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 Table of Contents  
Year : 2020  |  Volume : 5  |  Issue : 1  |  Page : 40-45

Piecewise hazard model for under-five child mortality

Department of Mathematics and Statistics, SRM Sikkim University, Gangtok, Sikkim, India

Date of Submission24-Sep-2019
Date of Decision12-Dec-2019
Date of Acceptance05-Feb-2020
Date of Web Publication08-Jul-2020

Correspondence Address:
Dr. Rakesh Kumar Saroj
Research Scholar, Department of Kayachikitsa, Institute of Medical Sciences, Banaras Hindu University, Varanasi, Uttar Pradesh
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Source of Support: None, Conflict of Interest: None

DOI: 10.4103/bjhs.bjhs_54_19

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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 co-factors. The aim of the study to describe the piece-wise constant hazard model and find the important factors in under-five child mortality data.
METHODS: For the research used the National Family Health Survey-IV 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 under-five child mortality.
CONCLUSION: Piecewise hazard model is found very important for the under-five 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 Survey-IV, piecewise hazard, R software, survival analysis, under-five child mortality

How to cite this article:
Saroj RK. Piecewise hazard model for under-five child mortality. BLDE Univ J Health Sci 2020;5:40-5

How to cite this URL:
Saroj RK. Piecewise hazard model for under-five child mortality. BLDE Univ J Health Sci [serial online] 2020 [cited 2020 Aug 13];5:40-5. Available from: http://www.bldeujournalhs.in/text.asp?2020/5/1/40/289200

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 product-limit 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 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 as follows:

The eβ1 is the hazard ratio and another constant over time t. β is the regression co-efficient 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,

λ0(t) is called the baseline hazard. There are several choices for the baseline hazard models include exponential hazard model, Weibull hazard model, and the Piecewise-constant hazard model.

Piecewise-constant hazard model

An essential expansion of the exponential model is known as a piecewise exponential model or piecewise-constant 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 piecewise-constant hazard function when the observed variables are subject to random censoring.[3] Research has shown multiple change points in piecewise-constant hazard functions.[4] This study suggested a piecewise-exponential 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 piecewise-constant 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 long-term survivors.[8] In the research article, it is shown that a survival analysis in the context of the new method suggests estimating the piecewise-constant hazard rate model.[9] Another study has been done to find the survival status of under-five child mortality in Uttar Pradesh.[10] The recent article has used the survival parametric models to estimate the factors of under-five child mortality data.[11] However, previous research studies did not determine the impact of factors on under-five child mortality with the help of a piecewise-constant hazard model.

The key interest of this article is to determine the potential determinants of under-five child mortality of data of Uttar Pradesh using a piecewise-constant hazard model. In this model, the various time points have used to find the crucial time points and explain the important factors for reducing under-five child mortality.

  Materials and Methods Top

This article investigates the determinants of under-five child mortality in Uttar Pradesh using the (National Family Health Survey [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 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 under-five 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 piecewise-constant PH 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) δi = 1 and if Xi≤Ci and zero otherwise. Consider the following change-point model:

Where 0 < τ1 < ... < τk, K is the number of change points in the model, and αj is the value of the hazard function between the time points τj-1 and τj. 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: 1-6 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.


The significant factors are calculated for under-five child mortality through the Cox–regression analysis. These important factors are used in the piece-wise constant hazard analysis in this article.

  Results Top

[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.
Table 1: Status of child survival of under 5 years

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Figure 1: Under-five child mortality survival status

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In this article, the hazard rate is calculated for under-five 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 6-month intervals; therefore, the 6-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 into seven consecutive intervals of 59 months length. The estimated parameters of piecewise constant hazards model of under-five 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 under-five child mortality in piece wise hazard model. [Figure 3] shows the detail of the model, including factors with a piecewise-constant hazard function, piecewise-constant cumulative hazard function, piecewise-constant density function, and piecewise-constant survivor function. In [Figure 3], the peak of the hazards continuously increases respect for increasing time.
Table 2: Hazard rate of the under 5 years age of children

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Table 3: Estimated parameters of the of piecewise constant hazards model of under-five child mortality

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Figure 2: Hazard rate graph of according to various month intervals

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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 Top

This article has analyzed the determinants and critical time point for under-five child mortality in Uttar Pradesh using the (NFHS-IV) data through a piecewise-constant hazard model. This is thefirst of its kind analysis for under-five 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 Top

The piece-wise constant hazard model has been used for finding the crucial month period and significant factors in under-five child mortality. In this study, it is found that proximate and biological factors are more critical for under-five 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 under-five child mortality. The results suggest focusing on different significant factors for achieving a decline in under-five child mortality. This model would be useful in health-care 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 under-five child mortality.

Financial support and sponsorship


Conflicts of interest

There are no conflicts of interest.

  References Top

Kaplan EL, Meier P. Non-parametric estimation from incomplete observations. J Am Stat Assoc 1958;53:457-81.  Back to cited text no. 1
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Gijbels I, Gürler U. Estimation of a change point in a hazard function based on censored data. Lifetime Data Anal 2003;9:395-411.  Back to cited text no. 3
Goodman MS, Li Y, Tiwari RC. Detecting multiple change points in piecewise constant hazard functions. J Appl Stat 2011;38:2523-32.  Back to cited text no. 4
Li Y, Gail MH, Preston DL, Graubard BI, Lubin JH. Piecewise exponential survival times and analysis of case-cohort data. Stat Med 2012;31:1361-8.  Back to cited text no. 5
Laaksonen MA, Knekt P, Härkänen T, Virtala E, Oja H. Estimation of the population attributable fraction for mortality in a cohort study using a piecewise constant hazards model. Am J Epidemiol 2010;171:837-47.  Back to cited text no. 6
Han G, Schell MJ, Kim J. Improved survival modeling in cancer research using a reduced piecewise exponential approach. Stat Med 2014;33:59-73.  Back to cited text no. 7
Zhang W, Qian L, Li Y. Semi parametric sequential testing for multiple change points in piecewise constant hazard functions with long-term survivors. Commun Stat Simul Comput 2014;43:1685-99.  Back to cited text no. 8
Bouaziz O, Nuel G. L0 Regularization for the estimation of piecewise constant hazard rates in survival analysis. Appl Math 2017;8:377-94.  Back to cited text no. 9
Saroj R, Murthy KN, Kumar M. Survival analysis for under-five child mortality in Uttar Pradesh. Int J Res Anal Rev 2018;5:3.  Back to cited text no. 10
Saroj RK, Murthy KH, Kumar M, Singh R, Kumar A. Survival parametric models to estimate the factors of under-five child mortality. J Health Res Rev 2019;6:82-8.  Back to cited text no. 11
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  [Figure 1], [Figure 2], [Figure 3]

  [Table 1], [Table 2], [Table 3]


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