INTRODUCTION Reducing child mortality is now a global concern. Globally, Under-five child mortality rate has decreased by 58%, from an estimated rate of 93 deaths per 1000 live births in 1990 to 39 deaths per 1000 live births in 2017 (W.H.O., 2017). A lot of work has been done in the literature of under-five child mortality for different regions of the world. In Indian context, studies on child mortality have been mainly addressed the role of maternal, socioeconomic and health-related determinants. As far as Uttar Pradesh, India is concerned, poor health delivery system, poor maternal and health care services are responsible for low infant and child mortality (N.F.H.S.-IV, U.P.).This is a matter of serious concern for the human as well as social development of the state. The study is found that the determinants of maternal health services data of Uttar Pradesh is analysed by logistic regression methods (Patel K.K. et. al., 2014 and Saroj R.K. et.al., 2016).The research work has been done on survival analysis of under-five child mortality data by Cox proportional hazard model and frailty models (Ayele D.G. et. al., 2016).The author has applied classical and Bayesian approaches to Cox proportional hazard model on under-five child mortality data (Nilima S., 2017). Another citation related to survival analysis of under-five mortality of children and its associated risk factors (Geachew Y. and Bekele S., 2016). All above citation are related to survival models are used on different type of large sample data of under developing countries. This is difficult task to compare the parametric survival models together in real life data due to complex mathematical assumptions and software suitability (Nasejje J.B.et.al., 2015). Cox proportional hazard model is very important in survival analysis; the advantage of this model is that it includes the nonparametric and parametric models both (Geachew Y. and Bekele S., 2016). These studies were restricted to the analysis of mortality risks in children through survival analysis. After review, it was found that whatever studies have been done on under-five child mortality, none of them studies have been done combined fitting of parametric survival models in the under-five child mortality data Through this study, we intend to emphasize those determinants which are nearer in time to the outcome. The objective of this study to identify the risk factors which affect under-five child mortality by using Cox proportional hazard model, parametric models and find out the best parametric model for under-five child mortality.

MATERIALS AND METHODS DATA SOURCE In this chapter the data set of under-five child mortality of Uttar Pradesh is extracted from data National Family Health Survey (N.F.H.S.-IV), published in 2016. From the data sets, we extracted the socioeconomic and demographic factors which include educational level, sex of household head, women’s age in years, current marital status, husband’s/partner’s education level, husband’s/partner’s occupation, number of respondents currently working, respondent’s occupation, district, religion, and caste etc. Proximate and biological factors included total number of children ever born, births in the last 5 years, number of living children, currently breastfeeding, anemia level, smokes cigarettes, chewing tobacco, desire for more children, sex of child, size of child at birth, birth weight (kg), delivery by caesarean section, ANC visits, birth order, media exposure, birth interval. Environmental factors which includes place of residence, wealth index, source of drinking water, slum designation by observation, type of toilet facility. DATA ANALYSIS The outcome variable of under-five child mortality is defined as mortality from the age of one month to the age of fifty-nine months. Therefore, the dependent variable in this study is “the risk of death occurring in an age interval in the 1–59 month period.” The outcome variable was thus survival time in months of children under the age of five (Nasejje J. B. et. al., 2015). The explanatory variables are total number of children ever born, births in the last 5 years, number of living children, currently breastfeeding, anemia level, smoking cigarettes, chewing tobacco, desire for more children, sex of the child, size of the child at birth, birth weight (kg), delivery by cesarean section, ANC visits, birth order, media exposure, and birth interval. SEMI-PARAMETRIC MODEL Proportional hazard models are a class of survival models. Survival models are related to the time that passes, before some event occurs, to one or more covariates that may be associated with that quantity of time. This model comprises two sections: the fundamental work, frequently significant as h 0 (t), portraying how the risk of the occasion per time unit changes after some time at baseline levels of covariates; and the impact of parameters depicting how the risk shifts in light of explanatory covariates(X) which is a vector of explanatory covariates and β is a vector of unknown regression parameters. The hazard function in Cox’s regression model is given in equation (4.2.1). The Cox PH model (for p independent variables x 1 ,x 2 ....x p ) is described by the equation: (4.2.1) (4.2.1) 0 (t) is called the baseline hazard function (the expected hazard without any effect of the considered factors), e is a base of the natural logarithm, β 1 , β 2 ,... β p , regression coefficients. The expression h(t)/h 0 (t) is called the hazard ratio (Cox D.R. and Oakes D., 1998). PARAMETRIC MODELS Different parametric models such as Weibull, exponential, log-normal, and log-logistic are used for this study. Exponential distribution is one-parameter lifetime distribution and a special case of Weibull distribution. In this distribution, hazard function is always constant (Epstein B., 1958). Weibull distribution is one of the flexible parametric models to study the lifetime data utilized frequently in health field, and hazard function of this distribution could be increasing, decreasing, or constant (Carroll K.J., 2003). Log-logistic is another parametric model. The hazard rate in this distribution first increases and then decreases (Bennett S., 1983). Log-normal is another distribution which is extensively used in medical sciences (Royston P., 2011).The Weibull, exponential, log-normal, and log-logistic parametric models are fitted on under-five child mortality data in this chapter.

MODEL FORMULATION Consider the model for parametric fitting is described as γ(x) = γ1Women’s age in years + γ2Education + γ3Religion + γ4Total children ever born+ γ5Birth Last five years+ γ6Number of living children+ γ7 Currently Breastfeeding + γ8Smokes + γ9 Desire for more children+ γ10Size of child + γ11Delivery by caesarean section+ γ12ANC Visits + γ13Birth order. By using this model we have fitted Weibull, exponential, log-normal and log- logistic model respectively in the Table 4.4, 4.5, 4.6 and Table 4.7. Akaike Information Criterion (AIC), criterion is used to measure the goodness of fit of estimated statistical models. The AIC value of the model is given as (Akaike H., 1974) AIC = -2*log(L) +2k (4.3.1) (4.3.1) Where k be the number of estimated parameters in the model, L is the maximum value of the likelihood function, and AIC compares the performance of parametric models. Minimum value of AIC gives the best fit model. WEIBULL REGRESSION MODEL The distribution of time to event, T, as a function of single covariate is written as In(T) = β 0 + β 1x + σε (4.3.2) (4.3.2) where β; 1 is the coefficient for corresponding covariate, ε follows extreme minimum value distribution G (0, σ) and σ is the shape parameter. This is also called the accelerated failure-time model because the effect of the covariate is multiplicative on time scale and it is said to “accelerate” survival time. The hazard function of Weibull regression model in proportional hazards form is given by (4.3.3) (4.3.3) ,where γ = e -λβ0 and θ 1 = e -λβ1 the baseline hazard function is h 0 (t) = λγtλ-1,Parameter θ 1 has a hazard ratio (HR) interpretation for subject-matter audience The accelerated failure-time form of the hazard function can be written as follows: (4.3.4) (4.3.4) Weibull regression model can be written in both accelerated and proportional forms. Weibull model has two parameters, where γ is the shape parameter and λ is scale parameter. At γ < 1 the failure rate decreases over time,

< 1 the failure rate decreases over time, At γ = 1 failure rate remains constant over time,

= 1 failure rate remains constant over time, At γ > 1 failure rate increases over time. After fitting the parametric models, it is found that the Weibull model is the best for under-five child mortality data hence, the Weibull regression model has been applied in this data.

RESULTS First the Cox regression analysis is fitted into the data to find factors affecting under-five child mortality. The results of the Cox regression analysis with socioeconomic and demographic factors are given in Table 4.1. From the results, it is found that educational level, women’s age in years, and religions have significant effects on under-five child mortality at the 5% of significance level. The Table 4.2 shows that the environmental factor does not play any significant role in the under-five child mortality. The results of the Cox regression analysis with proximate and biological factors in the under-five child mortality are given in Table 4.3. From the results, it is found that children ever born, births in the last 5 years, number of living children, currently breastfeeding, smoking cigarettes, desire for more children, delivery by cesarean section, ANC visits, and birth order have significant effects on under-five child mortality (P < 0.05).After finding the significant factors in under-five child mortality through Cox regression, various parametric models like Weibull distribution, Exponential distribution lognormal distribution and log-logistic are applied on the under-five child mortality data. The results of the Weibull distribution model are shown in Table 4.4. The estimate of the parameters, standard errors, and confidence intervals for under-five child mortality and result is found that eleven independent factors have significant effects on under-five child mortality except education and religion. The Table 4.5 shows the output of the Exponential distribution model and this model result found that all the factors have significant effects on under-five child mortality. The Table 4.6 summarizes the estimates of the parameters for lognormal distribution model and the result showed that education, religion, and delivery by cesarean have no significant effects on under-five child mortality out of thirteen factors. The results of the log-logistics distribution model are given in Table 4.7.From the results it is found that maximum factors have significant effects on under-five mortality at the 5% of significance level except education, religion, and delivery by cesarean. The Weibull, exponential, log-normal, and log-logistic parametric models are fitted to find factors affecting under-five child mortality data and compared to among. The Table 4.8 summarizes the result that of all compared model fitting and on the basis of AIC criteria Weibull model is found best model out of four models with minimum AIC value (32985.3). Table 4.9 shows the result of the Weibull regression model and based on the model found that all factors significantly affected the under-five child mortality except education, religion and delivery by caesarean section. The Figure 4.1 expresses the graphical representation of Weibull regression model with factors.

DISCUSSION In this chapter compared the performance of various parametric models of the under-five child mortality survival status. There are numerous studies over the past decade have found a positive association between education of mother and child’s survival. The risk of under-five child mortality is higher in developing countries as compared to developed nations. In this chapter, the national representative data (NFHS-IV) has been used for the study. The objective is to detect the important factors which are related to under-five child mortality through parametric survival models and to find the best parametric model for the study. First use the Cox proportional model is applied for finding the effect of different variables in under-five child mortality. The variables include socioeconomic, demographic, environmental, and proximate and biological factors. Similar findings are found in this article, factors such as education, age, total number of children, and family size were found to be significant (Ayele D.G. et. al., 2016).The author analyzed the data through survival analysis approaches and performed Cox proportional hazard model. It is found that mother’s education, type of residence, birth order, mother’s occupation, and type of birth played a significant role in child survival (Nasejje J. B. et. al., 2015). Survival analysis of under-five mortality of children and its associated factors was done by different survival methods such as Kaplan–Meier estimates, Cox proportional hazard regression model, and stratified Cox regression model. Variables such as region, socioeconomic status, contraceptive use, twin child, birth interval, and breastfeeding were found to be significant (Geachew Y. and Bekele S., 2016). In a recent article on survival analysis for under-five child mortality in Uttar Pradesh done by (Saroj et. al., 2018) the role of significant variables such as educational level, women’s age in years, religion, children ever born, births in the last 5 years, number of living children, breastfeeding, smoking cigarettes, desire for more children, size of child at birth, delivery by cesarean section, ANC visits, and birth order was analyzed and found to be significant. It is clear and verified that almost same variables were found to be significant in study. These are more or less similar to the findings of our study. In the chapter, exponential and Weibull models, log-normal and log-logistic model are chosen for under-five child mortality. The Weibull distribution shows the best fit to the data on the under-five child mortality. The performance of parametric model is checked by AIC value and Weibull model was found to be the most suitable model with minimum AIC (32985.3) value among all parametric models. Therefore, the Weibull regression model is preferred to find the role of various factors in under-five child mortality data, and it is found that women’s age in years, total number of children ever born, births in the last 5 years, number of living children, currently breastfeeding, smoking, desire for more children, size of child, delivery by cesarean section, ANC visits, and birth order covariates are statistically associated with the under-five child mortality.