Analysis of Influencing Factors on Survival Time of Patients with Heart Failure

To explore the influencing factors of survival time of patients with heart failure, a total of 1789 patients with heart failure were collected from Shanghai Shuguang Hospital. The Cox proportional hazards model and the mixed effects Cox model were used to analyze the factors on survival time of patients. The results of Cox proportional hazards model showed that age (RR = 1.32), hypertension (RR = 0.67), ARB (RR = 0.55), diuretic (RR = 1.48) and antiplatelet (RR = 0.53) have significant impacts on the survival time of patients. The results of mixed effects Cox model showed that age (RR = 1.16), hypertension (RR = 0.61), lung infection (RR = 1.43), ARB (RR = 0.64), β-blockers (RR = 0.77) and antiplatelet (RR = 0.69) have a significant impact on the survival time of patients. The results are consistent with the covariates age, hypertension, ARB and antiplatelet but inconsistent with the covariates lung infection and β-blockers.


Introduction
Heart failure is a syndrome with symptoms and signs caused by cardiac dysfunction, resulting in reduced longevity [1].The prevalence of heart failure in western countries is 1% -2% of the adult population and 5 -10 per 1000 population per year, respectively [2] [3].In China, the prevalence of heart failure in Chinese population aged 35 -74 is 0.9% and the population significantly increases with age [4] [5].With the acceleration of population aging in China, it is foreseeable that the burden caused by heart failure will become heavier in the near future.So it is important to study and analyze the influencing factors of the survival time In medical research, follow-up is the common way to study the law of things; for instance: study the efficacy of a drug, study the survival time after surgery, study the lifetime of a medical device [6] [7].The common ground of the above studies is that it will take some time to trace the research objects, which was called the survival time in statistics.The study of the distribution and influencing factors of survival time is the so-called survival analysis [8] [9] [10].Proportional hazard regression model has become the most common used procedure for modeling the relationship of covariates to a survival or other censored outcome since this model was proposed by D.R. Cox in 1972 [11].In clinical practice, many studies collect both longitudinal data [12] [13] (longitudinal data are data in which a response variable is measured at different time points over time) and survival-time data.In this paper, Cox proportional hazards model was used to model the survival-time data and mixed effects Cox model [14] [15] was used to model the survival-time and longitudinal data.

Cox Proportional Hazards Model
The Cox proportional hazards model was proposed by British statistician D.R.
Cox in 1972, which has been widely applied to analyze the effect of exposure and other covariates on patient's survival.The Cox model specifies the hazard for individual i as: where , , , , , × vector of covariates for subject i, and ( ) λ is an unspecified nonnegative function of time called the baseline hazard, describing how the risk of event per time unit changes over time at baseline levels of covariates.Since the hazard ratio for two subjects with fixed covariate vectors i X and j X ( ) ( ) is constant over time, the model is called proportional hazards model.
Let the event be observed to have occurred with subject i at time i t .The probability that happened can be written as where ( ) = and the summation is over the set of subjects j who is still under observation at time i t , the set is called risk set and denoted by ( this is the partial likelihood for subject i.So taking the product of Equation ( 3) yields the partial likelihood function: where i δ is 1 if the event is happened to subject i and 0 otherwise.

Mixed Effects Cox Model
In clinical practice, some subjects may be observed more than once during the ≤ ≤ is called longitudinal data, which is also called panel data in econometrics [16].This type of data is different from cross-section data and time series data.The linear mixed effects model is a common model to dealing with the longitudinal data [17].It adds individual difference as random effects into the regression model.These random effects describe how every object's measurement changes over time and reflect the internal structure of the longitudinal data.In matrix notation a mixed model can be represented as: where X and Z are the design matrices for the fixed and random effects respectively, β is the vector of fixed-effects coefficients and b is the vector of random effects coefficients and ε is the random error.The random effects distribution is modeled as Gaussian with mean zero and a variance matrix Σ .
Combining Equation ( 1) and (3) yields the mixed effects Cox model: Coefficients can be estimated based on the partial likelihood: where is the linear score for subject i at time t and ( ) i is still under observation at time t and 0 otherwise [18] [19].

Data
We collected patient basic information, laboratory information, medical records, doctor's advice information and other information from Shanghai Shuguang Hospital database during January shown in Figure 1.
Statistics for other binary variables are shown in Table 2.

Results
Firstly, we use the Cox proportional hazards to model the survival-time data with all covariates.The results are shown in Table 3.
Table 1.Variables description in heart failure dataset.Secondly, we use the mixed effects Cox model to model the survival-time data and longitudinal data with all the covariates and variable day as the covariate for random effects.The results are shown in Table 4.
time from first hospitalization to death.The number of hospitalizations and the days between two hospitalizations varies from patient to patient in the heart failure set.The Cox proportional hazards model only uses the survival-time data, which inevitably lose some useful information.The data obtained from multiple measurements of a series of experimental individuals over time are called longitudinal data.More precisely, suppose there are m individuals in an experiment where each individual is measured over time. 1 2, , , 1, , Medicine? 1 = yes, 0 = no Binary RBC Red blood cells in mg/ml Numeric HGB Hemoglobin in mg/ml Numeric hypertension Presence of hypertension, 1 = yes, 0 = no Binary coronary Presence of coronary heart disease, 1 = yes, 0 = no Binary diabetes Presence of diabetes, 1 = yes, 0 = no Binary lung_infe Presence of lung infection, 1 = yes, 0 = no Binary bronchitis Presence of chronic bronchitis, 1 = yes, ? 1 = yes, 0 = no Binary digitalis Whether used digitalis? 1 = yes, 0 = no Binary anti-platelet Whether used anti-platelet? 1 = yes, 0 = no Binary nitrate Whether used nitrate? 1 = yes, 0 = no Binary Open Journal of Statistics Cox proportional hazards model showed that age, hypertension, ARB, diuretics and antiplatelet have a statistically significant effect on the survival time of patients.Age (RR = 1.32) and diuretic (RR = 1.48) were risk factors.Hypertension (RR = 0.67), ARB (RR = 0.55) and antiplatelet (RR = 0.53) were protective factors.The mixed effects Cox model showed that age, hypertension, lung infection, ARB, β-blockers, and antiplatelet have statistically significant effects on the survival time of patients.Age (RR = 1.16) and lung infection (RR = 1.43) were risk DOI: 10.4236/ojs.2018.84042656 Open Journal of Statistics

Figure 1 .
Figure 1.Distribution of heart failure patients' age.
hospital date or the date of death or the end date of the study.According to the guidance of the doctor formed the heart failure dataset used in this paper.This dataset contains data from 1789 patients with heart failure, for a total of 8332 observations and 23 covariates.See Table1for details.
1, 2003 to December 31, 2013.The start point of survival analysis is the first time in hospital date and the end point is the last DOI: 10.4236/ojs.2018.84042654 Open Journal of Statistics time out of

Table 2 .
Statistics for binary variable in hear failure set (total = 1789).

Table 3 .
Result of Cox proportional hazards model with all covariates.

Table 4 .
Results of mixed effects Cox model.