The impact of the violation of the proportional hazards assumption on confirmatory analysis of survival data using delayed entry
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In many clinical trials with staggered entry, interim analyses are conducted for monitoring purposes. If some unexpected results are found during the interim analyses, performing confirmatory analyses may be desirable. However, a completely new study with different patients in different locations requires a lot of additional resources such as time and money. Furthermore, there may be ethical concerns regarding recruitment of patients for a new study if a harmful treatment effect was found in a previous study. In this situation, the result illustrated in Keiding et al. (1987) may be helpful. It suggests use of recurrence-free survivors among the initial cohort as delayed entry when performing confirmatory analyses. Therefore, we can increase the sample size and power of confirmatory analyses by combining recurrence-free survivors with patients enrolled who have not contributed to the interim analyses. The result from Keiding et al. (1987) requires several assumptions but, our study focuses on the proportional hazards assumption. In simulations the validity of the proportional hazards assumption depends on the way we generate the datasets and the models we fit. We simulate two binary covariates, Treatment and Status: one indicating either the treatment group or the control group and the other indicating whether patients developed their disease a long or short time before enrollment. Patients who have had their disease for a long time are called “Early patients” because, in our simulations, we assume that they are enrolled during the earlier part of hypothetical studies. Patients who developed the disease shortly before enrollment are called “Late patients” because we assume that they are enrolled during the later part of the hypothetical studies. Therefore, there are four types of patients, early patients in either the treatment or the control group, and late patients in either the treatment or the control group. We use an interaction effect between the two factors and a change in patient mix (Early and Late patients) over time to create a time dependent treatment effect in a model that does not account for the interaction effect. We focus on investigating the power of confirmatory analyses that follow an interim anal- ysis which detected a significant harmful effect. We compare the power of confirmatory analyses that only use participants enrolled after the interim analysis to the power of con- firmatory analyses that use participants enrolled after the interim analysis combined with recurrence-free survivors from the initial cohort. For recurrence-free survivors only the time of observation after the interim analysis is used and it is used as delayed entry. We fit three models: (1) a model that includes only a treatment effect, (2) a model that includes a treatment effect and a status effect and (3) a model that includes a treatment effect, a status effect and an interaction effect. The likelihood ratio test with one-degree of freedom is performed to test for the treatment effect in (1) and (2) and the likelihood ratio test with two degrees of freedom (main and interaction effect) is performed to test for a treatment effect in (3). For each of these three models we ensure (separately) that the model has a significant treatment effect at the interim analysis. We choose time frames of enrollment and follow-up and sample sizes that are similar to the setting of Keiding et al. (1987). We use exponentially distributed hazard functions for each of the four patient groups. For each of the three models, the treatment effect is estimated using four different analysis cohorts: (i) the initial cohort prior to the interim analysis only, (ii) the cohort that does not contribute to the interim analysis (only), (iii) the cohort that does not contribute to the interim analysis combined with recurrence-free survivors but each contributing only up to one year of data, (iv) the latter, but contributing up to four years of data. For models 1 and 2, the power for the confirmatory analysis that uses the recurrence-free survivors can be higher, about the same or lower than the power of the analysis that does not use them depending on the strength of the interaction effect. The use of recurrence-free survivors can typically not overcome issues of fitting an incorrect model. In contrast, when fitting model 3, the power of the analysis that includes the recurrence-free survivors is always higher than the power of the analysis not using the recurrence-free survivors regardless of the size of the interaction effect and the treatment effect. As expected, in simulations where the proportional hazards assumption holds (either because there is no time trend or the time trend is included in the model) using recurrence- free survivors as delayed entry always improves power regardless of the size of the treatment and interaction effect.
- Biostatistics