Methods in parametric cluster analysis commonly assume data can be modelled by means of a finite mixture of distributions. However, associating each mixture component to one cluster is frequently misleading because different mixture components can overlap, and then, associated clusters can overlap too suggesting a unique cluster. A number of approaches have already been proposed to construct the clusters by merging components using the posterior probabilities. This article presents a generic approach for building a hierarchy of mixture components that integrates and generalizes some techniques proposed earlier in the literature. Using this proposal, two new techniques based on the log-ratio of posterior probabilities are introduced. Moreover, to decide the final number of clusters, two new methods are presented. Simulated and real datasets are used to illustrate this methodology.

This article introduces a robust extension of the mixture of factor analysis models based on the restricted multivariate skew- t distribution, called mixtures of skew- t factor analysis (MSTFA) model. This model can be viewed as a powerful tool for model-based clustering of high-dimensional data where observations in each cluster exhibit non-normal features such as heavy-tailed noises and extreme skewness. Missing values may be frequently present due to the incomplete collection of data. A computationally feasible EM-type algorithm is developed to carry out maximum likelihood estimation and create single imputation of possible missing values under a missing at random mechanism. The numbers of factors and mixture components are determined via penalized likelihood criteria. The utility of our proposed methodology is illustrated through analysing both simulated and real datasets. Numerical results are shown to perform favourably compared to existing approaches.

We thank the discussants for their kind comments and their insightful analysis and discussion that has substantially added to the contribution of this issue. Overall, it seems the discussants have affirmed many of our primary points, and have also raised a number of other relevant and important issues that we did not emphasize in the paper. Several common threads emerged from these discussions, including the importance of software development, appropriate dissemination, and close collaboration with biomedical scientists and technology experts in order to ensure our work is relevant and impactful. Each discussant also mentioned other areas of bioinformatics that have been impacted by statistical researchers that we did not highlight in the original paper. In response, we will first summarize discuss these general themes, and then respond to specific comments of each discussant, and finally talk about the additional areas of bioinformatics impacted by statisticians that were mentioned by the reviewers.

Abstract: We introduce a class of dynamic regression models designed to predict the future of growth curves based on their historical dynamics. This class of models incorporates both baseline and time-dependent covariates, start with simple regression models and build up to dynamic function-on-function regressions. We compare the performance of the dynamic prediction models in a variety of signal-to-noise scenarios and provide practical solutions for model selection. We conclude that (a) prediction performance increases substantially when using the entire growth history relative to using only the last and first observation; (b) smoothing incorporated using functional regression approaches increases prediction performance; and (c) the interpretation of model parameters is substantially improved using functional regression approaches. Because many growth curve datasets exhibit missing and noisy data, we propose a bootstrap of subjects approach to account for the variability associated with the missing data imputation and smoothing. Methods are motivated by and applied to the CONTENT dataset, a study that collected monthly child growth data on 197 children from birth until month 15. R code describing the fitting approaches is provided in a supplementary file.

Overpayment estimation using a sample of audited medical claims is an often used method to determine recoupment amounts. The current practice based on central limit theorem may not be efficient for certain kinds of claims data, including skewed payment populations with partial overpayments. As an alternative, we propose a novel Bayesian inflated mixture model. We provide an analysis of the validity and efficiency of the model estimates for a number of payment populations and overpayment scenarios. In addition, learning about the parameters of the overpayment distribution with increasing sample size may provide insights for the medical investigators. We present a discussion of model selection and potential modelling extensions.

This is a discussion on the article ‘Regularized Regression for Categorical Data’ by Tutz and Gertheiss.

Considering the case of the School of Economics of the University of Florence, the paper investigates whether the pre-enrolment assessment test is an effective tool to predict student performance. The analysis is tailored to evaluate the additional information yielded by the test beyond the background characteristics of the candidates already available from administrative records, such as the high school type and final grade. The student performance is measured by the number of gained credits after one year, which is a count variable with an irregular distribution and a peak in zero. These features pose a challenge in statistical modelling, which is solved by a two-part model with a logit specification for the zeros, while positive values are analyzed by quantile regression for counts. To disentangle direct and indirect effects of background variables, the result of the pre-enrolment assessment test is treated as an intermediate variable in a regression chain graph. The results show that the pre-enrolment test adds some information to predict student performance, which can be exploited for tutoring.