IIPS Talk

Introduction
Background and Motivation
Goals of the Study
Data and Methods
Results and Conclusion
Estimation of TFR through Bayesian Approach
Reetabrata Bhattacharyya
PhD Student, BIRU
Indian Statistical Institute, Kolkata
Email: breetabrata@gmail.com
Arni S.R. Srinivasa Rao
Assistant Professor, BIRU
Indian Statistical Institute, Kolkata
Email: arni@isical.ac.in.
Reetabrata Bhattacharyya IIPS National Seminar, Kolkata Estimation of TFR through Bayesian Approach 16th
March, 2012

Introduction
Goals of the Study
Data and Methods
Overview
1 Introduction
2 Background and Motivation
3 Goals of the Study
4 Data and Methods
Data
Methodology
5 Results and Conclusion
Results
Conclusion
March, 2012

Introduction
Goals of the Study
Data and Methods
Total Fertility Rate (TFR)
Total Fertility Rate (TFR) is the average number of children born
to a women in her entire child bearing period according to the
age speciﬁc fertility schedule.
It is the most widely accepted index of current fertility.
Fertility is an important factor in the study of population
dynamics as well as one of the key component in population
projection.
During last two centuries rapid fertility declines in India.
March, 2012

Introduction
Goals of the Study
Data and Methods
Bayesian Approach
In Bayesian Inference we use current observed data and past
information which led to the observed data and blend these two
to arrive an estimate of the parameter which we call a Bayes
estimate.
This approach helps us in updating the prior information and
continuously update prior probability densities.
Bayesian estimation expresses the uncertainty about unknown
model parameter.
March, 2012

Introduction
Goals of the Study
Data and Methods
Necessity for Estimating TFR
Most of the developing countries are handicapped due to lack of
complete and reliable vital statistics.
Some of these countries started registration of births and deaths
as early as the beginning of nineteen century, but still far from
satisfactory in respects of its completeness and reliability.
The countries where births and death registrations are not
adequately maintained, it is necessary to estimate reproduction
rates by indirect way.
March, 2012

Introduction
Goals of the Study
Data and Methods
Techniques
Brass (1953) ﬁrst time developed an indirect methods of
estimating fertility and reproduction rates from data on the
reproductive histories of women.
Henry (1953), Brass (1968) and Chowdhury(1982) introduced
the concept of Parity Progression Ratio, P/F Ratio and Pregnancy
Prevalence method for estimation of fertility rate.
The mean completed family size and the fertility estimates have
been derived by Brass Birth Order Ratio technique(1971, 1975).
Srinivasa (1980), Yadava and Kumar (2002) estimated TFR
through Open Birth Interval Technique.
March, 2012

Introduction
Goals of the Study
Data and Methods
Models
Coal and Trussell (1974) developed a model in estimating
fertility measures from data on children everborn.
Brass (1980) showed that, if pattern of fertility can be expressed
by Gompertz function, then it provides better estimate of TFR
than Coal-Trussell model.
Schmertmann(2003) proposed a new models for age-speciﬁc
fertility schedules in which three index ages determine the
schedule’s shape.
Alkema et al.(2011) describe a Bayesian Projection Model to
produce country speciﬁc projections of TFR.
March, 2012

Introduction
Goals of the Study
Data and Methods
Objectives
1 Estimating the Total Fertility Rate (TFR) in India by using
Bayesian Approach.
2 Compare with Conventional Fertility Rate to test the validity of
our method.
March, 2012

Introduction
Goals of the Study
Data and Methods
Data
Methodology
Data
Third rounds of National Family Health Survey(NFHS-3) were
conducted during November 2005-August 2006.
We extracted each birth information by mothers age-group from
NFHS-3.
March, 2012

Introduction
Goals of the Study
Data and Methods
Data
Methodology
Deﬁne Variable
We have constructed dichotomous random variable xij.
xij : Birth status of the jth child within the ith age-group of
mother;
∀ i=1(1)k and ∀ j = 1(1)ni.
where,
k : Total age-group of mother within the sample.
ni : Number of children in the ith age-group of mother .
xij =



1 if jth child of the ith age-group of mother was born
during last one years preceding to the survey.
0 otherwise
Member’s of the individual age-group are homogeneous.
March, 2012

Introduction
Goals of the Study
Data and Methods
Data
Methodology
Likelihood function
Likelihood is a function of model parameters (unknown).
L(xi|pi) =
k
i=1
p
ni
j=1 xij
i (1 − pi)ni−
ni
j=1 xij
; xij = 0, 1 (1)
where,
xi = (xi1 , xi2 ,....., xini ), obtained data from the sample;
pi : probability of child born within the ith age-group
of mother, ∀ i= 1(1)k.
In our frame work pis, are the model parameter and
pi ∈ (0, 1).
March, 2012

Introduction
Goals of the Study
Data and Methods
Data
Methodology
Prior Elicitation
We collect prior information’s from our believe of literature.
This prior information s are the age-speciﬁc fertility rates within
mother’s age-group among neighborhood countries of Indian
during the speciﬁed time periods.
This information helps us to construct prior distribution which
measures the uncertainty about the true value of parameters.
Here parameters of the prior distribution are estimated from
Indian neighborhood countries data by method of moments.
March, 2012

Introduction
Goals of the Study
Data and Methods
Data
Methodology
Prior Distribution
Let pi ∼ Beta(αi , βi), ∀ i = 1(1)k.
Where αi > 0 and βi > 0 are unknown parameters.
Our believe on literature of fertility rate provides us the values of
αi and βi with in the ith age-group of mother.
Also, pi’ s are independent because they came from different
age-group of mother where as mother’s age-group are
independent.
March, 2012

Introduction
Goals of the Study
Data and Methods
Data
Methodology
Posterior Distribution
Posterior distribution of pi is
π(pi|xi) =
p
ni
j=1 xij
i (1 − pi)ni−
ni
j=1 xij
.
p
ˆαi−1
i (1−pi)
ˆβi−1
B( ˆαi, ˆβi)
1
0 p
ni
j=1 xij
i (1 − pi)ni−
ni
j=1 xij
.
p
ˆαi−1
i (1−pi)
ˆβi−1
B( ˆαi, ˆβi)
dpi
=
p
ni
j=1 xij+ ˆαi−1
i (1 − pi)ni−
ni
j=1 xij+ ˆβi−1
B( ni
j=1 xij + ˆαi, ni − ni
j=1 xij + ˆβi)
, 0 < pi < 1
March, 2012

Introduction
Goals of the Study
Data and Methods
Data
Methodology
Bayes Estimates
Mean of the Posterior Distribution is
Epi|x(pi) =
B( ni
j=1 xij + ˆαi + 1, ni − ni
j=1 xij + ˆβi)
B( ni
j=1 xij + ˆαi, ni − ni
j=1 xij + ˆβi)
(2)
= Bayes Estimate of pi, ∀i = 1(1)k
March, 2012

Introduction
Goals of the Study
Data and Methods
Results
Conclusion
Table: Parameter Estimate, Credible Region and Bayes
Estimte within Age-group
Age Parameter HPD Credible Bayes
Estimates Region Estimates
ˆαi
ˆβi LCL UCL
15-19 1.55222 21.84802 0.420 0.469 0.4445
20-24 8.86752 45.92662 0.178 0.194 0.18596
25-29 16.04513 80.80676 0.065 1.000 0.07014
30-34 10.79503 75.56520 0.022 1.000 0.02528
35-39 11.20322 145.12070 0.007 1.000 0.00901
40-44 4.56989 159.61770 0.001 1.000 0.00234
45-49 1.28768 174.30530 0.000 1.000 0.00068
TFR - 0.693 5.663 0.73796
March, 2012

Introduction
Goals of the Study
Data and Methods
Results
Conclusion
Figure: Comparison of NFHS-III and Bayes Estimate
0
0.1
0.2
0.3
0.4
0.5
0.6
15-19 20-24 25-29 30-34 35-39 40-44 45-49
Age-specificFertilityRate
NFHS-III Bayes Estimate
Mother's Age-group
March, 2012

Introduction
Goals of the Study
Data and Methods
Results
Conclusion
Discussion
The Table and Figure provides satisfactory results.
Bayesian type of estimation procedure adopted here for the
fertility data worked well.
There is a convergence in the credible region values at the higher
ages(Table).
We have understood that there could be an effect of truncation in
the computation of Bayesian credible region and Bayes
estimates.
March, 2012

Introduction
Goals of the Study
Data and Methods
Results
Conclusion
References
Alkema L., Raftery A.E., Gerland P., Clark S.J. , Pelletier F. ,
Buettner Thomas and Heilig (2011), Probabilistic Projections of
the Total Fertility Rate for all Countries , Demography; 48, pp.
815-839.
Schmertmann Carl P.(2003), A system of model fertility
schedules with graphically intuitive parameters, Demographic
Research; 9(5), pp.81-110.
Yadava R. C. and Kumar Anupam (2002), On an Indirect
Estimation of Total Fertility Rate from Open Birth Interval
Data, Demography India; 31(2), pp.211-222.
March, 2012

Introduction
Goals of the Study
Data and Methods
Results
Conclusion
Contd. References
Chowdhury A. K. M. Alauddin (1982), Pregnancy Prevalence :
A Direct Method of Estimating Fertility , Sankhya: The Indian
Journal of Statistics, Series B; 44(3), pp.330-342.
Srinivasan K.(1980), Birth interval analysis in fertility surveys:
An illustrative application to data from Fiji fertility survey ,
Scientiﬁc Report; 7, International Statistical Institute,
Netherlands.
Brass W (1980), The relational Gompertz model of fertility by
age of women. World Fertility Survey, Regional Workshop on
Techniques of Analysis of World Fertility Survey Data; London,
World Fertility Survey.
March, 2012

Introduction
Goals of the Study
Data and Methods
Results
Conclusion
Contd. References
Coal Ansley J. and Trussell T. James(1974), Model Fertility
Schedules: Variation in the Age Structure of Childbearing in
Human Populations, Population Index; 40(2), pp.191-228.
Brass Willam and Coal A.J.(1968), Fertility Analysis through
extension of Stable Population Concepts. , Population
Monography;2, Institute of International Studies. Barkeley:
University of California.
Brass Willam (1953), The Derivation of Fertility and
Reproduction Rates from Restricted Data on Reproductive
Histories, Population Studies; 7(2), pp.137-166.
March, 2012

Introduction
Goals of the Study
Data and Methods
Results
Conclusion
THANK YOU
FOR
YOUR ATTENTION
March, 2012

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