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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 is a syndrome with symptoms and signs caused by cardiac dysfunction, resulting in reduced longevity [

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 [

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.

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 [

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 [

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

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

Statistics for other binary variables are shown in

Firstly, we use the Cox proportional hazards to model the survival-time data with all covariates. The results are shown in

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 |

variables | N (%) | |
---|---|---|

status | alive | 1531 (85.6) |

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) |

no | 989 (55.3) | |

diuretic | yes | 383 (21.4) |

no | 1406 (78.6) | |

digitalis | yes | 1117 (62.4) |

no | 672 (37.6) | |

anti-platelet | yes | 709 (39.6) |

no | 1080 (60.4) | |

nitrate | yes | 892 (49.9) |

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

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

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.

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

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.

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

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