Interval Mapping of Quantitative Trait Loci for Time-to-Event Data with the Proportional Hazards Mixture Cure Model

Firth adjusted score function for monotone likelihood in the mixture cure fraction model

Models for situations where some individuals are long-term survivors, immune or non-susceptible to the event of interest, are extensively studied in biomedical research. Fitting a regression can be problematic in situations involving small sample sizes with high censoring rate, since the maximum likelihood estimates of some coefficients may be infinity. This phenomenon is called monotone likelihood, and it occurs in the presence of many categorical covariates, especially when one covariate level is not associated with any failure (in survival analysis) or when a categorical covariate perfectly predicts a binary response (in the logistic regression). A well known solution is an adaptation of the Firth method, originally created to reduce the estimation bias. The method provides a finite estimate by penalizing the likelihood function. Bias correction in the mixture cure model is a topic rarely discussed in the literature and it configures a central contribution of this work. In order to handle this point in such context, we propose to derive the adjusted score function based on the Firth method. An extensive Monte Carlo simulation study indicates good inference performance for the penalized maximum likelihood estimates. The analysis is illustrated through a real application involving patients with melanoma assisted at the Hospital das Clínicas/UFMG in Brazil. This is a relatively novel data set affected by the monotone likelihood issue and containing cured individuals.

Semiparametric methods for survival data with measurement error under additive hazards cure rate models

It is well established that measurement error has drastically negative impact on data analysis. It can not only bias parameter estimates but may also cause loss of power for testing relationship between variables. Although survival analysis of error-contaminated data has attracted extensive interest, relatively little attention has been paid to dealing with survival data with error-contaminated covariates when the underlying population is characterized by a cured fraction. In this paper, we consider this problem for which lifetimes of the non-cured individuals are featured by the additive hazards model and the measurement error process is described by an additive model. Unlike estimating the relative risk in the proportional hazards model, the additive hazards model allows us to estimate the absolute risk difference associated with the covariates. To allow the model flexibility, we incorporate time-dependent covariates in the model. We develop estimation methods for the two scenarios, without or with measurement error. The proposed methods are evaluated from both the theoretical view point and the numerical perspectives. Furthermore, a real-life data application is presented to illustrate the utility of the methodology.

Concordance measure and discriminatory accuracy in transformation cure models

Many populations of early-stage cancer patients have non-negligible latent cure fractions that can be modeled using transformation cure models. However, there is a lack of statistical metrics to evaluate prognostic utility of biomarkers in this context due to the challenges associated with unknown cure status and heavy censorship. In this article, we develop general concordance measures as evaluation metrics for the discriminatory accuracy of transformation cure models including the so-called promotion time cure models and mixture cure models. We introduce explicit formulas for the consistent estimates of the concordance measures, and show that their asymptotically normal distributions do not depend on the unknown censoring distribution. The estimates work for both parametric and semiparametric transformation models as well as transformation cure models. Numerical feasibility of the estimates and their robustness to the censoring distributions are illustrated via simulation studies and demonstrated using a melanoma data set.

Proportional hazards under Conway–Maxwell-Poisson cure rate model and associated inference

Cure rate models or long-term survival models play an important role in survival analysis and some other applied fields. In this article, by assuming a Conway–Maxwell–Poisson distribution under a competing cause scenario, we study a flexible cure rate model in which the lifetimes of non-cured individuals are described by a Cox’s proportional hazard model with a Weibull hazard as the baseline function. Inference is then developed for a right censored data by the maximum likelihood method with the use of expectation-maximization algorithm and a profile likelihood approach for the estimation of the dispersion parameter of the Conway–Maxwell–Poisson distribution. An extensive simulation study is performed, under different scenarios including various censoring proportions, sample sizes, and lifetime parameters, in order to evaluate the performance of the proposed inferential method. Discrimination among some common cure rate models is then done by using likelihood-based and information-based criteria. Finally, for illustrative purpose, the proposed model and associated inferential procedure are applied to analyze a cutaneous melanoma data.

A sequential threshold cure model for genetic analysis of time-to-event data

In analysis of time-to-event data, classical survival models ignore the presence of potential nonsusceptible (cured) individuals, which, if present, will invalidate the inference procedures. Existence of nonsusceptible individuals is particularly relevant under challenge testing with specific pathogens, which is a common procedure in aquaculture breeding schemes. A cure model is a survival model accounting for a fraction of nonsusceptible individuals in the population. This study proposes a mixed cure model for time-to-event data, measured as sequential binary records. In a simulation study survival data were generated through 2 underlying traits: susceptibility and endurance (risk of dying per time-unit), associated with 2 sets of underlying liabilities. Despite considerable phenotypic confounding, the proposed model was largely able to distinguish the 2 traits. Furthermore, if selection is for improved susceptibility rather than endurance, the error of applying a classical survival model was nonnegligible. The difference was most pronounced for scenarios with substantial underlying genetic variation in endurance and when the 2 underlying traits were lowly genetically correlated. In the presence of nonsusceptible individuals, the method provides a novel and more accurate tool for utilization of time-to-event data, and has also been proven successful when applied to zero-inflated longitudinal binary data.