Sample size calculation for multicenter randomized trial: Taking the center effect into account
Introduction
A multicenter randomized trial offers increased recruitment rate and better external validity [1], [2]. In multicenter trials, patients from the same center share common characteristics: they may be more similar than those among centers, they benefit from a common way of dispensing health care and implementing the features of the protocol, and they are assessed in the same way (which is important for subjective outcomes) [1], [2], [3]. Observations from the same center are therefore more similar (i.e., more correlated) than those from different centers; this phenomenon defines the center effect. Since the hypothesis of independence of the data is questionable, one should consider the center effect at any statistical stage of a multicenter trial.
When analyzing data from a multicenter trial, the estimation of the main treatment effect must take into account the center differences [1], [4], [5]. This requirement is widely accepted, although no real consensus exists on the statistical model to use [6], [7], [8], [9], [10]. However, the selected method depends on the survey's specific question and repercussions. If conclusions apply across the participating centers of the trial only, or if the centers cannot be considered as a random sample from a population, the analysis of data will involve a fixed effects regression model. On the contrary, if one wants to extend the results to all the centers that could be concerned by the experimental treatment, the analysis of data will involve a mixed effects model [5], [8], [10].
When planning a trial, randomization is usually stratified on the center [11]. However, regarding sample size calculation, no adjustment is performed, and, presently, sample sizes are fixed the same for multicenter trials as for monocenter trials.
The objective of this work was to derive a sample size formula taking into account the center effect for a continuous outcome, in the framework of a mixed effects analysis. The next parts of this paper display the statistical model used and some estimates of the center effect from three multicenter trials. Then, we derive a sample size formula for multicenter trials, which robustness is assessed through a simulation study. Finally, a conclusion reports the main results of this work and discusses their limits.
Section snippets
Statistical model
Let us consider the following mixed effects model for the two-way layout without interaction [12], [13]:where Yijk denotes the response from the kth subject, receiving the ith treatment in the jth center. The overall response mean is denoted by μ. The treatment effects αi's are fixed, with α1 + α2 = 0. We assume that the centers are a random sample from a large population of centers, so the Bj's are considered to be random, distributed as N (0, σB2). The
ICC estimate: examples
We estimated the center effect [14] in three multicenter trials [15], [16], [17] for several outcomes used in these trials (Table 1). Results show that the ICC lies between 0 and 0.15, indicating that up to 15% of the observation's variance may be due to the variability between the centers. Since this intra-center correlation can be of great magnitude, it is important to take it into account at the planning stage of a multicenter trial.
Analytical issue
The effect size (ES) pre-specified when planning a trial is defined by the ratio of the expected difference between the two treatments' mean responses to the standard deviation of the outcome: , where Δ = |μ2 − μ1|. Treatment i's mean response is defined as μi = μ + αi and the variance σ2 refers to the whole variance, with the sum of the center effect's variance σB2 and the residual variance σε2.
It can be shown (see Appendix) that the required sample size for each treatment arm equals:
Simulation study
The sample size formula (3) was derived assuming an equal center size. To validate this formula, we carry out a simulation study with variable center sizes. We also assess the consequence of using a multivariate fixed effects model rather than a mixed effects one when analysing the data. Simulations are performed considering nominal type I and type II errors at 5% and 20% respectively.
Results
Table 2 displays power estimates when 20% of the centers recruit 80% of the subjects. Power slightly varies around 80%, whatever the number of centers and ICC and whatever the statistical model (mixed or fixed effects model). All these results remain valid for the three other center size distributions considered (data not shown). The ICC range (from 0.01 to 0.50) is intentionally large to cover a wide range of situations. However, we limited this range to the maximal value of 0.50 since a
Discussion
We derived a sample size formula taking into account the center effect. Thus, we obtained a new sample size formula correcting the classical one by a (1 − ρ) factor, where ρ is the ICC. Our formula remained valid even if there is an important imbalance among center sizes or in case a fixed effects model is used rather than a mixed effects model for the analysis.
The imbalance between the sizes of the centers must be avoided as much as possible since it is known that it may lead to biased
Acknowledgements
We are indebted to Drs Luc Sensebé and Xavier Mariette and to Pr Jean-Pierre Valat for permission to use data from their studies. We are also grateful to the two anonymous referees for their helpful and constructive comments.
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