Suppose that a classes of ARCH (p) models is generated by the following equations

X t = { σ t ( θ X ) ε t , σ t 2 ( θ X ) = θ X 0 + ∑ i = 1 p X θ X i X t − i 2 for t = 1 , ⋯ , m 0 , for t = − p X + 1 , ⋯ , 0 (2.1.1)

where { ε t } is a sequence of i.i.d.(0,1) random variables with fourth-order cumulant κ 4 X , θ X = ( θ X 0 , θ X 1 , ⋯ , θ X p X ) ′ ∈ Θ X ⊂ ℝ p X + 1 is an unknown parameter vector satisfying θ X 0 > 0 , θ X i ≥ 0 , i = 1 , ⋯ , p X − 1 , θ X p X > 0 , and ε t is independent of X s , s < t . Denote by F ( x ) the distribution function of ε t 2 and we assume that f ( x ) = F ′ ( x ) exists and is continuous on ( 0 , ∞ ) .

Suppose that another class of ARCH(p) models, independent of { X t } , is generated similarly by the equations

Y t = { σ t ( θ Y ) ξ t , σ t 2 ( θ Y ) = θ Y 0 + ∑ i = 1 p Y θ Y i Y t − i 2 for t = 1 , ⋯ , m 0 , for t = − p Y + 1 , ⋯ , 0 (2.1.2)

where { ξ t } is a sequence of i.i.d. (0,1) random variables with fourth-order cumulant κ 4 Y , θ Y = ( θ Y 0 , θ Y 1 , ⋯ , θ Y p Y ) ′ ∈ Θ Y ⊂ ℝ p Y + 1 is an unknown parameter vector satisfying θ Y 0 > 0 , θ Y i ≥ 0 , i = 1 , ⋯ , p Y − 1 , θ Y p Y > 0 , and ξ t is independent of Y s , s < t . The distribution function of ξ t 2 is denoted by G ( x ) and we assume that g ( x ) = G ′ ( x ) exists and is continuous on ( 0 , ∞ ) . For (2.1.1) and (2.1.2), we assume that θ X 1 + ⋯ + θ X p X < 1 and θ Y 1 + ⋯ + θ Y p Y < 1 for stationarity (see [14] ).

Now we are interested in the two-sample problem of testing

H 0 : F ( x ) = G ( x ) forall x agains t H A : F ( x ) ≠ G ( x ) forsome x .

In this article, F ( x ) and G ( x ) correspond to the distribution for the expression data of samples driven by ER+ and ER−, individually.

For this testing problem, we consider a class of rank order statistics including, such as Wilcoxon’s two-sample test. The form is derived from the empirical residuals ε ^ t 2 = X t 2 / σ t 2 ( θ ^ X ) , t = 1 , ⋯ , n and ξ ^ t 2 = Y t 2 / σ t 2 ( θ ^ Y ) , t = 1 , ⋯ , n . Because Lee and Taniguchi [11] developed the asymptotic theory for not squared empirical residuals, we may apply the results to ε ^ t and ξ ^ t .

To obtain the empirical residuals as mentioned in 2.1, the ARCH model is applied to a vector { x i 1 , x i 2 , ⋯ , x i L } for the i th sample, where L is the total number of genomic locations in the microarray data. Assuming that the ER+ and ER− samples correspond to distributions F ( x ) and G ( x ) as shown in 2.1, orders p X and p Y of the ARCH model are identified by model selection using the Akaike Information Criterion (AIC), where the model with the minimum AIC is defined as the best fit model [15] (see 1. in Figure 1). According to those responses, the empirical residuals are grouped as ε i j + and ξ i j − . Wilcoxon statistic is applied as order statistic to those two groups for each genomic location j , and the p-value is calculated (see 2. in Figure 1). The p-values obtained for all genes are adjusted for multiple testing corrections using false discovery rate (FDR) [5] (see 3. in Figure 1).

For comparisons with the ARCH model’s performance, the two-group comparison testing to original array data and applying the test to the residuals obtained by ordinal AR (autoregressive) model.AR model represents the current

value using the weighted average of the past values as x i j = ∑ k = 1 K β i x i j − k + w i j , where β i , k , and w i j are the AR coefficient, the AR order, and the error terms. The AR model is widely applied in time series analysis and the signal processing of economics, engineering, and science. In this article, we apply it to the expression data for two ER+ and ER− groups. The AR order for the best fit model is identified by AIC. Empirical residuals w i j + for ER+ and w i j − for ER− are subtracted from the data by predictions.

These procedures are finally summarized as follows: 1) take the original microarray and the clinical data for ER from one study cohort; 2) apply the ARCH and AR models to the original data for each sample and identify the best fit model among the model candidates within 1 - 10 time lags; 3) subtract the residuals from the data by the prediction for the best fit model; 4) apply Wilcoxon statistic to the original data and to the empirical residuals by ARCH and AR; 5) list the p-values and identify the significant FDR (5%) corrected genes. 6) apply Gene Ontology analysis and pathway analysis (see the details in 2.3 and 2.4) for biological interpretation to the obtained gene list (see 4. in Figure 1).

The computational programs were done by the garchFit function (in “fGARCH”) for ARCH fitting, by the ar.ols function for AR fitting, the wilcox.test as a rank- sum test, and fdr.R for the FDR adjustment in the R package.

To investigate the gene product attributes from the gene list, Gene Ontology (GO) analysis was performed to find specific gene sets that are statistically associated among several biological categories. GO is designed as a functional annotation database to capture the known relationships between biological terms and all the genes that are instances of those terms. It is widely used by many functional enrichment tools and is highly regarded both for its comprehensiveness and its unified approach for annotating genes in different species to the same basic set of underlying functions [16] . Many tools have been developed to explore, filter, and search the GO database. In our study, Gorilla [17] was used as a GO analysis tool. GOrilla is an efficient web-based interactive user interface that is based on a statistical framework called minimum hypergenometric (mHG) for enrichment analysis in ranked gene lists, which are naturally represented as functional genomic information. For each GO term, the method independently identifies the threshold at which the most significant enrichment is obtained. The significant mHG scores are accurately and tightly corrected for threshold multiple testing without time-consuming simulations [17] . The tool identifies enriched GO terms in ranked gene lists for background gene sets which are obtained by the whole genomic location of microarray data. GO consists of three hierarchically structured vocabularies (ontologies) that describe gene products in terms of their associated biological processes, cellular components, and molecular functions. The building blocks of GO are called terms, and the relationship among them can be described by a directed acyclic graph (DAG), which is a hierarchy where each gene product may be annotated to one or more terms in each ontology [16] . GOrilla requires a list of gene symbols as input data. The obtained significant Etrez gene lists by FDR correction are converted into gene symbols using a web-based database called SOURCE [18] , which was developed by the Genetics Department of Stanford University.

As well as for GO analysis, the identified genes are mapped to the well-defined- biological pathways. Pathway analysis determines which pathways are overrepresented among genes that present significant variations. The difference from GO analysis is that pathway analysis includes interactions among a given set of genes. Several tools for pathway analysis have been published. In this study, we used a web-based analysis tool called REACTOME, which is a manually curated open-source open-data resource of human pathways and reactions [19] . REAC- TOME is a recent fast and sophisticated tool that has grown to include annotations for 7088 of the 20,774 protein-coding genes in the current Ensembl human genome assembly, 15,107 literature references, and 1421 small molecules organized into 6744 reactions collected in 1481 pathways [19] .

For the simulation data and ARCH residuals, we summarized the average of the number of the identified 5% FDR significant parts and the number of the overlapped parts in Table 1. In the case of the simulation data generated by normal

distribution, the significant number for original series and ARCH residuals in array sets A and B was not differ. The parts identified in A were mostly same as ones in B. On the other hand, in the simulation data generated by ARCH model, the ARCH residuals identified more significant parts from the data than the original series. The number of significant parts for ARCH residuals was about 30% less than the number of significant parts for the original series. The overlapped number was less than the case A, however over 50% parts were covered.

Based on the method explained in 3.2, the best fit AR and GARCH models were selected by AIC for each sample. The estimated orders of all the best fit models of all the studies are summarized in Supplementary Table 1. Figure 3 summarizes the ratio of the sample numbers for each selected order against the total number of samples. These figures suggest that the most often selected orders were one while ER+ samples tended to take more complicated models than for the ER− samples.

Using residuals obtained by the best fit ARCH and AR models and the original data, we applied Wilcoxon statistic to compare DEs between two groups divided by ER+ and ER−. The significant genomic locations were assessed by a FDR. The locations were mapped on Entrez gene IDs according to the Affy probes presented in the original microarray data and converted into gene symbols by SOURCE. The identified genes in the original data and the ARCH residual analyses are listed in Supplementary Table 2. Based on these gene lists, we investigated the overlapped significant genes for the original data with significant genes for the ARCH and AR residuals and summarized the results in Table 2. About 200 - 280 significant DE genes in the studies of Des, Mil, Min, and Chin were identified with FDR correction. For Loi, the significant genes were fewer than the other studies in all cases. Except for Loi, the number of significant genes for the ARCH residuals was reduced by about 20% - 35% less than the number of genes for the original data. The estimated genes for all the datasets (except for Loi’s case) overlapped 100% with the estimated genes for the original data. The number of significant genes for the AR residuals in the cases of Mil, Min, and Loi was less than the number of genes by ARCH and resulted in about a 20% - 35% reduction of significant genes for the original data except for Loi. The reduction rate was similar to the rate shown in the case B of our simulation study. Furthermore, these genes for the AR residuals did not completely 100% overlap with the genes for the original data unlike the case of the ARCH residuals. The results suggest that the real array data might be generated by a

similar structure as the ARCH process and empirical ARCH residuals might be more effective to specify important genes from a list of long genes than AR residuals.

To investigate the overlapping genes by the ARCH residuals with genes by the original data, the corresponding cytoband and gene symbols are summarized in Table 3. The total numbers of common genes by the original and ARCH residuals in four studies were 132 and 99. If we take into account Loi’s case, the total numbers of common genes across all studies for the original and ARCH residuals are 12 and 9. The genes obtained by the ARCH residuals were completely covered by the genes obtained by the original data. The results by the ARCH residuals covered several important genes for breast cancer, such as TP53 in the

chromosome 1q region, ERBB2 in the chromosome 17q region, and ESR1 in the chromosome 6q region, even if the number of identified Entrez genes was less than the number of identified genes from the original data.

Next, we performed GO enrichment analysis using significant DE gene lists for the original data and ARCH’s residual analyses in all studies. To correctly find the enriched GO terms for the associated genes, a background list was prepared of all the probes included in the original microarray data. The Entrez genes in the background list were converted into 13,177 gene symbols without any duplication by SOURCE. As the input gene lists to GOrilla, the numbers of summarized unique genes are shown in the parentheses of Table 2. All the associated GO terms for the original and ARCH residuals in all the studies are summarized in Supplementary Table 3. Since the estimated gene symbols in Loi’s case were less than half of the amount taken in other studies, few associated GO terms were identified in the biological process and cellular component and no GO terms in the molecular function. Also, significant DE genes for the ARCH residuals contributed to finding additional associated GO terms that did not appear in the GO terms for the original data, e.g., mammary gland epithelial cell proliferation for Des, a single-organism metabolic process for Des, an organonitrogen compound metabolic process for Mil and Min, and a single-organism developmental process for Min and Chin, all of which are related to meaningful biological associations like cellular differentiation, proliferation, and metabolic pathways in cancer cells [16] . Table 3 summarizes the common GO terms of the biological processes for Des, Mil, Min, and Chin and presented 13 terms for the original data. The terms for the ARCH residuals mostly overlapped with them except for Mil’s case. As shown in Supplementary Table 3, two terms in the molecular function and eight in the cellular components were commonly identified by the original data. The GO terms for the ARCH residuals covered them, and more terms were shown in the molecular function.

Furthermore, to investigate the consistency of the refined significant gene

lists, we applied pathway analysis to the significant DE genes for the original and ARCH residuals listed in Table 4. All the identified pathways with Entities FDR (<1.0) and associated genes are summarized in Supplementary Table 4. In the pathway components shown in Supplementary Table 3, ERBB2 signaling, EGFR, cell-cycle, immune system, metabolic pathway, AKT signaling and Wnt pathway are well- known important breast cancer-signaling pathways [31] . We took them to be representative of important pathways and counted the number of identified pathways related to these components in the case of the original and ARCH residuals. The number and associated gene symbols are summarized in Table 5. The representative pathways were mostly covered by the significant DE genes for the ARCH residuals. This result supports that the refined gene lists obtained by the ARCH residuals generally captured the differentiating breast tumors based on ER status and did not overlook any important biological information by the limited DE gene lists for the ARCH residuals.