Choosing a Cluster Sampling Design for Lot Quality Assurance Sampling Surveys

Lot quality assurance sampling (LQAS) surveys are commonly used for monitoring and evaluation in resource-limited settings. Recently several methods have been proposed to combine LQAS with cluster sampling for more timely and cost-effective data collection. For some of these methods, the standard binomial model can be used for constructing decision rules as the clustering can be ignored. For other designs, considered here, clustering is accommodated in the design phase. In this paper, we compare these latter cluster LQAS methodologies and provide recommendations for choosing a cluster LQAS design. We compare technical differences in the three methods and determine situations in which the choice of method results in a substantively different design. We consider two different aspects of the methods: the distributional assumptions and the clustering parameterization. Further, we provide software tools for implementing each method and clarify misconceptions about these designs in the literature. We illustrate the differences in these methods using vaccination and nutrition cluster LQAS surveys as example designs. The cluster methods are not sensitive to the distributional assumptions but can result in substantially different designs (sample sizes) depending on the clustering parameterization. However, none of the clustering parameterizations used in the existing methods appears to be consistent with the observed data, and, consequently, choice between the cluster LQAS methods is not straightforward. Further research should attempt to characterize clustering patterns in specific applications and provide suggestions for best-practice cluster LQAS designs on a setting-specific basis.     

Fuzzy clustering of fuzzy ecological data

Besides the problem of searching for effective methods for data analysis there are some additional problems with handling data of high uncertainty. Uncertainty problems often arise in an analysis of ecological data, e.g. in the cluster analysis of ecological data. Conventional clustering methods based on Boolean logic ignore the continuous nature of ecological variables and the uncertainty of ecological data. That can result in misclassification or misinterpretation of the data structure. Clusters with fuzzy boundaries reflect better the continuous character of ecological features. But the problem is, that the common clustering methods (like the fuzzy c-means method) are only designed for treating crisp data, that means they provide a fuzzy partition only for crisp data (e.g. exact measurement data). This paper presents the extension and implementation of the method of fuzzy clustering of fuzzy data proposed by Yang and Liu [Yang, M.-S. and Liu, H-H, 1999. Fuzzy clustering procedures for conical fuzzy vector data. Fuzzy Sets and Systems, 106, 189-200.]. The imprecise data can be defined as multidimensional fuzzy sets with not sharply formed boundaries (in the form of the so-called conical fuzzy vectors). They can then be used for the fuzzy clustering together with crisp data. That can be particularly useful when information is not available about the variances which describe the accuracy of the data and probabilistic approaches are impossible. The method proposed by Yang has been extended and implemented for the Fuzzy Clustering System EcoFucs developed at the University of Kiel. As an example, the paper presents the fuzzy cluster analysis of chemicals according to their ecotoxicological properties. The uncertainty and imprecision of ecotoxicological data are very high because of the use of various data sources, various investigation tests and the difficulty of comparing these data. The implemented method can be very helpful in searching for an adequate partition of ecological data into clusters with similar properties.     

Evaluation and comparison of gene clustering methods in microarray analysis

MOTIVATION: Microarray technology has been widely applied in biological and clinical studies for simultaneous monitoring of gene expression in thousands of genes. Gene clustering analysis is found useful for discovering groups of correlated genes potentially co-regulated or associated to the disease or conditions under investigation. Many clustering methods including hierarchical clustering, K-means, PAM, SOM, mixture model-based clustering and tight clustering have been widely used in the literature. Yet no comprehensive comparative study has been performed to evaluate the effectiveness of these methods. RESULTS: In this paper, six gene clustering methods are evaluated by simulated data from a hierarchical log-normal model with various degrees of perturbation as well as four real datasets. A weighted Rand index is proposed for measuring similarity of two clustering results with possible scattered genes (i.e. a set of noise genes not being clustered). Performance of the methods in the real data is assessed by a predictive accuracy analysis through verified gene annotations. Our results show that tight clustering and model-based clustering consistently outperform other clustering methods both in simulated and real data while hierarchical clustering and SOM perform among the worst. Our analysis provides deep insight to the complicated gene clustering problem of expression profile and serves as a practical guideline for routine microarray cluster analysis.     

LSimpute: accurate estimation of missing values in microarray data with least squares methods

Microarray experiments generate data sets with information on the expression levels of thousands of genes in a set of biological samples. Unfortunately, such experiments often produce multiple missing expression values, normally due to various experimental problems. As many algorithms for gene expression analysis require a complete data matrix as input, the missing values have to be estimated in order to analyze the available data. Alternatively, genes and arrays can be removed until no missing values remain. However, for genes or arrays with only a small number of missing values, it is desirable to impute those values. For the subsequent analysis to be as informative as possible, it is essential that the estimates for the missing gene expression values are accurate. A small amount of badly estimated missing values in the data might be enough for clustering methods, such as hierachical clustering or K-means clustering, to produce misleading results. Thus, accurate methods for missing value estimation are needed. We present novel methods for estimation of missing values in microarray data sets that are based on the least squares principle, and that utilize correlations between both genes and arrays. For this set of methods, we use the common reference name LSimpute. We compare the estimation accuracy of our methods with the widely used KNNimpute on three complete data matrices from public data sets by randomly knocking out data (labeling as missing). From these tests, we conclude that our LSimpute methods produce estimates that consistently are more accurate than those obtained using KNNimpute. Additionally, we examine a more classic approach to missing value estimation based on expectation maximization (EM). We refer to our EM implementations as EMimpute, and the estimate errors using the EMimpute methods are compared with those our novel methods produce. The results indicate that on average, the estimates from our best performing LSimpute method are at least as accurate as those from the best EMimpute algorithm.     

Bagging to improve the accuracy of a clustering procedure

MOTIVATION: The microarray technology is increasingly being applied in biological and medical research to address a wide range of problems such as the classification of tumors. An important statistical question associated with tumor classification is the identification of new tumor classes using gene expression profiles. Essential aspects of this clustering problem include identifying accurate partitions of the tumor samples into clusters and assessing the confidence of cluster assignments for individual samples. RESULTS: Two new resampling methods, inspired from bagging in prediction, are proposed to improve and assess the accuracy of a given clustering procedure. In these ensemble methods, a partitioning clustering procedure is applied to bootstrap learning sets and the resulting multiple partitions are combined by voting or the creation of a new dissimilarity matrix. As in prediction, the motivation behind bagging is to reduce variability in the partitioning results via averaging. The performances of the new and existing methods were compared using simulated data and gene expression data from two recently published cancer microarray studies. The bagged clustering procedures were in general at least as accurate and often substantially more accurate than a single application of the partitioning clustering procedure. A valuable by-product of bagged clustering are the cluster votes which can be used to assess the confidence of cluster assignments for individual observations. SUPPLEMENTARY INFORMATION: For supplementary information on datasets, analyses, and software, consult http://www.stat.berkeley.edu/~sandrine and http://www.bioconductor.org.     

Modeling and Visualizing Uncertainty in Gene Expression Clusters Using Dirichlet Process Mixtures

Although the use of clustering methods has rapidly become one of the standard computational approaches in the literature of microarray gene expression data, little attention has been paid to uncertainty in the results obtained. Dirichlet process mixture (DPM) models provide a nonparametric Bayesian alternative to the bootstrap approach to modeling uncertainty in gene expression clustering. Most previously published applications of Bayesian model-based clustering methods have been to short time series data. In this paper, we present a case study of the application of nonparametric Bayesian clustering methods to the clustering of high-dimensional nontime series gene expression data using full Gaussian covariances. We use the probability that two genes belong to the same cluster in a DPM model as a measure of the similarity of these gene expression profiles. Conversely, this probability can be used to define a dissimilarity measure, which, for the purposes of visualization, can be input to one of the standard linkage algorithms used for hierarchical clustering. Biologically plausible results are obtained from the Rosetta compendium of expression profiles which extend previously published cluster analyses of this data.     

Red–blue plots for detecting clusters in count data

A new index and four new graphical displays, termed "red–blue" plots, are presented to study and measure clustering in spatially referenced count data. The index can detect clusters in the form of patches, comprising several nearby large counts, and in the form of gaps, comprising several nearby small counts. The new methods quantify the degree to which the count for each sample unit contributes towards the overall degree of clustering, either as part of a patch or as a gap; provide tests of nonrandomness to detect clustering; and facilitate a comprehensive definition of the size and dimension of a cluster. The methods are illustrated using aphid field data.     

Braun‐Blanquet's legacy and data analysis in vegetation science

Abstract. This article investigates whether the Braun‐Blanquet abundance/dominance (AD) scores that commonly appear in phytosociological tables can properly be analysed by conventional multivariate analysis methods such as Principal Components Analysis and Correspondence Analysis. The answer is a definite NO. The source of problems is that the AD values express species performance on a scale, namely the ordinal scale, on which differences are not interpretable. There are several arguments suggesting that no matter which methods have been preferred in contemporary numerical syntaxonomy and why, ordinal data should be treated in an ordinal way. In addition to the inadmissibility of arithmetic operations with the AD scores, these arguments include interpretability of dissimilarities derived from ordinal data, consistency of all steps throughout the analysis and universality of the method which enables simultaneous treatment of various measurement scales. All the ordination methods that are commonly used, for example, Principal Components Analysis and all variants of Correspondence Analysis as well as standard cluster analyses such as Ward's method and group average clustering, are inappropriate when using AD data. Therefore, the application of ordinal clustering and scaling methods to traditional phytosociological data is advocated. Dissimilarities between relevés should be calculated using ordinal measures of resemblance, and ordination and clustering algorithms should also be ordinal in nature. A good ordination example is Non‐metric Multidimensional Scaling (NMDS) as long as it is calculated from an ordinal dissimilarity measure such as the Goodman & Kruskal γ coefficient, and for clustering the new OrdClAn‐H and OrdClAn‐N methods.     

Markov clustering versus affinity propagation for the partitioning of protein interaction graphs

Background  

Genome scale data on protein interactions are generally represented as large networks, or graphs, where hundreds or thousands of proteins are linked to one another. Since proteins tend to function in groups, or complexes, an important goal has been to reliably identify protein complexes from these graphs. This task is commonly executed using clustering procedures, which aim at detecting densely connected regions within the interaction graphs. There exists a wealth of clustering algorithms, some of which have been applied to this problem. One of the most successful clustering procedures in this context has been the Markov Cluster algorithm (MCL), which was recently shown to outperform a number of other procedures, some of which were specifically designed for partitioning protein interactions graphs. A novel promising clustering procedure termed Affinity Propagation (AP) was recently shown to be particularly effective, and much faster than other methods for a variety of problems, but has not yet been applied to partition protein interaction graphs.     

Statistical inference for simultaneous clustering of gene expression data

Current methods for analysis of gene expression data are mostly based on clustering and classification of either genes or samples. We offer support for the idea that more complex patterns can be identified in the data if genes and samples are considered simultaneously. We formalize the approach and propose a statistical framework for two-way clustering. A simultaneous clustering parameter is defined as a function theta=Phi(P) of the true data generating distribution P, and an estimate is obtained by applying this function to the empirical distribution P(n). We illustrate that a wide range of clustering procedures, including generalized hierarchical methods, can be defined as parameters which are compositions of individual mappings for clustering patients and genes. This framework allows one to assess classical properties of clustering methods, such as consistency, and to formally study statistical inference regarding the clustering parameter. We present results of simulations designed to assess the asymptotic validity of different bootstrap methods for estimating the distribution of Phi(P(n)). The method is illustrated on a publicly available data set.     

A reduction factor in goodness-of-fit and independence tests for clustered and weighted observations

Situations often arise in a large-scale household survey where a complex probability sample of clusters rather than of individuals is drawn from a large population. Typically, the clusters of such complex samples include a number of correlated members. The responses of these members are then weighted to obtain estimates for the population. Such weighted data are commonly published by the National Center for Health Statistics and other U.S. federal agencies. Frequently, problems arise when such data are tested by usual chi-square test statistics for goodness of fit or independence. Researchers have discovered that the usual chi-square tests provide spuriously inflated results when applied to cluster samples and that new methods are required to correct such problems. This paper proposes a strategy for a goodness-of-fit or independence test based on correlated and weighted data arising in cluster samples, and provides a factor that validly reduces the inflation of the usual chi-square statistics. This method is applied to the chronic condition data collected from the St Paul-Minneapolis, Minnesota, primary sampling unit (PSU) during the 1975 National Health Interview Survey (NHIS). This analysis, together with simulation studies presented elsewhere, provides evidence that the usual chi-square statistics from such data can be corrected for the impacts of clustering and weighting by use of the proposed reduction factor.     

Ensemble clustering for step data via binning

This paper considers the clustering problem of physical step count data recorded on wearable devices. Clustering step data give an insight into an individual's activity status and further provide the groundwork for health‐related policies. However, classical methods, such as K‐means clustering and hierarchical clustering, are not suitable for step count data that are typically high‐dimensional and zero‐inflated. This paper presents a new clustering method for step data based on a novel combination of ensemble clustering and binning. We first construct multiple sets of binned data by changing the size and starting position of the bin, and then merge the clustering results from the binned data using a voting method. The advantage of binning, as a critical component, is that it substantially reduces the dimension of the original data while preserving the essential characteristics of the data. As a result, combining clustering results from multiple binned data can provide an improved clustering result that reflects both local and global structures of the data. Simulation studies and real data analysis were carried out to evaluate the empirical performance of the proposed method and demonstrate its general utility.     

Missing value estimation methods for DNA microarrays

MOTIVATION: Gene expression microarray experiments can generate data sets with multiple missing expression values. Unfortunately, many algorithms for gene expression analysis require a complete matrix of gene array values as input. For example, methods such as hierarchical clustering and K-means clustering are not robust to missing data, and may lose effectiveness even with a few missing values. Methods for imputing missing data are needed, therefore, to minimize the effect of incomplete data sets on analyses, and to increase the range of data sets to which these algorithms can be applied. In this report, we investigate automated methods for estimating missing data. RESULTS: We present a comparative study of several methods for the estimation of missing values in gene microarray data. We implemented and evaluated three methods: a Singular Value Decomposition (SVD) based method (SVDimpute), weighted K-nearest neighbors (KNNimpute), and row average. We evaluated the methods using a variety of parameter settings and over different real data sets, and assessed the robustness of the imputation methods to the amount of missing data over the range of 1--20% missing values. We show that KNNimpute appears to provide a more robust and sensitive method for missing value estimation than SVDimpute, and both SVDimpute and KNNimpute surpass the commonly used row average method (as well as filling missing values with zeros). We report results of the comparative experiments and provide recommendations and tools for accurate estimation of missing microarray data under a variety of conditions.     

Fuzzy C-means method for clustering microarray data

MOTIVATION: Clustering analysis of data from DNA microarray hybridization studies is essential for identifying biologically relevant groups of genes. Partitional clustering methods such as K-means or self-organizing maps assign each gene to a single cluster. However, these methods do not provide information about the influence of a given gene for the overall shape of clusters. Here we apply a fuzzy partitioning method, Fuzzy C-means (FCM), to attribute cluster membership values to genes. RESULTS: A major problem in applying the FCM method for clustering microarray data is the choice of the fuzziness parameter m. We show that the commonly used value m = 2 is not appropriate for some data sets, and that optimal values for m vary widely from one data set to another. We propose an empirical method, based on the distribution of distances between genes in a given data set, to determine an adequate value for m. By setting threshold levels for the membership values, genes which are tigthly associated to a given cluster can be selected. Using a yeast cell cycle data set as an example, we show that this selection increases the overall biological significance of the genes within the cluster. AVAILABILITY: Supplementary text and Matlab functions are available at http://www-igbmc.u-strasbg.fr/fcm/     

Fast and accurate population admixture inference from genotype data from a few microsatellites to millions of SNPs

Model-based (likelihood and Bayesian) and non-model-based (PCA and K-means clustering) methods were developed to identify populations and assign individuals to the identified populations using marker genotype data. Model-based methods are favoured because they are based on a probabilistic model of population genetics with biologically meaningful parameters and thus produce results that are easily interpretable and applicable. Furthermore, they often yield more accurate structure inferences than non-model-based methods. However, current model-based methods either are computationally demanding and thus applicable to small problems only or use simplified admixture models that could yield inaccurate results in difficult situations such as unbalanced sampling. In this study, I propose new likelihood methods for fast and accurate population admixture inference using genotype data from a few multiallelic microsatellites to millions of diallelic SNPs. The methods conduct first a clustering analysis of coarse-grained population structure by using the mixture model and the simulated annealing algorithm, and then an admixture analysis of fine-grained population structure by using the clustering results as a starting point in an expectation maximisation algorithm. Extensive analyses of both simulated and empirical data show that the new methods compare favourably with existing methods in both accuracy and running speed. They can analyse small datasets with just a few multiallelic microsatellites but can also handle in parallel terabytes of data with millions of markers and millions of individuals. In difficult situations such as many and/or lowly differentiated populations, unbalanced or very small samples of individuals, the new methods are substantially more accurate than other methods.Subject terms: Population genetics, Evolutionary ecology     

Binary tree-structured vector quantization approach to clustering and visualizing microarray data

MOTIVATION: With the increasing number of gene expression databases, the need for more powerful analysis and visualization tools is growing. Many techniques have successfully been applied to unravel latent similarities among genes and/or experiments. Most of the current systems for microarray data analysis use statistical methods, hierarchical clustering, self-organizing maps, support vector machines, or k-means clustering to organize genes or experiments into 'meaningful' groups. Without prior explicit bias almost all of these clustering methods applied to gene expression data not only produce different results, but may also produce clusters with little or no biological relevance. Of these methods, agglomerative hierarchical clustering has been the most widely applied, although many limitations have been identified. RESULTS: Starting with a systematic comparison of the underlying theories behind clustering approaches, we have devised a technique that combines tree-structured vector quantization and partitive k-means clustering (BTSVQ). This hybrid technique has revealed clinically relevant clusters in three large publicly available data sets. In contrast to existing systems, our approach is less sensitive to data preprocessing and data normalization. In addition, the clustering results produced by the technique have strong similarities to those of self-organizing maps (SOMs). We discuss the advantages and the mathematical reasoning behind our approach.     

Combining multiple classifications of chemical structures using consensus clustering

Consensus clustering involves combining multiple clusterings of the same set of objects to achieve a single clustering that will, hopefully, provide a better picture of the groupings that are present in a dataset. This Letter reports the use of consensus clustering methods on sets of chemical compounds represented by 2D fingerprints. Experiments with DUD, IDAlert, MDDR and MUV data suggests that consensus methods are unlikely to result in significant improvements in clustering effectiveness as compared to the use of a single clustering method.