In clinical or epidemiological follow‐up studies, methods based on time scale indicators such as the restricted mean survival time (RMST) have been developed to some extent. Compared with traditional hazard rate indicator system methods, the RMST is easier to interpret and does not require the proportional hazard assumption. To date, regression models based on the RMST are indirect or direct models of the RMST and baseline covariates. However, time‐dependent covariates are becoming increasingly common in follow‐up studies. Based on the inverse probability of censoring weighting (IPCW) method, we developed a regression model of the RMST and time‐dependent covariates. Through Monte Carlo simulation, we verified the estimation performance of the regression parameters of the proposed model. Compared with the time‐dependent Cox model and the fixed (baseline) covariate RMST model, the time‐dependent RMST model has a better prediction ability. Finally, an example of heart transplantation was used to verify the above conclusions.

In the field of clinical chronic diseases, common prediction results (such as survival rate) and effect size hazard ratio (HR) are relative indicators, resulting in more abstract information. However, clinicians and patients are more interested in simple and intuitive concepts of (survival) time, such as how long a patient may live or how much longer a patient in a treatment group will live. In addition, due to the long follow-up time, resulting in generation of longitudinal time-dependent covariate information, patients are interested in how long they will survive at each follow-up visit. In this study, based on a time scale indicator—restricted mean survival time (RMST)—we proposed a dynamic RMST prediction model by considering longitudinal time-dependent covariates and utilizing joint model techniques. The model can describe the change trajectory of longitudinal time-dependent covariates and predict the average survival times of patients at different time points (such as follow-up visits). Simulation studies through Monte Carlo cross-validation showed that the dynamic RMST prediction model was superior to the static RMST model. In addition, the dynamic RMST prediction model was applied to a primary biliary cirrhosis (PBC) population to dynamically predict the average survival times of the patients, and the average C-index of the internal validation of the model reached 0.81, which was better than that of the static RMST regression. Therefore, the proposed dynamic RMST prediction model has better performance in prediction and can provide a scientific basis for clinicians and patients to make clinical decisions.

Competing risks data are common in medical studies, and the sub-distribution hazard (SDH) ratio is considered an appropriate measure. However, because the limitations of hazard itself are not easy to interpret clinically and because the SDH ratio is valid only under the proportional SDH assumption, this article introduced an alternative index under competing risks, named restricted mean time lost (RMTL). Several test procedures were also constructed based on RMTL. First, we introduced the definition and estimation of RMTL based on Aalen-Johansen cumulative incidence functions. Then, we considered several combined tests based on the SDH and the RMTL difference (RMTLd). The statistical properties of the methods are evaluated using simulations and are applied to two examples. The Type I errors of combined tests are close to the nominal level. All combined tests show acceptable power in all situations. In conclusion, RMTL can meaningfully summarize treatment effects for clinical decision making, and three combined tests have robust power under various conditions, which can be considered for statistical inference in real data analysis.

The proportional hazards (PH) assumption is often violated in clinical trials. If the most commonly used Log-rank test is used for trial design in non-proportional hazard (NPH) cases, it will result in power loss or inflation, and the effect measures hazard ratio will become difficult to interpret. To circumvent the issue caused by the NPH for trial design and to make the effect measures easy to interpret and communicate, two simulation-free methods about restricted mean survival time group sequential (GS-RMST) design are introduced in this study: the independent increment GS-RMST (GS-RMSTi) design and the non-independent increment GS-RMST (GS-RMSTn) design. For the above two designs, the corresponding analytic expression of the variance-covariance matrix, the calculations of the stopping boundaries and sample size are given in the study. Simulation studies show that both designs can achieve the corresponding nominal type I error and nominal power. The GS-RMSTn simulation studies show that the Max-Combo test group sequential design is robust in different NPH scenarios and is suitable for discovering whether there is a treatment effect difference. However, it does not have a corresponding easy-to-interpret effect measure indicating effect difference magnitude. GS-RMST performs well in both PH and NPH scenarios, and it can obtain time-scale effect measures that are easy to understand by both physicians and patients. Examples of both GS-RMST designs are also illustrated.

The almost sure local central limit theorem is a general result which contains the almost sure global central limit theorem. Let $\{X_{k},k\geq 1\}${Xk,k≥1} be a strictly stationary negatively associated sequence of positive random variables. Under the regular conditions, we discuss an almost sure local central limit theorem for the product of some partial sums $(\prod_{i=1}^{k} S_{k,i}/((k-1)^{k}\mu^{k}))^{\mu/(\sigma\sqrt{k})}$(∏i=1kSk,i/((k−1)kμk))μ/(σk) , where $\mathbb{E}X_{1}=\mu$EX1=μ , $\sigma^{2}={\mathbb{E}(X_{1}-\mu)^{2}}+2\sum_{k=2}^{\infty}\mathbb{E}(X_{1}-\mu)(X_{k}-\mu)$σ2=E(X1−μ)2+2∑k=2∞E(X1−μ)(Xk−μ) , $S_{k,i}=\sum_{j=1}^{k}X_{j}-X_{i}$Sk,i=∑j=1kXj−Xi .