Survival analysis and reliability analysis are the fields of statistics that deal with data related to the time to a specific event for an individual, such as the patient's survival time or the failure time of devices. This paper reviews the historical background and statistical properties of failure time models used in survival and reliability analyses. The paper is a tutorial, tailored for beginners. Basic survival analysis tools and concepts such as survival time, survival functions and hazard functions are defined, and their statistical properties and interpretations are explained. The general parametric models, such as the exponential, Weibull, and lognormal distributions, as well as regression models, such as the Cox proportional hazard model and accelerated failure time model, are also explained. Competing risk models are also briefly discussed. The data used in this paper, the derivation of the equations, and the R code for the data analysis are given in Appendix.

High-dimensional data with many and many covariates are available because of drastic progress in data-collecting technology. Hence statistical analysis of such high-dimensional data has been a very important issue for many years. We can find typical methods such as the Lasso even in textbooks for undergraduate students. In additon, in some areas we usually use ultra-high dimensional data such that p~exp(n^c), where p is the number of covariates, n is the sample size, c is a constant. In this paper, we review important studies on survival time data with (ultra-)high dimensional covariates from a perspective of my own studies on high-dimensional data. We focus on the Cox regression model which is one of the most important models for survival analysis. In addition, we refer to related topics.

Assuming the event hazard function to be of a piecewise constant form, survival analysis can be performed by Poisson regression. While this approach is less common compared to, for example, Cox proportional hazard regression, it offers flexible parametric modeling of the baseline hazard, and extension to models containing multiple time-dependent covariates and random effects can be conducted under the framework of generalized linear/nonlinear models. Due to these advantages, long-term follow-up analysis of a cohort study often relies on Poisson regression. The aim of this article is to formulate Poisson survival regression analysis and investigate its estimation properties by simulation. In particular, focus is on the impact of stratification of the main time-scale factor, the estimation performance under various situations including time-dependent covariates or random effects, and comparison with other methods such as Cox regression. A real data application to a large-scale cohort study is shown, which is followed by discussion about situations where Poisson survival regression might be most effective and future extensions.

In survival analysis, there exist various studies and applications that deal with a single event-time. On the other hand, although there are many potential needs to analyze multiple event-times simultaneously, there are not so many applications yet. We discuss copula modelling which is a useful approach to address multiple event-times in survival analysis. Focusing mainly on applications in biostatistics, we describe the main issues behind the copula, the fundamentals of copulas, and the history of their development. We discuss the usefulness of copula as an approach to solving problems in this area, such as addressing dependent censoring under semi-competing risk, in which we propose a new computational method on the estimation of survival function for a non-fatal event. As illustrative examples which deal with two event-times, we provide applications to bivariate Cox regression analysis and study design.

The dependence among multiple components (subsystems and components) that construct large-scale and complex systems should be considered for reliability analysis of the system. In this article, we introduce a method of reliability analysis by using multivariate Farlie-Gumbel-Morgenstern（FGM）copulas. We first explain the properties of the FGM copula. Next, to assess statistical dependence among components, we provide two likelihood-based estimation methods, the method of simultaneous estimation and the method of inference functions for margins, for parameters of the FGM copula. In particular, we derive standard errors and confidence intervals of estimators and investigate their properties. Finally, we demonstrate reliability analysis for an actual dataset of the reliability of ball bearings.

In survival analysis, dealing with truncated or competing risks data was a classical but, an important topic. Recently, Kundu et al. (2017) proposed a new statistical method for analyzing left-truncated and right-censored competing risks data, which require an assumption that the competing risks are independent of each other. However, the assumption may not hold in many applied studies. Therefore, I studied performances of the proposed method when it was applied to dependent competing risks data. Furthermore, I considered a new approach for analyzing such a dependent data.

Survival analysis plays a major role in the research of clinical oncology. Standard survival analysis, where only one event is analyzed, is still used in the majority of studies. By using competing risks and multistate models, which are the extensions of standard survival analysis, more useful information can be derived from the data and more clinical questions can be tackled. In this article, we provide an introductory explanation for the theory and application of multistate models for clinicians and biostatisticians. We explain how to use R in multistate models with the dataset ‘mgus2’ from the survival package. Key points for the application of multistate models are presented using the data from a previously published multicenter prospective observational study on palliative radiotherapy for gastric cancer. We would emphasize that standard survival analysis, competing risks analysis, and multistate models are analyzed in the same framework.

In marketing, interpurchase-timing models have been studied to analyze consumers' purchase timings using survival analysis. In this study, we propose a competing risk model considering dynamic frailty and apply it to the purchase behaviors on multi-channels. We consider interpurchase-timings in both EC sites and real stores using a competing risk model. Furthermore, in the competing risk model for interpurchase-timing, we propose a model that captures dynamic changes of unobserved heterogeneity using a state space model and consumers' heterogeneity using a hierarchical Bayesian model. In addition, we construct a joint model that simultaneously deals with purchase amounts in addition to interpurchase-timings. The proposed model is applied to single-source data recording purchase behavior in both EC sites and real stores. The results show that the proposed model performs better than models that deal with only a single channel or models that do not consider dynamic changes of consumers' heterogeneity. Furthermore, differences in the effects of marketing variables between channels were also revealed, indicating the usefulness of the proposed model.

The Mann-Whitney test is a nonparametric test for detecting the difference of two groups, which can be estimated by right-censored survival data. However, the traditional test based on Efron's estimator is invalid when the independent censoring assumption fails to hold. Recently, researchers discuss “dependent censoring”, where the independence assumption is violated. In this article, we review a method for studying the asymptotic bias of Efron's estimator under a copula-based dependent censoring model. We also review an asymptotically unbiased (consistent) estimator of the Mann-Whitney effect, which adopts the copula-graphic estimator to adjust for the effect of dependent censoring. This leads to a valid two-sample test when the structure of dependent censoring is correctly specified by a copula. We also derive the asymptotic distribution of the copula-based estimator under possible misspecification on an assumed copula. The method is illustrated by analyzing a real dataset. We provide the R code to reproduce the data analysis results.

The Net Benefit and the Win Ratio are the effect measures used in clinical trials with survival endpoints such as oncology and cardiology. These measures are based on generalized pairwise comparisons. In a randomized trial, the statistical methods estimate the ‘win’ probability that a subject randomly selected from the new treatment group has a better outcome than a subject from the control group and the ‘loss’ probability in the opposite situation. The Net Benefit is the difference between these probabilities, and the Win Ratio is the ratio between them. In this study, we review several estimators of the win/loss probability in terms of generalized pairwise comparisons, and the variance of those estimators based on U-statistic theory. Especially, we explain how to deal with the censored data to estimate those probabilities. Finally, we illustrate using the actual data to implement those statistical methods by R-packages.

Log-rank test is one of typical nonparametric test procedures in the two-sample problem for time-to-event data and is currently widely used in various fields. However, since the significance probability from log-rank test is usually computed using asymptotic chi-square approximation, it has been pointed out that the approximation precision is poor in situations where there is a large bias in the small sample size or imbalanced group allocation. In this paper, we propose a new test procedure that works well in such situations using computational algebraic statistics. The main feature of our algorithm is to estimate p-value by Markov Chain Monte Carlo method using a Markov basis on contingency tables. In order to evaluate the usefulness of the proposed method, simulation comparisons are also performed and the results show that the proposed method maintains an appropriate level of significance even in the settings where the asymptotic approximation becomes poor.

A decision tree is a statistical model constructed by recursively partitioning samples into several groups. A decision tree based on survival data (survival tree) can classify patients into different risk groups that are useful to predict patient prognosis. To test the significance of partitions, survival analysis methods are used, such as the log-rank test. However, the log-rank test may be unstable for small samples, and hence, the significance of partitions could be difficult to interpret. Furthermore, the R package for a decision tree, rpart, may overcorrect the significance for multiple testing under high-dimensional covariates. In this article, we introduce a method that alleviates these problems by the “stabilized score test” for constructing a survival tree. The proposed method also yields a simple tuning method by the P-value of the test. We illustrate the proposed method using a lung cancer dataset. The proposed method can be implemented by the R package “uni.survival.tree”. The R code for the data analysis is given in Appendix.