Suchergebnisse

We refine the classical Lindeberg-Feller central limit theorem by obtaining asymptotic bounds on the Kolmogorov distance, the Wasserstein distance, and the parameterized Prokhorov distances in terms of a Lindeberg index. We thus obtain more general approximate central limit theorems, which roughly state that the row-wise sums of a triangular array are approximately asymptotically normal if the array approximately satisfies Lindeberg's condition. This allows us to continue to provide information in nonstandard settings in which the classical central limit theorem fails to hold. Stein's method plays a key role in the development of this theory. ; Ben Berckmoes is postdoctoral fellow at the Fund for Scientific Research of Flanders (FWO). Geert Molenberghs gratefully acknowledges financial support from the IAP research network # P7/06 of the Belgian Government (Belgian Science Policy).

Analysis of longitudinal count data has, for long, been done using a generalized linear mixed model (GLMM), in its Poisson-normal version, to account for correlation by specifying normal random effects. Univariate counts are often handled with the negativebinomial (NEGBIN) model taking into account overdispersion by use of Gamma random effects. Inherently though, longitudinal count data commonly exhibit both features of correlation and overdispersion simultaneously, necessitating analysis methodology that can account for both. The introduction of the combined model (CM) by (Molenberghs, Verbeke, and Dem´etrio 2007) and (Molenberghs, Verbeke, Dem´etrio, and Vieira 2010) serves this purpose, not only for count data but for the general exponential family of distributions. Here, a Poisson model is specified as the parent distribution of the data with a normally distributed random effect at the subject or cluster level and/or a gamma distribution at observation level. The GLMM and NEGBIN are special cases. Data can be simulated from (1) the general CM, with random effects, or, (2) its marginal version directly. This paper discusses an implementation of (1) in SAS software (SAS Institute Inc. (2011)). One needs to reflect on the mean of both the combined (hierarchical) and marginal models in order to generate correlated and/or overdispersed counts. A pre-specification of the desired marginal mean (in terms of covariates and marginal parameters), a marginal variance-covariance structure and the hierarchical mean (in terms of covariates and regression parameters) is required. The implied hierarchical parameters, the variance-covariance matrix of the random effects, and the variance-covariance matrix of the overdispersion part are then derived from which correlated Poisson data are generated. Sample calls of the SAS macro are presented as well as output. ; The authors gratefully acknowledge support from IAP Research Network P7/06 of the Belgian Government (Belgian Science Policy).

Iddi and Molenberghs (2012) merged the attractive features of the so-called combined model of Molenberghs et al (2010) and the marginalized model of Heagerty (1999) for hierarchical non-Gaussian data with overdispersion. In this model, the fixed-effect parameters retain their marginal interpretation. Lee et al (2011) also developed an extension of Heagerty (1999) to handle zero-inflation from count data, using the hurdle model. To bring together all of these features, a marginalized, zero-inflated, overdispersed model for correlated count data is proposed. Using an empirical dataset, it is shown that the proposed model leads to important improvements in model fit. ; The authors gratefully acknowledge the financial support from the IAP research Network P7/06 of the Belgian Government (Belgian Science Policy).

Iddi and Molenberghs (2012) merged the attractive features of the so-called combined model of Molenberghs et al. (2010) and the marginalized model of Heagerty (1999) for hierarchical non-Gaussian data with overdispersion. In this model, the fixed-effect parameters retain their marginal interpretation. Lee et al. (2011) also developed an extension of Heagerty (1999) to handle zero-inflation from count data, using the hurdle model. To bring together all of these features, a marginalized, zero-inflated, overdispersed model for correlated count data is proposed. Using two empirical sets of data, it is shown that the proposed model leads to important improvements in model fit. ; The authors gratefully acknowledge the financial support from the IAP research Network P7/06 of the Belgian Government (Belgian Science Policy).

Overdispersion and correlation are two features often encountered when modeling non-Gaussian dependent data, usually as a function of known covariates. Methods that ignore the presence of these phenomena are often in jeopardy of leading to biased assessment of covariate effects. The beta-binomial and negative binomial models are well known in dealing with overdispersed data for binary and count data, respectively. Similarly, generalized estimating equations (GEE) and the generalized linear mixed models (GLMM) are popular choices when analyzing correlated data. A so-called combined model simultaneously acknowledges the presence of dependency and overdispersion by way of two separate sets of random effects. A marginally specified logistic-normal model for longitudinal binary data which combines the strength of the marginal and hierarchical models has been previously proposed. These two are brought together to produce a marginalized longitudinal model which brings together the comfort of marginally meaningful parameters and the ease of allowing for overdispersion and correlation. Apart from model formulation, estimation methods are discussed. The proposed model is applied to two clinical studies and compared to the existing approach. It turns out that by explicitly allowing for overdispersion random effect, the model significantly improves. (c) 2011 Elsevier B.V. All rights reserved. ; The authors gratefully acknowledge the financial support from the IAP research Network P6/03 of the Belgian Government (Belgian Science Policy).

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).

Statistical models often extend beyond the data available. First, in coarse data, what is actually observed is less detailed than what might be, owing to incompleteness, censoring, grouping, or a combination thereof. Second, in augmented data, the observed data are hypothetically supplemented with random effects, latent variables/classes, or component membership in mixture distributions. The two settings together will be referred to as enriched data. Reasons for modelling enriched data encompass mathematical and computational convenience, advantages in interpretation, and substantive plausibility. Models for enriched data combine evidence coming from empirical data with unverifiable model components, resting entirely on assumptions. This has acute consequences for enriched data, but knowledge about this issue is somewhat scattered. We provide a unified framework for enriched data and show, generally and with focus on incomplete-data models and random-effects models on the other hand, that to any given model an entire class of models can be assigned, with all of its members producing the same fit to the observed data but arbitrary regarding the unobservable parts of the enriched data. The concepts developed are illustrated using a clinical trial in toenail dermatophyte onychomycosis and a developmental toxicity study conducted in mice. ; The authors gratefully acknowledge support from IAP Research Network P6/03 of the Belgian Government (Belgian Science Policy).

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).

COVID-19 has developed into a pandemic, hitting hard on our communities. As the pandemic continues to bring health and economic hardship, keeping mortality as low as possible will be the highest priority for individuals; hence governments must put in place measures to ameliorate the inevitable economic downturn. The course of an epidemic may be defined by a series of key factors. In the early stages of a new infectious disease outbreak, it is crucial to understand the transmission dynamics of the infection. The basic reproduction number (R(0)), which defines the mean number of secondary cases generated by one primary case when the population is largely susceptible to infection ('totally naïve'), determines the overall number of people who are likely to be infected, or, more precisely, the area under the epidemic curve. Estimation of changes in transmission over time can provide insights into the epidemiological situation and identify whether outbreak control measures are having a measurable effect. For R(0) > 1, the number infected tends to increase, and for R(0) < 1, transmission dies out. Non-pharmaceutical strategies to handle the epidemic are sketched and based on current knowledge, the current situation is sketched and scenarios for the near future discussed.

COVID-19 has developed into a pandemic, hitting hard on our communities. As the pandemic continues to bring health and economic hardship, keeping mortality as low as possible will be the highest priority for individuals; hence governments must put in place measures to ameliorate the inevitable economic downturn. The course of an epidemic may be defined by a series of key factors. In the early stages of a new infectious disease outbreak, it is crucial to understand the transmission dynamics of the infection. The basic reproduction number (R-0), which defines the mean number of secondary cases generated by one primary case when the population is largely susceptible to infection ('totally naive'), determines the overall number of people who are likely to be infected, or, more precisely, the area under the epidemic curve. Estimation of changes in transmission over time can provide insights into the epidemiological situation and identify whether outbreak control measures are having a measurable effect. For R-0 > 1, the number infected tends to increase, and for R-0 < 1, transmission dies out. Non-pharmaceutical strategies to handle the epidemic are sketched and based on current knowledge, the current situation is sketched and scenarios for the near future discussed. ; Molenberghs, G (corresponding author), Hasselt Univ, I BioStat, Martelarenpl 42, B-3500 Hasselt, Belgium; Katholieke Univ Leuven, Martelarenpl 42, B-3500 Hasselt, Belgium. geert.molenberghs@uhasselt.be

We analyse the problem of two clinically inseparable, repeatedly measured responses of ordinal type by also incorporating their missingness process. In our application these are the therapeutic effect and extent of side effects of fluvoxamine. In the case of a composite end point, the scientific questions addressed can be answered only when the responses are modelled jointly. As an extension of the methodology, several missingness not at random models were fitted to a set of observed data and shown to yield approximately the same result as their missingness at random counterparts, although it affects precision. In addition, the effect of various identifying restrictions on multiple imputation is investigated. An alternative numerical approximation method is suggested to reduce computational time. ; Financial support from the Interuniversity Attraction Pole research network P7/06 of the Belgian Government (Belgian Science Policy), the Flemish Supercomputer Project and the Institute for the Promotion of Innovation through Science and Technology in Flanders, in which Intel and Janssen Pharmaceutica are partners, is gratefully acknowledged. We are also grateful to Dr Kris Bogaerts of I-BioStat for his expert advice.

In this paper, we propose a method to cluster multivariate functional data with missing observations. Analysis of functional data often encompasses dimension reduction techniques such as principal component analysis (PCA). These techniques require complete data matrices. In this paper, the data are completed by means of multiple imputation, and subsequently each imputed data set is submitted to a cluster procedure. The final partition of the data, summarizing the partitions obtained for the imputed data sets, is obtained by means of ensemble clustering. The uncertainty in cluster membership, due to missing data, is characterized by means of the agreement between the members of the ensemble and fuzziness of the consensus clustering. The potential of the method was brought out on the heart failure (HF) data. Daily measurement for four biomarkers (heart rate, diastolic, and systolic blood pressure, weight) were used to cluster the patients. To normalize the distributions of the longitudinal outcomes, the data were transformed with a natural logarithm function. A cubic spline base with 69 basis functions was employed to smooth the profiles. The proposed algorithm indicates the existence of a latent structure and divides the HF patients into two clusters, showing a different evolution in blood pressure values and weight. In general, cluster results are sensitive to choices made. Likewise for the proposed approach, alternative choices for the distance measure, procedure to optimize the objective function, choice of the scree-test threshold, or the number of principal components, to be used in the approximation of the surrogate density, could all influence the final partition. For the HF data set, the final partition depends on the number of principal components used in the procedure. ; The authors gratefully acknowledge support from IAP research Network P7/06 of the Belgian government (Belgian Science Policy).

In applied statistical data analysis, overdispersion is a common feature. It can be addressed using both multiplicative and additive random effects. A multiplicative model for count data incorporates a gamma random effect as a multiplicative factor into the mean, whereas an additive model assumes a normally distributed random effect, entered into the linear predictor. Using Bayesian principles, these ideas are applied to longitudinal count data, based on the so-called combined model. The performance of the additive and multiplicative approaches is compared using a simulation study. ; The authors gratefully acknowledge support from IAP research Network P7/06 of the Belgian Government (Belgian Science Policy). The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation - Flanders (FWO) and the Flemish Government – department EWI.