In reliability and survival analysis, the time-to-failure data play an important role in the development of the reliability and life characteristics of the products. These kinds of data are sometimes modeled using a competing risks model. Several inferences are frequently derived based on censored data in many practical investigations such as clinical trials, industrial experiments, mortality analysis, material strength, etc. Al-Essa et al. described that, when the life of the unit is distributed using the Gompertz distribution, noting that the units come from two lines of production and two independent causes of failure are activated [1] . There are different types of censored data such as Type-I censored data, Type-II censored data, Hybrid censored data, Joint censored data, etc. Competing risk models are applied to different types of censored data and compute statistical inference.

In this study, censored data have been used to correctly figure out and analyze death rates. When it comes to the topic of child mortality, the term “censored data” refers to circumstances in which the whole survival information of a child is not observed [2] . One of the most important indices of a nation’s socioeconomic development is the mortality rate of its children. This concern is also of the highest priority in developing nations worldwide. Despite Bangladesh’s notable advancements in reducing child mortality in recent years, the rate continues to remain high. Dina A. Ramadan et al. described that they take into consideration statistical inferences in competing risk models with Akshaya sub-distributions based on the type-II censoring scheme [3] .

In 2017, 85 percent of the 6.3 million deaths that occurred worldwide occurred in the first five years of life. More than eighty percent of the deaths that occurred in children under the age of fifteen in the first five years of their lives, regardless of the mortality rate [4] . Survival rates have risen for all age groups since 2000, but the gains were not spread out evenly. The most significant improvements in survival rates were observed among children between the ages of 1 and 4 years. The mortality rate for individuals in this age bracket experienced a 60% decrease between the years 2000 and 2017. The rate of newborn mortality decreased by 41 percent during this period, while the mortality rate among children aged 1 - 11 months, known as the post-neonatal period decreased by 51 percent. Between 2000 and 2017, there was a 37 percent decrease in mortality rates among children aged 5 to 14 [4] .

It is possible to evaluate life potentiality assumptions at various points in time through survival analysis. One of the primary challenges in life data analysis is comparing the lifetime distributions of different groups that have been studied [5] . Regression models are commonly used to model the impact of explanatory variables on the dependent variable of longevity. The models mentioned, specifically in references [6] , play a crucial role in survival analysis. Thus, this specific model is referred to as the Cox regression model or the proportional hazards regression model [7] . Alghamdi et al. described that they developed statistical inference of competing risks samples collected under a joint Type-II censoring scheme of products with Weibull lifetime distributions [8] . Amulya Kumar Mahto et al. described that inference for a competing risks model is studied when latent failure times follow the Kumaraswamy distribution and causes of failure are partially observed [9] . Chandrakant Lodhi et al. described that statistical inference for a competing risks model when latent failure times belong to a general family of inverted exponentiated distributions [10] . Chandrakant Lodhi et al. described that they study a competing risks model using the Gompertz distribution under progressive Type-II censoring when probability distributions of failure causes are identically distributed with scale and different shape parameters [11] .

The main objective of this study is to use censored data to describe the effect of categorical or quantitative variables on infant mortality by using the Cox proportional hazards regression model. The proportional hazard assumption is met by comparing the survival curves for two different groups of subjects. For example, compare the survival pattern of subjects who live in rural areas. Also, compare the survival pattern of people who practice Religion. Also, find out the determinants of infant mortality.

The study is an analytical cross-sectional study through secondary data from the Bangladesh Demographic Health Survey (BDHS 2017-2018). To date, eight rounds of BDHS have been conducted in Bangladesh by the National Institute for Population Research and Training under the Ministry of Health and Family Welfare of Bangladesh. The dataset has 12 variables such as Age, ANC visit, Father’s Education, Education, Religion, Infant Mortality, Child Mortality, Residence, Wealth Index, Birth Order, Mother age at First Birth, Highest Education level, and 7511 observations.

The data collected from the Bangladesh Demographic and Health Survey (BDHS) was analyzed using the COXPH and survival analysis tool using the R programming language. The Cox model was selected due to its ability to estimate the effect of multiple covariates simultaneously without assuming a specific baseline hazard, making it more suitable than the Kaplan-Meier or Weibull models for this multivariate analysis. While the Cox model included key variables such as maternal education, wealth, and antenatal care, potential interactions or confounding between them were not explicitly tested. Future work could include the interaction terms or stratified analysis to better understand these interdependencies. To apply this COXPH, the proportionality assumption has also been met.

This study utilized the Cox proportional hazard ratios for the factors influencing newborn and child mortality. The proportional hazard model, derived from Cox’s (1972) research, postulates that for an individual with a set of Covariates represented by $x_i$, the hazard rate at time $t_i$ can be expressed as:

$h_i\left(t_i;x_i\right)=h_0\left(t_i\right)\exp \left({\beta }^i{x}_i\right)$ (1)

$h_i\left(t_i;x_i\right)$ is the hazard function for the comparison group at time $t_i$, $x_i=1,$ $\beta$ represents a vector containing coefficients that are unknown and represent the effects of covariates, $x_i$ represents a recognized vector of predictor variables linked to the individual. The baseline hazard function, $h_0\left(t_i\right)$ represents the danger for an individual with $x_i=0$ at time $t_i$. Also, chi-square was used to assess the relationship between all the categorical variables. P-values below 0.05 were considered significant in this study.

Table 1 represents the age group of 21 - 30 years old, accounting for 59.9% of the population. 19% and 19.8% of the population, respectively, are under 20 years and between 31 to 40 years old. There is a relatively balanced distribution between individuals who received less than 4 ANC visits (34.6%) and those who received more than 4 ANC visits (32.1%). The majority of parents (Fathers (83.4%) and Mothers (93%)) received their education, and most of them belong to the religion of Islam. The observed infant mortality rate (1.9%) is consistent with national estimates from BDHS 2017-18, which reported an infant mortality rate of approximately 28 per 1000 live births (2.8%). In wealth distribution, 20.9% are from the richest family, whereas 19.4% are from poorer families. The birth order number is highest from [1, 2), which is 36.4%. 78.4% of the respondents had their first child during their teenage years (12 - 20 years). About 20.9% of respondents had their first child between the ages of 21 and 30. Only 0.7% of respondents had their first child between the ages of 31 and 41. 17.2% of respondents attained a higher education level, whereas only 7% of respondents had no education.

In Table 2 , compared to other age groups, mothers between the ages of 21 and 30 had a higher percentage of infant deaths. Mothers who had fewer than four antenatal care visits had a greater death rate (30.8%) than mothers who had more than four ANC visits (46.1%). Compared to infants with educated fathers, infants whose fathers had no formal education appear to have higher death risk, and the mother’s schooling is also a similar instance. Mortality rates between rural and urban areas show that rural areas (61.7%) have slightly higher infant mortality rates. Compared to infants from wealthier households, infants from the poorest households appear to have significantly higher mortality rates. Infant death rates according to birth order indicate a higher infant mortality rate for those with higher birth orders. Mothers who gave birth to their first child between the ages of 12 and 20 had a greater mortality rate. Compared to respondents with greater levels of education, infants who had uneducated parents appear to have slightly higher mortality rates.

In this study, the aim is to find out the causes and factors of infant mortality by using COXPH. Findings that the highest mortality occurred in the age group 20 - 30, and less than 4 ANC had higher mortality than more than or equal to 4 ANC. The Number of rural mortalities is higher than urban, but their percentage is lower. While Table 2 shows slightly higher mortality among mothers whose first birth occurred between 21 and 30, the difference is minimal. Therefore, the claim that early first birth leads to higher infant mortality requires further statistical testing and may not be conclusive in this dataset. Religion was treated as a categorical variable to capture potential differences in health practices and access to services. However, the results show that non-Muslims had a significantly higher hazard of infant mortality because the hazard rate is high that is 0.58158, indicating better survival outcomes among Muslims in this sample. This finding suggests that non-Islamic religious affiliation is associated with an increased risk of mortality, Muslims were more concise about health.

The model indicates that rural residents have a significantly higher mortality ratio compared to urban residents. A similar result was found in a prior study that urban areas have a higher survival rate than rural areas [3] . This finding is somewhat counterintuitive, as urban areas are typically associated with better access to healthcare facilities and services. Age is one of the best factors, found that 20 - 34-year-old mothers’ child mortality is lower than others. Similar results proved that urban areas had a lower.

From this study, we suggest that we need to stop early births and increase the health facilities in rural areas. This study used secondary data from BDHS 2017-2018, which may not reflect more recent trends. While BDHS offers high-quality, nationally representative data, limitations include potential recall bias and lack of control over variable definitions. Future studies could benefit from using updated datasets such as BDHS 2022-2023 if available.