The Change Point of the Survival Curves
A change point is the location where the distribution abruptly changes in a data sequence [9].Due to the mechanism of the IO anti-cancer drugs, the delayed separation of Kaplan-Meier survival curves is usually observed, sometimes the curves are converged before the change point (Figure1A), sometimes the curves are very close (Figure1B) before the change point. When the delayed separation of survival curve is present, it violates the fundamental study design assumption of the proportional hazards, and also results in a potential loss of statistical power to demonstrate the difference between two treatment groups because of the long invalid period before the separation change point.
Landmark analysis
With the exponential distribution assumption, the survival function is defined as:
where λ0, group and tg are baseline hazard, treatment group and the time point indicator, tCP is the time of change point. The hazard ratio before change point and after change point under the exponential distribution in Figure 1A should be estimated separately as:
For the landmark analysis, only the subjects survived in the period tg>tCP would be involved in the analysis. We recommend to pre-define the change point as an objective choice of landmark time of landmark analysis for the IO trials with delayed effect to avoid additional biases caused by the choice of landmark. We called this method change point landmark analysis (CPLA) in our research, and then change point was named as change point landmark in CPLA.
Simulation studies
Two simulation studies were conducted in our research.
Power and type I error evaluations
The first simulation study was performed to evaluate test power (scenarios A and B) and type I error (scenario C) of the change point landmark analysis. We conducted Monte Carlo simulations for landmark analyses and traditional full data Cox regression using delayed effect type survival data with change point at CP=2.5 month, three landmarks 1, 2.5 and 3.5months were considered, one was before the true change point, one was after the true change point. Equal sample sizes (N1=N2=20,30,40) simulated data were generated. A simulation of time-to-event data was performed based on a randomly censored model [10, 11], we generated individual lifetime X following the survival functions in Table1. To test the power of landmark analyses, the scenarios A were the simulated situations for the survival curves as Figure 1A, and the scenarios B were for Figure 1B. Different median survival times, different hazard ratios in the before and after change point periods were also considered in the power tests. Additionally, the scenario C was two overlapping survival curves to test type I errors. The censoring time S in two samples was generated from uniform distributions U (0, a) and U (0, b), where varying the values of a and b may result in censoring rates of approximately 10%, 20%, or 30% in the two samples. Because the lifetime X followed different distributions in each group, it was necessary for the values of a and b to be unequal to keep the average censoring rates in each group approximately equal to the given censoring rates. Each individual was assigned an observed survival time T=min(X, S) and an event indicator Δ=I [X≤S]. The exact power and size of test statistics were estimated by determining the proportion of samples for which the null hypothesis was rejected at the α = 0.05 significance level, based on 1000 simulations.
Simulated clinical trial
We performed simulations of trial designs to evaluate the advantages of change point recognition on the landmark analysis. Our initial simulations use the similar scenario of Chen[4] and Korn[12]: 680 patients are accrued and randomly assigned 1:1 over 34 months and final analysis occurred at 48 months after the first patient was randomly assigned. If there is no delay in treatment effect, this design has 90% power to detect a hazard ratio (HR) of 0.75 (using a two-sided 0.05-level log-rank test) with the 512 events in the final analysis, and there is no loss to follow-up. As is well known, the sample size and study duration are always fixed according to the sponsor’s budget during the study design, then we fixed the total duration as 48 months accordingly. Because it is also hard to know the actual treatment effect delay length in advance, to assess the impact of the delayed clinical effect, we considered the 3 scenarios which covered 1/12, 1/8, 1/6 delay of the total study duration, then the change points were at the 4th month, the 6th months and the 8th months (Figure 2), where the hazard ratio was 1.0 before the change points and 0.75 thereafter. As the whole study duration was pre-defined as 48 months, the observed events were decreased for the impact of the delayed effect. No interim analysis was considered in the simulated trials. Each scenario was evaluated based on 1000 simulations, the power and mean of the conditional HRs with 95% confidence interval (CI) were calculated for change point landmark, other landmarks and full data Cox regression.