Experimental Design

We perform experiments for each algorithm, varying the number of clusters in [k, ⌈$\sqrt{n}$⌉], where k represents the actual number of classes in a given data set with n samples. The recovery of the cluster structure is measured using the corrected Rand (cR) index by comparing the actual classes of the tissues samples (e.g., cancer types/subtypes) with the cluster assignments of the tissue samples. For each combination of clustering method, proximity measure and data transformation procedure, we calculate the mean of the cR over all data sets in two different contexts: (1) taking into account only the partition with the number of clusters equal to the number of actual classes in the respective data set; and (2) considering the partition that presents the best cR for each data set, regardless of its number of clusters. The results of these experiments are illustrated in Figure 1 (Affymetrix data sets) and Figure 2 (cDNA data sets). We also investigate the influence of reduced coverage on the algorithms [29]. The idea is to identify the methods that often generate their best partition, as calculated by the cR, with more clusters than the actual number of classes in the corresponding data set. In order to do so, for a given algorithm, we measure the difference between the number of clusters of the partition with best cR and the actual number of classes for each data set. The mean of the difference of these values for Affymetrix and cDNAs data sets are displayed in Figures 3 and 4, respectively.

Finally, we perform paired t-tests, adjusted for multiple comparisons with the Benjamini and Hochberg procedure [39], to determine whether differences in the cR between distinct methods (as displayed in Figure 1 and Figure 2) are statistically significant. More specifically, given that we are mainly interested in comparing the clustering methods, we select only the results with the proximity measure that displays the largest mean of cR for each method. As some data sets are harder to cluster than others, the cR values exhibited a large variance for a given clustering method and proximity measure (Figures 1 and Figure 2). Nonetheless, as all methods are applied to the same data sets, we can use the paired t-test, which measures the difference in the cR value between two methods for each data set. In this case, the null hypothesis is that the difference in the mean between paired observations is zero. We can therefore assess whether the difference in the performance of one method and another is statistically significant. More detailed results are found out in our supplementary material [38].

Recovery of Cancer Types by Clustering Method

Based on Figures 1 and 2, our results show that, among the 35 data sets investigated, the FMG exhibited the best performance – best corrected Rand (cR) -, followed closely by KM, in terms of the recovery of the actual structure of the data sets, regardless of the proximity measure used.

For Affymetrix data sets, the paired t-test (p-value < 0.05) indicated that FMG achieved a larger cR than SL, AL and CL; and KM a larger cR than SL and AL, when the number of cluster is set to the actual number of classes. Considering the partition that presents the best cR for each data set, regardless of its number of clusters, KM and FMG achieved a larger cR than SL, AL and CL (p-value < 0.05).

In the case of cDNA data sets, the paired t-test (p-value < 0.05) also indicated that KM and FMG achieved a larger cR than SL, AL, CL and SNN for both contexts investigated. Also, KM achieved a larger cR than SPC (p-value < 0.05). This holds for both contexts: the number of cluster sets to the actual number of classes and for the best cR, regardless of the number of clusters.

Furthermore, KM and FMG achieved, on average, the smallest difference between the actual number of classes in the data sets and the number of clusters in the partition solutions with the best cR (Figures 3 and 4). Likewise, SNN exhibited consistent behavior in terms of the values of cR in the different types of proximity measures, although with smaller cR values than those obtained with FMG and KM. In fact, this method, on average, returned cR values compatible to those achieved by the SPC.

Note that a good coverage alone (Figure 3 and Figure 4) do not imply accuracy in classes recovery (Figures 1 and 2). For example, according to Figure 3, SPC with ${\text{E}}_{Z_0}$ and KM with C respectively present a mean of the difference between the actual number of classes and number of clusters found in the partitions with best cR of 0.71 and 0.76 clusters. However, the latter led to a cR of 0.50, while the former achieved a cR of only 0.12.

Our results show that the class of hierarchical methods, on average, exhibited a poorer recovery performance than that of the other methods evaluated. Moreover, as expected, within this class of algorithms, the single linkage achieved the worst results. The paired t-test (p-value < 0.05) indicated that it led to the smallest cR, when compared to those of the other methods. Regarding the use of proximity measures with hierarchical methods, Pearson's correlation and cosine yielded best results. This is also in agreement with the overall common knowledge from clinical/biological studies [22]. In order to present cR values compatible to those obtained with KM and FMG, such a class of methods generally required a much more reduced coverage: a larger number of clusters than that in the underlying data. For example, according to Figure 1, the AL with P, achieves a cR of 0.27 for the actual number of classes. In contrast, if one considers partition solutions with a larger number of clusters, the cR increases to 0.38. Such an increase was achieved with partitions that had, on average, 1.48 more clusters than in the underlying structure. One surprising result is the good performance achieved with Z₃, mainly with the hierarchical clustering methods. In this context, the use of such a data transformation, together with the Euclidean distance, led to results very close to those obtained with P, C and SP, especially for the Affymetrix data sets. One reason for this behavior is the presence of outliers in the data sets, as Z₃ reduces their impact [29]. Thus, this is further evidence that such a class of algorithms is more susceptible to outliers than the other methods investigated.

Spectral clustering, in turn, is quite sensitive to the proximity measure employed. For example, the partitions generated with this method achieved large cR values (e.g., cR > 0.40) only for the cases of C, SP and P, but smaller cR values (e.g., cR ≤ 0.15) otherwise. It is well known in the literature that spectral clustering is susceptible to the similarity matrix used [40]. Moreover, there is as yet no rule to assist in making such a selection.

In another kind of analysis, we investigated the impact of reduced coverage on the performance of the algorithms. As previously mentioned, this impact was more significant for the case of the hierarchical clustering methods. As there are many data sets with an extremely unbalanced distribution of samples within the classes, this behavior also occurred with the KM, although to a smaller degree. In fact, all methods, even the ones that are supposed to deal well with clusters with an unbalanced number of samples, profited from a reduced coverage.

Comparison of Hierarchical Clustering and k-means

To better illustrate the issues discussed in the previous section, we analyzed the results obtained with k-means and hierarchical clustering for a single data set. More specifically, we employed a version of the data set in [4] (Alizadeth-V2), which has samples from three cancer types: 42 diffuse large B-cell lymphoma (DLBCL), 9 follicular lymphoma (FL) and 11 chronic lymphocytic leukemia (CLL). In the case of hierarchical clustering, we used the same algorithm as in [4], which is hierarchical clustering with average linkage and Pearson's correlation. In order to improve visualization of the red and green plot [22], the leaves of the resulting tree were rearranged according to a standard technique [41].

The partitions with three clusters for hierarchical clustering and k-means are illustrated in Figure 5(a) and Figure 5(b), respectively. The k-means yielded a partition very close to the underlying structure in the data. With the exception of a single DLBCL sample that was wrongly assigned to Cluster 1 (all other samples in this cluster are FL), the other clusters in the partition have samples of only the given class. In the case of hierarchical clustering, cutting the tree at level three (three sub-trees or clusters), one can observe the following: the red sub-tree contains only DLBCL samples and the blue sub-tree presents a combination of FL and CLL samples. The same DLBCL sample wrongly assigned in the case of k-means appeared in the blue sub-tree rather than the red sub-tree. The third sub-tree (black) is formed by a single sample from DLBCL (top). Indeed, in the tree generated, the FL and CLL samples are only separated if we cut the tree at level 25, which yields 25 sub-trees (clusters). Note that, although our tree differs slightly from the one in [4], as their analysis included additional non-cancer tissue samples; in that paper the FL and CLL samples would also only be in separate sub-trees when the tree is cut at a lower level.

One of the reasons for this kind of problem is that hierarchical clustering is based on local decisions, merging the most "compact" cluster available at each step [24]. The compactness criterion (or how close together objects are) is defined by the linkage criteria and proximity measure used. On the other hand, k-means maximizes a criterion, which is a combination of cluster compactness and cluster separation. The problem can be illustrated in a two-dimensional representation of the Alizadeth-V2 data set, after selecting the two-largest components using principal component analysis (PCA) – Figure 6. Based on this figure, we can see that samples from distinct classes form natural clusters. More specifically, although the cluster with DLBCL samples (red dots) is well-separated from the other two clusters, it lies within a non-compacted region. Thus, as hierarchical clustering has a bias towards compacted clusters, if we simply follow the hierarchical tree, it would first suggest the sub-division of the cluster with DLBCL samples before sub-dividing the groups with FL and CLL samples. In a hypothetical scenario where there is no a priori knowledge on the distinction between FL and CLL samples, the use of hierarchical clustering alone would not indicate the existence of these two classes. In contrast, this would be detected with k-means or clearly visualized with a simple PCA analysis.

Clustering Methods and Recovery Measure

Seven clustering algorithms are used to generate partition solutions and form a single factor in the overall experiment design: single linkage, complete linkage, average linkage, k-means, mixture of multivariate Gaussians, spectral clustering and shared nearest neighbor-based clustering (SNN). These algorithms have been chosen to provide a wide range of recovery effectiveness, as well as to give some generality to the results. In our analysis, when applicable, methods are implemented with Pearson's Correlation, cosine and Euclidean distance. We also use single linkage, complete linkage, average linkage and spectral clustering with the Spearman's correlation coefficient – the implementation that we used for k-means (Matlab) and SNN [28] do not support such a proximity measure.

Hierarchical clustering methods, more specifically the agglomerative ones, are procedures for transforming a distance matrix into a dendrogram [24]. Such algorithms start with each sample representing a cluster. The methods then gradually merge these clusters into larger ones; they start with trivial clustering in which each sample is in a unique cluster, ending with trivial clustering in which all samples are in the same cluster. There are three more widely used variations of hierarchical clustering, which are used in this paper: complete linkage (CL), average linkage (AL) and single linkage (SL). These variations differ in the way the distance between two clusters is calculated. For SL, the distance between two clusters is determined by the two closest samples in different clusters. In contrast, CL employs the farthest distance of a pair of samples to define the inter-cluster distance. In AL, the distance between two clusters is calculated by the average distance between the samples in one group and the samples in the other group. This method has been used extensively in the literature on gene expression analysis [31, 32, 46, 47], although experimental results have shown that, in many cases, the complete linkage outperforms it [5]. Another widely used method for gene expression data analysis is k-means [46, 47]. k-means (KM) is a partitional iterative algorithm that optimizes the best fitting between clusters and their representation using a predefined number of clusters [24, 25]. Starting with prototype values from randomly selected samples, the method works in two alternate steps: (1) an allocation step, where all samples are allocated to the cluster containing the prototype with the lowest dissimilarity; and (2) a representation step, where a prototype is constructed for each cluster. One problem with such an algorithm is its sensitivity to the selection of the initial partition. This could lead the algorithm to converge to a local minimum [24]. In order to prevent the local minimum problem, a number of runs with different initializations is often performed. The best run based on some cohesion measure is then taken as the result. Another characteristic of this method is its robustness to noisy data.

Finite mixture of Gaussians (FMG) is a convex summation of k multivariate Gaussian density functions or Gaussian components [26]. In mixture model-based clustering, each component in the mixture is assumed to model a group of samples. In order to obtain the probabilities of a sample belonging to each cluster (group), density functions can be used and mixing coefficients can be defined for the mixture. For a given number k, mixture models can be efficiently computed with the Expectation-Maximation (EM) algorithm. The EM works interactively by calculating the probability of each sample for the components and then recomputing the parameters of the individual Gaussian densities until convergence is reached. A mixture model has a number of advantages: it returns the uncertainty of a given cluster assignment and the models estimated can be seen as descriptors of the clusters found. In practice, k-means can be regarded as an oversimplification of a mixture of Gaussians, where every sample is assigned to a single cluster. In this simplified context, clusters tend to have the same size and lie within spherical regions of the Euclidean space. With the mixture of Gaussians using unit covariance matrices employed in this paper, clusters will also lie within spherical regions, but can have any arbitrary size.

Spectral clustering (SPC) is a general class of algorithms characterized by employing the spectrum of similarity matrices to reduce the dimensionality of a data set and then applying a basic clustering algorithm, such as k-means or graph cut-based methods, on this lower dimension data [27]. For example, assume that we have a graph, where nodes are the samples and edges are weighed by the similarity between two nodes. Spectral clustering methods can be interpreted as performing a random walk in this graph, finding clusters by ignoring edges that are rarely transversed in the walk. More specifically, in this paper, for a given similarity matrix S obtained with a Gaussian similarity function, we (1) calculate its normalized Laplacian matrix and (2) perform an eigenvalue decomposition of this matrix. We then select the k eigenvectors related to the k lowest eigenvalues and use them to perform k-mean clustering. Among other interesting characteristics, a spectral clustering method makes no assumptions on the data distribution at hand. It is also able to find clusters that are not in convex regions of the space.

The shared nearest-neighbor algorithm (SNN) is a recent technique. We have added it to our analysis because this method can robustly deal with high dimensionality, noise and outliers [28]. SNN searches for the nearest neighbors of each sample and uses the number of neighbors that two points share as the proximity index between them. With this index, SNN employs a density-based approach to find representative samples and build clusters around them. Along with the number of nearest neighbors (NN), two other kinds of parameters are considered: those regarding the weights of the shared nearest neighbor graph (strong, merge and label) and others related to the number of strong links (topic and noise). These parameters are thresholds on which each step of the algorithm is based.

Regarding the index for measuring the success of the algorithm in recovering the true partition of the data sets, as in [29], we use the corrected Rand [24, 29]. The corrected Rand index takes values from -1 to 1, with 1 indicating a perfect agreement between the partitions and values near 0 or negatives corresponding to cluster agreement found by chance. Unlike the majority of other indices, the corrected Rand is not biased towards a given algorithm or number of clusters in the partition [24, 48].

where (1) n _ij represents the number of objects in clusters u _i and v _j ; (2) n_i.indicates the number of objects in cluster u _i ; (3) n. _j indicates the number of objects in cluster v _j ; (4) n is the total number of objects; and(5) $\left(\begin{array}{c}a\\ b\end{array}\right)$ is the binomial coefficient $\frac{a!}{b!(a-b)!}$.

Data Transformation

In many practical situations, a data set could present samples the attribute or feature values of which (in our case, genes) lie within different dynamic ranges [24, 29]. In this case, for proximity measures such as Euclidean distance, features with large values will have a greater influence than those with small values. However, this will not necessarily reflect their importance in defining the clusters. Such a problem is often addressed by transforming the feature values so that they lie within similar ranges. There are several ways to perform transformations of attribute values [24, 29]. As the data sets used in our studies have no categorical features, we consider only the case involving numeric values. More precisely, for the Euclidean distance, we analyze three different forms of feature (gene) transformation.

The first two have been widely used in clustering applications [24, 29]: one is based on the z-score formula (standardization) and the other scales the gene values to [0, 1] or [-1, 1]. The third procedure transforms the values of the attributes into a ranking. This type of transformation is more robust to outliers than the first two procedures [29]. A large-scale study on the impact of these procedures for cancer gene expression data has been recently presented in [30].

The transformation that uses the z-score formula translates and scales the axes so that transformed feature (gene) will have zero mean and unit variance. Hereafter, for short, we will refer to this transformation as Z₁. The second procedure involves the use of the maximum and minimum values on the gene. Assuming non-negative values, a gene transformed with this procedure is bounded by 0.0 and 1.0, with at least one observed value at each of these end points [29]. For short, we will refer to this procedure as Z₂. If there are negative values, a gene transformed with Z₂ is bounded by -1.0 and 1.0. Unlike Z₁, the transformed mean and variance resulting from Z₂ will not be constant across all features.

The above procedures could be adversely affected by the presence of outliers in the genes. This is especially true for Z₂, which depends on the minimum and maximum values. A different approach, which is more robust to outliers, converts the values of the genes to rankings. Given a data set with n samples, this procedure yields a transformed gene value with mean of (n + 1)/2, range n - 1 and variance (n + 1) [((2n + 1)/6) - ((n + 1)/4)] for all genes [29]. For short, hereafter, we will refer to this procedure as Z₃.

Experimental Design

Empirical Study

The experiments compare seven different types of clustering algorithms: SL, AL, CL, KM, FMG, SPC and SNN. Each of these algorithms (with the exception of FMG, which already has the computation of the similarity as a built-in characteristic) was implemented with versions of four proximity measures: Pearson's Correlation coefficient (P), Cosine (C), Spearman's correlation coefficient (SP) and Euclidean Distance (E). As the implementation of the KM and SNN used in this work does not support Spearman's correlation coefficient, these algorithms were only tested with P, C and E. Furthermore, regarding the Euclidean distance version, experiments are performed with the samples in four forms, namely, original (Z₀), standardized (Z₁), normalized (Z₂) and ranked (Z₃). We also applied the data with these transformations for the case of FMG.

We run the algorithms for the configurations described in the previous paragraph. Recovery of cluster structure was measured via the corrected Rand index with regard to the actual number of classes known to exist in the data. That is, the number of clusters is set to be equal to the true number of the classes in the data. The known class labels were not used in any way during the clustering. As performed by [29], in order to explore the effects of reduced coverage, recovery was also measured on different levels preceding the correct solution partition. Reduced coverage implies that more clusters are present in the partition obtained than actually exist in the underlying structure. For example, if k represents the number of classes in a data set with n samples, we then developed experiments varying the number of clusters in [k, ⌈$\sqrt{n}$⌉]. In order to build the partition from hierarchical methods, recovery was measured based on the hierarchical solution that corresponds to each value in the range [k, ⌈$\sqrt{n}$⌉]. As KM is non-deterministic, we run the algorithm 30 times for each configuration pair (data set, proximity measure) and select the one with lowest inter-class error. For further analysis, we then choose the partition with the best corrected Rand (cR). This procedure is also used for SPC and FMG. We use default parameters in both cases, with the exception of the values for the number of clusters.

For SNN, we execute the algorithm with several values for its parameters (2%, 5%, 10%, 20%, 30% and 40% for NN), topic (0, 0.2, 0.4, 0.6, 0.8 and 1) and merge (0, 0.2, 0.4, 0.6, 0.8 and 1). Preliminary experiments reveal that variations in the other parameters did not produce very different results. Thus, the default values were used for the parameter strong and the value 0 was used for the parameters noise and label (in order to have all points assigned to a cluster). From the partitions created with such parameters values, we choose the partition with best cR and with k in the interval of interest for further analysis.