1. Introduction

OJS

Open Journal of Statistics

2161-718X

Scientific Research Publishing

10.4236/ojs.2018.84042

OJS-86068

Articles

Physics&Mathematics

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

Jianwei

Sheng

¹Xiyuan

Qian

¹^*Tong

Ruan

School of Information Science and Engineering, East China University of Science and Technology, Shanghai, China

School of Science, East China University of Science and Technology, Shanghai, China

* E-mail:xyqian@ecust.edu.cn(XQ);

18072018

08046516596, May 201816, July 2018 19, July 2018

2014

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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.

Heart Failure Survival Analysis Longitudinal Data Mixed Effects Cox Model

1. 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 of patients with heart failure.

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.

2. Models2.1. 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:

λ i ( t ) = λ 0 ( t ) exp ( β 1 X i 1 + β 2 X i 2 + ⋯ + β p X i p ) = λ 0 ( t ) exp ( X i ( t ) β ) (1)

where β = ( β 1 , β 2 , ⋯ β p ) T is a p × 1 column vector of coefficients, X i = ( X i 1 , X i 2 , ⋯ , X i p ) is a 1 × p vector of covariates for subject i, and λ 0 ( t ) 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 X i and X j

λ i ( t ) λ j ( t ) = λ 0 ( t ) exp ( X i β ) λ 0 ( t ) exp ( X j β ) = exp ( ( X i − X j ) β ) (2)

is constant over time, the model is called proportional hazards model.

Let the event be observed to have occurred with subject i at time t i . The probability that happened can be written as

L i ( β ) = λ ( t i | X i ) ∑ : t j ≥ t i λ ( t i | X j ) = θ i ∑ : t j ≥ t i θ j (3)

where θ j = exp ( X j β ) and the summation is over the set of subjects j who is still under observation at time t i , the set is called risk set and denoted by R ( t i ) , this is the partial likelihood for subject i. So taking the product of Equation (3) yields the partial likelihood function:

P L ( β ) = ∏ i = 1 n [ exp ( X i β ) ∑ j ∈ R ( t i ) exp ( X j β ) ] δ i (4)

where δ i is 1 if the event is happened to subject i and 0 otherwise.

2.2. Mixed Effects Cox Model

In clinical practice, some subjects may be observed more than once during the 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. Y i 1 , Y i 2 , ⋯ Y i n i , i = 1 , ⋯ , m are the measured data for the individual i at time t i 1 < t i 2 < ⋯ < t i n i , then { Y i k : 1 ≤ k ≤ n i , 1 ≤ i ≤ m } 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:

Y = X T β + Z T b + ε , b ∼ N ( 0 , Σ ) (5)

where and 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:

λ ( t ) = λ 0 ( t ) exp ( X T β + Z T b ) , b ∼ N ( 0 , Σ ) (6)

Coefficients can be estimated based on the partial likelihood:

ln [ P L ( β , b ) ] = ∑ i = 1 n ∫ 0 ∞ { Y i ( t ) η i ( t ) − ln [ ∑ j Y j ( t ) exp ( η j ( t ) ) ] } d t (7)

where η i ( t ) = X i ( t ) β + Z i ( t ) b is the linear score for subject i at time t and Y i ( t ) = 1 if subject i is still under observation at time t and 0 otherwise [18] [19] .

3. Data

We collected patient basic information, laboratory information, medical records, doctor’s advice information and other information from Shanghai Shuguang Hospital database during January 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 time out of 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 Table 1 for details.

Most are categorical variables, but age is a multi-variable. Its distribution is shown in Figure 1.

Statistics for other binary variables are shown in Table 2.

4. 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

variables	Description	Data Type
Id	Patient id	Categorical
num	Hospitalization number of patients	Categorical
Status	1 = dead, 0 = alive	Binary
day	Number of days between first hospitalization and death or last out of hospital	Numeric
days	Number of days between first hospitalization and this in hospitalization date	Numeric
age	1 = (0,40], 2 = (41,50], 3 =(51,60], 4 = (61,70], 5 = (71,80], 6 = (81,90], 7 = (91,100]	Multi-category
sex	1 = male, 0 = female	Binary
Chin_Med	Whether used Chinese 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, 0 = no	Binary
ACEI	Whether used angiotensin converting enzyme inhibitors? 1 = yes, 0 = no	Binary
ARA	Whether used aldosterone receptor antagonists? 1 = yes, 0 = no	Binary
ARB	Whether used angiotensin receptor blocker? 1 = yes, 0 = no	Binary
Blocker	Whether used β blocker? 1 = yes, 0 = no	Binary
diuretic	Whether used Diuretic? 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

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

variables		N (%)
status	alive	1531 (85.6)
status	death	258 (14.4)
sex	male female	955 (53.3) 834 (46.7)
chin_med	yes no	1337 (74.7) 834 (25.3)
coronary	yes no	501 (28) 1288 (72)
hypertension	yes no	1119 (62.6) 670 (37.4)
diabetes	yes no	498 (27.8) 1291 (72.2)
lung_infe	yes no	215 (12) 1574 (88)
bronchitis	yes no	246 (13.7) 1543 (86.3)
ACEI	yes no	392 (21.9) 1937 (78.1)
ARA	yes no	373 (20.8) 1416 (79.2)
ARB	yes no	361 (20.2) 1428 (79.8)
Blocker	yes	800 (44.7)
Blocker	no	989 (55.3)
diuretic	yes	383 (21.4)
diuretic	no	1406 (78.6)
digitalis	yes	1117 (62.4)
digitalis	no	672 (37.6)
anti-platelet	yes	709 (39.6)
anti-platelet	no	1080 (60.4)
nitrate	yes	892 (49.9)
nitrate	no	897 (50.1)

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.

5. Conclusions

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

Table 3 Result of Cox proportional hazards model with all covariates

variables	coef	RR	Se (coef)	z	p-value
sex	0.222649	1.249383	0.178811	1.245	0.21307
age	0.275551	1.317256	0.097916	2.814	0.00489
Chin_med	−0.31796	0.727633	0.200295	−1.587	0.11241
RBC	−0.1807	0.834684	0.244466	−0.739	0.4598
HGB	−0.00859	0.991447	0.007816	−1.099	0.27175
hypertension	−0.40512	0.6669	0.196386	−2.063	0.03913
coronary	0.029494	1.029934	0.203818	0.145	0.88494
diabetes	−0.01215	0.987926	0.22145	−0.055	0.95625
lung_infe	−0.26373	0.768185	0.307327	−0.858	0.39082
bronchitis	0.218949	1.244768	0.21796	1.005	0.31512
ACEI	−0.24764	0.780638	0.240374	−1.03	0.3029
ARA	−0.27402	0.760313	0.21431	−1.279	0.20102
ARB	−0.60086	0.54834	0.266228	−2.257	0.02401
Bblocker	−0.19269	0.824737	0.186844	−1.031	0.3024
diuretic	0.389164	1.475747	0.191756	2.029	0.04241
digitalis	0.305065	1.356714	0.184673	1.652	0.09855
anti-platelet	−0.64137	0.526573	0.206546	−3.105	0.0019
nitrate	0.319543	1.376498	0.173849	1.838	0.06605

*coef is the estimation of the coefficients; RR is relative risk; Se (coef) is the standard error of the estimation.

Table 4 Results of mixed effects Cox model

variables	coef	RR	Se (coef)	z	p-value
sex	0.301405	1.351757	0.161594	1.87	0.062
age	0.144165	1.155074	0.067629	2.13	0.033
Chin_med	−0.02249	0.977757	0.082878	−0.27	0.79
RBC	−0.1126	0.893511	0.169517	−0.66	0.51
HGB	−0.01085	0.989209	0.005333	−2.03	0.042
hypertension	−0.49125	0.611863	0.127701	−3.85	0.00012
coronary	−0.1687	0.844765	0.140132	−1.2	0.23
diabetes	−0.23967	0.786885	0.161708	−1.48	0.14
lung_infe	0.356836	1.428802	0.124253	2.87	0.0041
bronchitis	0.250458	1.284613	0.148653	1.68	0.092
ACEI	−0.32509	0.722463	0.154382	−2.11	0.035
ARA	0.069231	1.071684	0.123429	0.56	0.57
ARB	−0.44209	0.642691	0.122451	−3.61	0.00031
Bblocker	−0.26293	0.768796	0.089191	−2.95	0.0032
diuretic	0.115389	1.12231	0.104295	1.11	0.27
digitalis	0.037052	1.037747	0.081806	0.45	0.65
anti-platelet	−0.3711	0.689975	0.101789	−3.65	0.00027
nitrate	0.029271	1.029703	0.086633	0.34	0.74

factors; hypertension (RR = 0.61), ARB (RR = 0.64), β blockers (RR = 0.77) and antiplatelet (RR = 0.69) were protective factors. Results of the two models are consistent with the covariates age, hypertension, ARB and antiplatelet. Further, age was risk factor, namely the older has lower survival rate. Hypertension, ARB, and antiplatelet were protective factors, namely patients with hypertension have higher survival rates than those without hypertension; patients who used ARBs had higher survival rates than unused patients; patients who used antiplatelet drugs had higher survival rates than those who did not. Survival distributions by these covariates are shown in Figure 2.

The difference is that there are another two covariates which have significantly effect on the survival rate in the mixed effects Cox model: one was risk factor lung infection (RR = 1.43), and the other was protective factor β blocker (RR = 0.67). In addition, the protective factor diuretic in the Cox proportional hazards model became insignificant in the mixed effects Cox model, which shows that the effect of diuretics on survival rate gradually reduces.

Acknowledgements

This work was partially supported by The National High-Tech R&D Program of China (863 Program) under Grant No. 2015AA020107.

Cite this paper

Sheng, J.W., Qian, X.Y. and Ruan, T. (2018) Analysis of Influencing Factors on Survival Time of Patients with Heart Failure. Open Journal of Statistics, 8, 651-659. https://doi.org/10.4236/ojs.2018.84042

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