Jung Sin-Ho, Kim Kyung Mann
Department of Biostatistics and Bioinformatics, Duke University, Durham, North Carolina 27710, USA.
Stat Med. 2004 Mar 30;23(6):881-96. doi: 10.1002/sim.1653.
Due to the optional sampling effect in a sequential design, the maximum likelihood estimator (MLE) following sequential tests is generally biased. In a typical two-stage design employed in a phase II clinical trial in cancer drug screening, a fixed number of patients are enrolled initially. The trial may be terminated for lack of clinical efficacy of treatment if the observed number of treatment responses after the first stage is too small. Otherwise, an additional fixed number of patients are enrolled to accumulate additional information on efficacy as well as on safety. There have been numerous suggestions for design of such two-stage studies. Here we establish that under the two-stage design the sufficient statistic, i.e. stopping stage and the number of treatment responses, for the parameter of the binomial distribution is also complete. Then, based on the Rao-Blackwell theorem, we derive the uniformly minimum variance unbiased estimator (UMVUE) as the conditional expectation of an unbiased estimator, which in this case is simply the maximum likelihood estimator based only on the first stage data, given the complete sufficient statistic. Our results generalize to a multistage design. We will illustrate features of the UMVUE based on two-stage phase II clinical trial design examples and present results of numerical studies on the properties of the UMVUE in comparison to the usual MLE.
由于序贯设计中的可选抽样效应，序贯检验后的最大似然估计器（MLE）通常存在偏差。在癌症药物筛选的II期临床试验中采用的典型两阶段设计中，最初会招募固定数量的患者。如果在第一阶段观察到的治疗反应数量过少，试验可能会因治疗缺乏临床疗效而终止。否则，会再招募固定数量的患者，以积累关于疗效和安全性的更多信息。对于此类两阶段研究的设计有许多建议。在此我们证明，在两阶段设计下，二项分布参数的充分统计量，即停止阶段和治疗反应数量，也是完备的。然后，基于拉奥 - 布莱克威尔定理，我们将一致最小方差无偏估计器（UMVUE）推导为一个无偏估计器的条件期望，在这种情况下，它仅仅是在给定完备充分统计量时仅基于第一阶段数据的最大似然估计器。我们的结果推广到了多阶段设计。我们将基于两阶段II期临床试验设计示例说明UMVUE的特征，并给出与通常的MLE相比UMVUE性质的数值研究结果。
