A mixture model-based approach to the clustering of microarray expression data

Motivation: This paper introduces the software EMMIX-GENE that has been developed for the specific purpose of a model-based approach to the clustering of microarray expression data, in particular, of tissue samples on a very large number of genes. The latter is a nonstandard problem in parametric cluster analysis because the dimension of the feature space (the number of genes) is typically much greater than the number of tissues. A feasible approach is provided by first selecting a subset of the genes relevant for the clustering of the tissue samples by fitting mixtures of t distributions to rank the genes in order of increasing size of the likelihood ratio statistic for the test of one versus two components in the mixture model. The imposition of a threshold on the likelihood ratio statistic used in conjunction with a threshold on the size of a cluster allows the selection of a relevant set of genes. However, even this reduced set of genes will usually be too large for a normal mixture model to be fitted directly to the tissues, and so the use of mixtures of factor analyzers is exploited to reduce effectively the dimension of the feature space of genes. Results: The usefulness of the EMMIX-GENE approach for the clustering of tissue samples is demonstrated on two well-known data sets on colon and leukaemia tissues. For both data sets, relevant subsets of the genes are able to be selected that reveal interesting clusterings of the tissues that are either consistent with the external classification of the tissues or with background and biological knowledge of these sets.

A Mixture model with random-effects components for clustering correlated gene-expression profiles

Motivation: The clustering of gene profiles across some experimental conditions of interest contributes significantly to the elucidation of unknown gene function, the validation of gene discoveries and the interpretation of biological processes. However, this clustering problem is not straightforward as the profiles of the genes are not all independently distributed and the expression levels may have been obtained from an experimental design involving replicated arrays. Ignoring the dependence between the gene profiles and the structure of the replicated data can result in important sources of variability in the experiments being overlooked in the analysis, with the consequent possibility of misleading inferences being made. We propose a random-effects model that provides a unified approach to the clustering of genes with correlated expression levels measured in a wide variety of experimental situations. Our model is an extension of the normal mixture model to account for the correlations between the gene profiles and to enable covariate information to be incorporated into the clustering process. Hence the model is applicable to longitudinal studies with or without replication, for example, time-course experiments by using time as a covariate, and to cross-sectional experiments by using categorical covariates to represent the different experimental classes. Results: We show that our random-effects model can be fitted by maximum likelihood via the EM algorithm for which the E(expectation) and M(maximization) steps can be implemented in closed form. Hence our model can be fitted deterministically without the need for time-consuming Monte Carlo approximations. The effectiveness of our model-based procedure for the clustering of correlated gene profiles is demonstrated on three real datasets, representing typical microarray experimental designs, covering time-course, repeated-measurement and cross-sectional data. In these examples, relevant clusters of the genes are obtained, which are supported by existing gene-function annotation. A synthetic dataset is considered too.

On a resampling approach for tests on the number of clusters with mixture model-based clustering of tissue samples

We consider the problem of assessing the number of clusters in a limited number of tissue samples containing gene expressions for possibly several thousands of genes. It is proposed to use a normal mixture model-based approach to the clustering of the tissue samples. One advantage of this approach is that the question on the number of clusters in the data can be formulated in terms of a test on the smallest number of components in the mixture model compatible with the data. This test can be carried out on the basis of the likelihood ratio test statistic, using resampling to assess its null distribution. The effectiveness of this approach is demonstrated on simulated data and on some microarray datasets, as considered previously in the bioinformatics literature. (C) 2004 Elsevier Inc. All rights reserved.

An EM-based Semi-Parametric Mixture Model Approach to the Regression Analysis of Competing-Risks Data

We consider a mixture model approach to the regression analysis of competing-risks data. Attention is focused on inference concerning the effects of factors on both the probability of occurrence and the hazard rate conditional on each of the failure types. These two quantities are specified in the mixture model using the logistic model and the proportional hazards model, respectively. We propose a semi-parametric mixture method to estimate the logistic and regression coefficients jointly, whereby the component-baseline hazard functions are completely unspecified. Estimation is based on maximum likelihood on the basis of the full likelihood, implemented via an expectation-conditional maximization (ECM) algorithm. Simulation studies are performed to compare the performance of the proposed semi-parametric method with a fully parametric mixture approach. The results show that when the component-baseline hazard is monotonic increasing, the semi-parametric and fully parametric mixture approaches are comparable for mildly and moderately censored samples. When the component-baseline hazard is not monotonic increasing, the semi-parametric method consistently provides less biased estimates than a fully parametric approach and is comparable in efficiency in the estimation of the parameters for all levels of censoring. The methods are illustrated using a real data set of prostate cancer patients treated with different dosages of the drug diethylstilbestrol. Copyright (C) 2003 John Wiley Sons, Ltd.

Fitting a mixture model to three-mode three-way data with missing information

When the data consist of certain attributes measured on the same set of items in different situations, they would be described as a three-mode three-way array. A mixture likelihood approach can be implemented to cluster the items (i.e., one of the modes) on the basis of both of the other modes simultaneously (i.e,, the attributes measured in different situations). In this paper, it is shown that this approach can be extended to handle three-mode three-way arrays where some of the data values are missing at random in the sense of Little and Rubin (1987). The methodology is illustrated by clustering the genotypes in a three-way soybean data set where various attributes were measured on genotypes grown in several environments.

Modelling High-Dimensional Data by Mixtures of Factor Analyzers

We focus on mixtures of factor analyzers from the perspective of a method for model-based density estimation from high-dimensional data, and hence for the clustering of such data. This approach enables a normal mixture model to be fitted to a sample of n data points of dimension p, where p is large relative to n. The number of free parameters is controlled through the dimension of the latent factor space. By working in this reduced space, it allows a model for each component-covariance matrix with complexity lying between that of the isotropic and full covariance structure models. We shall illustrate the use of mixtures of factor analyzers in a practical example that considers the clustering of cell lines on the basis of gene expressions from microarray experiments. (C) 2002 Elsevier Science B.V. All rights reserved.

Data analysis in multiple-frequency bioelectrical impedance analysis

The performance of three analytical methods for multiple-frequency bioelectrical impedance analysis (MFBIA) data was assessed. The methods were the established method of Cole and Cole, the newly proposed method of Siconolfi and co-workers and a modification of this procedure. Method performance was assessed from the adequacy of the curve fitting techniques, as judged by the correlation coefficient and standard error of the estimate, and the accuracy of the different methods in determining the theoretical values of impedance parameters describing a set of model electrical circuits. The experimental data were well fitted by all curve-fitting procedures (r = 0.9 with SEE 0.3 to 3.5% or better for most circuit-procedure combinations). Cole-Cole modelling provided the most accurate estimates of circuit impedance values, generally within 1-2% of the theoretical values, followed by the Siconolfi procedure using a sixth-order polynomial regression (1-6% variation). None of the methods, however, accurately estimated circuit parameters when the measured impedances were low (<20 Omega) reflecting the electronic limits of the impedance meter used. These data suggest that Cole-Cole modelling remains the preferred method for the analysis of MFBIA data.

Long-term survivor mixture model with random effects: application to a multi-centre clinical trial of carcinoma

A mixture model incorporating long-term survivors has been adopted in the field of biostatistics where some individuals may never experience the failure event under study. The surviving fractions may be considered as cured. In most applications, the survival times are assumed to be independent. However, when the survival data are obtained from a multi-centre clinical trial, it is conceived that the environ mental conditions and facilities shared within clinic affects the proportion cured as well as the failure risk for the uncured individuals. It necessitates a long-term survivor mixture model with random effects. In this paper, the long-term survivor mixture model is extended for the analysis of multivariate failure time data using the generalized linear mixed model (GLMM) approach. The proposed model is applied to analyse a numerical data set from a multi-centre clinical trial of carcinoma as an illustration. Some simulation experiments are performed to assess the applicability of the model based on the average biases of the estimates formed. Copyright (C) 2001 John Wiley & Sons, Ltd.

Monitoring the composition and form of urban environments based on the vegetation-impervious surface-soil (VIS) model by sub-pixel analysis techniques

The majority of the world's population now resides in urban environments and information on the internal composition and dynamics of these environments is essential to enable preservation of certain standards of living. Remotely sensed data, especially the global coverage of moderate spatial resolution satellites such as Landsat, Indian Resource Satellite and Systeme Pour I'Observation de la Terre (SPOT), offer a highly useful data source for mapping the composition of these cities and examining their changes over time. The utility and range of applications for remotely sensed data in urban environments could be improved with a more appropriate conceptual model relating urban environments to the sampling resolutions of imaging sensors and processing routines. Hence, the aim of this work was to take the Vegetation-Impervious surface-Soil (VIS) model of urban composition and match it with the most appropriate image processing methodology to deliver information on VIS composition for urban environments. Several approaches were evaluated for mapping the urban composition of Brisbane city (south-cast Queensland, Australia) using Landsat 5 Thematic Mapper data and 1:5000 aerial photographs. The methods evaluated were: image classification; interpretation of aerial photographs; and constrained linear mixture analysis. Over 900 reference sample points on four transects were extracted from the aerial photographs and used as a basis to check output of the classification and mixture analysis. Distinctive zonations of VIS related to urban composition were found in the per-pixel classification and aggregated air-photo interpretation; however, significant spectral confusion also resulted between classes. In contrast, the VIS fraction images produced from the mixture analysis enabled distinctive densities of commercial, industrial and residential zones within the city to be clearly defined, based on their relative amount of vegetation cover. The soil fraction image served as an index for areas being (re)developed. The logical match of a low (L)-resolution, spectral mixture analysis approach with the moderate spatial resolution image data, ensured the processing model matched the spectrally heterogeneous nature of the urban environments at the scale of Landsat Thematic Mapper data.

Ecoregion classification using a bayesian approach and centre-focused clusters

Ecological regions are increasingly used as a spatial unit for planning and environmental management. It is important to define these regions in a scientifically defensible way to justify any decisions made on the basis that they are representative of broad environmental assets. The paper describes a methodology and tool to identify cohesive bioregions. The methodology applies an elicitation process to obtain geographical descriptions for bioregions, each of these is transformed into a Normal density estimate on environmental variables within that region. This prior information is balanced with data classification of environmental datasets using a Bayesian statistical modelling approach to objectively map ecological regions. The method is called model-based clustering as it fits a Normal mixture model to the clusters associated with regions, and it addresses issues of uncertainty in environmental datasets due to overlapping clusters.

Equational Reasoning as a Tool for Data Analysis

A combination of deductive reasoning, clustering, and inductive learning is given as an example of a hybrid system for exploratory data analysis. Visualization is replaced by a dialogue with the data.

A landscape-scale test of the predictive ability of a spatially explicit model for population viability analysis

1. Although population viability analysis (PVA) is widely employed, forecasts from PVA models are rarely tested. This study in a fragmented forest in southern Australia contrasted field data on patch occupancy and abundance for the arboreal marsupial greater glider Petauroides volans with predictions from a generic spatially explicit PVA model. This work represents one of the first landscape-scale tests of its type. 2. Initially we contrasted field data from a set of eucalypt forest patches totalling 437 ha with a naive null model in which forecasts of patch occupancy were made, assuming no fragmentation effects and based simply on remnant area and measured densities derived from nearby unfragmented forest. The naive null model predicted an average total of approximately 170 greater gliders, considerably greater than the true count (n = 81). 3. Congruence was examined between field data and predictions from PVA under several metapopulation modelling scenarios. The metapopulation models performed better than the naive null model. Logistic regression showed highly significant positive relationships between predicted and actual patch occupancy for the four scenarios (P = 0.001-0.006). When the model-derived probability of patch occupancy was high (0.50-0.75, 0.75-1.00), there was greater congruence between actual patch occupancy and the predicted probability of occupancy. 4. For many patches, probability distribution functions indicated that model predictions for animal abundance in a given patch were not outside those expected by chance. However, for some patches the model either substantially over-predicted or under-predicted actual abundance. Some important processes, such as inter-patch dispersal, that influence the distribution and abundance of the greater glider may not have been adequately modelled. 5. Additional landscape-scale tests of PVA models, on a wider range of species, are required to assess further predictions made using these tools. This will help determine those taxa for which predictions are and are not accurate and give insights for improving models for applied conservation management.