (phases) and
improving the interviewing effort, some representation of the latter
group can be managed.
This has the advantage that by appropriate mod-
ifications to the usual estimators of population values, the subsample
results can be used to provide unbiased estimates and, in some cases, to
reduce mean square error.
In most surveys and even for particular questions in multi-purPOse
surveys the value of the measured characteristic of the sampling units
·-e
will be correlated with, if not have a direct effect on the likelihood
of response.
Such a situation makes it imperative that measures be
taken to elicit a higher proportion of responses from the selected
units •
Only in this manner can bias be entirely eliminated from sub-
sequent estimates of population values.
Reduction of bias can sometimes
be accomplished empirically by utilizing accumulated data on both respondents and non-respondents.
for Itadjustment" purPOses.
Thus, historical data are sometimes used
other methods include better interviewers
and training methods, continued recall, improved questionnaires or, any
combinations of these and similar techniques.
When methods for collecting information from human populations are
modified, there is the possibility of introducing further biases into
the responses themselves.
In fact, the non-response segment of a given
4
population may actually change.
cated in this thesis.
tual non-existence.
Such problems as these are merely indi-
The results obtained herein postulate their virMuch work has been done in this area and is to be
found in the literature of statistics, psychology, and sociology.
By
proper attention to the techniques developed in these disciplines, the
effective bias introduced by attempts to increase the response rate
through modification of procedure can be minimized.
As of the present writing all published work on the statistical
aspects of the non-response problem has been confined to single stage
sampling plans.
(1)
It is the primary purpose of this thesis to:
examine the effects of non-response on the most
common estimators in light of some non-response
population models.
(2)
extend present procedures to both stratified and
unstratified two-stage
sam:Pli~.This will
include
the optimum allocation of effort to various sample
stages and phases.
(3)
develop some unbiased estimators of variance for
the two-stage one-phase case.
In the main, it is the work of Hansen and Hurwitz (1946) and
El-Badry (1956) which will be extended.
Where earlier authors have
neglected to find variance estimators, such will be derived where
possible.
It should be stressed that the plans considered in this study are
intended to utilize the information available to the investigator in
5
order to achieve maximum economic efficiency; i.e., to secure estimates
of population values which are as free as possible of non-response bias
and which have maximum precision for the available funds or which minimize cost while attaining a required precision.
1.2 Review of Literature
In conducting sample surveys of human populations in which the
sampling units (households, persons) under study have the choice of not
responding, the risk of bias is ever present.
However, choice on the
part of the individual selected is not the only criterion demanding
consideration of bias.
Individuals who are not at home or who live in
out of the way places also present the possibility of non-response.
·,e
A
number of studies are available showing some of the personal, educational and other traits which have been found correlated with the willingness to respond or ability to contact certain units.
Several of these
investigations will be discussed below followed by a discussion of
studies concerned with the statistical and design aspects of the problem.
Pace (1939) made a study of former students to discover the attitudes of a representative group of young adults 5 to 13 years out of
school.
Comparison of. the respondents and non-respondents showed that
whether the respondent actually graduated and the number of years completed were both important factors influencing the decision to respond
iJIDIlediately.
Stanton (1939) studied 11,169 school teachers, inquiring about
their ownership and use of classroom receiving equipment and other radio
facilities.
He found that a higher proportion of those having such
6
equipment answered the initial inquiry.
Those not having such facili-
ties responded mostly to follow-up questionnaires.
In a survey of listeners to child training broadcasts in Iowa City,
Suchman and McCandless (1940) found that willingness to respond was
highest among regular listeners and from individuals most interested in
the subject of the inquiry.
ingness to respond.
Educational level was also related to will-
The study itself was carried out in three phases:
first, a mail questionnaire; second, a follow-up mail questionnaire;
and, finally, a telephone interview of a random subsanrp1e of the nonrespondents to the earlier inquiries.
Gaudet and Wilson (1940) reported on a study consisting of initial
interviews on 2,800 individuals followed by reinterview of a random sample 01'1,800 of the original sample.
They concluded that the refusal
rate was negatively correlated to the educational level.
A second non-
responding group consisting mainly of industrial workers was not
avail~
able for interview.
Shuttleworth (1940) conducted a study on the employment status of
technology and chemistry graduates in New York City.
The survey con-
sisted of first and second mail questionnaires and personal interviews
of the non-respondents.
On the basis of the results, it was concluded
that unemployment was more prevalent amorig the non-respondents than
among the respondents at the initial phase.
Reporting on a study of over 4,900 compIeted interviews of consumer requirements, Hilgard and Payne (1944), have indicated the magnitude of bias resulting from the respondents successfully contacted at
7
the initial phase.
Subsequent interviews were made on different days
of the week and at various hours of the day.
In this manner, it was
found that housewives with young children and those who worked only in
the home were contacted more easily than those without children or who
worked·outside the home.
Wallace (1947) reported on a study of Time subscribers comparing
the characteristics of mail respondents and personal interview results
on the non-respondents.
It was tound that there were few important
differences between the two groups.
There were, however, a significant-
ly higher proportion of college educated persons among the mail respondents.
Because no statistically significant differences were found
between the groups with respect to marital status, income, residence,
ownership of appliances and other features, he conCluded that the sample
of mail respondents was in fact a random sample of all Time subscribers.
It should be noted that even though a population is homogeneous
with respect to some. characteristics (i.e., respondent characteristics non-respondent characteristics) others may differ substantially.
It is
in the latter instance that elimination of bias becomes essential.
Up to this point, the literature reviewed was concerned primarily
with pointing up the existence or nonexistence of bias.
The articles
reviewed below are mostly concerned with the theoretical aspects of
bias and its control.
Generally speaking, one of three devices has
been used to adjust results from mail and personal interview surveys.
These ar.e:
8
(1)
enumeration by initial contact and then recall of a subsample
of the non-respondents
(2)
obtaining auxiliary information on both respondents and nonrespondents that is correlated With items to be estimated
(3)
postulate a mathematical relationship (model) of the population and use information from successive interviews to estimate the unknowns of the model and use this to describe nonrespondents.
In
1948, Ferber considered the use of tests for random order as a
basis for measuring the correlation of sample unit characteristics to
Willingness to respond.
The responding questionnaires in a mail survey
were ranked according to the value of the measured characteristics.
Another ranking was also made on the chronological order of receipt.
By calculating the Spearman rank correlation coefficient (r) one could
test for correlation.
A significant value of E implies that premature
termination of the study or that a complete disregard of non-respondents
could introduce bias.
Naturally, such a test is made on the null hy-
pothesis of independence of the value of the measured item and the
promptness of response.
Such a null hypothesis would not be applicable
if one of the items measured were geographical location.
However, the
procedure seems appropriate in many instances.
Commenting on the procedure suggested by Ferber
Zeisel
(1948), Ford and
(1949) questioned its usefulness as a predictor of characteris-
tics of non-respondents.
They showed that projection of trend differ-
ences among early and late responses were unreliable in particular cases.
9
Their data were taken from a survey of employee attitudes with respect
to previous employment.
They found that willingness to respond. as
measured by earlier returns was correlated with the ratings made by the
present supervisors.
Projection of results based on an estimate of the
trend was found to include a disproportionately large number of unsatisfactory employees when the results were compared to personnel records
and the responses of follow-up respondents.
Probably the first published work on the statistical aspects of
non- response in sample surveys was by Hansen and Hurwitz (1946).
They
studied sample design, including estimation, appropriate to a single
stage mail survey.
Follow-up, Le., the first phase, is made by all
possible means to insure response.
··e
This is usually accomplished by
personal interview and intensive recall.
The follow-up interviews are
made on a randomly selected subsample of the non-respondents.
The
authors proposed the following unbiased estimator of the population
total for this plan:
, N (-'
-")
x=-mx+sx
n
where N
=
total number of addresses in the population
n
= number of questionnaires mailed
m
= number of mail replies
s
= number in the sample of n who do not respond to mail
questionnaires
X' =
sample mean of the
x" =
sample mean.of the.!: respondents to the personal interviews.
~
respondents to the mail questionnaires
10
Also derived in the above paper was the variance of' the proposed
estimator.
authors.
However, an estimate of' this variance was not given by the
Application of' the method to several survey conditions was
presented.
Formulae f'or optimum allocation of' ef'f'ort between mail and
f'ield interviews were derived.
These values are optimum in the sense
that they provide minimum expected cost f'or specif'ied precision (variance) •
Comparisons were made with plans calling f'or complete enumera-
tion of' all non-respondents.
In this thesis, the greater ef'f'iciency of'
subsampling of' non-respondents will be proven in generaL
The authors stressed that their sample design was most ef'f'icient
when the response rate to the mail questionnaireishighandthedif.f.erence between the cost of' obtaining this response and the personal inter-
··e
view costs is large.
This last comment should be borne in mind when
reading Durbin's 1953 paper.
In 1949, -Hendricks suggested f'inding general mathematical laws
which would enable one to project trends based on results obtained f'rom
at least three successive mailings.
He postulated the f'ollowing rela-
tionship f'or responses to successive mailings.
where X
x-
2
Ln
~ : N (0,
=
mailing number
(l,2,.~.)
=
average number of' mailings required f'or response.
x
0-
)
Using the data made available by a North Carolina f'arm study, the
variance of' this postulated distribution was estimated.
Further, he
approximated the relationship between the items of' interest by a
11
quadratic function in X, the number of mailings.
Using this equation,
values of the items of interest as well as the number responding to
future :mailings may be predicted and used to adjust the results.
Another method proposed by Politz and Simmons (1949) was aimed at
reducing non-response bias and at the same time eliminating expensive
recalls.
The technique is applicable to human populations and then only
where the method of contact is instantaneous, as in personal interviews
or telephone interviews.
The technique consists of contacting each
individual in the sample at a "random" time in the interview period.
From each person successfully contacted, information is obtained as to
whether or not that person was at home at certain other "random" times
during the interviewing period.
This information was
t~en
used to esti-
mate the proportion of time that person was at home during the interviewing period.
For practical purposes, the popuiation was divided into
six categories; those at home for 1/6, 2/6, 3/6, etc., of the interview
period.
The average response in each of these categories was then
weighted by the reciprocal of the associated proportion.
This provided
quasi-unbiased estimates, since the category 0/6 was not sampled at all.
The practical shortcomings of this plan include the selection of "random"
times for interviewing and the reliance on interviewer memory for estimating his "at home" record.
In 1955, Durbin compared a method utilizing continued recall to
the technique suggested by Politz and Simmons.
He also showed that the
former method, which is essentially the Hansen and Hurwitz (1946) approach, gave little increase in efficiency when compared to a 100 per cent
12
recall scheme unless the relative cost of obtaining non-respondents was
at least an order of magnitude greater than the cost of obtaining initial responses.
(This conclusion seems to be in keeping with the appli-
cation as made to mail surveys with personal recall by Hansen and
However, the suggestion made by Cochran (1953) that the
Hurwitz .)
Hansen and Hurwitz method be used even when initial contact is by personal interview is not encouraged by Durbin I s results.
Durbin found that the relative efficiency of the Politz-Simmons
approach to a recall method on all non-respondents is approximately
equal to
1
+~
2
, where
e
is the correlation between the response
characteristic and the probability of successful contact and response
during the interviewing period.
This result is true only for the case
where the relation between the value of the response variable and the
probability of successful contact is linear.
The distribution of the
probabilities was assumed, by Durbin, to be triangular.
The principle
objection to the conclusions is that the author assumed that interviewing costs were equal at all phases.
This, of course, is not true in
most practical cases, particularly, in mail-personal recall methods.
Birnbaum and Sirken (1950) considered the problem of bias due to
non-response when sampling from large binomial type populations, i.e.,
responses of the type "yes or no".
approximations to the binomial.
The results are based on normal
The authors present tables giving
limits on the bias and expected cost of the survey.
Variance of the
estimators are tabulated as functions of sample size and the number of
call-backs made on the non-availables.
Their method of sampling differs
13
from the Hansen and Hurwitz type in that each 'lUlit is contacted up to
k times.
If a response is obtained on the ith call (i ~ k), the 'lUlit
is considered to be a respondent.
Tables of minimum sample size re-
quired for a given precision are presented for values of
number. of call-backs, from one to five.
~,
the maximum
The techniques and tables pre-
sented do not apply to responses which are not of the "yes-no" type.
In the development of the theory of sampling with non-response,
Deming (1953) states
"the bias of non-response is probably so serious in many if not
most surveys that the specification of the number of recalls,
and the adjustment of the original size of sample to permit
either the use of the Politz plan or the requisite number of
recalls to balance the bias of non-response against the variance, and to stay within the allowable budget, are an essential
part of sample design where the aim is to produce as much
information as possible per 'lUlit cost."
Deming's sampling plan consists of making successive recalls on a
fixed proportion of non-respondents to the previous call.
This is in
lieu of procuring information on the proportion of time home as in the
Politz plan.
However, in assessing the relative bias, mean square
error and cost for varying number of recalls, the author constructs a
hypothetical popul~tion in the same manner as did Politz (1949).
He
concludes, on the'basis of several contrived examples, that most recall
plans should allow for at least 4 or 5 recalls.
Only when the preci-
sion requirements were high, was the Politz plan as efficient as the
Deming recall plan.
Yates (1949) suggested that,
"the simplest way of dealing with non-response is to regard
the non-respondents as similar to the remainder of the sample,
14
i.e., to treat the sample as if it were a sample on a smaller
number of units."
In this instance, one may take the approach due to Durbin (1957).
The theory outlined by Durbin (1957) is an extension to stratified
and multistage sampling of standard ratio methods of estimation.
The
problem of non-response is essentially ignored by definition of a domain of study consisting of respondents only.
Its application to sam-
pIing in the presence of non-response is therefore only valid under the
conditions implied by Yates above.
Another plan for utilizing population information to achieve maximum efficiency was put forth by EI-Badry (1956).
The method of sampling
involves using waves (phases) of mail questionnaires each to a subsample
of the non-respondents to the previous mailing.
,.A
·w
The final phase of
interviewing is carried out by personal interview on a subsample of the
non-respondents remaining after ,all the mailings have been made.
Clearly, this is an extension of the Hansen-Hurwitz (1946) plan which
involved using only one mail attempt followed by personal interview.
El-Badry considered the following unbiased estimator 'of the population total:
~l + .••
k k x31
2 3
+
n
ml
m
II k.
i=2 J.
n
m2
m
II k.
i=2 J.
15
Where
~
= number in the original mailing
Nl
nil
=
= number of sampling units responding to the i th mailing
nm2
=
ki +1
= sampling rate to be used at the i th mailing
m
=
number of sampling un1 ts in the population
number of sampling un1 ts not responding to mth mailing
number of mail attempts.
The final phase is carried out by interviewing a fraction,
n
m2
non-respondents at the mth mailing.
~,
of the
El-Badry derives optimum
value~
for the sampling fractions and the original sample size based on a
generalized Hansen-Hurwitz cost function.
A study of the results of a stratified one-stage sample survey on
morbidity carried out in Nashville, Tennessee, was presented by
··e
Vaivanijkul (1961).
The study consisted of comparing the responses of
2,564 respondents to the response of a subsample of 85 non-respondents.
The latter were made to respond by an intensive recall procedure.
It
was found that of 34 items studied there were significant differences
between the respondents and non-respondents in only six.
It was con-
cluded that the bias due to non-response would have been negligible had
the non-respondents not been considered at all.
Some important conclusions which can be drawn from the above
studies and from the existing literature on the problem of non-response
are as folloW's.
(1)
There is a certain amount of risk in putting confidence in
survey results that neglect the possible effects of nonresponse.
16
(2)
It is inefficient and even futile to strive for complete
coverage of, or merely resorting to larger samples as a
means of avoiding non-response error.
(3)
It is important to utilize past experience about a survey
or similar surveys in planning.
(4)
Present theory of non-response needs to be developed and extended to more commonly used sample designs.
1.3 Notation
When possible standard notation has been used throughout this
thesis.
For reading ease and to maintain continuity many of the sym-
bols used are defined where introduced.
the following tables are presented.
However, for ready reference
17
Table 1.
Notation relevant to unstratified single phase sampling
Spibol
Population
Sample
Definition
Number of first stage units (FSUs)
M
m
Within the i th FSU:
Number of times PSU i appears in sample
Selection probability
Inclusion probability
Number of second stage units (SSUs)
Number of responding SSUs
Number of non- responding SSUs
Number of first phase SSUs
Resample rate at first phase
Proportion of respondents
R
Proportion of non-respondents
i\
Measure of responding SSU
i
.J.
Y
Ylij
Y
Y2ij
Mean of all SSUs
Yi
Yi
Mean of all responding SSUs
Yli
Yli
Mean of all non- responding SSUs
Y2i
Y2i
Measure of non-responding SSU
lij
.J.
2ij
.!
Mean of first phase SSUs
Y2i
Total of all SSUs
Yi
Total of all responding SSUs
Yli
Total of all non-responding SSUs
Y2i
Total of first phase SSUs
Y2i
18
Table 1 (continued)
Definition
Symbol
Population Sample
Mean square of allSSUs
2
(Ni-l)ai
2
(ni-l)si
Mean square of all responding SSUs
2
(Nli-l)ali
2
(~i-l)sli
Mean square of all non-responding SSUs
(N2i-l)a~i (~i-l)S~i
2
(h2i -1)s2i
Mean square of first phase SSUs
For the case of stratified single stage sampling, all the notation
of Table 1 is made applicable by adding to each symbol the subscript
as, for example, N
2ki
··e
!
representing the total number of non-responding
SSUs in FSU i of stratum k.
Table 2.
Notation relevant to unstratified multi-phase sampling
Symbol
Population Sample
Definition
Within the i th FSU
~
Number of SSUs selected at second stage
Number of responding SSUs at phase
~
N(U+l)l
(u=1,2, ••• ,p-l)
Number of non-responding SSUs at phase
Sampling rate at phase
~
Sampling rate at phase
~
Mean of all responding SSUs at phase
~
.
i
N(U+l)2
n(u+l)2
i
[f(U+ll-
wi
~
-
Y(u+i)l
1
19
Table 2 (continued)
Symbol
Population Ssmple
Definition
Mean of all non-responding SSUs at phase
.&
~
y(u+l)2
Mean of uth phase SSUs
y(u+l)2
Fraction of responding SSUs at phase
i
~
F(u+l)l
Fraction of non-responding SSUs after phase
Mean square of responding SSUs at phase
~
~
i
F(u+l)2
2
CT(u+l)l
Mean square of non-responding SSUs after
phase
.·e
2
CT(u+l)2
~
For the case of stratified multistage sampling all the notation of
Table 2 is made applicable by adding to each symbol the subscript
.!
as,
i
for example, N(u+l)2k representing the total number of non-responding
th
th
u
phase SSUs in the i
FSU of stratum .!.
In defining inclusion probabilities, consider a particular sample
of n units drawn out of N.
If the n units are drawn without replacement
there are '(:) different samples.
Further, if order of draw is consid-
ered each sample may be drawn in n! ways.
Associated with each of the
(:) samples is the unconditional probability of its being drawn.
particular for the sample s, this probability is denoted by P.
s
In
It is
computed according to the elementary laws of probability from the selection probabilities defined at each draw.
If sampling is performed
20
with replacement, the definition of Ps remains the same except that.!
now ranges over ( N+n-l)
N-l
different possible samples.
The definition of inclusion probabilities is as follows:
where the first summation is over probabilities P associated with
s
sample ! which contains FSU
.!
and the second summation over probabil-
i ties Ps associated with sample ! containing FSUs
.!
and
J..
21
2.0 THE THEORY OF SAMPLING WITH NON-RESPONSE
AT THE SECOND STAGE
In sampling from a fim te population, the previously drawn units
mayor may not be replaced after each draw.
stage or phase in the sampling procedure.
This may apply at any
In this chapter we shall
consider sampling plans in which the following methods of selection
apply at each step in sampling.
1.
2.
First stage units are selected with unequal probabilities
( a)
without replacement
(b)
with replacement.
Second stage units are selected with equal probability and
without replacement.
··e
3.
First phase units are selected from among non-responding second
stage units with equal probability and without replacement.
For the development of the theory, it is assumed that the frame
from which sampling with replacement at the first stage is made is such
that N
i
~
ron for all!.
i
In practice, this premise should
p~esent
no
problem, since the likelihood of repeated selection of a particular
first stage unit, say up to
~
times, is extremely small for reasonable
size M, the total number of first stage units in the population.
A second point worthy of mention is that the development is restricted to only one type of estimator of the population total.
is done for two reasons.
Firstly, it is an extended version of the
estimator already considered by Hansen-Hurwitz
(1956).
This
(1946)
and El-Badry
Secondly, the results may be extended to other estimators
22
either by application of the same methodology developed herein or, by
substitution of appropriate expressions in the results obtained below
for this estimator.
2.1
2.1.1
Sampling with Unequal Probabilities and
Without Replacement at the First Stage
The Sampling Procedure.
Let us assmne the existence of a
population frame consisting of M first stage units.
tion a sample of
~
first stage units is selected.
From this populaThe method of selec-
tion is such that for the i th primary unit the inclusion probability is
Pi •
The i th first stage unit consists of N second stage units.
i
the second stage n
i
:s ni :s Ni )
units (where 1
At
are drawn from among N
i
with equal probability and without replacement.
··e
Initial contact is then made with the selected second stage units.
From the i th first stage unit ~i second stage units (where ~i
respond and the remaining units ~i (where ~i
spond.
= ni
:s ni )
- ~i) do not re-
In practice, the second stage units might represent human
beings, firms, drugstores, farms, etc., some of which respond to an
initial inquiry and some of which do not.
Subsequent to this initial attempt at soliciting a response the
following procedure is carried out.
Select from among the n
non2i
respondents in the i th primary unit a subsample, again with equal probability and without replacement.
This step in the process of selection
may be designated as second-stage first-phase sampling.
units selected at this step is
bol
f
i
~i·
Let
f i ~i =
~i.
The number of
Thus, the sym-
is seen to be the reciprocal of the sampling rate used at the
first phase among the non-respondents.
Each of the selected h units
2i
is now recalled until response is attained. It is at this phase where
extraordinary means are adopted to seek out and solicit responses.
The
cost per schedule at this phase is many times higher than at the initial
contact.
Having obtained responses from (~i + ~i) second stage units we
are not in a position to estimate the population total for the measured
characteristic.
The n
initial respondents measure the characteristics
li
of respondents while the h initial non-respondents estimate the char2i
acteristics of that part of the population which is not amenable to
initial contact.
2.1.2 An Estimator of the Total.
.···e
Consider the following estimator
of the population total for the measurable characteristic 'l:
n,i
~i!l,..i
( r Ylij + h
.'if' Y2ij )
j=l
2i J=l
(2.1)
This estimator provides an unbiased estimate of
M Ni
T = E E Yij
i=l j=l
the population total.
The proof is as follows:
on a given set of
all
(~~)
sets of
~
Taking expectation of (2.1) conditional
primary units and fixed sample sizes
~i
~1' ~i'
units each, produces the following results:
over
24
....
E(Tll m'~i,h2i)
•
where Y2i
(2.2 )
is the average value for all
non-respondents in the
~i
sample.
Now taking expectations over all sets [~i' ~J such that
~i
+ ~i
=
E( ~i) :: ni Ri
ni and making use of the results
and
Finally, taking expectations over all possible sets of m first
stage sampling units:
....
E(T ) ::
.l
~
6
P
6
m Y
i
)
1 Pi s
(~
(
M
~
6)i
p)=~
s
1
Y
i
Pi
.p
i
(2.4)
The probability, Ps ' is the unconditional probability of selecting
a particular set, namely the set~, of ~ out of Mfirst stage units.
P is calculable from the probabilities defined for the selection of
s
the first stage units. The selection probabilities are arbitrary and
the drawing made without replacement.
The notation 1::
indicates
sJ i
summation over all P associated with sets that include the first stage
s
unit 1.
25
A
Variance of the Estimator,
2.1.3
A
~l'
The variance of
~l
is
found to be:
M
YiYj
+ EE P P
ifj
i j
-.2
Pij - r
.
It can be seen that this expression has three discernible parts.
The first is a quadratic function of the first phase sampling units,
Le., the variance among the non-respondents.
The second is the cor-
responding variance among the responding second stage sampling units.
The third part represents a quadratic function of the first stage unit
··e
totals.
It will be shown later that for certain selection schemes the
third part reduces to the familiar expression for the between primary
unit variance for finite populations.
The proof of (2.5) is accomplished most directly by utilization
of a theorem formalized by Madow (1949) which is
(2.6)
where
~
represents a particular sample configuration consisting of
first stage units.
The i th unit contains n
li
out of n
i
~
responding
second stage units and ~i responding second stage (first phase) units
out of
~i
second stage units which did not respond.
called that ni
= ~i
from first stage unit
+
~i
.!.
It will be re-
is the number of second stage units drawn
26
Consider the first term on the right hand side of (2.6):
where the quantity
•
~iY2i
...
has been added and subtracted from the origi-
nal expression for T given in (2.1).
l
Since the sampling in each first stage unit is independent of the
sampling carried out in other FSUs for the given set of ,!!; first stage
units, (2.7) may be written
(2.8)
For the second stage sample of
!,
continued sampling at a rate (f
i
~i
rl
non-respondents in primary unit
and the process of taking expec-
tations relevant to second-stage first-phase sampling leads to replacing
s~i
by
O"~i'
Here the expected value of
its properties as a binomial variate.
~i
is easily determined from
Thus the expression becomes:
1
~[. (1 - -!)
n. -O"~ + R (f -1) 0"2
VeT... Is) = m
E ----
.1
1
~
i
ni
Ni
ni
i
i
J
2i
•
Finally, taking expectations relevant to first stage sampling gives:
(2.10)
which represents the first expression in (2.5).
27
The second term on the right hand side of (2.6) can be evaluated
as follows:
m
2
r, m
~2
Y Ip) - LE(.E Yi/Pi)J
1 i i
1
= E(.E
(2.11)
The notation .E
s:> i,j
indicates summation over all probabilities P
s
which are associated with sets including both first stage units
Adding
(2.10)
and
(2.11)
yields the desired result
1 and 1.
(2.5).
In the particular case where the sampling is in one stage, 1.e.,
dropping the subscripts
1
and setting Pi
= 1 we
find that
(2.5)
to:
This result was first obtained by Hansen and Hurwitz (1946).
reduces
28
2
It should be noted that when f i = 1, 0"2i disappears from the variance formula (2.5). In effect, non-response has been eliminated by
making the recall rate unity.
2.1.4 An Unbiased
Esti~tor
A
of V(Tl ). An estimator for (2.5)
and, incidentally, for (2.12) has not been given so far. One possible
estimator will be developed below.
A
In order to develop an unbiased estimator for V(Tl ), consider the
following sample statistics for the i th first stage sampling unit:
(2.14)
Expanding
~
and
~
and adding them together we obtain:
(2.16)
Taking expectation of this quantity conditional on fixed n , over
2i
all first phase samples of size ~i drawn from the selected ~i
29
non- respondents we have:
where Y2i is the average response of all ~i non-respondents.
Note
that sums taken over non-respondents, e.g., Y2i are not over all
~i
values.
To the right hand side of (2.17) add the quantity
.L (
ni
1
n
~i
i=l
.:
2
+
2iJ
y'
2
(~iY2i) •
~t
y.
y. /)
Jrf 2iJ 2i')
and subtract the identical quantity
After some algebra we optain the expression:
i
(2.18)
where the subscripts 1 and 2 have been dropped on the first term on the
right hand side of (2.18) indicating summation over all (~i + ~i)
observations.
In like manner ~i is the average of all (n1i + ~i) values.
30
Taking expectations over all samples of size
~i
out of all possi-
ble N items (2.18) becomes
2i
(2.19)
where
2
(ni-l) si
=
ni
E (Yij j=l
• 2
Y
i)
•
Finally, taking expectation of (2 .19) over all ~i ~ n
i
gives
(2.20)
Let us now return to (2.15).
~i' ~i;
Taking expectation conditional on
followed by expectation conditional on
expectations over all values of
~i
~i;
and then taking
we obtain:
(2.21)
Now combine (2.15) and (2.16) linearly to obtain an unbiased estimation
of the quantity:
Bi
= (1
ni
2
2
- N ) ~i + Ri(f i - 1) ~2i •
i
This is accomplished by using the coefficients
(2.16) and
(f -l)(N -1)
i
i
Ni(ni-l)
for (2.15), respectively.
estimator of (2.22) is given by the sample statistic:
(2.22)
Ni
- n
i
Ni(ni-l)
for
Thus, an unbiased
31
It should be noted that the terms considered involve second stage
sampling units.
The expectations have been taken over second stage and
first phase sampling units.
To complete the derivation of an estimator
for V(Tl ) consider the quantity:
V(Tl ) = m1E
1
Pi
n~
~PirPiPj)
m (l-Pi ) 2
m
(b i ) + E -:2 Yi + E E P P P
i
1
irj
ij i j
F1
YiY j
,.
which shall be shown to be an unbiased estimator of V(T ).
l
Taking expectations of (2.24) relevant to those steps in sampling
beyond the first stage for the given set of
terms of
~
first stage units, the
(2.24) are
The last expression is true because of the independent sampling carried
out in the first stage units.
Inserting these results in
taking expectations over all sets of
becomes:
~
(2.24) and
FSUs out of M, the expression
32
This may be rewritten:
(2.26)
Substituting Pi for I: Ps and Pij for I:
Ps
s' i s ; ) i, j
final result,
Equation
in (2.26) gives the
(2.27) is identical with (2.5) and hence, (2.24) is an
,.
unbiased estimate of V(T ).
l
2.2 Sampling with Unequal Probabilities and with
Replacement at the First Stage
2.2.1 The Sampling Procedure. The population is as defined above
in 2 .1.
The only difference in the sampling scheme is that each of the
first stage units is replaced prior to selection of the next unit.
Thus, it is possible for the i th FSU to be selected up to ~ times.
Each time a particular FSU is drawn into the sample, a different set of
second stage units is selected.
Hence, it is possible that as many as
th
SSUs maybe required from the i
first stage unit. For this reai
son it is necessary that Ni ~ Inni for all ~.
th
In the following development, the i
FSU has probability Pi of
Inn
being drawn into the sample at any draw.
The symbol ., i represents the
33
number of times the i th primary unit is drawn in a sample consisting of
~
units.
Thus, 7i may take on the values 0, 1, 2, ••• , m.
Further, it
M
is obvious that
2.2.2
i~
7i = m.
An Estimator of the Total.
For the above case of replace-
ment at the first stage of selection, it can be shown that
is an unbiased estimator of the population total T.
...
~2
In (2.28)
given by:
Yli
is
the sample average of the respondents in the i th first stage unit and
~i is t1.le average number of respondents for the 7i times this unit is
drawn into the sample.
and ~i'
Similar definitiona hold for the quantities
It is thus seen that
ability that 7
i
ii.li + ~i = ni
must hold.
Y2i
The prob-
= k is given by:
Also
(2.29)
Proof:
= 7i~iRi •
But
_
7i
(
)
7 n
- EILand
the proof of 2.29 is complete.
i li - f=l .Lif
A
Taking expectation of ~2 conditional on fixed ~i and ~i results
in the expression
E(T I i i ) =
2 ~i' li
N
1:m ~..!
p
1
i
-
~i
[~i
Y +- Y
li
n
~i
2iJ
(2.30)
i
Averaging over all values ~i conditional on ~i and using (2.29)
the following expression is obtained:
Finally, taking the expected value of (2.31) over all integral
values of
~i
:s m
and using the fact that
it is seen that
(2.32 )
A
which shows that T
2
is an unbiased estimator of T.
A
2.2.3
Variance of the Estimator, T2 •
is found to be:
where
A
The sampling variance of T2
35
Direct application of each part of Madow's theorem to the estimator as written below leads to the simplest proof.
Write T as:
2
.&
where the quantity Y2i represents the unknown sample mean based on
"i~i non- respondents. _ _
_
~
.&
We may replace '(~iYli n+ ~iY2i) by Y
, the true sample mean.
i
i
Thus, for the first part of Madow's theorem the expression is given by
since there is zero covariance between ~i and
(Y2i
- ~2i).
The condi-
tion indicated by ! on the left hand side is on a given set of "i.
Taking expectation on first phase sampling the equation becomes:
M(N.)2 2 [
+
E p~ "i V(Yi ) E
~i
2
(fi-I)
n·
"i
The expected value of this expression taken over
~i ~
1
i
D-
~i
i
on "i' gives
and over "i
A
~
.
E V( Tis)
2
m it reduces to
1
=-m
M1
E -
1 Pi
~
[( n i )
1--.
ni
Ni
-
This completes the first term of Madow's theorem.
2]
0"2i •
ni , conditional
The second term in the theorem yields the following expression:
Now it is well known that for multinomial variables:'
= m Pi
(a)
veri)
(b)
cov(ri,rj)
(l-Pi )
=-
m PiP j •
Thus
.--
~-.
which is the same as
1 2
m by
=-0:
(2·35)
•
Adding the formulae (2.34) and (2.35) produces the desired expression (2.33).
2.2.4 An Unbiased Estimator of V(T" ).
2
biased estimator of the sampling variance of
square between first stage units defined by:
" 2
(m-l) £- = ~
i _ T2 )
oy 1 i Pi
m
r (y
where
In order to derive an un-
"
T.2'
consider the mean
37
We may rewrite (2.36) in the f'orm
2
i)
"
M (Y
2
(m-l) s.. = 1:
oy
1 Pi
.. 2
- mT
i2
•
Taking expectations relevant to the second stage and second stage
f'irst phase sampling this expression becomes:
(m-l.)
E(~y)
= E
[~ 7 ~ I 7~J
i,
i
Using the relation:
=
E(x)
2
+ V(x)
and noting that
2
E
(y~ li'''i)
=
Pi
(2.37) becomes
(m-l-) E(s..2y)
o
=
E~l
p"2i . [(.--.2
L~ _ 1:i
i
- N ...,2) +
iVi
_2(...1- ...,2 + RNi. "n vi i
i
l
(f'i- )...,2 )~l_ mE(T.... )2
ni"i v2i~J
.2·
Consider the f'ollowing term which is included in the square brackets
on the right hand side of' (2.38):
•
It can be written in the alternate f'orm
38
where the symbol m' indicates surnxnation taken over the different units
appearing in the sample.
But the above ,expression can be written
(2.39)
where the sum is taken over all first stage units drawn into the sample
and where
.,;.
can only take on values in the range 1 through
m'
E.,;.
the restriction that
~
With
= m.
1
Substituting (2.39) into (2.38), noting that
and taking expectations relevant to first stage sampling, the folloWing
result is obtained after some algebra.
M
21
M 1
~
(2.40)
+ E NiCJ"i - E
B
1
. m-l 1 Pi n i i
It is observed that the terms on the right hand' side of
(2.40)
have the
folloWing relations to the expressions shown below:
2
0:
by
=
~ Pi~ rI-
1
(2.41)
39
(2.43 )
(2.44)
In (2.42) and (2.43), the statistic
s
where
~, ~
tively.
2
s~ is defined by
=
i
and ~ are given by (2.13), (2.14) and (2.15), respec-
After substitution of (2.41), (2.42), (2.43) and (2.44) into
(2.40) and after rearrangement of terms, it is found that,
where the operation of expectation is over all relevant steps as above.
Substituting (2.41) in the right hand side of (2.33) defines the left
hand side of (2.45).
A
mV( T
)
2
t
= E s..2oy
Thus (2.45) can now be written
1
--
m-1
and using (2.43) and (2.44) this may be rewritten, after some algebra,
as
40
,..
V(T2 )
m 6
1
~2 +1- I: -~-i (1 - ::T"")
m
y m-l 1 2
r;J.
Pini
1
=-E~
m
-Hi si~ •
1 Pi
(2.46)
- I:
Therefore,
,.. ,..
1 2
1
V(T2 ) = iii ~y +
m(m-l)
b
m
..u
(1 71)
2
i
I:
I
1
Pini
m Nisi2
1
-I: -,m
Pi
l
,..
is an unbiased estimator of V(~2)'
This form has a structure similar to the variance estimator given
by Sukhatme (1954), page 388, for the case of complete response.
,..
,..
2.3 A Comparison of V(~l) and V(~2) where the FSUs
Are Selected with Equal Probability for Each Case
One case which may have much utility in practice is where the first
stage selection probabilities are equal.
In such instances, selection
schemes featuring replacement of sample units prior to each draw are
generally thought to possess a larger variance than non-replacement
,..
schemes.
This matter will now be considered for the estimators
~l
and
,..
T given earlier in the chapter. Their variances for arbitrary selec2
tion probabilities are given by (2.5) and (2.33), respectively. For
the special case of equal selection probability, that is, in effect,
where the inclusion probability P. is equal to m/M and the selection
~
probability Pi is equal to l/M the formulae become:
V(T1 ) =
~ ~ ~ [(1 - :~) cr~ + Ri(fi-1) cr~iJ + ~ (1 - ~) cri
41
and
,..
MM
V(T )
2
= -m E
l
respectively.
In order to compare these quantities consider the difference
,..
,..
f:::,.
= V(Tl ) - V(T2 ) which is equal to:
""
D
~
(1 -
~) lT~ + <,";.1)
f
NilTi -
~
(1 -
~) lT~
which reduces to the equation
m
M
2
5 = ( - ) f:::,. = I: N 0"
m-l
1 i i
-
2
M cry
(2.48)
Now 5 is a quadratic form in Y , the values of the- character
iJ
being measured, for all individuals in the population, both respondents
and non-respondents.
If the matrix of such a form is examined according
to well known determinantal laws, it can be shown that 5 is an indefinite quadratic form.
As such, it may, depending on the particular
values of the variables, i.e., Y , N and M, be either positive,
i
iJ
negative or zero. Thus no general statement as to the relative magnitudes of these variances can be made.
3.0
OPrIMUM SAMPLE ALLOCATION FOR NON-RESPONSE
The first aim of this chapter will be to develop cost functions
appropriate to each of the sample designs considered in Chapter I.
These cost functions will then be used in combination with the corresponding variance formulae to achieve optimum allocation of the available resources.
The solutions will be given to satisfy the following
criteria:
(1)
expected cost to be minimized subject to a preassigned precision (variance)
(2)
variance to be minimized subject to a preassigned expected
cost.
Next an investigation will be made into the comparative efficiencies of:
(1)
optimum allocation of resources with complete enumeration at
the first phase and,
(2)
optimum allocation of resources with enumeration of a subsample of rion-respondents at the first phase.
In addition, a set of graphs are presented which simplify the calculation of optimum allocation values.
Finally, the above results will be extended to stratified sample
designs.
3.1 A Cost Function
A general cost function may be developed for two-stage one-phase
surveys with non-response by considering the variable costs introduced
into the sampling program.
For the sample designs of the previous
chapter, consider the following costs: '
(1)
the average cost associated with first stage un1 ts •
This is
necessary if the follow-up interview involves personal contact.
(2)
the average cost of sending out the first questionnaire or
otherwise making the initial contact.
This averaae cost will
be allowed to vary from FSU to FSU but, in general it will
usually be constant for mail questionnaires.
C~)
the average cost of processing respondents from the initial
contact.
( 4)
".
~'
A
This will be different for each first stage un1 t.
the average cost of obtaining and processing responses by
personal interview or by other follow-up techniques.
These average costs per schedule for the i th FSU are denoted by
co' c , c
and c , respectively. Using these definitions and those
1i
2i
3i
outlined in the first chapter we may define the following cost function:
m
m
m
C = m Co + ~ ni c1i + ~ ~ic2i + ~ ~ic3i/fi •
The first term on the right hand side of (3.1) is the total cost
associated with the various first stage units selected.
The second
term is the total cost of the initial mailing (or contact).
The third
term is the cost of processing the initial responses and the last term
represents the total cost of soliciting and processing responses from
a subsample of the non-respondents.
As (3.1) stands, C is a random variable since the last two right
hand terms vary from survey to survey.
To avoid this
complic~tion,
the
44
expected cost associated with the sampling plan will be considered as
the criterion.
If R is the average proportion of responding SSUs in the i th FSU
i
and, if R = 1 - R is the average proportion of non-responding SSUs,
i
i
then the expected cost for the case of selection without replacement of
the FSUs is found to be:
In (3.2), Pi is an inclusion probability.
For replacement of each FSU with general selection probabilities
Pi' the expected cost equation is found to be:
C2 = m Co
M
+
f m Pini (c1i + Ri c2i +
i\c3i
f
i
)
The problem is to determine fi' ni (i = 1,2, ••• ,M) and m such that
the expected cost function (e.g., 3.2) is a minimum subject to a fixed
variance or conversely.
For reasons given in the introduction to Section 3.4, the sampling
plan involving the selection of FSUs with replacement will be considered
first.
3.2
Sampling with Replacement at the First Stage
3.2.1 Fixed Variance and Minimum Cost.
Consider the problem of
determining the optimum allocation for the sample design described in
Section 2.2.
The estimator of the population total, T, is given by:
For this estimator the appropriate sampling variance was found to
be
To find values of ni , f i , and
~
such that the expected cost func-
,.
tion is a minimum subject to a fixed variance, V(T )
2
of Lagrange can be used.
=
V ' the method
20
However, to use the Lagrange method it is
necessary to evaluate the definiteness of a matrix of order (2M+2) in
,
order to ascertain the nature of the stationary solution, i. e., maximum,
minimum or neither.
Because of its generality as well as its ease of
application, an alternative procedure which makes use of the Cauchy Inequality will be used in all derivations of this nature.
This approach is described in detail by Stuart (1954).
The Cauchy
Inequality may be written:
(:5.6)
where a
i
and b
i
Furthermore, equality in (3.6) being
are real numbers.
necessarily a minimum of the left hand side), is attained, if and only
if,
a
b
i
=
K
=
constant for all i.
i
To make use of this inequality, identify the term (~ a~) with the
,.
2
J.
variance, V( T ), and the term -( 1:: bi) with the expected cost function
2
i
c2 •
46
~(T2 ~
If the product
[C2]
is minimized by insuring certain relation-
ships among the allocation constants, then exact solutions may be found
...
by requiring, in this instance, that V( ~2)
= V20.
This requirement is
met by suitable definition of K, the constant of proportionality given
above.
,..
To apply this technique rearrange the formula for V( ~2) and write
the expression:
~
~
ME
1 - · - [2
- 2J +M
- r2
+
cr -Rcr
E f-i - · - R c
1 mni
Pii
i 2i
1 mni
Pi i 2i •
".
'\.'a
The corresponding expected cost function in this sampling situation is
given by
(3.3)
and it may be rewritten as follows:
Now define the following relationships between the terms in equation
(3.7)
a
a
and
(3.8)
2
2
[0:
- ~
m by i=l t~:1) Nicr~J
0
J.
J.
=!
2
1
=i
mn
i
2
a(i+M)
=
(3.6):
and the a. and b. of
[~Pi (cr~
f
i
mni
- R
cr~i)J
[~R1cr~iJ
.
Pi
,
J.
i = 1,2, ••• ,M
(3.9)
47
and
b
2
=m c
00'
, i
= 1,2, ••• ,M.
(3.10)
Applying the conditions required for a minimum produces the following equations:
a
-,'..e
0
.!
b0
m
a
b
i
i
=
tby2 1=1M(Pi)Pi 12f
a:
-
I:
c
1
mn
i
NO"
-
i
= K
0
r
2]'
~;J
O"i - Ri 0"2i
+ c2i Ri
c
li
=K
~
i
= 1,2, ••• ,M.
,These equations are readily solved for the variables ~, mn , and fi/mn
i
i
in terms of K.
The solutions may then be substituted into the right
hand side of (3.7).
The constant K is thus given by the relation:
48
Eliminating
mIl
i
K from equations C~ .12) and C~ .13) yields the
solutions:
i = 1,2, ••. ,M.
Eliminating mK from equation (3.11) and (3.12) yields the solutions:
,
Using the value of K given by (3.14) in equation (3.11) produces
the result:
- ·e
These results are identical to those found by use of the Lagrange
method.
But in addition, it is assured that these give a minimum ex-
pected cost subject to the fixed variance, V .
20
The minimum expected cost is now calculable as a function of the
fixed variance V •
20
Substituting K, from C~.14), and the equations
(3 .15) through (3.17) into the formula for C gives for the minimum
2
(3.18)
It should be noted that the solutions given by equations
(3.16), (3.17),
and
(3.18)
(3.15),
can only be evaluated by use of a priori
2
2
information regarding the population, Le., O"i' 0"2i' R , etc.
i
3.2.2
Fixed Cost and Minimum Variance.
the cost of a survey
~
There are instances where
be specified in advance.
Under these condi-
tions the sample design, and in particular the over-all sample size
can be so chosen as to insure a minimum variance.
This section gives
these solutions for the sample design outlined in Section 2.2.
The solutions fO\lD.d by applying the Cauchy Inequality are:
i = 1,2, ••• ,M,
(3.19 )
i = 1,2, ••• ,M,
(3.20)
50
where Co is the predetermined expected cost.
It appears that only the solution for
~,
the number of first stage
units to be selected, is affected by the constraint.
The variance, under the constraint that the expected cost be equal
to Co' is found to be:
-e
M[2
~ N
i
(O'i -
3.3
Connnents on the Optimmn Solutions
The most important consideration required to make
t~e
entire pro-
cedure outlined above applicable is that:
(3.23)
In-the general case, there is no assurance that this will hold since
the left hand side of (3.23) is an indefinite quadratic form.
it must be evaluated in eacb particular application.
Thus,
51
Consider the solutions for f i which may be put in the form:
~ =
i
c3i (O"~/O"~i - i\)
(cli + c2i Ri J
,
i
= 1,2, ••• ,M.
It is interesting to note that these solutions do not depend on the
size of the first stage units, on the number of first stage units
selected, nor on cost or variance criteria.
However, since f
i
is the reciprocal of the sampling rate for FSU
!,
the solutions must be such that
This implies that,
··e
must
hold for aJ.l -i.
,
However, this condition does not seem too restric-
tive, since usually the right hand side of (3.25) is less than one from
c » c
and c > c
in practice and the ratio of
li
2i
3i
3i
the over-all within FSU variance to the within FSU non-response varithe fact that
ance is usually one or greater.
Only in those instances where the
tendency for non-response occurs at both extremes of the response range
will the variance of non-respondents tend to be greater than the overall variance.
This might occur if the question involved, say, income
and individuals earning either very high or very low salaries were reluctant to respond.
Solutions are optimum, in the uncomplicated case, if the constant,
K, determined by a restriction on either the cost or variance, provides
52
feasible solutions in the sense that the following inequalities are
satisfied:
m < M,
ni
~
i
N
1 < f
i
J,
i
= 1,2, ••• ,M.
(3.28)
In general, nothing can be done to modify the solutions f
in the event that (3.28) is not satisfied.
and n
i
i
Only by emending the frame
which, in effect, changes the population values
2
, Yi'
i
CT
2
2i , etc., can
CT
(3.28) be satisfied, if-at all.
Some situations may also arise where the constant K gives a solu-
··e
tion for
~
which is not possible, i. e., m > Mfor sampling FSUs without
replacement.
In this case, acceptable results may still be achieved.
The solution provided by (3.35) below requires that:
mK =
r
(M i : ~ Nicrif =
cr
o
If this requires an m > M, set m = M and redefine K
mKJ. = Cl •
Thus (3.29) is satisfied.
= K1
such that
This has the effect of nullifying
the restriction used to determine the original value, K.
The effect of
such a change on the expected cost may be evaluated from (3.33) and
using the relation:
53
where K =
K1
and C = computed expected cost, to determine the variance.
A judgement as to the acceptability of this precision and or cost can
then be made with the assurance that for this cost, precision is a maximum and, for this precision, cost is a minimum.
In the numerical example given in Section (3.6) the situation
discussed above does, in fact, arise.
The suggested procedure is
followed and an adequate sampling program derived.
3.4 Sampling without Replacement at the First Stage with
Equal First Stage Inclusion Probabilities
There is difficulty in applying the methods of Section 3.2 to the
general case of sampling without replacement at the first stage.
This
is due to the fact that, in general, the inclusion probabilities, Pi'
are functions of m.
Since
~
itself is to be determined in the procedure,
explicit optimum allocation solutions cannot be obtained unless the functional relationship between Pi and .!!!: is known.
The optimum solutions
for f i (i = 1,2, ••• ,M) remain unchanged since the inclusion probabilities do not enter into the solutions.
However, the n and m remain
i
undetermined in the general case.
Because it is commonly used in practice, the special case of equal
probability selection of the first stage units with the estimator (2.1)
is considered in this section.
Assuming that the inclusion probabilities are equal implies Pi = ~
Substituting this relation into the appropriate formulae gives:
[n,
M M
Ni r y
i
+ -roy:
~i %i
]
=
1::
3
n
j
=1
lij
~i
k=l
2ik
'
m i=l
i
,..
T
where
2
cry
M
=
E
ii - (~Yi)2
M(M-l) .
These formulae represent the estimator, its variance and the ·expected cost, respectively.
Application of the Cauchy Inequality produces the following results:
i
= 1,2, ••• ,M (3.34)
i
=.1,2, ••• ,M (3.35)
m=
for the case where V(T" ) is fixed at V and the expected cost
10
l
minimized. The expected cost in this situation is found to be:
55
As in the case of sampling with replacement the equations (3.34)
and (3.35) also hold for the case of fixed expected cost equal to C
o
where the variance is to be minimized.
~
In this case the equation for
is:
and the variance for cost fixed at Co is found to be:
-e
3.5 Relative Efficiency
In order to eliminate bias the sample schemes thus far discussed
have employed the technique of recall on some subsample of initial nonrespondents.
Another, and possibly more connnon, technique used in
personal interview surveys is to attempt to enumerate all initial nonrespondents.
The relative efficiency of these proposed techniques can
be assessed for any estimator.
formula (3.31).
Let us consider the estimator given by
Suppose a sample survey is conducted as follows: m FSUs are
.
th
selected from M without replacement, the i
FSU being drawn with inm
th
elusion probability M. From the i
selected FSU, n SSUs are chosen
i
from among Ni without replacement and with equal probability. To each
of the selected second stage units questionnaires are mailed and after
a fixed time period all non-responding units are personally interviewed.
The recall rate, l/f
i
=L
The expected cost function associated with
this plan is assumed to be:
It should be noted that f
i
:: 1, substituted into equation (3.33), gives
this equation.
"e
The variance under this plan is given by:
1 _ ni )
(
N
i
0-2
i
+
M2m
(1 _
~)
M
0".2
Y
c
' the optimum
Assuming that the expected cost is set equal to
30
th
number of SSUs to be selected from the i
first stage unit is given
by:
·2 =
n
i
and the optimum number of first stage units to choose is given by:
m=
57
Substitution of these results into the formula for the variance
<:3.41 )
gives the following variance for this plan:
Now an alternate plan utilizing subsampling of non-response with
the subsampling rate in FSU
1.
given by l/f
found in Section (3.2.2) for fixed cost.
formulae are given by
(3.34), (3.35)
and
i
would give the results
The appropriate allocation
(3.38)
where Co is replaced by
C30 •
The variance under this plan is given by (3 ~39) again with C reo
placed by C •
30
This formula is:
Accepting the criterion that the more efficient estimator is that
possessing the smaller variance we find on comparing
that they differ only in the second term in brackets.
(3.44)
and
(3.45)
For the sampling
plan requiring recall in the entire non-responding group, this term is
while the subsampling scheme has the term
58
Thus if it can be shown that A
recall is more efficient
> B then the sampling plan with partial
than the one calling for complete recall.
Consider the following argument:
If the general term in the
aggregate composing A can be shown greater than the corresponding term
Now both sides of this expression are positive and so is the term
"e
(CT~
-
RiCT~i)
as can be seen from the demonstrations (3.25) and (3.26).
2
Thus it is sufficient to show that A
> B2 which, if true, implies
But it is obvious that (3.47) is a perfect square.
Hence, A
> B as was
to be proven.
Similar results can be shown to hold for the estimator discussed
in Section 3.2.1.
3.6 Application of Theory to Data
In order to illustrate the application of the theory developed
above, consider two alternative sampling designs as applied to the
59
numerical data given in Table 3.
It was shown above that a plan call-
ing for subsampling of non-respondents was more efficient than a plan
calling for comPlete coverage of the non-respondents.
The extent of
this increased efficiency will be indicated by the example.
"e
..
60
Table
··e
3.
Population data
2
FSU
N
i
Y
i
NiO"i
NiO"i
1
2
3
4
5
6
7
8
9
10
11
17
14
18
19
18
16
73
21
44
62
42
72
25
39
84
81.919
23·777
65.984
75.428
35.466
85·666
30.035
46.313
84.698
90.283
160.85
58·958
71.270
74.520
53.041
65.189
50.242
75 ·972
45.007
44.928
57.066
0
41.839
66·356
67.638
1,552.445
13
14
15
16
17
18
19
20
21
22
23
24
25·:
16
18
19
14
15
13
"4
15
16
23
56
64
158
394.75
40.38
241.88
299.44
69.88
458.67
75.18
112.89
597.82
627.00
1,293.63
347.60
282.18
241.45
234.45
265.60
140.24
303.78
144.69
134.57
250.50
0
116.71
275.20
198.91
TOTAL
394
1,443
7,147.40
12
12
19
12
13
20
10
18
23
12
48·
101
41
69
70
43
52
44
74
45
54
58
'·4
61
The data in Table 3 are from a population of 25 first stage sampling un1 ts •
The number of second stage sampling un1 ts (N ) in each,
i
the total for the characteristic (Y ), the product NiCT~, and NiCT are
i
i
given in columns 2, 3,
4 and
5, respectively.
In addition, the follow-
ing costs, inclusion probabilities, variances and non-response rates
will be assumed:
Co
= 500
c
Pi
= 1/25
c
R
= R = .5
c
CTi
= CT2i
c
3i
0
=5
li
= .5
2i
=1
= 10
It is assumed that the cost and response rates are identical for all
first stage un1 ts •
This simplifies considerably the computational work
while illustrating the efficiency increase, even in such a special case.
It is also assumed that the variance among all SSUs is the same as the
A.
variance among non-respondents.
Plan I.
The estimator of the total is
Complete recall at first phase.
~l.
For optimum allocation
of sampling effort the following formulae are appropriate;
62
where
Using the given data we find that:
25
E
1
25
"e
Y
i
= 1,443
,
E'Ni~i
= 1,552.44 ,
e
= 14,455.766 ,
~
= 3,802.67 •
1
Therefore:
Now,
m= 26.117
> 25, which is not a feasible solution. Let us
therefore redefine!::: by setting m
= 25.
From the equation:
63
we find that
A.
= 10.7539 •
Using this in the formula for n we find that
i
NiO"i
Checking with the N in Table 3, one finds that all solutions for
i
ni
are feasible (i. e ., Iii > N ) except for the fact that they must be
i
rounded to integral values.
The solutions as found can be used to
check out the new expected cost caused by a change in
~.
= (25)(5) + (.037962)(1,552.44)(6)
.
Co
= 125
+
353.60 = 478.60 •
Thus the cost is not too much altered from
500. We may now com-
" to be expected under, this plan from the equation:
pute the variance of T
l
2
25
V = A. C - t
o
Plan II.
0
i=l
2
NiO"i •
Subsampling at first phase.
For optimum allocation with
the same cost and response rates as for plan I, we must utilize the
following equations:
64
For the data given we find, in addition to the totals found in
Plan I, the value
M
~
= 2.28826
.E Ni~i
= 3,552.386.
J.=l
Therefore:
500
m = 5 + 13.2134' = 27.4523 •
This, too, is not feasible.
that for m
=
25,
~
=
10.7539.
Therefore, recompute
~
as before and find
Subject to this new constraint, we find
that
It should be noted that the number of second stage' uni ts required to
contact imtially in this plan is greater than for Plan I.
Also, all
these solutions are feasible subj ect to rounding to nearest integral
value.
Similarly we may solve for f
as required.
i
to find
Thus all conditions are satisfied simultaneously.
Com-
puting the expected cost of this plan we find
E(C)
= Co = (25)(5)
Co
= 125
+
+ (.06575)(1,552.44)(3.23606)
330.31 = 455.31 •
Again this is less than the
'" for this
500 specified. The variance of T.l
plan is
To compare the two plans we shall use the inverse ratio of the
.'e
products:
(Cost) (Variance)
..
for the two plans.
Thus the degree' of efficiency, in the sense of
this problem, is given by
or, in percentage terms, an
11 per cent increase in efficiency
:is acl:devei
by sub-sampling a"llong the non-respondents.
It should be pointed out that these calculations have not taken
into account the effects of rounding off n
i
and f
i
to usable values.
Assuming a rounding scheme allowing for both upward and downward changes,
the compensation should not affect the results to any significant degree.
66
3.7 Extension to Stratified Case
The theory derived earlier in Section 2.,,2 will be extended to
stratified populations.
Also, the Cauchy Inequality will be applied
to the corresponding optimum allocation theory.
Consider an extension of the results of Section 2.2.
the population now consists of S strata.
first stage units.
Suppose that
th
The k
stratum contains
The sampling plan consists of selecting
~
l\
FSUs
from stratum! with replacement and with selection probability Pki for
the i th first stage unit.
At the second stage one selects ~i second
stage sampling units (stratum!, FSU .!) from among Nki with equal probability and without replacement.
Consider the estimator
"
T
s2
= S"
I: T
k=12k
m Nki
[i\kiYlki + ~kiY2k~
I: - 1
k=l ~ i=l Pki ki
~ki
= I:S -1
which is clearly unbiased, since it is merely the sum of individual
unbiased estimators of strata totals.
The variance of this estimator is constructed similarly and by
extending (2.33) is found to be
~ ~
k=l i=l
J
N (J2 + ~ .!-- [(J2
- ~ (l-Pki) N (J2
ki ki
k=l ~ kby i=l Pki
ki ki
since sampling in each stratum is performed independently.
67
The corresponding expected cost equation is written as
(3.50)
Applying the Cauchy Inequality it is found that the solutions
guaranteeing a minimum of the quantity
(3.51)
must satisfy the set of equations
"e
1
~co
[2O"kby -
~
i=l
(PPkiki) NkiO"ki2~
= >"1 '
k = 1,2,3, ••• ,8,
(3.52 )
1
'2
2 - ~0"2ki
- 2J
O"ki
rkij
I!1tI1ti Pki [ckl.i+ C1<2il \i
1
= >"1
'
i = 1,2, ••. ,~,
(3.53 )
f
ki
I!1tI1ti
[N 2ki
~
P
ki
CT
k = 1,2,3, .•• ,8,
= >"1 '
fc:'k3i ki
where >"1 is the constant determined by the particular restriction imposed.
For the restriction
the constant >"1 is determined by the equation
68
where
(3.56)
Optimum solutions for f
ki
and
~i
are given by:
,
i
= 1,2, ••• ,~,
k
= 1,2, ..• , S
•
(3.58)
Substitution of ~1 into equation'
and
(3.52) and use of identities (3.55)
(3.56) gives:
1
~
=
~
(c)
ko
"2
S
[E
k=l
1.
(ak c kO )2 +
S
S
M,~
E ~
k=l i=l
N
i
Nkib
e
:ki
2
V
+ E ~- Nk,CTk ,
so
k=l i=l
~ ~
kJ ,
k = 1,2, ... ,S •
(3.59)
The cost of attaining this variance is found to be
(3.60)
69
Similar solutions are obtained when the restriction is made on the
expected cost as, for example, CS2 = Cs20 '
(3.59) and (3.60), respectively, become
In this case the formulae
,
~=
(3.61)
The procedure is extended in a similar manner for sampling without
··e
replacement and with equal inclusion probabilities.
These results can
easily be written down using the appropriate formulae from earlier
sections of this chapter.
These equations will not be reproduced here.
3.8 Some Graphic Solutions for Optimum Recall Rates
.
-1
It has been shown that the recall rate f i
in FSU
1: (a second
subscript is implicit in stratified cases) is independent of over-all
cost or variance constraints.
Write the equation for ~ as:
,
2
where
T
O"i
- --- , which is a linear function of the relative variable
i - ~i
costs at the various phases of sampling.
Further, for instances in
70
which the initial contact is via mail and the recall is on a personal
interview basis, usually,
In practice these relative costs ordinarily will lie in the ranges
and
respectively.
r·e
If, in addition, we can specify the approximate initial response
rate (R ) and the ratio of the variances (T ), simple linear graphs
i
i
can be used to provide rapid solutions for f • Having these values and
i
writing the formulae for ni and ~ in terms of f i lead to rapid solutions for the allocation constants.
Sample graphs providing contours of equal f i for
T
= 1 and
Ri = .25, .50 and .75 are shown in Figures 1, 2 and 3, respectively.
Since the appropriate modifications necessary to express n and ~ as
i
functions of f are relatively straight forward, such modified forms
i
will not be included here.
71
.20
.10
o
Figure 1.
Contours of fi for
~
= .25
72
.60
.50
.40
·-e
.20
.10
.00
Figure 2.
Contours of f i for T' == 1 and Ri == .50
73
..50
.40
··e
.20
.10
o
Figure 3.
C1i
c3i
Contours of fi
.25
for" = 1 and Ri = 075
74
4.0
TWO STAGE NON-RESPONSE THEORY EXTENDED
TO MULTIPHASE SAMPLING
In this chapter El-Badry's (1956) multiphase extension of Hansen
and Hurwitz's (1946) theory will be generalized.
This theory assumes
that an individual or unit will respond after a definite number of
contacts.
The sampling plan is as follows for the single stratum case:
(1)
A sample of !!! first stage units is selected from among M
according to some probability system.
(2)
From the i
~
~
i
chosen FSU, ~ SSUs are s~lected from among W
with equal probability and without replacement.
(3)
.-e
Mail questionnaires are sent to those selected individuals.
(4) Successive waves of questionnaires are mailed in attempts to
reach the more reluctant individuals.
A random sample of
the non-respondents to the previous mailing are chosen at
each mail phase.
(5)
Finally, the units of a random sample of the non- respondents
remaining at the p
th
phase are contacted in person and con-
tinued recall made as necessary to secure response.
For this plan the following population frame will be considered.
It is assumed that for any FSU, the proportion of SSUs requiring 1, 2,
••• , (p-l) contacts prior to response is known.
In other words, a
subset of SSUs exists which responds to the first mailing (contact),
a subset which responds to the second mailing, etc.
It should be noted
that any one mailing secures information from only one of these ·sets.
Non-response occurs because contact has been made with SSUs not
75
belonging to the corresponding set.
It will appear subsequently that
the responses to any particular mailing will furnish an unbiased estimate of the characteristic being studied for the corresponding set in
any FSU in the population.
In the development which follows the problem of proper identification of symbols by sub- and super-scripts arises.
In order to simplify
the presentation, consider Table 4 which defines terms relating to the
i
th
first stage unit.
Symbols referring to the i
by the appropriate superscript as in Fil'
th
FSU will be made
However, this superscript
will be omitted in the tabular presentation, it being understood to
th
apply to the i
FSU.
.-e
In addition, some symbols relating to the non-responding groups
to each of the
~-mail
attempts must be defined.
Corresponding to the
set E, there is a well defined group comprising all the members of the
FSU who would not respond even after Emailings.Obviously.this group
involves all the sets from the (u+l)th through the (p+l)th.
The pro-
portion of the Ni SSUs in that group is denoted by F where, clearly,
u2
u
= 1,2, ••• ,p-l.
Thus it represents the sum of the fraction answering at the next mailing and the fraction not answering the next mailing.
These sets of
non-respondents also have corresponding means, Y ; totals, Y ; and
u2
2
variance, CTu2 '
u2
76
Table 4.
Population values of respondent SSUs to various mailings for
th
the i
FSU
Mailing
number
Number
Fraction
1
Nll
Fll
1'11
Yll
2
N21
1'21
Y21
3
N
31
F21
F
31
1'31
Y
3l
Mean
Total
-
2
0"11
2
0"21
2
0"31
2
u
..
Variance
O"ul
(·p-l)
N(p_l)l
F(p_l)l
1'(P_l)l
Y(p-l)l
2
O"(p_l)l
P
Npl
Fpl
1'pl
Y
pl
2
O"pl
N(p+l)l
F(p+l)l
1'(P+l)l
Y(p+l)l
2
O"(p+l)l
(p+l)
r:J
!:/The set (p+l) consists of SSUs responding only to personal interviews and recalls after not responding to ~ mailed questionnaires.
Certain parallel notation is also needed to define sample values.
These are:
~
= size of imtial mailing,
=
sample number responding to the uth mailing where u
= 1,2,
.. . ,p,
n
u2
= sample number not responding to the uth mailing where n = 1,
2, ••• ,p,
77
f
u
=
reciprocal of sampling rate to be applied to the nonrespondents to the (u_l)th mailing and to be sent a uth
wave of questionnaires, where u
= 2, 3, •.. , p and f 1
== 1.
Corresponding sample values for means and variances will carry similar
""'2
"'2
subscripts as Yul' crul ' Yu2 and cru2 •
4.1
Sampling with Unequal Probabilities and
without Replacement at the First Stage
In this sampling plan, .!!! first stage units are selected without
replacement and with general inclusion probability
FSU 1.
From the i
th
FSU,
i
~
-
associated with
second stage units are selected with equal
probability and without replacement.
..
p~
From the
ni questionnaires mailed
i i i
we obtain ~l responses. A random sample of ~2/f2 are drawn and sent
a second mailing.
From the
i
i
~2
non-respondents to the second mailing,
i
randomly select ~2/f3 units and send them a third mailing, etc.,
through ~ mailings.
Finally, at the last phase, n~2/wi units are
chosen and personally interviewed until these all have been successfully contacted.
4.1.1
An Estimator of the Total.
Under the plan outlined above
" is an estimate of the population total
TIE
M
T = E
·rt
E Yio
i=l j=l
J
and is given by the equation:
+ ••. +
( 4.1)
78
A.
(T ) can be shown to be unbiased.
1E
The proof is as follows:
It is obvious that the quantity on the right hand side is equal to
m
E !:
i=l
1
P
i
rr ~"'-iYll ~l + Y2lf 2En2l
i i
+ •.•
.,..-1
i
n
l
(p
i) i
+ Y 1 II fjEn 1
p
j =2
P
+
and where, from binomial sampling theory,
Hence, 4.2 reduces to
1 [P (.
m p
E(TA.1E ) =E !:
i=l
i
m
= E !:
i=l
!:
u=l·
i i .,..-1
i
NFulY~ +NF
1 [P (
T
)~
1
i .,..-1
_.-1 -i ~
m
Nulr ul + Ir
Yi
u2 Y 2 = E !: P
u=l
P
i=l i
!:
1
-
Y
P2 P2
= T •
A.
4.1.2
Variance of the Estimator, T •
1E
In order to find the vari-
A.
ance of T , apply Madow's Theorem to the estimator which, after adding
1E
and subtracting appropriate identical terms, may be written in the
following form:
79
'"
The first term in the theorem, V(T1Els),
(~ indicating a particular set
of m first stage units), is seen to be
+ •••
where the expected value of the bracketed quantity in
(rr)
(4.4), conditional
on i, is
and the dot notation yi is the sample mean of the SSUs
u·
th
u mailing including both respondents and non-respondents. Squaring
the quantity in brackets as indicated and taking expectations relevant
to the phases, the cross product terms vanish since the non-respondents
of the uth mailing constitute the population sampled at the (u+l) th
attempt.
This gives
+ ••• +
(4.6)
80
Now by adding and subtracting yiu2 and noting that the cross product
th
term vanishes, the u term becomes:
2 2 2 2 2
i (-i
-i )
i (-1
~~ )2
i (-1 ~~ )
E nu2 Y(u+l). - Yu2
= E nu2
Y(u+l). - ~
- E nu2 yu2 -ru2
Taking conditional expectations one obtains
and
Substituting these expressions into
(4.7) gives
Thus
where
~ is the population variance of the i ~ primary sampling unit.
~
81
Finally by the same reasoning used in Section 2.1.3 in obtaining formula (2.10) the first term in the theorem becomes
=
M 1
E
-
i=l Pi
p-1
+ E
u=2
Since [T1EI~ is an unbiased estimator of T, the evaluation of the
second term in the theorem, viz., V[E(T1Els~ is exactly the same as
-
(2.11), that is:
(4.10 )
..
The sum of (4.9) and (4.10) results in the desired formulae:
( 4.11)
where
4.1.3
Optimum Allocation.
Define the following cost function
appropriate to the sampling plan considered in this section:
82
i
u2
+ ••• + - - +
i
n
iU+l)
~i
i·
i
+ m
~ c2i ~l + ~l + ••• + nul +
i=l
.
m
i
+ n( -1)1) + ~ c~i
p
i=l ~
i
n 2
~
w
•
The terms in this cost function are easily interpreted by extending the
description of the two-stage single-phase design.
Now consider the
expected value of the cost which has the form:
+
F
i
u2
+. • •
u+l i
n
f
j=2
j
F)
(R-l)2
i
II f
j
j=2
p
+ •••
Since the inclusion probabilities, in general, are functions of
each situation must be handled separately.
consid~r
~
As in Section 3.4.1 we shall
m'
the special case where Pi = M'
that is the inclusion prob-
abilities are eq-qal.
In this 6ituation the expected cost function is
given by (4.13 ) with Pi set equal to m/M.
From the application of Cauchy's Inequality to minimize the product,
(4.14 )
[n;:J,
The optimum solutions for m,
[wi] and [f~ must satisfy the
equations:
= K,
mlCo
=
1
i (
+
F)2
m n eli c 2i 11
K,
(4.15 )
i = 1,2, ••• ,M ,
( 4.16)
l
1
2
-
(+1 i) ( i
i
)
Su- S(u+l)
M Ni~fj
u
i
l
m ~ (cli Fu2 + c2iF(u+l)1)2
= K,
=
1,2, ••• ,p-l,
(4.17)
i = 1,2, ••• ,M,
..
M
i
NiW
(irj=2f~)
Si
p
= K,
m ~i IC.
c3i hi
Fp2
i
= 1,2, ••• ,M
(4.18)
These equations lead to the folloWing solutions for the fits, wi,s,
and ni,s which are independent of any restriction on cost or presicion:
c2iF~i
2
si)
i2· = (CliFi2 +
) (O"i ._
f2
+
iii'
cli
c2iF11
Sl - S2
i=1,2, ••• ,M,
(4.19 )
(4.20 )
84
i
= 1,2, ••• ,M,
( 4.21)
i
=
1,2, .•• ,M.
(4.22 )
At this point, define the following relationships:
c:;.
=
M(M
cri - i=l~ rrcr~),
"e
The only remaining allocation constant
depend on the restriction.
is~.
The two possible solutions
If the expected cost is fixed at ClEO the
solution for m is found to be
ClEO
m=
1.
c
+
c
o
(~)
~
2
M
E
rrf3i
i=l
and the least variance attained by the estimator is:
(4.24 )
85
For a fixed variance VlEO ' the solution for
~
.!.
~ 2
+
(c)
o
M
E
i=l
~
is:
rt-13i
and the minimum. expected cost is:
(4.26 )
It is easy to verify that the solutions are generalizations of
the earlier ones. obtained for two-stage uni-phase sampling.
4.2
Sampling with unequal Probabilities and with
Replacement at the First Stage
The sampling plan considered below is the same as qut1ined in the
introduction to this chapter and in Section 4.1 except that the first
stage sampling is made with arbitrary selection probabilities and with
replacement.
4.2.1 An Estimator of the Total.
The estimator considered here
is an extension of (2.38) and is given by
This estimator is unbiased; however, the proof will be omitted as
it follows closely the development in Sections 2.2.2 and 4.1.1.
86
A
4.2.2
A
Variance of the Estimator, ~2E.
The variance of ~2E is
derived by an extension of the methods used in Sections (2.2.3) and
(4.1.2) and is found to be:
A
V(T )
2E
~2
= -1
ni) cri 2 -
M 1
[(
E -1 - m i=l Pi
~
ni
i
P-l(U+l i) ( i
i
)
+ E II f
S - Sf
1
u=l j=2 j
U
"u+l)
S
(4.28 )
Comparing formula (4.28) with the variance formula for the singlephase two-stage plan given by (2.33) the only differences are in terms
composed solely of phase constants.
4.2.3
Optimum Allocation.
That is S~, f~, wi, etc.
In defining a suitable cost function,
it is apparent tha:t (4.12) is appropriate to this sample design.
How-
ever, in obtaining the expected cost we must use the a priori selection
probabilities and make use of the fact that first stage sampling is
with replacement.
On
finding the expected value of· the cost as given
by (4.12) and taking these modifications into consideration the following cost function is obtained:
(4.29 )
Using Cauchy's Inequality leads to the following solutions via the
steps used in Section (4.1.3).
i i i
For f , f (u = 2,3, ••• ,p) and 101' the
u
2
87
solutions are identical with equations (4.19), (4.20), and (4.21),
respectively.
For
ni we have:
,
~
~
=
P1[cr~y -
. i
~N
r~i 2
2
8i ]
l
f (::1) ~lJ r-Cl=i-+-C-2-i-F~i~':::::
(4.30 )
At·this point define:
Then fixing the cost at C2EO the solution for the remaining variable m is as follows:
(4.31)
The least variance attained by the estimator is
V(T ) =...!.- ~Cl c )~ + ~
2E
~
~ 2 0
1
2EO
2
r~ilJ -
The solution for !- in case of fixed
'"
V(~2E)
set, say, at V2EO is:
(4.33 )
88
The least expected cost of the sample plan is
r
(4.34 )
Again, these formulae are extensions of the single-phase formulae
when first stage sampling is made with replacement.
4.3
Comments on the Solutions
If we consider the solutions for fiu and wi for the i th first . stage
~
tUlit as given by equations (4.19), (4.20) and (4.21), we find that the
f
i
u
will be greater than or equal to unity if
'-e
>
..
~~-i) - S~
(4.35 )
cliF(u_l)+c2iFul
u
=
2,3, ••• ,p-l,
and if
(4'.36 )
These conditions are direct extensions of those discovered by .
El-Badry (1956), when sampling is carried on only at one stage.
Also,
note that these conditions, as well as the restraint imposed by (4.22)
for sampling FSUs without replacement, namely:
<
( 4.37)
Mc:0
cannot be altered through cost or variance constraints imposed on the
solutions.
Only through modification of the frame and/or the. probabil-
ity system can these be varied.
However, the constraint, m < M, can be met by following a procedure
similar to that suggested in Section 3.3.
As
for the population information required for the calculation of
fi and wi, it is apparent from formulae (4.19), (4.20) and (4.21) that
u
i i i
one must know F(u-2)2' F(u-l)2 and F(u_l)l as well as the variances
of the non- responding groups in the (u-l)th and uth attempt relative
to the non-responding
....
group
in the (u-2 )th attetnPt.
While a priori
knowledge of the relative variances is required, the proportions
i
i
F(u-2)2, F(u-l)2 and F(u-l)l can be estimated from the results of the
previous two phases (i. e. mail responses).
By
considering the optimum
variance or cost functions (i.e., (4.24), (4.26), (4.32) or (4.34) for
(p+l) stages), the expected cost or variance can be reduced if the
following inequality holds:
This merely states that the added terms necessitated by using the
(P+l)th phase (i.e., those on the left hand side of (4.38»
are less
90
than the term replaced.
This too is a direct extension of the results
found by El-Badry (1956).
Thus, (4.38) can be evaJ..uated after the pth phase by considering
both the response information obtained at that attempt and by using any
other information to estimate the probability of response at the next
i
/ Fi ) and the ratio of the variances of the non-respondphase ( F(pTl)l
p2
2
2
i
ing.groups, (O"(P+l)l/O"p2).
i
Now (4.38) is not a theoreticaJ..ly sufficient
indicator of the phase at which to start personaJ.. interviewing.
Theo-
reticaJ..ly, one should consider extensions of (4.38) to include the
phases (p+2), (p+3), ••• , (p+k).
However, for practicaJ.. applications
these considerations do not seem to warrant consideration.
4.4 Extension of the Theory to Stratified Sampling
In this section, the estimators (4.1) and (4.27) will be extended
to a stratified sampling situation.
Since this is accomplished by mere-
ly adding a stratification subscript to the earlier notation and then
summing the results over aJ..l S strata, the estimators retain their property of unbiasedness.
Also, because sampling is independent from
stratum to stratum the variance of the stratified estimators is simply
the sums of the variances of the individuaJ.. strata.
An extension of the cost function is aJ..so easily accomplished by
incorporating strata subscript notation and adding the individuaJ..
stratum cost functions.
a stratified plan.
This covers all the variable costs implicit in
Because of the, straightforward' changes outlined
above, only the optimum solutions for
Ilk
and the associated minimum
91
costs and variances will be given.
All proofs will be omitted and
where results are similar to earlier formulae except for strata subscripts, these earlier results will be cited and the appropriate notational changes indicated.
Each plan outlined below will be used to draw
from stratum. ! of the S strata in the population.
~
first stage units
At the second stage
~ secondary sampling units are drawn from among N~ in FSU i," stratum.
k, with equal probability and without replacement.
Plan I.
First stage units selected according to arbitrary selec-
tion probabilities and without replacement.
ing the i
The probability of includ-
~.
primary unit in stratum. ! in a sample of size
~
i
is P •
k
The estimator is constructed by adding the subscript! to each
symbol in formula (4.1).
Summing this over! (the stratum. identifica-
tion index) from 1 to S gives the desired estimator.
done with formula (4.11) and the result
is the appropriate formula for
the variance of the stratified estimator.
ried out. on the cost function (4.13).
This may also be
A similar procedure is car;..
This cost function consists only
of variable costs denoted by C •
IES
•
_i
i
Under these conditions the optimum. solutions for ~, f"ku' wk and
i
~
are given by (4.19), (4.20), (4.21) and (4.22), respectively, with
the subscript ! added.
Defining
'1u
i
and 13k as in Section 4.1.3 with appropriate ! sub-
scripts added, the formula for
~
is found to be:
~
Ilk
~
(ex. _c
c ko k=l
··kL
ko
)~
S
=
VSO + r:
k=l
+
~ }.
k=l i=l k
1\
rt ~ik
2
cry
, k
= l,2, ••• ,S,
k
where Pk sampling phases are carried out in each selected FSU of
stratwn k, and where VSO is the fixed variance required of this estimate.
The least expected cost of this plan is given by:
(4.40)
Plan II.
First stage units selected with arbitrary selection
probabilities and with replacement.
i
th
FSU at any draw in stratum
~
The probability of selecting the
i
is Pi.
The estimator is found by adding the subscript
formula (4.27), and summing over ~ from 1 to S.
~
to each symbol in
The appropriate vari-
ance for this estimator is found by carrying out a similar procedure. on
formula (4.28).
Similarly, we obtain the variable cost portion of the
associated cost function.
For this situation the optimum solutions for
f~, f~, w~ and ~
are given by (4.19), (4.20), (4.21) and (~.22), respectively, with the
subscript
~
added.
Defining
0k2
the formula for
as in Sect~on 4.2.3 by adding appropriate ~ subscripts
Ilk
subject to a fixed cost constraint is:
9'
1I1t =
8
E
k
= 1,2, ••• ,8,
(4.42)
k=l
where '0
is the cost desired.
80
The expected variance of this plan is given by
-
8
E
~ _-1.
.r:'"
k=l i=l
Nk
i2
cr •
k
(4.4,)
It should be noted that for :plan I, optimum allocation values for
1I1t
were given for the fixed variance constraint while for Plan II these
constants were given for a fixed cost constraint.
Either
reault can
be obtained for both plans by interchanging the terms which differ in
the equations (4.40) and (4.42) and the equation (4.41) and (4.43).
5.0
SUMMARY AND CONCLUSIONS
5.1
Summary of Results
Estimators of the population mean and variance have been derived
appropriate for two stage sampling with non-response at the second
stage.
Both possess the property of unbiasedness.
These were derived
for the case of:
(1)
sampling without replacement and with arbitrary selection
probabilities of the first stage units.
(2 )
sampling with replacement and with arbitrary selection probabilities of the first stage units.
In all cases the selection of second stage units was with equal
probability and without replacement.
For general cost functions appropriate to the above sample designs,
optimum solutions for the constituent sample sizes and the sampling
rates for non-respondents were derived.
These are optimum subject to
a fixed expected cost or fixed variance.
Results were also extended
to the stratified two stage uni-phase case.
The efficiency of single phase plans calling for complete recall
of the non-respondents is compared with that for subsampling of nonrespondents.
cient.
It has been shown that the latter procedure is more effi-
An actual example of the application of both methods indicated
that subsampling was 11 per cent more efficient.
An extension of multiphase sampling is given following the model
proposed by El-Badry (1956).
Expressions for optimum first stage, sec-
ond stage, first phase, second phase, etc., sample sizes have been
95
obtained for both stratified and unstratified cases when first stage
uni ts are sampled with or without replacement.
The conditions required
for the applicability of the procedure are also derived and discussed.
5.2
Summary of Conclusions
As evidenced by the numerical example given in the text, applica-
tion of the results seems to produce worthwhile gains in efficiency.
The restrictions imposed on the frame constants necessary for using
this procedure in the single phase case do not appear to detract from
its applicability.
For multiphase sampling, the amount of a priori information needed
for proper application of the results derived seems to be rather exces-
de
sive, except for two or three phases.
5.3 Suggestions for Further Research
The research reported in this thesis covers only one approach to
the non-response problem.
As evidenced by the review of literature,
many other approaches have been suggested.
Thus, the suggestions for,
further research will emphasize extension of the work presented in this
thesis, and development of other approaches to the problem of nonresponse.
Some of the areas of research suggested by this thesis are;
(1)
the development of "best" variance estimators for single and
multiphase two-stage sampling plans.
(2)
methods for utilizing auxiliary population information so as
to construct frames to which the procedures suggested by this
thesis are amenable.
96
Researchers have generally restricted their studies to single
stage sampling plans.
An especially fruitful area for research appears
to be the possible development
o~ mod~ls
for specific populations which
utilize auxiliary information to eliminate non-response bias •
.0_
97
LIST OF REFERENCES
Birnbaum, Z. W., and Sirken, M. G. 1950. Bias due to non-availability
in sampling surveys. Jour. Amer. Stat. Assoc. 45 :99-111.
Cochran, W. G.
1953.
Sampling Techniques.
John Wiley, Inc., New York.
Deming, W. E. 1953. On a probability mechanism to attain an economic
balance between the resultant error of response and the bias of
non-response. Jour. Amer. Stat. Assoc. 48:743-772.
Durbin, J. 1955.
Inst. Stat.,
Non-response and call-backs in surveys.
2!!:: 72- 86.
Bull. Int.
Durbin, J. 1957. Sampling theory for estimates based on fewer individuals than the number selected. Bull. Int. Inst. Stat.,2§.:113.
El-Badry, M. A. 1956 . A sampling procedure for mailed questionna.ires.
Jour. Amer. Stat. Assoc . .'21 :209-227 •
Ferber, R. 1948. The problems of bias in mail returns:
Publ. Opin. Quart. 12: 669-676.
Ford. R. N., and Zeisel, H.
Quart. 13:495-501.
1949.
Bias in mail surveys.
a solution.
Pub1. Opin.
Gaudet, H., and Wilson, E. C. 1940. Who escapes the personal investigator? Jour. App1. Psycho1. 24:773-777.
'
Hansen, M. H., and Hurwitz, W. N. 1946. The problem of non-response
in sample surveys. Jour. Amer. Stat. Assoc. 41:517-529.
Hendricks, W. A. 1949. Adjustment for bias caused by non-response in
mailed surveys. Ag. Econ. Res. 1:52-56.
Hilgard, E. R. and Payne, S. L. 1944. Those not at home:
pollsters. Pub1. Opin. Quart. ~:254-261.
riddle for
Madow, W. G. 1949. On the theory of systematic sampling II.
Math. Stat. 20 :333-354.
Ann.
Neyman, J. 1938. Contributions to the theory of sampling human populations. Jour. Amer. Stat. Assoc. 33:101-116.
Pace, C. R. 1939. Factors influencing questionnaire returns from
former university students. Jour. App1. Psycho1. ~ :388-397.
Politz, A., and Simmons, W. 1949. An attempt to get the "not at
.homes" into the sample without call backs. Jour. Amer. Stat.
Assoc. 44:9-31.
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Shuttleworth, F. K. 1941. Sampling errors involved in incomplete
returns to mail questionnaires. Jour. App1. Psycho1. ~: 588-591.
Stanton, F.
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1939. Notes on the validity of mail questionnaire reJour. Appl. Psycho1. ~:95-l04.
stuart, A. 1954. A simple presentation of optimum sampling results.
Jour. Royal Stat. Soc. l6B: 239-241.
Suchman, E. A., and McCandless, B. 1940.
Jour. App1. Psycho1. 24:758-769.
Who answers questionnaires?
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Iowa State College Press and the Indian Society of Agricultural
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Vaivanijkul, N. 1961. A comparison of respondents and non-respondents
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Edition. Charles Griffin and Co., London, 1953.
Second
INSTITUTE OF STATISTICS
NORTH CAROLINA STATE COLLEGE
(Mimeo Series available for dHribution at cost)
265. Eicker, Friedheim.
266.
Consistency of parameter-estimates in a linear time-series model.
October, 1960.
Eicker, Friedheim. A necessary and sufficient condition for consistency of the LS estimates in linear regression. October,
1960.
267. Smith, W. L.
On some general renewal theorems for nonidentically distributed variables. October, 1960.
268. Duncan, D. B. Bayes rules for a. common multiple comparisons problem and related Student-t problems.
1960.
269. Bose, R. C. Theorems in the additive theory of numbers. November, 1960.
270. Cooper, Dale and D. D. Mason.
November,
Available soil moisture as a stochastic process. December, 1960.
271. Eicker, FriedheIm. Central limit theorem and consistency in linear regression. December, 1960.
272. Rigney, Jackson A. The cooperative organization in wildlife stati~tics. Presented at the 14th Annual Meeting, Southeastern Association of Game and Fish Commissioners, Biloxi, Mississippi, October 23-26, 1960. Published in Mimeo Series,
January, 1961.
273. Schutzenberger, M. P. On the definition of a certain class of automata. January, 1961.
274. Roy, S. N. and J. N. Shrizastaza. Inference on treatment effects and design of experiments in relation to such inferences.
January, 1961.
275. Ray-Chaudhuri, D. K. An algorithm for a minimum cover of an abstract complex. February, 1961.
276. Lehman, E. H., Jr. and R. L. Anderson. Estimation of the scale parameter in the Weibull distribution using samples censored by time and by number of failures. March, 1961.
277. Hotelling, Harold.
The behavior of some standard statistical tests under non-standard conditions.
February, 1961.
278. Foata, Dominique. On the construction of Bose-Chaudhuri matrices with help of Abelian group characters.
1961.
279. Eicker, Friedheim. Central limit theorem for sums over sets of random variables. February, 1961.
280. Bland, R. P.
A minimum average risk solution for the problem of choosing the largest mean.
281. Williams, J. S., S. N. Roy and C. C. Cockerham.
282. Roy, S. N. and R. Gnanadesikan.
April, 1961.
283. Schutzenberger, M. P.
285. Patel, M. S.
286. Bishir,
J.
May, 1961.
Equality of two dispersion matrices against alternatives of intermediate specificity.
April, 1961.
A coding problem arising in the transmission of numerical data.
Investigations on factorial designs.
W.
March, 1961.
An evaluation of the worth of some selected indices.
On the recurrence of patterns.
284. Bose, R. C. and I. M. Chakravarti.
February,
April, 1961.
May, 1961.
Two problems in the theory of stochastic branching processes.
May, 1961.
287. Konsler, T. R. A quantitative analysis of the growth and regrowth of a forage crop. May, 1961.
288. Zaki, R. M. and R. L. Anderson. Applications of linear programming techniques to some problems of production planning over time. May, 1961.
289. Schutzenberger, M. P.
A remark on finite transducers.
June, 1961.
= b·+mcU' in a free group.
290. Schutzenberger, M. P.
On the equation a2+ n
291. Schutzenberger, M. P.
On a special class of recurrent events. June, 1961.
June, 1961.
292. Bhattacharya, P. K. Some properties of the least square estimator in regression analysis when the 'independent' variables
are stochastic. June, 1961.
293. Murthy, V. K.
On the general renewal process.
294. Ray-Chaudhuri, D. K.
295. Bose, R. C.
June, 1961.
Application of geometry of quadrics of constructing PBIB designs. June, 1961.
Ternary error correcting codes and fractionally replicated designs.
May, 1961.
296. Koop, J. C. Contributions to the general theory of sam pling finite populations without replacement and with unequal
probabilities. September, 1961.
297. Foradori, G. T.
Some non-response sampling theory for two stage designs. Ph.D. Thesis.
298. Mallios, W. S.
Some aspects of linear regression systems. Ph.D. Thesis.
299. Taeuber, R. C.
On sampling with replacement: an axiomatic approach. Ph.D. Thesis.
J.
Srivastava, J.
300. Gross, A.
On the construction of burst error correcting codes.
301.
N.
303. Roy, S. N.
November, 1961.
August, 1961.
Contribution to the construction and analysis of designs.
302. Hoeffding, Wassily.
November, 1961.
November, 1961.
The strong laws of large numbers for u-statistics.
August, 1961.
August, 1961.
Some recent results in normal multivariate confidence bounds.
August, 1961.
304. Roy, S. N. Some remarks on normal multivariate analysis of variance. August, 1961.
305. Smith, W. L. A necessary and sufficient condition for the convergence of the renewal density.
306. Smith, W. L.
A note on characteristic functions which vanish identically in an interval.
307. Fukushima, Kozo.
308. Hall, W.
J.
A comparison of sequential tests for the Poisson parameter.
Some sequential analogs of Stein'S two-stage test.
309. Bhattacharya, P. K.
August, 1961.
September, 1961.
September, 1961.
September, 1961.
Use of concomitant measurements in the design and analysis of experiments.
November, 1961.