Clustering.

Clustering techniques are used to arrange genes in some natural way, that is, to organize genes into groups or clusters with similar behavior across relevant tissue samples (or cell lines). These techniques can also be applied to tissues rather than genes. Methods such as hierarchical agglomerative clustering, k-means clustering, the self-organizing map, and model-based methods have been used. Here we focus on mixtures of normals to provide a model-based clustering of tissue samples (gene signatures) and of gene profiles, including time-course gene expression data.

A benchmark for evaluation of algorithms for identification of cellular correlates of clinical outcomes.

The Flow Cytometry: Critical Assessment of Population Identification Methods (FlowCAP) challenges were established to compare the performance of computational methods for identifying cell populations in multidimensional flow cytometry data. Here we report the results of FlowCAP-IV where algorithms from seven different research groups predicted the time to progression to AIDS among a cohort of 384 HIV+ subjects, using antigen-stimulated peripheral blood mononuclear cell (PBMC) samples analyzed with a 14-color staining panel. Two approaches (FlowReMi.1 and flowDensity-flowType-RchyOptimyx) provided statistically significant predictive value in the blinded test set. Manual validation of submitted results indicated that unbiased analysis of single cell phenotypes could reveal unexpected cell types that correlated with outcomes of interest in high dimensional flow cytometry datasets.

Clustering of gene expression data via normal mixture models.

There are two distinct but related clustering problems with microarray data. One problem concerns the clustering of the tissue samples (gene signatures) on the basis of the genes; the other concerns the clustering of the genes on the basis of the tissues (gene profiles). The clusters of tissues so obtained in the first problem can play a useful role in the discovery and understanding of new subclasses of diseases. The clusters of genes obtained in the second problem can be used to search for genetic pathways or groups of genes that might be regulated together. Also, in the first problem, we may wish first to summarize the information in the very large number of genes by clustering them into groups (of hyperspherical shape), which can be represented by some metagenes, such as the group sample means. We can then carry out the clustering of the tissues in terms of these metagenes. We focus here on mixtures of normals to provide a model-based clustering of tissue samples (gene signatures) and of gene profiles.

Testing for group structure in high-dimensional data.

With the use of finite mixture models for the clustering of a data set, the crucial question of how many clusters there are in the data can be addressed by testing for the smallest number of components in the mixture model compatible with the data. We investigate the performance of a resampling approach to this latter problem in the context of high-dimensional data, where the number of variables p is extremely large relative to the number of observations n. In order to be able to fit normal mixture models to such data, some form of dimension reduction has to be performed. This raises the question of whether a practically significant bias results if the bootstrapping is undertaken solely on the basis of the reduced dimensional form of the data, rather than using the full data from which to draw the bootstrap sample replications.

Application of gene shaving and mixture models to cluster microarray gene expression data.

Researchers are frequently faced with the analysis of microarray data of a relatively large number of genes using a small number of tissue samples. We examine the application of two statistical methods for clustering such microarray expression data: EMMIX-GENE and GeneClust. EMMIX-GENE is a mixture-model based clustering approach, designed primarily to cluster tissue samples on the basis of the genes. GeneClust is an implementation of the gene shaving methodology, motivated by research to identify distinct sets of genes for which variation in expression could be related to a biological property of the tissue samples. We illustrate the use of these two methods in the analysis of Affymetrix oligonucleotide arrays of well-known data sets from colon tissue samples with and without tumors, and of tumor tissue samples from patients with leukemia. Although the two approaches have been developed from different perspectives, the results demonstrate a clear correspondence between gene clusters produced by GeneClust and EMMIX-GENE for the colon tissue data. It is demonstrated, for the case of ribosomal proteins and smooth muscle genes in the colon data set, that both methods can classify genes into co-regulated families. It is further demonstrated that tissue types (tumor and normal) can be separated on the basis of subtle distributed patterns of genes. Application to the leukemia tissue data produces a division of tissues corresponding closely to the external classification, acute myeloid leukemia (AML) and acute lymphoblastic leukemia (ALL), for both methods. In addition, we also identify genes specific for the subgroup of ALL-Tcell samples. Overall, we find that the gene shaving method produces gene clusters at great speed; allows variable cluster sizes and can incorporate partial or full supervision; and finds clusters of genes in which the gene expression varies greatly over the tissue samples while maintaining a high level of coherence between the gene expression profiles. The intent of the EMMIX-GENE method is to cluster the tissue samples. It performs a filtering step that results in a subset of relevant genes, followed by gene clustering, and then tissue clustering, and is favorable in its accuracy of ranking the clusters produced.

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. AVAILABILITY: A Fortran program blue called EMMIX-WIRE (EM-based MIXture analysis WIth Random Effects) is available on request from the corresponding author.

A simple implementation of a normal mixture approach to differential gene expression in multiclass microarrays.

MOTIVATION: An important problem in microarray experiments is the detection of genes that are differentially expressed in a given number of classes. We provide a straightforward and easily implemented method for estimating the posterior probability that an individual gene is null. The problem can be expressed in a two-component mixture framework, using an empirical Bayes approach. Current methods of implementing this approach either have some limitations due to the minimal assumptions made or with more specific assumptions are computationally intensive. RESULTS: By converting to a z-score the value of the test statistic used to test the significance of each gene, we propose a simple two-component normal mixture that models adequately the distribution of this score. The usefulness of our approach is demonstrated on three real datasets.

Mixture modelling for cluster analysis.

Cluster analysis via a finite mixture model approach is considered. With this approach to clustering, the data can be partitioned into a specified number of clusters g by first fitting a mixture model with g components. An outright clustering of the data is then obtained by assigning an observation to the component to which it has the highest estimated posterior probability of belonging; that is, the ith cluster consists of those observations assigned to the ith component (i = 1,..., g). The focus is on the use of mixtures of normal components for the cluster analysis of data that can be regarded as being continuous. But attention is also given to the case of mixed data, where the observations consist of both continuous and discrete variables.

Modelling the distribution of ischaemic stroke-specific survival time using an EM-based mixture approach with random effects adjustment.

A two-component survival mixture model is proposed to analyse a set of ischaemic stroke-specific mortality data. The survival experience of stroke patients after index stroke may be described by a subpopulation of patients in the acute condition and another subpopulation of patients in the chronic phase. To adjust for the inherent correlation of observations due to random hospital effects, a mixture model of two survival functions with random effects is formulated. Assuming a Weibull hazard in both components, an EM algorithm is developed for the estimation of fixed effect parameters and variance components. A simulation study is conducted to assess the performance of the two-component survival mixture model estimators. Simulation results confirm the applicability of the proposed model in a small sample setting.

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.

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. AVAILABILITY: EMMIX-GENE is available at http://www.maths.uq.edu.au/~gjm/emmix-gene/

Patient-specific analysis of sequential haematological data by multiple linear regression and mixture distribution modelling.

Automated storage and analysis of the results of serial haematologic studies are now technically feasible with present-day laboratory instruments and devices for data storage and processing. In current practice, physicians mentally compare a laboratory result with previous values and use their clinical judgement to determine the significance of any change. To provide a statistical basis for this process, we describe a new approach for the detection of changes in patient-specific sequential measurements of standard haematologic laboratory tests. These methods include hierarchical multiple regression modelling, with a weighted minimum risk criteria for model selection, to choose models indicating changes in mean values over time. This study is the first to analyse sequential patient-specific distributions of laboratory measurements, utilizing mixture distribution modelling with systematic selection of starting values for the EM algorithm. To evaluate these statistical methods under controlled conditions, we studied 11 healthy human volunteers who were depleted of iron by serial phlebotomy to iron-deficiency anaemia, then treated with oral iron supplements to replete iron stores and correct the anaemia. Application of sequential patient-specific analyses of haemoglobin, haematocrit, and mean cell volume showed that significant departures from past values could be identified, in many cases, even when values were still within the population reference ranges. Additionally, for all subjects sequential alterations in red blood cell volume distributions during development of iron-deficiency anaemia could be characterized and quantified. These methods promise to provide more sensitive techniques for improved diagnostic evaluation of developing anaemia and serial monitoring of response to therapy.

Distribution of transferrin saturation in an Australian population: relevance to the early diagnosis of hemochromatosis.

BACKGROUND & AIMS: An elevated transferrin saturation is the earliest phenotypic abnormality in hereditary hemochromatosis. Determination of transferrin saturation remains the most useful noninvasive screening test for affected individuals, but there is debate as to the appropriate screening level. The aims of this study were to estimate the mean transferrin saturation in hemochromatosis heterozygotes and normal individuals and to evaluate potential transferrin saturation screening levels. METHODS: Statistical mixture modeling was applied to data from a survey of asymptomatic Australians to estimate the mean transferrin saturation in hemochromatosis heterozygotes and normal individuals. To evaluate potential transferrin saturation screening levels, modeling results were compared with data from identified hemochromatosis heterozygotes and homozygotes. RESULTS: After removal of hemochromatosis homozygotes, two populations of transferrin saturation were identified in asymptomatic Australians (P < 0.01). In men, 88.2% of the truncated sample had a lower mean transferrin saturation of 24.1%, whereas 11.8% had an increased mean transferrin saturation of 37.3%. Similar results were found in women. A transferrin saturation threshold of 45% identified 98% of homozygotes without misidentifying any normal individuals. CONCLUSIONS: The results confirm that hemochromatosis heterozygotes form a distinct transferrin saturation subpopulation and support the use of transferrin saturation as an inexpensive screening test for hemochromatosis. In practice, a fasting transferrin saturation of > or = 45% identifies virtually all affected homozygous subjects without necessitating further investigation of unaffected normal individuals.

On the EM algorithm for overdispersed count data.

In this paper, we consider the use of the EM algorithm for the fitting of distributions by maximum likelihood to overdispersed count data. In the course of this, we also provide a review of various approaches that have been proposed for the analysis of such data. As the Poisson and binomial regression models, which are often adopted in the first instance for these analyses, are particular examples of a generalized linear model (GLM), the focus of the account is on the modifications and extensions to GLMs for the handling of overdispersed count data.

An analysis of valve re-replacement after aortic valve replacement with biologic devices.

Biologic valve re-replacement was examined in a series of 1343 patients who underwent aortic valve replacement at The Prince Charles Hospital, Brisbane, with a cryopreserved or 4 degrees C stored allograft valve or a xenograft valve. A parametric model approach was used to simultaneously model the competing risks of death without re-replacement and re-replacement before death. One hundred eleven patients underwent a first re-replacement for a variety of reasons (69 patients with xenograft valves, 28 patients with 4 degrees C stored allograft valves, and 14 patients with cryopreserved allograft valves). By multivariable analysis younger age at operation was associated with xenograft, 4 degrees C stored allograft, and cryopreserved allograft valve re-replacement. However, this effect was examined in the context of longer survival of younger patients, which increases their exposure to the risk of re-replacement as compared with that in older patients whose decreased survival reduced their probability of requiring valve re-replacement. In patients older than 60 years at the time of aortic valve replacement, the probability of re-replacement (for any reason) before death was similar for xenografts and cryopreserved allograft valves but higher for 4 degrees C stored valves. However, in patients younger than 60 years, the probability of re-replacement at any time during the remainder of the life of the patient was lower with the cryopreserved allograft valve compared with the xenograft valve and 4 degrees C stored allografts.

Parametric estimation in a genetic mixture model with application to nuclear family data.

The apparent conflict between the biometrician and Mendelian genetics has been recently resolved by the introduction of a genetic mixed model to analyze continuous traits measured on human families and to elucidate the mechanism of underlying major genes. The mixed model formulated by Elston and Stewart (1971, Human Heredity 21, 523-542), extended by Morton and MacLean (1974, American Journal of Human Genetics 26, 489-503), and reviewed, with further extensions, by Boyle and Elston (1979, Biometrics 35, 55-68) has become an extremely useful tool of wide applicability in the field of genetic epidemiology. This model allows for segregation at a major locus, a polygenic effect, and a sibling environmental variation. The main concern of this paper is with estimating the model parameters by the method of maximum likelihood. The expectation-maximization (EM) algorithm is developed to derive the estimates iteratively. An approximation of the information matrix when using the EM algorithm is given. We illustrate the methodology by fitting the model to the arterial blood pressure data collected by Miall and Oldham (1955, Clinical Science 14, 459-487).

On the role of finite mixture models in survival analysis.

In this paper we review the role of finite mixture models in the field of survival analysis. Finite mixture models can be used to analyse failure-time data in a variety of situations. In particular, they provide a way of modelling time to failure in the case of competing risks.

An analysis of risk factors for death and mode-specific death after aortic valve replacement with allograft, xenograft, and mechanical valves.

From September 1967 to January 1990, a total of 2100 patients underwent 2366 aortic valve replacements with a variety of allograft, xenograft, and mechanical valves. Concomitant procedures were performed in 764 patients. Actuarial survival at 12 years was 59.6% (70% confidence limits 57.8% to 61.4%). Hazard function for death was highest immediately after operation, falling to merge with a slowly rising phase of risk at approximately 3 months. Actuarial freedom from sudden death at 12 years was 88.0% (70% confidence limits 86.7% to 89.3%). The shape of the hazard function for sudden death was similar to that for death. Actuarial freedom from death with cardiac failure at 12 years was 87.9% (70% confidence limits 86.5% to 89.2%). The shape of the hazard function for death with cardiac failure was also similar to that for death. Risk factor analysis revealed the important deleterious impact on long-term survival resulting from impaired left ventricular structure and function because of aortic valve disease. No current-era valve used in this study (allograft, xenograft, or mechanical) was a risk factor for death. Both aortic wall disease and endocarditis necessitating aortic valve replacement substantially decreased long-term patient survival. Aortic valve replacement is advisable much earlier in the natural history of aortic valve disease before secondary left ventricular damage occurs.

Aortic valve infection. Risk factors for death and recurrent endocarditis after aortic valve replacement.

Patients (n = 195) undergoing aortic valve replacement (n = 209) for native or prosthetic valve endocarditis were studied to determine risk factors for death and recurrent endocarditis and also to determine the valve type least likely to be associated with recurrent endocarditis. Ten-year survival was 60%, the highest risk of dying occurring within the first 3 postoperative months. Risk factors for death in this early phase included increased urea concentration, higher New York Heart Association functional class, prosthetic valve endocarditis, infection status (lower in patients with healed endocarditis), longer duration of cardiopulmonary bypass, and nonuse of an allograft valve. In the late phase (beyond 3 months), risk factors included age at operation and Staphylococcus aureus infection (only in New York Heart Association functional class V). Ten years after aortic valve replacement, 79% of valves were free of recurrent endocarditis. The highest risk of recurrence was in the first 4 months. Longer duration of cardiopulmonary bypass was a weak risk factor for recurrent endocarditis in the early phase, and in the late phase risk factors were S. aureus infection (only in New York Heart Association functional classes III, IV, and V) and the use of now discontinued biologic valves. Allograft aortic valve replacement was shown to be associated with a low and constant risk of recurrent endocarditis, whereas other valve types were associated with a high early risk. The allograft valve should be the preferred replacement device for aortic root infection.

Cluster analysis and related techniques in medical research.

In this paper we review methods of cluster analysis in the context of classifying patients on the basis of clinical and/or laboratory type observations. Both hierarchical and non-hierarchical methods of clustering are considered, although the emphasis is on the latter type, with particular attention devoted to the mixture likelihood-based approach. For the purposes of dividing a given data set into g clusters, this approach fits a mixture model of g components, using the method of maximum likelihood. It thus provides a sound statistical basis for clustering. The important but difficult question of how many clusters are there in the data can be addressed within the framework of standard statistical theory, although theoretical and computational difficulties still remain. Two case studies, involving the cluster analysis of some haemophilia and diabetes data respectively, are reported to demonstrate the mixture likelihood-based approach to clustering.