Original Article

Analysis of the Factors Affecting the Interval between Blood Donations Using Log-Normal Hazard Model with Gamma Correlated Frailty

Citation: Tavakol N, Kheiri S, Sedehi M. Analysis of the Factors Affecting the Interval between Blood Donations Using Log-Normal Hazard Model with Gamma Correlated Frailty. J Res Health Sci. 2016; 16(2):76-80.

Introduction

Blood and blood products play a big part in saving patients’ lives. Regarding the increase in the consumption of blood because of various reasons, it is necessary that donors increase as well ¹ . Providing sufficient and healthful blood is up to blood transfusion centers. Unless enough healthy blood is provided, the society’s health will be at stake. Therefore, those in charge of health care in society are looking for ways of supplying blood and preventing lack of it in blood banks. For this end, detecting and attracting constant donors is of great importance ² . Constant blood donors, according to the standards of the Blood Transfusion Organization, are those who donate blood at least twice a year. Therefore, the blood transfusion centers try to increase the number of constant donors in order to ample blood supply for the patients in need of it ³ .

When it is possible for a person to experience something several times, a recurrent event takes place. The data obtained from the repetition of these events are called recurrent events data. Blood donation is considered a recurrent event in survival analysis ⁴ . In recurrent events survival data, due to individual differences and the effect of previous events, there is a correlation between survival time intervals ⁵ . Owing to the correlation between the independent survival recurrent times, hazard regression models with independent data cannot be applied and it is necessary that a model capable of covering this correlation is used ^6-8 . One way of handling this problem is to use a frailty models. In this model, a common random effect called frailty effect is given to the members of each group later multiplied in the hazard function of each group member ⁸ .

Since the log-normal hazard model with correlated frailty is very complex, Bayesian approach was used to estimate parameters of this model. In this approach, the prior knowledge about the parameters and likelihood is used to produce the posterior distribution, which represents total information about the parameters after the data have been observed ⁹ . Due to the complexities of frailty models, it is not possible to calculate the posterior distribution of the parameters in an analytic way ¹⁰ . Therefore, in order to do a Bayesian analysis, it is necessary to estimate the posterior distribution of parameters using Markov Chain Monte Carlo (MCMC) method. Successive sampling of full conditional distributions of parameters produces a Markov Chain; so that after convergence these samples can be assumed as dependent samples from marginal posterior distributions of parameters and based on them inference about parameters of interest can be done. In recent years, by using MCMC methods, complex and extensive models for any sample size have been conducted and precise estimations have been yielded ¹¹ .

Log-normal hazard model is one of important applicable model in survival analysis which belongs to the generalized gamma parametric survival model ¹² . In the log-normal model, the hazard rate increase from zero rapidly to a peak and then decrease gradually, so that it is a unimodal with comparatively long tail in the right ^{13, 14} . In a primary study on some parametric models without frailty, the log-normal model showed a better fit on our data compare to exponential, Weibull, log-logistic and generalized gamma models. Getting the log-normal model to fitting the interval between blood donations is reasonable because immediately after a successful donation, the chance to another donation is zero, then the donation chance increase with time, and after passing a long time from previous donation, the chance decline.

The aim of this study was to determine the effective factors on the interval between the blood donations in a sample consisting of first-time blood donors based on log-normal hazard model with correlated frailty.

Methods

A series of unknown factors can influence the time intervals between donations including donor's related and specific donation related. Therefore, a correlation between each person’s survival times is expected, which is the time interval between two donations for that person. Accordingly, a random variable called correlated frailty is considered for each person in order to recognize the correlation of the time interval between two donations of that person then the hazard function is multiplied ¹⁶ . The correlated frailty (Yij) consists of shared frailty (Wi) plus individual frailty (Zij), which are independent from each other; moreover, by considering their values, it can be said to what extent the survival time of each sample has been influenced by either common unknown factors or specific ones ¹⁶ . In this study, it was supposed that frailties were of gamma distributions with θ scale; hence, given the independent nature of frailty terms from each other, correlated frailties will also have gamma distributions. As a result, shared frailty is of a gamma distribution with φ,θ parameters and individual frailty is of a gamma distribution with θ-φ,θ parameters provided that 0<φ<θ. Then, the correlated frailty (Yij) has a gamma distribution with θ,θ parameters with mean one and variance of θ-1. Since the survival times for each person are correlated, the correlated frailty model was used. Therefore, regarding the frailty in the model, the survival times for each person were supposed to be independent (from each other) ¹⁶ . A full presentation of correlated frailty model for baseline log-normal hazard model is given in the appendix.

In order to modeling the baseline hazard function, log-normal hazard model was used. For analyzing return to donation, data three models were used as log-normal hazard model without frailty, log-normal hazard model with gamma-shared frailty, and log-normal hazard model with gamma correlated frailty. Furthermore, the estimation of parameters was done using Bayesian analysis and Markov Chain Monte Carlo (MCMC) method. Considering the complexity of posterior distribution calculations, MCMC method is used to estimate the parameters for any sample size ¹¹ .

For prior distribution of the model parameters, first the log-normal hazard model applying non-informative prior of regression coefficient was carried out and later their estimate was used for determining informative prior distribution in the shared frailty and correlated frailty models. Therefore for the parameters of regression, the prior for the coefficient of age assume as normal distribution with mean 1 and variance 100, sex as normal distribution with mean 1 and variance 4, body weight as normal distribution with mean 0 and variance 100, education as normal distribution with mean 1 and variance 4, job as normal distribution with mean 1 and variance 4, marital status as normal distribution with mean 0.1 and variance 4, stay as normal distribution with mean 1 and variance 2 and a β0 (constant value) as normal distribution with mean 1 and variance 4 were used. The prior distribution for frailty parameter of (φ) as uniform distribution on (0,θ) and for frailty parameters of (θ) as gamma distribution (α=0.5,β=10) were used as well. The parameters were estimated using OpenBUGS ¹⁷ software version 3.2.3. In order to make sure of the coverage of Monte Carlo simulations, Gelman-Rubin convergence criteria via BOA (Bayesian Output Analysis) program in R software version 3, 2, 0 was used ¹⁸ . Comparing between log-normal hazard model without frailty, log-normal hazard model with gamma shared frailty, and log-normal hazard model with gamma correlated frailty was done based on deviance information criterion (DIC) ¹⁹ .

Results

The time between recurrent donations was a response variable. The number of donations in each interval, the mean of time interval between donations and the censoring rate for each time interval is given in Table 2.

The results of fitting the log-normal hazard model with gamma correlated frailties, including mean, median, standard deviation, and 95% credible intervals, based on 30,000 simulated values after considering 5000 samples as burn-in period, is shown in the Table 3. Since there was a very high autocorrelation in the successive values of the simulated observation, every 50-th sample was monitored. The estimate of Gelman-Rubin convergence criteria is shown in the last column of Table 3. These values are very close to one ensuring the convergence of all parameters of model. In our data, DIC for log-normal hazard model without frailty was 32814800, for the log-normal hazard model with gamma shared frailty was 32810000, and for the log-normal hazard model with gamma correlated frailty was 32210000, which showed a better fitting for the log-normal hazard model with gamma correlated frailty. Based on estimate of φ and θ parameters, the estimate of variance of frailty random effect is 0.41, the mean of shared frailty is 0.66, the mean of individual frailty is 0.34, and the coefficient correlation of shared and individual frailty is 0.66.

Discussion

In the transfusion data framework, the correlated effect is sum of donor random effect (known as shared frailty) and specific donation random effect (known as individual frailty). The mean estimate of donor random effect in our data was almost 2 times of specific donation random effect. This results show that effect of donor's unknown factors are higher than specific donation's unknown factors. The correlation of interval times between donations of each donor means that each donor returns to donation in a roughly equal interval which show stability in his/her return to donation behavior.

The correlated frailty was used in many applications, ²⁸ . Here we used it to explain unknown random effect in the recurrent event survival data. Log-normal hazard model for survival analysis is complex especially in present of censoring, ¹³ . Its complexity increased by applying the random effect in the model. This problem causes the difficulty in convergence of simulated samples and encountering a high autocorrelation between subsequent samples of simulated values. In running of MCMC algorithm, one sample was monitored from every 50 produced samples to reduce autocorrelation between subsequent samples and obtaining convergence. Although a large development in computational speed has been created, Bayesian analyzing of such model is very time-consuming presently.

In this study, it was presupposed that components of frailty were of gamma distribution. Research can also be done on log-normal hazard model with correlated frailty having Inverse Gaussian and Positive Stable distributions.

Highlights

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