TAYLOR SERIES VARIANCE ESTIMATION
FOR SELECTED INDIRECT DEMOGRAPHIC ESTIMATORS
by
Ophelia M. Mendoza
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1412
August 12, 1982
Taylor Series Variance Estimation
For Selected Indirect Demographic Estimators
by
Ophelia M. Mendoza
A Dissertation submitted to the faculty of the
University of North Carolina at Chapel Hill in
partial fulfillment of the requirements for
the degree of Doctor of Public Health in the
Department of Biostatistics
Chapel Hill
1982
Approved by:
ABSTRACT
MENDOZA, OPHELIA M. Taylor Series Variance Estimation for Selected
Indirect Demographic Estimators. (Under the direction of WILLIAM D.
KALSBEEK.)
The numerous demographic studies done in countries with deficient
vital registration systems and population censuses have led to the
extensive use of indirect demographic estimators at present.
However,
while these estimators are frequently computed from survey data, their
application is often complicated by the fact that they are usually
complex, non-linear estimators obtained from complex sampling designs,
for which measures of precision are not easily formUlated.
In this dissertation, the mean square errors
(mse)
of four com-
monly used indirect fertility and mortality estimators are derived
using the Taylor series linearization method.
The estimators con-
sidered are the Coale-Demeny total fertility rate
the own-children method of estimating
child mortality estimators.
TFR,
(TFR)
estimator,
Sullivan's, and Trussell's
The derivations are applicable to any
multi-stage sampling design with stratification in the first stage.
Estimates of the mean square errors of these estimators are then computed using data from the East Java Population Survey, which has a complex sampling design.
In addition, design effects are computed in
which results are compared with
mse's
random sampling was the assumed design.
the effect of using an
mse
that one would obtain if simple
Such a comparison demonstrates
estimate of these measures which is naive,
since it is not consistent with the actual design used.
Since a large
number of design effects are computed, a descriptive analysis is done
to determine the conditions under which they considerably deviate
from 1.
Finally, for measures which are commonly and erroneously
interpreted as "proportions" like Sullivan's and Trussell' 5 child
mortality estimators, the binomial variance,
pq/n,
and compared with that of the estimated
using the actual samp-
ling design.
mse
is computed
This latter,comparison demonstrates the effect of
using an even more naive estimator which disregards both the form
of the function and the sampling design used.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS
LIST OF TABLES
LIST OF FIGURES
CHAPTER 1 - MOTIVATION AND REVIEW OF LITERATURE
1.1
1.2
1.3
1.4
1.5
On the Problem of Variance Estimation in
Demographic Surveys
The Need for Indirect Fertility and
Mortali ty Est imators
Objectives of the Study
A Review of Variance Estimation Techniques
1.4.1 Taylor Series Linearization Method
1.4.2 Balanced Repeated Replications
1.4.3 Jackknife Approach
1.4.4 Comparison of Three Approaches to
Variance bt imation
1.4.5 Computer Programs for Computing
Variances From Complex Samples
A Review of Indirect FertIlity and Mortality
Measures From A Single Survey
1.5.1 Indirect Ferti 1 i ty Est imatoY5
1.5.2 Indirect Mortali ty EstimatoY5
1
.4
~
6
8
8
11
14
15
17
23
23
,
29
CHAPTER 2 - APPLICATION OF TAYLOR SERIES LINEARIZATION METHOD
TO INDIRECT DEMOGRAPHIC ESTIMATORS
2.1
2.2
2.3
2.4
Introduction
The Woodruff Linearization Procedure
The Ultimate Cluster Variance Estimator
Derivation of Mean Square Errors
2.4.1 Notation
2.4.2 Sullivan's Child Mortality Estimator
2.4.3 Trussell's Child Mortality Estimator
2.4.4 Coale-Demeny TFR Estimator
2.4.5 Own-Children Method of Estimating TFR
ii
34
34
36
39
39
40
46
48
52
CHAPTER 3 - APPLICATION OF DERIVED ESTIMATORS TO ACTUAL DATA
3.1
3.2
3.3
3.4
Obj ectives of Analysis
The East Java Population Survey
3.2.1 Sampling Design
3.2.2 Sampling Weights
Description of Computer Programs to Compute
Mean Square Errors
Sample Size
55
59
59
61
63
65
CHAPTER 4 - FINDINGS
4.1
4.2
4.3
Results for Indirect Fertility Estimators
Results for Indirect Mortality Estimators
4.2.1 Effect of Selected Variables on Design
Effects
4.2.2 The pq/n Variance Estimator
Discussion of Results
68
72
,
76
84
88
CHAPTER 5 - SUMMARY OF RESULTS AND SUGGESTIONS FOR FUTURE RESEARCH
5.1
5.2
Summary of Results
Suggestions for Further Research
BIBLIOGRAPHY
APPENDIX
iii
"
94
96
ACKNOWLEDGEMENTS
I am especially endebted to my advisor, Dr. William D. Kalsbeek,
for his patient guidance and strong encouragement and support at
each step of this research project.
The valuable suggestions of the
other members of my committee, Dr. Krishnan Namboodiri, Dr. Barry M.
Popkin, Dr. Chirayath M. Suchindran and Dr. Jeremiah M. Sullivan are
also gratefully acknowledged.
The rigors of writing a dissertation have been made more bearable
by the moral support and assistance of friends especially the Filipino
community in Chapel Hill.
I am also grateful to the University of the Phillippines, in particular, the Department of Biostatistics of the Institute of Public
Health, for giving me the
training.
~pportunity
to undertake further graduate
The financial support provided by the World Health
Organization and the research assistantship granted by the U.S. Public
Health Service (Grant No. 5 004 AH 01816-02) are gratefully
~cknow­
ledged.
Many thanks are due to the Central Bureau of Statistics of
Indonesia for grating me permission to use their data for this research.
I also thank Jackie O'Neal for her skillful typing of the manuscript.
Finally, very special thanks are due to my parents, Mr. and Mrs.
Domingo L. Mendoza and to my aunts, Tiya Nening and Mama Fel ing, whose
unwavering support, patience and understanding, I have always leaned on.
iv
LIST OF TABLES
TABLE
PAGE
1
Balanced Half-Samples for H-S Strata
13
2
Summary of Comparative Evaluation of Different
Variance Estimation Techniques
18
Selected Characteristics of Variance Estimation
Programs
20
Trussell's Child Mortality Estimators and
Corresponding Taylor Series Linearization
49
5
Summary of EJPS Sampling Design
60
6
TFR Estimates Using the Coale-Demeny Estimator
and Corresponding Mean Square Errors and Design
Effects: East Java Population Survey, 1980
69
ASFR and TFR Estimates Using the Own-Children
Method and Corresponding Mean Square Errors and
Design Effects: East Java Population Survey, 1980
71
Estimates of q(a) Using Sullivan's Estimator and
Corresponding Mean Square Errors and Design
Effects for the Total Sample: East Java Population
Survey, 1980
73
Range of Values (and Percentages Occurring in Certain
Ranges) for Different Statistics Computed Among 34
Different Domains for Sullivan's and Trussell's Estimators: East Java Population Survey, 1980
74
3
4
e
7
8
9
10
11
Estimates of q(a) Using Trussell's Estimator and
Corresponding Mean Square Errors and Design Effects
for the Total Sample: East Java Population Survey,
1980
75
Estimates of Trussell's q(l) Estimator for Both
Sexes Combined and Corresponding Mean Square
Errors and Design Effects for the Total Sample
and for Different Domains: East Java Population
Survey, 1980
77
v
12
13
14
Frequency Distribution of the Design Effects for
The Estimated Mean Square Errors of Mortality
Estimators
78
Estimated mse's and mse Ratios for Sullivan's
and Trussell's Estimators Using the Naive
Estimator, pq/n, with Number of Births a Years
Ago as Denominator: Total Sample, East Java
Population Survey, 1980
86
Estimated mse's and mse Ratios for Sullivan's and
Trussell's Estimators Using'the Naive Estimator,
pq/n, with the Number of Children Ever Born to
Women as Denominator: Total Sample, East Java
Population Survey, 1980
87
vi
LIST OF FIGURES
FIGURE
1
2
3
4
5
6
PAGE
Distribution of Design Effects of Mortality
Est imaton by Author
80
Distribution of Design Effects of Mortality
Estimators According to Sex For Which the
Probability is Computed
80
Distribution of Design Effects of Mortality
Est imators by Type of Estimator
81
Distribution of Design Effects of Mortality
Estimators According to the Number of Random
Variables in the Estimator
81
Distribution of Design Effects of Mortality
Estimators by Sample Size
82
Distribution of Design Effects of Mortality
Estimators for the Total Population and by
Urban-Rural Grouping
82
vii
CHAPTER 1
Motivation and Review of Literature
1.1
On the Problem of Variance Estimation in Demographic Surveys
The role of sample surveys in demographic research cannot be under-
stated.
They are often the only means to get detailed information on
fertility, mortality and migration patterns and differentials.
They
are widely used to supplement established data collection systems and
to evaluate the quality of data collected in such systems (Lunde,
1976).
Sample surveys are also extensively used as a means of col-
lecting data for the estimation of fertility and mortality in places
where population censuses and vital registration systems are highly
deficient.
The quality of the estimates derived from sample surveys depends,
among other factors, on the sampling design and its implementation.
The utility of the statistics obtained is enhanced by a knowledge of
their degree of reliability, as measured by the variance of the estimates.
In addition to providing a measure of precision for the esti-
mates, a correct estimate of the variance of a demographic estimator
is also necessary to make valid inferences about a population, as in
the construction of confidence intervals and testing of hypotheses.
In particular, it is needed in applying statistical tests to make comparisons of fertility and mortality levels over time or between subgroups within populations.
2
The problem of variance estimation for estimators derived from
sample surveys is two-dimentional:
it depends on the form of the esti-
mator and the complexity of the design.
non-linear.
Estimators can be linear or
A linear estimator is one which can be expressed as a
linear function of random variables - i.e., it is of the form
e=
where
and the
e
is the estimator,
u.
J
t
So
So
+
and the
are the random variables.
the form of (1.1) are non-linear.
(1. 1)
L S.u.
J J
j
S.
J
are known constants,
Estimators which do not take
Means, totals and percentages are
examples of simple linear estimators; a ratio computed as one estimated total divided by another estimated total is an example of a nonlinear estimator.
or complex.
On the other hand, sampling designs can be simple
A simple design is one in which elements in the sample
are chosen independently.
Stratification mayor may not be involved.
Simple random sampling without replacement is an example of a simple
design.
A complex design is one which induces correlations between
element values.
An example of a complex design is a multi-stage sam-
pIing procedure where one first selects a sample of clusters and then
selects a sample of elements within each of the chosen clusters.
Since elements within clusters tend to be homogeneous, clustering
induces positive correlations between element values, which result in
an increase in the variance (Kish and Frankel, 1974).
The methodology of variance estimator for linear estimators
derived from simple and complex surveys is well developed and is discussed in most sampling textbooks (Hansen, Hurwitz and Madow, 1953;
3
Kish, 1965; Cochran, 1977).
Among the non-linear estimators however,
only a few have been studied extensively, especially for complex sampling designs.
Among these are the ratio, as well as regression coef-
ficients estimated from a sample.
In the specific case of demographic
estimators, there is a dearth of knowledge on their variance estimation for both simple and complex designs.
The existing work on the sampling variance of demographic
e~tima­
tors have been mostly on life table parameters, especially the expectation of life (Wilson, 1938; Irwin, 1949; Chiang, 1960; Keyfitz, 1966).
The most recent work on this field is by O'Brien (1981) who investigated four methods of estimating the variances of two life table functions,
hqt
and
tPo'
when using complex survey data.
The variance
of the net reproduction rate as a result of sampling for deaths or for
births was worked out by Keyfitz (1966).
He also derived the variance
of the instrinsic rate of natural increase when births, deaths and
population are derived from samples.
A formulation of the sampling
variance of the gross reproduction rate was worked out by Koop (1951).
While all of this work on variance estimation dealt with classical demographic measures which were developed more than fifty years
ago, practically no work has been done on the sampling variance of
demographic estimators developed in the last decade.
An example of
these estimators are the indirect fertility and mortality estimators
which are extensively used at present in areas with deficient population censuses and vital registration systems, and which are mostly
non-linear complicated functions of random variables, derived by multistage sampling procedures.
The only attempt found in the literature
to compute the variance of an indirect estimator was by Retherford and
4
Bennett (1977) who derived the sampling variance of own-children fertility estimates assuming a systematic random sampling of women.
Unfortunately, this sampling design is not often used in demographic
surveys.
Because of this gap in knowledge, survey results are often
published without the corresponding variance estimates; and studies
have generally been descriptive in nature.
There is, therefore, a
felt need for more work in the area of variance estimation for demographic measures based on sampling surveys, to enable demographers
and other researchers to undertake analytic studies and hence more
fully utilize sample survey results.
1.2
The Need for Indirect Fertility and Mortality Estimators
Fertility and mortality statistics are essential tools for the
demographer and the public health worker.
Like other types of demo-
graphic data, they serve three broad types of use:
for management
and administration, policy formulation and evaluation, and for
scientific research and analysis (POPLAB, 1976).
Measures derived
from them are used to analyze population changes and to make population projections, which, in turn, are needed to develop plans for
housing, educational and other facilities, and to provide and allocate resources to various groups in the population.
They are essen-
tial to health planning since they reflect the health status of the
population and provide a basis for structuring and evaluating health
programs.
The direct measurement of fertility and mortality requires knowledge of two types of information:
the number of births and deaths
5
that occured over some time period, and the size of the corresponding
population at risk.
From this information, crude and specific ferti-
lity and mortality rates can be computed.
The basic sources of these
statistics have traditionally been the vital registration system and
the population census.
In many areas however, there is a substantial
underregistration of births and deaths.
A study using data published
in the 1971 Demographic Yearbook showed that births are completely
registered in Africa for only 16% of the population while deaths are
complete for only 10% (Powell, 1975).
In a different study, only 9
out of 36 countries in the Asian and Pacific regions were considered
to maintain "reasonably complete" vital statistics (Arnold and
Kuhner, 1980).
Population censuses, especially in developing coun-
tries also suffer from coverage errors which result from persons being
missed or counted more than once, or content errors, which are errors
in the characteristics of the persons counted (e.g., age misreporting).
For example, an evaluation of the age distribution of the Philippine
population based on the 1960 census results showed considerable heaping
for ages which are multiples of five (Shryock and Siegel, 1976).
In the light of these deficiencies of vital registration systems
and population censuses, other data collection systems have been
developed to provide statistics for the direct measurement of fertility
and mortality.
These measures include single-round retrospective sur-
veys, multiround follow-up surveys, dual record systems and sample
vital registration systems.
However, considerable problems have also
been encountered in their implementation and administration.
This
problem has spurred the development of tehcniques that allow the
6
indirect estimation of fertility and mortality.
To date, several
of these estimators have appeared in the demographic literature using
different approaches like the application of census survival rates,
stable population analysis, and responses to questions on fertility
and mortality in sample surveys.
The rising interest in the demography
ofcountrie3with deficient vital registration systems have led to their
extensive use at present.
1.3
Objectives of the Study
From the above discussion, three conclusions can be reached con-
cerning indirect fertility and mortality estimators:
1.
Indirect fertility and mortality estimators are important
and are widely used at present, especially in developing
countries.
2.
Sample surveys play an important role in the use of indirect methods.
3.
Based on published materials, practically no work has been
done on the problem of estimating the variance of these
estimators, especially for complex sampling designs, which
is the type of design most commonly used in demograpic
surveys.
In view of these considerations, this dissertation proposes to
do the following:
1.
To derive estimators for the mean square error (mse) of
indirect fertility and mortality measures using the Taylor
series linearization method, and using data from a single
sample survey.
Specifically, four measures will be
7
considered, namely the Coale-Demeny total fertility rate
(TFR) estimator, the own-children method of estimating TFR,
Sullivan's child mortality estimator and Trussell's child
mortality estimators.
The Taylor series linearization
method will be used to derive and compute mean square errors
because of its extensive use in sample surveys, is applicability to all sampling designs and statistics, and its economic and computational efficiency.
2.
To compute the mean square error estimates of these indirect
fertility and mortality measures using data from an actual
survey with a complex sampling design.
The data that will
be used will be from the East Java Population Survey (EJPS)
which has a three stage sampling design with stratification
at the first stage.
3.
To compare the computed mse's using the actual sampling
design of the EJPS, with variances that one would obtain if
simple random sampling was the assumed design.
Such a com-
parison will demonstrate the effect of using an estimate of
the variance of these measures which is naive, since it is
not consistent with the actual design used.
In addition,
for measures which are commonly and erroneously interpreted
as "proportions" 1ike Sullivan's and Trussell's child mortality estimators, the binomial variance,
pq/n,
will be
computed, and compared with that of the estimated mse using
the actual sampling design.
This latter comparison will
demonstrate the effect of using an even more naive estimator
8
which disregards both the form of the function as well as the
sampling design used.
1.4
A Review of Variance Estimation Techniques
Three approximate methods have been developed for estimating the
mean square errors of non-linear estimators in complex surveys:
the
Taylor series linearization method, balanced repeated replications and
the jackknife approach.
The methodology for each method is reviewed in
the following sections.
Results of studies comparing these three
approaches are also presented, as well as a discussion of existing computer software packages for their computation.
1.4.1
Taylor Series Linearization Method
The use of the Taylor series to obtain an approximate formula
for the variance of a function of random variables has been discussed
in textbooks (Chiang,
1968; Kendall and Stuart, 1963).
However, it
was Tepping (1968) who was first to apply the method to a variety of
estimators from complex surveys, with special reference to regression
coefficients.
Woodruff (1971) later presented a general algorithm for
the method, which involved no restrictions on the form of the estimator, the number of random variables involved. or the complexity of the
sampling design used.
Under this method, a linear approximation
to the estimator of
interest is obtained by means of a first-order Taylor series approximation.
An approximate variance of the estimator is calculated by
deriving the variance of the linear approximation.
~ =
(x ,x ' ... ,x ),
l 2
t
In general, let
be a vector of sample statistics and let
9
~ =
(X 'X ' ... ,X ),
l 2
t
E(~) = ~.
be a vector of parameters such that
Suppose the population parameter of interest is a function
is to be estimated by
f(~).
f(~)
and
The Taylor series expansion about the
vector of expected values is
0.2)
+ .• ,
Using only terms through the first degree in
series approximation to
t
e
f(~)
J
J
the Taylor
is
f(~)
-
f(~)
(x.-X.),
+
L (x.-X.)
J J
j
af(~)
ax.
(1. 3)
J
or
t
f(~)
f(~)
-
The mean square error of
-
f(x)
L (x.-X.)
J J
j
af(~)
ax.
(1.4)
J
can therefore be expressed as
MSE [f (~)] - E[f(~)_f(~)]2
=
EU.
J
t
-
(x.-X.)
J J
~f(x)r
L ax~
j
afwj
ax.
J
E(xj-X j )
(1. 5)
10
_I L
~~X)l2
J
j
t
0..
j
where, in general, we let
JJ
df(~)
t
df(~)
Ij kI -----:-"dX j
0\
(1. 6)
oJ' k
denote the covariance of
Using matrix notation, if we let
tives and
+ 2
X.
J
and
F be the vector of partial deriva-
V be the variance-covariance matrix, i.e.,
~f(~~
dX
a
l
12
•••• a
It
df(D
dX
F =
2
V =
~tl
then, in matrix form, (1.6) can be written as
MSE[f(~)] =
The estimated mean square error of
F'VF
f(~)
0.7)
is produced by substituting
sample estimates for population values in (1.7).
in
Estimates for entries
V are obtained by the method of ultimate cluster variance developed
by Hansen, Hurwitz and Madow (1953).
In the context of sample surveys, the Taylor series approach, in
essence, involves calculating a linear combination of totals for each
primary sampling unit (PSU) and then estimating the variance of the sum
of those linear combinations (Tepping, 1968).
Therefore, this method
enables one to compute the variance of any complex
sample estimate as
11
long as the estimator can be expressed as a function of totals and its
partial derivatives can be expressed in closed form.
It is however,
dependent on the sample size being large enough for the linear approximation to be good.
~fuch
of the existing literature on the problem of estimating vari-
ances for non-linear estimators used the Taylor series linearization
method.
In
a
paper designed for a non-technical audience, Kish and
Hess (1959) used a Taylor series variance estimator in presenting a
detailed computational procedure for the variances of ratios and their
differences for multi-stage samples.
Other statistics for which this
method was applied in deriving the variance estimators include the ratio
of ratios (Smith, 1966) as well as several statistics of importance to
economic and social surveys like indexes which are usually sums of
ratios (Kish, 1963).
In the field of demography, Keyfitz (1966) used
this method to derive the variance of the stationary population
(nLx)'
the probability of dying
(Oe )'
x
the net reproduction rate (NRR), and the intrinsic rate of
natural increase
(r).
veys is also extensive.
(nqx)'
the expectation of life
The application of this method in actual surAmong the surveys to make use of this techni-
que of variance estimation are the Current Population Survey (Banks
and Shapiro, 1971), the Canadian Labor Force Survey (Fellegi and Gray,
1971) and the World Fertility Survey (International Statistical
Institute, 1978).
1.4.2
Balanced Repeated Replications
The use of replications for computing estimates of sampling errors
have been described in a number of papers (McCarthy, 1966,1969;
12
Simmons and Baird, 1968; Kish and Frankel, 1968, 1970).
One version
of this technique applied to samples with paired selection of PSUs
can be described as follows:
an estimate of the population parameter
of interest is computed using the whole sample.
Another estimate of
the same population parameter is computed using a randomly selected
half-sample or replicate.
The squared difference between the two
estimates provides an estimate of the variance.
The precision of
such an estimate is increased by repeated replications and computing
the average of the variances estimated by each replicate.
If the subset of all the possible half-samples that can be formed
is selected randomly, the number necessary to produce a stable variance
estimate is large.
McCarthy (1966) developed a method which reduces
the number of replicates necessary to estimate the variance.
By esti-
mating the mean from a stratified sampling design with two primary
sampling units per stratum, he showed that the variability among the
half-sample estimates of variance comes from the between-strata contributions to these estimates.
Cross-product terms in the formula for
the variance which represent these between-strata contributions can be
eliminated by choosing a relatively small subset of orthogonally
balanced half-samples.
This yields an unbiased estimator of the true
variance of the linear estimator which, in fact, is equal to the value
that would be obtained if all the possible half-samples were formed
to calculate the variance estimate.
This method of choosing half-
samples is called balanced repeated replications (BRR).
with
H
=
5
For example,
strata, one possible balanced subset of half-samples
would be as described by the following diagram:
13
TABLE I
Balanced Half-Samples for H=S Strata
Half-Sample
(g)
I
Stratum (h)
3
4
2
I
+
+
2
+
+
3
+
+
+
+
+
+
+
+
+
+
+
4
5
+
6
+
+
7
5
+
+
8
In the above, a
"+"
indicates that the first primary sampling unit
(PSU) in the primary stratum is used for the half-sample while a
indicates that the second PSU in the primary stratum is used.
"_"
Also,
any pair of columns in Table I are orthogonal - i.e., for any two
columns, the number of half-sample pairs with the same sign ie equal
to the number of pairs with different signs.
half-samples
(G)
for which orthogonality can be established is any
multiple of 4 such that
If we form
ance estimator of
The smallest number of
G
~
H.
G half-samples
f(~)
SI,S2"" ,SG'
one
BRR
type vari-
can be produced as follows:
(1. 8)
14
where
f
S
is the estimator of
(;s)
f(X)
computed from the
g
S th
g
half-sample.
The US Bureau of the Census was the first major survey organization to apply this method.
It was used in the estimation of sampling
errors of ratio estimators produced from the Current Population
Survey.
Other surveys to which this method was applied are those
of the National Health Examination Survey (McCarthy, 1966) and the
National Survey of Family Growth (French, 1978), both conducted by
the National Center for Health Statistics.
1.4.3
Jackknife Approach
The jackknife approach to variance estimation is similar to replication in that it is based on variation among sub-samples chosen from
the entire original sample.
The technique was suggested by Quenouil1e
(1956), and was so named by Tukey (1958) who saw it as a rough-andready statistical tool.
Under this method, a variance estimator is computed by repeatedly
eliminating a replicate defined by the following example:
simple random sample of size
each of size
m
samples of size
=
n/d.
m.
entire sample of size
n
is randomly divided into
This is equivalent to having
If
n
f(;S)
and
is the estimator of
f~(x)
J -
puted from the entire sample after the
deleted, a variance est imator of
f(~)
d
f(2$)
is the estimator of
. th
J-
suppose a
d
groups,
replicate
from the
f(2$)
com-
replicate has been
would be
(1. 9)
IS
Frankel (1971) applied this approach to stratified multi-stage
samples with two PSUs per stratum, and called his procedure jackknife
repeated replications (JRR).
estimator of
pIe.
h
th
f(~)
f(~)
Using his method, suppose
fh(~)
is an
derived by using all but one PSU from the full sam-
The PSU not used in
primary stratum.
fh(~)
is one randomly deleted from the
One version of the JRR variance estimation of
can be formulated as
(1.10)
The jackknife approach has not been used in large surveys to the same
extent as the Taylor series linearization method and the
1.4.4
BRR.
Comparison of Three Approaches to Variance Estimation
Most of the literature comparing the different approaches to
variance estimation have been based on empirical rather than theoretical studies.
Kish and Frankel (1974) empirically evaluated and com-
pared all three variance estimation methods using data from the Current
Population Survey.
They used three stratified and clustered sample
designs with 6, 12 and 30 strata, respectively, and with two primary
units per stratum.
The three methods, i.e., Taylor's,
BRR
and jack-
knife were used to compute sampling errors of several statistics:
ratio means, simple correlations and multiple regression coefficients.
Their results showed that using the relative bias of the mean square
error - i. e. ,
Bias(MSE)
MSE
as the criterion for evaluation, all three
methods performed well for variances of ratios and differences of
ratios.
BRR
was the worst variance estimator for regression
16
coefficients, though it was better than both Taylor and jackknife for
simple correlations.
Another criterion used was the degree to which
the studentized statistic
fex) - f(X)
t
Ivar
followed a t-distribution with
of strata).
BRR
f(x)
H degrees of freedom
(H
=
the number
was found to perform consistently better than jack-
knife and Taylor's method in that order.
On the other hand,
BRR
was
found to be the least stable among the three with the Taylor estimator
showing the lowest variability.
They recommended the use of the Taylor
method for relatively simple statistics like ratio5 and the
BRR
for
more complex statistics such as correlation and regression coefficients.
Another empirical investigation was conducted by Bean (1975) who
compared the
BRR
and the Taylor approaches for a sample design that
included stratification, two stages of sampling, and post-stratified
ratio estimation.
Both the
BRR
and Taylor methods were found to be
highly satisfactory methods for estimating variances of post-stratified
ratio estimators.
As in the Kish and Frankel study, the Taylor esti-
mator had a slightly lower variance and mean square error than the
Neither method showed substantial bias.
BRR.
It was also found that the pro-
portions based on the ratio
f(x) - f(X)
I var
as estimated by the
BRR
f(x)
were closer to the corresponding normal values
than the proportions with the Taylor's method as the variance estimator.
17
A recent theoretical work by Krewski and Rao (1978) established
the asymptotic normality of both linear and non-linear statistics and
the consistency of the variance estimators obtained by using the
Taylor,
BRR
and jackknife methods in stratified samples.
Their
result provides theoretical support for the empirical studies discussed above.
A different approach was used by Shah (1978) in comparing the
different techniques of variance estimation.
Basing his evaluation
on practical considerations, he used four criteria as follows: val idi ty
or number of assumptions required, restrictions on sample design, computational problems for large data sets and flexibility of applications.
A summary of his evaluation is shown in Table 2.
Considering all fac-
tors altogether, he suggests the use of Taylor method since it is
applicable to all designs and statistics, provides "good" answers for
"large" samples and is economically and computationally feasible.
1.4.5
Computer Programs For Computing Variances From Complex
Samples
In recent years, there has been a tremendous growth in statistical
software packages, yet most of them
(BMDP, SAS, OSIRIS, and PSTAT)
do
not provide procedures that take into account the sampling design for
computing sanpling errors (Shah, 1978b).
The widespread use and accep-
tance of these programs for survey data analysis could probably be due
to
(1) a lack of awareness on the part of the researchers of the con-
sequencies of ignoring the sampling design in the computation of sampIing errors,
(2) the dearth of programs which provide for a proper
analysis of sample survey data, and
existing programs.
(3) the limited availability of
Yates (1975) points to the curious fact that the
TABLE 2
Sun@ary of Comparative Evaluation of Different Variance Estimation Techniques
l
CRITERIA
TECHNIQUE
Assumptions
Restrictions on
Sample Design
Computational
Problems
Flexibility
Independent Replications
Minimal
Severe
Simple
Pseudoreplication
(i. e., BRR)
Independence of
complimentary half
replicates
2 PSUs per
stratum
Significant
Taylorized Deviations
General Central
Limit Theorem
None
Not difficult
Can be used for
variance components
Jackknife
Intuition
None
Greater than
Taylorized
deviation
May be useful for
some designs
IReproduced from Shah, B.V.. Variance Estimates for Complex Statistics From MUltistage Sample Surveys ~n
Survey Sampling and Measurement (ed., N. Krishnan Namboodiri), New York: Academic Press, p. 32.
......
00
e
e
e
19
only general survey program widely available on British university computer installations in SPSS, which, to his mind, has many defects and
is not well suited even to the needs of social scientists.
A review of different computer programs for computing variances
of estimators from complex sample surveys was done by Kaplan, Francis
and Sedransk (1979).
They identified 18 such programs, developed both
in the US and abroad.
Using information provided by the program
developers, the programs were evaluated according to capability,
availability, generality, documentation and the user command language.
A program was considered to be available if, and only if it can be
acquired, and is currently being used in at least two institutions.
Three categories of generality were considered:
particular survey.
or
(b) useful
(a) specific to a
to a particular kind of sample design,
(c) useful to several kinds of sample designs.
A summary of some
of their findings is shown in Table 3.
All the programs except the Table Producing Language (TSL) of the
Bureau of Labor Statistics can produce estimates of means and their
variances for the entire population.
The programs with the largest
capability are OSIRIS IV and CLUSTERS. both of which can produce estimates of means, ratios and differences of ratios and their corresponding variances, as well as coefficients of variation and design effects
for the entire population, individual strata and subpopulations.
The
program SUPERCARP has the same capability except for the computation
of coefficients of variation and design effects; however, it can also
produce regression estimates for multistage stratified samples
(Hidiroglou, Fuller and Hickman, 1980).
TABLE 3
Selected Characteristics of Variance Estimation Programs
PROGIWl
1. HES Variance and Cross
Availabi 1i ty
.
I
Genera I lty
B
A
C
I
I
Documentation
Users' Guide
tabulation Program
(National Center for
Health Statistics
2. STDERR-SESUDANN
(Research Triangle Inst.)
I
I
I
4. KOTAB2-VTAB
(National Central Bureau of
Statistics, Sweden)
I
5. Current Population Survey
Variance Progam
(US Bureau of the Census
6. SUPER CARP
lfVery little lf
I
3. Consumer Expenditure
Survey Variance Program
Users' Guide
In Swedish
Tech. Report; Internal
Documentation
I
I
Manual
I
I
Part of OSIRIS IV
Manual
(Iowa State University)
7. OSIRIS IV: PSALMS
(Univ. of Michigan)
e
e
N
0
e
e
e
--
TABLE 3 (continued)
8. OSIRIS IV: PREP
(Univ. of Michigan)
I
I
I
9. Variance/Covariance System
for the Canadian Labor
Force Survey
(Statistics Canada)
Part of OSIRIS IV
Manual
Technical Report
10. GENVAR
(US Bureau of the Census)
I
I
"Draft"
11. CLUSTERS
(World Fertility Survey)
I
I
Users' Manual
12. Consistent System (CS)
(Laboratory of Architecture
and Planning, MIT)
I
I
(Manuals, Reference,
Tutorial, Technical)
I
13. JES Summary System
(E.S.C.S. - U.S.D.A.)
14. System 4204
(E.S.D.S. - U.S.D.A.)
15. Table Producing Language (TPL)
(Bureau of Labor Stat.)
16. Generalized Survey System (GSS)
(Australian Bureau of Statistics)
I
Users' Manual
I
Users' Manual
I
Users' Manual
I
Manuals (users', operations, external reference specification)
~-.l
TABLE 3 (continued)
I
17. SPSS-Splithalf Procedure
(Australian Bureau of Statistics)
18. Rothamsted General Survey
Program (RGSP)
(Rothamsted Expt'l. Station)
I
Users' and systems
documentation
I
Manuals, Introductory
Guide
lA = specific to a particular survey
B = useful to a particular kind of sample design
C = useful for several kinds of sample designs
N
N
e
e
e
23
Other programs developed very recently are GENVAR80 and QTJACK80,
both by the US Bureau of the Census.
GENVAR80 computes sampling vari-
ances of functions of estimated population sums based on the first order
Taylor expansion while QTJACK80 estimates functions of population means
and their sampling variances based on the Quenouille-Tukey jackknife
procedure (Bureau of the Census, 1980).
Both programs are available
from the US Bureau of the Census.
1.5
A Review of Indirect Fertility and Mortality Measures From A
Single Sample Survey
Several indirect fertility and mortality measures have been
developed to date.
The following sections present a review of the
methodology and some problems associated with the most commonly used
ones.
For the section on indirect mortality estimators, only child
mortality measures are included since these are the ones considered in
this dissertation.
1.5.1
Indirect Fertility Estimators
The simplest kind of indirect fertility estimator derived from
sample surveys is the number of children ever born.
This estimator
represents cumulative fertility and estimates the total fertility rate
if computed for women who have completed childbearing - i.e., women
aged 50 or over.
It also measures the extent of childlessness in a
population and provides information on the distribution of mothers by
number of children (Shryock and Siegel, 1976).
The main advantage of this type of measure is that there is no
time element involved in the question asked of the respondent, and
24
hence no possibility of errors of time reference.
However,
error~
in
the information given may arise as a result of faculty memory of the
woman, especially an older woman who bore her children a long time ago,
or because of lack of knowledge on the part of a respondent who may be
a person other than the woman involved (Shryock and Siegel, 1976).
There is some evidence that the accuracy of reporting the number of
children ever born tends to vary inversely with the age of the women,
with the older women tending to under-report their number of children
(Brass et al., 1968).
Another method of estimating the total fertility rate was
developed by Coale and Demeny, who based their calculations on the
young group of females for whom information is more reliable and more
recent (Brass, 1975).
lity rate,
P,
According to their approach, the total ferti-
is expressed by
F
where
P3
and
P
2
=
(1.11)
are the mean number of children ever born to
women aged 25-29 and 20-24, respectively.
This estimator provides a
fertility measure under the following conditions:
(1) the fertility
at ages 15-29 has been constant in the recent past, and
(2) the age
pattern of fertility in the population conforms to the "natural" fertility pattern found in populations practicing little birth control.
William Brass (Brass, et al., 1968; United Nations, 1967)
developed a technique
from data on
of~tjmating
the age-specific fertility rates
(1) the number of children ever born, and
tive data on number of children born during
(2) retrospec-
a specified interval of
25
of time, usually one year, prior to the taking of the census or the
survey.
His method was designed primarily for use in Africa and is
applicable under the following assumptions (Lingner, 1974):
1.
Fertility patterns have been constant over time.
2.
The level of fertility is accurately reflected in the number
of children ever born as reported by females under 30.
3.
The age pattern of fertility is accurately depicted by the
data on number of children born in the past year, or other
intervals.
The method is based on the fact that the number of children ever born
to a woman of a given age should be equal to the sum of her agespecific fertility at all previous ages.
A description of the method
is given in UN Manual IV ((1967, p. 32) as follows:
The method requires the estimation of the average value of
cumulative fertility by age over the same age intervals
(usually 15-19, 20-24, 25-29, etc.) for which the average
number of children ever born is reported. It is then
assumed that the source of the difference between the
estimated average value of cumulated fertility at the
younger ages (such as 20 to 24 or 25 to 29) and the
average number of children ever born at these ages is an
erroneous perception of the reference period by the respondents. The multiplier that would be needed to bring cumulative fertility at the younger ages in line with the
reported average number of children ever born is determined,
and the reported numbers of births at all ages are multiplied by this factor.
Computationally, the Brass fertility estimator is of the form
f~ = f.
1
1
P.
x
1
F.
(1.12)
1
where
f~
1
=
adjusted age-specific fertility rate for age interval
i',
f.
1
=
average number of births in preceeding year per woman in
26
age interval
P.
1
=
i;
average number of children ever born among women in age
interval i',
i -1
i-I
f. + w. f. , where
f. IS the cumulative fertility
F. =
1
1 1
j=O J
j =0 J
at the beginning of the interval and w.1 is the multiply-
I
I
ing factor for estimating average value fertility.
The ratio
is u5ually substituted for
P./F.
1
1
in (1.12), and
serves as a correction factor that makes the fertility rates consistent with the average number of children ever born reported by women
20 to 25.
The Brass method has been found to produce reasonably accurate
results only when net age misstatement is small; distortions in the
age distribution resulting from misstatements of age can seriously
bias the results.
Furthermore, the estimates are valid only to the
extent that the assumptions hold; in many real situations, they will
not (Lingner, 1974).
Since methods fore5timating fertility rates by using data on children over born classified by age are not robust to the presence of substantial age misreporting, Coale
Hill and Trussell (1975) developed
a technique which would overcome this problem.
Their method provides
the estimates of age-specific fertility rates from a single census or
survey in which a question is asked on the number of children ever
born per woman and in which the responses are tabulated by duration
of marriage.
The robustness of the method rests on two bases:
(1) the
apparent lesser distortion that occurs in some populations in reporting
27
duration of marriage than in reporting age, and
(2) the greater uni-
formity of fertility structures based on duration of marriage than of
age-specific fertility structures.
This procedure is however appli-
able only to populations in which there is little practice of voluntary birth control and in which only a small proportion of births occur
outside of marriage.
An additional limitation is that the method is
not applicable to populations like that of India in which many marriages occur before the age of menarche and where there is no close
relationship between the age at marriage and the age at which exposure to the risk of pregnancy begins (Hill, Zlotnick and Trussell,
1981).
Another approach to fertility measurement from survey data
requires data on the number of own children under 10 years living in
the household, tabulated by age of children and age of mother.
procedure was suggested by Grabill and Cho (1965).
uses the fact that children at each age,
of births which occurred in the year
the survey was taken.
x,
t-x,
This
Their approach
represent the survivors
where
t
is the year when
By applying the appropriate reverse survival
ratio and adjusting for the proportion of children not living with
their mothers, the number of births occurring in each of the 10 years
preceeding the survey can be estimated.
data for calculating birth rates.
This provides the numerator
Denominator data is obtained in
the same manner - i.e., by reverse surviving the number of women at
each single year of age.
Following the notation of Hill, Zlotnick and
Trussel (1981), the number of children born in year
is the time of the survey) to women aged
mula
a-x
t-x
(where
t
is estimated by the for-
28
(1.13)
where
a
C = number of children aged
x
x
whose mothers were aged
a
at
the time of the survey;
U = total number of unmatched children aged
x
C
total number of children aged
x
IL
X
=
x;
probability of surviving from birth to age
However, since
the year
x·,
[x,x+I).
represents the number of births occurring during
(t-x-l,t-x)
to women aged
a
and
a+ I
at time
have to consider births to women whose ages ranged between
a-x+l
(exclusive) during the year
a-I
t-x
done by taking the average of
M
(t-x-l, t-x).
Ma - x .
and
t,
we
a-x-l
and
This adjustment is
This average is
t-x
Na-x .
t-x
Correspondingly, the number of women whose ages range between
denoted by
a-x
and
a-x+l
at time
t-x
a-x
t-x =
W
where
a x
wt-x
is estimated by
IV
a
t (lL a-x IlL)
x
a
(1.14)
=
number of women aged
at time
=
reverse survival ratio for women.
t
of the survey;
Since the number of women used as denominator for births in the
COill-
putation of age-specific fertility rate is the mid-year population during the year
taken.
(t-x-l,t-x),
the average of
wa - x
t-x-l
If this mid-year population is denoted by
age-specific fertility rate for women aged
a
and
a-x
Pt -x '
wa - x
t-x
is also
then the
during the year
29
[t-x-l,t-x)
can be estimated by
a-x
t-x
f
(a) =
a-x
t -x
P
t-x
N
(1.15)
A limitation of the above method is that it is useful only where
data on own children living in the household are available.
In addi-
tion, since "own" children may include adopted children or stepchildren, practices in regard to marriage and adoption may produce variations in the number of own children by age of mother.
Recent surveys
which used this method of estimatng fertility rates are the 1978
Colombia National Household Survey (POPLAB Staff, 1980a) and the 1979
Mexico National Fertility and Mortality Survey (POPLAB Staff, 1980b).
1.5.2
Indirect Mortality Estimators
William Brass (1968) was the first to develop a procedure for converting statistics on the proportion dead of children ever born
reported by women in the age intervals 15-19, 20-24, etc., into estimates of the probability of dying before attaining certain exact childhood ages.
Following the notation in the literature, he developed a
procedure to convert
D(i)
values into
q(a)
where
D(i)
is the
proportion dead of children ever born to women in successive five-year
age intervals
(i=l refers to the interval 15-19, etc.) and
the probability of dying between birth and exact age
a.
q(a)
is
The basic
form of the Brass relationship is
q(a)
=
k(i)D(i)
(1.16)
30
where
k(i)
affecting
is a multiplier to adjust for non-mortality factors
D(i).
ing pairs of
Brass found that the relation between correspond-
D(i)
and
q(a)
is primarily influenced by fertility
conditions, particularly the age at onset of childbearing.
To evalu-
ate his mUltipliers, he generated fertility schedules by using the
third-degree polynomial
f(x)
where
f(x)
= c(x-s) (s
(l.17)
is the age-specific fertility rate at age
mines the fertility level and
in the population.
k(i)
+ 33_x)2
s
x,
c
deter-
is the earliest age at childbearing
Accordingly, the selection of a Brass multiplier
is keyed to a fertility parameter,
P/P ,
2
the ratio of the
average parity of women aged 15-19 to women aged 20-24.
This parity
ratio was found to be highly correlated with the age at onset of childbearing and is easily obtained from the data necessary for the calculation of the proportion dead among children ever born to women of fiveyear age intervals.
The Brass technique assumes constant age-specific
fertility and infant and childhood mortality in recent years.
Another approach to estimating childhood mortality was developed
by Sullivan (1972).
He showed that if actual fertility schedules
instead of the polynomial used by Brass were used to generate fertility
schedules, multiplying factors based on linear regression yielded better results than the Brass multipliers.
As in the Brass technique,
the method assumes constant fertility and mortality in recent years.
It is designed for use with data on the proportion dead of children
ever born tabulated by age intervals of mother (age model) and by
marital duration intervals (duration model).
31
The data for Sullivan's age model were generated by pairing 10
life tables from each of 4 mortality patterns given in Coale and Demeny
(1966) with 65 fertility schedules.
A regression analysis was done
relating the ratio of selected pairs of
q(a)/D(i)
parameter.
for fixed values of
and
i)
and
D(i)
(i.e.,
to a fertility schedule
The ratio of the average parity of women aged 20-24 to
women aged 25-29
q(a)/D(i)
a
q(a)
(P2/ P3)
was found most highly correlated with
and was therefore the fertility parameter used in the model.
Regression coefficients were then calculated according to the equation
(l.18)
where
q(a), D(i), P
2
and
P
3
are as defined earlier.
Sullivan also developed a duration model, based on data on the
proportion dead among children ever born tabulated by marital duration
intervals.
For this model, the ratio of the average parity of women
aged 15-19 to women aged 20-24
(P /P )
l 2
was found to have greater
explanatory power in the regression equation and was therefore the fertility parameter used in the model.
For the duration model, the
regression equation used in estimating
q(a)
is of the form
(1. 19)
where
D(i)
is the proportion dead of children ever born to women of
marriage duration
i.
A third set of multipliers for the estimation of childhood mortality was developed by Trussell (1975).
Using the same approach as
32
that of Sullivan but a different and larger data base, Trussell
derived a regression equation of the form
(1.20)
Like the Brass and Sullivan techniques, this method of estimation
assumes that fertility and childhood mortality have remained constant
in the recent past.
No considerable differences were found between
the resulting estimates using the three techniques.
CHAPTER 2
Application of Taylor Series
Linearization Method to Indirect Demographic Estimator5
2.1
Introduction
This chapter deals with the derivation of the mean squareerror3 of
the four estimates considered, using the Taylor series linearization
method.
In particular, it illustrates the use of the Woodruff lineari-
zation procedure which provides a computational shortcut in the computation of the mse's.
The ultimate variance estimation technique is used
to derive the mse of the estimators when a complex design is used. All
derivations are applicable to any multi-stage sampling design with stratification of the first stage.
2.2
The Woodruff Linearization Procedure
One of the problems encountered with the application of the Taylor
5eries linearization method of variance stimation is the evaluation of
the approximate variance when the number of random variables defining
f(~)
is large.
squared terms (variances) and
~~en
t
random variables in
then the variance estimator as shown in (1.6) will consist of
f(~),
t
For example, if there were
t
t(t-l)
cross-products (covariances).
is say, greater than 3, this entails very tedious computation5.
A computation short-cut to evaluate
MSE[f(~)]
using the
Taylor series method was developed by Woodruff (1971).
Given the
34
vectors
x
and
in cases when
X defined in Section 1.4, his procedure is useful
X.
J
in
is a total and each component of
~
~
is the
simple unbiased estimator,
(2.1)
where
x
is the observation used to produce the
jhaS
total from the
stratum, and
element in the
IT
haS
a
th
PSU in the
. th
J
h
th
estimated
primary
is the selection probability (i.e., the reciprocal
of the sampling weight) for the
haS
th
element.
From (1.5) and (1.6), we have
MSE [f(D] -
E~
(x.-x.)
.
J J
J
= tI.
J
where
(J • .
JJ
and
respectively.
tf(~T
3X .
af(vr
3X.
J
t t
JJ
J
2
+
(J ••
I I
df(~) 3f(~)
3X 3X
j k
j <k
0
are the variance and covariance of
jk
X.
J
and
By definition,
MSE [f(~)] ;
var~
af(~)
(2.2)
ax.
J
Substituting (2.1) in (2.2), we get
MSE [fCD] -
var~
3f(~)
ax.
=
Var
r
Ih aI
b
b
ha
I
S
ha
I I I
h a
J
h
H~
Zha~
IT
haS
13
Xjh"~
IT
haB
(2.3)
e
35
L L L ZhaJ
~~
b
ha
- Var
h a
(2.4)
S
where
"
t
=
ZhaS
af(~)
L
j
ax.
x
J
(2.5)
jhaS
and
ZhaS
ZhaS = IT
haS
The variable
ZhaS
in (2.5) is the Taylor series linearization
(TSL) computed for each observation.
square error of
f(~),
(2.6)
To calculate the estimated mean
we use the estimated
TSL
from the sample-
i. e. ,
t
Z
haS
=
L
j
af(~)
(2. 7)
aX j
and
(2.8)
which leads to
mse[f(~)]
(2.9)
Hence, after the estimated values of
zhaS
has been computed for each
member of the sample, the estimated mean square error of
f(~)
can be
derived and computed by simply applying the formula for the variance
estimator of the single variable
zhaS'
corresponding to the sampling
36
design used in the survey.
books.
This is found in standard sampling text-
For example, if we assume simple random sampling on the set
we have
of
mse[f(~)]SRS = n(1-f)5
where
f
n/N,
(2.10)
z
the finite population correction, and
nln Z7 2 - [nI
s
2.3
2
2
Z
.
1
1
.
1
z~
)2
1
(2.11)
=
n(n-l)
The Ultimate Cluster Variance Estimator
The estimation of the variance of complex, non-linear estimators
for multi-stage designs involves two steps.
First, the complex non-
linear estimator is reduced to a simpler, linear and approximate form
by the use of the linear terms of its Taylor series expansion.
Second,
the ultimate cluster variance is obtained for the approximate linear
form of the estimator.
The ultimate cluster variance estimator was developed by Hansen,
Hurwitz and Madow (1953).
An ultimate cluster consists of all the
listing units in the sample from a sample primary sampling unit (PSU),
regardless of how sample elements are chosen within the PSU, as long
as the method of selection is probability sampling.
If we let
a = the number of PSUs in the sampling
a x
x =
a
a
Ya
a
y =
a
I
I
a
a
e
37
where
Y
a
.
x
a
and
y
a
(a
= 1,2, ... ,a)
are unbiased estimators of
X
a
and
respectively, it can be shown that for a multi-stage sample where
PSUs
are independently selected with unequal probability and with
replacement,
a (x -x)(y
cov(x,y) = L
a
a
-y)
a(a-1)
is an unbiased ultimate cluster covariance estimator for
For the special case where
x
=
Cov(x,Y).
y,
- 2
a (xa-x)
var(x) = L a(a-l)
=
a
is an unbiased ultimate cluster variance estimator of
Var(x) =
Cov(x,x).
For the same design but with stratification, it can be shown that
N
cov(x' ,y')
=
L COy (~, Yh)
h
~
N
=
L
L (x hCt -x h ') cYhCt -Yh')
Ct
a h (~-1)
h
(2.12)
is unbiased for
N
Cov(x',y')
L Cov(xh,yh )
h
where
(2.13)
38
N
Xl
=
L x'h ,
the estimated sample total,
h
Xl
h
II
ha
~ x'ha
=L
=
II
,
the estimated total for the
h
th
stratum
ha
a
the selection probability for the
th
in the
PSU
h
th
stratum
b
x' =
ha
ha
L
S
x
ha13
,
the estimated total for the
IIS(ha)
the
IIS(ha)
=
h
th
Nth
PSU
in the
th
PSU
in
stratum
selection probability for the
~
a
h th
13
th
element, given the
stratum was chosen .
The same definitions hold for
y.
Similarly,
H
var(x ' )
=
I
var(~)
h
a
H
~
h
L (xha -xh)2
a
ah(~-l)
(2.14)
is unbiased for
H
Var(x ' ) =
L Var(Yh)
.
h
Applying the above results to derive the estimator for the mean square
error of the complex function,
lineari zat ion,
for
f(x),
we substitute our Taylor series
in the above equations and get
39
~ ~I
= var ~
mse [f(~) ]
ha
b~
J
ZhaB
a
h
I(z* -Z*I)
H
ha h
a
= I a (~-1)
h
h
.
(2.15)
Expanding terms and expressing (2.15) in computational form, we get
ah
mse [f (~)]
=
I __aI zh~ {Ia Zha}
H ah
where
;
(2.16)
ha
I
zha =
2
~-1
h
b
ah
is the selection probability for the
B
element.
2.4
Derivation of Mean Square Errors
In the following sections, the estimators of the mean square
errors of four selected indirect fertility and mortality estimators
will be derived, using the concepts discussed above.
2.3.1
Notation
The following notations will be used in the succeeding sections:
the number dead among children ever born to the
women in the
x1haB,x4haB
x7haB,x10haB
a
th
PSU
in the
th
B
stratmn if the
woman is in the age-group 15-19, 20-24, 25-29 and 30-31,
=
respectively
o
if the woman is not in the age-group 15-19, 20-24,
25-29 and 30-34, respectively
40
the number of children ever born to the
in the
X2haS'X5haB,
,x
x
ShaB llhaB
a
th
PSU
in the
h
th
woman
stratum, if she is
in the age-group 15-19, 20-24, 25-29 and 30-34,
..
respectively
o
if the woman is not in the age-group 15-19, 20·· 24,
25-29 and 30-34, respectively
if the
h
X3haS'X6haS, _
th
woman in the
a
th
PSU
in the
stratum is in the age-group 15 - 19,
20-24,
25-29 and 30-34, respectively
x9haS,x12haB
o if the woman is not in the age-group 15-19, 20-24,
25-29, and 30-34, respectively
2.4.2
Sullivan's Child Mortality Estimator
As shown in (1.18), Sullivan's child mortality estimator is of the
form
q (a)
D(i) = Ai
+
Bi [
r;-2J
p
or
(2.17)
where
q(a)
probability of dying between birth and age
D(i)
proportion dead of children ever
age-interval
etc.)
i
(i = 1
a
born to women in
refers to age-group 15-19,
41
P2
=
no. of children ever born to women 20-24
no. of women aged 20-24
P3
=
no. of chi ldren ever born to women 25-29
no. of women aged 25-29
A. ,8. = constants (regression coefficients) corresponding
1
1
. th
to women in the
1
age-group
Sullivan established estimators for three ages, namely
q(S).
and
These three estimators are similar in form and differ only in the
age-group of the women providing information on
q(2),
q(2), q(3)
O(i).
In the case of
the corresponding equation is
(2.18)
Using the notation presented in Section 2.2 and Section 2.3.1, an
estimator for
q(2)
for a multi-stage sample with stratification at
the first stage will be
X'
q (2) =
X'
where
0(2)
x'
5
(2.19)
X'
5
4
=
4
XI
5
P2
= XI'
X'
8
and
x'9
6
Taking partial derivatives with respect to each
3q (2)
Clx'
4
aq(2)
ax I
5
1
xI
4
= -
=
~A
x'4
+ 8
2 XI
2
L 5
(
X')
-1P
-4
x I 2 x' J
5
5
=
-1
XI5
x'x'x~
5 9
X'X'X '
568
t
=
giQ
x'
4
C21 -8 20(2)
~~
P3
X! ,
J
we get
42
aq(2)
ox'6
aq (2)
ax I
=:
=:
8
aq (2)
=:
dX 9'
x 5'X')
9
x'6 [B 2 X'X'
6 8
-1
X'X')
5 9
-1
[B 2 X'x'
XI
8
6 8
B x5I X')
9
x'9 [ 2 x 6xS
1
=:
P2
-1
0(2)
-,
B
x6 2
P
A
3
=:
P
2
0(2)
-,
Xs B2
P
-1
A
3
=:
P
2
1
-, B2 0(2)
Xg
A
P
3
To determine the Taylor series linearization for each observation in
the sample, we substitute the partial derivatives in (2.7) and get
ZhaS
q (2)
t
4has
x'4
-
P
+
B 0(2)
2
2
A
P
3
X5ha~
x'
t
5
5haB _ x 6hoB _ x ShoB ,
x'5
x'6
An estimate of the mean square error of
X'S
q(2)
X9ha~
x
'·
9
(2.20)
is then derived by
determining the ultimate cluster variance estimator of
zhaS
for a
multi-stage sample with stratification in the first stage and with
replacement.
Applying (2.16) we have
ms e [ q ( 2)] c 5
b
IazhaS
and where zhaS is as defined in (2.20). In
S TI haS
most surveys however, PSUs are drawn without replacement rather than
where
Z*
ha
=:
with replacement.
The effect of using the above formula under this
assumption is to slightly overestimate the mean square error.
43
For a simple random sample, the corresponding estimator for the
mean square error is
mse[q(2)]
where
s
2
z
= n (I -f) s
51'S
2
z
is given by (2.11).
To estimate
q(3), Sullivan uses information provided by women
aged 25-29 for the proportion dead among children ever born.
The
corresponding equation is
P2~
q (3)
P
Using notations presented earlier, the
(2.21)
.
3
estimator for a multi-
q(3)
stage sample with stratification at the first stage is
e
where
0(3)
=
x'
7
-I'
X
s
7
= -,
X
s
~A
3
+
x.x~
5 9
8 -,-,
3 x x
6 S
(2.22)
Taking partial derivatives, we get
3q(3)
3x I
5
= x'
3q(3)
3x'
6
= x'
3q(3)
3x I
7
= XI
39(3)
3x!
-1
x'
S
1
e
x·
A
q(3)
1
5
-1
6
1
7
=
x.x.xj
579
2
x'x'
6 S
[3
I XSX7X~
3
, ,2
x x
6 S
L
t
t
x'
7
x'
3
-+8
8
x'
7
x'
S
3
-+
1
= -x , 8 3 0(3)
5
" .:.l
B
x
6 3
x'x.xj
579
3 XIX' 2
6 S
28
3
x.x.x~
579
XIX'
6 S
2
0(3)
P2
A
P
3
P
2
A
P
3
_ 1
- XI q(3)
7
A
44
= X'
ft
-1
t
-1
R
=
3
A
)
r-
z
A + B
3
('3) + 8
XI8
p,~
P
3
3
A
P
3
x'x'xl
3
0(3)
P
3
~~
0(3)
3q (3)
S 7 9
1
=
X' IB 3
2
3x'
9
9
x'x'
6 S
+ 8
P
3
1
= -, B
L
x
9
3
0(3)
P
2
A
P3
Substituting these partial derivatives in (2.7), the Taylor series
linearization for each sample observation is
Z
haS
q(3)
:=
IX 7haS
1
-
L
x'
7
_
X8ha~
x'
8
t
~
ShaS _ x 6haS _ x 8hct,f3_ + x 9haS
x'
x'
x'
x'
5
6
8
9
As in
q(2),
an estimate of the mean
square error of
(2.23)
.
q(3)
is
derived by determining the ultimate cluster variance estimator of
as well as its simple random sample variance estimator.
ZhaS'
is done by applying (2.16) and (2.10), where
zhaS
This
is as defined in
(2.23) .
For
q(5),
Sullivan's estimator takes the form
(2.24)
where
0(4)
is the proportion dead among children ever born to women
aged 30-34, and
A
4
to this age-group.
and
8
4
are regression coefficients cooresponding
Using our notation, an estimator of
q(S)
for a
4S
multi-stage sample with stratification is
(2.25)
where
0(4)
After taking the partial derivative of
q(S)
with respect to each
x!
J
and substituting them in (2.7), we get the following Taylor series
linearization for each sample observation
Zhetl3
= q(S)
t
Shet13 _ x6ha13 _ xShet13 + X9het~
x'
x'
x'
x'
5
6
S
(2.26)
9
~
The determination of
mse[q(S)]
dome samples proceed as in
q(2)
for both multi-stage and simple ranand
q(3).
It is interesting to note that the linearizations for Sullivan's
estimators follow a definite pattern as one goes from
q(2)
to
The first term of the linearizations is always a product of
q(S).
q(a)
an expression which is the difference of the variables defining
and
D(i).
The second term of the linearizations in always the product of
D(i)B P2/ P3'
i
which is the second term of the estimator itself, and
an expression made up of variables defining
P2
and
P .
3
Since the
form of the linearization is closely related to the estimator itself,
this makes the linearization formulas easier to remember and more convenient to use.
46
Sullivan also developed estimators of
q(a)
using the duration
model, which is based on data on the proportion dead among children
ever born tabulated by marital duration intervals.
Since the form of
his estimators using this model is exactly the same as that of the
age model discussed above, then the formulas for computing the mean
square error of the estimators will be exactly the same, except that
some variables will have different meanings.
2.
~.3
Trussell's Child Mortality Estimator
As given in (1.20), Trussell's child mortality estimator is of
the form
q(a)
D(i)
where
C.
1
and
q(a), D(i), P
2
and
is the average parity of women 15-19.
1
.th
are regression coefficients corresponding to the
PI
A., B.
are as defined earlier,
1
and
1
age-group,
Although Trussell
established estimators for four ages, the mean square errors of only
three estimators will be derived here, namely those of
and
q(S).
q(2), q(3),
Like Sullivan's, Trussell's estimators are similar in form
and differ only in the age-group of women providing information on the
number dead among children ever born
(D(i)).
In the case of
q(2),
the corresponding equation is
(2.27)
Using notations established earlier, an estimator of
q(2)
for a
47
multi-stage sample with stratification is
(2.28)
0(2)
where
=
x'4
x'
2
XI'
5
PI
= XI'
3
x'
5
P2 = x'
6
x'8
x'9
and
Taking the partial derivatives of (2.28) with respect to each
we get
aq(2)
ax'2
ag (2)
ax'3
ag (2)
ax'4
-0(2)
x'5
ag (2)
ax'5
ag (2)
0(2)
ax'6
X6 l
aq (2)
ax'8
=
-c
2
x'8
aq(2)
ax'9
Applying (2.7), the Taylor series linearization for each sample
observation will be
x! ,
J
48
A
= q
(
+
2 ) [X4haf3
x'
4
XShaf3 ]
x'
5
X2haf3 _ x3haf3 _ xShaf3
[ x'
x'
x'
235
0(2)
+
X6haf3 J
x'6
X~haf3]~
XShaf3 _ x 6haf3 _ x Shaf3 +
x'6 8
x' 9
x'
[ x'5
The estimates for the mean square error of
q(2)
(2.29)
for both multi-stage
and simple random samples can be evaluated by computing
values
based on the above formula, and then applying Equation (2.16) and
(2.10).
For Trussell's
q(3)
and
procedures are applied as in
las for
zha(3
q(S)
q(2).
estimators, exactly the same
The resulting estimators and formu-
are summarized in Table 4.
Like Sullivan's estimators,
the linearizations for Trussell's estimators follow a definite pattern
as one goes from
q(2)
to
q(S),
and has a form very closely related
to the estimator itself.
2.4.4
Coale-Oemeny TFR Estimator
The Coale-Oemeny estimator of the total fertility rate,
TFR,
is
of the form
(2.30)
Where
P2
and
are as defined in the preceeding sections.
our earlier notation, an estimate of
with stratification is
Using
TFR _
for a multi-stage design
C D
e
e
e
TABLE 4
Trussell's Child Mortality Estimators and Corresponding Taylor Series Linearization
"
t
q(a)T
x'
q(2) =
-4
Xs
ZhaS
x'x'!
5 9 I
x 2 x6
A2 + B2 Xtx,+ C2 x'x' I
3 5
=haB = qt2)
6 8J
shas
[X 4h
,as _ X
]
x
x'
4
+ D(2)
5
f2
~ [X 2haS _ x3ha13 _ x ShaS
X6haB]
,+
,
A
,
P
X
2
p
,
X
2
X ShaB
+ C
2
2-;:;--
x'
[
X
q(3)
I
-2
x'
8
~
A
+B
3
x' x'
x' x 'i
2 6+ C ~I
3 x'x'
3 x'x'. I
3 5
6 8~
ZhaS
x6
s
_ x 6haB _ x ShaB + x 9haB
x'
5
P3
X
3
x'
6
Xl
8
]~
9
8has
=q t3) [X 7h
,as _ X
]
x
x'
7
+
0(3) [B
8
3
X
PI
A
P
[
x
x'
C3-;:;-2
P
3
x'
2
2
P
+
x
x
2haB _ 3haS _ ShaS + 6ha(3
X shas
[
x'
3
x'5
5
x'
6
l
J
_ x6haS _ x 8haS + x9has
x'6
x'8 9
x'
j-j
.;:,.
\0
TABLE 4 (continued)
X10
i
q(S) = - -
xiI
~
A +B
4
, x ,
xZ
6
--+
3S
4 x x
C
3
xrx~
5 9
ZhaB = qrS) [XlOrhaB _ Xllhas1
xl o
x'11
)
XiX'
6 8
+ D(4)
I.
t
4
+C
i\P
X ZhaB
A
z
[
Pz
4
~
p3
_ x 3haS _ X ShaB + x 6ha(3 ]
X'
X'
Z
rXShaS
l
,-
X
X'
X'
356
s
x6ha (3
,
x6
xShaB
-
,
x8
X9has1-J
+
,
x9 )
lil
o
e
e
e
Sl
(x I Ix I )
8 9
2
TFR _ = ~-:--;--:--::-- =
(x I Ix I )
C O
S
Taking the partial derivatives of
6
XI
S
I
2
X
I
6
(2.31)
12
xSx g
TFR _
with respect to each
C O
x! ,
J
we get
=
-1
x'5
1
=X'
6
2
=
x'S
=
x'9
-2
Substituting thepartial derivatives in (2.7), the resulting Taylor
series linearization is
t
Shas _ x6haS _ 2xShaS
x'
x'
x'
5
6
S
+
2X9has~
x'
(2.32)
9
As with other complex estimators, an estimate of the mean square error
of
TFR _
for a multi-stage sample with stratification is derived by
C O
computing
zhaS
values for each element in the sample based on the
above formula, and then applying (2.16).
For the mean square error of
a simple random sample, Equation (2.10) will be applied.
52
2.4.5
Own-Children Method of Estimating TFR
The estimation of the total fertility rate using the own-children
method involves the estimation of the age-specific fertility rates of
women aged 15-49 by single years of age, and then summing them up i. e. ,
49
TFR
= L
f(a).
a=15
However, whereas
f(a)
is simply a ratio when computed directly, it
takes a more complex form when estimated by the own-children approach.
From (1.15), the age-specific fertility rate corresponding to
women aged
a
during the year
[t-x-l, t-x)
is
Na
ft_x(a)
where
a
Nt
-x
and
pa-x
t-x
=
t-x
a-x
p
t-x
(2.33)
are as defined earlier.
After making further
substitutions, (2.33) could be expressed as
f
a
=
t-x
L
1 x
ta-
a-x[1 La-~IJ
x [1 La_x] + W
t-x-l
L
1 a
(2.34)
t-x 1 La+l J
To simplify the notation in deriving the linearization, we drop the
subscripts for the moment and use the following substitutions:
S3
Notation in Original
Equation
New Notation
f(a)
x
. Uo
C , Ca , Ca+1
x
x
x
CO' C1 , C2
U
a-x
Wa-x
Wt _x _1 ' t-x
L
WI' W2
L
1 a-x 1 a-x
1Lx'
L
' r
1a
l'a+l
Applying the above notation, (2.34) can be written as
f(a)
where
k ' k
O 1
and
k
2
(2.35)
=
are constants. Taking the partial derivatives
with respect to each random variable, we have
-U
-Uo(Uo+C o) (C 1+C 2 )
af(a)
0
=
=
de
2
Co(Uo+C
o
o)
koCo(klw1+k2w2)
r, 1 + Co
Uo)
af(a)
f(a)
=
=
C
+C
dCl
ko(klw1+k2W2)
1 2
f(a)
S4
Uo)
Co
f(a)
afea)
=
= cl+c
ac
kO(klw1+kZwZ)
z
z
[1 +
-k
-k l (Uo+C o ) (C +C )
af(a)
1 Z
1
=
=
z
k
w
+k
w f(a)
aWl
1 l z Z
koCo(klwl+kzwz)
-k
-kZ(UO+C O) (Cl+C Z)
af(a)
Z
fCa)
=
=
z
aw z
klwl+kzw z
koCo(klwl+kzwz)
Using the same notation system defined earlier, and applying the above
derivatives to (Z.7), the Taylor series linearization will be
ZhaS
=
f(a)
(2.36)
The estimation of the mean square error of
fCa)
for both multi-stage
and simple random samples proceed exactly as with other estimates.
Since the total fertility rate is the sum of the age-specific fertility rates, then the estimated mean square error of
49
I
TFR _
will be
OC
A
var[f(a) ]
a=15
under the assumption that variables are independent among age-groups.
CHAPTER 3
Application of Derived E5timators to Actual Data
3.1
Objectives of Analy5is
After the mean square errors of the selected indirect demographic
estimator5 have been derived, the next objective of this dissertation
is to compute them u5ing data from an actual 5urvey.
The Ea5t Java
Population Survey (EJPS) data is used for this study.
Three type5 of
mean square error estimators are computed as follows:
1.
mse[f(x)] cs - considers both the complex form of the estima~
tor and the actual design used in the 5urvey;
2.
mse[f(x)] srs - considers the complex form of the estimator
~
but assumes simple random sampling;
3.
mse[f(~)]bin
- disregards both the complex form of the esti-
mator and the sampling design used.
Since most of the esti-
mators considered here are commonly and erroneously interpreted as proportions, the binomial variance,
pq/n,
will
be computed to represent this estimator.
It should be noted that while the mean square error of an estimator,
f(~),
is defined as
(3.1)
the bias referred to in (3.1) includes only the sampling bias, which
56
is equal to zero when the estimator for a given sampling design is
unbiased.
It does not include biases arising out of non-sampling
errors like measurement errors.
In addition, since the indirect esti-
mators, especially the mortality estimators considered here, are
based on mathematical models, errors in the fitting of these models
are also note considered.
In particular, the regression coefficients
for Sullivan's and Trussell's estimators are treated as constants
since their values were derived independently of the survey for
which data for this study is taken.
To determine the extent to which the naive estimates of the mean
square error deviate from
which is considered here as
mse[f(x)]
- cs
the "correct" estimator, the results are compared by computing the
ratios
mse ratio
mse ratio
l
z
=
mse[f(x)]
- cs
mse [f (x)] srs
(3.2)
~
=
mse[f(x)] cs
~
mse[f(x)]b'In
(3.3)
~
The ratio in Equation (3.1) is also called design effect (deff), which
is defined by Kish (1965) as the ratio of the actual sampling variance,
taking into account the complexity of the sampling design, to the variance under assumptions of simple random sampling with the same sample
size,
n.
The estimates of the mean square error and the design effects are
computed for the total population, as well as for different domains.
Since the main objective in computing design effects is to determine
the conditions under which the effect of violations in sampling
design and form of estimator is greatest, the domains considered are
57
selected to ensure that different sample sizes and types of domain
(i.e., cross-class vs. segregated) are represented in the analysis.
Cross-class domains refer to sub-groups in the population which cut
across PSUs.
Examples of cross-class domains are categories of age,
sex and educational status since samples belonging to these sub-groups
are expected to be found in all
PSUs. Segregated domains are sub-
groups of the population in which each
PSU
members or all non-members of the sub-group.
consists of either all
For example, an analy-
sis according to urban-rural classification deals with a segregated
domain, since a
respondents.
PSU
would consist entirely of either urban or rural
The distinction between cross-class and segregated
domains is needed since each has different statistical implications
for survey estimates.
For the Coale-Demeny estimator, 11 domains are considered,
which include 9 regions and 2 for urban-rural grouping.
For the own-
children method of estimating TFR however, only the TFR for urban
areas is estimated because of the very tedious computations required
to obtain the estimate.
In the case of Sullivan's and Trussell's
q(a)
estimators.
separate computations are done to estimate male and female probabilities of dying, as well as children of both sex combined.
different domains are considered, broken down as follows:
Thirty-four
58
No. of
Categories
Domain
Urban-Rural
2
Educational Attainment
of Mother
5
Urban-Rural x Educ.
Attainment of Mother
10
Main Source of HOU3ehold Income
8
Region
9
TOTAL
Estimates of
34
q(2), q(3)
and
q(5)
and their corresponding mean
square errors and design effects are computed for each domain, as
well as for the total population.
Altogether, 630 different estimates
and design effects are computed for both Sullivan's and Trussell's
estimators combined.
All estimates are done using the West variant
of SUllivan's and Trussell's model, with the regression coefficients
being treated as constants in the derivations.
Lastly, a descriptive analysis of the design effects is done to
determine the conditions under which they considerably deviate from
1.
Among the variables deemed to affect the design effects are sample
size, the number of random variables in the estimator, the type of
estimator (i.e.,
V5.
q(2)
V5.
q(3)
vs.
q(5)),
the author (Sullivan
Trussell) and the sex (both sexes vs. male vs. female probabili-
ties).
The distribution of design effects according to these vari-
abIes are determined and possible relationships between them are
observed.
S9
3.2
The East Java Population Survey
The East Java Population Survey (EJPS) is a demographic survey
conducted by the Central Bureau of Statistics of Indonesia, in collaboration with the International Population Laboratories of the
University of North Carolina.
Its objectives are two-fold:
1) to
estimate the fertility and mortality levels for the province of East
Java and
2) to evaluate the completeness of birth and death report-
ing in the provincial civil registration system.
involve three rounds of field work.
The survey will
The first round was conducted
from May to June 1980, and was designed to provide baseline information from a sample of households.
It is data from this round that
was used for this study.
3.2.1
Sampling Design
The sampling design for the
EJPS
is a stratified three-stage
sampling design with desa (villages), census blocks and households
as primary, secondary and tertiary sampling units, respectively.
Stratification was done only for the first stage.
This process
involved explicit classification of desa into 33 primary strata and
implicit classification into substrata or zones of equal size.
each substratum, two
For
PSUs were selected by systematic sampling with
probabilities proportional to size.
A total of 1,238 desa were
selected of which 200 were in urban areas and 1038 were in rural areas.
From each selected desa, one census block was randomly picked.
Sample
households were then systematically drawn from each selected census
block.
The total number of households selected in the sample was
19,782. A summary of the
EJPS
sampling design is shown in Table S.
TABLE 5
Summary of EJPS Sampling Design
Sampling
Stage
Sampling
Unit
Primary
Desa
Sencondary
Census
Tertiary
Household
Block
Sample
Size
Stratification
Variable
1239 (200 urban,
1038 rural)
Kabupaten/Kotamadya
at primary level:
populationa densi ty implici ty
1238 (200 urban,
1038 rural)
None
Randomly select 1 census
block per sample desa
19782
None
Systematic sampling
(K* = 3.5 for urban,
for rural)
Sample Selection
Procedure
PPS systematic with
stratum
2
4.7
*k = systematic sampling interval for households within census blocks
Q'I
o
e
e
e
61
Sampling Weights
3.2.2
An ideal and desirable attribute of any sampling design is for
it to be self-weighting - i.e., one in which the population elements
have equal probabilities of selection.
In the real world of sample
surveys, however, designs which are precisely self-weighting are nonexistent and hence some degree of variation among selection probabilities is common and often tolerable.
However, once selection probabi-
lities begin to vary significantly, the design ceases to become selfweighting and the task of computing weights becomes necessary lest one
risks producing biased estimates.
The selection probabilities for the households covered by the
EJPS are not equal and had sufficient variability to warrant the computation of sampling weights.
Given the sampling design used for the
EJPS, the probability of selection for the
desa and
th
6
census block of the
IT
ha6y
2~a
Fha
h
th
y
th
household in the
ry,
th
stratum is
1
1
Nha
~a
=--x--x
(3.3)
where
the probability of selecting a
PSU
(desa) in a primary
stratum;
=
the second stage selection
block in the
1
~a
=
a
th
probability for a census
desa of the
stratum;
the third stage selection probability for any household in
a selected census block within the
stratum;
a
th
desa of the
62
the measure of size for the
a
th
PSU in the
stratum.
This number represents the target number of census block
for the
a
th
desa in the
h
th
stratum;
a.
F
h
N
ho:
the zone size, computed as
=
ha
where
~a/Gh
a
number of zones to be formed in the
h
th
G
h
h
th
is the
stratum;
the number of census blocks actually formed in the
desa of the
K
r
a
th
stratum;
the systematic sampling interval; this was equal to 3.5
for urban desa and 4.7 for rural desa.
Since the sampling weight is the reciprocal of the selection probability for a sample observation, it can then be computed as
1
IT
(3.5)
haSy
The computed weights for the
EJPS
varied considerably, with values
ranging from 59.5 to 1073.8.
Ninety-two percent of the weights, how-
ever, had values ranging from 300 to 400.
The extreme values occured
in cases where the target number of census blocks in the desa
fered considerably from the actual number.
dif-
Since the respondents of
the survey included every eligible member of each sample hou.'Sehold,
the sampling weight of every household member included in the survey
is the same, and is equal to that of the household weight.
sampling weights were applied in the computation of
PSU
These
totals.
63
3.3
Description of Computer Programs to Compute Mean Square Errors
The computer programs developed to compute the mean square errors
for the estimators considered here were all written in SAS. Generally,
the programs for all estimators have six basic steps as follows:
1.
Creation of dummy variables.
There should be as many dummy
variables as there are totals needed in the equation for the
f(~).
estimator,
For example, Sullivan's
q(2)
estimator
needs 5 dummy variables to be defined while Trussell's
needs 8.
q(5)
These dummy variables correspond to the notation
defined in Section 2.3.1.
2.
for each dummy vari-
Computation of weighted totals,
able, where
H ah
L LX!
h
and
w
ha13
=
J
is the sampling weight.
It should be noted that
irrespective of the number of stages used in the sampling
design, one needs only to keep tab of the stratum and the
PSUs
to compute the mean square error using the ultimate
cluster variance estimator approach.
3.
Computation of the estimate,
For example, for Sullivan's
f(~),
q(2)
using the weighted totals.
estimator, this step cor-
responds to the application of (2.19).
The use of the
weighted totals is necessary when the sampling design of the
survey is not self-weighting.
64
4.
Computation of the Taylor series linearization,
each observation.
ZhaS'
For example, for Sullivan's
q(2)
for
esti-
mator, this corresponds to the application of (2.20)
s.
Computation of
w
is as defined
haS·
where
earlier.
6.
mse[f(~)].
Computation of
For multi-stage designs. the following steps are followed:
a.
Sum up
over
zhaS
PSUs within each stratum - 1. e. , deter-
mine
b
b.
*2
zha
Determine
ha
z*
ha
I(3
and
(zha)2
zhaB
e
and compute
[f z*he. ) 2
a
(3.6)
a-I
h
for each of the
in the
c.
h
th
H strata, where
is the number of PSUs
stratum.
Sum up (3.6) over all strata.
mse[f(~)]cs
~
The resulting figure is
for a multi-stage sampling design.
For simple
random sampling.
For simple random sampling.
a.
Determine
and
and
compute
5
2
Z
using (2.11).
65
b.
Compute the sampling fraction,
f
=
n/N.
If a value for
N
is not readily available especially for domains, this can be
approximated by taking the sum of the weights.
c.
Compute
by applying (2.10).
mse [f(x)]
- srs
In the case of the own-children method of estimating
TFR,
two
other steps are necessary before the creation of dummy variables.
This
includes matching of mothers and children so that the data file would
include both mother's age and child's age on the same record. Secondly,
survival probabilities,
lL '
x
have to be computed by single years of
age for both women and children.
This is necessary for the reverse
survival of women and children to the specific year for which the
TFR
is estimated.
3.4
Sample Size
Since for the estimators considered here, the observational
units are women in different domains (i.e., age-groups), the sample
size varies from estimator to estimator.
of Sullivan's
for Trussell's
q(5)
For example. the computation
includes only women in the age-group 20-29 while
q(5), women in the age-group 15-29 are included.
The
same process is carried through in the computation of their mean square
errors.
Here, a new random variable,
Equation (2.7).
computation of
The values of
mse[f(~)].
Since
zhaB'
is defined as shown in
become the observations in the
zhaB
is equal to zero for women
outside of the age-groups involved in the computation of
the sample size is the total number of women with non-zero
f(~),
zhaB
then
66
values - i. e., the total number of women in the domain defined by the
estimator.
This is true for both
mse[f(x)]
- cs
and
mse[f(x)]
.
- srs
The different domains corresponding to each estimator are shown below.
Estimator
TFR _
C D
TFR _
DC
Domain
Women 20-29
Women 15-49*
q(2)S' q(3)S
Women 20-29
q(5)S
Women 20-34
q(2)T' q(3)T
Women 15-29
q(5)T
Women 15-34
*Women aged 16-50 at the time
of the survey were the ones
actually considered, but they
were reverse-survived for 1
year to determine the number
of women 15-49 in 1979, which
is the year for which TFR is
computed.
For the estimation of
mse[f(x)]b' = pq/n,
in
a different sample
size is considered, depending on the variable used for its denominator,
n,
For the estimation of
mse [ASFR (a)]b' ,
in
number of women in each age-group, a.
used in estimating
estimation of
mse [ASFR (a)]cs
mse[q(a)]b'
in
the denominator is the
This is the sample sample size
and
mse [ASFR (a)]srs'
however, the denominator can be defined
in a number of ways, depending on how one interprets
portion.
For the
q(a)
Here, two different denominators are considered.
as a proIn the
67
first case, the number of births 2, 3 and 5 years ago are estimated
by reverse-surviving the number of children aged 2, 3 and 5 years old
at the time of ths survey using the model life tables.
These numbers
represent the original size of each cohort of children and are used
as the denominators in estimating the naive
q(S),
respectively.
mse
of
q(2), q(3)
and
In the second case, the total number of chil-
dren ever born to women in the age-groups 20-24, 25-29 and 30-34 are
used as denominator for the estimated
respectively.
and
mse[q(a)]
mse
of
q(2), q(3)
Hence, while the sample size in estimating
srs
refers to the number of woemn defined by
the corre5ponding value of
n
for
mse[q(a)]b'
In
and
q(S),
mse[q(a)]
cs
q(a),
refers to either the
number of births or the number of children ever born.
The rationale
for using these variables is given in greater detail in Section 4.2.2.
CHAPTER 4
Findings
4.1
Results for Indirect Fertility Estimators
The estimated total fertility rate for East Java using the Coale-
Oemeny estimator is
4.38.
For the different domains considered, the
resulting figures vary from 3.96 of Region 2 to 5.39 of Region 4.
There is not much difference between urban and rural total fertility
rates, with the urban rates being slightly higher than that of the
rural areas.
These figures are shown in Table 6.
As expected, the mean square error of
TFR _
computed for both
C O
the total sample and the different domains is inversely related to
sample size.
This is true for both the actual design of the survey
and under the assumption of simple random sampling.
mse (TFR
C-
0)
cs
For example,
increases from .0163 for the total sample with a sample
size of 7789, to .5027 for. Region 9, with a sample size of 362.
Cor-
respondingly, the coefficients of variation vary from 3% for the
total sample, to 16% for Region 9 which has the smallest sample size.
The resulting design ,effects, comparing
mse(TFR
C-
0)
srs
mse(TFR
are low, varying from. 79 to 1.44.
C- 0) cs
and
Of the 12 design
effects computed for this estimator, four are below 1.
For the own-children method of estimating
TFR,
it is necessary
to first compute age-specific fertility rates (ASFR) by single years
69
TABLE 6
TFR Estimates Using the
Coale-Demeny Estimator and Corresponding Mean Square
Errors and Design Effects: East Java Population Survey, 1980
A
A
Domain
mse (TFR)
Complex
SRS
deff
Sample Size
n
TFR
Total Population
7789
4.38
.0163
.0153
1. 06
Urban
Rural
1874
5915
4.54
4.35
.0925
.0201
.0996
.0178
0.93
1.13
776
725
794
1102
1307
1194
785
744
362
4.55
3.96
4.16
5.39
4.54
3.86
3.98
4.71
4.46
.1240
.1214
.1375
.1301
.1054
.0734
.2055
.3620
.5027
.1564
.1183
.1200
.1389
.0910
.0684
.1432
.3154
.6122
0.79
1. 03
1.14
0.94
1.16
1. 07
1. 44
1.15
0.82
Region
Region
Region
Region
Region
Region
Region
Region
Region
1
2
3
4
5
6
7
8
9
of age from age 15 to 49, and their corresponding mean square errors.
Because of the very tedious computations and the high cost involved in
arriving at these estimates, computations are done only for women in
urban areas.
In addition, only children aged 1 at the time of the
survey are considered, hence the computed
is the year preceeding the survey.
TFR
refers to 1979, which
The survival probabilities used in
reverse-surviving the children and the women are taken from Level 16,
West model, of the Coa1e-Demeny model life tables, and are treated as
constants in the derivations.
70
The resulting estimates for the own-children method are shown in
Table 7.
responding
approach.
The estimated
TFR
TFR
is 3.15, which is lower than the cor-
estimate for urban areas using the Coale-Demeny
The two figures however, are not directly comparable since
they refer to different time periods.
The mean square'errors for both
the complex design and under the assumption of simple random sampling
are also lower.
The coefficients of variation for the age-specific
fertility rates are high, ranging from 11% to 100%.
This reflects
the great instability of the estimates by single years of age.
The
very high coefficients of variation occur at the younger and older
reproductive ages, where the number of births used as a basis for the
estimates are very few.
The coefficient of variation for the total
fertility rate however, is 3.26%, which is an acceptable level.
Although the resulting values of the design effects for the agespecific fertility rates range from .77 to 2.00, 22 out of the 35
design effects are between .9 and 1.1.
than 1, 17 are between 0.8 and 1.0.
to the estimated
TFR
Of the 18 design effects less
The design effect corresponding
is 0.98.
Since age-specific fertility rates are comonly misinterpreted as
1
proportions , a third type of
variance,
q
~
l-p,
pq/n,
and
n
mse
is also computed.
is the
nw~ber
estimator which is the binomial
Here,
is the estimated
ASFR,
of women in each age-group at the time
of the survey, reverse-survived to 1979.
IAn
p
Except for that of age 15,
age-specific fertility rate is actually a ratio of the number of
births to women in a specific age-group to the number of women in that
age-group.
71
TABLE 7
ASFR and TFR Estimates Using
the Own-Children Method and Corresponding Mean Square
Errors and Design Effects: East Java Population Survey, 1979
I
...
Age
Sample Size
n
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
479
486
477
472
476
429
380
362
406
467
408
31;\
276
306
347
290
223
188
228
289
252
224
225
285
349
268
175
126
214
275
213
160
155
202
321
TFR
5600
1
ASFR I
m<;A
(aC>PRl
1
deff
Complex
SRS
.0049
.0143
.0304
.0688
.1125
.1456
.1606
.1628
.1811
.1533
.1587
.2032
.1946
.1677
.1421
.1227
.1554
.1575
.1169
.1028
.0961
.0796
.0683
.0848
.0591
.0414
.0381
.0345
.0167
.0094
.0092
.0234
.0196
.0130
.0047
0.2400
0.3417
0.6718
2.0507
2.5116
3.0456
4.4268
4.1852
4.3331
3.1455
3.9352
4.9675
6.0189
5.8012
4.4062
3.2519
5.3152
5.7291
7.0424
3.8807
3.3675
4.0458
3.7053
5.3179
1. 5617
1. 0453
2.2529
2.2668
1. 4023
0.4596
0.4109
1. 7690
1. 3249
0.9189
0.2334
0.1199
0.3356
0.6934
1.6101
2.3908
3.0570
4.0850
4.3161
4.9468
3.3445
3.3123
5.5510
6.2667
6.2679
3.9546
3.6251
6.3645
7.4085
6.8028
4.0702
3.7582
3.9992
3.1753
4.2093
1.7834
1.3137
2.3452
2.3274
1.3760
0.4358
0.4190
1. 7890
1.1234
0.9077
0.2235
2.00
1.02
0.97
1. 27
1. 05
1.00
1. 08
0.97
0.88
0.94
1. 19
0.89
0.96
0.93
1.11
0.90
0.84
0.77
1. 04
0.95
0.90
1. 01
1.17
1. 26
0.88
0.80
0.96
0.98
1. 02
1.05
0.98
0.99
1. 18
1.01
1.00
3.1536
105.3725
107.7026
0.98
Estimates need to be multiplied by 10
-4
.
72
the resulting
mse(TFR
O-
C)
srs
mse's
,
are greater than either
mse(TFR
O-
C)
cs
or
resulting in design effects considerably lower than
1.
4.2
Results for Indirect Mortality Estimators
The resulting values for Sullivan's estimators are shown in
Table 8.
The estimate of
of both sexes is .1220.
q(2)
for East Java, considering children
This is interpreted as the probability of
dying between birth and age 2.
The corresponding probabilities for
ages 3 and 5 are slightly higher.
As expected, the female probabili-
ties of dying are lower than those of the males.
The detailed results for different domains and type of estimator
are shown in Table A.I of the Appendix.
For Sullivan's estimators,
results show strong differentials, especially among educational groups.
For example, the estimated values of
q(2), q(3)
and
q(5)
for chil-
dren whose mothers have no education are 0.1476, 0.1604 and 0.1635,
respectively, while the corresponding values for those whose mothers
have reached at least jllnior high are 0.0698. 0.0645 and 0.0650.
The estimated mean square error of
q(2)
for both the complex
design and under the assumption of simple random sampling is consistently higher than that of
q(3)
and
q(5).
As expected, there is an
inverse relationship between sample size and mean square error.
The
range of values for different statistics computed for each estimator
among the 34 domains considered is shown in Table 9.
The results for Trussells' estimators are shown in Table 10. They
fo11O\\I the same pattern as those of Sullivan's.
the estimates of
q(a)
Although in general,
and the mean square errors are slightly higher
e
e
e
TABLE 8
Estimates of q(a) Using Sullivan's Estimator and Corresponding Mean
Square Errors and Design Effects for the Total Sample: East Java Population Survey, 1980
a
No. of
Variables in
Estimator
Children Both Sexes
1
mserq(a)]
q(a)
deff
Complex SRS
~
n
Male Children
1
mse[q(a)]
q(a)
deff
Complex SRS
A
Female Children
1
mse[q(a)]
q(a)
deff
Complex SRS
A
2
5
7789
.1220
0.39
0.33 1.18
.1382
0.76
0.66 1.15
.1049
0.56
0.53 1.06
3
5
7789
.1243
0.24
0.20 1.22
.1359
0.42
0.36 1. 16
.1123
0.36
0.32 1.12
5
6
10539
.1320
0.26
0.21 1. 21
.1391
0.41
0.35 1.16
.1243
0.42
0.36 1.16
1Estimates need to be multiplied by
2
10- .
---J
~
TABLE 9
Range of Values (and Percentages Occurring
in Certain Ranges) for Different Statistics Computed Among 34
Different Domains for Sullivan's and Trussell's Estimators: East Java Population Survey, 1980
Type of
Estimator
n
Sull ivan! s
q(2)
199-5915
.0230-.2338
q(3)
199-5915
q(5)
c. v.
deff
(%.9-1.1)
(% < 30%)
0.50-38.97
0.57-2.03
(66.02)
5.56-99.33
(75.73)
.0384-.2002
0.34-25.56
0.62-1. 86
(41.75)
4.36-59.84
(91. 26)
260-8043
.0407-.2346
0.34-21.13
0.52-1.54
(56.31)
4.18-48.80
(93.20)
Trussell's
q(2)
276-9385
.0229-.2315
0.51-34.94
0.59-1.97
(66.02)
5.57-99.53
(76.70)
q(3)
276-9385
.0389-.2035
0.36-26.40
0.65-1.78
(45.63)
4.42-56.99
(90.29)
q(5)
344-11513
.0420-.2447
0.36-22.66
0.85-1.59
(55.34)
4.17-48.28
(93.20)
q(a)
lEstimates need to be multiplied by
mse [q (a)]
1
10- 4 .
..,.
~
e
e
e
e
e
e
TABLE 10
Estimates of q(a) Using Tlussell's Estimator and Corresponding Mean
Square Errors and Design Effects for the Total Sample: East Java Population Survey,
1980
a
No. of
Variables in
Estimator
n
2
7
12435
.1229
0.40
0.34 1.18
.1394
0.78
0.68 1.14
.1055
0.57
0.54 1.06
3
7
12435
.1264
0.27
0.20 1.33
.1382
0.44
0.37 1. 21
.1142
0.39
0.32 1. 20
5
8
15185
.1362
0.28
0.23 1. 22
.1283
0.45
0.38 1.18
.1283
0.45
0.38 1.18
Children Both Sexes
1
mser q (an
deff
q(a)
Complex SRS
A
A
lEstimates need to be multiplied by
Male Children
1
mse[q(a)]
deff
q (a)
Complex SRS
A
A
Female Children
1
mserq(a)]
deff
q(a)
Complex SRS
A
A
4
10- .
---J
(Jl
76
than Sullivan's, there are no considerable differences between
them.
Trussell also developed an estimator for
probability of dying between birth and age 1.
as his other
q(a)
q(l),
which is the
This has the same form
estimators, except that childhood survivorship
data is provided by women 15-19.
Table 11 shows the resulting esti-
mates for the total sample as well as for the different domains considered.
A comparison of the
mse
estimates for
q(l)
with those of
other ages shows that its mean square error is considerably larger
than those of
q(2), q(3)
and
q(5).
the fact that the estimation of
q(l)
This could be attributed to
is based on the proportion dead
among children ever born to women 15-19 which is very unstable, is
highly sensitive to defects in the data, and is subject to large sampling fluctuations due to the very small number of births to women in
this age-group.
In fact, the
UN
manual on indirect estimators
(UN, 1967) has categorically stated that the level of infant mortality
derived from child survival reported by women 15-19 should not be
regarded with much confidence.
mean square errors of
q(l)
The very high and extremely unstable
derived here lend empirical support to
this claim.
4.2.1
Effect of Selected Variables on
D~~i$~f~ct~
To determine the conditions under which the design effects
deviate from
1,
a descriptive analysis of the data is done.
The
distribution of design effects according to author, sex for which the
probabilities are computed, type of estimator, number of random
77
TABLE 11
Estimates of Trussell's q(l) Estimator
for Both Sexes Combined and Corresponding
Mean Square Errors and Design Effects of the Total
Sample and for Different Domains: East Java Population Survey, 1980
1
mse[Q(l)l
Complex SRS
A
A
deff
Sample Size
n
q(l)
12,435
0.1172
1.84
1. 76
1. 04
Urban
Rural
3,050
9,385
0.1300
0.1144
8.70
2.31
9.83
2.13
0.88
1.08
No Education
Primary Grades, 1-3
Primary Grades, 4-6
Primary Grade Completed
Junior High +
2,677
2,101
2,749
2,716
2,192
0.1536
0.1137
0.1142
0.0867
0.1021
10.22
7.99
7.55
5.19
16.88
9.88
7.96
8.50
4.33
17.26
1. 04
1.00
0.89
1. 20
0.98
No Education, Urban
Primary Grades, 1-3
Primary Grades, 4-6
Primary Grades Completed
Junior High +
320
301
548
609
1,272
0.1577
0.1752
0.0762
0.1283
0.1103
28.59
75.65
4.28
27.63
44.19
61.58
96.90
25.62
27.37
49.82
0.46
0.78
0.17
1. 01
0.89
No Education, Rural
Primary Grades, 1-3
Primary Grades, 4-6
Primary Grade Completed
Junior High +
2,357
1,800
2,201
2,210
920
0.1533
0.1053
0.1216
0.0764
0.0991
12.82
8.46
10.36
6.41
27.78
11. 52
8.08
11. 69
4.83
27.34
1.11
1.05
0.92
1.33
1.02
Agriculture
Manufacturing/Handicraft
Trade
Transportation
Services
Government
Income Recipient
Other
7,341
517
1,141
276
989
877
471
790
0.1259
0.0488
0.0900
0.0816
0.0379
0.1450
0.2143
0.0858
2.93
20.62
24.10
58.07
6.07
59.82
85.84
21. 91
2.74
23.35
23.77
68.44
6.41
54.73
92.91
21. SO
1. 07
0.88
1. 01
0.85
0.95
1.09
0.92
1.02
Region
Region
Region
Region
Region
Region
Region
Region
Region
1,254
1,205
1,242
1,808
2,044
1,836
1,239
1,195
612
0.1782
0.1809
0.1452
0.0670
0.0790
0.1033
0.1190
0.1068
0.2298
41.60
35.34
38.29
9.00
7.01
5.26
12.30
30.67
40.46
34.38
31.96
33.19
9.01
6.11
6.04
13.09
21. SS
83.37
1. 21
1.10
1.15
1.00
1. 15
0.87
0.94
1.42
0.48
Domain
Total Sample
1
1
2
3
4
5
6
7
8
9
Estimates need to be mUltiplied by
10
-4
.
78
variables in the estimator, sample size and domain is done and possible
relationships are observed.
Six hundred and thirty design effects are computed which could be
broken down into 34 domains plus those of the total sample, 3 estimators for each of two authors, and 3 sex categories for which
values are estimated.
q(a)
The design effects are generally clustered
around 1, with 52% of the total number computed having values between
0.9 and 1.10.
The range of values however, is from 0.57 to 2.03.
The
frequency distribution of design effects is shown in Table 12.
TABLE 12
Frequency Distribution of the Design Effects
for the Estimated Mean Square Errors of Mortality Estimators
Design Effect
Frequency
%
< .80
6
0.95
.80- .89
30
4.76
.90- .99
124
19.68
1. 00-1. 09
206
32.70
1.10-1.19
117
18.56
1. 20-1. 29
85
13.49
1. 30-1. 39
31
4.92
1. 40-1. 49
10
1. S9
1. 50-1. 69
13
2.06
8
1. 27
630
100.00
:0:1. 7
TOTAL
e
e
79
The
di~tribution
variable~ con~idered
no
con~iderable
are
de~ign effect~
pre~ented
according to the different
in Figure 1 to Figure 6.
difference between the
of Sullivan's and
Figure 1.
of
di~tribution
Tru~~ell's e~timator~
There
i~
of design effects
taken altogether, as shown in
However, when the estimators are grouped according to the
sex for which the probabilities are computed, the proportion of design
effects between .90 and 1.09 for female as well as for male children
is much higher than when the probability of death being estimated
for both sexes combined.
i~
The pattern however reverses for higher
design effects, with the estimators for both sexes combined having a
consistently larger proportion of design effects for all categories
~
1.10.
On the other hand, when the design effects are grouped accord-
ing to the type of estimator (i.e.,
values for
q(2)
q(2), q(3)
q(3)
and
q(5)
The corresponding percen-
are 41% and 52%, respectively.
has a higher proportion of design effects
pattern reverses for higher design effects.
q(a)
q(5)), the
tend to be lower, with 63% of the total design
effects having values between .90 and 1.09.
tages for
and
~
.90.
q(2)
also
As with sex, the
Since both
q(2)
and the
estimators computed for each sex separately are the more unsta-
ble estimators relative to other groups due to the smaller number of
births or deaths reported for these categories, these results seem to
indicate the tendency of unstable estimators to have lower design
effects.
These results are
con~istent
with those presented earlier
regarding the design effect of Trussell's
q(l)
estimator, wherein 15
out of 35 design effects computed are below 1, a proportion higher than
those of other estimators, and with much lower
magnitude~.
80
100
80
;; 60
Legend
n
c:::J
Sullivan
315
~
Trussell
315
C
"u
"
~
0-
40
<.9'0
.9U-I.09
1.10-:.:~
1.30-1.49
~
1.50
DEFF
Fig. 1:
Distribution of Design Effects of
Mortality Estimators by Author
100
Legend
c::J
~
[:\.;.;::.J
80
i'<
Both Sexes
210
Females
210
Males
210
60
~
u
~
"
0-
40
20
< .90
.90-1.09
1.10-1.29
1.30-1.49
~
DEFF
Fig. 2:
Distribution of Design Effects of Mopality
Estimators According to Sex for Whicl the
Probability Is Computed
n
1. 50
81
100
80
n
q (2)
210
~
qO)
210
f""·:::'l
q(4)
210
Legend
n
c::J
N
~
Legend
60
c:
OJ
...u
'
..
.',:
OJ
0-
::.:
40
.'
::.
.':
':.
20
....
.:~ :
(.90
.90-1.09
1. 30-1.49
1.10-1.29
~
1.50
DEFf
fig. 3:
Distribution of Design Effects of Mortality
Estimators by Type of Estimator
100
c::J
~
80
...
var.
210
var.
105
k::;:!.;N
var.
210
IlIIlIIm
var .
105
60
~
OJ
~
OJ
0-
40
'.
20
;'.:
.'
."
,,'
<.90
.90-1.09
1.10-1. 29
1. 30-1. 49
~
1. 50
DEff
fig. 4:
Distribution of Design Effects of Mortality Estimators
According to the Number of Rando: Variables in the
Estimator
82
100
Legend
n
c:J c 500
~
111
500-999
153
f::\··.·;·····:A
1000-4999
318
I1lllIlIlIlIIlI
~5000
80
48
N
60
~
'"
u
~
'"
'"
40
.~. :",
20
<.90
.90-1.09
1.10-1.29
1.30-1.49
.. 1.. SO
DUF
Fig. S:
Distribution of Design Effects of Mortality Estimators
According to Sample Size
100
Legend
n
Total
18
80
c=J
~
N
60
~
...u'"
'""
Rural
18
f':';'..;.;'-;:j Urban
18
'~.:
:,:
40
".
'":
"
.;-
20
<.90
.90-1.09
1.10-1.29
1.30.. 1.49
~1
.50
DEFF
Fig. 6:
Distribution of Design Effects of Morta1itv Estimators
For the Total Population and by Urban-Rural Grouping
83
There are four categories when the basis of grouping is the number of random variables in the estimator.
involves 5 different totals while his
and
q(3)
has 8.
Sullivan's
q(S)
has 6.
q(2)
and
q(3)
Trussell's
q(2)
on the other hand have 7 different variables while his
q(5)
The distribution of design effects according to the number of
random variables is shown in Figure 4.
The graph shows no marked dif-
ferences in the distribution of design effects according to the number
of random variables in the estimator, implying the absence of an effect.
For the distribution of design effects according to sample size,
four categories of sample sizes are considered:
1000-4999
and
~
5000.
<
500, 500-999,
These are presented in Figure 5.
The graph
shows that when the sample size is really large, i.e., 5000 and over,
the design effects are all within
0.9
and 1.49, with the proportion
of design effects between 1.10 and 1.29 more than twice as high as
those of other categories.
The 6 design effects falling within the
.90-1.09 category are all greater than 1.
However, when the sample
sizes are lower than 5000, the design effects have a larger range,
taking on values both below and above 1.
Of the total number of design
effects below 1, 56% occur when the sample size is below 1000.
It is
also when the sample sizes are small that a slightly larger proportion
of higher design effects, i.e.,
~
1.5,
are found.
The distribution of design effects according to the different
domains is also considered.
Figure 6 shows the distribution of design
effects of estimators for the total population and for urban and rural
areas.
Since these domains have large sample sizes, with the esti-
mators for the total population and the rural areas having sample
84
4It
sizes of at least 5000, the patterns exhibited in Figure 6 are similar
to those of large sample sizes in Figure 5.
Except for this tendency
of estimators for real large domains to have design effects between
1.0 and 1.49, no other consistent patterns can be established.
4.2.2
The
Although
pq/n
q(a)
Variance Estimator
is actually a complex non-linear estimator of the
probability of dying between birth and age
interpreted as a proportion.
a,
it is very often mis-
In view of this, it is very highly con-
ceivable for somebody who does not have enough background of statistics
and sampling theory to compute its variance as
pq/n,
which is the
variance for a proportion presented in basic statistics textbooks.
Such an estimator represents a very simplistic approach as it disregards both the actual sampling design of the survey and the complex
non-linear form of the estimator.
A
To determine the effect of using the naive estimator
= pq/n,
these are computed from the data for the total sample, as well
as for urban-rural grouping.
ffise[q(a)]
ffise[q(a)]b'In
cs
The results are then compared with
by taking their ratios, as defined in (3.2).
One of the problems associated with the use of this estimator is
the determination of the value of the denominator,
n.
could differ greatly, depending on how one interpret3
portion. For example, since Sullivan's
q(a)
bability of dying between birth and age
a,
estimator
It's value
q(a)
15
as a prothe pro-
one could think of it as
a proportion dead among the origianl cohort of children born
ago.
a
years
Hence if the number of children 2, 3 and 5 years old at the time
85
of the survey is reverse-survived for 2, 3, and 5 years, respectively,
one gets an estimate of the original size of these cohorts at the time
of birth.
These numbers can be used as the denominator in estimating
the naive
mse
of
q (2) , q(3)
and
respectively.
q (5) ,
~
Table 13 shows the estimated values of
sponding
a
mse
and corre-
ratios for the total sample, using the number of births
years ago a5 the denominator.
is
mse[q(a)]b"In
substituted for
p,
mean square error of
q = l-p.
and
q(2)
Here, the estimated value of
q (a)
Like earlier results, the
is consistently greater than that of
q(3)
~
and
In addition,
q(5).
mse[q(a)]cs
mse[q(a)]b'
In
for all values of
are all less than
1,
a.
is consistently greater than
This results in
mse
ratios which
implying an overestimation of the mean square
error when using a naive estimator defined in this manner.
results are derived for urban-rural grouping.
observed between sample size and
mse
The same
No relationship is
ratios.
Another way of determining the value of the denominator,
by looking at the functional form of the estimator.
and Trussell's
q(a)
aged
D(i)
i,
is
Both Sullivan's
estimators can be expressed as
q(a)
where
n,
=
D(i) • k
is the proportion dead among children ever born to women
and
k
Since the equation suggests
is a correction factor.
a close relationship between
q(a)
and
D(i),
and
D(i)
a proportion, one can useits denominator to substitute for
is in fact,
n,
in
~
estimating
mse[q(a)]b'In .
Table 14 shows the estimated values of
sponding
mse
illse[q(a)]b"
In
and corre-
ratios for the total sample, when the denominator is the
e
e
e
TABLE 1.3
Estimated mse's and mse Ratios for Sullivan 1 sandTrussell's
Estimators Using the Naive Estimators, pq/n, with Number of Births
a Years Ago as Denominator: Total Sample, East Java Population Survey, 1980
TRUSSELL's
Sex
And Type
n
1
A
SULLIVAJ\J'S
A
A
2
mse
ratio
n
1
A
A
2
mse
ratio
q(a)=p
pq/n
2169
2633
2647
0.1220
0.1243
0.1320
0.4939
0.4134
0.4329
0.79
0.58
0.60
0.72
0.51
0.54
1102
1379
1348
0.1382
0.1359
0.1391
1.0808
0.8516
0.8884
0.70
0.49
0.46
0.64
0.48
0.52
1068
1254
1299
0.1049
0.1123
0.1243
0.8792
0.7950
0.8379
0.64
0.45
0.50
q(a)=p
pq/n
2169
2633
2647
0.1229
0.1264
0.1362
0.4970
0.4297
0.4445
0.80
0.63
0.63
1102
1379
1348
0.1394
0.1382
0.1283
1. 0886
0.8637
0.8297
1068
1254
1299
0.1055
0.1142
0.1283
0.8836
0.8067
0.8610
2
2
Both Sexes
q (2)
q(3)
q(5)
I
Males
q(2)
q(3)
q(S)
Females
q (2)
q(3)
q(5)
I
I
IThe sample size, n, for q(2), q(3) and q(5) correspond to the number of births 2, 3 and 5 years
ago, respectively.
2
Estimates need to be multiplied by
10
-4
.
(X)
0\
e
e
e
e
e
e
TABLE 14
Estimated mse's and mse Ratios for Sullivan's and Trus~ell's
Estimators Using the Naive Estimator, pq/n, with the Number of Children
Ever Born to Women as Denominator: Total Sample, East Java Population Survey, 1980
Sex
And Type
TRUSSELL'S
n
1
SULLIVAN'S
2
mse
ratio
q(a)=p
pqjn
4178
7621
8397
0.1229
0.1264
0.1362
0.2580
0.1449
0.1401
1. 5S
2144
3879
4350
0.1394
0.1382
0.1283
2034
3742
4047
0.1055
0.1142
0.1283
2
mse
ratio 2
n
q(a)=p
pq/n
1. 86
2.00
4178
7621
8397
0.1220
0.1243
0.1320
0.2564
0.1428
0.1364
1. 52
1.68
1. 90
0.5596
0.3070
0.2571
1. 39
1.43
1. 75
2144
3879
4350
0.1382
0.1359
0.1391
0.5555
0.3027
0.2753
1.37
1. 39
1.49
0.4640
0.2703
0.2764
1. 23
1.44
1.63
2034
3742
4047
0.1049
0.1123
0.1243
0.4616
0.2664
0.2690
1. 21
2
Both Sexes
q (2)
q(3)
q(5)
Males
q(2)
q (3)
q(5)
Females
q (2)
q(3)
q(5)
1.35
1.56
IThe sample size, n, for q(2), q(3) and q(5) correspond to the number of children ever born to
women 20-24, 25-29 and 30-34, respectively.
2Estimates need to be multiplied by
4
10- .
co
'-I
e
e
e
88
total sample, when the denominator is the number of children ever
born to women 20-24, 25-29 and 30-34, respectively.
values of
p
Here, the same
are used as in Table 12, but with considerably larger
~
sample sizes.
This results in
mse[q(a)]b'
In
those obtained earlier, with the
mse
defined in this manner.
ratios all greater than
mse
implying an underestimation of the
values much lower than
1,
when the naive estimator is
This is consistent with the expected pattern
if the values substituted for
p
are in fact, simple proportions.
The degree of underestimation is however large, ranging from 20% to
100%.
4.3
Discussion of Results
The findings in this study which are of greatest practical signi-
ficance to a researcher are the magnitudes and patterns exhibited by
the
mse
ratios, since they measure the extent of errors that would
arise through violations in the sampling design and the form of the
estimator in the computation of the
mse.
This is important since
the assumption of simple random sampling is frequently made for ana1ysis, even when a more complex design has been employed.
In addition,
the additional computer time and money needed for a more complicated
analysis of survey data which takes into account the complexity of
the design and the estimator may be considerable.
The use of the binomial variance,
pq/n,
as the
mse
estimator
for some of the demogrphic measures considered here represents the
simplest and cheapest approach since one does not even need a computer
to
evaluate
it, as long as the value of
p
is already known.
In
89
addition, its simplicity of form makes it
50
appealing to a researcher
who feels the need and the importance of computing a variance for his
estimator, but does not have sufficient knowledge of sampling theory
to consider the complexity of the sampling design and the form of the
estimator in computing its variance.
The results in this study how-
ever show the problems associated with this kind of estimator, the
greatest of which is the determination of the denominator,
case of a simple proportion,
tor used in computing
of the denominator.
p
and
n
is well defined:
q,
n.
In the
it is the denomina-
with the numerator being a subset
For complex estimators however, the numerator is
often unrelated to the denominator, or they may not even be explicity
defined, hence the ambiguity in defining
denominators used here in computing
ferent approaches in determining
a
n.
n
arises.
The two
kind~
of
represent two dif-
mse[q(a)]b'In
The use of the number of births
years ago as the denominator is based on the intuitive interpreta-
tion of
q(a)
while the use of the number of children ever born to
women is based on its mathematical formulation.
The first approach is
likely to be used by a researcher who is not mathematically oriented
while the latter approach is more likely to be used by those with a
certain level of mathematical sophistication.
Of the two approaches,
it is the latter which has results consistent with the expected pattern
if the values substituted for
p
However, the estimated
using this approach are underestimated
by at least 20%.
the
mse
mse's
are in fact, simple proportions.
The intuitive approach leads to an overestimation of
by the same extent, illustrating the possibility of erring
in both directions when using the naive estimator.
In both instances,
90
the value of
n
used in estimating
with the value of
that the
pq/n
n
mse[q(a)]b'In
used in estimating
q(a).
is not consistent
It is unfortunate
estimator is highly sensitive to the value of
n,
for
which there is no clear-cut definition for the complex, indirect demographic
pq/n
estimator~
considered here.
For this reason, the use of the
estimator is not recommended, in spite of its simplicity.
An interesting result in this study is the occurrence of design
effects
<
1.
This is something unexpected, considering what is cur-
rently known in sampling theory about design effects.
mean,
x,
the design effect is approximately
deff ::: 1
where
roh
For the sample
+
roh(b-l)
(4. I)
is the estimated intraclass correlation coefficient, which
measures the degree to which values for variables are homogeneous
within sample clusters, and
Roh
6 is the average size of sample clusters.
is generally regarded to be positive, in which case, the design
effect is at least equal to 1.
Although
(4, 1)
was derived for
x,
j t
has traditionally been assumed to also apply to complex estimators, in
the absence of any theoretical work done in this field.
However, the
occurrence of design effects <1 in 183 out of 677 design effects computed
in this study raises questions about this assumption.
In another study
done by O'Brien (1981), 28% out of the 68 design effects computed for
the variance of life table conditional and survival probabilities using
the Taylor series linearization method had values less than 1.
One possible explanation for the occurrence of design effects less
than 1 is the sample size, as shown in Figure 5.
the sample size is
~
5000,
The fact that when
the values of the design effects have lower
91
variability and are all
~
1
seems to imply that for the complex esti-
mators considered here, real large sample sizes are needed before the
resulting estimates attain a certain level of stability.
•
If this is
true, then perhaps the design effects for the complext estimators considered here are also generally
~
1,
but large sample sizes are
needed before this pattern is exhibited.
Another factor deemed to affect the magnitude of the design
effects computed here is the fact that in the computation of
mse[f(x)] srs ,
unequal sampling fractions are used, instead of a con-
~
stant selection probability.
By definition of a simple random sample,
elements are supposed to have equal probabilities of selection.
the computation of the estimates of
q(a)
In
however, it was necessary
to apply unequal sampling weights since the actual design is not selfweighting.
We were then faced with the problem of choosing the appro-
priate selection probability to use in the estimation of
mse[f(x)] srs .
~
The choice is between the unequal sampling weights which is consistent
with those applied in computing the estimate but which is not in the
spirit of simple random sampling in the real sense, or the use of a
constant,
N/n,
for
mse estimation even if unequal weights were
actually used in computing the estimates.
It was deemed best to be
consistent, and hence unequal sampling weights were applied to compute
mse[f(x)]
mse[f(x)]
srs
.
were recomputed using a constant sampling fraction for
srs
Trussell's
mse[q(2)T]
To determine the effect of doing this, values of
q(2)
srs
estimator.
values.
The result was a general increase in the
However, the difference between the resulting
estimates using the two approaches is not large and design effects
still persisted even after the modification.
<
1
92
Above all these conjectures, the question still remains as to
whether the magnitude of the error arising out of the violation of the
sampling design is big enough
to warrant the additional computer time
and money necessary for a more complicated analysis of the data.
Results of this study show that in 48% of the cases,
deviate from
mse[f(x)] cs
mse[f(x)]
.
srs
~
by at least 10%, in both directions.
~
Of
course, the degree of tolerable error is subjective, and varies, among
other factors, according to the personal preference of the researcher
or to the use for which the estimators are derived.
noted however, that since
mse[f(x)]
- srs
It should be
as it is defined here, con-
siders the complex, non-linear form of the estimator, its computation
is a relatively complex procedure which also entails the
tion of the Taylor series linearization,
z
netS'
jU5t like
determinamse[f(x)]
.
- cs
Since these two estimators differ only in the final formula to be
applied as illustrated in Section 3.3, then one might as well use the
"correct"
mse
estimator which considers both the sampling design
and the form of the estimator.
The other finding worth noting in this study is the relatively
large coefficients of variation derived for domain estimates. for all
types of
mse
estimators.
This strongly supports the need for larger
sample sizes when estimates for domains are to be computed, especially
for complex estimators.
An interesting finding is that for the same
sample size, the coefficients of variation for domain estimates as
well as for the total sample are considerably higher \'1hen the estimator
is more complex, as shown in Table 14.
Here it is shown that although
the same number of women is used in the computation of Coale-Demeny
TFR
and Sullivan's
q(3)
estimators, the coefficients of variation
'5
93
TABLE 14
Coefficients of Variation and Corresponding
Sample Sizes for Selected Indirect Estimators
Domain
Sample Size
(n)
TFR _
C D
q(3)S
Sample Size
(n)
q(3)T
Total
7789
2.92
3.94
12,345
4.11
Urban
Rural
1874
5915
6.69
3.26
10.64
4.36
3,050
9,385
10.82
4.42
776
725
794
1102
1307
1194
785
744
362
7.73
8.79
8.91
6.70
7.16
7.02
11.39
12.78
15.90
14.00
11.25
14.90
11.65
8.68
7.90
9.57
14.86
29.20
1,254
1,205
1. 242
1,808
2,044
1,836
1. 239
1,195
612
13.89
10.65
15.89
11. 75
8.43
8.64
11. 73
14.70
30.03
Region
Region
Region
Region
Region
Region
Region
Region
Region
1
2
3
4
5
6
7
8
9
are in general, considerably higher for
domains.
Note that
q(3)S'
especially for small
TFR _
has a less complex form and has only 4
C D
random variables compared to 6 random variables in
other hand, Trussell's
q(3)
q(3)S'
On the
estimator which has the most complex form
with 7 random variables in it, has coefficients of variation slightly
larger than those of Sullivan's
q(3).
These however, were derived
with sample sizes considerably larger than those of
q(3)S
and
CHAPTER 5
Summary of Results and
Suggestions for Future Research
5. I
Summary of Results
In this dissertation, estimators of the mean square errors
of four commonly used indirect fertility and mortality
derived and computed.
TFR
mea~;ures
(mse)
are
The measures considered are the Coale-Demeny
estimator, the own-children method of
TFR
estimation,
Sullivan's and Trussell's child mortality estimators.
All of these
measures are complex, non-linear estimators derived from sample surveys.
The method of estimation used in deriving the mean square errors
is the Taylor series linearization method.
It is chosen among other
variance estimation techniques because of its extensive use in sample
surveys, its applicability to all sampling designs and statistics, and
its economic and computational efficiency.
In particular, the Woodruff
linearization procedure is applied, since it provides a computational
shortcut in the evaluation of the
mse's.
All derivations done are
applicable to any multistage sampling design with stratification in
the first stage.
The mean square errors are computed using data from the East Java
Population Survey (EJPS), which has a three stage sampling design with
stratification at the primary level.
Estimates for the total sample
as well as for domains are computed.
In addition, mean square errors
95
under the assumption of simple random sampling are also determined to
enable the computation of design effects.
For measures which are com-
monly and erroneously interpreted as proportions, a third type of estimator equaivalent to the binomial variance,
compared with that of the
mse
pq/n,
is computed, and
obtained taking into account the actual
sampling deisgn used for the EJPS.
Such a comparison demonstrates the
effect of using a very naive estimator which disregards the complexity
of both the sampling design and the form of the estimator.
Results of the computations show that for estimates referring to
the total sample, relatively precise estimates of the four measures
considered are obtained, with coefficients of variation ranging from
3% to 7%.
For domain estimates however, the coefficients of variation
are considerably higher.
As expected, both mean square errors and
coefficients of variation are inversely related to sample size.
The design effects for the four estimators considered varied
between .57 and 2.03.
Since values both below and above 1 are obtained,
this implies that the assumption of simple random sampling when in fact
a more complex
design has been employed could lead to either an over-
estimation or an underestimation of the
known to be unstable like Trussell's
mse.
q(l)
Measures which are
estimator or age specific
fertility rates by single years of age, as well as those computed from
domains with small sample sizes (i.e., 1000)
of design effects less than 1.
h~ve
a larger proportion
On the other hand, estimates computed
from doamins with real large sample sizes (5000) have design effects
all greater than 1.
96
Finally, estimates of the mean square error using the binomial variance,
pq/n,
are computed using two different denominators.
When the number of births
a
years ago is used, computed
mse's
are
overestimated by at least 20%; when the number of children ever born
is used, the
mse's
are underestimated by the same extent.
Because
of the sensitivity of the naive estimator to changes in the value
of
n,
for which there is no clear-cut definition for the indirect
demographic estimators considered here, the use of the
pq/n
estima-
tor is not recommended.
5.2
Suggestions for Further Research
The above results have probably generated more questions about
complex estimators from complex surveys, rather than answer them.
Because of the very little that is known about these estimators at
present, this empirical study is basically exploratory in nature,
hence its replication using the same estimators but different populations and different sampling designs is called for to validate results.
It would also be interesting to compare the results in this study
with those obtained using other methods of variance estimation.
It is
possible to arrive at more precise estimates using other methods especially for small domains where the Taylor series approximation does
not work very well.
As mentioned earlier, the occurrence of design effects less than
I in 28% of the cases considered here is something unexpected in the
light of what is currently known in sampling theory about them.
While
it is hypothesized in this study that this phenomenon is an effect of
97
of sample size, it could very well be due to other factors.
An in-depth
analytic study of the conditions under which design effects less than I
occur will be helpful.
Lastly, the expansion of this empirical work to other demographic
techniques, like the adult mortality estimators is suggested.
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APPENDIX
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