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In this study a model is conceptualized to measure the child mortality under different parity of women such that a better strategy can be formulated to bring down mortality rates. In the estimation of probability of child mortality some socio demographic variables are taken in consideration. The estimates are obtained under Bayesian procedure. Two different models are formulated for it and model fitting is observed by graphical approach along with the chi square test. First model is betabinomial and second is binomial regression model. Second model shows the better fit on the data. The estimate of probability of child mortality at higher parities namely, parity 3, parity 4 and parity 5 were obtained as 0.06, 0.09 and 0.13 respectively on the basis of the second model.

Children are important assets of a nation and reduction in infant and child mortality is one of the key factors which are taken into consideration while planning for the growth of the nation. Child mortality is also a good indicator of level and quality of health care as well as socio-economic condition of community. Recognizing the important role that child health plays in the overall health of societies, the Millennium Development Goals include a goal explicitly aimed at reducing child mortality by two-thirds, between 1990 and 2015.

Child mortality, also known as under-5 mortality, refers to the death of infants and children under the age of five. The trend in child mortality is declining over time but rate of decline is not fast. Child Mortality rate is the highest in low-income countries. Still many deaths in the third world go unnoticed since many poor families cannot afford to register their babies in the government registry.

Under five year mortality rate for the world was 91 per 1000 live birth in 1990 which decreased to 43 in 2015 (http://www.who.int/). The under five mortality for India was 126 in 1990 as compared to 48 in 2015 (http://www.childmortality.org/). No doubt there is significant reduction in child mortality in India but as per UNICEF report, India is still in the list of high child mortality countries.

The infant and child mortality studies have long been of interest to demographers and person concerned with public health problem. The most common problem in such studies is the error in data of deaths during infancy and childhood. In this condition, development of stochastic models is a good choice to minimize the effect of such errors.

Perhaps the first attempt to model the infant mortality was made by Keyfitz [

The availability of computers based computational technique; Bayesian method of estimation is getting popularity as well as acceptance in all modeling situations. Various authors have been used this technique in the estimation of infant and child mortality [

The key features of Bayesian modeling are the likelihood function, which reflects information about the parameters contained in the data, and the prior distribution, which quantifies what, is known about the parameters before observing data. The prior distribution and likelihood are further combined to get the posterior distribution of parameter, which represents total knowledge about the parameters after the data have been observed. Simple summaries of this distribution can be used to isolate quantities of interest and ultimately to draw substantive conclusions whereas the classical modelling approaches uses only the likelihood function and hence sampling distribution of estimator, which reflects information about the parameters contained in the data.

The present study is an attempt to model child death in such a way that we get estimate of child mortality according to the parity of mothers. The aim behind it was to provide a platform for monitoring agencies to re-plan their strategies for bringing faster decline in child mortality. In present study an attempt is also made to find out the relationship of different socio-economic variables with child mortality in total reproductive life span of women according to different parity. Two models have been taken into account to estimate the mortality rate among different parity.

Let us consider the population of N women who have given birth to n children during their reproductive period. Define a random variable Z_{ij} (i = 1, 2, ⋯ , N and j = 1, 2, ⋯ , n) as given below:

Z i j = { 1 | if j t h child of i t h women dies 0 | if j t h child of i t h women survives

Thus Z i j is a Bernoulli random variable. Further, we can define another random variable Y i (i = 1, 2, ⋯ , N) such that

Y i = ∑ j = 1 n Z i j

with possible values 0, 1, 2, ∙∙∙, n and it shows the number of child death experienced by i^{th} women. If we assume that deaths of children experienced by i^{th} women are independent and with same unknown probability say p_{i} then Y i assumed to follow binomial distribution. Here n will denote parity (total children born in life span) of a women and let pi is probability of experiencing child death by i^{th} women of parity n.

The model -I is defined as below according to above explanation of the study:

Let

f ( Y i ) = C y i n p i y i ( 1 − p i ) n − y i (1)

here n denotes parity of women and we have considered n = 3, 4 and 5, i denotes i^{th} women of parity n (i − 1, 2, ⋯ , N). The prior distribution for parameter p i is taken as beta distribution with probability density function:

f ( p i ) ∝ p i α − 1 ( 1 − p i ) β − 1 (2)

Combining it by Bayes theorem the posterior probability density function will have a beta distribution. We use non-informative beta (1, 1) prior which is flat and reflect no prior information regarding the parameter being estimated.

In this model, while estimating the probability of child death to particular women the consideration of different socio-economic and demographic variables was incorporated in model. For each n (n = 3, 4, 5) pi is unknown parameter of interest. Since p_{i} is affected by various socio-economic variables. Then we consider pi as a function of a few explanatory variables X_{m}, (m = 1, ⋯ , 4) and propose to use generalized linear model(GLM). The general linear model (logit model) is popularly used to see the association between Bernoulli response variable and the factors affecting it. In modeling by general linear model, the dependent variable is transformed into continuous form by using link function. Dichotomous dependent variable like Z_{ij} with probability of death of j^{th} (i = 1, 2, ⋯ , n) child born to i^{th} (i = 1, 2, ⋯ , N) women as p_{i}, a link function called logit link function is used to make dependent variable continuous. It is actually expected value of logarithm of odds of experiencing a child death to not experiencing a child death. Thus GLM defines the relationship as

log i t ( p i ) = log ( p i 1 − p i ) (3)

Now

log ( p i 1 − p i ) ~ N ( μ i , σ 2 )

Thus expected value of log i t ( p i ) = μ i and if we take

μ i = β 0 + β 1 X 1 + β 2 X 2 + ⋯ + β p X p .

So,

log i t ( p i ) = β 0 + β 1 X 1 + β 2 X 2 + ⋯ + β p X p (4)

where, X 1 , X 2 , ⋯ , X p (generally called explanatory variables) are the variables associated with log i t ( p i ) known as logistic regression.

The unknown regression coefficients in (4) are β 0 , β 1 , β 2 , ⋯ , β p which are estimated from the data and for the Bayesian approach they are estimated from their joint posterior distribution. The proposed model is linear for the regression coefficients on the logit scale.

In present study four explanatory variables are taken for the estimation of probability of child death experienced by women. The four explanatory variables are age at first birth (X_{1}), education of mother (X_{2}), religion (X_{3}) and type of house (X_{4}). The variable X_{1} quantitative and continuous in nature and measured in years, and X_{2}, X_{3}, X_{4} are categorical variables. X_{2} has four levels as illiterate, primary, middle and above middle. There are two levels of X_{3} as Hindu and non-Hindu. Finally the fourth explanatory variable X_{4} has three levels as kaccha, semi pucca and pucca. Thus the proposed model is:

l o g i t ( p i ) = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + β 4 X 4 (5)

where,

p i = exp ( β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + β 4 X 4 ) 1 + exp ( β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + β 4 X 4 )

Further, for the Bayesian analysis, priors are required for all the parameters before data in hand.

In present study independent non-informative priors have been used considering that we have no strong a priori idea about regression parameters and hence following prior distribution is considered:

p ( β i ) = 1 2 π σ 2 e − 1 2 σ 2 ( β i − β ¯ ) 2 i f − ∞ ≤ β i ≤ ∞

for i = 1, 2, 3 and 4. We put β ¯ = 0 a n d σ 2 = 1 0.0001 which shows a negligible

faith on prior knowledge and has uniform type shape. Further, suppose that we draw the random samples from binomial distribution then the joint probability function can be written as p ( y | β 0 , β 1 , β 2 , β 4 ) and thus the joint posterior distribution can be obtained as:

p ( β 0 , β 1 , β 2 , β 4 | y ) p ( β m ) p ( y | β m ) ∫ p ( β m ) p ( y | β m ) d β m (6)

After obtaining the posterior distribution of regression coefficients we obtain the Bayesian estimate of regression coefficients from this distribution under squared error loss function with use of Monte Carlo Markov Chain process. Here Gibbs sampling procedure is used for this purpose.

To check the suitability of proposed methodology in estimating the situation of child mortality, data was taken from District Level Household and Facility Survey (DLHS-3, 2007-2008) [

Parity | Number (%) of Child Death | Total | |||||
---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | ||

3 | 5535 (82.76%) | 1083 (16.19%) | 68 (1.02%) | 2 (0.03%) | - | - | 6688 (100%) |

4 | 4259 (68.19%) | 1669 (26.72%) | 296 (4.74%) | 22 (0.35%) | 0 | - | 6246 (100%) |

5 | 2789 (52.23%) | 1839 (34.44%) | 570 (10.7%) | 125 (2.33%) | 17 (0.3%) | 0 | 5340 (100%) |

Education | Parity 3 | Parity 4 | Parity 5 | |||
---|---|---|---|---|---|---|

No | % | No | % | No | % | |

Illiterate | 3429 | 51.3 | 4064 | 65.1 | 3946 | 73.9 |

Primary | 939 | 14.0 | 898 | 14.4 | 669 | 12.5 |

Middle | 1017 | 15.2 | 675 | 10.8 | 452 | 8.5 |

Above Middle | 1303 | 19.5 | 609 | 9.8 | 273 | 5.1 |

Religion | ||||||

Hindu | 5766 | 86.2 | 5109 | 81.8 | 4184 | 78.4 |

Non Hindu | 922 | 13.8 | 1137 | 18.2 | 1156 | 21.6 |

Type of house | ||||||

Kaccha | 1592 | 23.8 | 1720 | 27.5 | 1600 | 30.0 |

Semi Pucca | 3117 | 46.6 | 3089 | 49.5 | 2750 | 51.4 |

Pucca | 1979 | 29.6 | 1437 | 23.0 | 990 | 18.6 |

The Bayesian analysis of both of the proposed models was performed on the basis of data given in

The results obtained from proposed model-I was tested for its fit by the help of graphs of observed frequency of death and expected frequency of death for each parity.

No. of Death | Parity 3 | Parity 4 | Parity 5 | |||
---|---|---|---|---|---|---|

Obs | Exp | Obs | Exp | Obs | Exp | |

0 | 5535 | 3480 | 4259 | 2866 | 2789 | 2098 |

1 | 1083 | 1971 | 1669 | 1786 | 1839 | 1464 |

2 | 68 | 935 | 296 | 994 | 570 | 933 |

3 | 2 | 303 | 22 | 458 | 125 | 526 |

4 | NA | 0 | 141 | 17 | 243 | |

5 | NA | NA | 0 | 76 | ||

Total | 6688 | 6688 | 6246 | 6246 | 5340 | 5340 |

Prob. of Death | 0.24 | 0.23 | 0.23 | |||

χ^{2} Value | 2715.85 | 1731.80 | 1056.43 |

No. of Death | Parity 3 | Parity 4 | Parity 5 | |||
---|---|---|---|---|---|---|

Obs | Exp | Obs | Exp | Obs | Exp | |

0 | 5535 | 5544 | 4259 | 4257 | 2789 | 2795 |

1 | 1083 | 1066 | 1669 | 1682 | 1839 | 1818 |

2 | 68 | 76 | 296 | 283 | 570 | 591 |

3 | 2 | 2 | 22 | 24 | 125 | 119 |

4 | NA | 0 | 1 | 17 | 15 | |

5 | NA | NA | 0 | 1 | ||

Total | 6688 | 6688 | 6246 | 6247 | 5340 | 5339 |

Prob. of Death | 0.06 | 0.09 | 0.13 | |||

χ^{2} Value | 1.09 | 1.76 | 2.56 |

HPD intervals | Odds ratio | ||||
---|---|---|---|---|---|

Estimate of β | SD | 5.00% | 95.00% | ||

Intercept | −3.014 | 0.1769 | −3.314 | −2.733 | |

AFB | −0.00783 | 0.07521 | −0.1306 | 0.1148 | 0.9922006 |

Education | |||||

Illiterate | 0.3908 | 0.2072 | 0.04988 | 0.7258 | 1.4781629 |

Primary | 0.1826 | 0.2477 | −0.2321 | 0.5934 | 1.2003342 |

Middle | 0.3621 | 0.2318 | −0.0242 | 0.7487 | 1.4363426 |

Religion | |||||

Non Hindu | −0.4745 | 0.1779 | −0.771 | −0.1779 | 0.6221961 |

Type of House | |||||

Kaccha | −0.2353 | 0.291 | −0.7336 | 0.2247 | 0.7903337 |

Semipucca | 0.07917 | 0.1549 | −0.1753 | 0.3339 | 1.0823883 |

HPD intervals | Odds ratio | ||||
---|---|---|---|---|---|

Estimate of β | SD | 5.00% | 95.00% | ||

Intercept | −2.354 | 0.0823 | −2.49 | −2.223 | |

AFB | 0.1155 | 0.02163 | 0.07997 | 0.1513 | 1.1224345 |

Education | |||||

Illiterate | 0.07167 | 0.08058 | −0.0628 | 0.2049 | 1.0743008 |

Primary | 0.004913 | 0.09589 | −0.1544 | 0.1624 | 1.0049251 |

Middle | 0.004717 | 0.101 | −0.1629 | 0.1731 | 1.0047281 |

Religion | |||||

Non Hindu | −0.4465 | 0.06618 | −0.5566 | −0.3367 | 0.6398638 |

Type of House | |||||

Kaccha | 0.09936 | 0.06672 | −0.0102 | 0.2076 | 1.1044638 |

Semipucca | 0.05445 | 0.05913 | −0.0435 | 0.1524 | 1.0559597 |

HPD intervals | Odds ratio | ||||
---|---|---|---|---|---|

Estimate of β | SD | 5.00% | 95.00% | ||

Intercept | −2.125 | 0.09566 | −2.281 | −1.969 | |

AFB | 0.02462 | 0.01966 | −0.0080 | 0.05772 | 1.0249256 |

Education | |||||

Illiterate | 0.03906 | 0.09447 | −0.1169 | 0.1939 | 1.0398329 |

Primary | 0.01981 | 0.1072 | −0.1538 | 0.1963 | 1.0200075 |

Middle | 0.08063 | 0.1134 | −0.1124 | 0.2649 | 1.0839698 |

Religion | |||||

Non Hindu | −0.3955 | 0.05367 | −0.4827 | −0.3083 | 0.6733433 |

Type of House | |||||

Kaccha | 0.2454 | 0.06245 | 0.1416 | 0.3483 | 1.2781325 |

Semipucca | 0.172 | 0.0563 | 0.08093 | 0.2656 | 1.1876778 |

For parity 5, the outcome of regression analysis is mentioned in

The aim of present study was to find out valid estimate of relationship between child mortality and selected socio-economic variable through mathematical modelling. Generalized linear model is used in Bayesian setup for this purpose, where risk of death was dependent on selected socio-economic factors. Data were used from DLHS-3 for UP state. Factors like education of mother were found decreasing effect on the risk of child death which is not significant but various studies have supported a direct causal relationship between mother’s education and child mortality [

The factor type of house has no significant effect for parity-3, 4 but for parity 5 it shows significant effect. But there are various studies on infant or child mortality namely Gyimah [

The authors declare no conflicts of interest regarding the publication of this paper.

Tripathi, A., Singh, G.P. and Singh, A. (2019) Model for Assessment of Child Mortality under Different Parity: A Bayesian Swatch. Journal of Biosciences and Medicines, 7, 113-126. https://doi.org/10.4236/jbm.2019.75014