Advances in Mixture Models

The importance of mixture distributions is not only remarked by a number of recent books on mixtures including Lindsay (1995), Böhning (2000), McLachlan and Peel (2000) and Frühwirth-Schnatter (2006) which update previous books by Everitt and Hand (1981), Titterington et al. (1985) and McLachlan and Basford (1988). Also, a diversity of publications on mixtures appeared in this journal since 2003 (which we take here as a milestone with the appearance of the first special issue on mixtures) including Hazan et al. (2003), Benton and Krishnamoorthy (2003), Woodward and Sain (2003), Besbeas and Morgan (2004), Jamshidian (2004), Hürlimann (2004), Bohacek and Rozovskii (2004), Tao et al. (2004), Vaz de Melo Mendes and Lopes (2006), Agresti et al. (2006), Bartolucci and Scaccia (2006), D’Elia and Piccolo (2005), Neerchal and Morel (2005), Klar and Meintanis (2005), Bocci et al. (2006), Hu and Sung (2006), Seidel et al. (2006), Nadarajah (2006), Almhana et al. (2006), Congdon (2006), Priebe et al. (2006), and Li and Zha (2006). In the following we give a brief introduction to the papers contributing novels aspects in this Special Issue. These come from a diversity of areas as different as capture–recapture modelling, likelihood based cluster analysis, semiparametric mixture modelling in microarray data, latent class analysis or integer lifetime data analysis—just to mention a few. Mixture models are frequently used in capture–recapture studies for estimating population size (Chao, 1987; Link, 2003; Böhning and Schön, 2005; Böhning et al., 2005; Böhning and Kuhnert, 2006). In this issue, Mao (2007) highlights a variety of sources of difficulties in statistical inference using mixture models and uses a binomial mixture model as an illustration. Random intercept models for binary data—as useful tools for addressing between-subject heterogeneity—are discussed by Caffo et al. (2007). The nonlinearity of link functions for binary data is blurred in probit models with a normally distributed random intercept because the resulting model implies a probit marginal link as well. Caffo et al. (2007) explore another family of random intercept models where the distribution associated with the marginal and conditional link function as well as the random effect distribution are all of the same family. Formann (2007) extends the latent class approach (as a specific discrete multivariate mixture model) for situations where the discrete outcome variables (such as longitudinal binary data) experience nonignorable associations and, in addition and most importantly, have missing entries as it is rather typical for repeated observations in longitudinal studies. The modelling also incorporates potential covariates. This is illustrated using data from the Muscatine Coronary Risk Factor Study. The contribution by Grün and Leisch (2007) introduces the R-package flexmixwhich provides flexible modelling of finite mixtures of regression models using the EM algorithm. Alfò et al. (2007) consider a semiparametric mixture model for detecting differentially expressed genes in microarray experiments.An important goal of microarray studies is the detection of genes that show significant changes in observed expressions when two or more classes of biological samples (e.g. treatment and control) are compared. With the c-fold rule a gene is declared to be differentially expressed if its average expression level varies by more than a constant (typically 2). Instead, Alfò et al. (2007) introduce a gene-specific random term to control for both dependence among genes and variability with respect to the probability of yielding a fold change over a threshold c. Likelihood based inference is accomplished with a two-level finite mixture model while nonparametric Bayesian estimation is performed through the counting distribution of exceedances. Mixtures-of-experts models (Jacobs et al., 1991) and their generalization, hierarchical mixtures-of-expert models (Jordan and Jacobs, 1994) have been introduced to account for nonlinearities and other complexities in the data.