Introduction: The application of piecewise hazard model in mortality data becomes more useful in survival methods. This method is used to find the number and location cut points and to estimate the hazard model. The objective of this paper is to illustrate the piecewise constant hazard model and to find significant factors for under-five child mortality.
Materials and Methods: The piecewise hazard model is illustrated on secondary data National Family health Survey (NFHS-IV). The piecewise constant hazard model is fitted on mortality data by using socio-demographic, biological and proximate co-factors. Free R software is used for analysis purpose.
Results: It is found that women’s age in years, total children ever born, present breastfeeding, smoking, size of child, delivery by caesarean section, ANC visits, and birth orders are important factors and six month time interval is very crucial for child till completing the five year of the age.
Conclusion: Piecewise hazard model is found very important for the under-five child mortality, through various times cut points. The Piecewise hazard model can be useful for clinicians, researchers and public health experts. Time is very important factor for reducing the child mortality.
he survival analysis is a technique to estimate the patient's survival that after treatment in a specifics infection. The survival analysis uses in life the survival status of patients. The survival probability S (t) also known as survivor function, and it described the individuals survival time in specified time period t. The second term hazard is represented by h (t) or λ (t) and hazard rate is defined as instantaneous failure rate for the survivors to time t during the next instant of time. The survival probability is evaluated by nonparametric methods from observed survival times for uncensored and censored cases with the help of KM or product-limit method. The Cox PH regression technique is used to find the association between survival time of patients and one or more indicator variables. The Cox PH regression models characteristic log of the risk at time t, denoted by h(t), as a function of the baseline hazard (h0(t))and a few indicator factors x1,x2.....xn The model is given by
Take the exponentiation both sides of the equation, and limit the right hand side to just a single categorical exposure variable (x1) with two groups (x1=1 for exposed and x1=0 for unexposed), the equation becomes:
After solving the equation estimate the hazard ratio, comparing the exposed to the unexposed individuals at time t is given by
The eβ1 is the hazard ratio and other constant over time t. The β is the regression co-efficient that estimate from model and represent the log(Hazard Ratio) for each unit increase in the corresponding predictor variable.
Let x be the row vector of explanatory variables, and β be the corresponding column vector of coefficients. The λ(t,x,β)hazard model is given by,
λ0(t) is called the baseline hazard. There are several choices for the baseline hazard models are exponential hazard model, Weibull hazard model and Piecewise-constant hazard model.
PIECEWISE-CONSTANT HAZARD MODEL
An essential expansion of the exponential model is known as piece-wise exponential model or piecewise- constant hazard model. This model originates from an appropriation whose hazard rate is a step function. The model, need to segment the survival into a numerous pieces. In this model accept that inside each segment, the hazard is constant but between segments, hazard could be different.
The hazard function can be written as:
In this study assessing the basic hazard function with possible time change points. It will demonstrate the survival pattern of the patients and which time points is more failure or more censored. Assessing the survival trend for entire population will give better understanding of how changing treatment, patients monitoring, health facilities of patients, and public health services. Past researches have proposed strategies for estimating a single change point in a piecewise constant hazard function when the observed variables are subject to random censoring. The research is shown multiple change points in piecewise constant hazard functions. The study suggested a piecewise-exponential methodology where Poisson regression model parameters are estimated from pseudo likelihood and comparing differences were determined by Taylor linearization strategies. In study, It is estimated that the population attributable fraction for mortality in a cohort study using a piecewise constant hazards model. Another study is demonstrated the cancer research using a reduced piecewise exponential approach. The research considers parameter estimation in the hazard rate with multiple change points in the presence of long-term survivors. In research article, it is shown that a survival analysis in context of new method suggest to estimate the piecewise constant hazard rate model. Another study has been done to find the survival status of under-five child mortality in Uttar Pradesh. Recent articles use the survival parametric models to estimate the factors of under five child mortality data.
However, previous studies did not determine the impact of factors on under-five child mortality with the help of piecewise constant hazard model.
The key interest of this paper is to determine the potential determinants of under-five child mortality of data of Uttar Pradesh using piecewise constant hazard model. In this model, the various time points are used to find the crucial time points and explain the important factors for reducing the under-five child mortality.
Material and Methods
This paper investigates the determinants of under-five child mortality in Uttar Pradesh using the (NFHS-IV) data. The piecewise hazards model is used to evaluate the comparative impact of the hypothesized factors on under-five child mortality. For analysis purpose, the age of the children in months is ascertained as pursues: age = V008 – B3, where V008 is the century month code (CMC) of the date of meeting, and B3 is the CMC for date of birth of the kid. The responsible factors for under-five child mortality are selected according by the citation. 
These components are sorted into the additional four categories; Social demographic and social economic, environmental and proximate or biological factors but in this paper used the analysis based on three social demographic and social economic, proximate and biological factors only. There are several kinds of hazard model available and a piecewise constant proportional hazards model is used in this study.
Let X1,X2.....Xn denote independent identically distributed survival times, and C1,C2.....Cn be the censoring times which are assumed to be independent of X. We have only observed the pairs, (Ti,δi),i= 1,2,....n,where Ti = min(Xi,Ci) and if and zero otherwise. Consider the following change point model:
,0 < τ1 <.....< τk, where, K is the number of change points in the model, αj and is the value of the hazard function between the time points τj-1 and τj. The τ0j. can be thought of as the order statistics for the change points in the hazard function. The time axis is split into sections in a piecewise constant model and a constant hazard is presumed within each section. In this analysis the time axis is divided into nine groups: 0-6 months, 6-12 months, 12-18 months, 18-24 months, 24-30 months, 31-36 months,36-42 months ,42-48 months and 48-59 months. The baseline hazard λ(t)describes how the mortality rate changes with the age of the child. This is parameterized as a piecewise constant function.
Remark: The significant factors are calculated for under-five child mortality through Cox –regression analysis. These important factors are used in the piece-wise constant hazard analysis in this paper.
Table 1.1 shows that survival status of under-five year children. In this table shows that total 93% cases censored because child death considered as main event. Total survival time of children is 59 months shows in Figure 1.1 and plot shows the survival probability and censor cases report of children, where plus (+) symbols represented as the censored.
In this paper, the hazard rate is calculated for under-five child mortality at different time points for piecewise hazard analysis. The time points have been distributed in six months, ten months, eleven months and twelve months in Table 1.2. The hazard rate variation shows in Figure 1.2 at the six, ten, eleven and twelve month’s interval. The figure clearly shows that maximum number of disparity of hazard rate in six month intervals therefore the six month time interval has been selected for piecewise hazard analysis. The piecewise constant hazards model is used at different time intervals and significant factors are examined and compared with the estimates obtained by means and hazard function at various seven time points. The hazard function value comes between defined pieces or interval time in months of all patients. For specifying the model with a smooth hazard function, the follow-up period is divided in seven consecutive intervals of fifty-nine months length. The first piecewise hazard model includes educational level, religion and women's age in years for the analysis of under-five child mortality status. This model breaks the data into pieces, where may fit constant hazards within these pieces. Table 1.3 shows the result of piecewise constant hazards model and it is found that educational level and women's age are found significant factor in piecewise hazard model. Figure 1.3 shows the detail of first model including factors with a piecewise constant hazard function, piecewise constant cumulative hazard function, piecewise constant density function and piecewise constant survivor seven consecutive intervals till fifty-nine months. In this figure peak of the hazards continuously increasing respect to increasing time. The second piecewise constant hazards model includes proximate and biological factor for under-five child mortality.
Table 1.4 shows the output of proximate and biological factors role in under-five child mortally status and this model result found that all the factors have significant effects on under-five child mortality.
Figure 1.4 shows the piecewise constant hazard function, piecewise constant cumulative hazard function, piecewise constant density function and piecewise constant survivor seven consecutive intervals till fifty-nine months including proximate and biological factors. Finally selected all significant factors and combined together to check effect of the factors in under-five child mortality status through piecewise constant hazard model. Table 1.5 shows that the factors women's age in years, total children ever born, birth in last five years, number of living children, present breastfeeding, smoking, desire for more children, size of child, delivery by caesarean section, ANC visits and birth orders have significant effect in under-five child mortality in the pricewise constant hazard model. Figure 1.5 shows the piecewise constant hazard function, piecewise constant cumulative hazard function, piecewise constant density function and piecewise constant survivor seven consecutive intervals till fifty-nine months including all factors.
This paper is analyzed the determinants and critical time point of under-five child mortality in Uttar Pradesh using the (NFHS-IV) data through piecewise constant hazard model. This is the first time this kind of analysis has been done and the results are really significant in gaining an overall insight into under-five child mortality differentials in Uttar Pradesh. This analysis shows that six months of the children increase the risk of death under age five year. The hazard rate is supposed to be constant on some pre-defined time intervals and plotting the hazard rate gives a quick idea of the progress of the event of interest through time. Different time periods for evaluation of mortality, extend from 28 days to a half year and including 35 days, 60 days, and 90 days. It has been utilized in earlier clinical trials , while this model used in a nonparametric setting, usually utilized in mix with covariates impacts.  The applicability in the situation for the popular Poisson regression model,  and to find the relative impact on the covariates and a piecewise constant hazard model for the baseline hazard.  Proposed an imaginative strategy to evaluate the hazard rate in a piecewise constant model. This technique gives a programmed strategy to locate the number of cut points and to estimate the hazard on each cut interval. This model is exceptionally helpful to find the patient hazard in various time points.
In conclusion, piece-wise constant hazard model has been used for finding the crucial month period and significant factors in under-five child mortality. In study, it is found that proximate and biological factors are more critical for under-five child mortality and half year or six months are extremely urgent for children till completing five years of age. These outcomes can advise the determination of time points for evaluating survival status of under-five child mortality. The results suggest to focus on different significant factors for achieving a decline in under-five child mortality. This model would be useful in health care policy makers and with an upgraded comprehension of the adjustments in population death rates can recognize gaps, seek solutions, improve performance, and ultimately, better the public’s health. The government should also put more emphasis on programmes aimed at population with under-five child mortality.
Funding: There is no financial support for this study/paper
Declaration of conflicting interests: The authors declare no conflict of interest.
Table 1.1 Status of child survival of under-five year
|Total N||N of Events (Deaths)||Censored|
Table 1.2 Hazard rate of the under-five year age of children
|Hazard Rate for 6 months intervals|
|Hazard Rate for 10 months intervals|
|Hazard Rate for 11 months intervals|
|Hazard Rate for 11 months intervals|
Table 1.3 Piecewise constant hazards model with socio-economic and demographic factors
|3||Women's age in years||28.53||-0.042||0.959||0|
Table 1.4 Piecewise constant hazards model with proximate and biological factors
|3||Desire for more children||4.2||-0.33||0.716||0|
|4||Size of child||3.12||0.15||0.16||0|
|5||Child Birth weight (kg)||6348.05||0||1||0.003|
|9||Delivery by caesarean section||0.08||0.27||1.31||0|
|10||Total children ever born/td>||3.23||0.19||1.21||0|
Table 1.5 Piecewise constant hazards model with combined variables
|1||Women's age in years||28.54||-0.04||0.95||0|
|4||Total children ever born||3.23||0.48||1.63||0|
|5||Birth in last five years||1.72||0.32||1.38||0.003|
|6||Number of living children||2.88||-0.99||0.36||0|
|9||Desire for more children||4.19||-0.16||0.85||0|
|10||Size of child||3.12||0.17||1.18||0|
|11||Delivery by caesarean section/td>||0.08||0.14||1.15||0.03|
Figure 1.1:Under-Five child mortality survival status
Figure 1.2 Hazard rate graph of according to various month intervals
Figure 1.3 Piecewise constant hazards, cumulative hazard, density and survivor function with covariates
Figure 1.4 Piecewise constant hazards, cumulative hazard, density and survivor function with covariates
Figure 1.5 Piecewise constant hazards, cumulative hazard, density and survivor function with covariates
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