

ORIGINAL ARTICLE 

Year : 2020  Volume
: 5
 Issue : 1  Page : 4045 

Piecewise hazard model for underfive child mortality
Rakesh Kumar Saroj
Department of Mathematics and Statistics, SRM Sikkim University, Gangtok, Sikkim, India
Date of Submission  24Sep2019 
Date of Decision  12Dec2019 
Date of Acceptance  05Feb2020 
Date of Web Publication  08Jul2020 
Correspondence Address: Dr. Rakesh Kumar Saroj Research Scholar, Department of Kayachikitsa, Institute of Medical Sciences, Banaras Hindu University, Varanasi, Uttar Pradesh India
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/bjhs.bjhs_54_19
OBJECTIVE: The application of piecewise hazard model in mortality data becomes more useful over classical survival methods. This method is used to find the number of location of cut points and estimate the hazard model. The piecewise hazard model is fitted on National family health survey (NFHS IV) from different variables like socio demographic, biological and proximate cofactors. The aim of the study to describe the piecewise constant hazard model and find the important factors in underfive child mortality data. METHODS: For the research used the National Family Health SurveyIV data of Uttar Pradesh. The Cox regression analysis is used for finding the important of factors through preliminary analysis. After that, apply the piecewise constant hazard model in those important factors. RESULTS: The piecewise model shows that six month time interval is very crucial for children till completing the five year of the age. The important factors are women's age in years, total children ever born, present breastfeeding, smoking, size of child, delivery by caesarean section, ANC visits, and birth orders in the underfive child mortality. CONCLUSION: Piecewise hazard model is found very important for the underfive child mortality, through the various times cut point. The Piecewise hazard model can be useful for clinicians, researchers and public health experts. Time is very important factor for reducing the child mortality.
Keywords: National Family Health SurveyIV, piecewise hazard, R software, survival analysis, underfive child mortality
How to cite this article: Saroj RK. Piecewise hazard model for underfive child mortality. BLDE Univ J Health Sci 2020;5:405 
The survival analysis is a technique to estimate patient's survival after treatment of any disease. The survival probability S (t) is also known as survivor function, and it describes the individuals survival time in a specified time period t. The second term hazard is represented by h (t) or λ (t), and the hazard rate is defined as the 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 Kaplan–Meier or productlimit method.^{[1]} The Cox proportional hazards (PH) regression technique is used to find the association between the survival time of patients and one or more predictor variables.^{[2]} 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 as follows:
Take the exponentiation on both sides of the equation and limit the righthand side to just a single categorical exposure variable (x_{1}) with two groups (x_{1}= 1 for exposed and x_{1}= 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 as follows:
The e^{β1} is the hazard ratio and another constant over time t. β is the regression coefficient that estimates from the model and represents 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 hazard model λ(t, x, β) is given by,
λ_{0}(t) is called the baseline hazard. There are several choices for the baseline hazard models include exponential hazard model, Weibull hazard model, and the Piecewiseconstant hazard model.
Piecewiseconstant hazard model
An essential expansion of the exponential model is known as a piecewise exponential model or piecewiseconstant hazard model. This model originates from an appropriation whose hazard rate is a step function. The model needs to segment the survival into numerous pieces. In this model, inside the each segment, the hazard is constant but between the segments, the hazard could be different.
The hazard function can be written as follows:
This study evaluates the basic hazard function with possible time change points. It demonstrates the survival pattern of the patients and which time points are more failure or more censored. Assessing the survival trend for the entire population will give a better understanding of how changing treatment, patient monitoring, health facilities of patients, and public health services. The previous research studies have proposed the strategies for estimating a single change point in a piecewiseconstant hazard function when the observed variables are subject to random censoring.^{[3]} Research has shown multiple change points in piecewiseconstant hazard functions.^{[4]} This study suggested a piecewiseexponential methodology where Poisson regression model parameters are estimated from pseudolikelihood and comparing the differences were determined by Taylor linearization strategies.^{[5]} In this study, it is estimated that the population attributable fraction for mortality in a cohort study used a piecewiseconstant hazards model.^{[6]} Another study demonstrated the cancer research using a reduced piecewise exponential approach.^{[7]} The research considers parameter estimation in the hazard rate with multiple change points in the presence of longterm survivors.^{[8]} In the research article, it is shown that a survival analysis in the context of the new method suggests estimating the piecewiseconstant hazard rate model.^{[9]} Another study has been done to find the survival status of underfive child mortality in Uttar Pradesh.^{[10]} The recent article has used the survival parametric models to estimate the factors of underfive child mortality data.^{[11]} However, previous research studies did not determine the impact of factors on underfive child mortality with the help of a piecewiseconstant hazard model.
The key interest of this article is to determine the potential determinants of underfive child mortality of data of Uttar Pradesh using a piecewiseconstant hazard model. In this model, the various time points have used to find the crucial time points and explain the important factors for reducing underfive child mortality.
Materials and Methods   
This article investigates the determinants of underfive child mortality in Uttar Pradesh using the (National Family Health Survey [NFHSIV]) data. The piecewise hazards model is used to evaluate the comparative impact of the hypothesized factors on underfive child mortality. For analysis purposes, 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 the meeting, and B3 is the CMC for date of birth of the kid. The responsible factors for underfive child mortality are selected according to the citation.^{[12]}
These components are sorted into the additional four categories: sociodemographic and social, economic, environmental, and proximate or biological factors, but in this article, the analysis used based on three social demographic and social economic, proximate, and biological factors only. There are several kinds of hazard models available, and a piecewiseconstant PH model is used in this study.
Let X_{1}, X_{2}..... X_{n} denote independent identically distributed survival times, and C_{1}, C_{2}..... C_{n} be the censoring times which are assumed to be independent of X. We have only observed the pairs (T_{i}, δ_{i}), i = 1, 2, ...n, where T_{i} = min(X_{i}, C_{i}) δ_{i} = 1 and if X_{i≤}C_{i} and zero otherwise. Consider the following changepoint model:
Where 0 < τ_{1} < ... < τ_{k}, K is the number of change points in the model, and α_{j} is the value of the hazard function between the time points τ_{j1} and τ_{j}. τ_{0j} 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 piecewiseconstant model, and a constant hazard is presumed within each section. In this analysis, the time axis is divided into nine groups: 16 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 piecewiseconstant function.
Remark
The significant factors are calculated for underfive child mortality through the Cox–regression analysis. These important factors are used in the piecewise constant hazard analysis in this article.
Results   
[Table 1] shows the survival status of under 5 years children. [Table 1] shows that a total of 93% of cases censored because child death considered the main event. The total survival time of children is 59 months, as shown in [Figure 1], and the plot shows the survival probability and censor cases report of children where plus (+) symbols represented as the censored.
In this article, the hazard rate is calculated for underfive child mortality at different time points for the piecewise hazard analysis. The time points have been distributed in 6 months, 10 months, 11 months, and 12 months, as shown in [Table 2]. The hazard rate variation shows in [Figure 2] at 6, 10, 11, and 12 months interval. [Figure 2] clearly shows that a maximum number of disparity of hazard rate in 6month intervals; therefore, the 6month time interval has been selected for piecewise hazard analysis. The piecewiseconstant 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 seventime 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 followup period is divided into seven consecutive intervals of 59 months length. The estimated parameters of piecewise constant hazards model of underfive child mortality result shows in [Table 3]. The model breaks the data into various time points cuts where fit the constant hazard within these time points. The result shows that Size of child at birth, ANC visits, Currently breast feeding and Birth interval have significant role of underfive child mortality in piece wise hazard model. [Figure 3] shows the detail of the model, including factors with a piecewiseconstant hazard function, piecewiseconstant cumulative hazard function, piecewiseconstant density function, and piecewiseconstant survivor function. In [Figure 3], the peak of the hazards continuously increases respect for increasing time.  Table 3: Estimated parameters of the of piecewise constant hazards model of underfive child mortality
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 Figure 3: Graphical display of the Piecewise constant hazard model's hazard function, cumulative hazard function, density and survivor function
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Discussion   
This article has analyzed the determinants and critical time point for underfive child mortality in Uttar Pradesh using the (NFHSIV) data through a piecewiseconstant hazard model. This is thefirst of its kind analysis for underfive child mortality in Uttar Pradesh. This analysis shows that 6 months of the children increase the risk of death before completing the 5 years of age. The hazard rate is supposed to be constant on some predefined 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 the evaluation of mortality, extend from 28 days to ½ year and including 35 days, 60 days, and 90 days. It has been utilized in earlier clinical trials,^{[13]} while this model used in a nonparametric setting, usually utilized in mix with covariates impacts.^{[14]} The applicability in the situation for the popular Poisson regression model ^{[15]} and to find the relative impact on the covariates and a piecewise constant hazard model for the baseline hazard.^{[16]} 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 at various time points.
Conclusion   
The piecewise constant hazard model has been used for finding the crucial month period and significant factors in underfive child mortality. In this study, it is found that proximate and biological factors are more critical for underfive child mortality, and ½ year or 6 months are extremely urgent for children until completing 5 years of age. These outcomes can advise the determination of time points for evaluating the survival status of underfive child mortality. The results suggest focusing on different significant factors for achieving a decline in underfive child mortality. This model would be useful in healthcare policymakers, 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 programs aimed at the population with underfive child mortality.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
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[Figure 1], [Figure 2], [Figure 3]
[Table 1], [Table 2], [Table 3]
