Md. Rafiqul Islam

Correspondence:
Dr. Md. Rafiqul Islam
Associate Professor and Chairman
Dept. of Population Science and Human Resource Development
University of Rajshahi
Bangladesh.
E-mail: rafique_pops@yahoo.com

Introduction

Bangladesh is placed in southern Asia, and borders the Bay of Bengal, between Burma and India and occupies a total of 1,44,000 square kilometers and it is situated latitudinally between 21⁰5/ and 26⁰4/ North and longitudinally between 88⁰5/ and 92⁰5/ East. The country has a tropical climate with cool, dry weather conditions from October to March. The monsoon season is characterized by heavy rainfall from June to October and summers are hot and humid. It is, at times, affected by severe natural disasters such as floods, cyclones, water logging and droughts. Bangladesh is hilly in the Southeast consisting mostly of flat alluvial plains. More or less around 73 percent of the land of Bangladesh is arable. The landscape has an extensive network of rivers that is very important in boosting the socioeconomic status of the nation. Ganges-Padma, Brahmaputra-Jamuna, and the Megna are the main ones among them.

The population has increased from 42 million in 1941 to 129.25 million, of which 65.84 million were male and 63.41 million are female in 2001, published by Bangladesh Bureau of Statistics (BBS) (BBS, 2001). Bangladesh is one of the most developing countries in modern times with accelerated population growth at a rate of 1.47 percent per year. It is one of the most densely populated countries. In fact, it is the ninth most populous country on this globe having a population density of 834 persons per square k.m. (Mitra and Associates, 2001).

Bangladesh, like many other developing countries, has not yet started completing a Vital Registration System (VRS) though the Bangladesh Demographic Health survey (BDHS) collects data on a sampling basis but they did not provide sufficient information of mortality data. For this, information on mortality fully depends on censuses as well as other sources using some sophisticated indirect techniques. A number of works on fertility has been carried out but mortality works are studied on a limited scale in our country. For this reason, an effort has been given attention to estimate indirectly some mortality measures 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 tool in Population Science, particularly, in Demography in the modern technological era. Life table is widely used to estimate various demographic parameters such as IMR, CDR, ASDRS, net reproduction rate (NRR), survival rate, and replacement index. It is also applicable to enumerate net migration rate. It is also broadly used in life insurance companies.

Two abridged life tables, one for males and other for females have been constructed applying Widowhood Method (Ali, 1990). Then, in this report, IMR, CDR and ASDRs for males, females and both sexes were also calculated from the constructed abridged life tables and showed that ASDRs follow a traditional U-shape pattern. In (Islam, 2006), ASDRs, life table CDR and IMR for males, females and both sexes of Bangladesh in 2005 have been estimated indirectly from the constructed abridged life tables in which these were showing a decreasing trend. So, it is seen that there have been changes in mortality levels in Bangladesh.

Therefore, the main objectives of this study are:

Data and Methodology

The life expectancy at birth for males is 63 and for females is65, taken from UN (2006) and is used as raw data in this study.

The linear interpolation technique (UN, 1983) is applied to estimate lx values for corresponding life expectancy at birth using South Asian Model Life Tables from the United Nations Model Life Tables for Developing Countries (UN, 1982), then to construct an abridged life table using the following functional relationships:
which can be approximated as where as L₀=0.3l₀+0.7l₁ and L₁=0.4l₁+0.6l₂ , that is equivalent to , and (Biswas, 1988; Keyfitz, 1968; Shryock, 1975). 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 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, q₀.

Model Fitting

It appears from the scattered plot of the number of persons surviving at an exact age x (l_x) for males and females by age groups (Fig.1) that l_x can be distributed by a polynomial model for different ages. Therefore, an nth degree polynomial model is treated and the structure of the model is given by

i) (Montgomery and Peck, 1982),
where, x is age group; y is l_x for male and female; a₀ is the constant; a_x is the coefficient of xⁱ (i =1, 2, 3, ..., n) and u is the error term of the model. Here, in both cases, an appropriate n has been selected such that the error summation of square is least.
Using the software STATISTICA, the mathematical models have been estimated.

Model Validation Technique

To test the stability of the model, 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 shrinkage of the model is the absolute value of the difference of and R². Moreover, the stability of R² of the model is equal to (1- shrinkage) (Stevens, 1996).

F-test

The F-test is applied to the model to verify the measure of the 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, 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 2006 have been estimated and presented in the respective tables. Moreover, ASDRs for both sexes of Bangladesh in 2006 have also been enumerated and presented 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-shape 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 2005-2006, ASDRs of Bangladesh in 2005 are taken from Islam (2006) and presented in Table 3. It is observed that ASDRs for males, females and both sexes in 2006 are strictly lower at every age, than that of ASDRs for males, females and both sexes in 2005, excepting the last age group. That is, they are indicating a decreasing trend with passing of time.

The CDR and IMR for males, females and both sexes of Bangladesh in 2006 are calculated using the information from the estimated abridged life tables and presented in Table 4. To estimate CDR for both sexes in 2006, life expectancy is estimated as 63.50675 years from the constructed male and female life Tables. To see the trend of CDR and IMR during 2005-2006, these have been taken from Islam (2006) and presented in the first row of Table 4. Here, it is observed that CDR and IMR for male, female and both sexes in 2006 are exhibiting a decreasing trend during 2005-2006.

The fitted model of lx values for male of Bangladesh in 2006 is:
y=93634.87-660.195x+24.27725x²-0.32979x³ ... (i)
t-stat (88.19) (-4.521) (5.49790) (-9.34711)
providing the coefficient of determination R² is 0.99246 and = 0.9892.

The fitted model of lx values for female of Bangladesh in 2006 is:
y=94361.12-842.215x+31.30959x²-0.3807x³ ... (ii)
t-stat (106.53) (-6.914) (8.49957) (-12.9340)
providing the coefficient of determination R² is 0.99384 and = 0.9912.

The information on model fitting has been shown in Table 5. From this table it is seen that the fitted models are highly cross-validated and their shrinkages are only 0.003279 and 0.002679 respectively. Furthermore, the fitted models will be stable more than 98% and 99%. Moreover, from this table, it is seen from the statistical view that the parameters of the fitted models are highly significant both explaining more than 99% of variance. 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 789.76 and 968.03 with (3, 18) d.f. 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 the R2 are highly statistically significant. Hence, the fit of both models is well.

Age associated with instantaneous force of mortality ( ) is calculated from the fitted model of lx values for male and female populations of Bangladesh in 2006 and that is presented in the 5^th and 6^th columns of Table 3 and shown in Fig. 1. From Table 3 and the figure, it is found that force of mortality for male and female is decreasing up to age group 25 and strictly increasing in the whole range, but rapidly increasing after the age interval 55 and above.


Click here to view table 1 - 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 ASDRs, CDR and IMR for male, female and both sexes are showing a decreasing trend over time during 2005-2006. It is seen 1_x that the values for male and female follows the 3^rd 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 and increasing in the remaining age interval. We hope the latest findings on the 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 by the updated information of life tables in this study to boost their plan of insurance.

References