Research Article: Randomization in survival studies: An evaluation method that takes into account selection and chronological bias

Date Published: June 3, 2019

Publisher: Public Library of Science

Author(s): Marcia Viviane Rückbeil, Ralf-Dieter Hilgers, Nicole Heussen, Vance W Berger.


The random allocation of patients to treatments is a crucial step in the design and conduct of a randomized controlled trial. For this purpose, a variety of randomization procedures is available. In the case of imperfect blinding, the extent to which a randomization procedure forces balanced group sizes throughout the allocation process affects the predictability of allocations. As a result, some randomization procedures perform superior with respect to selection bias, whereas others are less susceptible to chronological bias. The choice of a suitable randomization procedure therefore depends on the expected risk for selection and chronological bias within the particular study in question. To enable a sound comparison of different randomization procedures, we introduce a model for the combined effect of selection and chronological bias in randomized studies with a survival outcome. We present an evaluation method to quantify the influence of bias on the test decision of the log-rank test in a randomized parallel group trial with a survival outcome. The effect of selection and chronological bias and the dependence on the study setting are illustrated in a sensitivity analysis. We conclude with a case study to showcase the application of our model for comparing different randomization procedures in consideration of the expected type I error probability.

Partial Text

One of the main purposes of randomization is to improve comparability between treatment groups by balancing observed and unobserved covariates in expectation [1]. Randomization furthermore helps to mitigate the risk of selection bias and, depending on the randomization procedure, can protect against imbalanced group sizes throughout the allocation process [1]. Despite the many benefits of randomization, there are also some limitations; for a comprehensive discussion, see [2]: One issue that cannot be addressed by randomization is that patients usually enter a clinical trial sequentially and are often treated immediately [1, 2]. Consequently, new patients will be enrolled and assigned to therapies, while others have already received treatment [2]. This delay in time entails several potential sources of bias: On the one hand, the treatment success itself may be affected by unobserved time trends (chronological bias) [3, 4]. These may result from, for example, improved treatment performance due to experience gain [5, 6], or changes in inclusion or exclusion criteria [4, 7]. On the other hand, the sequential enrollment creates the risk for selection bias whenever blinding cannot be fully attained [8, 9].

We consider a randomized, two-arm, parallel group trial where a control (C) and an experimental (E) treatment are compared with regard to a survival outcome. With an intended 1:1 allocation ratio, a total of n patients are enrolled over an accrual period of length A ≥ 0. The maximum duration of follow-up is of length F, with F > A, so that all patients who have not yet had an event until then are regarded as right-censored. We assume that throughout the accrual period, patients enter the trial sequentially according to a uniform distribution. Furthermore, there is a random censoring mechanism that can be modeled by a probability distribution Scen which is independent of the survival distributions. Let d denote the number of observed events, where d ≤ n, with the distinct ordered event times τ1 < τ2 < … < τd. The following bias model is a generalization of the model described in [9] which also incorporates time trends and can be applied not only to exponentially distributed survival times. For a randomization sequence z, let hi(t, z) denote the hazard function of patient i at time t. We assume that the hazard function is affected by a biased selection of patients and an unobserved time trend such that hi(t,z)=hC(t)exp(ziln(HR)+ηi(z)+θi),(2) where ηi(z) and θi are functions that quantify the strength of selection and chronological bias. In contrast to the additive bias model for continuous outcomes [10], this newly defined bias model for survival outcomes has a multiplicative effect on the hazard functions. If the survival distributions of the patients are influenced by selection or chronological bias as introduced in our bias model from Eq (2), this also affects the distribution of the log-rank statistic. We derive an approximation formula to compute the rejection probabilities in the presence of bias depending on the randomization sequence. The dependence of the type I error probability on the sample size, the bias effects and the randomization procedure is showcased in a sensitivity analysis. We conclude with a case study to illustrate how our evaluation method can be applied to select a suitable randomization procedure on the basis of the expected type I error probability. All computations were performed on a computer with an Intel i7-4710MQ CPU Quad-core (2.5 GHz) and 16 GB RAM under a Windows 7 (64-bit) operating system. The code was written in R version 3.5.1. [33], using the randomizeR package version 2.0 [34]. Although it is widely acknowledged by regulatory authorities that well-conducted randomized controlled trials yield a higher level of evidence compared to observational studies [37–39], the importance attributed to randomization itself is usually limited to good implementation rather than a well conceived design [10]. This is reflected in the predominant use of permuted block randomization [40], which is usually chosen without a sound explanation [7, 23]. For this reason, we strive to strengthen awareness of the fact that the choice of randomization procedure affects the extent to which a study is susceptible to certain types of bias. This was confirmed both in our sensitivity analysis and in the considered case study. In order to enable scientists to select a randomization procedure that is suitable for their individual study, appropriate bias models and easy-to-use evaluation methods are needed. The presented results enable researchers in the planning phase of a survival study to make a scientifically sound choice of a randomization design. Due to the frequent use of the log-rank test and Cox’s proportional hazards model, our approach is applicable in most scenarios.   Source: