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Year , Volume 4 , Issue 1, Pages 0 - 0 Zotero Mendeley EndNote. The Cox regression model which is commonly used in survival analysis is established under the proportional hazards assumption. However cases in which the data shows heterogeneity come across in studies.

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In this case, heterogeneity should be explained in order to make the interpretations more effective which were obtained depending on the model. Frailty models are one of the survival analysis methods which were developed for explaining heterogeneity. In this study, frailty models are examined theoretically and were applied to the lung cancer data.

The unshared frailty model has been used to explain the difference between general risk and momentary risk of individuals in the data set. As for comparing the momentary risk between individuals with various levels of explanatory variables with other individuals, shared frailty models have been used. References Ata, N. Babiker A. A simple frailty model for family studies with covariates.

Statistics in Medicine, 13, Clayton, D. Biometrika, 65, — Multivariate generalisations of the proportional hazards model with discussion. Congdon, P. Modelling Frailty İn Area Mortality. Statistics in Medicine, 14, Cox, D. Regression models and life-tables. Duchateau, L. Economou, P. Graphical tests for the assumption of gamma and inverse Gaussian frailty distributions. Lifetime Data Analysis 11, — Guo, G.

Estiamting amultivariate proportional hazards model for clustered data using the EM algorithm with an application to child survival in Guatemala. Journal of the American Statistical Association, 87, Gutierrez, R. Parametric frailty and shared frailty survival models. The Stata Journal, 2, 22— Hougaard, P. Life table methods for heterogeneous populations. Biometrika, 71, 75— Survival models for heterogeneous populations derived from stable distributions. Biometrika 73, — Frailty models for survival data. Lifetime Data Analysis, 1, — Ibrahim, J. Keiding, N. The role of frailty models and accelerated failure time models in describing heterogeneity due to omitted covariates.

Statistics in Medicine 16, — Kheiri, S.

Hamadan University of Medical Sciences

Bayesian analysis of an inverse Gaussian correlated frailty model, Computational Statistics and Data Analysis 51, — Klein, J. Semiparametric estimation of random effects using the Cox model based on the EM algorithm. Regarding the frailty distribution, we assumed gamma and inverse Gaussian distributions. For comparison of different distributions, the AIC criteria were used, but for comparing nested models, likelihood ratio test were used.

The data set used in this study were obtained from Wolisso St. Luke catholic hospital, Wolisso, south west Ethiopia. A total of children who came from kebeles around Wolisso district were considered. Kebeles that contribute only a child were omitted since the shared frailty model should be done on at least two children per kebele. Therefore a total of children with severe acute malnutrition from kebeles were considered.

The response variable time-to-cure from SAM were obtained by calculating the difference in day from the start of treatment until the child were cured recovered or censored. Children were considered to be cured and discharged, which is our event of interest, if they fulfilled the discharging criteria for SAM [ 15 ]. However, the time-to-cure were censored for those children transferred to other hospital, dropped treatment, died, did not cure at January 31, at the end of study. The following variables were considered for their influence on the time-to-cure from SAM; sex , age , type of malnutrition and co-infection.

For age we used six categories; 0—5 months, 6—11 months, 12—23 months, 24—35 months, 36—47 months and 48—59 months. Types of malnutrition were categorised as Marasmus, Kwashiorkor and Marasmic-kwashiorkor. Co-infection was categorized based on whether the child has co-infection such as malaria, anemia, pneumonia, measles and giardiasis or not. Whereas an alternative if the proportional hazards assumption does not hold is the accelerated failure time frailty model which assumes. The survival time is assumed to be conditionally independent with respect to the shared common survival times frailty.

This shared frailty is the cause of dependence between survival time within the clusters. In order to investigate the effect of the candidate covariates on the time-to-cure from SAM, we first did a univariable analysis by fitting a separate model for each candidate covariates. Covariates that were found to be significant in the univariable analysis were included in the multivariable analysis. The multivariable survival analysis in the study was done by assuming the exponential, weibull and log-logistic distributions for the baseline hazard function; and the gamma and inverse Gaussian frailty distributions.

It was performed using the three most significant covariates in the univariable analysis namely age , type of malnutrition and co-infection. However, we excluded sex which was not significant in univariable analysis. Of all malnourished patients, Using all the multivariable frailty models, the covariate co-infection was significant, indicating that it was the most important prognostic factor for the time-to-cure from SAM. Age group was significant in the three models namely, weibull-gamma, weibull-inverse Gaussian and log-logistic-inverse Gaussian frailty models.

Type of malnutrition was not a significant factor for time-to-cure from SAM using all the models. The AIC value of the log-logistic-inverse Gaussian model Analysis based on log-logistic-inverse Gaussian frailty model showed that age group of the children and presence of co-infection were significant. An acceleration factor of greater than 1 indicates prolonging the time-to-cure from SAM. Therefore, children aged between 12 and 23 months had prolonged cure time by a factor of 1. The confidence interval of the acceleration factor of co-infection was 1. This value greater than unity indicates that the shape of hazard function is unimodal, i.

The predicted frailty values increases with range of 0. That is, these values are lower for lower values of event times and higher for higher value of event times. The median value of the frailty distribution is around 1. Prediction of frailties for the SAM dataset as given by the parametric log-logistic-inverse Gaussian frailty model.


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Conditional hazard rates of the log-logistic- inverse Gaussian frailty model for the SAM dataset. The conditional hazard functions given the 75 th quartile frailty values was greater, followed by the conditional hazard functions given by the 50 th median and 25 th quartile frailty values of the clusters respectively.

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But the gap widens through time, specifically at mid time. To check the adequacy of our baseline hazard, the exponential has been plotted by the cumulative hazard function with time-to-cure from SAM. Similarly, the weibull has been plotted by the logarithm cumulative hazard function with the logarithm of time-to-cure from SAM and log-logistic has been plotted by the logarithm of the failure odds with the logarithm of time-to-cure from SAM Figure 3.

The plot of log-logistic was more linear than the other plots, though only few observations were scattered at the beginning time. The patterns suggested that the log-logistic hazard function was appropriate in the model. The Cox-Snell residuals together with their cumulative hazard function were obtained by fitting the exponential, weibull and log-logistic models to our dataset, via maximum likelihood estimation Figure 4.


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  • The plots showed that the Cox-Snell residuals fitted to assess the log-logistic model for the dataset were nearest to the line through the origin as compared to the other models, again indicating that this model described the SAM dataset well. Graphical evaluation of the exponential, weibull and log-logistic assumptions. Cox-Snell residuals obtained by fitting exponential, weibull and log-logistic models to the SAM dataset. A quantile-quantile or q-q plot was made to check if the accelerated failure time provided an adequate fit to the data using two different groups of population.

    We checked the adequacy of the accelerated failure-time model by comparing the significantly different age groups children in the age group 0—5 months and 12—24 months ; as well as the co-infected and non co-infected groups of patients Figure 5. The figures appear to be approximately linear for both covariates age group and co-infection with slopes equivalent to the acceleration factors 1.

    Therefore the log-logistic baseline model used for time-to-cure from SAM was accelerated failure time model. The main aim of the study was to model time-to-cure from SAM using appropriate survival model among various parametric frailty models. The comparison of distributions of the models was performed using the AIC criteria, where a model with minimum AIC is accepted to be the best [ 13 ].

    Accordingly, the log-logistic-inverse Gaussian frailty model which has AIC value of This study also showed that there was a clustering frailty effect on modeling time-to-cure from SAM which might be due to the heterogeneity in kebele from which the child came, assuming children coming from the same kebele share similar risk factors related to SAM. Therefore, it was important considering the clustering effect in modeling the hazard function.

    Clusters with minimum median time have smaller frailties, so that these clusters are predicted to have a high hazard [ 9 ], more probable to cure in this case. Our data demonstrated that the nuisance frailty terms modified the hazard function, and therefore the the hazard function was evaluated conditionally on this effect. Kebeles that frail more were more likely to cure than the less frail kebeles since the event is positive.

    The inverse Gaussian frailty generates very strong dependence at mid time [ 16 ]. According to the conditional hazard function given the 25 th , 50 th and 75 th quantile frailty values, the hazard function depends on these values especially at the mid time Figure 1. This frailty distribution introduced as an alternative to the gamma distribution [ 17 ] was better in this dataset compared to the gamma frailty distribution.

    Nonetheless, the most acknowledged parametric model is the weibull, which allows the proportional hazards and accelerated life time model [ 8 ]; the SAM data set was best described by the log-logistic baseline as compared to the exponential and weibull hazard functions. According to the diagnostic plots the log failure odds of log-logistic baseline with log time was more linear as compared to the plots of exponential cumulative hazard versus time and weibull log cumulative hazard versus log time , showing the SAM dataset was best described by the log-logistic baseline.

    This result was also confirmed by the cumulative hazard plots for the Cox-Snell residuals of the exponential, weibull and the log-logistic models. The plot was more approached to the line in case of the log-logistic model, indicating that the log-logistic was best. A q-q plot was done to check if the accelerated failure time provided an adequate fit to the dataset and the log-logistic as baseline was accelerated failure time model.

    Hence, a survival model need not be chosen arbitrarily to fit event times, the baseline hazard function as well as the frailty distribution should be compared and the most appropriate model should be selected for appropriate inference. The prognostic factors considered were the age group of the child, type of malnutrition and presence or absence of co-infection , which were significant covariates using univariable analysis.

    Analysis using the best model, log-logistic-inverse Gaussian frailty model showed that the age group of children and presence of co-infection s were the determinant factors for the time-to-cure from SAM. Children aged between 12—23 months had prolonged cure time as compared to older age groups.

    This age group is known to be the time when the prevalence of SAM is the highest [ 19 — 21 ]. This may be related to the fact that they start sub-optimal complementary feeding and compromise breast feeding practice which is important in preventing malnutrition among children [ 22 ].

    Frailty models for survival data | SpringerLink

    Literatures like [ 23 — 25 ] identified infection as a prognostic indicators, likewise, co-infection s prolonged the time-to-cure from SAM in this study. However, our findings showed that the time of curing did not depend on type of malnutrition. Similarly, Efrem et al. Luke Catholic hospital was 14 days with maximum cure time of 63 days of which This implies that the program was acceptable as per the above standard.

    We classified SAM cases as marasmus, kwashiorkor and marasmic-kwashiorkor, a naming might imply that the cause is protein and calorie deficiency perse. However, the intent was to see if there are differential in recovery time between edematous and non-edematous cases of SAM. We acknowledge the limitation of our study for not being able to isolate SAM cases with micronutrient deficiency. The most appropriate statistical model for our dataset among various parametric frailty models, which well described the time-to-cure from SAM of the patients who were diagnosed in Wolisso St.

    Luke Catholic hospital is the log-logistic-inverse Gaussian frailty model. There is a frailty clustering effect on time-to-cure from SAM that arises due to heterogeneity between the kebeles of the children. The median curing time of the children is about 14 days with maximum cure time of 63 days of which These values show acceptable functioning of the program in the hospital. Lancet , Lancet , — Asayehegn T, Mekitie W, Girma A, Kebede D: Cost effectiveness of community-based and in-patient therapeutic feeding programs to treat severe acute malnutrition in Ethiopia.

    Biomed Central Kerac M: Improving the treatment of severe acute malnutrition in childhood. Zelalem T, Tsinuel G, Ayalew T: Treatment outcome of severe acute malnutrition in children with and without hiv infection: a historical cohort study in South-West Ethiopia.

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    Ethiop J Pediatr Child Health , 6: 14— Cox D: Regression models and lifetables with discussions. J Roy Stat Soc , — Demography , — New York: Springer-Verlag; Sastry N: A nested frailty model for survival data, with an application to the study of child Survivalin Northeast Brazil. J Am Stat Assoc , — Comput Stat Data Anal , — Munda M: Parametric frailty models in R.

    Am Stat Assoc , 1—