This dataset was first presented by [24
]. It includes expression values of 7129 genes of 49 breast tumor samples. The expression data were obtained using the Affymetrix gene chip technology and are available at [26
]. The response describes the lymph node (LN) status, which is an indicator for the metastatic spread of the tumor, a very important risk factor for the disease outcome. Among the 49 samples, 25 are positive (LN+) and 24 are negative (LN-). We threshold the raw data with a floor of 100 and a ceiling of 16000. Genes with max(expression
) < 10 and/or max(expression
) – min(expression
) < 1000 are also excluded [1
]. 3332 (46.7%) genes pass the first step screening. A base 2 logarithmic transformation is then applied. The Nodal data have also been studied by [22
Although there is no limitation on the number of genes that can be used in the proposed approach, we first identify 500 genes for each dataset based on marginal significance to gain further stability as in [23
]. Compute the sample standard errors of the d
and denote their median as med.se
. Compute the adjusted standard errors as 0.5(se(1)
). Then the genes are ranked based on the t
-statistics computed with the adjusted standard errors. The 500 genes with the largest absolute values of the adjusted t
-statistics are used for classification. The adjusted t
-statistic is similar to a simple shrinkage method discussed in [27
For the Colon and Nodal data, clusters are constructed using the K-means approach and the Gap statistic is used to select the optimal number of clusters. We show in Figure the Gap statistic as a function of the number of clusters. 9 clusters are constructed for the Colon data and 20 clusters are constructed for the Nodal data. Details of the clustering information are available upon request. With the generated clusters, we apply the proposed SGLasso approach. Tuning parameters are chosen using 3-fold cross validation. Summary model features are shown in Table . For the Colon data, 22 genes are present in the final model, representing 8 clusters. For the Nodal data, 66 genes are selected, representing 17 clusters. We list the identified genes in Tables and .
Gap statistics as a function of number of clusters. Red solid line: Colon data; Green dashed line: Nodal data.
Comparison of estimation and prediction performance of different approaches.
Colon data: genes with nonzero estimates from SGLasso.
Nodal data: genes with nonzero estimates from SGLasso.
For the Colon data, gene Has.1039 has also been identified to be associated with Colon cancer in [28
]. Gene Hsa.42949 is estrogen sulfotransferase. Research show that certain compounds, such as soy, have protective effect for colon cancer. The protective role of these compounds could be due to an ability to inhibit competitively the activation of promutagenic estrogen metabolites into carcinogens by estrogen sulfotransferases. The official symbol of gene Hsa.1454 is CSNK1E. Studies have revealed a negative regulatory function of CK1 in the Wnt signaling pathway, where CK1 acts as a negative regulator of the LEF-1/beta-catenin transcription complex, thereby protecting cells from development of cancer. Gene Hsa.8214 has official symbol DCGR6. It has been shown to be associated with mammary cancer and tumor cell proliferation in general. Gene Hsa.462 (official symbol SERPINC1) has been shown to be related to cancer cell proliferation. Gene Hsa.627 is also identified as a Colon cancer biomarker in [28
]. Gene Hsa.696 has official symbol BTN1A1. RT-PCR analysis has revealed strongest expression of BTNL3 in small intestine, colon, testis, and leukocytes. Tristetraproline (gene Hsa.1682) has been reported to negatively regulate tumor necrosis factor alpha (TNF-alpha) production by binding the AU-rich element within the 3' noncoding sequences of TNF-alpha mRNA. Gene Hsa.3016 is S100P protein. 100P is expressed in human cancers, including breast, colon, prostate, and lung. In colon cancer cell lines, its expression level was correlated with resistance to chemotherapy.
For the Nodal data, gene U27185_at has official Symbol RARRES1. Also known as TIG1, the expression of this gene is upregulated by tazarotene as well as by retinoic acid receptors. Silencing of TIG1 promoter by hypermethylation is common in human cancers and may contribute to the loss of retinoic acid responsiveness in some neoplastic cells. The role of the matrilins (gene U69263) in tumorigenesis has not been studied. However, a related family of proteins (fibulins) has been implicated in cancer. Increased fibulin expression is seen in breast cancer, lung adenocarcinoma, colon cancer, and other solid tumors, suggesting that these proteins might play a role in tumor formation or progression. Gene U07223 is a member of the chimerin family and encodes a protein with a phorbol-ester/DAG-type zinc finger, a Rho-GAP domain and an SH2 domain. Decreased expression of this gene is associated with high-grade gliomas and breast tumors, and increased expression of this gene is associated with lymphomas. Findings suggest that CEACAM1 (gene X16354) participates in immune regulation in physiological conditions and in pathological conditions, such as inflammation, autoimmune disease, and cancer. Gene D87071_at is a confirmed breast cancer biomarker. Ras (gene J00277) and c-Myc play important roles in the up-regulation of nucleophosmin/B23 during proliferation of cells associated with a high degree of malignancy, thus outlining a signaling cascade involving these factors in the cancer cells. GNA11 (gene M69013) is involved in signaling of gonadotropin-releasing hormone receptor, which negatively regulates cell growth.
Down-regulation is suggested to be involved in human breast cancers. Gene D59532 encodes a member of the C-type lectin/C-type lectin-like domain (CTL/CTLD) superfamily. Members of this family share a common protein fold and have diverse functions, such as cell adhesion, cell-cell signaling, glycoprotein turnover, and roles in inflammation and immune response. Gene X53587 is human mRNA for integrin beta 4. Colonization of the lungs by human breast cancer cells is correlated with cell surface expression of the alpha(6)beta(4) integrin and adhesion to human CLCA2 (hCLCA2), Tumor cell adhesion to endothelial hCLCA2 is mediated by the beta(4) integrin, establishing for the first time a cell-cell adhesion property for this integrin that involves an entirely new adhesion partner. This adhesion is augmented by an increased surface expression of the alpha(6) beta(4) integrin in breast cancer cells selected in vivo for enhanced lung colonization but abolished by the specific cleavage of the beta(4) integrin with matrilysin. Wnt-5a (gene L20861) has been shown to influence the metastatic behavior of human breast cancer cells, and the loss of Wnt-5a expression is associated with metastatic disease. NFAT1, a transcription factor connected with breast cancer metastasis, is activated by Wnt-5a through a Ca2+ signaling pathway in human breast epithelial cells. Endogenous RXR beta (gene M84820) contributes to ERE binding activity in nuclear extracts of the human breast cancer cell line MCF-7. Detailed microscopic analysis of the morphology of MCF7 breast cancer cells lacking CtBPs (gene U37408) reveals an increase in the number of cells containing abnormal micronucleated cells and dividing cells with lagging chromosomes, indicative of aberrant mitotic chromosomal segregation. Methyl-CpG-binding domain protein-2 (gene X99687) mediates transcriptional repression associated with hypermethylated GSTP1 CpG islands in MCF-7 breast cancer cells. The nmt55/p54nrb protein (gene U02493) is post-transcriptionally regulated in human breast tumors leading to reduced expression in ER- tumors and the expression of an amino terminal altered isoform in a subset of ER+ tumors. Expression of elastin (gene U77846) in breast carcinoma cells has been demonstrated by immunohistochemistry and in situ hybridization. Cytochrome P450 1B1 (CYP1B1, gene X07618) is active in the metabolism of estrogens to reactive catechols and of different procarcinogens. The CYP1B1 gene polymorphisms do not influence breast cancer risk overall but may modify the risk after long-term menopausal hormone use. Genetic deficiency of DPYD enzyme (gene U09178) results in an error in pyrimidine metabolism associated with thymine-uraciluria and an increased risk of toxicity in cancer patients receiving 5-flourouracil chemotherapy.
We evaluate the prediction performance of the proposed approach via Leave-One-Out (LOO) cross validation. For the Colon and Nodal data, we compute the LOO cross validation errors. In this evaluation process, tuning parameters are computed using 3-fold cross validation for each reduced set. For comparison purposes, we also consider the following alternative approaches.
1. Lasso: we ignore the clustering structures and apply the Lasso directly. This approach has been considered in [18
] for Cox survival analysis and [23
] for logistic binary classification.
2. GLasso: we ignore the first step supervised selection and apply the GLasso directly. For binary classification, the GLasso has been investigated in [20
3. Simple clustering: with the generated clusters, we compute the median of the gene expression level for each cluster. The medians are used as covariates. Since the number of "covariates" is less than the sample size, logistic/Cox models can be fit directly. This mimics the approach in [5
For the alternative approaches, we also compute the LOO cross validation errors. Tuning parameters when presented are also chosen via 3-fold cross validation. Comparison results are shown in Table .
We can see from Table that the SGLasso is capable of feature selection at both the cluster level and the within cluster gene level. The number of genes selected is much less than its counterpart from the simple clustering approach and GLasso. The Lasso is also capable of selecting a small number of genes. Especially we note that the number of genes selected by Lasso is smaller than by SGLasso. For simple data sets such as the Colon data, the Lasso prediction error is the same as the SGLasso. However for data sets that are more difficult to classify (Nodal), the SGLasso prediction error is much smaller. Both data sets have also been analyzed by other approaches. For the Colon data, ROC based approach has prediction error 0.14 [23
]; LogitBoost has classification errors 0.145, 0.194 and 0.161 [22
]; and classification tree has classification error 0.145 [22
]. We note that since different sets of genes are used in those studies, Table only provides rough comparisons. For the Nodal data, in [22
], LogitBoost yields prediction error 0.184, 0.265 and 0.184, while classification tree has prediction error 0.204 and 1-nearest neighbor has prediction error 0.367.