Regression Estimation of Bongaart Indices from the Childbearing Indices: A Study of India/States/Districts

In a series of research articles El-khorazaty, Horne and Suchindran have showed how one can derive for any given population indirectly various childbearing and Bongaart fertilityinhibiting indices using only given information on the ASFRs, and the mathematical and regression models suggested by them. Very recently Bongaart revised his old model and suggested a set of new revised formulae to estimate various fertility-inhibiting indices. Following El-Khorazaty and Horne it is aimed to show in the present paper how one can derive various Bongaart revised fertility-inhibiting indices from the given information on various childbearing indices which were further seen derived from the only given information on TFR and a set of regression models that were earlier suggested by the first author and it is shown that the present study succeed in giving meaningful estimates for India its States, UTs, and Districts. Various regression models referring to estimation of childbearing indices used in this study were developed earlier by Ponnapalli using the state level time series of ASFRs overtime of the SRS of India and Horne et al., mathematical model. The regression models used in indirect estimation of the fertility-inhibiting indices from the TFR and also from the childbearing indices were developed by Ponnapalli using the Bongart indices of the DHS surveys earlier given by Bongaart in his revised recent study.


INTRODUCTION
In an interesting study El-khorazaty (1992) proposed a 'multivariate regression model' to estimate Bongaart fertility-inhibiting indices of Cm, Cc, and Ci from the information on childbearing indices. Horne and El-khorazaty (1996) further suggested a methodology to estimate childbearing and Bongaart indices from the ASFRs derived using the Coale-Trussell model fertility schedules. To be precise, in a series of research articles El-khorazaty, Horne and Suchindran showed a way how to derive indirectly various childbearing and Bongaart indices from the given information on ASFRs and the models derived by them.
In a recent study Ponnapalli (2016), following the above researchers, suggested a regression methodology to estimate the childbearing and Bongaart indices even from the total fertility rate (TFR), without the need of ASFRs. So far no researcher tried to provide and study district level variations in fertility using the estimates of childbearing and Bongaart indices due to obvious reasons. So the present study is such an attempt.
The two specific objectives of the present paper are: 1) For districts in India and states, for 1997 and 2011, to derive systematically various Bongaart revised indices indirectly, from the given set of childbearing indices using a regression approach earlier suggested by the first researcher.
2) To study the progress in the fertility transition in India during 1997 to 2011 using the estimate thus obtained and based on the study results to suggest some policy implications.
Details of the data used, methodology followed, analysis of the results, discussion of the results and conclusions brought out from this study followed by references are provided in the follow sections.

DATA
Data required for the present study are a set of childbearing indices for India, its states and 640 districts that refers to 1997 and 2011.
This study borrows the above information from another study made by the present authors (See Ponnapalli and Akash, 2019). To state, these childbearing indices were derived using a set of regression models. For convenience, the relevant regression models of the above study are presented here as Appendix Table 1. For details of the models one may refer the above study by Ponnapalli and Akash (2019). It is seen from Appendix Table 1 Ponnapalli, 2017). Secondly, the present paper they derived a set of Bongaart indices for each of the State, UT, and 640 Districts in India for 1997 and 2011. Validity of the present indices obviously depends on the validity of the TFR (RSM) that was the main basic input for deriving the earlier childbearing and Bongaart indices here. To state, validity of the TFR (RSM) was tested by the researchers by comparing them with the another set of TFR estimates indirectly derived by Guilmoto, and Rajan (2013) who further used a modified reverse survival method earlier suggested by Bhat (1996), christened here as (MRSM). It is seen that estimates made by both the methods were highly correlated with an R-Square value of 0.96. It is to state, the validity of the agesex distribution data which was used in deriving the TFR (RSM) were further verified by means of calculating age-sex accuracy indices such as Whipple's Index, and Myers Index at district level in India. (See Akash and Ponnapalli, 2017). El-khorazaty (1992) in his paper entitled "estimation of fertility-inhibiting indices using vital registration data" proposed a 'correspondence model' to derive the Bongaart indices. It uses Bongaart (1978Bongaart ( , 1982Bongaart ( , 1983) PD model (old version) and provided indices relevant to Cm, Cc and Ci. However, Ponnapalli (2016) model proposed here to estimate Bongaart indices, unlike El-khorazaty (1992) model, uses the very recent modified version of the Bongaart (2015) model that allows even to estimate a value for the index Ca.

Models
Information provided in this section of the paper is heavily borrowed from the Bongaart recent paper entitled "Modeling the fertility impact of the proximate determinants: Time for a tune-up" (See Bongaart, 2015).
As we know already, like mortality and migration, fertility is also determined by a number of factors. The factors that determine fertility are traditionally divided into background and proximate determinants. Background determinants (such as sociocultural, economic and environmental factors) are theorised to effect fertility only through their effect on proximate determinants (behavioural factors such as use of contraception and abortion, prevailing practices of marriage and breastfeeding). The interrelationship between these background and behavioural factors were first recognised by Davis and Blake (1956) and they proposed a set of 3 of behavioural factors which again consists of 11 variables in total.
Davis and Blake (1956) christened these 11 variables as "Intermediate fertility variables" as background variables affects the fertility only through these selected variables. Several researchers further attempted to understand and simplify the process suggested by Davis and Blake. Fortunately, in late 1970s, Bongaart succeeded in suggesting a set of 7 intermediate variables which were christened by Bongaart as "proximate determinates (PDs)" of fertility (See Bongaart, 1978, 1982, 1983. Bongaart (1978) further developed a 'simple' model to quantify the effect of various PDs, especially of four crucial PDs, on fertility (TFR/CBR). His model was widely used in studying and understanding levels, trends and differentials in fertility in terms of proximate determinants and their effect of many a number of countries in the world (Bongaart, 2015).

Stover (1998) followed by other researchers made a number of attempts to revise this
Bongaart late 1970s PD model. Certain drawbacks related to various assumptions made by Bongaart in the original model and the recent changes taken place in the factors that determine fertility with a progress in the demographic transition of world countries, genuinely led to the suggestions and modifications for the original model. Interestingly Bongaart (2015) himself made a further attempt to revise his original model incorporating 'six adjustments' and tried to show that his 'new revised model provides an improved assessment of the roles of the proximate determinants. This was made possible through his experiments using the recent DHS survey data of 36 countries carefully chosen by him. (Bongaart, 2015, page 554). Bongaart (1978) study suggested two different models namely 'aggregate model' and 'age-specific model'. The aggregate model which is widely used is as given below: states TF is recognised to be the same as the TFR but is a 'hypothetical' one that is assumed to be 'around 15 births per woman' seen in any population when Cm = Cc = Ci = Ca = 1.
Using the DHS survey information and the following formulae of Cm, Cc, Ci, Ca earlier suggested by Bongaart (1978Bongaart ( , 1982Bongaart ( , 1983 further given in        Table 4, of Bongaart (2016).
A summary picture of the Bongaart age-specific PD model and relevant equations of the indexes originally proposed and given by Bongaart and Potter (1983) are shown below as table 2, that which is again extracted from Bongaart (2015, page 539). The age-specific PD model however is having an advantage over the aggregate model, its use is observed to be limited as it demands more detailed data to calculate various indexes. Bongaart (2015) realized that each of the above proximate determinants require a revision in their calculation

Regression Models for Indirect Estimation of Bongaart Revised Indices from the Childbearing Indices
After an understanding of the Bongaart original and revised models it is easy to understand and appreciate the indirect estimate procedure proposed by Ponnapalli, the first author, which is given below: The above regression models observed to have highest percent of variation explained (i.e., R-Square). These models were further seen to be derived using a set of regression models earlier derived by Ponnapalli (2016). Using the childbearing indices and the regression models 1 to 4 given above, a set of proximate determinants are derived for the years 1997 and 2011 (These estimates may be obtained from the researchers).

Understanding fertility transition in India at the state level during 1997 to 2011: Using various indirect estimates of Bongaart indices
Fertility transition taken place during 1997 to 2011 at the major state level is studied here by means of finding the relative contribution of a change in a particular PD over time from 1997 to 2011 to the overall change in the TFR during the same time period of 1997 to 2011.
The following formulae are used for the same:
Here it is realized TFR = CmCcCaCiTF Appendix Table 2  For convenience the AARG is shown in Figure 1 in positive terms for ease of presentation but when it is noticed in Appendix Table 2 the same is shown negatively as the formulae used gives it in negative terms. Both are observed to be the same except for the sign used and not to be confused.
After a keen observation of figure 1, figure 2 and from the results given Appendix Table 2 below panel, it is concluded that: 1) The AARG in TFR from 1997 to 2011 of the 5 states of : Uttar Pradesh, followed by West Bengal, Uttarakhand, Bihar, Assam seen to be more than the other states during the said period.
2) The AARG in TFR from 1997 to 2011 of the 2 states of: Kerala and Tamil Nadu observed to be very low, obviously it is seen that these are the two states with very low fertility during the study period. As fertility transition is almost completed we cannot expect much change over time in these and other low fertility states.

Understanding fertility transition in India at the district level during 1997 to 2011: Using various indirect estimates of Bongaart indices
We may summarize the district level analysis results in this section of the paper again by means of depicting the progress in PDs by box plots as shown in figure 4.   Bongaart (2015). To be specific Bongaart (2015, page 549) states that "As expected, the indexes Cm, Cc, Ca decline as countries move from high to low fertility and the inhibiting effects of these PDs become stronger. In contrast, Ci rises as countries move through the transition because breastfeeding and postpartum abstinence decline." Thus said, the above two sections of the paper well proved the usefulness of the Bongaart indices derived indirectly and their validity as reliable estimates also as the results corroborate the finding from the study by Bongaart (2015).

CONCLUSIONS
In conclusion it may be stated that the Bongaart indices derived here, indirectly from the childbearing indices, seem to be quite acceptable and also found to be useful in an Pradesh.

ACKNOWLEDGEMENTS
The above authors sincerely thank both the reviewers for their helpful comments and the editors of the present journal for giving an opportunity to publish their article in one of the prestigious journals of the Mekelle University, Mekelle, Ethiopia.  (2019), models prepared by Ponnapalli.

Appendix
Note: (1): All the coefficients of various independent variables and constant terms given in various models above are observed to be statistically significant having a t -statistic value of more than 2.0.   1997-2011 1997-2011 1997-2011 1997-2011 1997-2011 1997-2011