Predicting Survival of Heart Failure Patients Using the Cox Proportional Hazards Model

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The main purpose of this study was to predict the survival of heart failure patients using the Cox Proportional Hazards Model. Method: In this paper, we analyze a dataset of 299 patients with heart failure collected in 2015, of whom 105 were women and 194 were men aged between 40 and 95 years old. With the support of the RStudio statistical Programme, the Cox Proportional Hazards Model was estimated to determine the survival of heart failure patients. Results: Each additional year of patient age increases the hazard (HR = 1.0446; p-value = 8.41e-07). As a result, survival decreases as the age of the heart patient increases. Most importantly, heart failure patients with hypertension (high blood pressure) had a worse survival than patients without hypertension (HR = 1.5948; p-value = 0.0284). Furthermore, when all other factors were held constant, increased ejection fraction was found to decrease the hazard (HR = 0.9495; p-value = 2.57 e-07) and improve survival, whereas increased creatinine was found to increase the hazard (HR = 1.4167; p-value = 1.05 e-07), hence reducing survival. Conclusions: These findings are expected to underscore the significance of the studied factors in predicting mortality from heart failure in Pakistan. Given the vital role of the heart, predicting heart failure remains a priority for medical practitioners and physicians, highlighting the importance of advancing predictive models for effective management. Heart failure Survival prediction Cox Proportional Hazards Model Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Background Heart failure (HF) occurs when the heart cannot pump enough blood to meet the needs of the body. A creatinine test is a measure of how well kidneys are performing their job of filtering waste from the blood. Ejection fraction (EF) refers to how well left ventricle (or right ventricle) pumps blood with each heartbeat. According to [ 1 ], cardiovascular diseases (CVDs) are the number one cause of death globally, taking an estimated 17.9 million lives each year, which accounts for 31% of all deaths worldwide. Heart failure is a complex clinical syndrome that results from a functional or structural heart disorder impairing ventricular filling or ejection of blood to the systemic circulation. In addition, it is a failure to meet the systemic demands of circulation [ 2 ]). Most cardiovascular diseases can be prevented by addressing behavioral risk factors such as tobacco use, unhealthy diet and obesity, physical inactivity and harmful use of alcohol using population-wide strategies. People with cardiovascular disease or who are at high cardiovascular risk (due to the presence of one or more risk factors such as hypertension, diabetes, hyperlipidemia or already established disease) need early detection and management wherein a machine learning model can be of great help [ 1 ]. Hospitalization with acute heart failure is an important cause of mortality in both resource rich and resource limited countries [ 3 , 4 ]. In USA, for example, in- hospital mortality following an acute heart failure hospitalization varies between 2.3 and 3.8% [ 5 ]. Though no data exists in resource-limited countries for comparison, heart failure (HF) is a major cause of socioeconomic and public health burden worldwide [ 5 ], in US alone about 6 million people are living with heart failure [ 5 – 7 ] while in sub-Saharan Africa, heart failure accounts for up to 7% of all hospitalizations [ 8 ]. In resource-rich settings, coronary artery disease, either alone or in combination with hypertension, is the leading cause of HF [ 9 ]. In contrast, resource-poor settings are undergoing an epidemiological shift from predominately non-ischemic etiologies of heart failure e.g., rheumatic heart disease and endemic cardiomyopathies [ 10 , 11 ] to ischemic etiologies as seen in resource rich-settings [ 8 , 12 ]. In Pakistan, recent epidemiological data highlights the substantial burden of cardiovascular disease (CVD) within the population. According to a 2019 study, the estimated age-standardized incidence of CVD stands at 918.18 per 100,000 individuals, underscoring the significant public health challenge posed by this condition [ 13 ]. Moreover, in 2020, coronary heart disease emerged as a leading cause of mortality, with 240,720 individuals in Pakistan succumbing to this ailment, representing approximately 16.49% of all recorded fatalities[ 13 – 15 ]. It’s against this background that this study aimed to investigate the prediction of survival of heart failure patients using the Cox Proportional Hazard Mode to help medical practitioners and physicians, emphasizing the significance of advancing predictive models for effective management. Methods We analyzed a dataset obtained from Kaggle, which contains the medical records of 299 heart failure patients. The data were originally collected at the Faisalabad Institute of Cardiology and the Allied Hospital of Faisalabad (Punjab, Pakistan) between April and December 2015 [ 1 ]. The dataset, made available by Chicco & Jurman (2020) on Kaggle, comprises records of patients with heart failure, including demographic information, clinical variables, and outcomes data. Of the patients included in the dataset, 105 were women and 194 were men, with ages ranging between 40 and 95 years old. Measures of outcome The dependent variable of the study was the death event of the patients caused by heart failure and it was measured in days. The death event indicated the death status of the patient. It was encoded as 1 for dead and 0 for censored (indicating that the outcome is not known or not recorded including those who survived). Measures of explanatory variables The independent variables were demographic, prognostic factor and risk factors. They included count and continuous variables such as age, creatinine phosphokinase, ejection fraction, platelets, serum creatinine, serum sodium, and time, as well as categorical variables such as sex (categorized as female and male), anaemia (categorized as negative and positive), diabetes (categorized as negative and positive), high blood pressure (categorized as negative and positive), smoking (categorized as nonsmoker and smoker), death event (categorized as survived and died), and age group (categorized as 40–49, 50–59, 60–69, and 70+). Statistical analysis. The data analysis was done in three phases. The first phase involved the generation of summary statistics for the variables under study at the univariate level of analysis. In the second phase, the study employed the Kaplan-Meier survival estimate at a bivariate level of analysis, followed by estimating the Cox Proportional Hazards Model for predicting the survival of heart failure patients at a multivariate level of analysis. The third phase accounted for the associated Cox Proportional Hazards Model assumptions. Also, the first phase involved the installation and loading of the usual R packages to enable smooth running of the models under consideration with the associated assumption tests. The study used RStudio programming and statistical software to estimate the summaries like minimum, maximum, median, and mean values for all the non-categorical variables under study, whereas frequencies and percentage compositions for all the categorical variables were also estimated. The Kaplan-Meier survival curves were employed to estimate whether there is a statistically significant difference in the survival times given the different levels of categorical covariates under study. The difference in survival time of heart failure patients for different levels of categorical variables was assessed using the log-rank Chi-Square test with a 5% significance level [ 16 ]. The Cox proportional hazards model is the most commonly used model for analyzing survival data. It was chosen for this study because it is a semiparametric model that makes a parametric assumption about the effect of predictors on the hazard function, namely linearity and proportional hazards. Furthermore, the justification for selecting this model was to evaluate the effect of several risk factors on the survival of patients with heart failure disease using the hazard rate at the same time. Particularly, this study follows a survival analysis approach where the response variable is the time between a time origin recorded for a heart failure patient and a time point of death. Although standard statistical techniques require that the data be normally distributed, it was sought to be easily corrected with a transformation that necessitates a more realistic data distribution approach, such as the Cox Proportional Hazards Model. The second problem with this survival data is that some of it was censored. An observation is censored when the end-point has not been reached when the subject is removed from study. This was because the study ended before the occurrence of the death event, and even some heart failure patients withdrew from active participation. Also, some heart patients died before the specified period or were moved, thus the need for censoring. Let T be the survival time. That is, T is the elapsed time from the beginning point, in this case the diagnosis of heart disease, to death due to that disease. The values of T can be thought of as having a probability distribution. Cox (1972) expressed the relationship between the hazard rate and a set of covariates using the model. Taking the exponential of both sides of the above equation gives the ratio between the actual hazard rate and the baseline hazard rate, sometimes called the relative risk. This can be rearranged to give the model where 𝑥 1 , 𝑥 2 , …, 𝑥 𝑝 are covariates, 𝛽 1 , 𝛽 2 , …, 𝛽 𝑝 are regression coefficients to be estimated; T is the elapsed time, and ℎ 0 (T) is the baseline hazard rate when all covariates are equal to zero. Strictly two cox proportional assumptions were tested in this study, that is to say, the proportional hazards assumption and the linearity assumption. In RStudio, just like in other statistical packages, the linearity assumption was estimated with the aid of the residual versus fitted plot using the "Martingale residuals" using the syntax "plot (predict (model), residuals (model, type="martingale")" while the proportional hazards assumption was estimated using the syntax "cox.zph (model)". When the proportional hazards assumption holds, it implies that the hazard rates for different groups or levels of a covariate are proportional (constant) over time. The linearity assumption should hold to ensure a linear relationship between the survival of heart failure patients (the outcome variable) and the independent covariates. Results Table 1 shows the average age for the heart failure patients was 60 years with the youngest and the oldest at 40 and 95 years. The mean age of the patients being so close to the median age indicates that the age of the study participant was centered. With regards to the conditions relating to heart failure patients, the average ejection fraction was 38%, this value was found to be below the borderline ejection fraction according to the American Heart Association. Therefore, it implied that the amount of blood pumped to the body by the left ventricle is less with each heartbeat. Serum creatinine which measures how well a patient’s kidneys are performing their roles, the mean value was found to be 1.394mg/dL which was slightly out of range for male patients. Table 1 Summary Descriptive Statistics for heart failure patients (Non categorical) Covariates Min 1st quartile Median Mean 3rd Quartile Max Age 40 51 60 60.83 70 95 Creatinine phosphokinase 23 116.5 250 581.8 582 7861.1 Ejection fraction 14 30 38 38.08 45 80 Platelets 25100 212500 262000 263358 303500 850000 Serum creatinine 0.5 0.9 1.1 1.394 1.4 9.4 Serum sodium 113 134 137 136.6 140 148 Time 4 73 115 130.3 203 285 Furthermore, the average serum sodium was 136.6 milliequivalents per liter, which was found to be within the normal range (135 to 145) milliequivalents per liter, an indication that the concentration of sodium was fine. The mean platelets volume (263358) was way above the normal which range from 150,000 to 450,000 per microliter of blood. The mean platelet volume (263358) was found to be falling in the normal range of 150,000 to 450,000 per microliter of blood. Table 2 Summary Descriptive Statistics for heart failure patient (Categorical) Covariates Frequency (N = 299) Percentage (%) Death Event Survived 203 67.9 Died 96 32.1 Sex Female 105 35.1 Male 194 64.9 Anaemia Negative 170 56.9 Positive 129 43.1 Diabetes Negative 174 58.2 Positive 125 41.8 High Blood Pressure Negative 194 64.9 Positive 105 35.1 Smoking Nonsmoker 203 67.9 Smoker 96 32.1 Age Group 40–49 47 15.7 50–59 82 27.4 60–69 93 31.1 70+ 77 25.8 Forty-three percent of the heart failure patients were anaemic; 42% were diabetic; and 35% were hypertensive. Of the patients, 65% were male, and the majority had a positive smoking status. The smoking patients appeared to have died (32%). 15.7% of the study participants were in the age group of 40–49 years, 27.4% were in the age group of 50–59 years, and 31.1% belonged to the age group of 60–69 years, whereas 25.8% were 70 + years old. Kaplan Meier survival curves Figure 1 bellow, shows non-anaemic heart failure patients exhibited a trend of longer survival compared to their counterparts. Although the difference did not reach statistical significance at the 5% level, the visual disparity in the survival experiences suggested that non-anaemic patients may have survived longer on average. According to the Kaplan-Meier model findings, it suggested that diabetic and non-diabetic heart failure patients exhibited similar survival experiences since there was no statistically significant survival difference between the two groups (p-value = 0.98). The p-value obtained from the Kaplan-Meier survival analysis in comparing the survival outcomes of hypertensive and non-hypertensive heart failure patients was 0.036, indicating a statistically significant difference in survival between these two groups. The visual disparity in the survival curves indicated that non-hypertensive heart failure patients survived longer than those who were hypertensive. Regarding the sex of the heart failure patient, the lack of statistical significance (p = 0.95) supports the conclusion that gender did not appear to influence patient survival, and therefore male and female heart failure patients exhibited similar survival experiences during the study period. The Kaplan-Meier survival analysis revealed that neither being a smoker nor a non-smoker produced a statistically significant difference in the survival of heart failure patients (p-value = 0.96). The Kaplan-Meier study also explored the survival outcomes across various age groups, including 40–49, 50–59, 60–69, and 70 + years, and the p-value obtained was 0.00069, signifying a robust and statistically significant difference in the survival among these age categories; hence, this demonstrated that age has a substantial impact on heart failure patient’s survival. Model for predicting survival of patients with heart failure Of the predictors of heart failure survival, only five (5) were significant. The likelihood ratio test confirmed a significantly improved fit of the Cox proportional hazards model for the data. The column marked z in the output records the ratio of each regression coefficient to its standard error, a Wald statistic that is asymptotically normal under the hypothesis that the corresponding β is 0. The covariates age, high blood pressure, ejection fraction, serum creatinine, and creatinine phosphokinase had highly statistically significant coefficients as related to the other covariates. The exponentiated coefficients in the second column were the multiplicative effects on the hazard. Using these exponentiated coefficients, the results from Table 3 revealed the following. Holding other covariates constant, a patient who was one year older had a higher risk of dying from heart failure than other heart failure patients; therefore, increased age implied lower survival for heart failure patients. When holding other factors constant, patients with high blood pressure had a higher risk of dying from heart failure compared to patients without high blood pressure. High blood pressure was reported to reduce the survival of heart failure patients. Ejection fraction produced a decreased risk of death from heart failure. Therefore, when all other covariates were held constant, an increase in the ejection fraction of a heart failure patient reduced the risk of death from heart failure, thus improving survival for heart failure patients. Holding other covariates constant, there was an increased risk of dying from heart failure for every 1 mg/dL increase in the level of creatinine in the heart failure patient’s blood. The creatinine phosphokinase produced a hazard ratio of 1, implying that there was neither an increase nor a decrease in the risk of death for a 1 mcg/L change in patients’ creatinine phosphokinase when other covariates were held constant. Moreover, creatinine phosphokinase levels were within the normal range for all study participants, indicating no significant hazard. Table 3 Model for predicting survival of patients with heart failure Covariate Hazard Ratio Z p-value 95% CI exp(coef) Pr(>|z|) Lower Upper Age 1.0446 4.926 8.41e-07 *** 1.0266 1.0629 Anaemia Positive 1.4818 1.847 0.0648 0.9762 2.2493 HBP Positive 1.5948 2.192 0.0284 * 1.0506 2.4209 Ejection Fraction 0.9495 -5.152 2.57e-07 *** 0.9310 0.9684 Serum Creatinine 1.4167 5.318 1.05e-07 *** 1.2460 1.6108 Creatinine Phosphokinase 1.0002 1.993 0.0462 * 1.0000 1.0004 Goodness of fit Tests Test df Chi-square p-value Likelihood ratio test 6 77.02 1e-14 Wald test 6 85.82 2e-16 Score (logrank) test 6 83.51 7e-16 df = degrees of freedom, p-value = probability value Diagnostic check Table 4 results from the Cox proportional assumption indicate strong evidence of proportional hazards for all covariates except for ejection fraction, while the global test was not statistically significant. This implies that these tests are sensitive to linear trends in the hazard. Results from the Cox proportional assumption indicate strong evidence of proportional hazards for all covariates except for ejection fraction, while the global test was not statistically significant. This implies that these tests are sensitive to linear trends in the hazard. Table 4 Test for the assumption of proportional hazards 𝒄𝒉𝒊 𝟐 (𝝌 𝟐 ) df p-value Age 0.1774 1 0.674 Anaemia 0.0062 1 0.937 High Blood Pressure 0.0105 1 0.918 Ejection Fraction 5.0657 1 0.024 Serum Creatinine 1.8459 1 0.174 Creatinine Phosphokinase 0.8805 1 0.348 GLOBAL 9.8324 6 0.132 Plotting the object returned by the cox.zph function returned graphs of the scaled Schoenfeld residuals against transformed time as can be seen in the figures below. From Figs. 2 , 3 and 4 the proportional hazards assumption appears to be supported for covariates such as age, anaemia, high blood pressure, serum creatinine, and creatinine phosphokinase, but there appears to be a trend in the plot for ejection fraction, with the ejection fraction effect declining with time; this effect was also detected in the test reported above. Linearity check Nonlinearity is a violation for the Cox Proportional Hazards model because it is an incorrectly specified functional form in the parametric part of the model. It is a problem in Cox regression, just as it is in linear and generalized linear models. The model used the martingale residuals to test for this assumption. The residual and component plus residuals plots showed that the significant predictors did not violate the linearity assumption. The residual scores appear to be linear, compacted, and not dispersed as illustrated in Fig. 5 bellow. Discussion The main purpose of this study was to predict the survival of heart failure patients using the Cox Proportional Hazards Model. Specifically, the study aimed to find out whether health problems such as diabetes, hypertension, anaemia as well as heart disease preconditions such as serum creatinine, ejection fraction, serum sodium, and creatinine phosphokinase, predict the survival of heart failure patients., we tested four hypotheses: First, Diabetes is unlikely to be associated with the survival of heart failure patients; Second, anemia is unlikely to be associated with the survival of heart failure patients; Third, hypertension is unlikely to be associated with the survival of heart failure patients; Fourth, age, sex, and smoking habits are unlikely to be associated with the survival of heart failure patients. Results from this study are different from the findings from a similar study by [17] who found out that one-third of all patients with heart failure have anaemia, and its presence is associated with more symptoms, increased rates of hospitalization, and increased mortality. Different from results from this study on diabetes as a predictor of heart failure, results in this study showed that diabetes was not a risk factor for heart failure. However, [18], demonstrated a robust correlation between diabetes and a significant rise in heart failure-related mortality [18]. Also, another study by [19] indicated that diabetes and heart failure are linked, that people with diabetes are more likely to develop heart failure, and that people with heart failure are more likely to get diabetes, which was completely contrary to this current study. As supported by the [20], longstanding hypertension ultimately leads to heart failure (HF), and as a consequence, most patients with heart failure have a history of hypertension. Conversely, the absence of hypertension in middle age is associated with lower risks for incident heart failure across the remaining life course. Besides, [20] stressed that diastolic dysfunction and heart failure with preserved ejection fraction are the most common cardiac complications of hypertension [20]. Also, pulmonary hypertension in heart failure patients is associated with high morbidity and mortality [20]. This confirms to the results from this study which revealed the existence of increased risk of mortality for heart failure patients with high blood pressure. Hypertension is the leading risk factor for numerous cardiovascular diseases, including stroke, coronary artery disease, atrial fibrillation, and peripheral vascular disease, according to [21]. Except for age, other demographic and prognostic factors did not predict death in heart failure patients. This is quite parallel to the finding by [7] who maintained that the various socioeconomic factors influence disease progression, treatment compliance, and hospitalization/rehospitalization rates [7]. The findings further suggest that a combination of clinical and laboratory parameters, as well as biomarkers, can help identify individuals at risk for heart failure in this patient population. This study reveals significant insights into the survival factors for heart failure patients. Firstly, age was identified as a crucial determinant, with older patients facing a higher risk of mortality [22]. On the other hand, a higher ejection fraction was associated with improved survival, and a lower ejection fraction was associated with worsened survival, highlighting the adverse outcomes of declining LVEF, which is associated with higher mortality [23]. Additionally, elevated creatinine levels were linked to an increased risk of death, underscoring the intricate relationship between heart and kidney health in heart failure [1]. Importantly, this study found that normal creatinine phosphokinase (CPK) levels had no significant impact on survival, indicating that CPK may not be a key prognostic marker for heart failure patients [24]. In summary, these findings reinforce existing research and emphasize the need for comprehensive, patient-specific approaches to heart failure management, addressing age, blood pressure control, ejection fraction improvement, and kidney function monitoring to enhance patient survival and quality of life. There are some limitations which can be addressed in future. The inability to estimate the median survival for the majority of the categorical factors. A significant proportion of the survival data used consists of censored observations. Censoring occurs when heart failure patients have not experienced the event of interest (death event) by the end of the study or were lost to follow-up before the event occurred. With the limited number of death events, it becomes difficult to reach the 50% survival probability point, which is used to estimate the median survival. Therefore, the study utilized the log-rank test to test whether there was a significant difference in the different levels of a given categorical covariate, for example, whether hypertensive and non-hypertensive heart failure patients’ survival times were significantly different. Abbreviations AHF: Acute Heart Failure; CHF: Chronic Heart Failure; CKD: Chronic Kidney Disease; CMO: Cardiomyopathies; CVDs: Cardiovascular Diseases; DBP: Diastolic Blood Pressure; Dplyr: Data.frame specific set of tools for plyr function in R; EF: Ejection Fraction; EPO: Erythropoietin Production; ESAs: Erythropoiesis-Stimulating Agents; HBP: High Blood Pressure; HF: Heart Failure; HFPEF: Heart Failure with Preserved Ejection Fraction; HHD: Hypertensive Heart Disease; HTN: Hypertension; IV: Intravenous; LV: Left Ventricular; LVH: Left Ventricular Hypertrophy; OPTIMIZE: Organized Program To Initiate Lifesaving Treatment in Hospitalized Patients; PH: Pulmonary Hypertension; RENAISSANCE: Randomized Etanercept North American Strategy to Study Antagonism of Cytokines; RHD: Rheumatic Heart Disease; SBP: Systolic Blood Pressure; SSA: Sub Saharan Africa. Declarations Ethical approval and consent to participate. Utilizing secondary data from Kaggle’s Heart Failure Clinical Records dataset, ethical considerations were addressed in accordance with the dataset’s terms of use and the guidelines set forth by the dataset creators. Moreover, informed consent was obtained from all participants or their legal guardian(s) as part of the data collection process. All experimental protocols were approved by Allied Hospital of Faisalabad and licensing committe. All procedures contributing to this research adhered to the ethical standards set by relevant national and institutional committees for human experimentation, as well as the Helsinki Declaration of 1975, amended in 2008. Consent for Publication. Not applicable Availability of Data and Materials. The dataset is available for access at Kaggle, via https://www.kaggle.com/andrewmvd/heart-failure-clinical-data Conflict of Interest. The authors have no conflicts of interest to declare. Funding. This study received no specific grant from any funding agency, commercial entity or not-for-profit organization. However, the corresponding author is grateful to the college of Business and Management Science (CoBAMS), Makerere University, for the institutional grant, as research facilitation support, for staff. Authors Contributions. Not applicable Acknowledgements. The authors extend their sincere gratitude to Kaggle, and Chicco & Jurma (2020) for allowing them to use the Patients Heart failure data. References Chicco, D., G.J.B. Jurman, and D. Making, Machine learning can predict survival of patients with heart failure from serum creatinine and ejection fraction alone. 20(1), 1-16. 2020. Malik, A., et al., Congestive heart failure. 2017. Abraham, W.T., et al., Predictors of in-hospital mortality in patients hospitalized for heart failure: insights from the Organized Program to Initiate Lifesaving Treatment in Hospitalized Patients with Heart Failure (OPTIMIZE-HF). J. Am. Coll. 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Yamaguchi, S., et al., Prognostic value of lactate dehydrogenase for mid-term mortality in acute decompensated heart failure: a comparison to established biomarkers and brain natriuretic peptide. 29(9), 1318-1327. 2020. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-3993213","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":276077910,"identity":"04c6f7e5-f7c5-4a5d-92d3-95bf59775069","order_by":0,"name":"Mulumba Jackson","email":"data:image/png;base64,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","orcid":"","institution":"Makerere University","correspondingAuthor":true,"prefix":"","firstName":"Mulumba","middleName":"","lastName":"Jackson","suffix":""},{"id":276077911,"identity":"c5c24999-cba4-436b-b45e-8edf4073f765","order_by":1,"name":"Leonard Atuhaire","email":"","orcid":"","institution":"Makerere University","correspondingAuthor":false,"prefix":"","firstName":"Leonard","middleName":"","lastName":"Atuhaire","suffix":""},{"id":276077912,"identity":"d32b78a4-611b-401b-96cc-8d2f40d87b26","order_by":2,"name":"Dick Nsimbe","email":"","orcid":"","institution":"Makerere University","correspondingAuthor":false,"prefix":"","firstName":"Dick","middleName":"","lastName":"Nsimbe","suffix":""}],"badges":[],"createdAt":"2024-02-27 07:33:49","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3993213/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3993213/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":52032519,"identity":"e0fded7f-8566-4f1c-aa35-7131d13d8629","added_by":"auto","created_at":"2024-03-05 16:37:26","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":100965,"visible":true,"origin":"","legend":"\u003cp\u003eKaplan-Meier plots showing the patients’ survival probabilities by selected demographic and socioeconomic factors: (a) anaemia, (b) Diabetes, (c) blood pressure, (d) sex, (e) Smoking, and (f) age group.\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3993213/v1/67e00e89b8aa09e41e83c357.jpg"},{"id":52032483,"identity":"6e178d9b-0141-4621-9156-e8c7f0e1496f","added_by":"auto","created_at":"2024-03-05 16:37:21","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":53224,"visible":true,"origin":"","legend":"\u003cp\u003eScaled Schoenfeld residuals for age and anaemia against time\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3993213/v1/86c8006da60450fe1d49c345.jpg"},{"id":52032485,"identity":"fc27e128-cd58-404c-9352-2de922c9290a","added_by":"auto","created_at":"2024-03-05 16:37:22","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":56993,"visible":true,"origin":"","legend":"\u003cp\u003eScaled Schoenfeld residuals for high blood pressure and ejection fraction against time.\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3993213/v1/6eed6abd2c374178f7614d94.jpg"},{"id":52032524,"identity":"fc97d458-0eed-40ae-ac47-cd9964d9b038","added_by":"auto","created_at":"2024-03-05 16:37:27","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":54207,"visible":true,"origin":"","legend":"\u003cp\u003eScaled Schoenfeld residuals for serum creatinine and creatinine phosphokinase\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3993213/v1/2480acdf73f83722def16225.jpg"},{"id":52032484,"identity":"26db669a-dd1b-44c3-9d97-4a6a1f7727b2","added_by":"auto","created_at":"2024-03-05 16:37:21","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":68374,"visible":true,"origin":"","legend":"\u003cp\u003eResidual and component-plus-residual plot\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3993213/v1/f9c639274aa613f1338358a1.jpg"},{"id":53609122,"identity":"b04beb79-bbfa-4162-8642-add2e2582e89","added_by":"auto","created_at":"2024-03-28 04:43:56","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":596487,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3993213/v1/ba486dcf-5c2c-47da-bfc7-ad582b97a311.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Predicting Survival of Heart Failure Patients Using the Cox Proportional Hazards Model","fulltext":[{"header":"Background","content":"\u003cp\u003eHeart failure (HF) occurs when the heart cannot pump enough blood to meet the needs of the body. A creatinine test is a measure of how well kidneys are performing their job of filtering waste from the blood. Ejection fraction (EF) refers to how well left ventricle (or right ventricle) pumps blood with each heartbeat. According to [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e], cardiovascular diseases (CVDs) are the number one cause of death globally, taking an estimated 17.9\u0026nbsp;million lives each year, which accounts for 31% of all deaths worldwide. Heart failure is a complex clinical syndrome that results from a functional or structural heart disorder impairing ventricular filling or ejection of blood to the systemic circulation. In addition, it is a failure to meet the systemic demands of circulation [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]). Most cardiovascular diseases can be prevented by addressing behavioral risk factors such as tobacco use, unhealthy diet and obesity, physical inactivity and harmful use of alcohol using population-wide strategies. People with cardiovascular disease or who are at high cardiovascular risk (due to the presence of one or more risk factors such as hypertension, diabetes, hyperlipidemia or already established disease) need early detection and management wherein a machine learning model can be of great help [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eHospitalization with acute heart failure is an important cause of mortality in both resource rich and resource limited countries [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. In USA, for example, in- hospital mortality following an acute heart failure hospitalization varies between 2.3 and 3.8% [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. Though no data exists in resource-limited countries for comparison, heart failure (HF) is a major cause of socioeconomic and public health burden worldwide [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e], in US alone about 6\u0026nbsp;million people are living with heart failure [\u003cspan additionalcitationids=\"CR6\" citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] while in sub-Saharan Africa, heart failure accounts for up to 7% of all hospitalizations [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. In resource-rich settings, coronary artery disease, either alone or in combination with hypertension, is the leading cause of HF [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. In contrast, resource-poor settings are undergoing an epidemiological shift from predominately non-ischemic etiologies of heart failure e.g., rheumatic heart disease and endemic cardiomyopathies [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] to ischemic etiologies as seen in resource rich-settings [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. In Pakistan, recent epidemiological data highlights the substantial burden of cardiovascular disease (CVD) within the population. According to a 2019 study, the estimated age-standardized incidence of CVD stands at 918.18 per 100,000 individuals, underscoring the significant public health challenge posed by this condition [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. Moreover, in 2020, coronary heart disease emerged as a leading cause of mortality, with 240,720 individuals in Pakistan succumbing to this ailment, representing approximately 16.49% of all recorded fatalities[\u003cspan additionalcitationids=\"CR14\" citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]. It\u0026rsquo;s against this background that this study aimed to investigate the prediction of survival of heart failure patients using the Cox Proportional Hazard Mode to help medical practitioners and physicians, emphasizing the significance of advancing predictive models for effective management.\u003c/p\u003e"},{"header":"Methods","content":"\u003cp\u003eWe analyzed a dataset obtained from Kaggle, which contains the medical records of 299 heart failure patients. The data were originally collected at the Faisalabad Institute of Cardiology and the Allied Hospital of Faisalabad (Punjab, Pakistan) between April and December 2015 [\u003cspan class=\"CitationRef\"\u003e1\u003c/span\u003e]. The dataset, made available by Chicco \u0026amp; Jurman (2020) on Kaggle, comprises records of patients with heart failure, including demographic information, clinical variables, and outcomes data. Of the patients included in the dataset, 105 were women and 194 were men, with ages ranging between 40 and 95 years old.\u003c/p\u003e\n\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\n \u003ch2\u003eMeasures of outcome\u003c/h2\u003e\n \u003cp\u003eThe dependent variable of the study was the death event of the patients caused by heart failure and it was measured in days. The death event indicated the death status of the patient. It was encoded as 1 for dead and 0 for censored (indicating that the outcome is not known or not recorded including those who survived).\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\n \u003ch2\u003eMeasures of explanatory variables\u003c/h2\u003e\n \u003cp\u003eThe independent variables were demographic, prognostic factor and risk factors. They included count and continuous variables such as age, creatinine phosphokinase, ejection fraction, platelets, serum creatinine, serum sodium, and time, as well as categorical variables such as sex (categorized as female and male), anaemia (categorized as negative and positive), diabetes (categorized as negative and positive), high blood pressure (categorized as negative and positive), smoking (categorized as nonsmoker and smoker), death event (categorized as survived and died), and age group (categorized as 40\u0026ndash;49, 50\u0026ndash;59, 60\u0026ndash;69, and 70+).\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\n \u003ch2\u003eStatistical analysis.\u003c/h2\u003e\n \u003cp\u003eThe data analysis was done in three phases. The first phase involved the generation of summary statistics for the variables under study at the univariate level of analysis. In the second phase, the study employed the Kaplan-Meier survival estimate at a bivariate level of analysis, followed by estimating the Cox Proportional Hazards Model for predicting the survival of heart failure patients at a multivariate level of analysis. The third phase accounted for the associated Cox Proportional Hazards Model assumptions. Also, the first phase involved the installation and loading of the usual R packages to enable smooth running of the models under consideration with the associated assumption tests. The study used RStudio programming and statistical software to estimate the summaries like minimum, maximum, median, and mean values for all the non-categorical variables under study, whereas frequencies and percentage compositions for all the categorical variables were also estimated. The Kaplan-Meier survival curves were employed to estimate whether there is a statistically significant difference in the survival times given the different levels of categorical covariates under study. The difference in survival time of heart failure patients for different levels of categorical variables was assessed using the log-rank Chi-Square test with a 5% significance level [\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e]. The Cox proportional hazards model is the most commonly used model for analyzing survival data. It was chosen for this study because it is a semiparametric model that makes a parametric assumption about the effect of predictors on the hazard function, namely linearity and proportional hazards. Furthermore, the justification for selecting this model was to evaluate the effect of several risk factors on the survival of patients with heart failure disease using the hazard rate at the same time. Particularly, this study follows a survival analysis approach where the response variable is the time between a time origin recorded for a heart failure patient and a time point of death. Although standard statistical techniques require that the data be normally distributed, it was sought to be easily corrected with a transformation that necessitates a more realistic data distribution approach, such as the Cox Proportional Hazards Model. The second problem with this survival data is that some of it was censored. An observation is censored when the end-point has not been reached when the subject is removed from study. This was because the study ended before the occurrence of the death event, and even some heart failure patients withdrew from active participation. Also, some heart patients died before the specified period or were moved, thus the need for censoring. Let T be the survival time. That is, T is the elapsed time from the beginning point, in this case the diagnosis of heart disease, to death due to that disease. The values of T can be thought of as having a probability distribution. Cox (1972) expressed the relationship between the hazard rate and a set of covariates using the model.\u003c/p\u003e\n \u003cdiv id=\"Equa\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 347px; height: 47.3182px;\" width=\"347\" height=\"47.3182\"\u003e\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eTaking the exponential of both sides of the above equation gives the ratio between the actual hazard rate and the baseline hazard rate, sometimes called the relative risk. This can be rearranged to give the model\u003c/p\u003e\n \u003cdiv id=\"Equb\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAckAAABhCAYAAAC9B1pGAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAFiUAABYlAUlSJPAAAABfaVRYdFNuaXBNZXRhZGF0YQAAAAAAeyJjbGlwUG9pbnRzIjpbeyJ4IjowLCJ5IjowfSx7IngiOjQ1OCwieSI6MH0seyJ4Ijo0NTgsInkiOjk4fSx7IngiOjAsInkiOjk4fV19kk6M+QAAGqhJREFUeF7tndurFtX/x8fffWjpVYhEdpGYFKkZUUFBnhIxqLS6CRJNi6CjtjO60KwdRTcdLAq6SFMqCKlMBYWvFZ0xKAxsh0R0pan5B/h7XmvPe7f2OPPMzH4O+3n2fr9gsWbWHJ5n1vrM57M+6zRTzjdIjDHGGHMB/5fGxvQNf/zxR7J06dJkypQpyTPPPBPSXnrppeSSSy4JYffu3SHNGGNaxZ6k6TvWrFmTbN++Pdm4cWPYnz9/frJs2bLkxhtvDMbzu+++S/75559wzBhjWsGepKnNl19+Gby4boYYPMXLL788bO/fvz+5++67g4GEW2+9NTl9+nTYNsaYVrGRNLW56KKL0q3xBY9xw4YNydVXX52mJMmJEyfSLWOMaR0bSVMbjNLg4GC6N8yCBQuSkydPJrTetyPs2LEjvXM+P//8c/AY77333jRlmAMHDoT/Yowx7cBG0oyJp556KlmyZEm6lyQ//PBD8uyzz6Z7rbN+/foLDHHMN998E+I5c+aEGGiGHRoaSh5//PE0xRhjWsNG0oyZnTt3JhdffHG6lyRvvvlmW0eWPvDAA+nWhRw+fDjEx44dC/Hnn38eBvLQ/MrAHmOMaQeVjGR2oAbD7bNwDiMLUVatwPB+lJ2G9ldFUwIUTOeZPn16snfv3nRvGMqOptB2wP1jbzVGzaorV64M5f3II48kL7zwQvLGG2+kZxhjTBs4X4EjR44wTSTEeXzwwQfnGwrr/NDQUJoyTMPLCNcRV+Ho0aPnZ8+ePep3dI9mIf5d/VfTPXbs2DGqPJCFkydPpkdbY3Bw8ILyRE5I++yzz9IUY4zpDC03t+JB4j188cUXI8PyheaqLV68OMTNOHXqVHLLLbck77333shwfrxKBmc0lOGowRwNJRn2BwYGkoZRveB3TXeh/3D16tXpXnv7J+n7pKxj1B955ZVXhtgYYzpFy0by/vvvTzZv3hyaxorA+JWBUr3uuutGDCRoiP/y5cvDPv1Q9IFpyD8TyD2SsTd4/fXXQ4VFtLt/Mkb9kfHvGWNMJ2jJSNL3xGjCO++8M00ZDV4mzJ07t3TZMNIwuDEMwIj7mPBQMKQCg9opRWzqQSXp448/TveGaWf/ZAxljneZ9TCNMabdtGQkWe2kWXPn119/HeJ9+/YlN9xwQ3L8+PGwT5NqDMaUZtXYAGah6RWDzIoqpjfBw4/nN1Kma9euDU3pxhjTj7RkJH/66afkiiuuSPcu5NChQyHWsmFqkp02bVqIxa+//hriZn2LNL0Cxtb0LvRP0kQu2j1/0hhjuklLRvLMmTPpVj54mvGyYVolJdtHefbs2XSrmP/9738hjvssTW+ydevWUX3FneyfNMaYTtLywJ0i1Be1YsWKEINGJV5//fUhrsP333/vQTrjBM2l9CnTalClj5EWg3feeWfUQgOd6p+caDDPmL54zfclz/PmJRvT69CNhiy3srgH8o/uoLttvGjJSGabTWNkEBctWhRiyI5OFVOnTk238kFJ02xXZSqJaS8YNsqQpnP6krNlVwTnxYOu3D9ZDsrg9ttvT6699tqRdXCR+U2bNtlQmr6CxWBY6ANZZuS74P2nVQlZxwBmK4NZ/cCiIYCDNG6tUY0XsZSixQSY6F20UMDq1avDwgAxS5YsCYGJ5hzXhHPdn0nieWiy+sDAQJpSjBcTaB/Kyw0bNqQp9eFa7qHQyr0mMiyMQP7kyTjpRe+ZMb0GOh55zdPnpEsPaBEYYhYgIZ04Dxas4Tg2p9u0ZCS18km84o0gMzCEMVo9hYzIZiDnkxFZ9NsKZaus2Ei2B8qUMkHgW4GKkF4AhbxynuxQoSS/81YqUr4Z0+tQyUNWs7ZCcCxPp6BvJOdF18p+dHulrZaMJPByt8O6U7NoVSGDjWR7kAdYJLB1oEKkGiSB7aJWg8mIvMg8L1vKI9sqY0yvIVkdqx6XfqDlMA8qkOiObr8LLQ/coZ/qxRdfbLmviRGRTPPQAgRm/KCTnBGp9AO0YzQx/ZPvv/9+uuf+ySyffvppiONBbuLdd98N8bp160JsTK/y8ssvh5iPDbQCi8/kwYDARkUyzJfvav9kaiyb0syThKIFzuuCd0EtoRXvpdc9SfKKmhb/kcDzZpsP2FcTJXHcBKc84hjncQxPXmnyRpQuD470vKa8PNQHXFSjGytqilHI85x6GfKeLgT+e9YbJm9VrlX6zmNUdtnyUT9MVgaM6UV4JwhjgXdJ71UzWdd52a68TlLLSCrkNa9yDkqi1fZiDC3Ks66iiQ0PoSrZ66oGrqsDBY+yQwjUJxcLhuD5uTfnqw1exopjnKv/wDkIi47LsCpdlQ0p9qrN4sqTViorReg/KrQqL92EfKHM9D7EL2r8XHWeiTLlGq4X/IYqFHUqN8aMF3on6upFIZ1TpWLOeYRu0bsu1wQiNpAowJhmBS4FKsHjHoQipanfkDIXEuBYETej2X9qlayhr2q4ew15f4BByyvbKmBQuY+8alWMuP9Y7mfMeCC5Hcv7rJarqi1LMqidqMTn0bHFBMx/0K/EPE++lhLPM9Tk+kahhzgLy/Q1lGXoq9VHqPkkWdEXV/gN+vvoE8ibz8jx8YZn4uPI0DAshYvj9zqas8vcLvpvmQNcdQ5pzFdffRXim2++OcR8GqxhHEPesDKVF2AwExnGoDz44IOhr7FnP5ieGkvTQfAyCHiABGpA8j7KPAZqVxRTMw8SmnmLveRJ8gw8N/fv56kg6i/kWco8PvK9KO9VK87eQ/kkuYlhn2vqdkkY0ynG4kki88h3mW7LYk9ygqEvnBBmzJgRwrZt28I+Nafff/+9qQdy2WWXhfiuu+4q9CBBi8QzajSLjuGVjjf33XdfePbVq1e3tFzVeDNz5swQ8+WaMg8SD37hwoXp3mhY3xiy96CsuTd5dezYsTR1GO3PmzcvxMb0G4ygp6UEndSsdawXKDWSWjaoX0MZS5cuzb2uLHBdFfS5sEYNC9csBIQCA1lmJGhqe/vtt8P2iRMnQlyEPkScty6ujq1atSrE48Vbb7018nm1eKmqfoOpK48++mhoEqWSUwZlnteUpOlORc3tRVNkmJbDPfu5kmEmL8g1leR+MJBQaiSl2Ps1lEEh5V1XFriuk2AgqWnhdaKMWeC9GVrjMOuRUGPbs2dPuMdtt92WplajnfMYeR76HoC5tb3+YhRBnlBBwjOkX3JoaGjM+SQPP+8bqdxTfciXXnppiI3pdap80YlP5/He8BGErB5g7EXZOsXt1EtVcHNrh5k1a1aI84QHwxF7pHgW7MtAagV9mt1QmEUr4ZNOs1weAwMDIWYyf1XDJM8m28w3VhBqNQPjUffj585oPZCBBBa/YPFmOHjwYIizcK5aHvKQhy8ZidEiApRF/J3Vui0ZoGvihTrYVnoMCqrq/X3fYTiHc7PKXfet8nt59NN99Z3fH3/8McRF8JUbBroxeK+oQl+GKo/d0iM2kh0G7w0vDsGQ8CEMCCiGMF6dAs+C5shrrrkmNEWoie6OO+4I8UcffTSiqGODqQ9Sg1ai4DgGFqHbsWNHsnz58pBeBf2emopbhZojgr1gwYIwerPfoNIC9CdTAyZPqXCoMoFnDHzZACUgaG3gmYuaU/Wyv/LKKyOyQfkiG3z5g2t37twZ0sXJkydDnOd9GjNeYLDQc+pjL0L6jlYlGV8FdB7vVzP0ntBc2y16xkjy8Cj/WMlUAcWEUum2C14VlCkeA01zN910UxAGtvEsjx8/Psp4ydtEqcbNuUyTQGGiOMmjLVu2jPIu9EFqvEYUrgQOjhw5kqxfvz5sV0XNsh9++GGIW0E1R16gKrXEXuTcuXMhJk8pS+U9NWGGrqMY+NQP+9nKCC/9/Pnz073/oBLDMcqVQVn0cVJuGGL6ofG48/pr/vrrrxBfddVVITamV1AfeZEOR8eXGcEy9u3bF+Kujq84X4OGogtDb4nbCcPpG8oiTDQX+q1mQeczGZXr6wwjnkjw7ORHO9HUk1aGWVM+Ksd+WlmnXTDEvejZtYhA3Wkcmng9WWXd9C6878hmo5KfprQXZB590qispindobZmJRPi5bhaBSXMg8cvvTJbykWKQXPJUCzZjJKhnGyQb+RNu59dBq6V+/Ky8N+qrqQx0Whm0JBhjtWdK8q7Nxnl3PQHkutOVIo7ee9mjMlI8vK3C4xddgIqiiNWrCgGFLbAsOYZau7VzxPUxwJ5QZl0whDJ2xnLvSlTrqVMJqvX08ygqQKh1pCq8B7U9T6N6SbIPHJatshGHbR4x3jIfi0jKYVMjBIkIwh5hgnFiHLluJRltgag5qgyRcG1VVx4MnCy1bJljDolPFoVg/yn3KsgOSG080XpN5BF8o13IVupI2/I1zqo1YAyJ19tLE2vIt2PrLZSScY2cC/kfrwcoFpGMlbIKEIeXgo0hgfDsKEkZAC5hmtjg8j9OK8ZnC/FUIY8n1YKpd+QRyKB7ATkp8oqLr88OJfzqpbZRIbnJx8oo7iyoEpEXmtIGSpvFMdkknPTfyDnyPhYWqIENoTry/ROJ6llJPWCxi88yjn7snMe6fFLLMUQN9VyHec2Q24215eh36hyrukMlCllUFauxhjTD9SaAsJQ94ZVH5kEyvwxJrEz30+QpvPKJq+fOXMm3SpG0xv6cQL6ZIM5mlrdJzu/zxhj+pHKRlITqlesWBFi+Oabb0IcrxeqVVqWLVsWYqH5LXlrizaD5dgaLne6Z3oV5IM5q1BndR9jjOllKhtJGcRFixaFGJhYjdcQLy+Ut/waE/2ZUL5kyZJR506bNi3dyofrWJVE3+4zvQvLztGqMDAwUGt1nzKQASbZs4hCFh3TtzaNMabdVDaSGMTZs2eP8hBoLmVdUZQVqy0Qa7WWXbt2hZiVRaTgBgcHQyxY+zJeUi0Ly7DV4d9//w3xnDlzQmy6A0ZKy8499thjaWp7+Pbbb0Oct2pNuz8ZpaX8smtaGmMmMWnfZCl5A3Q0eo8RSPFgHkaZaoQj1xWNTuIazsk7pkE4ClUmkDKClv9iukdcTrEMtAuNiu7kSFkGmOl3Ov1bxpj+ovZiAu0GY9oupcS9JttiAuMJxoVKEIalnQtMxHRjOgmVP+SG57GRNMbEjPsC53xB4cUXXwxNta1AExn9o1pk13Sehx56KPRD0tdcdxH1Mmj6ZCDQULog8tSpU0Ms6Iss6qusC6NykRsPNjLGZBl3I8nUDj4JhbJDMY4FvnjPFys6/SFk8x/kub7qwZQfGa12Bfq/Gewl5s6dm24N0/D6QuxPRhljOsm4G0mgFv/qq69e8D2+KnANI2rzPitkOgPTPfge3HhS9MkoWhTyjG42GGNMFabQ5ppuG1MJKjXd/jbkkSNHRi0ogSeLocajbGflCAM6ODjYlx+HNsa0n57wJE1/QR8edatuhuyKS0xJYsqJWw+MMZ3ERtL0JQcOHMhdZMLNrcaYdmIjafoORkIzqpYRr/SPxivu0Eya54lmQx54yHDo0KGWR1sbYyYG7pM0fQmjobWQ/tatW1tqdsX73LRpU7o3Gr8exkxubCSNMcaYAtzcaowxxhRgI2mMMcYUYCNpjDHGFGAjaYwxxhRgI2mMMcYUYCNpjDHGFGAjaYwxxhRgI2mMMcYUYCNpjDHGFGAjaYwxxhRgI2mMMcYUYCNpjDHGFGAjaYwxxhRgI2mMMcYUYCNpjDHGFGAjaYwxxhRgI9nD8MX8K664IpkyZUoIfI3/559/To9OLE6dOpVs3LgxueSSS0ael33STe/gcjKTjvOmq5w8efL86tWrz5dl/ZIlS8I5GzZsCPsDAwNhf/bs2WE/y2effXb+4osvPj84OJimjB/8F56R/8N/JixYsCCk50Ge6NwPPvggpHF+/PycwzHliwK/c/To0XCO6SxVysmYiYaNZBdBmWPkpOCLQAFxPDaIGL+i62RACeNtJI8cORL+B8p0aGgoKFYpUgJpWVCwHIsVrYwhMej52eee5KUUdlHFwbSXKuVkzETDza1d4q233kquueaapGEk0pRitmzZEuInn3wyxHDixIkQN5RRiOGPP/5IFi5cmGzfvj1N6R02b96cXH755cn06dOTu+66K01Nkr///jvdGoZnePPNN8P2+vXrQwy///57iG+99dYQC/KGe1599dXJddddF9Kq5KlpjbrlZMyEITWWZozg9WWboJSm2jXeE9vEsdeXBx6SjqsZUZ4lHhlelOBeO3bsCGm6ptOepP4/z6f/p7Si35YXyDXx/wf+v46JPI8lD3sx+ZDHyhvJTJyGt0+Qh0+TNVBOkuVsWVYtp7r3NaYqyC+ylQU9FHfvECOTea1WUHSfImwkW4BCoHDoa6NwaPaLjVyeQiBNx/OQQaSgUWxqnpUBLkL37KQS4j+gLGUUEUQpTwIKMo9mzyBFSz5yPc/N+UX3EuQ913G+jLUZhjwlT2SsyHfyl20CciUlojTKlDyP5ZfzRJVyGst9jamC9GZWL0hfEpBHDKD0TZFukCxW1ZU2km1CBUXBYDiKKDOSsTICDcih4PMKXOienTSSAkHlt/hfhKzgxuh5ea48JNDkGcpTRhdhb6ZM5RWVVR4mM8p7ygiDma1Zx4ZLZUieZ9OgTjnVua8xZSC3yDCyFoNMkY5MZXWu9CjHs/IJOt5MpwobyTahWnu2sLKUGUndJzZ2MggynHnont0wkrHCayZkquUVGbIixam0orxEwBH+KgI+mYlbOPIUhQxd3PREOSj/dU3dcqp6X2OqIIOWrVxJzrLGU6hilzeqXq0dRdfGeOBOG2DuYiPTw/bZs2dDPFZ++OGHEN9www0hjjlz5ky6Nb4cPHgw3UqSc+fOpVuj+fLLL5N77rknaRjIZM2aNWnqaI4dO5ZuJcmNN96Ybv1HXl4yd3T37t3J4cOHw+AdU8ynn34aYmSTwU5ZNOd28eLFIYZ9+/aFuKFgRq6pW05V72tMGRow1qhwXSB7f/75Z5CnO+64I00ZDXPM4ZdffglxDIMKG05Hsn///tK55zaSBaDkUe7xpGlGkn7++efpGcNw3i233DIy0vKnn34K8VjgXiIWCI0gpGA7ARPBMT48n56V586bJM4oXdIRWvj1119DHINgr1y5Mml4taMMJNv8jvj6669D3KjNhRi4VsyaNSvdGoa837RpU7J3794RA8n/42WI826iQt6Q9/ECE2zHeQqa8P/9998nDY87pOUpAo7DvHnzQgx79uwJ8bp160IMdcup6n2NKePdd98N8dq1a0Mc8/zzzwfdGI+2FrwD3333XdiO5TZm1apVIUanNSX1KE0KTUG44GQN7rpc/HjgA818cfMRgxPUD0MzIK58XtMo6bo3Idt8EDfFqikxblsval6MO6/53TrNWVzLvbmW5+BaPYvuB/F/o/kiburgOfKah/NC3nnkM79LUHMzcfwcbOt/5oVsXk40NFiKPFDzpsqAoDTlaSyH7JPvBOUT+alrVSaStbiZFOqWU9X7GlMGMofsFOm+LMgfOk3XIXtF8H5wDu9KM/rGSDZTvM0C19Uh/p24YHjhlY6ikYJCeQkpJJRBtlB1bV6Q4oqvVyFTgKTnCUn8n7KhynPzuzpfxlBk78Nx/ov6FxFG/UeOSVHG98wLelbgfqTxG9rmngh2rHih2bMSJjLxs8sYQpzXMkjkI3koeeEc0gixrOpanc82MffJ5r3Kpko51bmvMc2QESNUQfpToZmBFJLRWC9lsScZEXtkcQbzciszCRReJ9BvyBB1mqJnij2UKoI2FuIXoFP5ORGI84nyiomVQl2ZkeEtq0zVLaeq9zWmDA08qypLvA+cq1YOQlyBz4PzOS+ufGZxn2SEVroB2rnpd2GQyKJFi8Lgh0btOGkUXEf6BmlD5zdg5syZIe4kPJd+ryFI4Zno16Mv68EHHxxJ37p1a9huN7/99lu61bm+1onAyy+/nG4Nr8CEnNA3y2L36udrGKbCwVFFHDp0KMTz588PcRF1y6nqfY0pI2/ATTPQaV988UXoE294hkFf84489NBD6RkXIjltNuDSRjIlHqEKLCHXqLkn7733Xhil16ipJ8ePH0+WL1+entFevv3223QrfwRhu/nkk0/SreFBFQwCuf/++0PFAKV79OjRIHSdGon41VdfhbhRkwuxyYcyEFReZsyYkTz33HPJtGnTkkbtN8jsU089lZ5RHQ1qmDp1aoiLqFtOVe9rTCdBh77//vthG/0WDzSLqSKnNpIp8TB3FELDyw6Bmskbb7wRauqdHLquWtOCdNRop9FUE6DWxbMyUoznRel2enrFjz/+GGJ7HMXwYp8+fTrdS0ZkkpoyxpPWjrF44VQIdd+33367UIFAnXKqc19jOk3s0GTXjK5D3xhJmpc07L1O4LoqMOdGjMdizRgmKcBuEHvN3fBcs2CMeV6GcZt84he7nR43FSAZXCpGzQxtnXKqc19jWgXdzlS1VqZ/VZnX3jdGUi9r3cB1dVG/Soz6gegTmmhkhYxnxHOOm/rM+MKk5yx4bshk3hxIY/qdq666Kt3Khy4HWi527dqVpoxG8x/pmyxyBNRS0qzZ1c2tKXGBoJBkOGgyYrL27bffPqEMZNys+9prr408W3ZQiBkGGchrmWCf9OyEfrVkFNVySS87Z86cOenWMPoNyooKDItYqA/QmInGlVdeGeK8CiI8/PDDIWZFnuyCAOixp59+Omzz2b4i9P5cf/31Ic6l4W2ZFA0Hzgscm0jzvDSfLS8wx61b01D6haKpDZIZjscoL4vmX8X5X3QO6HfzAlNC8ubPTibK8tHH+/u4pqkVyXk8bY9zeR91DaHZFLahdHoT+q4Z9iQjdu7cGUZ2MqpVNDI9jGyl2XYirTlJ80NDKMM0D8FzN4QqDOqpO6XAdAb6qpG/2PNnGzllRLTXrzUTGS1jWORNoqekx/75559wHjH7pDPosgitQV2m66ZgKdNtY4wxpmegu4vKO0FrWLcL1qrGIWAQY7NBZvYkjTHG9CQYL1q3MGRFffdjAeOLgaSlsGwUto2kMcaYnuWJJ54II1S3bduWprSOVrKi26IMG0ljjDE9C54eI1TjWQetgBfJiNiBgYFKffrukzTGGNPzaJ56qwuu6D5VB2PaSBpjjDG5JMn/AziGN7Q10TY1AAAAAElFTkSuQmCC\" style=\"width: 370px; height: 78.5339px;\" width=\"370\" height=\"78.5339\"\u003e\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere 𝑥\u003csub\u003e1\u003c/sub\u003e, 𝑥\u003csub\u003e2\u003c/sub\u003e, \u0026hellip;, 𝑥\u003csub\u003e𝑝\u003c/sub\u003e are covariates, 𝛽\u003csub\u003e1\u003c/sub\u003e, 𝛽\u003csub\u003e2\u003c/sub\u003e, \u0026hellip;, 𝛽\u003csub\u003e𝑝\u003c/sub\u003e are regression coefficients to be estimated; T is the elapsed time, and ℎ\u003csub\u003e0\u003c/sub\u003e(T) is the baseline hazard rate when all covariates are equal to zero. Strictly two cox proportional assumptions were tested in this study, that is to say, the proportional hazards assumption and the linearity assumption. In RStudio, just like in other statistical packages, the linearity assumption was estimated with the aid of the residual versus fitted plot using the \u0026quot;Martingale residuals\u0026quot; using the syntax \u0026quot;plot (predict (model), residuals (model, type=\u0026quot;martingale\u0026quot;)\u0026quot; while the proportional hazards assumption was estimated using the syntax \u0026quot;cox.zph (model)\u0026quot;. When the proportional hazards assumption holds, it implies that the hazard rates for different groups or levels of a covariate are proportional (constant) over time. The linearity assumption should hold to ensure a linear relationship between the survival of heart failure patients (the outcome variable) and the independent covariates.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"Results","content":"\u003cp\u003eTable\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e shows the average age for the heart failure patients was 60 years with the youngest and the oldest at 40 and 95 years. The mean age of the patients being so close to the median age indicates that the age of the study participant was centered. With regards to the conditions relating to heart failure patients, the average ejection fraction was 38%, this value was found to be below the borderline ejection fraction according to the American Heart Association. Therefore, it implied that the amount of blood pumped to the body by the left ventricle is less with each heartbeat. Serum creatinine which measures how well a patient\u0026rsquo;s kidneys are performing their roles, the mean value was found to be 1.394mg/dL which was slightly out of range for male patients.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\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\u003eSummary Descriptive Statistics for heart failure patients (Non categorical)\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"7\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eCovariates\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMin\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e1st quartile\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMedian\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMean\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e3rd Quartile\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMax\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\"\u003e\n \u003cp\u003eAge\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e40\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e51\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e60.83\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e70\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e95\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCreatinine phosphokinase\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e23\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e116.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e250\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e581.8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e582\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e7861.1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eEjection fraction\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e38\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e38.08\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e45\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e80\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePlatelets\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e25100\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e212500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e262000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e263358\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e303500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e850000\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSerum creatinine\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.394\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSerum sodium\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e113\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e134\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e137\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e136.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e140\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e148\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eTime\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e73\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e115\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e130.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e203\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e285\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\u003eFurthermore, the average serum sodium was 136.6 milliequivalents per liter, which was found to be within the normal range (135 to 145) milliequivalents per liter, an indication that the concentration of sodium was fine. The mean platelets volume (263358) was way above the normal which range from 150,000 to 450,000 per microliter of blood. The mean platelet volume (263358) was found to be falling in the normal range of 150,000 to 450,000 per microliter of blood.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eSummary Descriptive Statistics for heart failure patient (Categorical)\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"3\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eCovariates\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eFrequency (N\u0026thinsp;=\u0026thinsp;299)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ePercentage (%)\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\"\u003e\n \u003cp\u003e\u003cstrong\u003eDeath Event\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSurvived\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e203\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e67.9\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDied\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e96\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e32.1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eSex\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eFemale\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e105\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e35.1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMale\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e194\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e64.9\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eAnaemia\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eNegative\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e170\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e56.9\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePositive\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e129\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e43.1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eDiabetes\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eNegative\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e174\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e58.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePositive\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e125\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e41.8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eHigh Blood Pressure\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eNegative\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e194\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e64.9\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePositive\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e105\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e35.1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eSmoking\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eNonsmoker\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e203\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e67.9\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSmoker\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e96\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e32.1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eAge Group\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e40\u0026ndash;49\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e47\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e15.7\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e50\u0026ndash;59\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e82\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e27.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e60\u0026ndash;69\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e93\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e31.1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e70+\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e77\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e25.8\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\u003eForty-three percent of the heart failure patients were anaemic; 42% were diabetic; and 35% were hypertensive. Of the patients, 65% were male, and the majority had a positive smoking status. The smoking patients appeared to have died (32%). 15.7% of the study participants were in the age group of 40\u0026ndash;49 years, 27.4% were in the age group of 50\u0026ndash;59 years, and 31.1% belonged to the age group of 60\u0026ndash;69 years, whereas 25.8% were 70\u0026thinsp;+\u0026thinsp;years old.\u003c/p\u003e\n\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\n \u003ch2\u003eKaplan Meier survival curves\u003c/h2\u003e\n \u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e bellow, shows non-anaemic heart failure patients exhibited a trend of longer survival compared to their counterparts. Although the difference did not reach statistical significance at the 5% level, the visual disparity in the survival experiences suggested that non-anaemic patients may have survived longer on average. According to the Kaplan-Meier model findings, it suggested that diabetic and non-diabetic heart failure patients exhibited similar survival experiences since there was no statistically significant survival difference between the two groups (p-value\u0026thinsp;=\u0026thinsp;0.98). The p-value obtained from the Kaplan-Meier survival analysis in comparing the survival outcomes of hypertensive and non-hypertensive heart failure patients was 0.036, indicating a statistically significant difference in survival between these two groups. The visual disparity in the survival curves indicated that non-hypertensive heart failure patients survived longer than those who were hypertensive.\u003c/p\u003e\n \u003cp\u003eRegarding the sex of the heart failure patient, the lack of statistical significance (p\u0026thinsp;=\u0026thinsp;0.95) supports the conclusion that gender did not appear to influence patient survival, and therefore male and female heart failure patients exhibited similar survival experiences during the study period. The Kaplan-Meier survival analysis revealed that neither being a smoker nor a non-smoker produced a statistically significant difference in the survival of heart failure patients (p-value\u0026thinsp;=\u0026thinsp;0.96). The Kaplan-Meier study also explored the survival outcomes across various age groups, including 40\u0026ndash;49, 50\u0026ndash;59, 60\u0026ndash;69, and 70\u0026thinsp;+\u0026thinsp;years, and the p-value obtained was 0.00069, signifying a robust and statistically significant difference in the survival among these age categories; hence, this demonstrated that age has a substantial impact on heart failure patient\u0026rsquo;s survival.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n \u003ch2\u003eModel for predicting survival of patients with heart failure\u003c/h2\u003e\n \u003cp\u003e\u003cem\u003eOf the\u003c/em\u003e predictors of heart failure survival, only five (5) were significant. The likelihood ratio test confirmed a significantly improved fit of the Cox proportional hazards model for the data. The column marked z in the output records the ratio of each regression coefficient to its standard error, a Wald statistic that is asymptotically normal under the hypothesis that the corresponding \u0026beta; is 0. The covariates age, high blood pressure, ejection fraction, serum creatinine, and creatinine phosphokinase had highly statistically significant coefficients as related to the other covariates. The exponentiated coefficients in the second column were the multiplicative effects on the hazard. Using these exponentiated coefficients, the results from Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e revealed the following. Holding other covariates constant, a patient who was one year older had a higher risk of dying from heart failure than other heart failure patients; therefore, increased age implied lower survival for heart failure patients. When holding other factors constant, patients with high blood pressure had a higher risk of dying from heart failure compared to patients without high blood pressure. High blood pressure was reported to reduce the survival of heart failure patients. Ejection fraction produced a decreased risk of death from heart failure. Therefore, when all other covariates were held constant, an increase in the ejection fraction of a heart failure patient reduced the risk of death from heart failure, thus improving survival for heart failure patients. Holding other covariates constant, there was an increased risk of dying from heart failure for every 1 mg/dL increase in the level of creatinine in the heart failure patient\u0026rsquo;s blood. The creatinine phosphokinase produced a hazard ratio of 1, implying that there was neither an increase nor a decrease in the risk of death for a 1 mcg/L change in patients\u0026rsquo; creatinine phosphokinase when other covariates were held constant. Moreover, creatinine phosphokinase levels were within the normal range for all study participants, indicating no significant hazard.\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable id=\"Tab3\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eModel for predicting survival of patients with heart failure\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"6\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eCovariate\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eHazard Ratio\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eZ\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ep-value\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003e95% CI\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\u0026nbsp;\u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eexp(coef)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\u0026nbsp;\u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ePr(\u0026gt;|z|)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eLower\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eUpper\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\"\u003e\n \u003cp\u003eAge\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.0446\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.926\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.41e-07 ***\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.0266\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.0629\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eAnaemia Positive\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.4818\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.847\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.0648\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.9762\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.2493\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eHBP Positive\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.5948\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.192\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.0284 *\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.0506\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.4209\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eEjection Fraction\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.9495\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-5.152\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.57e-07 ***\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.9310\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.9684\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSerum Creatinine\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.4167\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5.318\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.05e-07 ***\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.2460\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.6108\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCreatinine Phosphokinase\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.0002\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.993\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.0462 *\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.0000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.0004\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eGoodness of fit Tests\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eTest\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003edf\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eChi-square\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003ep-value\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eLikelihood ratio test\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e77.02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1e-14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eWald test\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e85.82\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2e-16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eScore (logrank) test\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e83.51\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e7e-16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" colspan=\"5\"\u003e\n \u003cp\u003e\u003cem\u003edf\u0026thinsp;=\u0026thinsp;degrees of freedom, p-value\u0026thinsp;=\u0026thinsp;probability value\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\n \u003ch2\u003eDiagnostic check\u003c/h2\u003e\n \u003cp\u003eTable\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e results from the Cox proportional assumption indicate strong evidence of proportional hazards for all covariates except for ejection fraction, while the global test was not statistically significant. This implies that these tests are sensitive to linear trends in the hazard. Results from the Cox proportional assumption indicate strong evidence of proportional hazards for all covariates except for ejection fraction, while the global test was not statistically significant. This implies that these tests are sensitive to linear trends in the hazard.\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable id=\"Tab4\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eTest for the assumption of proportional hazards\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"4\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\u0026nbsp;\u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e𝒄𝒉𝒊\u003csup\u003e𝟐\u003c/sup\u003e(𝝌\u003csup\u003e𝟐\u003c/sup\u003e)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003edf\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ep-value\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\"\u003e\n \u003cp\u003eAge\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1774\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.674\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eAnaemia\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.0062\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.937\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eHigh Blood Pressure\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.0105\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.918\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eEjection Fraction\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5.0657\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.024\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSerum Creatinine\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.8459\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.174\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCreatinine Phosphokinase\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.8805\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.348\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eGLOBAL\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.8324\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.132\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\u003ePlotting the object returned by the cox.zph function returned graphs of the scaled Schoenfeld residuals against transformed time as can be seen in the figures below. From Figs. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e, \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e the proportional hazards assumption appears to be supported for covariates such as age, anaemia, high blood pressure, serum creatinine, and creatinine phosphokinase, but there appears to be a trend in the plot for ejection fraction, with the ejection fraction effect declining with time; this effect was also detected in the test reported above.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\n \u003ch2\u003eLinearity check\u003c/h2\u003e\n \u003cp\u003eNonlinearity is a violation for the Cox Proportional Hazards model because it is an incorrectly specified functional form in the parametric part of the model. It is a problem in Cox regression, just as it is in linear and generalized linear models. The model used the martingale residuals to test for this assumption. The residual and component plus residuals plots showed that the significant predictors did not violate the linearity assumption. The residual scores appear to be linear, compacted, and not dispersed as illustrated in Fig. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e bellow.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"Discussion","content":"\u003cp\u003eThe main purpose of this study was to predict the survival of heart failure patients using the Cox Proportional Hazards Model. Specifically, the study aimed to find out whether health problems such as diabetes, hypertension, anaemia as well as heart disease preconditions such as serum creatinine, ejection fraction, serum sodium, and creatinine phosphokinase, predict the survival of heart failure patients., we tested four hypotheses: First, Diabetes is unlikely to be associated with the survival of heart failure patients; Second, anemia is unlikely to be associated with the survival of heart failure patients; Third, hypertension is unlikely to be associated with the survival of heart failure patients; Fourth, age, sex, and smoking habits are unlikely to be associated with the survival of heart failure patients. \u0026nbsp;Results from this study are different from the findings from a similar study by \u0026nbsp;[17] who found out that one-third of all patients with heart failure have anaemia, and its presence is associated with more symptoms, increased rates of hospitalization, and increased mortality.\u003c/p\u003e\n\u003cp\u003eDifferent from results from this study on diabetes as a predictor of heart failure, results in this study showed that diabetes was not a risk factor for heart failure. However, [18], demonstrated a robust correlation between diabetes and a significant rise in heart failure-related mortality [18]. Also, another study by [19] indicated that diabetes and heart failure are linked, that people with diabetes are more likely to develop heart failure, and that people with heart failure are more likely to get diabetes, which was completely contrary to this current study.\u003c/p\u003e\n\u003cp\u003eAs supported by the [20], longstanding hypertension ultimately leads to heart failure (HF), and as a consequence, most patients with heart failure have a history of hypertension. Conversely, the absence of hypertension in middle age is associated with lower risks for incident heart failure across the remaining life course. Besides, [20] stressed that diastolic dysfunction and heart failure with preserved ejection fraction are the most common cardiac complications of hypertension [20]. Also, pulmonary hypertension in heart failure patients is associated with high morbidity and mortality \u0026nbsp;[20]. This confirms to the results from this study which revealed the existence of increased risk of mortality for heart failure patients with high blood pressure. Hypertension is the leading risk factor for numerous cardiovascular diseases, including stroke, coronary artery disease, atrial fibrillation, and peripheral vascular disease, according to [21].\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eExcept for age, other demographic and prognostic factors did not predict death in heart failure patients. This is quite parallel to the finding by \u0026nbsp;[7] who maintained that the various socioeconomic factors influence disease progression, treatment compliance, and hospitalization/rehospitalization rates [7]. The findings further suggest that a combination of clinical and laboratory parameters, as well as biomarkers, can help identify individuals at risk for heart failure in this patient population. This study reveals significant insights into the survival factors for heart failure patients. Firstly, age was identified as a crucial determinant, with older patients facing a higher risk of mortality [22]. On the other hand, a higher ejection fraction was associated with improved survival, and a lower ejection fraction was associated with worsened survival, highlighting the adverse outcomes of declining LVEF, which is associated with higher mortality [23]. Additionally, elevated creatinine levels were linked to an increased risk of death, underscoring the intricate relationship between heart and kidney health in heart failure [1]. Importantly, this study found that normal creatinine phosphokinase (CPK) levels had no significant impact on survival, indicating that CPK may not be a key prognostic marker for heart failure patients [24].\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn summary, these findings reinforce existing research and emphasize the need for comprehensive, patient-specific approaches to heart failure management, addressing age, blood pressure control, ejection fraction improvement, and kidney function monitoring to enhance patient survival and quality of life. There are some limitations which can be addressed in future. The inability to estimate the median survival for the majority of the categorical factors. A significant proportion of the survival data used consists of censored observations. Censoring occurs when heart failure patients have not experienced the event of interest (death event) by the end of the study or were lost to follow-up before the event occurred. With the limited number of death events, it becomes difficult to reach the 50% survival probability point, which is used to estimate the median survival. Therefore, the study utilized the log-rank test to test whether there was a significant difference in the different levels of a given categorical covariate, for example, whether hypertensive and non-hypertensive heart failure patients\u0026rsquo; survival times were significantly different.\u003c/p\u003e"},{"header":"Abbreviations","content":"\u003cp\u003eAHF: Acute Heart Failure; CHF: Chronic Heart Failure; CKD: Chronic Kidney Disease; CMO: Cardiomyopathies; CVDs: Cardiovascular Diseases; DBP: Diastolic Blood Pressure; Dplyr: Data.frame specific set of tools for plyr function in R; EF: Ejection Fraction; EPO: Erythropoietin Production; ESAs: Erythropoiesis-Stimulating Agents; HBP: High Blood Pressure; HF: Heart Failure; HFPEF: Heart Failure with Preserved Ejection Fraction; HHD: Hypertensive Heart Disease; HTN: Hypertension; IV: Intravenous; LV: Left Ventricular; LVH: Left Ventricular Hypertrophy; OPTIMIZE: Organized Program To Initiate Lifesaving Treatment in Hospitalized Patients; PH: Pulmonary Hypertension; RENAISSANCE: Randomized Etanercept North American Strategy to Study Antagonism of Cytokines; RHD: Rheumatic Heart Disease; SBP: Systolic Blood Pressure; SSA: Sub Saharan Africa.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eEthical approval and consent to participate.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eUtilizing secondary data from Kaggle\u0026rsquo;s Heart Failure Clinical Records dataset, ethical considerations were addressed in accordance with the dataset\u0026rsquo;s terms of use and the guidelines set forth by the dataset creators. Moreover, informed consent was obtained from all participants or their legal guardian(s) as part of the data collection process.\u003c/p\u003e\n\u003cp\u003eAll experimental protocols were approved by Allied Hospital of Faisalabad and licensing committe. All procedures contributing to this research adhered to the ethical standards set by relevant national and institutional committees for human experimentation, as well as the Helsinki Declaration of 1975, amended in 2008.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent for Publication.\u003c/strong\u003e Not applicable\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAvailability of Data and Materials.\u003c/strong\u003e The dataset is available for access at Kaggle, via https://www.kaggle.com/andrewmvd/heart-failure-clinical-data\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflict of Interest.\u003c/strong\u003e The authors have no conflicts of interest to declare.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding.\u003c/strong\u003e This study received no specific grant from any funding agency, commercial entity or not-for-profit organization. However, the corresponding author is grateful to the college of Business and Management Science (CoBAMS), Makerere University, for the institutional grant, as research facilitation support, for staff.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthors Contributions.\u0026nbsp;\u003c/strong\u003eNot applicable\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgements.\u003c/strong\u003e The authors extend their sincere gratitude to Kaggle, and Chicco \u0026amp; Jurma (2020) for allowing them to use the Patients Heart failure data.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eChicco, D., G.J.B. Jurman, and D. Making, \u003cem\u003eMachine learning can predict survival of patients with heart failure from serum creatinine and ejection fraction alone. 20(1), 1-16.\u003c/em\u003e 2020.\u003c/li\u003e\n\u003cli\u003eMalik, A., et al., \u003cem\u003eCongestive heart failure.\u003c/em\u003e 2017.\u003c/li\u003e\n\u003cli\u003eAbraham, W.T., et al., \u003cem\u003ePredictors of in-hospital mortality in patients hospitalized for heart failure: insights from the Organized Program to Initiate Lifesaving Treatment in Hospitalized Patients with Heart Failure (OPTIMIZE-HF). J. Am. Coll. Cardiol, 52(1), 347\u0026ndash;356.\u003c/em\u003e 2008.\u003c/li\u003e\n\u003cli\u003eAgbor, V.N., et al., \u003cem\u003eHeart failure in sub-Saharan Africa: a contemporaneous systematic review and meta-analysis. 257, 207-215.\u003c/em\u003e 2018.\u003c/li\u003e\n\u003cli\u003eO\u0026rsquo;Connor, C.M. and L.A. Allen, \u003cem\u003eManagement of acute decompensated heart failure. Can. Med. Assoc., J 176, 797\u0026ndash;805.\u003c/em\u003e 2007.\u003c/li\u003e\n\u003cli\u003eAllen, L.A., et al., \u003cem\u003eRates and predictors of 30-day readmission among commercially insured and Medicaid-enrolled patients hospitalized with systolic heart failure. Circ. Heart Fail 5 (2012) 672\u0026ndash;679, 5(1), 672\u0026ndash;679.\u003c/em\u003e 2012.\u003c/li\u003e\n\u003cli\u003eCuyjet, A.B. and O. Akinboboye, \u003cem\u003eAcute heart failure in the African American patient. J. Card. Fail, 20(1), 533\u0026ndash;540.\u003c/em\u003e 2014.\u003c/li\u003e\n\u003cli\u003eDamasceno, A., et al., \u003cem\u003eHeart failure in sub- Saharan Africa: time for action. J. Am. Coll. Cardiol, 50, 1688 \u0026ndash;1693.\u003c/em\u003e 2007.\u003c/li\u003e\n\u003cli\u003eMcMurray, J.J. and S. Stewart, \u003cem\u003eEpidemiology, aetiology, and prognosis of heart failure,.Heart, 83, 596\u0026ndash;602.\u003c/em\u003e 2000.\u003c/li\u003e\n\u003cli\u003eMcMurray, J.J. and S. Stewart, \u003cem\u003eEpidemiology, aetiology, and prognosis of heart failure,. Heart, 83, 596\u0026ndash;602.\u003c/em\u003e 2000.\u003c/li\u003e\n\u003cli\u003eBloomfield, G.S., et al., \u003cem\u003eHeart failure in sub- Saharan Africa. Curr. Cardiol., 157.\u003c/em\u003e 2013.\u003c/li\u003e\n\u003cli\u003eCho, D.J., et al., \u003cem\u003eCharacteristics, outcomes and predictors of long-term mortality for patients hospitalized for acute heart failure: a report from the Korean heart failure registry. Kor. Circ. J, 363\u0026ndash;371.\u003c/em\u003e 2011.\u003c/li\u003e\n\u003cli\u003eAshraf, T., T. Ahmed, and A. Nadeem, \u003cem\u003eGenetics and Ischemic Heart Disease: Should We Opt for Genetic Testing for Primary Prevention?. Pakistan Heart Journal, 56(3), 193-194.\u003c/em\u003e 2023.\u003c/li\u003e\n\u003cli\u003eZahid, F.M., et al., \u003cem\u003eGender based survival prediction models for heart failure patients: A case study in Pakistan. PloS one, 14(2), e0210602.\u003c/em\u003e 2019.\u003c/li\u003e\n\u003cli\u003eAhmad, T., et al., \u003cem\u003eSurvival analysis of heart failure patients: A case study. PloS one, 12(7), e0181001.\u003c/em\u003e 2017.\u003c/li\u003e\n\u003cli\u003eKaplan, E.L. and P. Meier, \u003cem\u003eNon applied estimation from incomplete observation. Journal of the American Statistical Association 53(1),457-481.\u003c/em\u003e 1958.\u003c/li\u003e\n\u003cli\u003eGrote Beverborg, N., D.J. van Veldhuisen, and v.d. Meer, \u003cem\u003eAnemia in heart failure: still relevant? , 6(3), 201-208.\u003c/em\u003e 2018a.\u003c/li\u003e\n\u003cli\u003eAaron, F., et al., \u003cem\u003eDiabetes in heart failure: prevalence and impact on outcome in the population. The American journal of medicine, 119(7), 591-599.\u003c/em\u003e 2006.\u003c/li\u003e\n\u003cli\u003eRosano, G.M., C. Vitale, and P.J. Seferovic, \u003cem\u003eHeart failure in patients with diabetes mellitus. 3(1), 52.\u003c/em\u003e 2017.\u003c/li\u003e\n\u003cli\u003eMesserli, F.H., S.F. Rimoldi, and S.J. Bangalore, \u003cem\u003eThe transition from hypertension to heart failure: contemporary update. 5(8), 543-551.\u003c/em\u003e 2017.\u003c/li\u003e\n\u003cli\u003eSlivnick, J. and B.C. Lampert, \u003cem\u003eHypertension and heart failure. 15(4), 531-541.\u003c/em\u003e 2019.\u003c/li\u003e\n\u003cli\u003eStrait, J.B. and E.G. Lakatta, \u003cem\u003eAging-associated cardiovascular changes and their relationship to heart failure. 8(1), 143-164.\u003c/em\u003e 2012.\u003c/li\u003e\n\u003cli\u003eLup\u0026oacute;n, J., et al., \u003cem\u003eDynamic trajectories of left ventricular ejection fraction in heart failure. 72(6), 591-601.\u003c/em\u003e 2018.\u003c/li\u003e\n\u003cli\u003eYamaguchi, S., et al., \u003cem\u003ePrognostic value of lactate dehydrogenase for mid-term mortality in acute decompensated heart failure: a comparison to established biomarkers and brain natriuretic peptide. 29(9), 1318-1327.\u003c/em\u003e 2020.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"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":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Heart failure, Survival prediction, Cox Proportional Hazards Model","lastPublishedDoi":"10.21203/rs.3.rs-3993213/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3993213/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u003cstrong\u003eBackground:\u003c/strong\u003e Cardiovascular diseases kill approximately 17 million people globally every year, and they mainly exhibit as myocardial infarctions and heart failures. The main purpose of this study was to predict the survival of heart failure patients using the Cox Proportional Hazards Model.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMethod:\u003c/strong\u003e In this paper, we analyze a dataset of 299 patients with heart failure collected in 2015, of whom 105 were women and 194 were men aged between 40 and 95 years old. With the support of the RStudio statistical Programme, the Cox Proportional Hazards Model was estimated to determine the survival of heart failure patients.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eResults:\u003c/strong\u003e Each additional year of patient age increases the hazard (HR = 1.0446; p-value = 8.41e-07). As a result, survival decreases as the age of the heart patient increases. Most importantly, heart failure patients with hypertension (high blood pressure) had a worse survival than patients without hypertension (HR = 1.5948; p-value = 0.0284). Furthermore, when all other factors were held constant, increased ejection fraction was found to decrease the hazard (HR = 0.9495; p-value = 2.57 e-07) and improve survival, whereas increased creatinine was found to increase the hazard (HR = 1.4167; p-value = 1.05 e-07), hence reducing survival.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConclusions:\u003c/strong\u003e These findings are expected to underscore the significance of the studied factors in predicting mortality from heart failure in Pakistan. Given the vital role of the heart, predicting heart failure remains a priority for medical practitioners and physicians, highlighting the importance of advancing predictive models for effective management.\u003c/p\u003e","manuscriptTitle":"Predicting Survival of Heart Failure Patients Using the Cox Proportional Hazards Model","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-03-05 16:37:05","doi":"10.21203/rs.3.rs-3993213/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"4711d53c-f5f0-4453-bdac-fe2e572b6d9b","owner":[],"postedDate":"March 5th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-06-17T03:45:52+00:00","versionOfRecord":[],"versionCreatedAt":"2024-03-05 16:37:05","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3993213","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3993213","identity":"rs-3993213","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}