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Although most models for incomplete longitudinal data are formulated within the selection model framework, pattern-mixture models have gained considerable interest in recent years [R.J.A. Little, Pattern-mixture models for multivariate incomplete data, J. Am. Stat. Assoc. 88 (1993), pp. 125-134; R.J.A. Lrittle, A class of pattern-mixture models for normal incomplete data, Biometrika 81 (1994), pp. 471-483], since it is often argued that selection models, although many are identifiable, should be approached with caution, especially in the context of MNAR models [R.J. Glynn, N.M. Laird, and D.B. Rubin, Selection modeling versus mixture modeling with nonignorable nonresponse, in Drawing Inferences from Self-selected Samples, H. Wainer, ed., Springer-Verlag, New York, 1986, pp. 115-142]. In this paper, the focus is on several strategies to fit pattern-mixture models for non-monotone categorical outcomes. The issue of under-identification in pattern-mixture models is addressed through identifying restrictions. Attention will be given to the derivation of the marginal covariate effect in pattern-mixture models for non-monotone categorical data, which is less straightforward than in the case of linear models for continuous data. The techniques developed will be used to analyse data from a clinical study in psychiatry. Ivy Jansen and Geert Molenberghs gratefully acknowledge the support from Fonds Wetenschappelijk Onderzoek-Vlaanderen Research Project G.0002.98 'Sensitivity Analysis for Incomplete and Coarse Data' and from IAP research Network P6/03 of the Belgian Government (Belgian Science Policy).

Recently, a lot of concern has been raised about assumptions needed in order to fit statistical models to incomplete multivariate and longitudinal data. In response, research efforts are being devoted to the development of tools that assess the sensitivity of such models to often strong but always, at least in part, unverifiable assumptions. Many efforts have been devoted to longitudinal data, primarily in the selection model context, although some researchers have expressed interest in the pattern-mixture setting as well. A promising tool, proposed by Verbeke et al. (2001, Biometrics57, 43–50), is based on local influence (Cook, 1986, Journal of the Royal Statistical Society, Series B48, 133–169). These authors considered the Diggle and Kenward (1994, Applied Statistics43, 49–93) model, which is based on a selection model, integrating a linear mixed model for continuous outcomes with logistic regression for dropout. In this article, we show that a similar idea can be developed for multivariate and longitudinal binary data, subject to nonmonotone missingness. We focus on the model proposed by Baker, Rosenberger, and DerSimonian (1992, Statistics in Medicine11, 643–657). The original model is first extended to allow for (possibly continuous) covariates, whereafter a local influence strategy is developed to support the model-building process. The model is able to deal with nonmonotone missingness but has some limitations as well, stemming from the conditional nature of the model parameters. Some analytical insight is provided into the behavior of the local influence graphs. Récemment, beaucoup de soucis se sont manifestés à propos des hypothèses nécessaires pour ajuster des modèles statistiques à des données longitudinales multidimensionnelles incomplètes. En réponse, des recherches sont consacrés au développement d’outils évaluant la sensibilité de tels modèles à des hypothèses souvent fortes et toujours invérifiables, aumoins partiellement. Beaucoup d’efforts ont été consacrésaux données longitudinales principalement dans le contexte d’un modèle de sélection bien que quelques chercheurs aient montrée de l’intérêt pour le modèle de mélange de profils. Unoutil prometteur, proposé par Verbeke et coll.(2001), reposesur l’étude de l’influence locale. Ces auteurs considèrent le modèle de Diggle et Kenward(1994) qui repose sur un modèle de sélection et qui combine un modèle mixte linéaire pour descritères d’évaluation continus et une régression logistique pourles abandons de traitement. Dans ce papier, nous montrons qu’une idée similaire peut être développé pour des données binaires longitudinales multidimensionnelles, sujettes à un profil non-monotone de valeurs manquantes. Nous nous concentrons sur le modèle proposé par Baker, Rosenberger et Der Simonian(1992).Le modèle d’origine est d’abord étendu à la prise en compte de covariables, possiblement continues, puis une stratégie d’influence locale est développée à l’appui de la construction du modèle. Ce modèle peuttraiter un profil non-monotone de valeurs manquantes, mais présente certaines limitations liées à la nature conditionnelle des paramètres du modèle. Un apercu analytique est fourni pour le comportement des graphes d’influencelocale. We gratefully acknowledge support from FWO-VlaanderenResearch Project “Sensitivity Analysis for Incomplete andCoarse Data.” The fourth author gratefully acknowledgessupport from a research grant of Vlaams Instituut voor deBevordering van het Wetenschappelijk-Technologisch Onder-zoek in de Industrie. The fifth author wishes to thank theVlaamse Interuniversitaire Raad for granting support. We aregrateful for support from “Interuniversity Attraction PolesProgramme P5/24—Belgian State—Federal Office for Scien-tific, Technical and Cultural Affairs.”

Using standard missing data taxonomy, due to Rubin and co-workers, and simple algebraic derivations, it is argued that some simple but commonly used methods to handle incomplete longitudinal clinical trial data, such as complete case analyses and methods based on last observation carried forward, require restrictive assumptions and stand on a weaker theoretical foundation than likelihood-based methods developed under the missing at random (MAR) framework. Given the availability of flexible software for analyzing longitudinal sequences of unequal length, implementation of likelihood-based MAR analyses is not limited by computational considerations. While such analyses are valid under the comparatively weak assumption of MAR, the possibility of data missing not at random (MNAR) is difficult to rule out. It is argued, however, that MNAR analyses are, themselves, surrounded with problems and therefore, rather than ignoring MNAR analyses altogether or blindly shifting to them, their optimal place is within sensitivity analysis. The concepts developed here are illustrated using data from three clinical trials, where it is shown that the analysis method may have an impact on the conclusions of the study.