A procedure for estimating a few demographic indicators of Assam state and its districts

Introduction

Life expectancy at birth ( 0 e ), i.e., the longevity of a new born based on age-specific death rates, is an important indicator to determine the mortality level of a population from which demographers can draw an idea about the socio-economic condition and health status of that particular population (Najafi, et al., 2018, Motlagh, et al, 2012) [7,6] . However, life expectancies at ages other than zero are also used for studying mortality pattern of cohort population. In the case of most developing countries and historical populations, high incidence of infant and child mortality results in lower values of 0 e than at other childhood ages, and in such populations, those surviving the hazards of early childhood have a higher life expectancy than new born and the highest life expectancy does not occurs at birth but at later age (Romo and Becker, 2011) [9] . Phukon and Ahamed (2019) [8] estimated life expectancy at birth by developing two polynomial regression models comprising child survivorship probabilities at ages one, two, three and four, i.e., l using 54 set of SRS based abridged life tables of Assam. Prior to this, Romo and Becker (2011) [9] and Sarma and Choudhury (2012) [11] had estimated the life expectancy at birth by developing the linear and quadratic regression models between 0 e and survivorship function at age one respectively. In some of the earlier literatures, it can always be found that 0 e is somehow inextricably related to life expectancies at childhood ages. In retrospect, Lederman (1969) [5] developed a series of one parameter and two parameters regression models, and thus a number of life tables were constructed based on such models; the author proposed model life tables by generating the logarithmic regression equation between probability of dying at ages x and (x+5).

A number of sets of regression coefficients for each equation based on different independent variables (e.g., 0 e , 10 q , 5 0 q , 15 0 q , 20 30 q , 20 45 q and 50  m ) or a pair of variables were obtained.

Hence, it can be observed that 0 e is also indispensably influenced by mortality conditions in childhood ages. Roy (1989) [10] obtained reliable estimates of child survivorship functions, l = 1 -q xx , for India using logit smoothing. In this paper, a five parameters polynomial regression model

, where l by fitting polynomial regression on aforesaid SRS data of Assam. Life expectancy at birth is estimated by using the fitted model as the estimating equation Assam. With the sequential application of Trussell's variant of Brass method, logit smoothing and fitting of Weibull survival function (Sarma and Choudhury, 2012, 2014) [11,12] at district level of Assam, estimates of

Materials and Methods

In this paper, 69 sets of SRS-based abridged life tables of Assam, which was published by office of the Registrar General of India, from 1970-75 to 2012-16, 23 each for male, female, and male and female combined are used. The polynomial regression model proposed in this paper is fitted to this SRS data of Assam. For estimating the demographic parameters such as, 0 e , IMR and U5MR for sub-state level of Assam, CEB, CS and total numbers of women in child bearing ages are taken from Indian Census data of 2001 and 2011 of Assam. Moreover, in selecting suitable standard model life table for Assam, each of five United Nations model life tables for developing countries (1983) [13] and four Princeton models-Coale & Demeny model life tables (1966) [3] is examined to find out which standard model is most appropriate for Assam.

Selection of a Model Life Table Family for Multipliers

The appropriate model life table family has been selected by using the following procedure. At first, in graphical method, the estimates of 1 4 q are plotted against those of 0 1 q (Kenneth, 2013) [4] on a graph showing the corresponding relationships in model life tables, shown in Figure 1. Although observed data of Assam reflects trivial nature in this graphical representation, somehow the graph of model life table which is closely fitted to that of observed data is selected. Secondly, we calculate

x m is the Age Specific Mortality Rate (ASMR) of observed data,

x M is the ASMR of standard life table, L is the level that corresponds to that of observed data. The model set that minimizes the sum is taken as appropriate model life. In both methods, Assam's SRS data is found to be most compatible with United Nations South Asian model life tables.

Trussell's Variant of Brass Method

Brass developed a method for converting the proportions dead of children ever born reported by women in childbearing ages into estimates of the probability of dying whereas translating proportions of children dead among children ever born into standard life table indicators was first pioneered by Brass and Coale (1968) [1] . If mortality has changed over time, the estimated probabilities of dying reflect the mortality rates that have prevailed at a range of ages and dates. The following steps are followed in estimating probability of dying. The average parity per woman is computed by using

, where CEB is the children ever born for age group i, i = 1,2,…7 for age-groups called child bearing age groups, i.e., 15-19, 20-24,… 45-49 respectively and N(i) is the total number of women of the age-groups. The proportion of children dead for each age-group i is estimated as

, where CS(i) is the number of children surviving for the age-group i and CD(i) is the children dead reported by the mothers belonging to the age-group i. The multipliers K(i) are computed according to the Trussell's variant of original Brass method. Equations are obtained as

, where a(i), b(i) and c(i) are coefficients for estimation of child mortality multipliers, which is taken from United Nations (1983) [13] South Asian model in this context. Multiplier is the factor to convert the proportion of children dead into a probability of dying which will be estimated by using,

Weibull Survival Function

The survival probability function for age x of Weibull distribution is

1981) [2] ; it can be written as

 

. We can also obtain (Roy, 1989) [10] . Using the smoothed values of ˆx P ,

x l can be finally estimated by using the relation 2 ˆ11

x P x l e   . These x l ˆvalues are used in the polynomial regression model fitted to SRS data of Assam to estimate life expectancies at birth for the state of Assam and her all districts.

Polynomial Regression Model

A second degree polynomial regression model has been derived by establishing the relationship between the life expectancy at birth, 0 e and the survivorship probabilities Becker (2011) [9] and Sarma and Choudhury (2012) [11] respectively as follows:

  is the probability of survivorship function at age one.

The equation ( 1) can be extended as

where  is the highest age attained by a member of population. Considering the effect of childhood mortality changes in age interval (0-3) on 0 e , the following equation can be obtained:

is a weighted linear function of 5. It is seen from this table that Pearson correlation co-efficient between   t e 3 and   3 lt in all model life table families are exceptionally high. It is also known that Pearson correlation coefficients are often used as an indicator of strength of linearity between two variables. Moreover, the consistency of this assumption is reflected in the final results given by the model constructed based on the assumptions.

Results

Table 1 is presents fitted polynomial regression models along with coefficients of determination (R 2 ) and standard errors (SE), in each case, in all five United Nations model life tables and four Princeton-Coale & Demeny's model life tables sexwise, and also for Assam sex-wise as well as total of male and female. The goodness of fit has been seen from the high coefficients of determination and low standard errors of fitting in all model life table families and SRS life tables of Assam. It is analytically justifiable that the polynomial regression model constructed based on certain assumptions and mathematical formulation of vital statistics works realistically in estimating life expectancy at birth. Tables 2 to 4 present the estimated life expectancies at birth, IMRs and U5MRs of Assam and its districts in 2001 and 2011 along with their percentage increase/decrease respectively.

References

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