When planning an oncology clinical trial, the usual approach is to assume an exponential distribution for the time-to-event endpoints. Often, besides the gold-standard endpoint overall survival, progression-free survival is considered as a second confirmatory endpoint. We use a survival multistate model to jointly model these two endpoints and find that neither exponential distribution nor proportional hazards will typically hold for both endpoints simultaneously. The multistate model approach allows us to consider the joint distribution of the two endpoints and to derive quantities of interest as the correlation between overall survival and progression-free survival. In this paper, we use the multistate model framework to simulate clinical trials with endpoints OS and PFS and show how design planning questions can be answered using this approach. In addition to the major advantage that we can model non-proportional hazards quite naturally with this approach, the correlation between the two endpoints can be exploited to determine sample size and type-I-error. We consider an oncology trial on non-small-cell lung cancer as a motivating example from which we derive relevant trial design questions. We then illustrate how clinical trial design can be based on simulations from a multistate model. Key applications are co-primary endpoints and group-sequential designs in pivotal clinical trials. Simulations for these applications show that the standard simplifying approach often leads to underpowered or overpowered clinical trials. Our approach is quite general and can be extended to more complex trial designs, further endpoints, and other therapeutic areas.

In many studies, probabilities of non-standard endpoints are of interest which are more complex than overall survival. One such probability is chronic GvHD- and relapse-free survival, the probability of being alive after stem cell transplantation, not suffering from chronic graft-versus-host disease (GvHD) and not having had a relapse with chronic GvHD being a recurrent event. Because the probabilities for such non-standard endpoints with recurrent events may not fall monotonically but may also rise again, one should not use a simple Kaplan-Meier estimator for the estimation of these probabilities, but the Aalen-Johansen estimator. One concern with this estimator is its consistency when the Markov assumption is not fulfilled, but Nießl et al. (2021) showed that the Aalen-Johansen estimator is in fact consistent even in non-Markov scenarios, as long as state occupation probabilities are estimated and the censoring is random. In some multistate models, it is also possible to estimate probabilities for complex, non-standard endpoints using linear combinations of Kaplan-Meier estimators. For these linear Kaplan-Meier combinations, we propose a wild bootstrap procedure for inference and to obtain confidence bands with the aim to compare the results with confidence bands obtained using the wild bootstrap technique for the Aalen-Johansen estimator in non-Markov scenarios (Bluhmki et al., 2018). In the proposed wild bootstrap procedure for the linear combinations of Kaplan-Meier estimators, the limiting distribution of the Nelson-Aalen estimator is approximated using the wild bootstrap and transformed via the functional delta method. This approach gives the same results as those of Liu et al. (2008) and Zhang et al. (2022) and is easily adaptable to different models. An advantage of the wild bootstrap is that the censoring may also be event-dependent, but then the Markov assumption must be fulfilled in the case of the Aalen-Johansen estimator. Using real data, confidence bands are generated using the wild bootstrap approach for the chronic GvHD- and relapse-free survival, since they provide an easy interpretive approach for e.g. two group comparisons. In addition, the coverage probabilities of confidence intervals and confidence bands generated by Efron’s bootstrap and by the wild bootstrap are examined with simulations.

References:
(1) A. Nießl, A. Allignol, J. Beyersmann and C. Mueller (2021): Statistical inference for state occupation and transition probabilities in non-Markov multi-state models subject to both random left-truncation and right-censoring. Econometrics and Statistics.
(2) T. Bluhmki, C. Schmoor, D. Dobler, M. Pauly, J. Finke, M. Schumacher and J. Beyersmann (2018): A wild bootstrap approach for the Aalen–Johansen estimator. Biometrics, 74: 977-985.
(3) L. Liu, B. Logan and J.P. Klein (2008): Inference for current leukemia free survival. Lifetime data analysis, 14(4): 432–446.
(4) X. Zhang, S.R. Solomon and C. Sizemore (2022): Inferences for current chronic graft-versus-host-disease free and relapse free survival. BMC Medical Research Methodology, 22: 318.

In clinical studies, multiple correlated survival times arise naturally when a patient can experience multiple events over the course of a study. Quantities of interests include probabilities that patients experience certain events, or amount of time elapsed before certain events occur. In the estimand framework, often one event is of interests, and many others are considered inter-current. Data analysis in such setups requires simplifying assumptions. For example, when the survival time of interest is a sum of multiple survival times, the intermediate events are usually ignored, and the total survival time is modeled directly. Moreover, when interests change to a different event, separate models must be fitted by reversing the roles of the event of interest and the inter-current events.

One solution to this is to model such data with multi-state models (MSM). MSMs support the estimand framework by defining all the relevant events—including all possible intercurrent events—as states in an MSM. Complete patient trajectories can be visualized through these states. MSMs also allow us to compare different estimand strategies for addressing the inter-current events by identifying corresponding states and their transitions. As a result, MSMs extend traditional univariate time-to-event analyses methods.

Many approaches to analyzing MSMs have been proposed. Markov processes have gained popularity for their algebraic simplicity. However, the Markovian assumption implies exponentially distributed survival times, which usually is too restrictive. Alternatively, non-parametric methods based on counting processes are a common choice; yet numerical trackability becomes an issue when the number of states increases.

In this presentation we introduce semi-Markov processes (SMPs), a versatile tool for MSMs. They generalize Markov processes by allowing survival times between states to have nearly any distributions—parametric or non-parametric.

The theory of SMPs was fully developed by the 1960s. One important mathematical concept that enables the derivations of the quantities of interests in MSMs is Laplace transform (LT). In particular, the close relationship between LTs and the characteristic functions of distributions allows complex computations from the time domain, for instance, convolutions of multiple integrals of distributions of survival times, to be translated to simpler algebraic manipulations in the frequency domain, in this case, products of LTs. The LTs of quantities of interests are then inverted back to the time domain analytically or numerically.

The numerical inversion of LTs had been an obstacle for the use of SMPs in the past. With the advances of computational powers, the SMPs have gained recognition, and have been utilized for analyzing MSMs for a variety of applications. In addition, a new R package “smproc” further alleviates the computational burden of implementing SMPs for data analysts.

As an illustration, we present an exploratory data analysis where we use an SMP to characterize different stages of treatment responses. The data were collected in a recently completed pivotal phase III randomized study, testing a new compound against a standard of care. The primary read-out of the study demonstrated the superiority of the new compound, which has since been approved by many health authorities including the FDA and EMA.

When interest lies in the progression of a disease rather than on a single outcome, multistate Markov models represent a natural and powerful modelling approach. Often the nature of the phenomenon itself renders constant monitoring unfeasible, thus leading to the process being observed only intermittently. This setting is challenging and existing methods and their implementations do not yet provide flexible enough mechanisms for fully exploiting the information contained in the data. To this end, we propose a closed-form expression for the local curvature information of the transition probability matrix, which has not been previously attempted. Building on this, we introduce a general framework that allows one to model any type of multistate process where the transition intensities are flexibly specified as functions of regression splines. Parameter estimation is carried out through a carefully structured, stable penalised likelihood approach. We exemplify our method by modelling cognitive decline in the English Longitudinal Study of Ageing. To support applicability and reproducibility, all developed tools are implemented in the R package flexmsm.

In survival data analysis, the stratified Cox model becomes a popular option when the proportional hazards assumption of the conventional Cox model does not hold for certain covariates. When survival times include interval-censored observations, the method of maximum partial likelihood is not viable, and thus cannot be applied. We present a penalized likelihood method for estimating the model parameters, including the baseline hazards. Penalty functions are used to produce smoothed baseline hazards estimates, and also to relax the requirement on optimal number and location of the knots used in the baseline hazards estimates. We also explain a large sample normality result for the estimates, which can be used to make inferences on quantities of interest, such as survival probabilities, without relying on computing-intensive resampling methods.