Estimation of number of ever born children using zero truncated count model: evidence from Bangladesh Demographic and Health Survey

Abstract

Fertility is an important demographic indicator for any country and there has always been a concern for number of ever born children to know the transition of fertility pattern intensively. Child ever born is the count response variable ranges from 1 to 15 and was originally collected by the Bangladesh Demographic and Health Survey (BDHS) considering the reproductive women who had given at least one birth. This study proposes zero truncated Poisson and zero truncated negative binomial regression models in order to find the best fitted model to estimate number of ever born children using BDHS 2014 dataset. Findings reveal that, the number of children increases with the increment of respondent’s age but number of children declines if education status of respondents as well as their husbands’ increases. Similarly, religion, wealth index and wanted last child have significantly influenced the number of child ever born. Surprisingly, the number of children ever born to a mother from rural area does not differ significantly from that of urban area in Bangladesh, though there exists a little fluctuation in the number of children ever born to a mother living in seven administrative divisions. Intension of contraceptive use has no influence on number of ever born children to a mother.

Introduction

Number of children ever born is a measure of number of children born alive among the married women of age 15 years or more and it includes all the live birth that are living or dead from married women up to the time of collection of data. Child ever born is also called summary of birth histories which quantify all the live births a woman experienced in her lifetime. Not only this, data on children ever born for successive age groups of women provide information about complete trends of childbearing which can be used top redict fertility behavior of that region if census and survey data are inadequate or missing. For instance, data on the number of births by age of mother before 1967 were not available in Japan, it was not possible to compute total fertility by the way of age specific birth rates, so a method was developed by Watanabe to estimate total fertility from children ever born data [1] and this means that Children ever born has direct effect on fertility rate which can be estimated from number of it. In past, up to 1965, global fertility rate was more than 5 children per woman. Now-a-days, due to modernization of societies, number of children per woman has decreased substantially throughout the world and reached just below 2.5 children per woman [2]. This rate of transition is very rapid and surprising not only for developed countries but also for the developing country like Bangladesh. According to World Bank report, Bangladesh took 20 years to decline fertility from more than 6 children per woman to less than 3 children per woman. Bangladesh is a densely populated country and has an estimate of populations 166.37 million in 2018 and projected just over 200 million within 2053 [3]. So, this is going to be an enormous trouble for the government of Bangladesh to ensure basic needs for these populations in the future. That’s why, government is trying to influence its people to control the live birth by campaigning with the slogan of “Not more than two children, one is better”. But due to the increasing speed of population growth as well as pros and cons of overpopulation, the introduction of population slogan is not enough. Moreover, Bangladesh is passing through the demographic dividend from 1980s and will continue until 2040 and if the country fails to take potential economic benefits from it then Bangladesh has to pay huge cost with unemployment, unbearable stain on education, health and old age security [4].This research is going to contribute by developing a statistical model to predict the children ever born as well as their influential factors so that government can response significantly by taking necessary steps.

Literature review

The decision of childbearing is mainly influenced by psychological, economical as well as social and demographic factors. It is examined that relationship among the partner influences first and higher order birth rates [5]. Similarly, from economic view, demand for children is similar like the demand of goods which was based on a model developed by economist Gary Becker [6] and he found that improving the women education status results less likely to want fewer children due to adjustment of opportunity cost is for higher education by cutting down opportunity of higher number of children. Moreover, when the demand of labor is high and supply is low then fertility rate is relatively higher among the young couples which suggest that timing of fertility is closely linked with economic condition [7].

From the demographic point of view, mostly women were chosen as the respondents in order to study about childbearing, as well as prenatal and postnatal care [8, 9]. From the prior research, it is found that, in Japan, younger women have lower fertility intensions but it is higher among them who live in rural areas with larger family members [10]. Moreover, age, income level and childcare share with spouses were found important childbearing determinants among the Korean working women [11]. Another research found that, there is a negative effect of mother’s education on average number of children ever born to the women in Botswana [12]. Similarly, in accordance with educational level, birth level was also found very significant effects on children ever born among the reproductive women in Semnan, Iran [13]. Not only this, age at first marriage, gap between births, infant mortality, wealth index were significantly associated with fertility of Sudanese women [14]. However, not only wife but also husband’s wishes were found significant on fertility intensions [15].

In Bangladesh, some studies were conducted on fertility influence and it is found that, high infant mortality as well as economic security at later life has influenced the decision about family size [16] and this is because, though, Bangladesh had achieved the target of reduction of infant mortality from 151 per 1000 live birth in 1991 to 41.01 per 1000 live birth in 2017 but still the infant mortality is high compared to the developed countries. Another study showed that, education, employment as well as food security was key responsible childbearing factors among the women [17].

Different statistical methods were applied to study about children ever born including logistic regression [17], Poisson regression model [12–14], multiple classification analysis [18]. Winkelmann, and Zimmermann proposed generalized event count model performed better than Poisson model irrespective to under or over dispersion [19]. On the other hand, Ahmed & Nasser compared Poisson model with Support Vector Machine to predict children ever born in Bangladesh but prediction power of both models was not satisfactory. Beside this, support vector machine gives no idea about influential factors associated with children ever born [20].

Regression models based on Poisson probability distribution are known as Poisson regression model. An alternative to Poisson regression model is the negative binomial regression model and it is used to remedy some of the deficiencies of Poisson regression model. Both the distributions are used to model count data. However, most of the prior research showed that Poisson regression model is one of the robust models for analyzing count data. In this paper, children ever born is the count response variable ranges from 1 to 15 and was originally collected by the BDHS considering the reproductive women who had given at least one birth. So this variable suggests a truncated model as zero has not been taken into consideration in the range of the response variable while collecting the data. By this truncation some complexity can be avoided such as women who yet have not given birth might be the result of physical infertility or other personal problems. However, estimating a Poisson regression model or negative binomial regression model without accounting this truncation might result in biased estimates of the parameter vectors as well as erroneous inferences. That’s why this research only focuses to apply zero truncated regression models. Though prior research on children ever born to the Bangladeshi women based on data of Bangladesh Demographic Health Survey (BDHS)2007 [20] and 2011 [17] did not consider zero truncated model, so this current research aimed to apply these regression models to the latest BDHS data. This research is strongly influenced by Curtin [21] and Aldieri and Vinci [22] as they successfully applied truncated model in their research paper for modeling for parity distribution as well as the number of children ever born.

Data and methods

Data

Data for this study was extracted from the Bangladesh Demographic and Health Survey (BDHS) 2014 dataset, which is the seventh national-level demographic and health survey conducted under the authority of the National Institute for Population Research and Training (NIPORT) of the Ministry of Health and Family Welfare. The Bangladesh Demographic and Health Survey (BDHS) 2014 is a nationwide cross-sectional survey that was implemented by Mitra and Associates, funded by USAID Bangladesh and ICF international. The survey was conducted from June to November 2014 and designed to provide information on fertility and infant mortality, fertility preferences, family planning, maternal health, and children’s health and nutritional status.

The survey was based on two stage stratified sample with consist of 7 divisions, 64 districts and 545 upazilas/thanas that also divide the whole country into rural and urban areas. The design selected total 18,245 ever married women aged 15–49 and among them finally 17,863 were interviewed. More details about sampling design and data collection methods are available at website [23].For this research, only socio-demographic characteristics of the ever married women aged 15–49 who had given at least one birth were considered.

Children ever born is the key variable in our study which was collected through the questionnaire by asking the ever married women of age 15 to 49 who had ever given birth during their lifetime along with other variables. It is observed that, children ever born to a woman in our study ranges from 1 to 15 but due to lower frequency of more than nine children to a woman, these observations were merged and finally recoded as 1 to 9 +.

Methods

One of the most standard regression models for count data is Poisson regression model and it is a non-linear regression model. As children ever born is a count response variable with all observations greater than zero, so this study aimed to construct regression model with zero truncated Poisson regression distribution and negative binomial distribution [24].

Zero truncated Poisson regression model

If $Y_i$ is the number of event of interest in which at least one event must occur then its probability distribution is called zero truncated Poisson distribution with mean ${\mu }_i$ and probability mass function of $Y_i$ is given by

$$f\left(Y_i;{\mu }_i\right)=P\left\{Y_i=y_i|y_i>0\right\}=\frac{exp(-{\mu }_i){\mu }_i^{y_i}}{\left[1-exp\left(-{\mu }_i\right)\right]y_i!}$$ 1

A regression model based on this distribution follows by conditioning the distribution of $y_i$ on a k-dimensional vector covariates $X_i=\left[x_{1i},x_{2i},\dots .,x_{\mathit{ki}}\right]$ and parameter ${\beta }_i$ can be express for the mean parameter by using log linear form

$${\mu }_i=exp(X_i^T{\beta }_i)$$ 2

The standard estimator for this model is the maximum likelihood estimator (MLE). For given independent observations, the log-likelihood function is

$$l\left(Y_i;{\mu }_i|y_i>0\right)=\sum \left\{y_i\left[X_i^T{\beta }_i\right]-exp\left(X_i^T{\beta }_i\right)-ln{y}_i!-ln\left[1-exp\{-exp(X_i^T{\beta }_i)\}\right]\right\}$$ 3

Differentiating (3) with respect to ${\beta }_i$ yields MLE of ${\widehat{\beta }}_i$

Zero truncated negative binomial regression model

Zero truncated Poisson regression model may not be trustworthy if some of its assumptions violated, then zero truncated negative binomial regression model is more suitable for count data. The probability mass function for the zero truncated negative binomial distribution for a random variable $Y_i$ is

$$f\left(Y_i;{\mu }_i,k\right)=P\left\{Y_i=y_i|y_i>0\right\}=\frac{\Gamma \left(\text{k}+y_i\right){\left(\frac{k}{\text{k}+{\mu }_i}\right)}^k{\left(1-\frac{k}{\text{k}+{\mu }_i}\right)}^{y_i}}{\left[1-{\left(\frac{k}{\text{k}+{\mu }_i}\right)}^k\right]y_i!\Gamma \left(\text{k}\right)}$$ 4

Then negative binomial regression model for mean ${\mu }_i$ can be express for $y_i$ with a k-dimensional vector covariates $X_i=\left[x_{1i},x_{2i},\dots .,x_{\mathit{ki}}\right]$ and parameter ${\beta }_i$ as

$${\mu }_i=exp(X_i^T{\beta }_i+{\epsilon }_i)$$ 5

where $\epsilon$ is the combine effect of unobserved variables. Log-likelihood function can be derived analogous to the zero truncated Poisson model by using equation Eqs. (4) and (5) in order to estimate the parameter.

Results and discussion

From a bunch of socio-demographic variables in the BDHS 2014 dataset, we considered only nine predictor variables namely age group (5 years), respondent’s education level, husband’s education level, wealth index, religion, residence type, wanted last child, region and contraceptive method which were found to have significant effects on the dependent variable. The ever born child of the women aged 15–49 of Bangladesh was taken as the dependent variable. In the main dataset, this variable had 15 categories (1–15). The data recorded the number of children from those women of aged 15–49 who had contributed to the population by giving birth and the least possible outcome is one child. There is no zero count observation and the characteristics of the data suggest that this is a count data which is truncated at zero.It is observed that the mean number of children per mother was 3.68 with variance 3.54 which is very close to the mean value. Thus zero-truncated Poisson regression model was suggested and tested for the data as well as zero truncated negative binomial models treated for further queries.

Descriptive statistics of children ever born

The Table 1 highlights the summary statistics of the response variable (children ever born).The range of the ever born children is 1–9. It is noticed that there were 43,772 observations and the mean and the standard deviation of the response variable were 3.68 and 1.88 respectively. The minimum number of children was 1 where the maximum number of children was 9.

Table 1.

Summary statistics of children ever born

Variable	N	Mean	SD	Minimum	Maximum
Children ever born	43,772	3.68	1.88	1	9

The number of ever born children and their frequencies along with the percentage has been represented in Table 2. From the table it is obvious that the 22.82% of the women had 3 children and 21.53% of the women had 2 children which was the second highest. It is also clear that in Bangladesh the number of women having 2–5 children were more than the women of having one child as well as the women of having more than 5 children.

Table 2.

Frequency distribution of number of children ever born

Children ever born	1	2	3	4	5	6	7	8	9
Frequency	3975	9424	9990	7520	5490	3618	1834	944	977
Percent	9.08	21.53	22.82	17.18	12.54	8.27	4.19	2.16	2.23

Socio-economic and demographic characteristics of the women

The number of ever born children of a woman maybe influenced by the age(5 years group),women(respondent’s) education level, residence type, religion, region, wealth index, wanted last child, contraceptive method intension and husband’s education level.

Table 3 shows the socio-economic and demographic characteristics of the women on child ever born. For age group, it is clear from the Table 3 that 81.74% women among the respondents had 1 child when they were in the group 15–19 and 16.38% of them had 2 children. In the group 20–24, the number of having more children increased and it is seen that 37.19% women had 1 child and 45.07% women had 2 children. At age group 25–29, 41.26% and 29.4% women had respectively 2 and 3 children. It is clear that when the ages of the respondents increased, the number of having more children also increased and for the group 45–49, it has been noticed that 9.9% women had 7 children and 6.92% had 9 children. Again, 26.09% of the respondents who were living in the urban area had 2 children and 24.22% had 3 children where 22.2% women who lived in the rural area had 3 children. Besides, 29.28% and 46.04% women whose education level was higher had 1 and 2 children respectively. Women without education had more children than those who had secondary and higher education. Besides, 22.59% women who were Muslim had 3 children and 28% who were non-Muslim had 2 children. The regional effect of having more or fewer children is also noticeable from the table and it highlights that, women from Barisal (20.67%), Chittagong (21.55%),Dhaka (24.52%) and Sylhet (16.41%) had 3 children and women from Khulna(28.81%), Rajshahi (27.37%) and Rangpur (24.95%) had 2 children. The average rate of having more children (e.g.; 4, 5 and 6) was higher in Sylhet than the women from other regions. The women who were poorest (20.97%), poorer (21.61%), and middle (22.96%) according to their wealth index, had 3 children and the richer (24.35%), and richest(30.75%) women had 2 children. The variable Wanted Last Child also influenced the average rate of child ever born. The women who had no curiosity about having the last child had more children than the women who wanted then or later. Again the women who were not intended to use contraceptive method were likely to have more children than the women who were using modern and traditional method. The respondents whose husbands’ were highly educated had 2 children where the respondents whose husbands’ were not educated had more than 2 children. Eventually from the results, it can be concluded that women who and whose husbands were highly educated were not interested to have more children. And thus higher education status of the respondents and their husbands helps in reducing the population of Bangladesh.

Table 3.

Socio-demographic characteristics of women by row percentage on child ever born

Variable	Child ever born
	1	2	3	4	5	6	7	8	9
Age group
15–19	81.74	16.38	1.54	0.34	0	0	0	0	0
20–24	37.19	45.07	13.59	3.49	0.48	0	0.17	0	0
25–29	11.51	41.26	29.4	11.75	4.14	1.4	0.31	0.23	0
30–34	3.58	24.27	31.24	21.25	11.53	5.13	1.91	0.76	0.32
35–39	2.34	15.76	25.01	21.87	15.46	11.19	4.57	2.09	1.72
40–44	1.48	9.57	21.06	20.42	18.42	14.02	6.92	4.35	3.76
45–49	1.21	6.87	15.35	19.5	20.74	14.78	9.9	4.74	6.92
Residence type
Urban	11.69	26.09	24.22	15.48	10.8	6.66	3.16	1.12	0.78
Rural	7.92	19.49	22.2	17.94	13.32	8.98	4.65	2.62	2.88
Women education level
No education	2.69	10.39	19.17	21.24	16.97	13.59	8.04	3.82	4.09
Primary	6.25	18.8	24.95	18.92	13.96	9.15	3.38	2.35	2.25
Secondary	16.4	33.65	25.46	12.65	7.6	2.33	1.19	0.32	0.39
Higher	29.28	46.04	18.91	4.11	1.34	0	0.31	0	0
Religion
Muslim	8.93	20.9	22.59	17.27	12.78	8.53	4.39	2.29	2.33
Non-Muslim	10.67	28	25.18	16.32	10.07	5.58	2.17	0.83	1.19
Region
Barisal	8.58	19.03	20.67	19.11	13.41	8.82	4.63	2.06	3.37
Chittagong	7.93	16.79	21.55	17.82	14.63	9.65	6.27	3.27	2.1
Dhaka	10.63	23.35	24.52	16.66	12.42	6.95	2.26	1.24	0.96
Khulna	10.51	28.81	27.93	15.73	8.73	4.55	2.22	0.85	0.69
Rajshahi	10.74	27.37	24.55	16.55	10.12	6.28	2.61	1.28	0.5
Rangpur	9.24	24.95	24.77	18.62	11.89	6.13	2.34	0.8	1.24
Sylhet	6.3	12.44	16.41	15.86	15.59	14.67	7.24	5.14	6.36
Wealth index
Poorest	6.17	16.85	20.97	18.34	13.43	11.85	5.44	3.54	3.41
Poorer	7.14	17.41	21.61	18.33	15.13	9.41	6.22	1.58	3.18
Middle	8.65	19.87	22.96	18.41	12.28	8.56	4.03	3.01	2.25
Richer	10.86	24.35	22.67	16.92	12.6	6.12	3.28	1.18	1.62
Richest	13.36	30.75	26.44	13.32	8.71	4.59	1.52	0.82	0.38
Wanted last child
Wanted then	24.41	31.07	20.23	11.58	8.85	3.7	1.62	0.62	0.92
Wanted later	18.52	44.12	23.29	7.99	5	0	1.08	0	0
Wanted no more	0	4.74	25.42	23.16	16.58	13.58	6.63	3.37	6.53
Contraceptive use intension
Using modern method	8.91	24.95	25.72	17.55	11.09	6.31	3.04	1.3	1.14
Using traditional method	5.14	19.16	22.65	18.28	12.71	9.86	5.03	3.7	3.47
Non-user(intend to use later)	21.53	28.3	20.63	12.74	8.5	5.1	1.77	0.58	0.85
Does not intend to use	4.78	12.33	18.21	18.21	17.3	13 k.11	7.34	4.09	4.65
Husband’s education level
No education	4.17	14.29	21.35	20.29	15.28	12.32	6.33	2.81	3.16
Primary	8.38	20.31	23.47	17.02	12.99	8.06	4.17	3.02	2.59
Secondary	13.45	27.51	25.05	15.19	10.88	4.95	2.43	0.68	0.87
Higher	17.59	35.6	23.31	11.66	5.91	2.66	1.03	1.01	1.22

The graphs in Fig. 1 have been plotted by displaying the dependent count variable with different type of regressors. All the graphs from the figure highlight that the number of ever born children increased or decreased with the regressors as expected: number of ever born children increased with increment of ages of the respondents, while decreased by higher level of education level of respondents as well as their spouses. The trends of lower childbearing continued among the respondents from poorer to higher wealth index. The urban mother had lower tendency of childbearing with compare to her rural counterpart. The median number of children ever born to a mother remained consistent between Muslims or other religious women; however this has been fluctuated for the other factors like region, wanted last child as well as contraceptive intention.

Fig. 1.

Number of children ever born plotted against the regressors used

Predicting the number of ever born children using zero-truncated Poisson and negative binomial regression model

The incidence rate ratio (IRR) along with its confidence interval (CI) and their standard errors from zero-truncated Poisson and negative binomial regression models for child ever born have been presented in Table 4 where the incidence rate ratio (IRR) is the exponential of estimated zero-truncated Poisson and negative binomial regression.

Table 4.

Results of zero-truncated Poisson and negative binomial regression model

Children ever born	Zero-truncated Poisson				Zero-truncated negative binomial
	IRR	SE	CI		IRR	SE	CI
			Lower	Upper			Lower	Upper
Intercept	0.48	0.07	0.42	0.56	0.85	0.06	0.76	0.95
Age group
15–19*
20–24	4.02	0.07	3.51	4.61	2.39	1.38	2.16	2.65
25–29	6.65	0.07	5.82	7.60	3.87	0.05	3.51	4.26
30–34	8.89	0.07	7.79	10.17	5.15	0.05	4.67	5.68
35–39	10.55	0.07	9.21	12.09	6.18	0.05	5.58	6.84
40–44	12.45	0.07	10.89	14.37	7.75	0.06	6.94	8.64
45–49	12.53	0.08	10.72	14.65	8.08	0.06	7.13	9.16
Respondent’s education level
No education*
Primary	0.98	0.02	0.94	1.01	0.97	0.02	0.94	1.00
Secondary	0.85	0.02	0.81	0.88	0.85	0.02	0.81	0.89
Higher	0.52	0.05	0.47	0.57	0.55	0.05	0.51	0.61
Husband’s education level
No education*
Primary	0.93	0.02	0.98	1.04	1.00	0.02	0.97	1.04
Secondary	0.95	0.02	0.89	0.97	0.92	0.02	0.89	0.96
Higher	0.89	0.03	0.89	1.01	0.93	0.03	0.87	0.99
Wealth index
Poorest*
Poorer	0.89	0.02	0.86	0.93	0.89	0.02	0.86	0.92
Middle	0.87	0.02	0.84	0.91	0.87	0.02	0.84	0.90
Richer	0.84	0.02	0.80	0.88	0.84	0.02	0.81	0.88
Richest	0.74	0.03	0.70	0.79	0.75	0.03	0.71	0.80
Religion
Muslim*
Non-Muslim	0.83	0.03	0.78	0.89	0.83	0.03	0.79	0.88
Residence
Urban*
Rural	1.01	0.02	0.98	1.05	1.01	0.02	0.98	1.05
Wanted last child
Wanted then*
Wanted later	1.13	0.02	1.08	1.19	1.14	0.02	1.09	1.19
Wanted no more	1.22	0.02	1.19	1.27	1.25	0.01	1.21	1.29
Region
Barisal*
Chittagong	1.15	0.02	1.08	1.21	1.14	0.02	1.10	1.20
Dhaka	0.99	0.03	0.94	1.04	0.99	0.02	0.95	1.04
Khulna	0.88	0.03	0.83	0.94	0.89	0.03	0.84	0.95
Rajshahi	0.91	0.03	0.86	0.96	0.92	0.03	0.87	0.97
Rangpur	0.94	0.03	0.89	0.99	0.95	0.03	0.90	1.01
Sylhet	1.22	0.02	1.16	1.28	1.22	0.02	1.16	1.28
Contraceptive intension
Using modern method*
Using traditional method	1.01	0.03	0.96	1.06	1.00	0.03	0.95	1.06
Non-user—intends to use later	1.01	0.02	0.98	1.04	1.01	0.02	0.98	1.04
Does not intend to use	1.04	0.02	0.99	1.09	1.04	0.02	0.99	1.09

Here (*) indicates the reference category

IRR Exp(estimate), CI Confidence interval, SE standard error

Results from the analysis revealed that the women in the age group 20–24 had about 4 times more children than women of age group 15–19 and it is 6 times more for the age interval 25–29 with compare to the reference group. This tendency of childbearing continued with a similar trend for the subsequent ages until the age group 40–45 where there was a lower tendency to give birth as IRR was almost similar for the women of the last two groups. This means that 20–39 is the most vulnerable age group for childbearing among the women in Bangladesh.

The education level of the respondents has significant effect on the number of ever born children. The women who completed primary education had about 2% fewer children than the women who had no education and it is seen that when the level of education of the women was becoming higher the number of having children was decreasing. The women who were highly educated had 48% fewer children than the illiterate women. The husbands’ education level of the respondents had also significant effect on the number of children for the women. The women whose husbands’ educational statuses were primary and secondary had about 7% and 5% fewer children respectively than the reference category. It is observed that women educational status played more vital role in controlling the number of children when their educational statuses were becoming higher compared to their husbands’ education level.

For wealth index, from the results it can be said that the richest women intended to have fewer children than the poorest women. The results visualized that the poorer women had about 11% [IRR = 0.89; CI (0.86,0.93)] fewer children than the poorest women and the rate of having fewer children was increasing with the advancement of wealth index status of the respondents and also it is observed that the women whose wealth index status was richest had about 26% [IRR = 0.74, CI (0.70;0.79)] fewer children than the poorest women.

The results also revealed that the non-Muslims women had about 17% [IRR = 0.83; CI (0.78,0.89)] fewer children than the Muslims women in Bangladesh. The rate of having children was almost same among the rural and urban women. There was a little bit effect of the variable wanted last child on the ever born children. Women who wanted last child later and who wanted no more had respectively 13% [IRR = 1.13; CI (1.08,1.19)] and 22% [IRR = 1.22; CI (1.19,1.27)] more children than the women who wanted last child then. Women from Chittagong [IRR = 1.15; CI (1.08,1.21)], and Sylhet [IRR = 1.22; CI (1.16,1.28)] had more children compared to the women of Barisal region. Women from Khulna [IRR = 0.88; CI (0.83,0.94)], Rajshahi [IRR = 0.91; CI (0.86,0.96)] and Rangpur [IRR = 0.94; CI (0.89,0.99)] had fewer children compared to the women of Barisal region. It can clearly be observed that the women of Dhaka and Barisal regions had the same number of children. It is a matter of wonderment that there was no significant effect of contraceptive intension method among the women of Bangladesh where women who did not intend to use contraceptive method had only 4% [IRR = 1.04; CI (0.99,1.09)] more children than the women who were using the modern method.

Like the zero-truncated Poisson model, the zero-truncated negative binomial model illustrates approximately the same outcome for all the variables except the age group only. Here the women of age group 20–24 had 2 times more children where in the Poisson model they had 4 times more than the women of age group 15–19. With the increment of the age of the women the average rate of ever born children was increasing and finally the women of aged 45–49 had 8 times more children where in Poisson model they had 12 times more than the women of age group 15–19.

Model selection criteria

To identify the best fitted model between zero-truncated Poisson and zero- truncated negative binomial, Table 5 illustrates the values of the well-known model selection criteria (AIC, BIC and log(L)). Although both the models had given almost the same results but their standard errors were different. The AICs, BICs and Log-Likelihoods for both of the models was computed and compared to finalize the best fitted model. The AIC’s were 27861.13 and 28055.93 respectively for the zero-truncated Poisson and the negative binomial models. Again the BIC’s were 28076.5 and 28278.48 respectively for the zero-truncated Poisson and the Negative Binomial models. The AIC and BIC criteria suggested the zero-truncated Poisson model as the best fitted model compared to the zero-truncated negative binomial model based on their principle that the lower the value of AIC and BIC of any model, the better the model fits. Similarly, the Log-Likelihood values of the zero-truncated Poisson and the negative binomial models were − 13900.56 and − 13996.97 and the criteria also suggested the zero-truncated Poisson model as the best fitted model for the data. Similar results were found in Fig. 2, which compare residuals to the predicted values. Though the residual plots were quite identical to each other, except the tail of truncated negative binomial model was seemed to be over fitted and support the higher values of AIC and BIC. For more investigation, predicted count values of both the models were graphically plotted into the histograms and it appeared that truncated negative binomial had over predicted the last few categories with compare to truncated Poisson regression model. Hence, considering the model fit and diagnostic indicator, between the two fitted models, the zero-truncated Poisson model was considered as the best fitted model in this research.

Table 5.

Model selection criteria for child ever born

Model	AIC	BIC	Log(L)
ZTPR	27861.13	28076.5	− 13900.56
ZTNBR	28055.93	28278.48	− 13996.97

Most fitted Zero Truncated Poisson model are given in bold

ZTPR Zero truncated Poisson regression, ZTPR zero truncated negative binomial regression

Fig. 2.

Residuals plot and density curve for two fitted models

Conclusion

The zero-truncated Poisson regression model was found to fit the data on number of children ever born than zero truncated negative binomial regression model. Age and education status of women significantly influenced the number of ever born children. Residence type of the respondents and using contraceptive method did not influence the number of ever born children in Bangladesh so much. The number of ever born children was found to increase with the rising of women age and the rate was found to fall when the level of education of the women was increasing. Besides, wealth index, wanting last child, religion and respondents’ husband’s education level also influenced the number of ever born children. As women are the main stakeholder in the population of Bangladesh, so government and local agencies should invest more on improving these socio-demographic determinates.

Funding

The authors like to thank Comilla University for supporting this research through grant.

Compliance with ethical standards

Conflict of interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and publication of this article.