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The generalized linear mixed model is commonly used for the analysis of hierarchical non-Gaussian data. It combines an exponential family model formulation with normally distributed random effects. A drawback is the difficulty of deriving convenient marginal mean functions with straightforward parametric interpretations. Several solutions have been proposed, including the marginalized multilevel model (directly formulating the marginal mean, together with a hierarchical association structure) and the bridging approach (choosing the random-effects distribution such that marginal and hierarchical mean functions share functional forms). Another approach, useful in both a Bayesian and a maximum likelihood setting, is to choose a random-effects distribution that is conjugate to the outcome distribution. In this paper, we contrast the bridging and conjugate approaches. For binary outcomes, using characteristic functions and cumulant generating functions, it is shown that the bridge distribution is unique. Self-bridging is introduced as the situation in which the outcome and random-effects distributions are the same. It is shown that only the Gaussian and degenerate distributions have well-defined cumulant generating functions for which self-bridging holds. ; The authors gratefully acknowledge support from IAP research Network P7/06 of the Belgian Government (Belgian Science Policy).

"This book is written with three main aims; to provide a thorough introduction to the general MI methods, to provide a detailed discussion of the practical use of the MI method and to present real-world examples drawn from the field of biostatistics"--Provided by publisher

Over the last decade a variety of models to analyse incomplete multivariate and longitudinal data have been proposed, many of which allowing for the missingness to be not at random, in the sense that the unobserved measurements influence the process governing missingness, in addition to influences coming from observed measurements and/or covariates. The fundamental problems that are implied by such models, to which we refer as sensitivity to unverifiable modelling assumptions, has, in turn, sparked off various strands of research in what is now termed sensitivity analysis. The nature of sensitivity originates from the fact that a missingness not at random (MNAR) model is not fully verifiable from the data, rendering the empirical distinction between MNAR and missingness at random (MAR), where only covariates and observed outcomes influence missingness, difficult or even impossible, unless we are willing to accept the posited MNAR model in an unquestioning way. We show that the empirical distinction between MAR and MNAR is not possible, in the sense that each MNAR model fit to a set of observed data can be reproduced exactly by an MAR counterpart. Of course, such a pair of models will produce different predictions of the unobserved outcomes, given the observed outcomes. Theoretical considerations are supplemented with an illustration that is based on the Slovenian public opinion survey, which has been analysed before in the context of sensitivity analysis. ; Ivy Jansen and Geert Molenberghs gratefully acknowledge support from Fonds Wetenschappelijk Onderzoek-Vlaanderen research project G.0002.98 'Sensitivity analysis for incomplete and coarse data' and from Interuniversity Attraction Pole research network P6/03 of the Belgian Government (Belgian science policy).

Even though models for incomplete longitudinal data are in common use, they are surrounded with problems, largely due to the untestable nature of the assumptions one has to make regarding the missingness mechanism. Two extreme views on how to deal with this problem are (1) to avoid incomplete data altogether and (2) to construct ever more complicated joint models for the measurement and missingness processes. In this paper, it is argued that a more versatile approach is to embed the treatment of incomplete data within a sensitivity analysis. Several such sensitivity analysis routes are presented and applied to a case study, the milk protein trial analyzed before byDiggle and Kenward (1994). Apart from the use of local influence methods, some emphasis is put on pattern‐mixture modeling. In the latter case, it is shown how multiple‐imputation ideas can be used to define a practically feasible modeling strategy.

The primary analysis of time‐to‐event data typically makes the censoring at random assumption, that is, that—conditional on covariates in the model—the distribution of event times is the same, whether they are observed or unobserved. In such cases, we need to explore the robustness of inference to more pragmatic assumptions about patients post‐censoring insensitivity analyses. Reference‐based multiple imputation, which avoids analysts explicitly specifying the parameters of the unobserved data distribution, has proved attractive to researchers. Building on results for longitudinal continuous data, we show that inference using a Tobit regression imputation model for reference‐based sensitivity analysis with right censored log normal data isinformation anchored, meaning the proportion of information lost due to missing data under the primary analysis is held constant across the sensitivity analyses. We illustrate our theoretical results using simulation and a clinical trial case study.

In statistical practice, incomplete measurement sequences are the rule rather than the exception. Fortunately, in a large variety of settings, the stochastic mechanism governing the incompleteness can be ignored without hampering inferences about the measurement process. While ignorability only requires the relatively general missing at random assumption for likelihood and Bayesian inferences, this result cannot be invoked when non-likelihood methods are used. A direct consequence of this is that a popular non-likelihood-based method, such as generalized estimating equations, needs to be adapted towards a weighted version or doubly-robust version when a missing at random process operates. So far, no such modification has been devised for pseudo-likelihood based strategies. We propose a suite of corrections to the standard form of pseudo-likelihood to ensure its validity under missingness at random. Our corrections follow both single and double robustness ideas, and is relatively simple to apply. When missingness is in the form of dropout in longitudinal data or incomplete clusters, such a structure can be exploited toward further corrections. The proposed method is applied to data from a clinical trial in onychomycosis and a developmental toxicity study. ; The authors gratefully acknowledge support from IAP research Network P6/03 of the Belgian Government (Belgian Science Policy).

Completeness means that any measurable function of a sufficient statistic that has zero expectation for every value of the parameter indexing the parametric model class is the zero function almost everywhere. The property is satisfied in many simple situations in view of parameters of direct scientific interest, such as in regression models fitted to data from a random sample with fixed size. A random sample is not always of a fixed, a priori determined size. Examples include sequential sampling and stopping rules, missing data and clusters with random size. Often, there then is no complete sufficient statistic. A simple characterisation of incompleteness is given for the exponential family in terms of the mapping between the sufficient statistic and the parameter, based upon the implicit function theorem. Essentially, this is a comparison of the dimension of the sufficient statistic with the length of the parameter vector. This results in an easy verifiable criterion for incompleteness, clear and simple to use, even for complex settings as is shown for missing data and clusters of random size. This tutorial exemplifies the (in)completeness property of a sufficient statistic, thereby illustrating our proposed characterisation. The examples are organised from more classical, simple examples to gradually more advanced settings. ; The authors acknowledge support from IAP research Network P7/06 of the Belgian Government (Belgian Science Policy). The research leading to these results has also received funding from the European Seventh Framework programme FP7 2007-2013 under grant agreement no. 602552. We gratefully acknowledge support from the IWT-SBO ExaScience grant.

We consider a conceptual correspondence between the missing data setting, and joint modeling of longitudinal and time-to-event outcomes. Based on this, we formulate an extended shared random e ects joint model. Based on this, we provide a characterization of missing at random, which is in line with that in the missing data setting. The ideas are illustrated using data from a study on liver cirrhosis, contrasting the new framework with conventional joint models. ; We gratefully acknowledge support from IAP research Network P7/06 of the Belgian Government (Belgian Science Policy). We also thank the referees for their helpful comments.

Clustered count data are commonly analysed by the generalized linear mixed model (GLMM). Here, the correlation due to clustering and some overdispersion is captured by the inclusion of cluster-specific normally distributed random effects. Often, the model does not capture the variability completely. Therefore, the GLMM can be extended by including a set of gamma random effects. Routinely, the GLMM is fitted by maximizing the marginal likelihood. However, this process is computationally intensive. Although feasible with medium to large data, it can be too time-consuming or computationally intractable with very large data. Therefore, a fast two-stage estimator for correlated, overdispersed count data is proposed. It is rooted in the split-sample methodology. Based on a simulation study, it shows good statistical properties. Furthermore, it is computationally much faster than the full maximum likelihood estimator. The approach is illustrated using a large dataset belonging to a network of Belgian general practices. ; Financial support from the IAP research network P7 = 06 of the Belgian Government (Belgian Science Policy) is gratefully acknowledged. Alvaro J. Flórez acknowledges funding from the European Seventh Framework programme FP7 2007-2013 under grant agreement Nr. 602552.

Convergence problems often arise when complex linear mixed-effects models are fitted. Previous simulation studies (see, e.g. [Buyse M, Molenberghs G, Burzykowski T, Renard D, Geys H. The validation of surrogate endpoints in meta-analyses of randomized experiments. Biostatistics. 2000;1:49–67, Renard D, Geys H, Molenberghs G, Burzykowski T, Buyse M. Validation of surrogate endpoints in multiple randomized clinical trials with discrete outcomes. Biom J. 2002;44:921–935]) have shown that model convergence rates were higher (i) when the number of available clusters in the data increased, and (ii) when the size of the between-cluster variability increased (relative to the size of the residual variability). The aim of the present simulation study is to further extend these findings by examining the effect of an additional factor that is hypothesized to affect model convergence, i.e. imbalance in cluster size. The results showed that divergence rates were substantially higher for data sets with unbalanced cluster sizes – in particular when the model at hand had a complex hierarchical structure. Furthermore, the use of multiple imputation to restore 'balance' in unbalanced data sets reduces model convergence problems. ; Financial support from the IAP research network #P7/06 of the Belgian Government (Belgian Science Policy) is gratefully acknowledged. Wim Van der Elst acknowledges funding from the European Seventh Framework programme FP7 2007 − 2013 under grant agreement Nr. 602552. Geert Molenberghs acknowledges funding from Intel, Janssen Pharmaceutica and by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT).

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 areas (Grambsch P. Sequential sampling based on the observed Fisher information to guarantee the accuracy of the maximum likelihood estimator. Ann Stat 1983; 11: 68–77; Barndorff-Nielsen O and Cox DR. The effect of sampling rules on likelihood statistics. Int Stat Rev 1984; 52: 309–326). 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 counterintuitive. 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-ancillarity of the sample size and incompleteness of the minimal sufficient 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 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. ; IAP research Network of the Belgian Government (Belgian Science Policy) - NIH

We combine conjugate and normal random effects in a joint model for outcomes, at least one of which is non-Gaussian, with particular emphasis on cases in which one of the outcomes is of survival type. Conjugate random effects are used to relax the often-restrictive mean-variance prescription in the non-Gaussian outcome, while normal random effects account for not only the correlation induced by repeated measurements from the same subject but also the association between the different outcomes. Using a case study in chronic heart failure, we show that model fit can be improved, even resulting in impact on significance tests, by switching to our extended framework. By first taking advantage of the ease of analytical integration over conjugate random effects, we easily estimate our framework, by maximum likelihood, in standard software. ; The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: We gratefully acknowledge support from IAP research Network P6/03 of the Belgian Government (Belgian Science Policy).