Survival Analysis and Frailty Modelling of Time-to-event Data: an Application to Infant Mortality in Malawi

Survival Analysis and Frailty Modelling of Time-to-event Data: an Application to Infant Mortality in Malawi | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Survival Analysis and Frailty Modelling of Time-to-event Data: an Application to Infant Mortality in Malawi Esther Khundi, Mavuto Mukaka This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6121147/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 12 You are reading this latest preprint version Abstract Introduction : Infant mortality rate is one of the important health and development indicators in a country or community. Malawi, like many countries in the sub Saharan Africa is a country that suffers from the highest rates of infant mortality across the globe. Methods : This study used the most recent survey data, 2015-16 Malawi Demographic and health survey, to identify the factors associated with infant mortality in Malawi by using survival analysis techniques and frailty modelling to control for unobserved heterogeneity. Results : A total number of 4232 infants were analysed for this study and the results showed that children who were the second multiple babies to be born had a higher risk of dying before reaching the age of one year than children who are born single with P-value <0.001, HR=3.26 and 95%CI=(1.639, 5.700). Infants whose mother’s age group 45-49 years had a risk of death 4.63 times higher than infants whose mother’s age group was 15-20 years(P-value<0.001). Furthermore, there were unmeasurable family effects which made infant deaths to cluster in some families. Conclusion : social demographic, environmental and biological factors all have an effect on a child’s survival up to 1 year and the household that a child was born in had some unobservable effects on the child’s survival up to 1 year. Infant mortality cox proportional hazard frailty models Malawi Figures Figure 1 Introduction Infant mortality refers to the death of a child dying before reaching one year of age, and it is a global burden especially in developing countries like Malawi ( 1 ). Infants are the most vulnerable and this is why this study focused on infant mortality. Infant mortality rate (IMR) is sensitive to general structural factors like socio-economic development and basic living conditions, as such it is regarded as an important national health indicator( 2 ). Research shows that there was a 2.5 percent (2.5%) annual decline in global child mortality between 1960 and 1990, with Sub-Saharan Africa (SSA) having the slowest decline, of about 1.0 percent (1.0%) in the 1960s, 2.0 percent (2.0%) between 1970 and 1985, and 1.0 percent (1%) between 1985 and 1990( 3 )( 4 ). Malawi, like many countries in the Sub-Saharan Africa is a country that suffers from the highest rates of infant mortality across the globe .Malawi's infant mortality rate was at 37.828 deaths per 1000 live births in 2020( 5 ). Studies have been conducted on infant mortality in Malawi but few or none used the most recent available data The aim of this study was to apply survival analysis techniques to identify factors associated with infant mortality in Malawi using the recent demographic and health survey dataset.. The objectives of this study were to identify factors associated with infant mortality in Malawi, to examine the effects of unobserved heterogeneity (frailty) on infant mortality both at family and community level and to find the best fit model for infant mortality data. Methods The data for this study was from the 2015-16 Malawi Demographic Healthy Survey (2015-16 MDHS) which was implemented by the National Statistical Office from 19 October 2015 to 17 February 2016. The survey was based on a nationally representative sample which provided estimates at the national and regional levels and for urban and rural areas with key indicator estimates at the district level. The survey included 26,361 households, 24,562 female respondents, and 7,478 male respondents ( 6 ). There are many classical modelling methods that are commonly used to examine factors associated with mortality and one of the methods is Logistic regression and this method does not determine the causal relationship between an independent variable and the outcome variable, but rather will allow for the describing of the variables associated to infant mortality ( 7 ). The cox-proportional hazard model is essentially a commonly used survival regression model in medical research for investigating the association between the survival time of patients and one or more predictor variables. The cox model was introduced in 1972 and has the form; $$\:{\lambda\:}\left(t|X\right)={\lambda\:}_{0}\left(t\right)\text{e}\text{x}\text{p}\left({X}^{T}\beta\:\right)$$ Frailty models aim at modelling the heterogeneity in the population, they can be used to account for the influence of unobserved covariates ( 8 ). In the cox PH frailty model also known as the mixed PH model, the hazard rate of subject j belonging to cluster i, conditionally on the covariates x ij and the shared frailty b i is given by; $$\:{\lambda\:}\left(t|{x}_{ij},{b}_{i}\right)={b}_{i}{\lambda\:}_{0}\left(t\right)\text{exp}\left({x}_{ij}^{T}\beta\:\right)\:\:\:\:\:\:\:\:\:$$ $$\:\:\:\:\:\:\:\:i=1,\dots\:,n,\:\:j=1,\dots\:,{n}_{i}$$ Where \(\:{b}_{i}\) is the frailty term and frequently assumed to follow a gamma distribution because of its mathematical convenience. There are two categories of frailty models which are the univariate frailty models that consider univariate survival times and the multivariate frailty models that take into account multivariate survival times ( 9 ). Univariate frailty models take into account that the population is not homogeneous. Unobserved heterogeneity comes about when important covariates have not been observed even though heterogeneity maybe explained by covariates. Given that an individual child under-five years of age i, i = 1, 2,…, n has a survival time denoted as \(\:{t}_{i}\) the covariate vector X i , with a frailty term denoted as \(\:{b}_{i}\) the survival function of individual i conditional on frailty is given by; $$\:{S}_{i}\left({t}_{i},{X}_{i}|{b}_{i}\right)=exp\left(-{b}_{i}{e}^{{x}_{i}^{T}\beta\:}{\int\:}_{0}^{{t}_{i}}{\lambda\:}_{0}\left(s,{X}_{i}|{b}_{i}\right)ds\right)=exp\left(-{b}_{i}{\lambda\:}_{0}({t}_{i}\right)exp\left({x}_{i}^{T}\beta\:\right)$$ Where \(\:{\lambda\:}_{0}\left({t}_{i}\right)=\sum\:_{0}^{{t}_{i}}{\lambda\:}_{0}\left(s\right)ds\) is the cumulative baseline hazard function. The shared frailty model is relevant to event times of related individuals, similar organs and repeated measurements. It is assumed that individuals in a cluster share the same frailty and that is why this model is called a shared frailty model .Given n clusters with n i individuals with an unobserved frailty b i which are assumed to be identically and independently distributed random variables, survival time T ij and associated vector X ij for individual j in cluster i ( 9 ) \(\:,\) the hazard function of the individual is given as: $$\:{\lambda\:}_{ij}\left(t\right)={b}_{i}{\lambda\:}_{0}\left(t\right)\text{e}\text{x}\text{p}\left({X}_{ij}^{T}\beta\:\right)$$ Results Infant mortality defined as the death of a child under the age of 1 year was the dependent variable in this study and was measured as a binary response: yes or no. The covariate/predictors included in the model were social-demographic, social economic, biological factors and environmental factors. These covariates are religion, area of residence, mothers’ highest education, mother’s age group, sex of child, place of delivery, size of child, source of drinking water, sex of household head(SHH), family economic status and type of birth. The plots in Fig. 1 displays the Kaplan Meier curves for SHH and Family economic status covariates. In the Fig. 1 b, the survival curve for infants who are born in male headed families is above the survival curve for infants born in female headed families which implies that infants from male headed households had a higher survival probability than infants born in families headed by females. It can also be seen from the Fig. 1 a that infants born in poorer families had a high probability of death compared to their counterparts The results of the Cox proportional hazard model are given in Tables 1 a and 1 b, from the results, it was found that SHH, mothers’ age group, source of drinking water, religion, type of birth and place of delivery had a significant association with infant mortality at the 5% of significance level. For place of delivery, only private hospital categorical was significant with P-value = 0.020 and hazard ratio HR = 2.97 (95%CI = 1.17, 7.52) which implied that children who were born in private hospitals had a higher risk of dying 2.97 times more than children born in respondent’s home before reaching the age of 1 year. The results showed that infants who were the second multiple babies to be born and first multiple babies had a higher risk of dying before reaching the age of one than children who were born single. This covariate, type of birth, was highly significant with p-values (95% CI) of < 0.001(1.639, 5.700) and 0.049 (1.004, 4.326) for 2nd and 1st multiple babies subcategories respectively. The results also indicated that children born in households whose head was a female had a higher risk of dying before reaching the age of 1 year than children born in households with a male head, P-value = 0.026, HR = 1.37 and 95%CI= (1.037, 1.803). As can be observed from the results, two groups were highly significant from the covariate mothers’ age group with p-values of < 0.001. Age group 40–44 and 45–49 had hazard ratios of 3.86, 95%CI = (2.148, 6.930) and 5.22, 95%CI= (2.630, 10.337) respectively, which indicated that infants whose mother’s age group was 40–44 and 45–49 had a higher risk of death compared to infants whose mother’s age group was 15–20. It was also found that the confidence interval for the 45–49 age category was wide, this could be because the sample size used for the analysis was small. Table 1 a: Cox proportional hazard model results Covariate Hazard ratio (95% conf. Interval) Std Error P-value Sex of child male(ref) female 1.000 1.13(0.881 1.443) 1.14 0.339 Family economic status Poorest (ref) Poorer Middle Richer richest 1.000 1.13(0.764, 1.569) 1.20(0.839, 1.764) 0.89(0.585, 1.357) 0.90(0.534, 1.579) 0.21 0.23 0.19 0.25 0.620 0.301 0.593 0.759 Place of delivery Respondents home(ref) Other home Govt hospital Govt health center Govt outreach Private hospital Cham/mission hospital Cham/mission health center Other 1.000 1.41(0.521, 4.809) 1.11(0.604, 2.537) 1.32(0.722, 2.871) 1.71(0.584, 6.415) 2.90(1.175, 7.524) 0.81(0.377, 2.227) 1.76(0.874, 4.653) 1.03(0. .276, 3.887) 0.79 0.40 0.46 1.04 1.36 0.36 0.74 0.69 0.417 0.560 0.301 0.280 0.021 0.847 0.100 0.958 Religion Catholic(ref) CCAP Anglican Seventh day Adventist Other Christian Muslim Other 1.000 0.85(0. 519, 1.343) 0.59(0.251, 1.300) 1.01(0. .558, 1.757) 0.99(0. 689, 1.384) 0.85(0 .533, 1.327) 10.22(1.374, 87.240) 0.21 0.25 0.29 0.18 0.20 10.84 0.457 0.183 0.974 0.895 0.457 0.024 Type of birth Single(ref) 1st of Multiple 2nd of multiple 1.00 2.27(1.004, 4.326) 3.26(1.639, 5.700) 0.84 1.03 0.049 < 0.001 Table 1 b: cox proportional hazard model results Source of drinking water Piped into dwelling Piped to yard Piped to neighbor Public tap Tube well/borehore Protected well Unprotected well Unprotected spring River/dam Not a dejure resident 1.00 0.64(0.281, 1.829) 0.17(0.046, 0.795) 0.33(0 .145, 1.029) 0.30(0.135, 0.892) 0.22(0.067, 0.856) 0.29(0.124, 0.964) 0.35(0.111, 1.297) 0.23(0.088, 0.829) 0.16(0.019, 1.620) 0.30 0.12 0.16 0.14 0.14 0.15 0.22 0.13 0.18 0.487 0.023 0.057 0.028 0.028 0.042 0.122 0.022 0.125 Mothers’ highest education No education(ref) Primary Secondary higher 1.00 1.17(0.790, 1.729) 0.93(0 .559, 1.594) 1.85(0.025, 1.604) 0.23 0.25 0.20 0.443 0.830 0.131 Sex of household head Male(ref) female 1.00 1.37(1.037, 1.803) 0.19 0.026 Mothers’ age group 15–19(ref) 20–24 25–29 30–34 35–39 40–44 45–49 1.00 1.36(0.910, 2.204) 1.31(0.842, 2.193) 1.31(0 .822, 2.217) 1.40(0 .900, 2.611) 3.49( 2.148, 6.930) 4.63(2.630, 10.337) 0.31 0.32 0.33 0.38 1.04 1.60 0.122 0.209 0.235 0.116 < 0.001 < 0.001 Area of residence Urban(ref) Rural 1.00 1.60(0.923, 2.686) 0.43 0.095 Size of child Very large(ref) Larger than average Average Smaller than average Very small Don’t know 1.00 0.89(0.527, 1.504) 0.97(0.594, 1.582) 1.70(0.985, 2.941) 1.01(0.497, 2.049) 2.46(0.815, 7.413) 0.24 0.24 0.48 0.37 1.39 0.667 0.904 0.056 0.979 0.110 The household/family frailty value had the chi-square test statistic ( \(\:\chi\:\) = 2.90) and a p-value of 0.04 at 0.05 level of significance, this result showed that there exist unmeasurable factors that have an effect on the hazard of infant death. However for the community effects, the model did not converge. In addition to these semi-parametric frailty models, Weibull and log-normal parametric frailty models were fit for both household and community effects and the results for both effects were not significant in infant mortality. Discussion and Conclusion This study used survival analysis and frailty modelling to examine the factors that are associated with infant mortality in Malawi. The mean age for infants who died was2.25 months which is consistent with what was reported by ( 10 ) that the first 28 days of life are when a child’s survival is most vulnerable. This study found that SHH, mothers’ age group, source of drinking water, religion, type of birth and place of delivery have a significant association with infant mortality. Particularly, the results indicated that FHH are at a higher risk of children dying before the age of 1 year compared to MHH and mothers who have single births have a lower chance of experiencing infant mortality compared to mothers who have multiple births. The results also indicated that women who have children at older ages are at a higher risk of experiencing infant mortality compared to women who birth children at younger ages. Furthermore, this study found significant evidence that some families/households experience more infant deaths than others which indicated that there do exists some unobservable family/household effects which tend to make infant deaths cluster in some families. Although Malawi has managed to achieve a significant reduction in infant and child mortality rates, the rates remain high compared to most African countries as such there is need for more effort to reduce these mortality rates. This study has provided insights into the risk factors of infant mortality in Malawi, which contains vital information for health policy makers in government and non-governmental organizations. In conclusion, this study revealed that SHH, mothers’ age group, source of drinking water, religion, type of birth and place of delivery have a significant association with infant mortality after controlling for the effects of other factors and also that there are unobservable family effects which make infant deaths to cluster in some families. These factors need to be considered when planning and developing policies against infant mortality in order to successfully work towards reducing infant mortality rate in Malawi. Limitations The study had some limitations which might have affected the results. One of the limitation is that the DHS survey collect data from women aged 15–49 who are alive in a given household which implies that no information is collected for mothers who have died which creates a bias in the results. It was also a challenge to control for community effects because the MDHS data doesn’t have specific community characteristics. Declarations Ethics approval and consent to participate Not applicable Not applicable Consent for publication Not applicable Availability of data and materials Data for the submitted work can be accessed through https://www.dhsprogram.com Competing interests The authors declare that they have no competing interests Funding The authors did not receive support from any organization for the submitted work Author’s contributions EK analysed and interpreted the data. MM offered guidance as a supervisor. All authors read and approved the final manuscript Acknowledgements The authors thank the DHS program for their contribution to creating and giving access to the data file. References Ndawala J. Infant and Child Mortality. DHS Progr. 2015;97–104. Sartorius K, Sartorius B. Global infant mortality trends and attributable determinants-an ecological study using data from 192 countries for the period 1990–2011. Popul Health Metr. 2014. Jahn A, Floyd S, Crampin A. Declining child mortality in northern Malawi despite high rates of infection with HIV. Bull world Heal Organ. 2010. Hill K, Amouzou A. Trends in child mortality, 19960 – 2000. world Bank. 2006. Ajaari J, Masanja H, Owusu-Agyei S. Impact of Place of delivery on Neonatal Mortality in Rural Tanzania. Glob Heal. 2012;49–59. National Statistical Office (NSO). Malawi Demographic and Health Survey 2015-16. 2017. Dube ZB. The relationship between mothers’ maternal age and infant mortality in Zimbabwe. UNIVERSITY OF WITWATERSRAND; 2012. Vaupel J, Manton K. The Impact of heterogeneity in individual frailty on te dynamics of mortality. JSTORI. 1979;439–54. Wienke A. Frailty Models. Max Planck Inst Demogr Res. 2003;49(0):0–13. UNICEF DATA. Neonatal mortality. 2020. Additional Declarations No competing interests reported. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6121147","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":427645424,"identity":"5c9b2fa6-d8ce-4094-a0fe-293e9d1c99e6","order_by":0,"name":"Esther Khundi","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3UlEQVRIiWNgGAWjYJACxsYGIHmY+QCQlJAhQgMzTAtbAkgLDwlaDvAYgLiEtfDznz/4ceaOw3l8x3k+v7pRY8HDwH746AZ8WiRnJDNLbjxzuFjyMO8265xjQIfxpKXdwKfF4AYzg+TDtsOJG4BajHPYgFokeMzwarE/f5j5J0QLzzPjnH9EaDFgSGaT3AjRwvw4t40ILRI3ks0sZ7alJ848zGbGnNsnwcNGyC/8/Qcf3+xts07sO3/48eecb3Vy/OyHj+HVggzYJMAkscpBgPkDKapHwSgYBaNg5AAACOlMRZC8lCoAAAAASUVORK5CYII=","orcid":"","institution":"University of Malawi","correspondingAuthor":true,"prefix":"","firstName":"Esther","middleName":"","lastName":"Khundi","suffix":""},{"id":427645425,"identity":"2e555154-701b-4cf2-91a7-1b75870a5679","order_by":1,"name":"Mavuto Mukaka","email":"","orcid":"","institution":"Mahidol University","correspondingAuthor":false,"prefix":"","firstName":"Mavuto","middleName":"","lastName":"Mukaka","suffix":""}],"badges":[],"createdAt":"2025-02-27 12:53:29","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6121147/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6121147/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":78518826,"identity":"d432a441-8a49-4e53-ba12-42bf9eeeaede","added_by":"auto","created_at":"2025-03-14 11:33:59","extension":"jpeg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":319181,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003ea family economic status and b sex of household head Kaplan Meier curves\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"floatimage1.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6121147/v1/dca6a190593ec424a998bfe5.jpeg"},{"id":78519753,"identity":"4257c218-000a-4889-8ac0-cc4dc9a2dd0c","added_by":"auto","created_at":"2025-03-14 11:50:00","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1199675,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6121147/v1/ff6277b1-831e-4ce4-aba8-8fee19be029c.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eSurvival Analysis and Frailty Modelling of Time-to-event Data: an Application to Infant Mortality in Malawi\u003c/p\u003e","fulltext":[{"header":"Introduction","content":"\u003cp\u003eInfant mortality refers to the death of a child dying before reaching one year of age, and it is a global burden especially in developing countries like Malawi (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e). Infants are the most vulnerable and this is why this study focused on infant mortality. Infant mortality rate (IMR) is sensitive to general structural factors like socio-economic development and basic living conditions, as such it is regarded as an important national health indicator(\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e). Research shows that there was a 2.5 percent (2.5%) annual decline in global child mortality between 1960 and 1990, with Sub-Saharan Africa (SSA) having the slowest decline, of about 1.0 percent (1.0%) in the 1960s, 2.0 percent (2.0%) between 1970 and 1985, and 1.0 percent (1%) between 1985 and 1990(\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e)(\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e). Malawi, like many countries in the Sub-Saharan Africa is a country that suffers from the highest rates of infant mortality across the globe .Malawi's infant mortality rate was at 37.828 deaths per 1000 live births in 2020(\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e). Studies have been conducted on infant mortality in Malawi but few or none used the most recent available data The aim of this study was to apply survival analysis techniques to identify factors associated with infant mortality in Malawi using the recent demographic and health survey dataset.. The objectives of this study were to identify factors associated with infant mortality in Malawi, to examine the effects of unobserved heterogeneity (frailty) on infant mortality both at family and community level and to find the best fit model for infant mortality data.\u003c/p\u003e"},{"header":"Methods","content":"\u003cp\u003eThe data for this study was from the 2015-16 Malawi Demographic Healthy Survey (2015-16 MDHS) which was implemented by the National Statistical Office from 19 October 2015 to 17 February 2016. The survey was based on a nationally representative sample which provided estimates at the national and regional levels and for urban and rural areas with key indicator estimates at the district level. The survey included 26,361 households, 24,562 female respondents, and 7,478 male respondents (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThere are many classical modelling methods that are commonly used to examine factors associated with mortality and one of the methods is Logistic regression and this method does not determine the causal relationship between an independent variable and the outcome variable, but rather will allow for the describing of the variables associated to infant mortality (\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e). The cox-proportional hazard model is essentially a commonly used survival regression model in medical research for investigating the association between the survival time of patients and one or more predictor variables. The cox model was introduced in 1972 and has the form;\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:{\\lambda\\:}\\left(t|X\\right)={\\lambda\\:}_{0}\\left(t\\right)\\text{e}\\text{x}\\text{p}\\left({X}^{T}\\beta\\:\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eFrailty models aim at modelling the heterogeneity in the population, they can be used to account for the influence of unobserved covariates (\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e). In the cox PH frailty model also known as the mixed PH model, the hazard rate of subject j belonging to cluster i, conditionally on the covariates x\u003csub\u003eij\u003c/sub\u003e and the shared frailty b\u003csub\u003ei\u003c/sub\u003e is given by;\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:{\\lambda\\:}\\left(t|{x}_{ij},{b}_{i}\\right)={b}_{i}{\\lambda\\:}_{0}\\left(t\\right)\\text{exp}\\left({x}_{ij}^{T}\\beta\\:\\right)\\:\\:\\:\\:\\:\\:\\:\\:\\:$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\:\\:\\:\\:\\:\\:\\:\\:i=1,\\dots\\:,n,\\:\\:j=1,\\dots\\:,{n}_{i}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{b}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the frailty term and frequently assumed to follow a gamma distribution because of its mathematical convenience. There are two categories of frailty models which are the univariate frailty models that consider univariate survival times and the multivariate frailty models that take into account multivariate survival times (\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e). Univariate frailty models take into account that the population is not homogeneous. Unobserved heterogeneity comes about when important covariates have not been observed even though heterogeneity maybe explained by covariates. Given that an individual child under-five years of age i, i\u0026thinsp;=\u0026thinsp;1, 2,\u0026hellip;, n has a survival time denoted as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{t}_{i}\\)\u003c/span\u003e\u003c/span\u003e the covariate vector X\u003csub\u003ei\u003c/sub\u003e, with a frailty term denoted as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{b}_{i}\\)\u003c/span\u003e\u003c/span\u003e the survival function of individual i conditional on frailty is given by;\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e\n$$\\:{S}_{i}\\left({t}_{i},{X}_{i}|{b}_{i}\\right)=exp\\left(-{b}_{i}{e}^{{x}_{i}^{T}\\beta\\:}{\\int\\:}_{0}^{{t}_{i}}{\\lambda\\:}_{0}\\left(s,{X}_{i}|{b}_{i}\\right)ds\\right)=exp\\left(-{b}_{i}{\\lambda\\:}_{0}({t}_{i}\\right)exp\\left({x}_{i}^{T}\\beta\\:\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\lambda\\:}_{0}\\left({t}_{i}\\right)=\\sum\\:_{0}^{{t}_{i}}{\\lambda\\:}_{0}\\left(s\\right)ds\\)\u003c/span\u003e\u003c/span\u003e is the cumulative baseline hazard function.\u003c/p\u003e \u003cp\u003eThe shared frailty model is relevant to event times of related individuals, similar organs and repeated measurements. It is assumed that individuals in a cluster share the same frailty and that is why this model is called a shared frailty model .Given n clusters with n\u003csub\u003ei\u003c/sub\u003e individuals with an unobserved frailty b\u003csub\u003ei\u003c/sub\u003e which are assumed to be identically and independently distributed random variables, survival time T\u003csub\u003eij\u003c/sub\u003e and associated vector X\u003csub\u003eij\u003c/sub\u003e for individual j in cluster i (\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e)\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:,\\)\u003c/span\u003e\u003c/span\u003e the hazard function of the individual is given as:\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e\n$$\\:{\\lambda\\:}_{ij}\\left(t\\right)={b}_{i}{\\lambda\\:}_{0}\\left(t\\right)\\text{e}\\text{x}\\text{p}\\left({X}_{ij}^{T}\\beta\\:\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003eInfant mortality defined as the death of a child under the age of 1 year was the dependent variable in this study and was measured as a binary response: yes or no. The covariate/predictors included in the model were social-demographic, social economic, biological factors and environmental factors. These covariates are religion, area of residence, mothers\u0026rsquo; highest education, mother\u0026rsquo;s age group, sex of child, place of delivery, size of child, source of drinking water, sex of household head(SHH), family economic status and type of birth.\u003c/p\u003e\n\u003cp\u003eThe plots in Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e displays the Kaplan Meier curves for SHH and Family economic status covariates. In the Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003eb, the survival curve for infants who are born in male headed families is above the survival curve for infants born in female headed families which implies that infants from male headed households had a higher survival probability than infants born in families headed by females. It can also be seen from the Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ea that infants born in poorer families had a high probability of death compared to their counterparts\u003c/p\u003e\n\u003cp\u003eThe results of the Cox proportional hazard model are given in Tables \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ea and \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003eb, from the results, it was found that SHH, mothers\u0026rsquo; age group, source of drinking water, religion, type of birth and place of delivery had a significant association with infant mortality at the 5% of significance level. For place of delivery, only private hospital categorical was significant with P-value\u0026thinsp;=\u0026thinsp;0.020 and hazard ratio HR\u0026thinsp;=\u0026thinsp;2.97 (95%CI\u0026thinsp;=\u0026thinsp;1.17, 7.52) which implied that children who were born in private hospitals had a higher risk of dying 2.97 times more than children born in respondent\u0026rsquo;s home before reaching the age of 1 year. The results showed that infants who were the second multiple babies to be born and first multiple babies had a higher risk of dying before reaching the age of one than children who were born single. This covariate, type of birth, was highly significant with p-values (95% CI) of \u0026lt;\u0026thinsp;0.001(1.639, 5.700) and 0.049 (1.004, 4.326) for 2nd and 1st multiple babies subcategories respectively. The results also indicated that children born in households whose head was a female had a higher risk of dying before reaching the age of 1 year than children born in households with a male head, P-value\u0026thinsp;=\u0026thinsp;0.026, HR\u0026thinsp;=\u0026thinsp;1.37 and 95%CI= (1.037, 1.803). As can be observed from the results, two groups were highly significant from the covariate mothers\u0026rsquo; age group with p-values of \u0026lt;\u0026thinsp;0.001. Age group 40\u0026ndash;44 and 45\u0026ndash;49 had hazard ratios of 3.86, 95%CI = (2.148, 6.930) and 5.22, 95%CI= (2.630, 10.337) respectively, which indicated that infants whose mother\u0026rsquo;s age group was 40\u0026ndash;44 and 45\u0026ndash;49 had a higher risk of death compared to infants whose mother\u0026rsquo;s age group was 15\u0026ndash;20. It was also found that the confidence interval for the 45\u0026ndash;49 age category was wide, this could be because the sample size used for the analysis was small.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003ea: Cox proportional hazard model results\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\" style=\"width: 44.8265%;\"\u003e\n \u003cp\u003eCovariate\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 22.08%;\"\u003e\n \u003cp\u003eHazard ratio (95% conf. Interval)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 13.9339%;\"\u003e\n \u003cp\u003eStd Error\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 19.7883%;\"\u003e\n \u003cp\u003eP-value\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003cth align=\"left\" style=\"width: 44.8265%;\"\u003e\n \u003cp\u003eSex of child\u003c/p\u003e\n \u003cp\u003emale(ref)\u003c/p\u003e\n \u003cp\u003efemale\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 22.08%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e1.000\u003c/p\u003e\n \u003cp\u003e1.13(0.881 1.443)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 13.9339%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e1.14\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 19.7883%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.339\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003cth align=\"left\" style=\"width: 44.8265%;\"\u003e\n \u003cp\u003eFamily economic status\u003c/p\u003e\n \u003cp\u003ePoorest (ref)\u003c/p\u003e\n \u003cp\u003ePoorer\u003c/p\u003e\n \u003cp\u003eMiddle\u003c/p\u003e\n \u003cp\u003eRicher\u003c/p\u003e\n \u003cp\u003erichest\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 22.08%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e1.000\u003c/p\u003e\n \u003cp\u003e1.13(0.764, 1.569)\u003c/p\u003e\n \u003cp\u003e1.20(0.839, 1.764)\u003c/p\u003e\n \u003cp\u003e0.89(0.585, 1.357)\u003c/p\u003e\n \u003cp\u003e0.90(0.534, 1.579)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 13.9339%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.21\u003c/p\u003e\n \u003cp\u003e0.23\u003c/p\u003e\n \u003cp\u003e0.19\u003c/p\u003e\n \u003cp\u003e0.25\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 19.7883%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.620\u003c/p\u003e\n \u003cp\u003e0.301\u003c/p\u003e\n \u003cp\u003e0.593\u003c/p\u003e\n \u003cp\u003e0.759\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 44.8265%;\"\u003e\n \u003cp\u003e\u003cstrong\u003ePlace of delivery\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003eRespondents home(ref)\u003c/p\u003e\n \u003cp\u003eOther home\u003c/p\u003e\n \u003cp\u003eGovt hospital\u003c/p\u003e\n \u003cp\u003eGovt health center\u003c/p\u003e\n \u003cp\u003eGovt outreach\u003c/p\u003e\n \u003cp\u003ePrivate hospital\u003c/p\u003e\n \u003cp\u003eCham/mission hospital\u003c/p\u003e\n \u003cp\u003eCham/mission health center\u003c/p\u003e\n \u003cp\u003eOther\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 22.08%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e1.000\u003c/p\u003e\n \u003cp\u003e1.41(0.521, 4.809)\u003c/p\u003e\n \u003cp\u003e1.11(0.604, 2.537)\u003c/p\u003e\n \u003cp\u003e1.32(0.722, 2.871)\u003c/p\u003e\n \u003cp\u003e1.71(0.584, 6.415)\u003c/p\u003e\n \u003cp\u003e2.90(1.175, 7.524)\u003c/p\u003e\n \u003cp\u003e0.81(0.377, 2.227)\u003c/p\u003e\n \u003cp\u003e1.76(0.874, 4.653)\u003c/p\u003e\n \u003cp\u003e1.03(0. .276, 3.887)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.9339%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.79\u003c/p\u003e\n \u003cp\u003e0.40\u003c/p\u003e\n \u003cp\u003e0.46\u003c/p\u003e\n \u003cp\u003e1.04\u003c/p\u003e\n \u003cp\u003e1.36\u003c/p\u003e\n \u003cp\u003e0.36\u003c/p\u003e\n \u003cp\u003e0.74\u003c/p\u003e\n \u003cp\u003e0.69\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.7883%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.417\u003c/p\u003e\n \u003cp\u003e0.560\u003c/p\u003e\n \u003cp\u003e0.301\u003c/p\u003e\n \u003cp\u003e0.280\u003c/p\u003e\n \u003cp\u003e0.021\u003c/p\u003e\n \u003cp\u003e0.847\u003c/p\u003e\n \u003cp\u003e0.100\u003c/p\u003e\n \u003cp\u003e0.958\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 44.8265%;\"\u003e\n \u003cp\u003e\u003cstrong\u003eReligion\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003eCatholic(ref)\u003c/p\u003e\n \u003cp\u003eCCAP\u003c/p\u003e\n \u003cp\u003eAnglican\u003c/p\u003e\n \u003cp\u003eSeventh day Adventist\u003c/p\u003e\n \u003cp\u003eOther Christian\u003c/p\u003e\n \u003cp\u003eMuslim\u003c/p\u003e\n \u003cp\u003eOther\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 22.08%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e1.000\u003c/p\u003e\n \u003cp\u003e0.85(0. 519, 1.343)\u003c/p\u003e\n \u003cp\u003e0.59(0.251, 1.300)\u003c/p\u003e\n \u003cp\u003e1.01(0. .558, 1.757)\u003c/p\u003e\n \u003cp\u003e0.99(0. 689, 1.384)\u003c/p\u003e\n \u003cp\u003e0.85(0 .533, 1.327)\u003c/p\u003e\n \u003cp\u003e10.22(1.374, 87.240)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.9339%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.21\u003c/p\u003e\n \u003cp\u003e0.25\u003c/p\u003e\n \u003cp\u003e0.29\u003c/p\u003e\n \u003cp\u003e0.18\u003c/p\u003e\n \u003cp\u003e0.20\u003c/p\u003e\n \u003cp\u003e10.84\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.7883%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.457\u003c/p\u003e\n \u003cp\u003e0.183\u003c/p\u003e\n \u003cp\u003e0.974\u003c/p\u003e\n \u003cp\u003e0.895\u003c/p\u003e\n \u003cp\u003e0.457\u003c/p\u003e\n \u003cp\u003e0.024\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 44.8265%;\"\u003e\n \u003cp\u003e\u003cstrong\u003eType of birth\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003eSingle(ref)\u003c/p\u003e\n \u003cp\u003e1st of Multiple\u003c/p\u003e\n \u003cp\u003e2nd of multiple\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 22.08%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e1.00\u003c/p\u003e\n \u003cp\u003e2.27(1.004, 4.326)\u003c/p\u003e\n \u003cp\u003e3.26(1.639, 5.700)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.9339%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.84\u003c/p\u003e\n \u003cp\u003e1.03\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.7883%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.049\u003c/p\u003e\n \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\u0026nbsp;\u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eb: cox proportional hazard model results\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\" style=\"width: 43.5864%;\"\u003e\n \u003cp\u003eSource of drinking water\u003c/p\u003e\n \u003cp\u003ePiped into dwelling\u003c/p\u003e\n \u003cp\u003ePiped to yard\u003c/p\u003e\n \u003cp\u003ePiped to neighbor\u003c/p\u003e\n \u003cp\u003ePublic tap\u003c/p\u003e\n \u003cp\u003eTube well/borehore\u003c/p\u003e\n \u003cp\u003eProtected well\u003c/p\u003e\n \u003cp\u003eUnprotected well\u003c/p\u003e\n \u003cp\u003eUnprotected spring\u003c/p\u003e\n \u003cp\u003eRiver/dam\u003c/p\u003e\n \u003cp\u003eNot a dejure resident\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 26.9197%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e1.00\u003c/p\u003e\n \u003cp\u003e0.64(0.281, 1.829)\u003c/p\u003e\n \u003cp\u003e0.17(0.046, 0.795)\u003c/p\u003e\n \u003cp\u003e0.33(0 .145, 1.029)\u003c/p\u003e\n \u003cp\u003e0.30(0.135, 0.892)\u003c/p\u003e\n \u003cp\u003e0.22(0.067, 0.856)\u003c/p\u003e\n \u003cp\u003e0.29(0.124, 0.964)\u003c/p\u003e\n \u003cp\u003e0.35(0.111, 1.297)\u003c/p\u003e\n \u003cp\u003e0.23(0.088, 0.829)\u003c/p\u003e\n \u003cp\u003e0.16(0.019, 1.620)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 15.2978%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.30\u003c/p\u003e\n \u003cp\u003e0.12\u003c/p\u003e\n \u003cp\u003e0.16\u003c/p\u003e\n \u003cp\u003e0.14\u003c/p\u003e\n \u003cp\u003e0.14\u003c/p\u003e\n \u003cp\u003e0.15\u003c/p\u003e\n \u003cp\u003e0.22\u003c/p\u003e\n \u003cp\u003e0.13\u003c/p\u003e\n \u003cp\u003e0.18\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 14.0529%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.487\u003c/p\u003e\n \u003cp\u003e0.023\u003c/p\u003e\n \u003cp\u003e0.057\u003c/p\u003e\n \u003cp\u003e0.028\u003c/p\u003e\n \u003cp\u003e0.028\u003c/p\u003e\n \u003cp\u003e0.042\u003c/p\u003e\n \u003cp\u003e0.122\u003c/p\u003e\n \u003cp\u003e0.022\u003c/p\u003e\n \u003cp\u003e0.125\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003cth align=\"left\" style=\"width: 43.5864%;\"\u003e\n \u003cp\u003eMothers\u0026rsquo; highest education\u003c/p\u003e\n \u003cp\u003eNo education(ref)\u003c/p\u003e\n \u003cp\u003ePrimary\u003c/p\u003e\n \u003cp\u003eSecondary\u003c/p\u003e\n \u003cp\u003ehigher\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 26.9197%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e1.00\u003c/p\u003e\n \u003cp\u003e1.17(0.790, 1.729)\u003c/p\u003e\n \u003cp\u003e0.93(0 .559, 1.594)\u003c/p\u003e\n \u003cp\u003e1.85(0.025, 1.604)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 15.2978%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.23\u003c/p\u003e\n \u003cp\u003e0.25\u003c/p\u003e\n \u003cp\u003e0.20\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 14.0529%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.443\u003c/p\u003e\n \u003cp\u003e0.830\u003c/p\u003e\n \u003cp\u003e0.131\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003cth align=\"left\" style=\"width: 43.5864%;\"\u003e\n \u003cp\u003eSex of household head\u003c/p\u003e\n \u003cp\u003eMale(ref)\u003c/p\u003e\n \u003cp\u003efemale\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 26.9197%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e1.00\u003c/p\u003e\n \u003cp\u003e1.37(1.037, 1.803)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 15.2978%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.19\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 14.0529%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.026\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 43.5864%;\"\u003e\n \u003cp\u003e\u003cstrong\u003eMothers\u0026rsquo; age group\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003e15\u0026ndash;19(ref)\u003c/p\u003e\n \u003cp\u003e20\u0026ndash;24\u003c/p\u003e\n \u003cp\u003e25\u0026ndash;29\u003c/p\u003e\n \u003cp\u003e30\u0026ndash;34\u003c/p\u003e\n \u003cp\u003e35\u0026ndash;39\u003c/p\u003e\n \u003cp\u003e40\u0026ndash;44\u003c/p\u003e\n \u003cp\u003e45\u0026ndash;49\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 26.9197%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e1.00\u003c/p\u003e\n \u003cp\u003e1.36(0.910, 2.204)\u003c/p\u003e\n \u003cp\u003e1.31(0.842, 2.193)\u003c/p\u003e\n \u003cp\u003e1.31(0 .822, 2.217)\u003c/p\u003e\n \u003cp\u003e1.40(0 .900, 2.611)\u003c/p\u003e\n \u003cp\u003e3.49( 2.148, 6.930)\u003c/p\u003e\n \u003cp\u003e4.63(2.630, 10.337)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 15.2978%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.31\u003c/p\u003e\n \u003cp\u003e0.32\u003c/p\u003e\n \u003cp\u003e0.33\u003c/p\u003e\n \u003cp\u003e0.38\u003c/p\u003e\n \u003cp\u003e1.04\u003c/p\u003e\n \u003cp\u003e1.60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.0529%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.122\u003c/p\u003e\n \u003cp\u003e0.209\u003c/p\u003e\n \u003cp\u003e0.235\u003c/p\u003e\n \u003cp\u003e0.116\u003c/p\u003e\n \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e\n \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 43.5864%;\"\u003e\n \u003cp\u003e\u003cstrong\u003eArea of residence\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003eUrban(ref)\u003c/p\u003e\n \u003cp\u003eRural\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 26.9197%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e1.00\u003c/p\u003e\n \u003cp\u003e1.60(0.923, 2.686)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 15.2978%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.43\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.0529%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.095\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 43.5864%;\"\u003e\n \u003cp\u003e\u003cstrong\u003eSize of child\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003eVery large(ref)\u003c/p\u003e\n \u003cp\u003eLarger than average\u003c/p\u003e\n \u003cp\u003eAverage\u003c/p\u003e\n \u003cp\u003eSmaller than average\u003c/p\u003e\n \u003cp\u003eVery small\u003c/p\u003e\n \u003cp\u003eDon\u0026rsquo;t know\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 26.9197%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e1.00\u003c/p\u003e\n \u003cp\u003e0.89(0.527, 1.504)\u003c/p\u003e\n \u003cp\u003e0.97(0.594, 1.582)\u003c/p\u003e\n \u003cp\u003e1.70(0.985, 2.941)\u003c/p\u003e\n \u003cp\u003e1.01(0.497, 2.049)\u003c/p\u003e\n \u003cp\u003e2.46(0.815, 7.413)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 15.2978%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.24\u003c/p\u003e\n \u003cp\u003e0.24\u003c/p\u003e\n \u003cp\u003e0.48\u003c/p\u003e\n \u003cp\u003e0.37\u003c/p\u003e\n \u003cp\u003e1.39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.0529%;\"\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003e0.667\u003c/p\u003e\n \u003cp\u003e0.904\u003c/p\u003e\n \u003cp\u003e0.056\u003c/p\u003e\n \u003cp\u003e0.979\u003c/p\u003e\n \u003cp\u003e0.110\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eThe household/family frailty value had the chi-square test statistic (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\chi\\:\\)\u003c/span\u003e\u003c/span\u003e = 2.90) and a p-value of 0.04 at 0.05 level of significance, this result showed that there exist unmeasurable factors that have an effect on the hazard of infant death. However for the community effects, the model did not converge.\u003c/p\u003e\n\u003cp\u003eIn addition to these semi-parametric frailty models, Weibull and log-normal parametric frailty models were fit for both household and community effects and the results for both effects were not significant in infant mortality.\u003c/p\u003e"},{"header":"Discussion and Conclusion","content":"\u003cp\u003eThis study used survival analysis and frailty modelling to examine the factors that are associated with infant mortality in Malawi. The mean age for infants who died was2.25 months which is consistent with what was reported by (\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e) that the first 28 days of life are when a child\u0026rsquo;s survival is most vulnerable. This study found that SHH, mothers\u0026rsquo; age group, source of drinking water, religion, type of birth and place of delivery have a significant association with infant mortality.\u003c/p\u003e \u003cp\u003eParticularly, the results indicated that FHH are at a higher risk of children dying before the age of 1 year compared to MHH and mothers who have single births have a lower chance of experiencing infant mortality compared to mothers who have multiple births. The results also indicated that women who have children at older ages are at a higher risk of experiencing infant mortality compared to women who birth children at younger ages. Furthermore, this study found significant evidence that some families/households experience more infant deaths than others which indicated that there do exists some unobservable family/household effects which tend to make infant deaths cluster in some families.\u003c/p\u003e \u003cp\u003eAlthough Malawi has managed to achieve a significant reduction in infant and child mortality rates, the rates remain high compared to most African countries as such there is need for more effort to reduce these mortality rates. This study has provided insights into the risk factors of infant mortality in Malawi, which contains vital information for health policy makers in government and non-governmental organizations.\u003c/p\u003e \u003cp\u003eIn conclusion, this study revealed that SHH, mothers\u0026rsquo; age group, source of drinking water, religion, type of birth and place of delivery have a significant association with infant mortality after controlling for the effects of other factors and also that there are unobservable family effects which make infant deaths to cluster in some families. These factors need to be considered when planning and developing policies against infant mortality in order to successfully work towards reducing infant mortality rate in Malawi.\u003c/p\u003e"},{"header":"Limitations","content":"\u003cp\u003eThe study had some limitations which might have affected the results. One of the limitation is that the DHS survey collect data from women aged 15\u0026ndash;49 who are alive in a given household which implies that no information is collected for mothers who have died which creates a bias in the results. It was also a challenge to control for community effects because the MDHS data doesn\u0026rsquo;t have specific community characteristics.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eEthics approval and consent to participate\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable\u003c/p\u003e\n\u003cp\u003eNot applicable\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent for publication\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAvailability of data and materials\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eData for the submitted work can be accessed through https://www.dhsprogram.com\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that they have no competing interests\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors did not receive support from any organization for the submitted work\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor\u0026rsquo;s contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eEK analysed and interpreted the data. MM offered guidance as a supervisor. All authors read and approved the final manuscript\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgements\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors thank the DHS program for their contribution to creating and giving access to the data file.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eNdawala J. Infant and Child Mortality. DHS Progr. 2015;97\u0026ndash;104.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSartorius K, Sartorius B. Global infant mortality trends and attributable determinants-an ecological study using data from 192 countries for the period 1990\u0026ndash;2011. Popul Health Metr. 2014.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJahn A, Floyd S, Crampin A. Declining child mortality in northern Malawi despite high rates of infection with HIV. Bull world Heal Organ. 2010.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eHill K, Amouzou A. Trends in child mortality, 19960\u0026thinsp;\u0026ndash;\u0026thinsp;2000. world Bank. 2006.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eAjaari J, Masanja H, Owusu-Agyei S. Impact of Place of delivery on Neonatal Mortality in Rural Tanzania. Glob Heal. 2012;49\u0026ndash;59.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eNational Statistical Office (NSO). Malawi Demographic and Health Survey 2015-16. 2017.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eDube ZB. The relationship between mothers\u0026rsquo; maternal age and infant mortality in Zimbabwe. UNIVERSITY OF WITWATERSRAND; 2012.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eVaupel J, Manton K. The Impact of heterogeneity in individual frailty on te dynamics of mortality. JSTORI. 1979;439\u0026ndash;54.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eWienke A. Frailty Models. Max Planck Inst Demogr Res. 2003;49(0):0\u0026ndash;13.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eUNICEF DATA. Neonatal mortality. 2020.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"bmc-public-health","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"pubh","sideBox":"Learn more about [BMC Public Health](http://bmcpublichealth.biomedcentral.com/)","snPcode":"","submissionUrl":"https://www.editorialmanager.com/pubh/default.aspx","title":"BMC Public Health","twitterHandle":"@BMC_series","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"em","reportingPortfolio":"BMC Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Infant mortality, cox proportional hazard, frailty models, Malawi","lastPublishedDoi":"10.21203/rs.3.rs-6121147/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6121147/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u003cstrong\u003eIntroduction\u003c/strong\u003e: Infant mortality rate is one of the important health and development indicators in a country or community. Malawi, like many countries in the sub Saharan Africa is a country that suffers from the highest rates of infant mortality across the globe.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMethods\u003c/strong\u003e: This study used the most recent survey data, 2015-16 Malawi Demographic and health survey, to identify the factors associated with infant mortality in Malawi by using survival analysis techniques and frailty modelling to control for unobserved heterogeneity.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eResults\u003c/strong\u003e: A total number of 4232 infants were analysed for this study and the results showed that children who were the second multiple babies to be born had a higher risk of dying before reaching the age of one year than children who are born single with P-value \u0026lt;0.001, HR=3.26 and 95%CI=(1.639, 5.700). Infants whose mother’s age group 45-49 years had a risk of death 4.63 times higher than infants whose mother’s age group was 15-20 years(P-value\u0026lt;0.001). Furthermore, there were unmeasurable family effects which made infant deaths to cluster in some families.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConclusion\u003c/strong\u003e: social demographic, environmental and biological factors all have an effect on a child’s survival up to 1 year and the household that a child was born in had some unobservable effects on the child’s survival up to 1 year.\u003c/p\u003e","manuscriptTitle":"Survival Analysis and Frailty Modelling of Time-to-event Data: an Application to Infant Mortality in Malawi","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-03-14 11:33:54","doi":"10.21203/rs.3.rs-6121147/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"editorInvitedReview","content":"","date":"2025-06-02T07:12:11+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"6168641212708456545835059517509030723","date":"2025-04-02T05:54:26+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"121577427146261203652945022077582243298","date":"2025-03-31T17:32:58+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-03-29T14:37:36+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-03-29T08:54:45+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"131895546143527614324438736414949202767","date":"2025-03-29T06:37:18+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"210802351173920630312724635608504069686","date":"2025-03-28T09:11:53+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-03-28T05:20:10+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-03-12T06:56:13+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-03-10T09:25:23+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-03-10T09:23:45+00:00","index":"","fulltext":""},{"type":"submitted","content":"BMC Public Health","date":"2025-02-27T12:41:18+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"bmc-public-health","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"pubh","sideBox":"Learn more about [BMC Public Health](http://bmcpublichealth.biomedcentral.com/)","snPcode":"","submissionUrl":"https://www.editorialmanager.com/pubh/default.aspx","title":"BMC Public Health","twitterHandle":"@BMC_series","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"em","reportingPortfolio":"BMC Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"1b488cf5-ffd8-4c37-9fa4-2e1d587be107","owner":[],"postedDate":"March 14th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2025-03-14T11:33:54+00:00","versionOfRecord":[],"versionCreatedAt":"2025-03-14 11:33:54","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6121147","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6121147","identity":"rs-6121147","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}