Confidence intervals for the first crossing point of two hazard functions

Comparing two hazard curves when there is a treatment time-lag effect

In cancer and other medical studies, time-to-event (eg, death) data are common. One major task to analyze time-to-event (or survival) data is usually to compare two medical interventions (eg, a treatment and a control) regarding their effect on patients' hazard to have the event in concern. In such cases, we need to compare two hazard curves of the two related patient groups. In practice, a medical treatment often has a time-lag effect, that is, the treatment effect can only be observed after a time period since the treatment is applied. In such cases, the two hazard curves would be similar in an initial time period, and the traditional testing procedures, such as the log-rank test, would be ineffective in detecting the treatment effect because the similarity between the two hazard curves in the initial time period would attenuate the difference between the two hazard curves that is reflected in the related testing statistics. In this paper, we suggest a new method for comparing two hazard curves when there is a potential treatment time-lag effect based on a weighted log-rank test with a flexible weighting scheme. The new method is shown to be more effective than some representative existing methods in various cases when a treatment time-lag effect is present.

Non-parametric treatment time-lag effect estimation

In general, the change point problem considers inference of a change in distribution for a set of time-ordered observations. This has applications in a large variety of fields, and can also apply to survival data. In survival analysis, most existing methods compare two treatment groups for the entirety of the study period. Some treatments may take a length of time to show effects in subjects. This has been called the time-lag effect in the literature, and in cases where time-lag effect is considerable, such methods may not be appropriate to detect significant differences between two groups. In this paper, we propose a novel non-parametric approach for estimating the point of treatment time-lag effect by using an empirical divergence measure. Theoretical properties of the estimator are studied. The results from the simulated data and the applications to real data examples support our proposed method.

Evaluation of the treatment time-lag effect for survival data

Medical treatments often take a period of time to reveal their impact on subjects, which is the so-called time-lag effect in the literature. In the survival data analysis literature, most existing methods compare two treatments in the entire study period. In cases when there is a substantial time-lag effect, these methods would not be effective in detecting the difference between the two treatments, because the similarity between the treatments during the time-lag period would diminish their effectiveness. In this paper, we develop a novel modeling approach for estimating the time-lag period and for comparing the two treatments properly after the time-lag effect is accommodated. Theoretical arguments and numerical examples show that it is effective in practice.

A practical divergence measure for survival distributions that can be estimated from Kaplan-Meier curves

This paper introduces a new simple divergence measure between two survival distributions. For two groups of patients, the divergence measure between their associated survival distributions is based on the integral of the absolute difference in probabilities that a patient from one group dies at time t and a patient from the other group survives beyond time t and vice versa. In the case of non-crossing hazard functions, the divergence measure is closely linked to the Harrell concordance index, C, the Mann-Whitney test statistic and the area under a receiver operating characteristic curve. The measure can be used in a dynamic way where the divergence between two survival distributions from time zero up to time t is calculated enabling real-time monitoring of treatment differences. The divergence can be found for theoretical survival distributions or can be estimated non-parametrically from survival data using Kaplan-Meier estimates of the survivor functions. The estimator of the divergence is shown to be generally unbiased and approximately normally distributed. For the case of proportional hazards, the constituent parts of the divergence measure can be used to assess the proportional hazards assumption. The use of the divergence measure is illustrated on the survival of pancreatic cancer patients. Copyright © 2016 John Wiley & Sons, Ltd.

Statistical inference methods for two crossing survival curves: a comparison of methods

A common problem that is encountered in medical applications is the overall homogeneity of survival distributions when two survival curves cross each other. A survey demonstrated that under this condition, which was an obvious violation of the assumption of proportional hazard rates, the log-rank test was still used in 70% of studies. Several statistical methods have been proposed to solve this problem. However, in many applications, it is difficult to specify the types of survival differences and choose an appropriate method prior to analysis. Thus, we conducted an extensive series of Monte Carlo simulations to investigate the power and type I error rate of these procedures under various patterns of crossing survival curves with different censoring rates and distribution parameters. Our objective was to evaluate the strengths and weaknesses of tests in different situations and for various censoring rates and to recommend an appropriate test that will not fail for a wide range of applications. Simulation studies demonstrated that adaptive Neyman’s smooth tests and the two-stage procedure offer higher power and greater stability than other methods when the survival distributions cross at early, middle or late times. Even for proportional hazards, both methods maintain acceptable power compared with the log-rank test. In terms of the type I error rate, Renyi and Cramér—von Mises tests are relatively conservative, whereas the statistics of the Lin-Xu test exhibit apparent inflation as the censoring rate increases. Other tests produce results close to the nominal 0.05 level. In conclusion, adaptive Neyman’s smooth tests and the two-stage procedure are found to be the most stable and feasible approaches for a variety of situations and censoring rates. Therefore, they are applicable to a wider spectrum of alternatives compared with other tests.

Model selection and diagnostics for joint modeling of survival and longitudinal data with crossing hazard rate functions

Comparison of two hazard rate functions is important for evaluating treatment effect in studies concerning times to some important events. In practice, it may happen that the two hazard rate functions cross each other at one or more unknown time points, representing temporal changes of the treatment effect. Also, besides survival data, there could be longitudinal data available regarding some time-dependent covariates. When jointly modeling the survival and longitudinal data in such cases, model selection and model diagnostics are especially important to provide reliable statistical analysis of the data, which are lacking in the literature. In this paper, we discuss several criteria for assessing model fit that have been used for model selection and apply them to the joint modeling of survival and longitudinal data for comparing two crossing hazard rate functions. We also propose hypothesis testing and graphical methods for model diagnostics of the proposed joint modeling approach. Our proposed methods are illustrated by a simulation study and by a real-data example concerning two early breast cancer treatments.

The consequences of proportional hazards based model selection

For testing the efficacy of a treatment in a clinical trial with survival data, the Cox proportional hazards (PH) model is the well-accepted, conventional tool. When using this model, one typically proceeds by confirming that the required PH assumption holds true. If the PH assumption fails to hold, there are many options available, proposed as alternatives to the Cox PH model. An important question which arises is whether the potential bias introduced by this sequential model fitting procedure merits concern and, if so, what are effective mechanisms for correction. We investigate by means of simulation study and draw attention to the considerable drawbacks, with regard to power, of a simple resampling technique, the permutation adjustment, a natural recourse for addressing such challenges. We also consider a recently proposed two-stage testing strategy (2008) for ameliorating these effects.

Nonparametric confidence intervals for the ratio of marginal hazard rates of paired survival times

Paired survival times with potential censoring are often observed from two treatment groups in clinical trials and other types of clinical studies. The ratio of marginal hazard rates may be used to quantify the treatment effect in these studies. In this paper, a recently proposed nonparametric kernel method is used to estimate the marginal hazard rate, and the method of variance estimates recovery (MOVER) is used for the construction of the confidence intervals of a time-dependent hazard ratio based on the confidence limits of a single marginal hazard rate. Two methods are proposed: one uses the delta method and another adopts the transformation method to construct confidence limits for the marginal hazard rate. Simulations are performed to evaluate the performance of the proposed methods. Real data from two clinical trials are analyzed using the proposed methods.