In survival analysis, modeling the hazard function, such as the Cox proportional hazards model, has been widely applied. However, there are some problems discussed in recent years. For example, the proportional hazards assumption is violated in some situations such as immunotherapies for cancer treatments. The other problem is the causal interpretability of hazard ratios discussed in the causal inference area. Therefore, as an alternative to modeling via hazard function, more flexible and interpretable methods for survival analysis are required. Pseudo-observations in survival analysis can allow for clinically interpretable analysis methods. It is applicable under a general framework on a generalized linear model if it is possible to compute pseudo-observations for the target of interest. In this paper, we explain the fundamental concept of pseudo-observations in survival analysis and the asymptotic theory for the estimator of regression coefficients while providing an overview of the related discussions.

It has been accepted worldwide that the cost as well as the effectiveness and toxicity is an essential part in the assessment of medical interventions. Following several countries, in Japan, the evaluation of cost-effectiveness was put into effect in 2019 . There are mainly two methods in cost-effective analysis: cost-effectiveness analysis alongside clinical trials and analysis using decision analytic modeling. Cost-effectiveness analysis alongside clinical trials, where utility and cost data are obtained from individual patients, has the advantage of high internal validity. In economic evaluation alongside trials, censoring frequently occurs in acquiring utility and cost data from individual patients and techniques of survival analysis is necessary to consider censoring into account. Methods using Kaplan-Meier survival estimator and inverse-probability weighting estimator are frequently used to account for censoring in the estimation of effectiveness and cost. In this paper, we provide an outline of the statistical methods on economic evaluation alongside trials, stressing especially the estimation of effectiveness and cost using survival analysis techniques to account for censoring

Dynamic prediction estimates the survival probability, conditioned on a subject not yet experiencing the event of interest at a specific time point. To improve the accuracy of dynamic prediction, one can incorporate baseline measurements and biomarkers measured during a follow-up. In this review article, we focus on two predominant approaches: joint modeling and landmarking. Joint modeling specifies the joint distribution of the biomarkers and the event times. Landmarking originally utilizes only the biomarkers at the onset of the start time point of prediction, but recent versions have begun to incorporate data observed after the prediction time point. We illustrate how the two approaches predict the survival probability and subsequently demonstrate the application of these methods through a primary biliary cholangitis dataset with the R codes provided.

A frailty model is a random effects model for event time variables, where the random effect acts on hazards. There are two main purposes for considering frailty models. The first purpose is to consider hazards that cannot be explained by observed covariates. Thus, unexplained individual differences are expressed as random effects (frailty). This allows us to clearly distinguish hazard behavior at the population level and the individual level. The other purpose is the construction of a multivariate survival distribution by assuming a common (shared) frailty within a cluster such as a medical center or a family. This shared frailty model can effectively account for intra-cluster dependence among event times. This paper provides an overview of the theory and practice of frailty models, especially univariate frailty models and shared frailty models, and discusses the problems to be addressed by frailty models.

Many clinical research studies evaluate a time-to-event outcome, illustrate survival functions and compare them between groups. Because actual relationships between survival curves are various, the use of conventional methods such as the log-rank test and a test of hazard ratio by Cox proportional hazards model are potentially underpowered compared with the planned power under the proportional hazards assumption, especially if a proportional hazards assumption is invalid. Therefore, when it is difficult to expect the proportional hazards assumption in the study planning phase, it is considered to use an alternative more efficient test method instead of the log-rank test or an alternative summary measure instead of the hazard ratio from statistical point of view. In this paper, we focus on the sample size determination and the information fraction, which are required for planning a study design and monitoring accrual progress, for each of the weighted log-rank test with Fleming– Harrington class of weights and a statistical significance test of the difference in the restricted mean survival time between groups as the summary measure of treatment effect. Also, they are illustrated by an application example.

When subjects may fail from several distinct causes but the occurrence of failure from one precludes the others, it is referred competing risks. While both event of interest and competing event types are unfavorable in many typical settings, there are some cases in which a favorable event and unfavorable event types are competing. We extend an application of competing risks model to a case in IgA nephropathy, in which two types of events were negatively created but not satisfy the condition of competing risks. Each event type does not necessarily preclude the other type of event and the exact order of time to events was often unclear due to irregular censoring intervals. We discuss interpretation of the result and generalization of our extension.

To elucidate the association between outcomes and covariates in long-term follow-up data, a time-to-event analysis (survival analysis) that incorporates time-dependent covariates (longitudinal data) is required. However, in real data applications, Cox models with time-dependent covariates are almost always forced to deal with incomplete data due to the missing of time-dependent covariates necessary for parameter estimation. The purpose of this article is to highlight this data missing problem and to introduce some relatively reasonable and convenient ways to address it. Multiple imputation method provides a persuasive framework in which uncertainty of imputed values for the missing time-dependent covariates is considered in a straightforward manner. We propose the usage of Gaussian process regression technique in the imputation step. This enables a flexible fitting to various longitudinal data easily and is compatible with the multiple imputation framework. In addition, Gaussian process regression can readily handle irregularly observed multiple time series data.