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ABSTRACT
The aim of the present
study is to estimate some mortality measures
such as age specific death rates (ASDRs), crude
death rate (CDR), infant mortality rate (IMR),
infant death rate (IDR) and child mortality
rate (CMR) of Bangladesh in 2007. For this purpose,
two abridged life tables for male and for female
have been constructed. The secondary data on
life expectancy at birth in Bangladesh is taken
from UN (2006). These estimates are compared
to the corresponding values of Bangladesh in
2005 and it is observed that these are showing
decreasing trends during 2005-2007. Moreover,
a mathematical model has been fitted to the
number of persons surviving at an exact age
x (lx) for male and for female. Age
associated with instantaneous force of mortality
( )
for male and female has been estimated. And
it is found that is exhibiting
decreasing trend up to age 25 for male and to
age 30 for female and increasing in the remaining
age group but rapidly increasing after age 55
years to infinity.
Key
words and Phrases: Life expectancy at
birth Interpolation method Life table Mortality
measures Modeling Force of Mortality Bangladesh.
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Introduction
Like
many other developing countries, Bangladesh has not
started complete Vital Registration System (VRS) yet.
Nevertheless, Bangladesh Demographic and Health Survey
(BDHS) procure data on a sampling basis but they do
not provide sufficient information on mortality data.
For this, information on mortality completely depends
on censuses as well as other sources by means of some
sophisticated indirect techniques. A number of good
research works on fertility has been carried out but
mortality works are studied in a very limited scale
in Bangladesh. On these grounds, an endeavor has been
given interest to estimate some mortality measures
indirectly from very simple data. In this study, firstly,
two abridged life tables have been constructed. It
is to be noted here that a life table is very sophisticated
and mathematical tools in Population Science, particularly,
in Demography in up to the minute technological period.
Life table is far and wide used to estimate various
demographic parameters such as IMR, IDR, CDR,CMR,
ASDRs, net reproduction rate (NRR), survival rate,
replacement index. It is also applicable to enumerate
net migration rate. It is also broadly used in life
insurance companies.
Abridged
life table for male has been constructed applying
Widowhood Method (Islam et al, 2003). Then, in this
report, IMR, CDR and ASDRs were calculated from the
constructed life table and showed that ASDRs follow
a traditional U-shape pattern. Abridged life table
for female has been constructed applying Widowhood
Method from male widowed information (Islam, 2005).
Islam (2006) estimated indirectly ASDRs, CDR and IMR
of Bangladesh in 2005 from the constructed abridged
life tables in which these were showing a decreasing
trend. So, it is found that there have been changes
in the declining trend in mortality levels in Bangladesh.
Therefore,
the main aims and objectives of this study are given
in the following:
i) to construct life tables for males and females
and hence to estimate IMR,
IDR, CDR, ASDRs, and CMR of Bangladesh in 2007, and
ii)
to fit a mathematical model to lx values for males
and females and then to
estimate age associated with instantaneous force of
mortality ()
for males and females of Bangladesh in 2007.
Data and Methodology
The
life expectancy at birth for maless=s63 and for femaless=s65
of Bangladesh in 2006 are taken from UN (2006). It
is noted that the life expectancy at birth in developing
countries is gaining at a rate of 0.5 (UN, 1967).
Therefore, it is assumed that the life expectancy
at birth for males = 63.5 and for females = 65.5 of
Bangladesh in 2007, that is used as raw material in
this study.
To
estimate lx values for corresponding life expectancy
at birth using South Asian Model Life Tables from
United Nations Model Life Tables for Developing Countries
(UN, 1982), the interpolation method (UN, 1983) is
employed here. Thereafter, abridged life tables are
constructed using the following mathematical relationships:
which can be approximated as 
where as L0=0.3l0+0.7l1
and L1=0.4l1+0.6l2  ,
that is equivalent to  .
Thereafter,
ASDRs are indirectly estimated from the constructed
life table using the formula ASDRs =  (Barclay,
1958). These are shown in Table 1 and Table 2. Moreover,
ASDRs for both sexes are estimated by using the formula
ASDRs =
from the constructed life tables of male and female
and presented in the last column of Table
2. Life table CDR is estimated using the formula
 ;
where e0 is the life expectancy at birth.
For this, the expectation of life at birth for both
sexes is estimated from the constructed life tables
using the formula  ;
where as 
is the expectation of life at birth for male, 
is the expectation of life at birth for female and
s is the sex ratio at birth. In the case of this study,
it is assumed to be 1.05 as a developing country (UN,
1967). Moreover, IMR is also calculated and in this
case it is, in fact, 1q0,
that is, the probability of dying before the first
birthday. Furthermore, CMR is estimated and it is
indeed 4q0 , that is, the probability
of dying between the first and fifth birthday.
The
force of mortality at age x is defined as the ratio
of instantaneous rate of decrease in lx to the value
of lx. It is denoted by and
is given by the following expression in Differential
Calculus:
 (Biswas,
1988; Keyfitz, 1968; Shryock, 1975).
Model Fitting
It appears from the scattered plot of the number of
persons surviving at an exact age x (lx) for males
and females of Bangladesh in 2007 by age groups that
lx can be distributed by polynomial model for different
ages. Therefore, an nth degree polynomial model is
treated and the structure of the model is given by
 (Montgomery
and Peck, 1982),
where
x is age group; y is lx for males and females of Bangladesh
in 2007; is the constant; is the coefficient of (i
=1, 2, 3, ..., n) and u is the error term of the model.
Here, in both cases, appropriate n have been selected
such that the error summation of square is the smallest
amount.
Using
the software STATISTICA, the mathematical models have
been estimated.
Model Validation
Technique
To
test how much the model is stable, the cross validity
prediction power (CVPP),  ,
is applied here. The method for CVPP is given by
 ;
where n is the number of cases, k is the number of
predictors in the model and the cross-validated R
is the correlation between observed and predicted
values of the dependent variable The absolute value
of the difference of
and R2 is the shrinkage of the model. Moreover,
the stability of R2 of the model is equal
to (1- shrinkage) (Stevens, 1996). Note that CVPP
was also employed as model validation by Islam (2003,
2004, 2005 and 2006) and Islam et al (2003).
F-test
The
F-test is applied to the model to verify the measure
of overall significance level of the model as well
as the significance of R2. The formula
for F-test is affirmed as
with (k-1, n-k) degrees of freedom (d.f.);
where k = the number of parameters to be estimated
in the model, n is the number of classes and R2
= the coefficient of determination of the model (Gujarati,
1998).
Results and Discussion
Two
abridged life tables for males and for females have
been constructed and shown in Tables 1 and 2 respectively.
ASDRs for males and females of Bangladesh in 2007
have been estimated and presented in the respective
tables. Moreover, ASDRs for both sexes of Bangladesh
in 2007 have also been enumerated and shown in the
last column of Table 2.
To
see the pattern of ASDRs of Bangladesh, these are
plotted in graph paper, then, it is seen that they
follow a traditional pattern, that is, U-shaped pattern.
It is to be noted here that traditional pattern of
ASDRs is a U-shape pattern ( Misra, 1995; Shryock
and Associates, 1975). To see the trend of ASDRs during
2005-2007, ASDRs of Bangladesh in 2005 is taken from
Islam (2006) and presented in Table 3. It is observed
that ASDRs for male, female and both sexes in 2007
are strictly lower at every age than that of ASDRs
for male, female and both sexes in 2005 excepting
the last age group. That is, they are indicating decreasing
trend with passing of time.
The
CDR, IMR, IDR and CMR for male, female and both sexes
of Bangladesh in 2007 are calculated using the information
from the constructed life tables and presented in
Table 4. To estimate CDR for both sexes in 2007, life
expectancy at birth is estimated as 63.99 years from
the constructed male and female life tables. To see
the trend of CDR, IMR, IDR and CMR during 2005-2007,
the CDR, IMR, IDR and child mortality rate in 2005
have been taken from Islam (2006) and presented in
Table 4. Here, it is observed that CDR, IMR, IDR and
CMR for male, female and both sexes in 2007 are exhibiting
decreasing trend
during 2005-2007.
The
fitted model of lx values for male of Bangladesh in
2007 is
y=93897.52-657.69x+24.5363x2-0.33271x3
... (i)
t-stat (91.76) (-4.673) (5.76525) (-9.78384)
providing coefficient of determination R2 is 0.99293
and =
0.989861.
The
fitted model of lx values for female of Bangladesh
in 2007 is given below:
y=94593.06-838.18x+31.48346x2-0.3825x3
... (ii)
t-stat (110.18) (-7.099) (8.81767) (-13.4087)
providing coefficient of determination R2
is 0.99413 and =
0.991577.
The
findings on model fitting have been shown in Table
5. From this table it is seen that the fitted models
are highly cross-validated and their shrinkages are
only 0.003073 and 0.002553 respectively. Furthermore,
the fitted models will be stable more than 98% and
99%. Moreover, from this table, it is seen in the
view of statistics that the parameters of the fitted
models are highly significant, both explaining more
than 98% and 99% of variance respectively. From t-statistics,
it is found that all the parameters of the model are
also highly significant. In both models, the stability
of R2 is more than 99%.
The
calculated value of F-test of the models are 842.6563
and 1016.15 with (3, 18) d.f. respectively whereas
the corresponding tabulated value is only 5.09 at
1% level of significance. Therefore, it seems from
the statistics that the overall measure of the fitted
models and its R2 are highly statistically
significant. Hence, the fit of both modelsis good.
Age associated with instantaneous force of mortality
()
is calculated from the fitted model of lx values for
the male and female population of Bangladesh in 2007
that is presented in the 5th and 6th columns of Table
3 and shown in Fig.1. From the Table 3 and figure,
it is found that force of mortality for males and
females is decreasing up to age group 25 for males
and to age 30 for females and strictly increasing
in the whole age range, but rapidly increasing after
the age interval 55 and above, that is, to infinity.
Click
here for Table 1 - Table 5
Fig
1. Force of Mortality for Male and Female of
Bangladesh in 2006. X: Age group and Y: Force of Mortality
Conclusion
It
is found that the CDR, IMR, IDR and CMR for males,
females and both sexes are showing a decreasing trend
over time during 2005-2007. It is seen that the lx
values for males and females follow a 3rd degree polynomial,
i. e. cubic polynomial model. It is observed that
force of mortality ()
for males and females is decreasing in the age interval
0 to 25 for male and zero to age 30 for female and
increasing quickly in the residual age interval.
It is expected that the updated results on life table
as well as mortality would encourage the government
and non-government organizations (NGOs), researchers,
academicians and planners to plan to bolster the socio-economic
development and health care program in the country.
Life insurance companies might be helped from the
most up to date information of life tables in this
study to boost their plan of insurance.
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