On random sample size, ignorability, ancillarity, completeness, separability, and degeneracy: Sequential trials, random sample sizes, and missing data

Abstract

The vast majority of settings for which frequentist statistical properties are derived assume a fixed, a priori known sample size. Familiar properties then follow, such as, for example, the consistency, asymptotic normality, and efficiency of the sample average for the mean parameter, under a wide range of conditions. We are concerned here with the alternative situation in which the sample size is itself a random variable which may depend on the data being collected. Further the rule governing this may be deterministic or probabilistic. There are many important practical examples of such settings, including missing data, sequential trials, and informative cluster size. It is well known that special issues can arise when evaluating the properties of statistical procedures under such sampling schemes, and much has been written about specific areas3-4. Our aim is to place these various related examples into a single framework derived from the joint modeling of the outcomes and sampling process, and so derive generic results that in turn provide insight, and in some cases practical consequences, for different settings. It is shown that, even in the simplest case of estimating a mean, some of the results appear counter-intuitive. In many examples the sample average may exhibit small sample bias and, even when it is unbiased, may not be optimal. Indeed there may be no minimum variance unbiased estimator for the mean. Such results follow directly from key attributes such as non-ancilliarity of the sample size, and incompleteness of the minimal su cient statistic of the sample size and sample sum. Although our results have direct and obvious implications for estimation following group sequential trials, there are also ramifications for a range of other settings, such as random cluster sizes, censored time-to-event data, and the joint modeling of longitudinal and time-to-event data. Here we use the simplest sequential group sequential setting to develop and explicate the main results. Some implications for random sample sizes and missing data are also considered. Consequences for other related settings will be considered elsewhere.

Clusters with random size: maximum likelihood versus weighted estimation

“…We note that the theorems are implications rather than equivalences. As has been shown in the sequential trial context, a lack of completeness does not preclude the existence of estimators with very good properties (Molenberghs et al, 2014). Liu and Hall (1999) established the incompleteness of the sufficient statistic for a clinical trial with a stopping rule, for the case of normally distributed endpoints.…”

“…Liu et al (2006) generalized this result to the entire exponential family. Molenberghs et al (2014) and Milanzi et al (2016) broadened it further to a stochastic rather than a deterministic stopping rule, hence encompassing the case of a completely random sample size. Indeed, it would seem at first sight that this latter case is standard, because the sample size is unrelated to the data, whether observed or not.…”

“…A consequence of our approach is the need for applying weights when combining results from the K strata. We establish how results on incomplete sufficient statistics in the context of weighted averages (Molenberghs et al, 2014;Hermans et al, 2015) imply that there may be no optimal set of weights. Given this, we propose pragmatically attractive weights, in terms of efficiency, bias, and computational ease.…”

Estimation After a Group Sequential Trial

“…Liu and Hall (1999) and Liu et al (2006) explored this incompleteness in group sequential trials, for outcomes with both normal and one-parameter exponential family distributions. For these distributions, Molenberghs et al (2012) and Milanzi et al (2012) embedded the problem in the broader class with random sample size, which includes, in addition to sequential trials, incomplete data, completely random sample sizes, censored time-to-event data, and random cluster sizes. In so doing, they were able to link incompleteness to the related concepts of ancillarity and ignorability in the missing-data sense.…”

“…Much work has been devoted to the inferential consequences of this design feature. Molenberghs et al (2012) and Milanzi et al (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.…”