Principal component tests: applied to temporal gene expression data
© Zhang et al; licensee BioMed Central Ltd. 2009
Published: 30 January 2009
Clustering analysis is a common statistical tool for knowledge discovery. It is mainly conducted when a project still is in the exploratory phase without any priori hypotheses. However, the statistical significance testing between the clusters can be meaningful in helping the researchers to assess if the classification results from implementing a clustering algorithm need to be improved, even after the cluster number has been determined by a well-established criterion. This is important when we want to identify highly-specific patterns through classification.
We proposed to use a principal component (PC) test, which is an implementation of an exact F statistic for the measures at multiple endpoints based on elliptical distribution theory, to assess the statistical significance between clusters. A challenge in the implementation is the choice of the number (q) of principal components to be considered, which can severely influence the statistical power of the method. We optimized the determination via validation according to a permutation test based on the clustering to be evaluated. The method was applied to a public dataset in classifying genes according to their temporal gene expression profiles.
The results demonstrated that the PC testing were useful for determining the optimal number of clusters.
Data clustering is a common technique for statistical data analysis used in many fields , including machine learning, data mining, pattern recognition, and image analysis. Theoretically, clustering analysis identifies and classifies objects (or individuals) based on the similarity of the characteristics they possess. It seeks to minimize within-group variation and maximize between-group variation and results in a number of heterogeneous groups with homogeneous contents. The general categories of clustering methods include tree clustering (hierarchical clustering), block clustering, k-means clustering, and model-based clustering . The evaluation of clustering analysis is a critical challenge in both theory and application.
The performance of clustering analysis can be assessed statistically in order to determine the appropriate clustering methods and cluster number . Pseudo F statistic  is widely used for partitioning clustering algorithms, such as k-means, and has been included in the procedure FASTCLUST of SAS software . BIC (Bayesian information criterion) is a well-established statistic based on standard statistical theory and fits model-based clustering procedures , which has been widely applied in bioinformatics [6–9]. Silhouette score  provides a measure of how well a data point was classified when it was assigned to a cluster according to both the tightness of the clusters and the separation between them. It has been used together with PAM (Partitioning Around Medoids) clustering algorithm . Recently, the so-called Gap statistic was proposed , which can use the output of any clustering algorithms for the optimization of cluster number. Furthermore, clustering algorithms are commonly assessed from other angles, such as robustness, stability, consistency, and functional congruence of the members of the same cluster [2, 12–18].
On the other hand, while clustering analysis is mainly conducted when we are still in the exploratory phase of our research and do not have any prior hypotheses, the statistical significance testing between the clusters can be meaningful. The testing can help us to assess whether the classification results from running a clustering algorithm need to be improved, even after the cluster number has been determined by a well-established criterion. This is important in the clustering of genes on the basis of the temporal expression profiles. In order to extract specific knowledge about gene function from the expression profiles [19–21], researchers usually hope to have the number of clusters as large as possible but the contrasts between the clusters, each of which corresponds to a co-regulation pattern, should be statistically significant in general.
The significance testing between the clusters can be done by using Hotelling's T2, the multivariate counterpart of Student's-t . But when the number of measurement points is large and the size of samples is relatively small, the results from Hotelling's test are usually unstable . Using the invariance of elliptical distribution theory, a type of exact t and F tests was proposed , which can be applied to high-dimension data with a small size of samples. The tests are based on the sum aggregates of original variables similar to O'Brien's method  but superior to the latter in maintaining the prescribed level of significance. Two direct implementations of the method are a one-fold principal component (PC) test corresponding to the exact t test and a multi-fold principal component test corresponding to the exact F test. The comparison of PC test and T2 clearly demonstrated the fact that the stabilizing effect of principal components and PC test made better use of the factor structure of the data of multiple end-points.
Microarray technology allows thousands of genes to be measured simultaneously on a single slide. Unsupervised learning on the basis of clustering analysis of microarray temporal gene expression data has been widely studied in order to discover classes of expression patterns and identify groups of genes that are regulated in a similar manner [7, 13, 19, 20]. However in literature the evaluation of clustering analysis was limited to the global assessment of clustering methods. In this paper, we proposed to use principal component tests based on the exact F test for multiple endpoint measures  to assess the significance of the contrasts between the gene clusters from different clustering algorithms and implemented it on a public data set. The testing can be conducted after the global evaluation for improving clustering analysis.
Clustering and patterns
A major difference between CL1 and CL2 was that the two big clusters in the former were divided into two or more smaller classes in the latter. For example, the aggregate of the cluster1 and cluster2 in CL1 approximately corresponded to the aggregate of the cluster2, cluster7, cluster13, cluster14, cluster20 and cluster23 in CL2. In CL3, the major (62%) 483 gens were classified into the first cluster, which largely corresponded to the first biggest cluster in CL2. But the expression patterns of genes of these clusters were very different. Therefore, implementation of PAM with silhouette score criterion does not seem fit for the addressed dataset.
Determination of q value
Statistical evaluation of clustering
Biological evaluation of clustering
Biologically functional enrichment analysis of the gene lists of eight clusters a
Enriched PANTHER biological Processes
lipid, fatty acid and steroid metabolism
steroid metabolism (GO: 0008202)
cholesterol metabolism (GO: 0008203)
blood clotting (GO: 0007596)
cell cycle (GO: 0007049)
nucleoside, nucleotide and nucleic acid metabolism (GO: 0006139)
embryogenesis (GO: 0009700)
cell cycle control (GO: 0000074)
mRNA transcription (GO: 0006366)
cell proliferation and differentiation
(GO: 0031054; GO: 0008283) cell structure and motility (GO: 0007010)
immunity and defense (GO: 0006952)
immunity and defense (GO: 0006952)
developmental processes (GO: 0007275)
nucleoside, nucleotide and nucleic acid metabolism (GO: 0006139)
lipid, fatty acid and steroid metabolism (GO: 0006629)
steroid metabolism (GO: 0008202)
Clustering analysis is a widely used tool for knowledge discovery. Moreover, it is applied as a routine method in biology in the post-genomic era. The evaluation of clustering is a problem in its application. In this study, we compared the results of different clustering algorithms from a unique angle by testing the statistical significance of the contrasts between the clusters. In our knowledge, this paper is the first investigation of this kind. We used q-fold PC test which is an implementation of Lauter's exact F tests  for the measures of multiple endpoints. The method is superior to Hotelling's T2  because of the stabilizing effects of the principal components, especially for the data with small sample size. This is important when we want to identify highly-specific patterns via clustering analysis.
The significance of the proposed clustering evaluation includes three aspects. Firstly, the results can tell us if the clustering is meaningful, at least from a statistical standpoint. A good clustering algorithm should meet a basic criterion, i.e., the clusters should be statistically distinguishable. In other words, all of the contrasts between the clusters should be statistically significant at a certain confidence level. Second, it can be helpful in the determination of cluster numbers. For example, in the analysis of temporal gene expression data mentioned above, both the BIC plot did not have the expected "U" shape. Thus, the determination based on a local minimum value may be equivocal and questionable. The results of the PC tests demonstrated that dividing the 483 genes (probes) into 18–20 clusters is appropriate. Finally, the method is extremely useful for the improvement of the results from a clustering analysis by demonstrating which clusters can be combined because of the lack of significant difference between them.
The number (q) of principal components to be considered is a challenge for the PC test. We optimized the determination via validation according to permutation test based on the clustering to be evaluated. In this way, the choice of q is determined by the data and clustering methods. It is superior to the choice based on cumulative energy content (CEC) because the latter needs an artificial threshold of the CEC percentage. More importantly, from the permutation test, we can assess the validity of the PC test itself in controlling type I error.
An alternative approach to the evaluation of clustering of genes based on the temporal expression profiling is biological validation. In this paper, we conducted biologically functional enrichment analysis of the gene lists of several clusters of interest. The results showed that the finer division of clusters from SSClust, a model-based clustering algorithm, can provide more specific relationships between clusters and biological functions.
It is worthy to note that the information from the biological validation is usually limited because the temporal gene expression profiles of the genes involved in a biological process can be very diverse, including, for instance, inverse co-regulation or co-regulation with a time lag or a combination of both [21, 27].
The proposed PCA test method was applied to a public dataset in classifying genes according to their temporal gene expression profiles. The results demonstrated that the PC testing were useful for determining the optimal number of clusters. We also anticipate that the method could be used for pattern identification and similarity analysis.
The initial data set, published by Iyer et al. , describes the transcription levels of genes detected by 517 gene probes, corresponding to 497 unique genes, during the first 24 h of the serum response in serum-starved human fibroblasts. By using an agglomerative hierarchical clustering method, the authors  detected 10 major gene expression profile clusters among the differentially expressed genes of the serum response. The ten classes contained 465 unique genes or 483 gene probes. Our work was focused on the data of these 483 gene probes with the log-transformed expression ratios as the variables. The gene symbols of 239 annotated genes were provided by Lagreid et al (2006).
Principal component tests
The q-fold principal test (PC) used in this paper is implemented on the basis of a type of t or F statistic for high-dimension data.
Assume there are n individuals (genes) and from each one we have p observations at different time points. Assume p-dimensional distribution for x i (i = 1, 2, n), i.e. x i ~ N(μ i , Σ). Denote , a p × n matrix representing the gene expression. We have
X~ Np× n(M, Σ ⊗ I n ),
where , Σ is a variance and covariance matrix.
exactly follows F distribution with q and n-q-1 as the degrees of freedom ,
where . For a given n and p, the power of this statistic is dependent on the choice of q. When q = 1, the statistic (5) has t-distribution with degree of freedom n-2.
Determination of q value
The number (q) of principal components to be considered is a challenge for the q-fold PC test. A solution is the choice based on cumulative energy content (CEC). However, the threshold of the CEC percentage has to be artificially determined. Here, we developed a permutation test based on the clustering to be evaluated. Let Ic be a vector containing the cluster IDs of the genes in the clustering. By shuffling, we get another vector which has all elements of Ic arranged in a random order. We, then, replace Ic with and carry our significance testing on the contrasts (k is the cluster numbers) between clusters using PC test with different q (q = 1, 2,...). For each q, we count the number (m) of the random contrasts with p-value smaller than the prescribed error of type-I at α, such as 0.05, and calculate the ratio R = m/M. Finally, we chose the minimum q which, R approximately equals to α. If the cluster number is small, the shuffling-testing procedure should be repeated several times.
The results from three clustering algorithms were evaluated in this paper. Following is a simple description of these methods.
Agglomerative hierarchical clustering
An agglomerative hierarchical clustering procedure produces a series of partitions of the data, Pn, Pn-1,..., P1, the first Pn consisting of n single object "clusters", the last P1, consisting of a single group containing all n cases. At each stage the method joins together two clusters which are closest together (most similar) . Differences between methods in this category arise because of the different ways of defining distance (or similarity) between clusters.
Model-based clustering with smoothing splines (SSClust)
where L is the likelihood for the mixture model, N is total gene number, k is the cluster number, and vi is the numbers of free parameters for ith cluster which is equivalent to the sum of the trace of the smoothing matrix . A small BIC score indicates strong evidence for the corresponding clustering.
Partitioning Around Medoids (PAM)
where a(i) is the average distance of gene i to other genes in the same cluster, b(i) is the average distance of gene i to genes in its nearest neighboring cluster. Like BIC, a small silhouette score indicates evidence for the corresponding clustering.
Functional enrichment analysis
A web-tool in PANTHER classification system  was used for the biologically functional enrichment analysis by comparing the lists of member genes contained in each cluster of interest with gene from H. Sapiens in NCBI. Only PANTHER biological processes, most of which can be exactly mapped to a Gene Ontology (GO) term , were investigated at detail. The p-values were firstly calculated on the basis of hyper-geometric distribution theory followed by correction for multiple testing using the Bonferroni method. Because the correction method is conservative, in the following text a biological process with adjusted p-value < 0.1 was considered as "significant".
The authors are grateful to the three reviewers for their constructive comments. This study was supported by a USDA-NRI grant (NRI Proposal No.2008-00870).
This article has been published as part of BMC Bioinformatics Volume 10 Supplement 1, 2009: Proceedings of The Seventh Asia Pacific Bioinformatics Conference (APBC) 2009. The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2105/10?issue=S1
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