Anurag Verma Bayesian Population Projection

21 Pages·2017·1.5 MB·English

by  Windows User

Probabilistic Approach of Population Projection for India and States
Anurag Verma1*, Abhinav Singh2 P.S. Pundir2
,
1. Dept. of Community Medicine, IMS, BHU, Varanasi, India
2. Dept. of Statistics, University of Allahabad, Allahabad, India
*presenting author, e-mail:[email protected]
Abstract
Projection of future population for a country by age and sex, are widely used for policy
development, planning and research. They are mostly done deterministically, but there is a
widespread need for probabilistic projections. In this paper we propose a Bayesian method
for probabilistic population projections for India. The total fertility rate (TFR) and life
expectancies at birth are projected probabilistically using Gompertz and logistic growth
models respectively under Bayesian paradigm. The estimates obtained from proposed two
models combined using cohort-component method to obtain age-specific projection of the
population by sex. The analysis has been made using Markov Chain Monte Carlo (MCMC)
technique with the software OpenBUGS. Convergence diagnostics techniques available with
the OpenBUGS software have been applied to ensure the convergence of the chains
necessary for implementation of MCMC. The method is illustrated by making 40- year
projection using Indian data for the period 1971-2011. The study will provide probabilistic
point estimates of parameter as well as the projection along with highest posterior density
(HPD) interval, which is derived from population number and vital events, includes age
specific death rates, life expectancies, age specific birth rate, total fertility rates and
dependency ratio.
Keywords: Demography, Probabilistic Population Projection, Bayesian Approach, Posterior
Distribution
Introduction
Population Growth has become one of the most important problems in the world [1]. The idea
of the future population is achieved with the help of projection. The demand of precise
projected figures is always requisite for government personals, actuaries, for their social,
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economic planning purposes. The size of the population and the growth of a country directly
affect the state of the economy, politics, culture, education and the environment, etc. In this
country, and determine the cost of exploring natural sources no one wants to wait until these
resources are depleted because of the population explosion. Therefore, the study of
population projection has been started earlier [2-4].
Currently, there are two main approaches in statistics, such as the Frequentist and the
Bayesian approach for data analysis [5]. The utilization of Bayesian approach in the field of
data analysis is moderately new and has discovered mass support throughout the previous
two decades to persons belonging to various disciplines. Probably the main reason behind the
growing support is its flexibility and generality that allows it to deal with the complex
situation. In addition, the Bayesian method is typically preferred by the classical method in
estimate parameters causing intractable from of the likelihood function [5]. Difficult situation
can be handled by BUGS (Bayesian analysis using Gibbs Sampling) software for its
flexibility and overall approach [6]. This study is based on a Bayesian way of data analysis.
Bayesian method, uncertainty in model choice is incorporated through averaging techniques.
Here the resulting predictive distributions from Bayesian forecasting models have two main
advantages over those obtained using more traditional stochastic models. First, uncertainties
in the data, the model parameters and model choice are explicitly represented using
probabilistic distributions. As a result, more realistic probabilistic population forecasts are
obtained. Second, Bayesian models formally allow the incorporation of expert opinion
including uncertainty into the forecasts [7, 8].
Most population projection are currently done deterministically, using the cohort component
method [9, 10], this is an age and sex-structured version of the basic demographic identity
that the population of a country at the next time point is equal to the population at the current
time point, plus the number of births minus the number of deaths, plus the number of
immigrants minus the number of emmigrants. It was formulated in matrix form by Leslie
[11].
Population Projection are currently produced by many organization, including national and
local governments and private companies. The main organizations that have produced
population projection for all states including India is Registrar General of India. In India’s
current method (RGI, 2006) [12] does not yield an assessment of uncertainty about future
population.
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Standard population projection methods are deterministic, meaning that they yield a single
projected value for each quantity of interest. However, Probabilistic projection that gives a
probability distribution of each quantity of interest and hence convey uncertainty about the
projections are widely desired [13-14].
In the recent past, researcher developed alternative methods which allowed for probabilistic
population projection, aimed for probabilistic interpretation of each demographic factor of
interest. Alho and Spencer (1985), Alho (1990) ,Cohen (1986, 1988), Pflanumer (1988), Lee
(1992) and Lee and Tuljapurkar (1994),Allho (1999),Keilman (2002) show a probabilistic
population projection. The comparison of deterministic and probabilistic method can be
found by Lee(1998), Alho& Spencer (2006) and Stillwell&Clarke (2011). In this study
expand on the work of Rahul(2007) up on is suggestion to integrate statistical and
demographic methodology in performing age-specific population projection.
The aim of the study is to improve current methodology in population projection for making
probabilistic population projection using cohort component method under Bayesian approach
for India and States. The total fertility rate and female and male life expectancies at birth are
projected probabilistically using Bayesian models estimated via Markov Chain Monte Carlo
under WinBUGS software using Indian population data. These are then converted to age-
specific rates and combined with a Cohort Component Projection model. This yields
Probabilistic projections of any population quantity of interest, the method is illustrated for
four Indian state of different demographic stages, continents and sizes.
In India’s current projection method dose not yield an assessment of uncertainty about future
population quantities. It is somewhat subjective because the model used have been selected
by the analyst from a small number of predetermined possibilities rather than estimated from
the data. It is also somewhat rigid in that the set of model used is small and may not cover a
full range of realistic future possibilities. To address these issues, we will develop a Bayesian
probabilistic population projection method. This involves building Bayesian models to
project the fertility and mortality rate, each of which produces a large number of possible
future trajectories from the posterior predictive distribution. These are then input to the
cohort component projection method to provide a posterior predictive distribution of any
future population quantity of interest.
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Methodology
Cohort Component Projection Method
The procedure for making cohort-component population projections was developed by
Whelpton in the 1930s. It is uses the components of demographic change to project
population growth. The technique projects the population by age groups, in addition to other
demographic attributes such as sex and ethnicity. This projection method is based on the
components of demographic change including births, deaths, and migration. It can be thought
of as an elaboration of the ideas encapsulated in the demographic balancing equation:
𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 = 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 +𝐵𝑖𝑟𝑡𝑕𝑠 −𝐷𝑒𝑎𝑡𝑕 +𝐼𝑚𝑚𝑖𝑔𝑟𝑎𝑡𝑖𝑜𝑛 −𝐸𝑚𝑖𝑔𝑟𝑎𝑡𝑖𝑜𝑛 (1)
𝑡+𝑛 𝑡 𝑡 𝑡 𝑡 𝑡
where, Population is the population at time t, Births and Death are number of births and
t t t
deaths occurring between t and t+n. Immigration and Emigration are the number of
t t
immigrants and of emigrants from the country during the period t to t+n. This equation
reminds us that there are only two possible ways of joining a population: one can be born into
it or one can migrate into it. Similarly, the only ways to leave a population are to emigrate or
to die. Cohort-component projections extend this concept to individual age cohorts. They
make use of the fact that every year of time that passes, every member of a population
becomes a year older. Thus, after 5 years the survivors of the cohort aged 0-4 years at some
baseline date will be aged 5-9 years, 5 years after that they will age 10-14 years, and so on.
The Age-specific fertility (the ability of an individual to give a livebirth) rates are required to
project the number of births in future fertility projections, which are made by simulate a large
number of trajectories of future values of the total fertility rate (TFR) and convert them to
age-specific fertility rates using model fertility schedules. Next, the Age and sex-specific
death rates required to project the total deaths in future mortality projections, which are made
by simulate an equal number of trajectories of life expectancy at birth for females and males
and convert them to age-specific mortality rates using a model life table. In this study, to
simulate future values of TFR by using Gompertz model and to simulate future values of life
expectancy at birth separately for both the sex by using Logistic model under the Bayesian
approach.

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Bayesian Inference
In Bayesian inference, uncertainty about the parameters 𝜃 of a statistical model is described
by its posterior probability distribution given observed data 𝑥 = 𝑥 ,𝑥 ,…..,𝑥 . The
𝑖 1 2 𝑖
probability density function of 𝑥 is obtained by using Bayes Theorem:
𝑖
𝑓(𝑥 |𝜃)𝑓(𝜃)
𝑓 𝜃 𝑥 = 𝑖 , (a)
𝑖
𝑓(𝑥 )
𝑖
where𝑓(𝑥 |𝜃) is the likelihood function and is defined by the model, 𝑓(𝜃) is the prior
𝑖
distribution for 𝜃 and 𝑓(𝑥 ) is a normalizing constant. The prior distribution 𝑓(𝜃) specifies
𝑖
the uncertainty about 𝜃 prior to observing any data.
Forecasting or prediction is particularly natural in a Bayesian framework. Uncertainty about
next N future values of 𝑥 (for n=i+1,…..,i+N) is described by the joint predictive probability
𝑛
distribution
𝑓(𝑥 ……,𝑥 |𝑥 = 𝑓(𝜃|𝑥 ) 𝑁 𝑓(𝑥 |𝑥 ,𝜃)𝑑𝜃. (b)
𝑖+1, 𝑖+𝑁 𝑖) 𝑖 𝑗=1 𝑖+𝑁 𝑖+𝑁−1
Note that the product term represent the joint predictive distribution in the case that parameter
𝜃 is known. The Bayesian predictive distribution simply averages or integrates this with
respect to the posterior probability distribution for 𝜃. Hence, uncertainty about 𝜃 in light of
the observed data is fully integrated.
In a Bayesian analysis we obtain forecasts and associated measures of uncertainty by
calculating marginal probability distributions for quantities of interest by integrating the
posterior distribution in (a) or the predictive distribution in (b). Performing these integrations
analytically is typically not possible for realistically complex models such as those described
above. Historically, this has been prevented demographers and others from taking advantages
of Bayesian methods for statistical inference. Recent developments in Bayesian computation
have focused on Markov chain Monte Carlo (MCMC) generation of samples from
distributions such as (a) and (b); see Gelman et al.(2004) for details. Once a samples has been
obtained from a joint distribution, then a sample from a distribution of any component or
function of components is readily available. To generate samples from the posterior and
predictive distribution in this study, used an MCMC sampling approach implemented using
the WinBUGS software.
Probabilistic Projection of Fertility
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Let Y to denote TFR of India in the year t (i=1,…..,n), where i refers to TFR data in the ith
i i
year in a country and its initial value is taken to be 1. The four parameter Gompertz model
use for the projection of Total fertility Rate may be described as follows. Assume general
regression equation:
𝑌 = 𝜇 +𝜀
𝑖 𝑖 𝑖
Where 𝑌 be the population size, in the year 𝑡 has been assumed to follow normal
𝑖 𝑖
distribution with respective mean µi and common precision (=1/variance)T. Here , µi is the
deterministic part and 𝜖 is the disturbance part. Therefore, the disturbance part 𝜖 ~iid N(0,𝜏).
𝑖 𝑖
For the implementation of four parameter Gompertz model, we assume deterministic part as:
(−𝜃3−𝜃4 𝑡𝑖𝜎−𝑡 )
𝜇 = 𝜃 +𝜃 𝑒−𝑒
𝑖 1 2
Where, 𝜃 is the lower asymptote and in our study this parameter is fixed at 1.8, 𝜃 is the
1 2
upper asymptote, 𝜃 is the parameter that determines the shape of the Gompertz curve, 𝜃 is
3 4
the rate at which the fertility increases.
To adopt Bayesian analysis, we need to provide prior distributions to all the parameters
present in the model, 𝜃 , 𝜃 , 𝜃 and 𝜏. We prefer to assign non-informative priors for all of
2 3 4
them as N(0,0.01), (variance=1/0.001) prior has been assigned to all of the parameters
𝜃 , 𝜃 , 𝜃 and Gamma(0.00001,0.00001) to the parameter 𝜏.
2 3 4
Probabilistic Projection of Mortality
Let 𝑌 denote the Life expectancy at birth of a country in the year 𝑡 (i=1,2,3,…..,n), where i
𝑖𝑗 𝑖
represents the time and j represents the sex agin its value is taken as 1. The four parameter
logistic model is used for the projection of Life expectancy at birth. Let us assume the general
regression equation as:
𝑌 = 𝜇 +𝜖
𝑖𝑗 𝑖𝑗 𝑖𝑗
Where, 𝑌 be the life expectancy at birth for males and females of a country in the year
𝑖𝑗
𝑡 (i=1,2,…,n), where i represents time and j represent the sex. 𝜇 is the deterministic part and
𝑖 𝑖𝑗
𝜖 assumes to be error term which follows iid N(0,𝜏 ), where 𝜏 is the precision. The
𝑖𝑗 𝑗 𝑗
deterministic part of our logistic model is:
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𝜃
2𝑗
𝜇 = 𝜃 + +𝜃 ;𝑗 = 1,2
𝑖𝑗 1𝑗 (1+𝑒𝑒2𝑗𝑒𝜃3𝑗 𝑡𝜎𝑖−𝑡𝑡 ) 4𝑗
Where 𝜃 is the lower asymptote, 𝜃 ,𝜃 , 𝜃 is the……….
1𝑗 2𝑗 3𝑗 4𝑗
In this model, the life-expectancy at birth 𝑌 in the year 𝑡 has been assumed to follow
𝑖𝑗 𝑖
normal distribution with respective means 𝜇 and common precisions 𝜏 . We need to assign
𝑖𝑗 𝑗
priors for parameter. We prefer to assign non informative priors have been assigned to all the
parameters of the model.Normal(0,0.01), (variance=1/0.001) prior has been assigned to the
parameters 𝜃 , 𝜃 , 𝜃 ,𝜃 and Gamma(0.00001,0.00001) prior to the parameter 𝜏 .
1𝑗 2𝑗 3𝑗 4𝑗 𝑗
Future migration is more difficult to project that fertility or mortality. Migration can be
volatile since short-term changes in economic, social, or political factors often play an
important role. In this study, we assumed that the population is closed, i.e no migration takes
place, or even if it does, net effect is zero.
As for the sex ration at births which divide the future number of newborn into male and
female, the female to male ratio is set 100:105 it is based on biological literature, and it
remains consistent from 2011 onward.
Data
In this section, illustrate projecting method with the data for India. These data represent a
case in which the counts of all population components by five year of age and sex are
available. The data used to produce projection represent the three year moving average from
period 1971-2013. The data on mortality and fertility rates were obtained from published
statistical report of SRS. The India population obtain for Census 2001 and 2011, were 2011
census used as a baseline for prediction, was also obtained from the office of Registrar
General India.
To test and implement the proposed model in this study selected four states namely Gujarat,
Kerala, Orissa and Uttar Pradesh besides the country as a whole. These states were selected
considering their geographical and demographic diversities. While Uttar Pradesh is the
largest populated state in India, Gujarat is an economically developed state but lacking far
behind in terms human development indicators. Kerala ranks one while Gujarat ranks 7, Uttar
Pradesh ranks 15 and Orissa ranks 17 among the states of India based on the report
“Inequality-adjusted Human Development Index for India’s States 2011” published by
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UNDP India. In addition to the diversities, all these selected states are situated in four
different corners of the country.
Results
A Win BUGS program was developed to make a Bayesian analysis for the data to provide
projection of the TFR and Life-expectancy at birth for India and states. WinBUGS codes of
the models are given in the appendix. During the implementation of the program, we have
taken two chains to run for each programme.
Projection of Fertility
In this study, for fertility model, we monitored four nodes 𝜃 , 𝜃 , 𝜃 and 𝜎 = 1/√𝜏
2 3 4
separately for different dataset. Here we present the various diagnostics for Indian TFR data
based on 45000 iterations after burning 5000 initial iterations.
The history plots of the sample values of four nodes 𝜃 , 𝜃 , 𝜃 and 𝜎 against iterations the
2 3 4
two chains of the model have been show in the Fig1(a-d). the mixing of the chains (in
different colors) for all nodes in this model looks quite good giving us a confidence of the
convergence of chains. Fig 2(a-d) presents autocorrelations for different lags for all the four
nodes which decling trend with increase in lags. The value of R in the bgr diagnostic for the
all nodes are also close to one as show in Fig3(a-d). the traces of the blue and green lines are
stable and the red on has converged to one for all the four parameters monitored. Fig 4(a-d)
shows smoothed curves of the posterior densities of the nodes. The appearance of all the
curves is bell shaped indicating asymptotically normal.
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Fig1 (a-d) History Plots for parameters from two chains

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(a):theta[2] (b):theta[3]
0 0
. .
1 1
0 0
. .
0 0
0 0
.1 .1
- -
0 50 0 50
lag lag

(c):theta[4] (d):sigma
0 0
. .
1 1
0 0
. .
0 0
0 0
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- -
0 50 0 50
lag lag

Fig2 (a-d) Autocorrelation plots for parameters from two chains
(a):theta[2] (b):theta[3]
0 0
. .
1 1
5 5
. .
0 0
0 0
. .
0 0
5225 10000 20000 5225 10000 20000
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Fig3 (a-d) bgr Plots for parameters from two chains
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