1.1.1 Time-to-Event Data and Cured Subjects
In many scientific studies, researchers are interested in time to the occurrence of a specific event of interest. When analyzing time-to-event data, it often happens that a certain proportion of subjects may never experience the event of interest and are considered to be cured. Cured subjects are immune or non-susceptible to the event. Time-to-event data with cured subjects are abundant in population health and medical studies. In some diseases, there is strong biological evidence that, when considering endpoints other than a natural death, the event would never occur for some fraction of the subjects. The assumption that some individuals will never experience the event may also be based solely on empirical considerations, such as the presence of a large number of long-term survivors. Recent progress in the diagnosis and treatment of cancer has led to substantial improvement in cancer survival. If patients are diagnosed early and receive effective treatment, e.g., the targeted therapy or immunotherapy, a subpopulation of the cancer patients eventually become cured from the disease, which leads to some long-term survivors if the follow-up time is long enough. The presence of cured subjects is often suggested by a Kaplan-Meier estimate of the marginal survival function, which shows a long and stable plateau with heavy censoring at the right extreme.
In sociology studies, the rates of life-long marriage and permanent employment are important concepts of social stability. An employee who has been with the same company for a sufficiently long time may be considered as unlikely to join other companies (Yamaguchi, 1992) and can be considered as cured or immune to the event of joining other companies. In criminal justice, the recidivism rate is an important feature to determining the effect of rehabilitation to stop offenders from committing crimes again and those subjects who will not commit crimes again after a long time may be considered as cured (Schmidt and Witte, 1989). The concept of cured subjects has applications in economics. For example, when considering the probability of default and the timing of default on loans, a company or subject who is not in default for a long time may be considered as cured (Tong et al., 2012). In reliability studies, one complex system is often considered as being composed of some subsystems in series. Usually, the failure of any subsystem would lead to the failure of the entire system. However, some subsystems’ lifetimes are long enough and even never fail during the life cycle of the entire system and they can be considered as cured. In political studies, the time to intervention over the course of a conflict and the conflicts that are never intervened for a long time may be considered as cured (Findley and Teo, 2006).
1.1.2 Survival Models and Cure Models
Standard statistical methods for time-to-event analysis, often called survival models, typically assume that every subject in the study population is susceptible to the event of interest and will eventually experience the event if the follow-up is sufficiently long. They do not account for cured subjects and can lead to biased estimates of the survival of susceptible subjects.
Survival models that take into account the possibility of cure are commonly referred to as cure models or cure rate models. A cure model allows a direct modeling of the cure proportion and the effect of covariates on the cure proportion. Additionally, the cure model also provides estimation and inference of the distribution of the survival time for uncured subjects or subjects who are not cured. Most of the existing cure models are modified survival models that include the probability of being cured. Based on how the cure proportion is introduced, the cure models can be roughly divided into two types: mixture cure models and non-mixture cure models.
Although the cure models have a shorter history compared to the standard survival models, they have been an area of active research in statistics since the early 1950s. The most widely used cure rate model is the mixture cure model which is also known as the standard cure rate model. This model was first introduced by Boag (1949). He introduced the traditional definition of a cured fraction for patients who have a specific illness, by a five-years survival rate. Although the definition of the cured fraction has changed and improved over the years, many authors developed and improved the original mixture model further. Berkson and Gage (1952) divided the population into two groups: susceptible individuals to the event of interest and insusceptible individuals. They suggested that a group of treated patients are considered to be cured if they have approximately the same survival distribution as the general population who have never had the disease of interest. Farewell (1982) used a mixture model as a combination of logistic model and Weibull distribution to model the toxicant and stress level for laboratory animals. Kuk and Chen (1992) proposed a semiparametric mixture cure model consisting of the logistic model for the probability of occurrence of the event of interest, and the proportional hazard model for the time to event of interest for uncured subjects. Maller and Zhou (1996) have collected a comprehensive account of mixture cure rate models with various survival functions. Goldman (1984), Taylor (1995), and Peng and Dear (2000), among others, have also investigated parametric, semiparametric, and nonparametric mixture cure rate models. Yu and Peng (2008) and Peng et al. (2007a) extended the mixture cure model to multivariate survival data. Both marginal survival models (Chen and Lu, 2012; Niu and Peng, 2014; Niu et al., 2018a) and random-effects frailty models (Peng and Taylor, 2011; Peng and Zhang, 2008a) have been proposed. More recently, Wu and Yin (2013) suggested quantile regression methods and martingale estimating equations for a better assessment of covariate effects on quantiles of a time-to-event of interest.
The non-mixture cure model is another type of cure models for modeling time-to-event data with a cure fraction. Non-mixture cure models were first introduced by Yakovlev et al. (1994a) and then discussed by Yakovlev et al. (1994b), Ibrahim et al. (2001), Chen et al. (1999), and Wang et al. (2012a). These models were motivated by an underlying biological mechanism for cancer cells, which assumes that the number of cancer cells after cancer treatment follows a Poisson distribution. Most of the current investigations on the non-mixture cure models are in the Bayesian context due to its special form. The non-mixture cure models can be expressed into the mixture cure models. However, the model interpretation, particularly the covariate effects on the distribution of the survival time of uncured subjects can have important differences between the two types of cure models.
Because of abundant time-to-event data with a cure fraction in many different disciplines, cure models have potentially wide applications in the disciplines. The novel applications of the cure models have help investigators understand the pattern and behavior of survival times after treatment or intervention. Particularly, in recent years there has been significant progress in the research and development of immune-oncology (I-O) therapy. Because the mechanisms of actions of immunotherapy are profoundly different from traditional chemotherapy and radiation therapy, the patient survival in the clinical trials of immunotherapy have demonstrated delayed clinical effects as well as long-term survival, and cure models can play an important role in revealing the complicated short-term and long-term effects of the therapy to cancer patients.
However, although there have been tremendous efforts and development in modeling of survival data with a cure fraction, there are very few books that provide a comprehensive coverage of the statistical concepts, methods, and applications of cure models. The applications of cure models, particularly those developed in recent years, in the disciplines are still limited, such as in the clinical literature (Othus et al., 2012; Yilmaz et al., 2013). The only published book that is devoted to cure models is Maller and Zhou (1996). However, it focused on large sample properties of estimation and tests for nonparametric and exponential-family based mixture cure models only and is not application-oriented. Moreover, the book was published more than 20 years ago and there has been no book since then that covers the rapid growth in research and applications of cure models in the last two decades. Ibrahim et al. (2001) devotes a chapter on specific Bayesian cure models. More recently there are a few review papers that cover more recent advances in cure models, such as Peng and Taylor (2014) and Amico and Van Keilegom (2018). Due to space limitations, the review papers do not include many details. Thus, a book that covers the rapid growth in research and applications of cure models in the last two decades and provides a comprehensive introduction of various cure models, their estimation methods and their applications using software programs that have been developed will provide a valuable resource for practitioners to apply cure models in various disciplines and for statistical researchers and students to further advance the investigation of analysis time-to-event data with a cure fraction.
