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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.”

1. Worldwide, the floristic composition of temperate forests bears the imprint of past land use for decades to centuries as forests regrow on agricultural land. Many species, however, display signi ...

Much research has been devoted to modelling strategies for longitudinal data with missingness, recently especially within the missingness not at random context. In this paper, the relatively unexplored but practically highly relevant domain of non-monotone missingness with multivariate ordinal responses is broached. For this, a dedicated version of the multivariate Dale model is formulated. Furthermore, we also assess the sensitivity of these models to their assumptions, by using the technique of global influence. 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 Belgian Interuniversity Attraction Pole network ‘Statistical techniquesand modeling for complex substantive questions with complex data.

The process of monoisotopic mass determination, i.e., nomination of the correct peak of an isotopically resolved group of peptide peaks as a monoisotopic peak, requires prior information about the isotopic distribution of the peptide. This points immediately to the difficulty of monoisotopic mass determination, whereas a single mass spectrum does not contain information about the atomic composition of a peptide and therefore the isotopic distribution of the peptide remains unknown. To solve this problem a technique is required, which is able to estimate the isotopic distribution given the information of a single mass spectrum. Senko et al. calculated the average isotopic distribution for any mass peptide via the multinomial expansion (Yergey 1983) [1], using a scaled version of the average amino acid Averagine (Senko et al. 1995) [2]. Another method, introduced by Breen et al., approximates the result of the multinomial expansion by a Poisson model (Breen et al. 2000) [3]. Although both methods perform well, they have their specific limitations. In this manuscript, we propose an alternative method for the prediction of the isotopic distribution based on a model for consecutive ratios of peaks from the isotopic distribution, similar in spirit to the approach introduced by Gay et al. (1999) [5]. The presented method is computationally simple and accurate in predicting the expected isotopic distribution. Further, we extend our method to estimate the isotopic distribution of sulphur-containing peptides. This is important because the naturally occurring isotopes of sulphur have an impact on the isotopic distribution of a peptide.

Commonly used methods to analyze incomplete longitudinal clinical trial data include complete case analysis (CC) and last observation carried forward (LOCF). However, such methods rest on strong assumptions, including missing completely at random (MCAR) for CC and unchanging profile after dropout for LOCF. Such assumptions are too strong to generally hold. Over the last decades, a number of full longitudinal data analysis methods have become available, such as the linear mixed model for Gaussian outcomes, that are valid under the much weaker missing at random (MAR) assumption. Such a method is useful, even if the scientific question is in terms of a single time point, for example, the last planned measurement occasion, and it is generally consistent with the intention-to-treat principle. The validity of such a method rests on the use of maximum likelihood, under which the missing data mechanism is ignorable as soon as it is MAR. In this paper, we will focus on non-Gaussian outcomes, such as binary, categorical or count data. This setting is less straightforward since there is no unambiguous counterpart to the linear mixed model. We first provide an overview of the various modeling frameworks for non-Gaussian longitudinal data, and subsequently focus on generalized linear mixed-effects models, on the one hand, of which the parameters can be estimated using full likelihood, and on generalized estimating equations, on the other hand, which is a nonlikelihood method and hence requires a modification to be valid under MAR. We briefly comment on the position of models that assume missingness not at random and argue they are most useful to perform sensitivity analysis. Our developments are underscored using data from two studies. While the case studies feature binary outcomes, the methodology applies equally well to other discrete-data settings, hence the qualifier ``discrete'' in the title., Comment: Published at http://dx.doi.org/10.1214/088342305000000322 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Models for incomplete longitudinal data under missingness not at random have gained some popularity. At the same time, cautionary remarks have been issued regarding their sensitivity to often unverifiable modeling assumptions. Consequently, there is evidence for a shift towards using ignorable methodology, supplemented with sensitivity analyses to explore the impact of potential deviations of this assumption in the direction of missingness at random. One such tool is local influence. It is shown that local influence tends to pick up a lot of different anomalies in the data at hand, not just deviations in the MNAR mechanism. This particular behavior is described and insight offered in terms of the non-standard behavior of the likelihood ratio test statistic for MAR missingness versus MNAR missingness within a model of the Diggle and Kenward type. (c) 2004 Elsevier B.V. All rights reserved. vy Jansen, Niel Hens, Geert Molenberghs and Marc Aerts gratefully acknowledge support from Fonds Wetenschappelijk Onderzoek-Vlaanderen Research Project G.0002.98 “Sensitivity Analysis for Incomplete and Coarse Data” and from Belgian IUAP/PAI network “Statistical Techniques and Modeling for Complex Substantive Questions with Complex Data”.

The authors analyze data on marital satisfaction, obtained from couples at two distinct moments in time (1990, 1995). The data are of a bivariate longitudinal type. Moreover, some couples provide incomplete records only, usually because the 1995 follow-up interview has not taken place. The authors propose a hierarchical modeling strategy that takes all these features into account and is more generally valid than a classical complete case or single imputation-based strategy.

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.

Between 2003 and 2010 two 3-year screening periods (I and II) were conducted to monitor the brown rat resistance to rodenticides in Flanders. Resistance was assessed by means of genetic mutations. The aim of the study was to estimate the degree of resistance in Flanders, to test whether there are differences between the 12 river basins, and whether the resistance is increasing over time. Multiple rats per location were caught.Several statistical analyses were performed on these data. On the one hand, aggregated results per location (presence/absence of resistance, defined in several ways) were analysed using logistic regression (glm), while on the other hand, mixed model logistic regression (glmm) was used on the rat-level to incorporate correlations between rats from a single location.The level of resistance differed strongly between river basins, some being almost completely resistant, while in others resistance was nearly zero. Correlation was strongest between rats within a location, but variograms also showed a strong correlation between nearby locations (The number of locations (resp. rats) per river basin ranged from 2 to 54 (resp. 2 to 151) for screening I, and from 4 to 65 (resp. 11 to 110) for screening II. Due to these small numbers, the estimate of resistance was inaccurate for some river basins. Also, evolution of resistance over time was not analysed, since data collection in both screening periods was not comparable. A more balanced follow-up monitoring program, with enough rats and locations per river basin, has been designed to answer all questions more accurately in the future.

We present an approach to construct a classification rule based on the mass spectrometry data provided by the organizers of the "Classification Competition on Clinical Mass Spectrometry Proteomic Diagnosis Data." Before constructing a classification rule, we attempted to pre-process the data and to select features of the spectra that were likely due to true biological signals (i.e., peptides/proteins). As a result, we selected a set of 92 features. To construct the classification rule, we considered eight methods for selecting a subset of the features, combined with seven classification methods. The performance of the resulting 56 combinations was evaluated by using a cross-validation procedure with 1000 re-sampled data sets. The best result, as indicated by the lowest overall misclassification rate, was obtained by using the whole set of 92 features as the input for a support-vector machine (SVM) with a linear kernel. This method was therefore used to construct the classification rule. For the training data set, the total error rate for the classification rule, as estimated by using leave-one-out cross-validation, was equal to 0.16, with the sensitivity and specificity equal to 0.87 and 0.82, respectively.

The use of anticoagulant rodenticides (AR) can lead to secondary poisoning in non-target wildlife species like various predators. As in Belgium each year approximately 600 tons AR are used we examined the liver of 150 polecats and 75 stone martens for the presence of 8 different AR-residues using a validated LC-HESI-MS⁄MS method: warfarin, coumatetralyl, chlorophacinone, bromadiolone, difenacoum, brodifacoum, flocoumafen and difethialone. Almost all animals were road kills, collected from 2006 to 2012. About 77% and 81% of the livers of respectively polecat and stone marten contained AR residues. The maximum (median) concentration was 3,813 µg/g (0,133 µg/g) for polecat and 1,370 µg/g (0,213 µg/g) for stone marten, while the maximum number of different AR residues detected simultaneously in one animal was six. 42% of the animals reached the cut-off of 0,2 µg/g from which survival probability starts to decrease and intoxication could be expected. Statistical analysis did reveal a borderline significant interaction effect between season and species (p=0.0409) on the sum of the residue levels, but no effect of sex. For a subset of adult male polecats (n=54) found dead in spring we analysed their residue level against the fitness of the animal expressed as body-condition (function of eviscerated weight and total length), mesenterial fat (g), kidney- and subcutaneous fat index. None of the observed variation in these condition variables could be explained by changes in residue concentrations. Hence we assume that the condition variables considered here are not altered by secondary poisoning.