Ranked Adjusted Rand: integrating distance and partition information in a measure of clustering agreement
- Francisco R Pinto^{1, 2}Email author,
- João A Carriço^{1, 2, 3},
- Mário Ramirez^{1} and
- Jonas S Almeida^{2, 4}
https://doi.org/10.1186/1471-2105-8-44
© Pinto et al; licensee BioMed Central Ltd. 2007
Received: 04 October 2006
Accepted: 07 February 2007
Published: 07 February 2007
Abstract
Background
Biological information is commonly used to cluster or classify entities of interest such as genes, conditions, species or samples. However, different sources of data can be used to classify the same set of entities and methods allowing the comparison of the performance of two data sources or the determination of how well a given classification agrees with another are frequently needed, especially in the absence of a universally accepted "gold standard" classification.
Results
Here, we describe a novel measure – the Ranked Adjusted Rand (RAR) index. RAR differs from existing methods by evaluating the extent of agreement between any two groupings, taking into account the intercluster distances. This characteristic is relevant to evaluate cases of pairs of entities grouped in the same cluster by one method and separated by another. The latter method may assign them to close neighbour clusters or, on the contrary, to clusters that are far apart from each other. RAR is applicable even when intercluster distance information is absent for both or one of the groupings. In the first case, RAR is equal to its predecessor, Adjusted Rand (HA) index. Artificially designed clusterings were used to demonstrate situations in which only RAR was able to detect differences in the grouping patterns. A study with larger simulated clusterings ensured that in realistic conditions, RAR is effectively integrating distance and partition information. The new method was applied to biological examples to compare 1) two microbial typing methods, 2) two gene regulatory network distances and 3) microarray gene expression data with pathway information. In the first application, one of the methods does not provide intercluster distances while the other originated a hierarchical clustering. RAR proved to be more sensitive than HA in the choice of a threshold for defining clusters in the hierarchical method that maximizes agreement between the results of both methods.
Conclusion
RAR has its major advantage in combining cluster distance and partition information, while the previously available methods used only the latter. RAR should be used in the research problems were HA was previously used, because in the absence of inter cluster distance effects it is an equally effective measure, and in the presence of distance effects it is a more complete one.
Background
Grouping individual entities into sets with identical properties is a recurrent task in bioinformatics, taxonomy and phylogeny studies. When there are a priori reasons that allow the identification of properties that define each group, it is possible to use classification algorithms to distribute individuals among the possible classes. In other situations, different classes are defined without the absolute knowledge of what properties (and values of those properties) could identify "natural" classes. The usual procedure is the collection of data characterizing each individual, relate every pair of individuals through a distance measure computed from the data and perform clustering algorithms to find a "natural" grouping structure of those individuals based on the collected data. For simplicity, most of the remaining manuscript will use the term clustering, but the problems and methods presented are also applicable to classifications.
In some well established fields, researchers may assume a "gold standard" classification. If such gold standard is available, clustering results based on a particular kind of data can then be evaluated against it. False positives and false negatives can be identified and counted, enabling the computation of several related statistics. Even when gold standards are not available, different clusterings still need to be compared. Facing two different data sources characterizing the same set of biological entities and producing two different clusterings, one may wish to know to what extent and under which conditions one can maximize agreement or disagreement between two clusterings. This information may be useful to decide if it is worthwhile to collect and analyse both data sources since if their results are in complete agreement, then it may be enough to collect data from a single source. On the other hand, if the two clusterings disagree, combining their results may offer additional information and discriminatory power. Additionally, if the two data sources carry independent information, clusters that have a good match in both clusterings can be more reliable than clusters resulting from one data source alone.
From previous measures to Ranked Adjusted Rand
Contingency Table (CT).
C' | ||||||
---|---|---|---|---|---|---|
C' _{1} | C' _{2} | ... | C' _{ K' } | C marginal totals | ||
C | C _{1} | ct _{1,1} | ct _{1,2} | ... | ct _{1,K'} | n _{1} |
C _{2} | ct _{2,1} | ct _{2,2} | ... | ct _{2,K'} | n _{2} | |
... | ... | ... | ... | ... | ... | |
C _{ K } | ct _{k,1} | ct _{k,2} | ... | ct _{K,K'} | n _{ K } | |
C' marginal totals | n' _{1} | n' _{2} | ... | n' _{ K' } | n |
Mismatch Matrix (MM).
c' | |||
---|---|---|---|
Match | Mismatch | ||
c | Match | a | b |
Mismatch | c | d |
Although the research in this area has produced many different methods, the classical methods are the most frequently referred, as an example, in a recent reference book on microarray data analysis, the only presented method to compare clusterings is equivalent to the Rand index [10]. On the other hand there is no general consensus on the choice of the method to compare clusterings, and active research on alternative methods was motivated by microarray and systems biology approaches [11]. It should be noted that the methods discussed here are not evaluating the quality or validating clustering algorithms. Instead the aim is to confront information of clusterings obtained from different data sources. It is also not a direct aim to achieve a combined better clustering closer to a hypothetical true classification. Nonetheless, these are possible secondary applications that are not tested in the present report. Additionally, researchers comparing clustering results should be aware that the measured levels of agreement could be strongly influenced by the inherent quality of the individual clusterings and by the type and quality of the datasets that originated the analysed clusterings.
The motivation to develop a new method stemmed from the observation that, for the available measures, when pairs of entities are in the same cluster on one clustering, and in different clusters on the other, it is considered irrelevant if these clusters are close neighbours or, on the contrary, very distant. A solution for such a problem was developed in a related subject, the quantification of the agreement of different observers performing a diagnostic test [12]. When the test has multiple possible categories with an ordinal relation (of disease severity, for example), weights are attributed to different degrees of disagreement. Minimal (when one observer chooses one category close to the one chosen by the other observer), intermediate and maximal disagreement (when the two observers choose categories in the two extremes of the ordinal scale), and these contribute proportionally to the overall measure of agreement computed.
The use of a similar weighting strategy in the comparison of clusterings is not directly applicable. First, and in contrast to the observer agreement case, the two clusterings are not forced to have the same number of groups. Second, there is no predetermined correspondence between the clusters in both clusterings. Third, the closeness relationships between clusters are frequently more complex (needing two or more dimensions to be correctly represented) than the simple ordinal scale of diagnostic categories (an unidimensional representation). The main achievements of the proposed measure are the solutions for these three problems. It consists on the definition of a new way to record pairwise agreements in a Ranked Mismatch Matrix (RMM), enabling the combined accounting of partition and intercluster distance information in the computation of an overall clustering agreement measure. The new measure was named Ranked Adjusted Rand (RAR), because it can be considered an expansion of the previous Hubert and Arabie (HA) adjusted Rand index and both measures are equivalent when there is no intercluster distance information available for both clusterings.
Results and discussion
Interpretation of MDD and RAR values
Ranked Mismatch Matrix (RMM).
C' | ||||||
---|---|---|---|---|---|---|
Match | Mismatch 1 | Mismatch 2 | ... | Mismatch _{ q } | ||
C | Match | rmm _{1,1} | rmm _{1,2} | rmm _{1,3} | ... | rmm _{1,q+1} |
Mismatch 1 | rmm _{2,1} | rmm _{2,2} | rmm _{2,3} | ... | rmm _{2,q+1} | |
Mismatch 2 | rmm _{3,1} | rmm _{3,2} | rmm _{3,3} | ... | rmm _{3,q+1} | |
... | ... | ... | ... | ... | ... | |
Mismatch _{ p } | rmm _{p+1,1} | rmm _{p+1,2} | rmm _{p+1,3} | ... | rmm _{p+1,q+1} |
RAR in the absence of intercluster distance information
When both clusterings being compared are flat, that is, when there is no intercluster distance information for any of them, two entities can only be either in the same cluster or in equally dissimilar clusters. RMM becomes identical to MM. In that situation, 1-MDD is equal to the Rand index. Analogously, RAR becomes HA, since the correction for chance agreement is similar for both measures in the absence of any intercluster distance information. Proof of this equivalence is presented in Additional file 1.
RAR with incomplete intercluster distance information
A major potential application of RAR is the comparison of a flat clustering with other for which interclusters distance information is available. One can use previously developed partition comparison measures to do this but the distance information available is neglected. However, RAR is able to compare both clusterings including the partial distance information available. The resulting RMM will have 2 × (q+1) (or (p+1) × 2) dimensions and it is possible to evaluate if the mismatches of the flat clustering tend to originate mismatches with larger rank differences in the other clustering than the flat matches.
This and the previous sections discussed the use of RAR when there is a partial or total absence of distance information. However, in most of the situations that Rand or Adjusted Rand indexes have been used, information about distance was indeed available. For an example see reference [11]. This is the most frequent situation when the partitions compared where produced by clustering algorithms. The clustering algorithm needs an inter-entity distance matrix, and this matrix is sufficient to derive the intercluster distances used for RAR computation. When the partitions are defined by classification methods, it may still be possible to have distance information, depending on the properties used to classify entities.
RAR with ties in intercluster distances
A positive feature of the R(i, j) is that it is unnecessary to define rules to deal with rank ties. If a cluster has two neighbour clusters at the same distance, they will have the same intercluster distance rank. The only consequence is that the maximum intercluster distance rank (p+1 or q+1) decreases with the number of ties. RMM will have p+1 = K rows and q+1 = K' columns in the absence of ties in intercluster distance ranks. Each tie in C will reduce one row and each tie in C' will reduce one column to RMM. A higher number of ties can be due to a more discrete intercluster distance function and will produce a reduced RMM that is more similar to MM. The existence of ties is then responsible for the approximation of RAR to HA. This is consistent with the fact that more ties are a consequence of a lower resolution of the metric used to define the intercluster distance function. The minimal resolution corresponds to a binary distance function that can be 0 (same cluster) or 1 (different cluster) – that is, when RAR is equal to HA, as discussed previously.
Design of small scale examples
Comparison of RAR with other measures applied to the small example of Figure 1.
Clusterings compared | |||
---|---|---|---|
Clustering comparison measures [reference] | A-B | A-C | A-D |
Rand [2] | 0.92 | 0.92 | 0.92 |
HA [4] | 0.77 | 0.77 | 0.77 |
Jaccard [5] | 0.70 | 0.70 | 0.70 |
Wallace forward [6] | 0.70 | 0.70 | 0.70 |
Wallace reverse [6] | 1.00 | 1.00 | 1.00 |
Larsen forward [7] | 0.95 | 0.95 | 0.95 |
Larsen reverse [7] | 0.81 | 0.81 | 0.81 |
MH [8] | 0.89 | 0.89 | 0.89 |
Variation of Information [8] | 0.11 | 0.11 | 0.11 |
Van Dongen [9] | 1.00 | 1.00 | 1.00 |
RAR | 0.38 | 0.29 | 0.67 |
Analysis of the small scale examples
The first point that these theoretical examples demonstrate is that RAR, contrary to the previous partition comparison measures, is able to detect a greater disagreement between clusterings if the entities causing the disagreement, besides changing the composition of the clusters change also the proximity relationships between clusters. That is shown by the difference in RAR value for the comparisons of A with B and A with C (Figure 1). All the other ten comparison measures consider B and C equally similar to A. In fact, the change in cluster composition from A to B is identical to the change from A to C. The difference is that the newly formed clusters in B are the closest neighbours, while in C they are the most distant clusters. As the discussed clusterings involve a small number of entities and clusters, and considering that the distance between points is proportional to the dissimilarity that was used to generate the clusterings, observation of Figure 1 clearly indicates that clustering B is more similar to A than C is.
On the contrary, RAR indicates that clustering D is more similar to A than B is. This arises because in D only one entity has a different location comparing with A. From A to B three entities changed position, although to the same relative location of the one entity cluster in D. Again, only RAR detected this difference, while all the other measures remained unchanged. This happens because RAR uses the intercluster distance ranks. The RMM comparing A with B will have more point pairs out of the diagonal than the comparison of A with D. In the first comparison three points changed their relative position, affecting the RMM position of 3 × 6 pairs of points. In the second comparison, only one point moved, hence, only 1 × 8 pairs of points can have different intercluster distance ranks than they would in a perfect match comparison. Consequently, RAR attributes more weight to the change from A to B than from A to D. From the point of view of partition comparison measures, B and D have equal differences relatively to A. They both result from A by splitting a cluster of 4 entities into one with 3 and other with 1 entity alone. This is the reason why the 10 partition comparison measures in Table 4 are not able to distinguish between the similarity of A with B and of A with D.
Simulation of large scale clusterings
The two small comparisons of the previous section are extreme cases where the advantages of RAR were demonstrated, since it was able to detect differences between clusterings that none of the previously available methods where able to detect. But in realistic data sets, it is expected that a variable number of entities change their cluster membership and their relative position simultaneously. Additionally, some entity changes may contribute to make clusterings more similar while others may differentiate them. For these reasons, larger clusterings where simulated, also with more extensive entity shuffling. A complete description of these simulations is provided in Additional file 2. Briefly, five factors were systematically varied in simulated clustering comparisons: 1) number of entities, 2) number of clusters, 3) cluster size distribution, 4) fraction of entities changing cluster membership and relative position and 5) extension of change in relative position. The only factor that had an effect on the final RAR values was the fraction of entities changing cluster membership and location, producing a linear correlation coefficient of r = -0.918. The number of entities (r = 0.077), number of clusters (r = -0.109), and the cluster size distribution (r = -0.032) had negligible impact on RAR values. These low correlations support the conclusion that RAR values are not systematically influenced neither by the number of entities and clusters being compared nor by the distribution of cluster sizes. As the entities changing cluster membership were randomly selected from every possible cluster, the change in relative position of some entities could be balanced by entities moving in the opposite direction. Consequently, varying the extension of change in relative position produced highly variable results and a low correlation with RAR values (r = -0.016). To evaluate more precisely the influence of this factor on RAR values, a partial correlation analysis was performed on the relation between RAR values, HA values (that can be interpreted as RAR values without intercluster distance information) and the net change in entity relative position (measured by the correlation coefficient between the distance matrices of the two clusterings being compared). The results, presented in detail on supplementary material, show that RAR integrates independent information contained in the HA Index and in the correlation coefficient between distance matrices. The partial correlation of RAR with both factors are strong and positive (0.758 and 0.720), which means that both a higher fraction of entities changing cluster membership and a higher net change in entity relative position independently induce higher RAR values.
Biological examples
To substantiate the general applicability of RAR, three examples with biological data are presented that compare 1) two microbial typing methods, 2) two gene regulatory network distances and 3) microarray gene expression data with pathway information. The first example is presented in the main text while the other two are included in Additional file 3.
Typing methods are major tools for the epidemiological characterization of bacterial pathogens, allowing the determination of clonal relationships between isolates based on their genotypic or phenotypic characteristics. Since typing schemes analyze different phenotypic or genotypic properties of bacteria, if some congruence between the methods is found, it suggests that a phylogenetic signal is being recovered by both methods, allowing greater confidence about evolutionary hypothesis or clonal dispersion of the strains under study. The same collection of bacterial isolates can be typed by different methodologies and it becomes of great epidemiological and evolutionary importance to understand the relationships between the clusters of isolates defined by the different methods. To this end we have recently evaluated the usefulness of a set of measures to quantitatively describe these relationships [13].
Data handling
We analyzed the data generated by the characterization of a collection of 325 macrolide-resistant Streptococcus pyogenes [14, 15]. This collection was characterized by emm sequencing, that generates groups of isolates differing by less than 92% in their DNA sequence and by comparison of the patterns generated after digestion of total DNA with the SmaI endonuclease and separation by pulsed-field gel electrophoresis (PFGE). Dice coefficient was used to compute dissimilarity between PFGE band patterns, enabling the subsequent hierarchical clustering with average linkage. Measurement of the agreement between the emm classification and PFGE clusterings for the same data set has already been done using HA and Wallace indices (W) [13]. Wallace index of clustering A relatively to B is the probability that two entities are in the same cluster in B, knowing they were in the same cluster in A. It is a pairwise asymmetric clustering agreement measure [6]. For the present work, PFGE clusterings were produced for 70 different Dice dissimilarity thresholds, covering all the possible values. Each of these clusterings was compared with the emm classification through RAR, HA and Wallace indices.
Practical example results and discussion
Wallace index
As the threshold increases, the number of PFGE clusters diminishes, resulting on a set of larger clusters. On this process, the Wallace index of emm classification relative to PFGE increases, meaning that the probability that two isolates are grouped in the same PFGE cluster if they share the same emm type increases. Also, the fact that PFGE clusters are larger raises the probability that any two isolates belong to the same cluster. The step like increases on the ascending curve corresponds to the collapse of clusters that had many isolates with the same emm type. The Wallace index of PFGE relative to emm type, which matches the probability that two isolates have the same emm type, knowing that they are on the same PFGE cluster, shows the opposite behaviour. On this case, the step like decreases on the curve correspond to the collapse of clusters rich in different emm types.
HA index
The HA curve reflects a compromise of the patterns of the two Wallace index curves, with a maximum around a 29% Dice similarity threshold, where both Wallace index curves present simultaneously relatively high values. HA is therefore similar to an average of the two Wallace indices, corrected by chance agreement.
RAR
RMM analysis for specific thresholds
Due to the fact that the emm classification does not offer distances between the different types, two isolates can only be labelled as being of the same emm type or not. This being so, RMM holds two colums (one with the isolates pairs with the same emm type and another with the isolates pairs with different emm types) and a number of lines corresponding to the maximum value of the PFGE inter cluster distance rank for each threshold. Figure 3 shows three plots where each of these two columns is represented by a curve. On these plots, the frequencies of the isolates pairs are relative so that the sum of all the represented points, including both curves, is 1. For the three studied thresholds, the frequency distributions of isolate pairs with the same emm type for different cluster distance ranks are very similar. The major difference is that the plots for thresholds 29% and 41% show a higher frequency for isolate pairs with the same emm type and a cluster distance rank of 0, meaning that the pair is in the same PFGE cluster. It is this fact that is responsible for the higher values of HA for these thresholds. On the other hand, HA is not able to detect the differences in the frequency distribution of isolate pairs with different emm types for different cluster rank distances. Compared with thresholds 29% and 41%, this distribution for threshold 20% is flatter, meaning that isolate pairs with different emm types are more homogeneously distributed throughout the cluster distance rank scale. For the higher thresholds there are stronger peaks in this distribution, and they occur in the first half of the cluster distance rank scale. This contributes to a weaker agreement. For the threshold 29%, the increase in the frequency of pairs with the same emm type and in the same PFGE cluster balances the effect of the peaks in the distribution of pairs with different emm type, thus RAR is practically identical to the one for the threshold of 20%. For the threshold 41% the peaks are stronger, occur at lower values of cluster distance rank and there is no counteracting effect, causing a significant decrease in RAR value that is not observed for HA. In fact, to compute HA the frequencies of isolate pairs with different emm type and cluster distance rank greater than 1 are grouped in just one class. This is not the case for RAR that uses all the values in the RMM for its computation. One can argue that the higher peaks in the frequencies of isolate pairs for higher thresholds are due to the lower number of clusters. Fewer clusters correspond to fewer degrees of freedom in clustering formation. With more clusters it becomes easier to build clusterings with a more perfect agreement. As RAR computes a weighted average over all isolate pairs, in all RMM positions, it is more sensitive to the shape of the distribution of frequencies along the different matrix elements than to the actual frequency values. If for the different thresholds studied, the frequency distributions of isolate pairs for different cluster distance ranks were the same, RAR would give similar results to HA, which is expectable since RAR is an extension of the HA method. The main difference is that RAR is sensitive to changes in the discussed distributions, or, in other words, it is sensitive to different levels of disagreement when a pair of isolates is not in the same class or cluster. This practical example shows the power of using RAR jointly with clustering comparison measures that only evaluate partition divergence, like HA or the Wallace coefficients. Using HA alone, it would be difficult to choose a Dice dissimilarity threshold in the interval of 28 to 41%. Between those values the partitions compared are almost equally similar, and the gains in W(emm-PFGE) are compensated by lower W(PFGE-emm). But RAR values are clearly higher for the 29% than for the 41% threshold. As the HA partition similarity is practically the same for both thresholds, it is safe to say that the change in RAR is due to an intercluster distance disagreement effect. Comparing the variation of RAR values with the corresponding variation of HA or other measures provides an easier way to infer the meaning of RAR values. RAR has another maximum at 20% dissimilarity threshold, and the corresponding HA value is not in the maximal plateau. This means that looking only at partition information, a 20% threshold would be inferior to 28% or 41%, but at 20% the entity pairs have a stronger tendency to be in equally separated clusters in both clusterings, which increases the RAR value. The existence of a RAR maximum at this value is actually confirming the empirically accepted Dice dissimilarity threshold of 20% to define PFGE clusters [13, 15–17], a value that does not correspond to the HA maximum.
Conclusion
As previously stated, comparing different clusterings for the same set of entities is a recurrent task. Hubert and Arabie's Adjusted Rand (HA) index is still commonly used to quantify these comparisons [11, 13, 18, 19]. The new method described here, the Ranked Adjusted Rand (RAR), can be useful in all instances where HA is applicable. RAR is an extension of HA, and produces identical results when there is no intercluster distance information. The novelty introduced by the RAR measure is the way the Ranked Mismatch Matrix (RMM) is built. The fact that the contribution of each entity pair in RMM is determined by the intercluster distance rank function allows the recording of different levels of disagreement circumventing the problem of pre-ordering clusters and of difference of number of clusters in both clusterings.
The artificial small examples highlighted the situations where HA and other available measures are not able to discriminate while RAR is. Namely, when from one clustering to another, a cluster is split in two and one or two of the child clusters change their localization relatively to the remaining clusters, only RAR is sensitive to differences in the relative distances of these new clusters as compared with the original clustering.
When applied to the comparison of larger clusterings, RAR proved to be robust to factors like number of entities and clusters, and also to different cluster density patterns. From the viewpoint of computation time needed to execute RAR no special problems are anticipated even with its application to very large clusterings. Simulated clustering comparisons clarified that the distance information that RAR integrates is not the same that is already implicit in the partition information. For constant partition information RAR is still sensitive to distance information changes. Analogously, for constant correlation between distance matrices, RAR is still sensitive to changes in partitions.
RAR was tested with experimental data from the field of molecular epidemiology. The test case was a comparison between one flat classification, without interclass distance information, the emm types, and a hierarchical clustering, from PFGE data, where there was inter cluster distance information for several clusterings originated from the same dendrogram. RAR produced different results from HA and Wallace indices. Analysis of RMM content proved to be helpful in the detection of what disagreements or agreements were causing changes in RAR and HA values. In conclusion, use of the RAR measure lead to a more informed decision on the best threshold to generate a PFGE clustering with a maximum agreement with the emm type classification. Although measures like Rand, Jaccard and Wallace indices continue to be useful, especially because the numbers generated have an associated intuitive meaning, we argue that RAR supersedes the previous indices when measuring the overlap between clusterings or classifications. The foundation of this argument lays in the fact that RAR is sensitive to the same partition differences that previous methods also detected, but in addition it is also sensitive to intercluster distance changes.
Methods
RAR description
A clustering C is a partition of the set of objects D, with n elements (identified below by the letters i and j), into sets (clusters) C_{1}, C_{2},...C_{ K }, with n_{1}, n_{2},...n_{ K }number of entities, all greater than 0. The task of measuring clustering agreement arises when, for the same set D, two different methods are used to produce two different clusterings, C and C', with K and K' clusters each. To evaluate the overlap of the two partitions, a contingency table is built, where every element of D contributes to the cell of the corresponding clusters in both C and C' as shown in Table 1. Focusing on the pairwise agreement, the information in CT can be further condensed in a mismatch matrix represented in Table 2, where a, b, c and d represent the counts of entity pairs that fall in each of the four possible categories. For example, entity pairs in the b category are in the same cluster in C but in different clusters in C'. The sum of a, b, c and d is n(n-1)/2, the total number of unique entity pairs.
Adjusted Rand, the RAR predecessor
Hubert and Arabie proposed an adjusted Rand index to quantify clustering agreement [4]:
Where n_{ c }is the correction for chance agreement, corresponding to the expected sum of a and d if C and C' where totally independent clusterings:
Milligan and Cooper [20] and more recently Steinley [21] performed comparative studies of several pairwise clustering agreement criteria. They found HA the criterion with the most desirable properties, especially the zero expected value in the case of independent clusterings and the robustness to changes in cluster number and cluster size heterogeneity. The basic principle of HA is to compute the fraction of entity pairs in the diagonal of MM, because those pairs are the ones contributing to the general agreement. The pairs in b and c have a null contribution to the agreement. This fraction must be corrected for the expected chance agreement.
Ranked Mismatch Matrix (RMM), a new format for the presentation of clustering data
To include the intercluster distance information, the entity pairs in a should continue to have a maximum contribution to the overall agreement, but b, c and d entity pairs should have different contributions according to the degree of mismatch in each of the two clusterings. First an intercluster distance rank function R is defined for every pair of entities (i, j) of a data set D (expression 3).
R(i, j) = (x, y): i, j ∈ {1,2,...n}; x ∈ {1,2,...K - 1}; y ∈ {1,2,...K' - 1} (3)
R(i, j) = (x, y), means that in clustering C, entity j is in the x^{ th }cluster closer to the one of entity i, and in clustering C', the cluster of entity j is the y^{ th }closer to the cluster of i. In the case i and j are in the same cluster in C, x will be 0. If i and j are in the same cluster in C', y will be 0. The distance between two clusters is here measured as the average distance between their entities. This is only possible when distances between every pair of entities are available. According to the problem, other intercluster distance function can be defined. For instance the standard single, complete or other linkage functions of hierarchical clustering can be used. In the absence of any distance information, the distance between a cluster and itself is 0 and between two different clusters is 1. Additionally, the intercluster distance definition does not have to be the same in the two clusterings being compared. These definitions allow the RAR method to be applied to any pair of clusterings. With the help of the intercluster distance rank function the Ranked Mismatch Matrix (RMM), represented in Table 3, can be computed, with the general element rmm_{x,y}defined as:
H(x) is a Heaviside step function that takes the value 1 when x is true and 0 otherwise. The double sum includes the equal entity pairs of type (i, i) and the repeated entity pairs of types (i, j) and (j, i). The pairs of the first type do not contribute to the final sum due to the Heaviside function H(i≠j). The repeated pairs (differing only by the order of the entities inside the pair) need to be accounted in the sum because, for each of the individual clusterings, the intercluster distance rank is not necessarily symmetric. As an example, cluster A may be the closest neighbour of cluster B, but the closest neighbour of cluster B may be C and not A. In RMM, intercluster distance rank information for every pair of entities is recorded without identifying which clusters are separated at what rank distance. This is important because for each cluster, the i^{ th }neighbour cluster can be different. In this way, the intercluster distance information can be integrated with the partition comparison without the need of a strict ordinal relationship between clusters (like the example of disease severity referred in the introduction), of a known cluster correspondence between clusterings or of an equal number of clusters in both clusterings.
Measuring clustering agreement
For two very similar clusterings, the majority of the entity pairs would contribute for elements close to the matrix diagonal. Even if RMM is not square, an alternative geometrical diagonal can be traced, linking the centre of the rmm_{1,1} element (with coordinates (0,0)) with the center of the rmm_{p+1,q+1}element (with coordinates (p, q)) If, on the contrary, the clusterings disagree to a large extent, most entity pairs will be far from the diagonal, concentrated around rmm_{p+1,1}and rmm_{1,q+1}. From these considerations it immediately follows that a good measure of cluster disagreement is the Mean Diagonal Deviation (MDD) for all the entity pairs in RMM.
The quantity inside the modulus is the normalized distance of the element (i, j) to the RMM diagonal, such that for the more distant elements (rmm_{1,q+1}and rmm_{p+1,1}) it takes the value of 1. Consequently, the maximum value of MDD is also 1. The modulus implies that MDD is always greater or equal to 0. To obtain a measure of agreement between clusterings it is enough to compute 1-MDD, although this quantity is not yet corrected for chance agreement. To perform this correction, the expected MDD value under independence of clusterings C and C' (conditional on the marginals of CT and on the intercluster ranked distances) must be known. To compute this MDD^{ ind }it is first necessary to build RMM^{ ind }according to:
MDD^{ ind }is then computed like MDD (expression 5), changing RMM elements by those of RMM^{ ind }. RAR is the correction of (1-MDD) for chance agreement and is the result of the following expression:
Functions to compute the RAR measure for any two clusterings were implemented in MATLAB (Release 14), and are available in Additional file 4 or at the toolbox's webpage [22].
Declarations
Acknowledgements
The authors would like to acknowledge Margarida Carrolo for precious help in manuscript preparation. Partial support for this work was provided by contract PREVIS (LSHM-CT-2003-503413 from the European Community) awarded to Jonas S. Almeida. Francisco R. Pinto and João A. Carriço were financially supported by the Portuguese Foundation for Science and Technology with the grants SFRH/BD/6488/2001, SFRH/BPD/21746/2005 and SFRH/BD/3123/2000.
Authors’ Affiliations
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