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Objective: To examine the potential difference in survival and risk of death between asymptomatic and symptomatic SARS-CoV-2 patients, controlled by age and gender for all the attendance in hospitals of Khyber Pakhtunkhwa (KP), Pakistan. Methods: In this retrospective study, the medical records of 6273 SARS-CoV-2 patients admitted to almost all hospitals in Khyber Pakhtunkhwa during the first wave of the coronavirus outbreak from March to June 2020 were analysed. The effects of gender, age, and being symptomatic on the survival of SARS-CoV-2 patients were assessed using cure-survival models as opposed to the conventional Cox proportional hazards model. Results: The prevalence of initially symptomatic patients was 55.8%, and the overall mortality rate was 11.8%. The fitted cure-survival models suggest that age affects the probability of death (incidence) but not the short-term survival time of patients (latency); symptomatic patients have a higher risk of death than their asymptomatic counterparts, but the survival time of symptomatic patients is longer on average; gender has no significant effect on the probability of death and survival time. Conclusion: The available data and statistical results suggest that asymptomatic and young patients are generally less susceptible to initial infection with SARS-CoV-2 and therefore have a lower risk of death. Our regression models show that uncured asymptomatic patients generally have poorer short-term survival than their uncured symptomatic counterparts. The association between gender and survival outcome was not significant. doi: https://doi.org/10.12669/pjms.40.8.8931 How to cite this: Asghar N, Khalil U, Iftikhar Uddin. Mixture and Non-Mixture Cure Models for the survival analysis of SARS-CoV-2 patients in Khyber Pakhtunkhwa, Pakistan. Pak J Med Sci. 2024;40(8):1841-1846. doi: https://doi.org/10.12669/pjms.40.8.8931 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Background HIV is one of the deadliest epidemics and one of the most critical global public health issues. Some are susceptible to die among people living with HIV and some survive longer. The aim of the present study is to use mixture cure models to estimate factors affecting short- and long-term survival of HIV patients. Methods The total sample size was 2170 HIV-infected people referred to the disease counseling centers in Kermanshah Province, in the west of Iran, from 1998 to 2019. A Semiparametric PH mixture cure model and a mixture cure frailty model were fitted to the data. Also, a comparison between these two models was performed. Results Based on the results of the mixture cure frailty model, antiretroviral therapy, tuberculosis infection, history of imprisonment, and mode of HIV transmission influenced short-term survival time ( p -value < 0.05). On the other hand, prison history, antiretroviral therapy, mode of HIV transmission, age, marital status, gender, and education were significantly associated with long-term survival ( p -value < 0.05). The concordance criteria (K-index) value for the mixture cure frailty model was 0.65 whereas for the semiparametric PH mixture cure model was 0.62. Conclusion This study showed that the frailty mixture cure models is more suitable in the situation where the studied population consisted of two groups, susceptible and non-susceptible to the event of death. The people with a prison history, who received ART treatment, and contracted HIV through injection drug users survive longer. Health professionals should pay more attention to these findings in HIV prevention and treatment.

Current status data occur in many biomedical studies where we only know whether the event of interest occurs before or after a particular time point. In practice, some subjects may never experience the event of interest, i.e., a certain fraction of the population is cured or is not susceptible to the event of interest. We consider a class of semiparametric transformation cure models for current status data with a survival fraction. This class includes both the proportional hazards and the proportional odds cure models as two special cases. We develop efficient likelihood-based estimation and inference procedures. We show that the maximum likelihood estimators for the regression coefficients are consistent, asymptotically normal, and asymptotically efficient. Simulation studies demonstrate that the proposed methods perform well in finite samples. For illustration, we provide an application of the models to a study on the calcification of the hydrogel intraocular lenses.

The authors propose a novel class of cure rate models for right-censored failure time data. The class is formulated through a transformation on the unknown population survival function. It includes the mixture cure model and the promotion time cure model as two special cases. The authors propose a general form of the covariate structure which automatically satisfies an inherent parameter constraint and includes the corresponding binomial and exponential covariate structures in the two main formulations of cure models. The proposed class provides a natural link between the mixture and the promotion time cure models, and it offers a wide variety of new modelling structures as well. Within the Bayesian paradigm, a Markov chain Monte Carlo computational scheme is implemented for sampling from the full conditional distributions of the parameters. Model selection is based on the conditional predictive ordinate criterion. The use of the new class of models is illustrated with a set of real data involving a melanoma clinical trial. /// Les auteurs proposent une nouvelle classe de modèles pour des durées de vie censurées à droite dans lesquels une possibilité de guérison est prise en compte. Cette classe est définie par le biais d'une transformation de la fonction de survie théorique inconnue. Les modèles dans lesquels la guérison est incorporée par voie de mélange et par temps d'apparition de tumeurs détectables en sont des cas particuliers. Les auteurs proposent une forme générale de structure de covariables qui satisfait automatiquement à une contrainte paramétrique inhérente au problème et qui inclut les structures binomiales et exponentielles correspondant aux deux principales formulations des modèles à taux de guérison. En plus de lier de façon naturelle les modèles à taux de guérison par mélange et à temps d'apparition de tumeurs détectables, la classe proposée suggère un grand nombre de nouvelles structures de modèles. Dans le cadre du paradigme bayésien, un algorithme de calcul de Monte-Carlo à chaîne de Markov est implanté pour l'échantillonnage à partir des lois conditionnelles complètes des paramètres. La sélection de modèles s'appuie sur un critère faisant intervenir des prévisions conditionnelles. L'emploi de la nouvelle classe de modèles est illustré au moyen de données issues d'essais cliniques sur le mélanome.

Cure models in survival analysis deal with populations in which a part of the individuals cannot experience the event of interest. Mixture cure models consider the target population as a mixture of susceptible and non-susceptible individuals. The statistical analysis of these models focuses on examining the probability of cure (incidence model) and inferring on the time to event in the susceptible subpopulation (latency model). Bayesian inference for mixture cure models has typically relied upon Markov chain Monte Carlo (MCMC) methods. The integrated nested Laplace approximation (INLA) is a recent and attractive approach for doing Bayesian inference but in its natural definition cannot fit mixture models. This paper focuses on the implementation of a feasible INLA extension for fitting standard mixture cure models. Our proposal is based on an iterative algorithm which combines the use of INLA for estimating the process of interest in each of the subpopulations in the study, and Gibbs sampling for computing the posterior distribution of the cure latent indicator variable which classifies individuals to the susceptible or non-susceptible subpopulations. We illustrated our approach by means of the analysis of two paradigmatic datasets in the framework of clinical trials. Outputs provide closing estimates and a substantial reduction of computational time in relation to those using MCMC.

In this paper, a new long term survival model called Nadarajah-Haghighi model for survival data with long term survivors was proposed. The model is used in fitting data where the population of interest is a mixture of individuals that are susceptible to the event of interest and individuals that are not susceptible to the event of interest. The statistical properties of the proposed model including quantile function, moments, mean and variance were provided. Maximum likelihood estimation procedure was used to estimate the parameters of the model assuming right censoring. Furthermore, Bayesian method of estimation was also employed in estimating the parameters of the model assuming right censoring. Simulations study was performed in order to ascertain the performances of the MLE estimators. Random samples of different sample sizes were generated from the model with some arbitrary values for the parameters for 5%, 1:3% and 1:5% cure fraction values. Bias, standard error and mean square error were used as discrimination criteria. Additionally, we compared the performance of the proposed model with some competing models. The results of the applications indicates that the proposed model is more efficient than the models compared with. Finally, we fitted some models considering type of treatment as a covariate. It was observed that the covariate  have effect on the shape parameter of the proposed model.

We propose a novel interpretation for a recently proposed Box–Cox transformation cure model, which leads to a natural extension of the cure model. Based on the extended model, we consider an important issue of model selection between the mixture cure model and the bounded cumulative hazard cure model via the likelihood ratio test, score test and Akaike’s Information Criterion (AIC). Our empirical study shows that AIC is informative and both the score test and the likelihood ratio test have adequate power to differentiate between the mixture cure model and the bounded cumulative hazard cure model when the sample size is large. We apply the tests and AIC methods to leukemia and colon cancer data to examine the appropriateness of the cure models considered for them in the literature.

We consider a class of cure rate frailty models for multivariate failure time data with a survival fraction. This class is formulated through a transformation on the unknown population survival function. It incorporates random effects to account for the underlying correlation, and includes the mixture cure model and the proportional hazards cure model as two special cases. We develop efficient likelihood-based estimation and inference procedures. We show that the nonparametric maximum likelihood estimators for the parameters of these models are consistent and asymptotically normal, and that the limiting variances achieve the semiparametric efficiency bounds. Simulation studies demonstrate that the proposed methods perform well in finite samples. We provide an application of the proposed methods to the data of the age at onset of alcohol dependence, from the Collaborative Study on the Genetics of Alcoholism.

In survival analysis it often happens that some subjects under study do not experience the event of interest; they are considered to be “cured.” The population is thus a mixture of two subpopulations, one of cured subjects and one of “susceptible” subjects. We propose a novel approach to estimate a mixture cure model when covariates are present and the lifetime is subject to random right censoring. We work with a parametric model for the cure proportion, while the conditional survival function of the uncured subjects is unspecified. The approach is based on an inversion which allows us to write the survival function as a function of the distribution of the observable variables. This leads to a very general class of models which allows a flexible and rich modeling of the conditional survival function. We show the identifiability of the proposed model as well as the consistency and the asymptotic normality of the model parameters. We also consider in more detail the case where kernel estimators are used for the nonparametric part of the model. The new estimators are compared with the estimators from a Cox mixture cure model via simulations. Finally, we apply the new model on a medical data set.

This paper presents estimates for the parameters included in long-term mixture and non-mixture lifetime models, applied to analyze survival data when some individuals may never experience the event of interest. We consider the case where the lifetime data have a three-parameter Burr XII distribution, which includes the popular Weibull mixture model as a special case. Classical and Bayesian procedures are used to get point estimates and confidence intervals for the unknown parameters. We consider a general survival model where the scale and shape parameters of the Burr XII distribution are dependent of some covariates. To illustrate the proposed methodology, we consider an application considering a leukaemia data set where the proposed model gives better fit for the data when compared to other existing models.