Consensus clustering and functional interpretation of gene-expression data
© Swift et al.; licensee BioMed Central Ltd. 2004
Received: 4 December 2003
Accepted: 13 September 2004
Published: 1 November 2004
Microarray analysis using clustering algorithms can suffer from lack of inter-method consistency in assigning related gene-expression profiles to clusters. Obtaining a consensus set of clusters from a number of clustering methods should improve confidence in gene-expression analysis. Here we introduce consensus clustering, which provides such an advantage. When coupled with a statistically based gene functional analysis, our method allowed the identification of novel genes regulated by NFκB and the unfolded protein response in certain B-cell lymphomas.
There are many practical applications that involve the grouping of a set of objects into a number of mutually exclusive subsets. Methods to achieve the partitioning of objects related by correlation or distance metrics are collectively known as clustering algorithms. Any algorithm that applies a global search for optimal clusters in a given dataset will run in exponential time to the size of problem space, and therefore heuristics are normally required to cope with most real-world clustering problems. This is especially true in microarray analysis, where gene-expression data can contain many thousands of variables. The ability to divide data into groups of genes sharing patterns of coexpression allows more detailed biological insights into global regulation of gene expression and cellular function.
Many different heuristic algorithms are available for clustering. Representative statistical methods include k-means, hierarchical clustering (HC) and partitioning around medoids (PAM) [1–3]. Most algorithms make use of a starting allocation of variables based, for example, on random points in the data space or on the most correlated variables, and which therefore contain an inherent bias in their search space. These methods are also prone to becoming stuck in local maxima during the search. Nevertheless, they have been used for partitioning gene-expression data with notable success [4, 5]. Artificial Intelligence (AI) techniques such as genetic algorithms, neural networks and simulated annealing (SA)  have also been used to solve the grouping problem, resulting in more general partitioning methods that can be applied to clustering [7, 8]. In addition, other clustering methods developed within the bioinformatics community, such as the cluster affinity search technique (CAST), have been applied to gene-expression data analysis . Importantly, all of these methods aim to overcome the biases and local maxima involved during a search but to do this requires fine-tuning of parameters.
Recently, a number of studies have attempted to compare and validate cluster method consistency. Cluster validation can be split into two main procedures: internal validation, involving the use of information contained within the given dataset to assess the validity of the clusters; or external validation, based on assessing cluster results relative to another data source, for example, gene function annotation. Internal validation methods include comparing a number of clustering algorithms based upon a figure of merit (FOM) metric, which rates the predictive power of a clustering arrangement using a leave-one-out technique . This and other metrics for assessing agreement between two data partitions [11, 12] readily show the different levels of cluster method disagreement. In addition, when the FOM metric was used with an external cluster validity measure, similar inconsistencies are observed .
The weighted-kappa guideline
0.0 ≤ K ≤ 0.2
0.2 <K ≤ 0.4
0.4 <K ≤ 0.6
0.6 <K ≤ 0.8
0.8 <K ≤ 1.0
Despite the formal assessment of clustering methods, there remains a practical need to extract reliably clustered genes from a given gene-expression matrix. This could be achieved by capturing the relative merits of the different clustering algorithms and by providing a usable statistical framework for analyzing such clusters. Recently, methods for gene-function prediction using similarities in gene-expression profiles between annotated and uncharacterized genes have been described . To circumvent the problems of clustering algorithm discordance, Wu et al. used five different clustering algorithms and a variety of parameter settings on a single gene-expression matrix to construct a database of different gene-expression clusters. From these clusters, statistically significant functions were assigned using existing biological knowledge.
In this paper, we confirm previous work showing gene-expression clustering algorithm discordance using a direct measurement of similarity: the weighted-kappa metric. Because of the observed variation between clustering methods, we have developed techniques for combining the results of different clustering algorithms to produce more reliable clusters. A method for clustering gene-expression data using resampling techniques on a single clustering method has been proposed for microarray analysis . In addition, Wu et al. showed that clusters that are statistically significant with respect to gene function could be identified within a database of clusters produced from different algorithms . Here we describe a fusion of these two approaches using a 'consensus' strategy to produce both robust and consensus clustering (CC) of gene-expression data and assign statistical significance to these clusters from known gene functions. Our method is different from the approach of Monti et al., in that different clustering algorithms are used rather than perturbing the gene-expression data for a single algorithm . Our method is also distinct from the cluster database approach of Wu et al . There, clusters from different algorithms were in effect fused if the consensus view of all algorithms indicated that the gene-expression profiles clustered independently of the method. In the absence of a defined rule base for selecting clustering algorithms, we have implemented clustering methods from the statistical, AI and data-mining communities to prevent 'cluster-method type' biases. When consensus clustering was used with probabilistic measures of cluster membership derived from external validation with gene function annotations, it was possible to accurately and rapidly identify specific transcriptionally co-regulated genes from microarray data of distinct B-cell lymphoma types .
Cluster method comparison
Initially we assessed cluster method consistency for HC, PAM, SA and CAST using the weighted-kappa metric and a synthetic dataset of 2,217 gene-expression profiles over 100 time points that partitioned into 40 known clusters. The weighted-kappa values derived from the metric indicate the strength of agreement between two observers (Table 1). To interpret two weighted-kappa scores, for example, from two cluster arrangements, the broad categories from Table 1 are used, together with an assessment of relative score differences. If the two scores in question were 0.2 and 0.4, one could say that the former is poor (worse) and the latter is fair (better), but not that one is twice as good as the other. To allow defined clusters to be extracted from the tree structure of HC we used the R statistical package  implementation of HC. This implementation uses the CUTTREE method to convert the tree structure into a specified number of clusters.
The partial agreement of the different clustering algorithms must reflect the clustering of highly similar gene-expression vectors regardless of the clustering methods used. Where algorithm-based inconsistency problems occur in other aspects of computational biology, such as protein secondary structure prediction, consensus algorithms are often used . These can either report a full or a majority agreement. This consensus strategy has also been applied to explore the effect of perturbing the gene-expression data for a single clustering algorithm . We have therefore designed a similar strategy to identify the consistently clustered gene-expression profiles in microarray datasets by producing a consensus over different clustering methods for a given parameter set (see Materials and methods). Extracting such consistently clustered robust data from a large gene-expression matrix is extremely useful, increasing overall analysis confidence.
Robust clustering uses the agreement matrix to generate a list, List, which contains all the pairs where the appropriate cell in the agreement matrix contains a value equal to the number of clustering methods being combined (that is, full agreement). Starting with an empty set of robust clusters RC, where RC i is the ith robust cluster, the first cluster is created containing the elements of the first pair in List. Then the pairs in List are iterated through and checked to see if one of the members of the current pair is within any of the existing clusters, RC i .
If one element of the current pair is found and the other element of the pair is not in the same cluster, then the other element is added to that cluster. If neither element of the pair is found in an existing RC i in RC, then a new cluster is added to RC containing each element of the pair. When the end of the list is reached, the set of robust clusters, RC, is the output. The robust clustering algorithm is as follows:
Input:Agreement Matrix (n× n), A
(1) Set List= all pairs (x, y) in the matrix, with agreement = the number of methods
(2) Set RC to be an empty list of clusters
(3) Create a cluster and insert the two elements (x, y) of the first pair in List into it
(4) For i = 2 to size of List -1
(5) For j = 1 to number of Clusters in RC
(6) If x or y of List i is found within RC j
(7) If the other member of the pair List i is not found in RC j
(8) Add the other member to RC j
(9) End If
(10) Else If the other member of the pair List i is not found in RC j
(11) Add a new cluster to RC containing x and y
(12) End If
(13) End For
(14) End For
Output:Set of Robust Clusters RC
Application of robust clustering
Number of robust clusters
% of variables assigned
Maximum robust cluster size
Minimum robust cluster size
Mean robust cluster size
The robust clustering algorithm is, by definition, subject to discarding gene-expression vectors if only one clustering method performs badly in the co-clustering. This effect of single method under performance on a given dataset has been previously observed for single linkage hierarchical clustering [10, 13]. Therefore, to generate clusters with high agreement across methods but not so restrictive as to discard majority consistent variables, we adapted the algorithm to generate consensus clusters, making use of the same agreement matrix.
Consensus clustering relaxes the full agreement requirement by taking a parameter, 'minimum agreement', which allows different agreement thresholds to be explored. Rather than grouping variables on the basis of full agreement only, consensus clustering maximizes a metric, which rewards variables in the same cluster if they have high cluster method agreement and penalizes variables in the same cluster if they have low agreement. Consensus clustering maximizes agreement using the function f(G i ) in Equation (1) to score each cluster of size s i
where A is the agreement matrix, G ij is the jth element of cluster i (G i ) and β is a user-defined parameter (the agreement threshold), which determines whether the score for the cluster is increased or decreased. The score for a clustering arrangement is the sum of the scores of each cluster, which consensus clustering attempts to maximize.
If β is equal to Min, the minimum value in A, then the function is maximized when all variables are placed into the same cluster (that is, a single large cluster). Alternatively, when β is equal to Max, the maximum value in A, the function is maximized when each variable is placed into its own cluster. Essentially all clusters produced by Consensus Clustering are scored by f(G i ), rewarding and preserving clusters with high agreement between members, while penalizing and discarding clusters containing low agreement between members. A value for β should lie between the minimum and the maximum agreement so as not to skew the scoring function. A suitable value for β is (Max + Min)/2, where Max is the maximum value in A and Min is the minimum. For a uniformly distributed agreement matrix, (Max + Min)/2 is the mean value; therefore we penalize values below the mean agreement and reward above it. For both the ASC and B-cell lymphoma data β was 2, as Max = 4 (four clustering algorithms giving complete agreement) and Min = 0 (no agreement). In order to maximize the scoring function for consensus clustering, a search over possible cluster membership is needed. There are many methods for performing a search and it was decided that SA was best because it is an efficient search/optimization procedure that does not suffer from becoming stuck in local maxima. The consensus algorithm is as follows:
Input: Agreement Matrix (n × n), A; Maximum Number of Clusters sought, m; Number of Iterations, Iter; Agreement Threshold, β; Initial Temperature, θ0; Cooling Rate, c
(1) Generate a random number of empty clusters (<m)
(2) Randomly distribute the variables (genes) 1.. n between the clusters
(3) Score each cluster according to Equation (1)
(4) For i = 1 to Iter do
(5) Either Split a cluster, Merge two clusters or Move a variable (gene) from one random cluster to another
(6) Set Δf to difference in score according to Equation (1)
(7) If Δ f < 0 Then
(8) Calculate probability, p, according to Equation (2)
(9) If p > random(0,1) then undo operator
(10) End If
(11) θ i = cθ i -1
(12) End For
Output:Set of Consensus Clusters
Note that random(0,1) (line 9) returns a random uniformly distributed real number between 0 and 1.
The 'split', 'merge' and 'move' operators (line 5) are as follows and used with equal probability:
Split a cluster:
Input: Cluster g of size n
(1) Randomly shuffle the cluster
(2) Set i to be a random whole number between 1 and n -1
(3) Create two empty clusters g 1 and g 2
(4) Add elements 1.. i of g to g 1
(5) Add elements i+1.. n of g to g 2
Output:Two new clusters g 1 and g 2
Here the old cluster is deleted and the two new clusters are then added to the set of clusters.
Merge two clusters:
Input: Two Clusters g 1 and g 2
(1) A new cluster g is created by forming the union of g 1 and g 2
Output: A new cluster g
Here the old clusters are deleted and new cluster is then added to the set of clusters.
Move a gene:
Input:A set of clusters G
(1) Two random clusters g 1 and g 2 are chosen where the size of g 1 is greater than one
(2) A random element of g 1 is moved into g 2
Output:The updated set of clusters G
The probability (p) (line 8) is calculated by:
In the following experiments we found θ0 = 100, c = 0.99994 and iter = 1,000,000 as the most efficient parameters for SA.
These parameter settings for SA are effectively determined by the iter setting. We denote the change in fitness during the SA algorithm as Δf and the starting temperature as θ0 which is always positive. From equation 2 it can be clearly seen that if Δf = θ0 then the (worse) solution will be accepted with probability 0.368 (e-1). As the temperature cools, this probability will reduce. Here we set θ0 to be the average of Δf over 1,000 trial evaluations, so that at the beginning of the algorithm, the average worse solution (Δf = θ0) will be accepted with the probability stated above.
It can be seen from the consensus algorithm that during the ith stage of the SA algorithm θ0 = θ0c i . The SA algorithm works by assuming that the temperature reduces to zero over an infinite number of iterations. As it is not practical to run the SA algorithm to infinity the method is usually terminated after a fixed number of iterations, (iter). At this time the temperature will not be zero, but very small and positive, say ε. Therefore,
Hence if some small positive value for ε is chosen, and the algorithm is to run for a defined number of iterations (iter), then the decay constant c is calculated as above.
Application of consensus clustering
As consensus clustering relaxes the 'complete agreement' criteria we would expect the majority but not necessarily all robust cluster members to be assigned to the same consensus clusters. This was indeed true for the B-cell data where consensus clustering of the datasets showed that 98.5% of the B-cell robust clusters were assigned correctly to their respective consensus clusters. With the more consistent ASC data 100% of the robust clusters were assigned to the correct consensus clusters.
Multiple runs of the stochastic clustering methods
Simulated annealing (SA)
Consensus clustering (CC)
Cluster partition weighted-kappa scores of two synthetic datasets
Interpretation of consensus clustering
Consensus clustering greatly improves the accuracy of identifying cluster group membership based solely on the gene-expression vector, but as with other clustering algorithms still produces essentially unannotated clusters which require further external validation by gene function analysis. To address this problem, we derived a probability score to test the significance of observing multiple genes with known function in a given cluster against the null hypothesis of this happening by chance. This identifies clusters of high functional group significance, aiding assignment of functions to unclassified genes in the cluster using the 'guilt by association' hypothesis.
The probability score is based on the hypothesis that, if a given cluster, i, of size s i , contains x genes from a defined functional group of size k j , then the chance of this occurring randomly follows a binomial distribution and is defined by:
where n is the number of genes in the dataset. As k j and x may potentially be very large, Pr from the above equation would be difficult to evaluate. Therefore the normal approximation to the binomial distribution can be used as defined by:
Large positive values of z mean that the probability of observing x elements from functional group j in cluster i by chance is very small, (for example z > 2.326 corresponds to a probability less than 1%). Note that we perform a one tailed test as we are only interested in the case where a significantly high number of co-clustered genes belong to the same functional group.
One potential confounding factor in this analysis is that consensus clustering achieves a lower probability score by finding smaller clusters. This would decrease the ability to associated new genes with a given functional group. In the worst case the number of genes defining a functional group (FG) would equal the cluster size (s i ) (FG/s i = 1). Alternatively, single clustering methods may produce lower probability scores by increasing the cluster size, thereby pulling many genes into the cluster resulting in a FG/s i ratio tending towards zero. This would also reduce the usefulness of the clusters. We determined the cluster size and functional group size for two representative functional groups where the consensus clustering probability was similar to the single method probability score, namely the endoplasmic reticulum (ER) stress response (also known as the unfolded protein response) (ER/UPR) functional group, or the markedly better nuclear factor-κB (NFκB) functional group (Figure 4b). Apart from SA, all single clustering methods tended to produce larger clusters, thereby decreasing the FG/s i ratio. In the most extreme case of the ER/UPR functional group, the HC cluster size was 310 compared to the consensus clustering size of 40. SA tended to produce similar cluster sizes as consensus clustering but with higher overall probabilities. Therefore, consensus clustering identifies significant functional clusters while achieving a workable balance between large or small cluster sizes.
This consensus cluster is predominantly overexpressed in the cell lines Raji, Pel-B, EHEB, Bonna-12 and L-428. These cell lines have in common the induction of the NFκB pathway, either through expression of Epstein-Barr virus LMP-1 protein (Raji, Pel-B, EHEB and Bonna-12) or the loss of function of the inhibitor of NFκB, namely IκB (L-428). This implies that many of these genes could be NFκB responsive. Twenty-four putative promoter regions were analyzed and NFκB-binding sites were identified in 12 of these. As expected, NFκB-binding sites were found in the CD40L receptor gene, Bfl-1, BIRC3, EBV-induced gene 3 (EBI3), and the genes for class I MHC-C and lymphotoxin α, as these have been previously characterized as NFκB responsive and were present in the initial NFκB-defined gene set. Interestingly, NFκB-binding sites were also found in six additional promoters for which accurate mapping of promoter transcription factor binding is not available (Figure 6a). All but four NFκB-binding sites conform precisely to the canonical consensus binding site (Figure 6b) [29, 30] and of the variants with T at position 1, two genes, lymphotoxin α and BIRC3 are known to be NFκB responsive. Overall, this indicates that the six additional genes identified are likely to be NFκB responsive.
Grouping data into sets based on a consistent property is a common occurrence in biological analysis. This has recently increased in importance with the production of large microarray datasets. Implicit in the experimental rationale is the fact that patterns of coexpressed genes should be identifiable in a gene expression matrix and these can be linked to shared biological processes. However, different clustering algorithms are known to partition data into different groups [10–13]. We also observe a similar lack of cluster-method concordance using a weighted-kappa metric. This metric effectively scores how well different cluster method pairs assign the same genes to the same clusters. The weighted-kappa metric readily shows that, even for highly correlated gene-expression profiles present in the ACS dataset, no two clustering algorithms have complete agreement, although the global search methods such as SA seem to produce the most consistent results. Overall this emphasizes that no single analysis method will identify all patterns in the gene-expression data; therefore multiple analyses should be performed and compared .
We and others recently described the use of consensus clustering to improve confidence  or as a re-sampling method for microarray analysis . It was suggested that a natural extension of Monti et al. was to use a meta-consensus across different clustering algorithms rather than to re-sample over the same algorithm. Our results represent this extension and confirm the validity of consensus clustering. We have developed both robust and consensus clustering, with these methods offering specific advantages over the use of individual clustering algorithms for microarray analysis. The robust clustering algorithm is useful for creating clusters of genes with high confidence and is extremely effective for reducing the dimensionality of large gene-expression datasets. However, robust clustering can be restrictive in discarding genes that do not have full agreement. Consensus clustering overcomes this problem, requiring a minimum-agreement parameter to generate clusters based on the combined results of a number of existing clustering methods. This strategy enables the effective identification of cluster groups that are of high reliability and cluster method independent.
The choice of clustering algorithms and parameter settings is a major stumbling block for all gene array cluster analysis. The effect of varying the parameters depends on the cluster method used. The performance of cluster methods has been extensively investigated . The authors show that model-based methods and certain partitional methods, when used with optimal distance matrices, perform well on synthetic and real-world data. From our study, SA, an optimization method, also performs well as a clustering method. Therefore, the individual algorithms used as input to consensus clustering should ideally consist of representative algorithms from optimization (for example, SA), graph theoretical (for example, CAST), model-based (for example, MCLUST ) and partitional (for example, HC). Some methods (for example, CAST, SA and MCLUST) can determine the number of clusters directly from the input data. However, some other methods require the number of clusters to be specified as a parameter (for example, PAM and HC). In principle, methods such as CAST, SA and MCLUST can be used to determine this parameter for methods such as PAM and HC.
Consensus clusters are likely to contain gene subsets that are co-regulated by common transcriptional control networks or are coexpressed to participate in cellular processes that together manifest a global phenotype of the cell or tissue. In either case, these clusters are of high biological value. To facilitate further analysis it is useful to know which clusters are involved in a given biological process. By supplying a list of genes from a given biological process or network, the use of the normal approximation to the binomial distribution of these genes over all consensus clusters, allows the identification of clusters of high functional significance. Similar statistical assignment of gene function based on cluster analysis was performed by Wu et al. using a database of clusters . To assign significance Wu et al. used the hypergeometric distribution. This distribution can be formally shown to asymptotically become the binomial distribution when the population size increases. Therefore, when used on large gene-expression datasets our methods are directly analogous to Wu et al. However, consensus clustering has the advantage over a database of clusters by producing low-probability clusters containing a significant percentage of known elements from functional groups.
Two functionally significant clusters, the NFκB-responsive cluster and the ER/UPR cluster, were investigated further here. Within the NFκB-responsive cluster 50% of the putative promoters of genes investigated had canonical NFκB-binding sites within 1,000 base pairs of the transcriptional start site, suggesting that they are NFκB responsive [29, 30]. The majority of these genes had NFκB-binding sites within 500 bp of the transcriptional start sites, consistent with the location in other NFκB-responsive genes . Of the remaining two genes with NFκB-binding sites greater than 800 bp from the transcription start site, one, Bfl-2, has been experimentally verified . Analysis of the ER/UPR consensus cluster also provided information on gene regulatory elements, but more interestingly provides insights into the control and effect of the ERSR/UPR.
In cells, the presence of unfolded proteins in the ER is associated with induction of the ER/UPR. However, during the maturation of B-cells to antibody-secreting plasma cells, expansion of the ER to accommodate increased secretion of immunoglobulins is thought to be coupled to the final stages of plasma-cell maturation. The induction of the ER/UPR occurs via the coordinated activation of the transcription factors ATF6 and XBP1 [31, 33]. ATF6 is normally maintained as an inactive, ER-resident, transmembrane protein that is cleaved, after translocating to the Golgi upon ER stress, by the site proteases S1P and S2P [37, 38]. The cleaved transcriptionally active ATF6 is then free to translocate to the nucleus, where it can activate target genes such as XBP1 and the ER chaperon protein GRP78/BiP . XBP1 mRNA is cleaved by the ER stress activated protein IRE1 to yield the transcriptionally active form of XBP1, inducing further genes of the UPR . The activation of both the ATF6 and IRE1/XBP1 pathways results in enhanced transcription of ESRE-responsive genes; however, only XBP1 appears able to transactivate the UPRE. The identification of ESRE binding sites in the promoter regions of genes for calcium-ion binding protein and UPRE binding sites in the promoter regions of N-glycan biosynthesis genes suggests that these genes are differentially regulated by ATF6/XBP1 and XBP1 respectively.
The only known UPRE target gene is ER degradation-enhancing α-mannoside-like protein (EDEM), whose induction depends solely on IRE1/XBP1 activity . Induction of the two UPRE-containing genes, UDP-GlcNAc:dolichol phosphate N-acetylglucosamine-1 phosphate transferase (DPAGT1) and UDP-GlcNAc:α-6-D-mannoside β-1-2-N-acetylglucosaminyltransferase II (MGAT2), which catalyze essential steps in the biosynthetic pathway of complex N-linked glycans, supporting a clear link between the dolichol pathway and the UPR . In addition, the ER/UPR functional group suggests that DPAGT1 and MGAT2 expression is regulated solely by the IRE1/XBP1 pathway. Altogether, these results show that consensus clustering and gene functional group analysis provide a highly accurate way of mining gene-expression data for novel insights into different genes within the cluster.
Robust and consensus clustering provide a platform for more efficient microarray analysis pipelines. There is effectively no limit to the number of different clustering algorithms that can be used to feed into the consensus clusters, and each clustering algorithm could be run under different parameter sets to fully explore a microarray dataset . In addition, different distance matrices could be used as input into the range of clustering algorithms. In each case the consensus clustering algorithm effectively acts as the collation and interpretation point for the different individual analysis methods. This environment is ideal for use in parallel processing computer farms and the GRID . In such an environment, each node of the farm could perform a range of analyses with a subset of clustering algorithms, with the master node compiling the consensus results. This would greatly increase computational speed and allow a thorough, single data entry point, access to an extensive range of clustering methods.
All areas of functional genomics that produce high-dimensional datasets with inherent patterns will require data partitioning to allow interpretation. Consensus clustering in the context of statistically defined functional groups could allow a consistent analysis platform for such diverse data types.
Materials and methods
We implemented and compared a representative sample of methods from the statistical, AI and data-mining communities. The methods used were average linkage HC, PAM, SA and CAST. As all the clustering techniques use correlation between variables, we used the Pearson's correlation coefficient, r, to measure the linear relationship between two variables, x1 and x2, where the variable can be either discrete or continuous . HC and PAM methods were implemented using the statistical package R , while CAST and SA were implemented locally in C++.
HC is an agglomerative method that produces a hierarchical (binary) tree or dendrogram representing a nested set of data partitions. It has been applied successfully to many gene-expression datasets . Sectioning a hierarchically clustered tree at a particular level leads to a partition with a number of disjointed groups, thereby yielding different clustering of the data. The tree was sectioned using the CUTTREE method, to yield 13 clusters for the ASC dataset and 40 for the B-cell dataset. The method PAM works by first selecting m out of n total objects that are the closest (according to a distance matrix) to the remaining (n - m) objects. The fitness of these medoids is calculated by placing the remaining (n - m) objects in a group according to the nearest medoid and summing the distances of the group members from this medoid. These m selected objects are the initial medoids. A Swapping procedure is then applied to reassort the objects until there is no improvement in the fitness of the medoids . As with HC, PAM is set to search for 13 and 40 clusters. The choice of 13 clusters for the ASC data was determined by the number of repeated genes, whereas 40 clusters for the B-cell data was based on previous exploratory data analysis .
SA  is an iterative improvement search technique that starts with a random solution to a given problem, and then tries to increase its worth by a series of small changes in cluster membership. If such a small change is better than the previous solution, then further changes are made from this new point. However, if the new solution is worse than the old one, it is not discarded, but accepted with a certain probability. The measured worth of the SA clustering arrangement is based here on the EVM metric . SA has recently been applied to the clustering of gene-expression data . The performance related parameters for SA were set as follows: θ0 = 100, c = 0.99994 and number of iterations = 1,000,000.
CAST  is a heuristic algorithm that uses an affinity measure to determine whether variables are assigned to clusters. It requires a threshold parameter, which determines whether variables are assigned or moved to new clusters. Once CAST is complete, a clean-up operation is applied to ensure that the affinity of every variable to its cluster is greater than a user defined threshold. The only parameter CAST needed was the affinity level, which was set to 0.5 as recommended .
Methods such as CAST and SA require the differences/relationships between a pair of observations, x1 and x2, to be expressed as binary (b). As Pearson's correlation coefficient is bounded, it provides a good basis for defining a binary relationship between two variables as defined by:
Datasets for evaluation
Two datasets were used for evaluating the cluster methods. The first is a set of multiply repeated control element spots relating to the Amersham Score Card (ASC) probe set on the Human Genome Mapping Project Human Gen1 clone set array . The ASC probes are present as a single row of 32 elements in each of the 24 array sub-grids. Of these elements, 13 gene probes consistently give signals above background in both the Cy5 and Cy3 channels. Therefore, each array has effectively 24 repeat measurements of 13 spots. After filtering for low signal-to-noise ratio (SNR) probes, a dataset of 30 arrays was examined by treating each positional repeat probe element across the 30 array set as an individual gene, which together with the remaining 23 same-gene probes per array, represents a highly correlated gene-expression profile. Therefore, we assume the repeated probes should cluster together; thus, this dataset becomes 308 genes/probe elements, which would cluster into 13 known groups, with each group having between 6 and 24 members after SNR filtering. In essence, the ASC data represent a semi-synthetic dataset for internal cluster method validation.
The second dataset consists of a series of 26 arrays (1,987 filtered genes) measuring gene-expression difference across a set of human B-cell lymphomas and leukemias . The dataset is available via the URL indicated in Jenner et al. . Each probe on the array detected a single gene transcript. This dataset contains a number of genes that correspond to known cellular functions, for example cell proliferation and NFκB response. The four clustering techniques described above were applied to both the datasets, with each method being set to find 13 clusters for the ASC and 40 clusters for the B-cell data.
Two synthetic datasets were generated using a vector autoregressive process (VAR) or a multivariate normal distribution (MVN). The two datasets contained 1,830 genes and 20 conditions (arrays). The VAR process of order p is a linear multivariate time series defined by
where x(t) is the n-dimensional vector of observations at time t, A i is the n × n autoregressive coefficient matrix at time lag i, and ε(t) is the zero mean n-dimensional noise vector at time t (drawn from a normal distribution). Therefore x(t) is a linear combination of the previous observations plus some random noise. For the synthetic dataset, each cluster was generated by a vector autoregressive model of order p = 1 and size n equal to the number of genes in the cluster.
For the MVN dataset, a vector of random variables x has a MVN distribution if every linear combination of that vector is also normal. Under such conditions we use the notation x ~ N(μ, Σ) to denote that x follows the MVN distribution, where μ is the mean vector and Σ is a positive definite matrix of covariance. The probability density function of x is given by
where |Σ| = det(Σ). For the synthetic dataset, each cluster was drawn from an MVN distribution with varying mean μ and covariance Σ.
To compare the resultant clusters for each method, a statistic known as weighted-kappa was used . This metric rates agreement between the classification decisions made by two or more observers. In this case the two observers are the clustering methods. The classification from each observer for each unique pairing of variables (within the clusters) is divided into a 2 × 2 contingency table. Rows and columns within this table are indexed according to whether the two variables are in the same group or in different groups. The total number of comparisons, N, is defined in the following equation, where Count ij is the number of elements in the matrix cell indexed by (i,j),
and n is the number of variables (genes) in the clusters as this represents the number of unique variable pairings.
The weighted-kappa metric is calculated from the contigency table by:
where, w ij is the weights for each category comparison; po(w) and pe(w) represent the observed weighted proportional agreement and the expected weighted proportional agreement; Count ij is the ith, jth element of the 2 × 2 contingency table; N is the sum of the elements within this table; Row(i) and Col(i) are the row and column totals for this table respectively and K w is the weighted-kappa value. The interpretation of weighted-kappa values indicates the strength of agreement between two observers (Table 1) is used to compare cluster method agreement in both datasets.
This work is supported in part by the BBSRC, the EPSRC and the MRC in the UK. We would also like to thank Richard Jenner for the viral gene expression dataset and Antonia Kwan for preparing the ASC dataset. These methods are available on request as functions in the statistical package, R.
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