Milanzi Elasma, Molenberghs Geert, Alonso Ariel, Kenward Michael G, Tsiatis Anastasios A, Davidian Marie, Verbeke Geert
I-BioStat, Universiteit Hasselt, B-3590 Diepenbeek, Belgium.
I-BioStat, Universiteit Hasselt, B-3590 Diepenbeek, Belgium ; I-BioStat, Katholieke Universiteit Leuven, B-3000 Leuven, Belgium.
Stat Biosci. 2015 Oct;7(2):187-205. doi: 10.1007/s12561-014-9112-6. Epub 2014 Feb 22.
Group sequential trials are one important instance of studies for which the sample size is not fixed but rather takes one of a finite set of pre-specified values, dependent on the observed data. Much work has been devoted to the inferential consequences of this design feature. Molenberghs (2012) and Milanzi (2012) reviewed and extended the existing literature, focusing on a collection of seemingly disparate, but related, settings, namely completely random sample sizes, group sequential studies with deterministic and random stopping rules, incomplete data, and random cluster sizes. They showed that the ordinary sample average is a viable option for estimation following a group sequential trial, for a wide class of stopping rules and for random outcomes with a distribution in the exponential family. Their results are somewhat surprising in the sense that the sample average is not optimal, and further, there does not exist an optimal, or even, unbiased linear estimator. However, the sample average is asymptotically unbiased, both conditionally upon the observed sample size as well as marginalized over it. By exploiting ignorability they showed that the sample average is the conventional maximum likelihood estimator. They also showed that a conditional maximum likelihood estimator is finite sample unbiased, but is less efficient than the sample average and has the larger mean squared error. Asymptotically, the sample average and the conditional maximum likelihood estimator are equivalent. This previous work is restricted, however, to the situation in which the the random sample size can take only two values, = or = 2. In this paper, we consider the more practically useful setting of sample sizes in a the finite set {, , …, }. It is shown that the sample average is then a justifiable estimator , in the sense that it follows from joint likelihood estimation, and it is consistent and asymptotically unbiased. We also show why simulations can give the false impression of bias in the sample average when considered conditional upon the sample size. The consequence is that no corrections need to be made to estimators following sequential trials. When small-sample bias is of concern, the conditional likelihood estimator provides a relatively straightforward modification to the sample average. Finally, it is shown that classical likelihood-based standard errors and confidence intervals can be applied, obviating the need for technical corrections.
序贯试验是样本量不固定而是取一组有限的预先指定值之一的研究的一个重要实例，该值取决于观测数据。大量工作致力于此设计特征的推断结果。莫伦伯格斯（2012年）和米兰齐（2012年）回顾并扩展了现有文献，重点关注一系列看似不同但相关的情形，即完全随机样本量、具有确定性和随机停止规则的序贯研究、不完全数据以及随机聚类大小。他们表明，对于广泛的停止规则和指数族分布的随机结果，普通样本均值是序贯试验后估计的一个可行选项。他们的结果在某种意义上有些令人惊讶，因为样本均值不是最优的，而且进一步说，不存在最优的甚至无偏的线性估计量。然而，样本均值在给定观测样本量的条件下以及对其进行边际化时都是渐近无偏的。通过利用可忽略性，他们表明样本均值是传统的最大似然估计量。他们还表明，条件最大似然估计量在有限样本中是无偏的，但比样本均值效率低且具有更大的均方误差。渐近地，样本均值和条件最大似然估计量是等价的。然而，此前的工作仅限于随机样本量只能取两个值，即(n = n_1)或(n = n_2)的情况。在本文中，我们考虑样本量在有限集({n_1, n_2, \ldots, n_k})中的更具实际用途的情形。结果表明，样本均值在联合似然估计的意义上是一个合理的估计量，并且它是一致的且渐近无偏的。我们还展示了为什么在根据样本量进行条件模拟时，模拟会给样本均值存在偏差的错误印象。其结果是，对于序贯试验后的估计量无需进行修正。当关注小样本偏差时，条件似然估计量为样本均值提供了一个相对直接的修正。最后，结果表明可以应用基于经典似然的标准误差和置信区间，从而无需进行技术修正。
