To understand why in ER^- breast cancer it has not been possible to derive a validated prognostic signature using conventional approaches, we compared the FDR for ER⁺ and ER^- disease. We used as cohorts integrated datasets obtained by merging together three of the largest profiled breast cancer cohorts 5918 using the z-score transformation, a procedure that we validated previously 17. Briefly, the z-score transformation shifts the mean of each gene expression vector in each cohort to zero, while scaling its variance to unity. The transformed gene expression vectors are then merged across cohorts. This merging step resulted in integrated expression matrices of 186 ER^- and 527 ER⁺ tumors profiled over a common set of 5,007 genes. To enable the comparison, we selected at random 186 ER⁺ tumors from the 527 available. We then used the univariate Cox proportional hazards model with time to distant metastasis (TTDM) and overall survival as end-points to obtain P values of significance for all the genes. Next, we estimated, for each choice of significance threshold, the number of false positives using the q-value approximation 20. The numbers of significant genes and false positives as a function of the significance threshold were then plotted for ER⁺ and ER^- breast cancer and for the two different end-points used (Figure 1). We verified that the curves for ER⁺ breast cancer were robust to random selections of the 186 ER⁺ tumors. This showed that the FDR is much higher in ER^- tumours and motivated us to develop a different feature selection approach based on a pattern recognition algorithm (PACK) 17.

If the aim is to identify subclasses within a tumor type, then it is natural that unsupervised methods be applied to sample sets that are composed entirely of this tumor type. In fact, given the hierarchical ER⁺/ER^- subdivision for breast cancer, it is questionable whether ER^- subgroups can be correctly defined based on clusters that were derived by using both ER⁺ and ER^- samples, as was done in other studies (for example 212223). To see this, assume that all tumor types and genes are used in the clustering algorithm. It is very likely that the clusters within ER^- tumors reflect not only the interesting variability of genes within ER^- tumors but also the variability of genes that are important for classification of ER⁺ tumors. The variability of these genes in ER^- tumors may represent undesired noise, which, if not removed, can affect the inferred clusters. Thus, in order to identify relevant subtypes of ER^- tumors more robustly, we decided to use ER^- tumors only. Moreover, in view of the relatively high FDR in ER^- disease, we decided to apply the PACK methodology, which has already been shown to provide a more reliable identification of molecular classifiers 17. Because PACK requires large sample sizes (for small sample sizes, kurtosis estimates can have large variance and clustering algorithms that aim to predict the optimal number of clusters have a high false negative rate), we applied it to the integrated data matrix derived previously. Two additional independent ER^- cohorts 719 were left out to serve as validation studies. The five microarray datasets used are summarized in Table 1 by platform type, number of ER^- samples, and number of poor outcome events.

Briefly, we review the concepts that underpin PACK (see Materials and methods, below, and Figure 2). The hypothesis is that genes that play an important role as classifiers or biomarkers are more likely to have expression profiles that are mixtures of gaussian distributions. On the other hand, false positives, in spite of their spurious association with a phenotype, are less likely to be described by a mixture of distributions. Thus, selecting genes based on whether they have structure in their expression profiles is likely to pick out the relevant markers from those that are just false positives. Next, we propose to focus on those genes that define the largest subgroups. Although genes that define small subgroups are also of interest, it is natural to identify first those genes that define the largest subclasses. While such features can be found from the inferred cluster sizes, it turns out that such features are generally also characterized by a negative (or close to zero) kurtosis profile (see Materials and methods, below) 17. As shown previously, negative kurtosis expression profiles are in effect a mixture of at least two (gaussian) distributions of approximately equal weights. Thus, by selecting those genes that have the most negative kurtosis expression profiles, we have identified the markers that define the largest subclasses within the sample set (Figure 2). It is clear that many of these features will be highly correlated because they define almost the same subclasses. It follows that further application of traditional clustering algorithms over these negative kurtosis profiles will enable reliable identification of the major subclasses within the sample set. We note that because we are interested in the most negative kurtosis profiles and because the clusters in the individual gene profiles are only needed to study the cluster distribution of phenotypes, the cluster inference step on the individual gene profiles (known as PAC) can be performed after computation of kurtosis (known as PAK) on the selected subset of negative kurtosis profiles (Figure 2).

Applying PAK to the integrated ER- data matrix of 5,007 genes, we found 813 genes with a negative kurtosis profile (Additional data file 1). Interestingly, applying the same analysis to ER⁺ breast cancer, we found only a much smaller number (193) of negative kurtosis profiles (Additional data file 2), despite there being roughly twice as many bimodal profiles in ER⁺ breast cancer (about 4500 in ER⁺ versus about 2,500 in ER^-). We verified by explicit simulation that the significantly lower proportion of negative kurtosis profiles in ER⁺ disease could not be explained by the larger sample size in ER⁺ (527) compared with ER^- disease (186; data not shown).

Having identified the relevant features, we next clustered the ER^- tumors over these. Using hierarchical clustering with Pearson correlation metric and complete linkage, we found that samples clustered into five main groups, each characterized by the expression patterns of four gene clusters that were found to be strongly enriched for specific gene ontologies (Figure 3a and Additional data file 3). One group was characterized by over-expression of ERBB2, the steroid hormone receptor AR, and genes related to steroid estrogen response (such as GATA3, TFF1, and DNALI1). This subtype is therefore similar to the apocrine subclass, recently proposed 2425, which is characterized by over-expression of AR and genes that are either direct targets of ER or responsive to estrogen 2425. Thus, we decided to call this ER^- subtype (over-expressing steroid response genes) 'SR⁺', although it is clear that it also defines the well known HER2⁺ subtype.

The other four groups were characterized mainly by absent or lower expression of these steroid response genes. One of these four clusters was characterized by over-expression of genes related to cell cycle and cell proliferation pathways (CC⁺), and another cluster also had over-expression of immune response genes (CC⁺/IR⁺). For the remaining two clusters, one was characterized by over-expression of extracellular matrix genes (ECM⁺), and the other was characterized by over-expression of immune response genes (IR⁺) only.

In order to relate the five identified molecular subtypes to the intrinsic breast cancer classification 2126, we used the recently validated single sample predictor (SSP) 23 to classify the 186 ER^- samples into the various intrinsic subtypes (Figure 3a and Table 2). In addition, we studied the expression profiles of recently validated basal markers 27 and genes in the ERBB2 amplicon across the five identified clusters (Figure 3a and Additional data file 4). Based on these figures and Table 2, we could draw the following conclusions. First, the SR⁺ cluster was highly correlated with the usual HER2⁺ intrinsic subtype. Second, the CC⁺/IR⁺ and CC⁺ clusters defined distinct subtypes of basal tumors. Third, the ECM⁺ cluster was mostly basal, but it contained a relatively high proportion of normal-like and luminal-A-like ER^- tumors; it also exhibited the most varied grade distribution, with most low-grade ER^- tumors falling into this class. Finally, the main constituents of the IR⁺ cluster were basal and HER2⁺.

It further follows from these observations that ER^- normal and luminal A samples were predominantly characterized by over-expression of ECM genes. ER^- basal tumors, on the other hand, exhibited a more complex pattern and appeared to divide into at least four subgroups (CC⁺/IR⁺, CC⁺, ECM⁺, and IR⁺).

To investigate further the relation of our five ER- subclasses with the intrinsic subtypes, we considered to which ER^- subclass ER⁺ samples of known intrinsic subtype were most similar. To this end we first constructed, for each of the five ER^- subclasses, mean centroids over the 813 negative kurtosis genes (Additional data file 5). To validate the centroids, the same ER^- samples were assigned a subclass using a nearest centroid criterion (samples for which the largest Pearson correlation coefficient was <0.25 were considered unclassified; see Materials and methods, below), which showed that 156 (84%) were classified, of which 143 (92%) were assigned the correct subclass. Next, using the SSP to classify the 527 ER⁺ samples into the intrinsic subtypes, we then assigned each of the 527 ER⁺ samples into an ER^- subclass based on the same nearest centroid criterion (Table 3). As expected, only 4% of ER⁺ tumors were classified as basal, whereas the majority (82%) of them were luminal. Moreover, the analysis showed that ER⁺ luminal B samples were most similar to CC⁺ and CC⁺/IR⁺ ER^- samples, which is consistent with the fact that all of these samples over-express cell cycle and cell proliferation genes. In contrast, ER⁺ luminal A samples were most similar to ECM⁺ (63%), IR⁺ (26%), and SR⁺ (8%) ER^- samples. Not surprisingly, almost all (16/19 [84%]) 'normal' ER⁺ samples were most similar to ECM⁺ ER^- samples. All basal ER⁺ samples had expression profiles most similar to CC⁺/IR⁺ and CC⁺ subtypes. Interestingly, only 16 of the 42 (38%) ER⁺ HER2⁺ tumors exhibited significantly correlated expression profiles to any one of the five ER^- subclasses, with most of these (11) mapping to the CC⁺/IR⁺ subtype.

We next considered whether the five identified clusters were associated with different prognostic groups. Because we had merged different cohorts, and it is questionable whether survival data can be also merged together, we decided to test first the clusters for association with clinical outcome by using a dichotomized outcome variable. Specifically, poor outcome was defined as any death or distant metastasis event, whereas good outcome was defined as a patient alive or with no distant metastasis. By studying the distribution of good and poor outcome events in the respective clusters, significant associations with prognosis could be found by means of Fisher's exact test (Table 2). Interestingly, this showed that the IR⁺ subgroup had better prognosis when compared with the ECM⁺ (P = 0.08), CC⁺ (P = 0.03), and SR⁺ (P = 0.005) subclasses.

Compared with the CC⁺/IR⁺ subgroup it also had better prognosis, although the difference was not statistically significant (P = 0.19). We thus combined the CC⁺/IR⁺ and IR⁺ subclasses together and evaluated the prognosis of this larger subclass relative to the rest of ER^- samples. Consistent with our previous result 17 we found that the ER^- tumors over-expressing immune response genes had better prognosis than ER^- samples that did not (P = 0.02).

To further confirm our findings, we also generated Kaplan-Meier survival curves for the five identified subclasses using TTDM as the end-point (Figure 3b). This showed that the SR⁺ and CC⁺ subclasses had worst prognosis, whereas the IR⁺ subclass was the group with best prognosis. Specifically, relative to the IR⁺ subclass the SR⁺ subgroup had a hazard ratio (HR) of 3.70 (95% confidence interval [CI] 1.49 to 9.24; P = 0.005), whereas the CC⁺ subgroup had an HR of 2.75 (95% CI 1.07 to 7.05; P = 0.035). Similarly, relative to the CC⁺/IR⁺ subclass, the SR⁺ subgroup had an HR of 2.35 (95% CI 1.13 to 4.88; P = 0.02), whereas for the CC⁺ subgroup it did not quite reach statistical significance (HR 1.80, 95% CI 0.83 to 3.89; P = 0.13). We verified the statistical significance of these survival differences by 10,000 random permutations of the samples, which showed that the theoretical P value estimates above were essentially identical to the empirically derived P values.

The better prognosis of the IR⁺ subclass is likely due to specific genes that are individually prognostic. In order to find these, we applied the PAC algorithm to the 813 genes with negative kurtosis expression profiles. Briefly, this procedure selects features based on whether their expression profiles are best described as a mixture of gaussians or not, and then evaluates whether the distribution of good and poor outcome events among the inferred clusters is statistically different from random or not. Applying PAC, we found 22 genes with a Fisher test P < 0.05 (Additional data file 1). A Gene Ontology (GO) enrichment analysis of these 22 genes using the GOTree Machine 28 showed that immune/defense response was the most enriched GO, with seven genes falling into this category (C1QA, IGLC2, LY9, TNFRSF17, SPP1, XCL2, and HLA-F). The expression profiles for these seven genes confirmed the presence of distinct clusters with nonrandom distributions of good and poor outcome samples (Figure 4 provides detailed profiles of C1QA and IGLC2).

In contrast, application of the same method to ER⁺ breast cancer yielded 29 genes (out of a possible 193) with P < 0.05 (Additional data file 2). In spite of there being only 29 genes in this list, there were many with mitotic cell cycle functions, notably UBE2C, MAD2L1, E2F1 and KIFC1, and GO enrichment analysis confirmed that the cell cycle GO was the most significantly enriched category followed by transcription regulatory activity. This result for ER⁺ breast cancer confirms findings reported elsewhere that poor prognosis in ER⁺ breast cancer is related mainly to over-expression of genes in cell cycle and cell proliferation pathways 357. Notably, none of the identified prognostic genes were related to immune response functions.

Having identified a prognostic module of seven immune response genes (henceforth called the IR module), we next confirmed that clustering over this module resulted in clusters significantly associated with clinical outcome. Specifically, the partitioning around medoids (pam) clustering algorithm 29 with two centers predicted one cluster with 52 good outcome and 56 poor outcome patients, and another cluster with 56 good outcome and only 19 poor outcome events, which was highly significant under Fisher's exact test (P = 0.0004; Figure 3c). Kaplan-Meier and Cox regression analyses further confirmed that the group not over-expressing the seven-gene module (out of the seven genes the majority [six] are not over-expressed) had a greater risk for distant metastasis (HR 2.02, 95% CI 1.2 to 3.4; P = 0.009; Figure 3d).

Next, we investigated the relation of the good prognosis subgroup identified here with the novel IFN regulated cluster identified recently 23. As shown previously 23, the IFN cluster, defined by over-expression of interferon regulated genes, including the transcriptional regulator STAT1, was most closely related to the basal subclass and had a prognostic performance similar to that of luminal B samples when compared with the luminal A and normal subtypes. Interestingly, when compared with the basal and HER2⁺ subclasses, the IFN group had better prognosis, although a formal comparison was not conducted by Hu and coworkers 23.

Among the 98 genes in the immune response gene cluster (Figure 3a and Additional data file 1), we identified a total of 14 with interferon related functions, including STAT1, SP110, NFKBI, IFI44, IFNGR1, ISGF3G, and IRF7. Interestingly, however, none of these genes showed association with prognosis except SP110. Thus, although it appears that our subclass is related to the IFN class discovered by Hu and coworkers 23, it is also distinct in that the associated prognostic markers are not in the IFN cluster.

Because our IR gene cluster included STAT1 and interferon-induced genes, and these genes have been shown to be associated with lymph node (LN) metastasis 30, we considered whether the subgroup of ER^- samples over-expressing the 98-gene IR cluster was significantly associated with LN status. Because all patients in the ECM cohort were LN negative, this analysis was only performed on the NKI2 and NCH cohorts (109 ER^- patients, of whom 42 had LN involvement). Using pam clustering with two centers over these 98 genes and 109 samples, we found subgroups with similar proportions of LN metastases (43 LN negative and 28 LN positive versus 24 LN negative and 14 LN positive; Fisher's exact test P = 0.84). Moreover, clustering only over the 14 genes involved in the IFN subcluster still did not show a significantly nonrandom distribution of LN metastases among the clusters (P = 0.23), although in agreement with Huang and coworkers 30 the cluster over-expressing the IFN genes had proportionally more LN metastases. Similarly, the distribution of LN metastases among the two clusters predicted by the IR module was not significantly different from random (P = 0.48). While LN status itself was a significant predictor of distant metastasis both in univariate (HR 2.28, 95% CI 1.36 to 3.85; P = 0.002) and multivariate (HR 2.16, 95% CI 1.28 to 3.64; P = 0.004) Cox regression analysis (this result is for a multivariate model including LN status and the IR module as predictors.), importantly, the IR module remained a prognostic predictor for TTDM (HR 1.93, 95% CI 1.14 to 3.26; P = 0.015) in the multivariate model that included LN status (Table 4). This showed that the IR module identified here adds independent prognostic value over LN status in ER^- breast cancer.

The upregulation of immune response genes in good prognosis tumors could be explained by the fact that these tumors elicit a stronger immune and inflammatory response, as measured for example by a higher degree of lymphocytic infiltration (LI). The association of high LI with good prognosis is well known 31323334, and although a few conflicting results have also been reported 353637, we thought it natural to consider whether upregulation of the identified immune response module conferred good prognosis independently of LI. To this end, we scored the samples in the NCH cohort for LI (see Materials and methods, below) and combined these with the available LI score information from the NKI2 cohort, yielding a total of 50 scored samples. We found that although there were proportionally more tumors with high LI scores in the group over-expressing the immune response genes (specifically there were 11 high LI and 9 low LI samples versus 8 high LI and 22 low LI in the under-expressed group; Fisher test, P = 0.07), a multivariate Cox regression with TTDM as the outcome variable and the seven-gene IR module and LI score as predictors showed that the immune response module was still a strong predictor of clinical outcome, independent of LI (HR 2.05, 95% CI 0.71 to 5.97; P = 0.19; Table 4). (The P value is only marginally below statistical significance, which is most likely due to the relatively small sample size.) Supporting this result further, we did not find in ER^- tumors any significant association between LI and clinical outcome (Table 4).

To further confirm the independent prognostic power of the IR module from LI, we investigated the relationship of the identified good prognosis tumors with medullary breast cancers (MBCs), which are characterized by high LI scores and relatively good prognosis 38. Thus, we considered to which ER^- subclass the 22 MBC expression profiles from the report by Bertucci and coworkers 38 were most similar (Table 3). (For completeness and reference, we also performed the analysis for the 44 ductal breast cancers [DBCs], also profiled by Bertucci and coworkers 38.) This showed that of the 22 MBCs, 20 showed reasonably strong correlation to one of the ER^- subclass centroids, 14 of which (70%) were most similar to the CC⁺/IR⁺ subtype, whereas only two (11%) were most similar to the IR⁺ subtype. In contrast, for the 33 DBC which could be classified, only four (12%) and three (9%) were most similar to the CC⁺/IR⁺ and IR⁺ subtypes. These results mirror the distribution of LI scores across the five ER^- subclasses (Figure 3a and Table 2) and further confirms that the best prognostic group (IR⁺) is not related to MBC, whereas CC⁺/IR⁺ is.

A similar analysis was performed for BRCA1 mutant tumors. Of the 16 BRCA1 mutants from the NKI2 cohort 3, which were also deemed classifiable based on a 0.25 correlation threshold, 13 (81%) had expression profiles most similar to the CC⁺/IR⁺ subtype (Table 3). None showed similarity to the IR⁺ subclass. Therefore, this suggests that ER^- BRCA1 mutants, in common with MBCs, are most similar to the CC⁺/IR⁺ subclass identified here.

Having identified a prognostic module related to immune response, we next attempted to validate this finding using two independent breast cancer cohorts 719. Specifically, the hypothesis to be tested was that over-expression of the identified prognostic markers is associated with good prognosis, except for SPP1, for which good prognosis is hypothesized to be associated with under-expression. Because of the relatively small sample size of the two external cohorts, an algorithm that attempts to learn the optimal number of clusters (as implemented in PACK) is unlikely to capture structure because of a large false negative rate 39. Hence, in order to define groups of over-expression and under-expression, we applied the pam algorithm with two centers to each of the seven genes in each of the two cohorts (Figure 4 and Additional data files 6 and 7). Because the small number of events, six, in each of the two external cohorts implied a highly discrete P value distribution, Fisher test P values would be too conservative and poor approximations for type I errors 40. To overcome this difficulty, we also considered the distribution of good and poor outcome samples over the combined clusters of over-expression and under-expression (Table 5). This showed that four of the seven genes (C1QA, IGLC2, TNFRSF17, and LY9) were also highly prognostic in these two external cohorts, thus confirming the validity of our finding. Moreover, as in the three original cohorts, over-expression of these genes in the two external cohorts was associated with good prognosis (Figure 4 and Additional data files 6 and 7). For the other three genes (HLA-F, SPP1, and XCL2), P values did not reach statistical significance (P about 0.2), yet their expression profile trends were entirely consistent with those found in the integrated cohort, thus confirming their role as members of a robust prognostic module (Additional data files 6 and 7).

To confirm that the good prognosis conferred by activation of the IR module is specific to ER^- breast cancer, we applied the same pam clustering algorithm over the seven genes to the integrated dataset of 527 ER⁺ breast tumors. This gave two clusters with unequal distributions of good and poor outcome samples (209 good and 99 poor for the cluster under-expressing the genes versus 163 good and 47 poor for the cluster over-expressing the genes; P = 0.02). Although this suggested to us that over-expression of this seven-gene module also conferred better prognosis in ER⁺ samples, the association was much weaker than for ER^- samples. Univariate Cox regression with TTDM as the outcome variable confirmed that under-expression of this seven-gene module conferred a much greater risk for distant metastasis in ER^- tumors (HR 2.02, 95% CI 1.2 to 3.4; P = 0.009) than in ER⁺ tumors (HR 1.25, 95% CI 0.9 to 1.7; P = 0.16; Table 4). It is also noteworthy that, in contrast to the ER^- case, in the multivariate model setting for ER⁺ tumors, a low LI score and LN involvement were stronger predictors of TTDM than the seven-gene module (HR 1.65, 95% CI 0.4 to 7.1 for LI score and HR 1.48, 95% CI 0.7 to 3.2 for LN involvement, versus HR <1; Table 4). The specificity of our prognostic module to ER^- breast cancer was confirmed by application of PAC, which showed that, with the exception of XCL2, none of the other six genes were individually prognostic.

In this context, it is worth noting once again the absence of immune response related genes among the 29 PAC derived prognostic genes in ER⁺ disease, which would suggest that a good prognosis IR related subtype is absent in ER⁺ breast cancer. To investigate this further we checked, by performing Wilcoxon rank sum tests on the 5,007 genes, that the absence of immune response GOs was not just an artefact of the small number of genes picked out by PACK. In fact, GO analysis (using GOTree Machine) of the top 500 genes obtained from this rank sum test (all with q-values < 0.05) showed that cell cycle and transcription regulator activity related GOs were the only categories with highly significant P values (uncorrected P < 10^-6), and that the only enriched immune response related GO was that of humoral immune response, with 11 represented genes (including BLM, FADD, C3, C7, BCL2, NFKB1, and the IR module member TNFRSF17), and which was only marginally enriched (uncorrected P = 0.005). Although we verified that this humoral immune response module was associated with prognosis in the integrated ER⁺ cohort independent of LN status and LI (HR 2.26, 95% CI 0.97 to 5.26; P = 0.06) we were unable to validate this prognostic module in the external UPP and JRH-2 cohorts (Additional data file 8). Moreover, the co-regulation patterns for the genes in this module were less coherent than those for the IR module in ER^- breast cancer (Figure 5 and Additional data file 8). Hence, independent of the methodology used, an IR related prognostic module in ER⁺ breast cancer could not be identified, which seems to suggest that a good prognosis subtype related to IR is specific to ER^- disease.

The microarray breast cancer datasets considered in this work are described elsewhere 5791819. For these cohorts we used the normalized data, which are available in the public domain (see 5791819). The retrieved datasets were further normalized, if necessary, by transforming them onto a common log2 scale and shifting the median of each array to zero. We also created an automated computational pipeline (Perl scripts on a Linux platform) to crosslink the annotation provided for each dataset with UniGene. For some datasets, the linkage relied on Ensembl 48 external database identifiers. Thus each probe was associated with a universal gene name. This procedure generated a nonredundant set of gene identifiers for the subsequent integrative analysis.

The hypothesis underlying PACK 17 is that genes that are true biologic or clinical markers have expression profiles that are generated by a mixture of two or more underlying distributions, whereas spurious features are more likely to have profiles generated by a single distribution. The biologic validity of this assumption was proved through a FDR analysis 17.

PACK can be viewed as a semi-supervised algorithm, consisting of two main steps: a feature selection criterion and a supervised step, in which the selected features are correlated to a phenotype (Figure 2). It is important to note that PACK is a flexible modular algorithm in that the feature selection step can be applied on its own. In this case, there are two possible versions of the algorithm: PAC and PAK. The precise way in which these two algorithms are used in PACK will depend on the purpose of the exercise. Below, we describe the PACK strategy implemented in this paper, which is slightly different from that applied previously 17.

Kurtosis is related to the fourth central moment and can conveniently be defined as follows 49:

K ( X ) ≡ E [ ( X − X ¯ ) 4 ] E [ ( X − X ¯ ) 2 ] 2 − 3 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8tuQ8FMI8Gi=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciGacaGaaeqabaqadeqadaaakeaacaWGlbGaaiikaiaadIfacaGGPaGaeyyyIO7aaSaaaeaaieaacaWFfbGaai4waiaacIcacaWFybGaeyOeI0Iab8hwayaaraGaaiykamaaCaaaleqabaGaaGinaaaakiaac2faaeaacaWFfbGaai4waiaacIcacaWFybGaeyOeI0Iab8hwayaaraGaaiykamaaCaaaleqabaGaaGOmaaaakiaac2fadaahaaWcbeqaaiaaikdaaaaaaOGaeyOeI0IaaG4maaaa@49C6@

where X is any random variable and E denotes the expectation. For a gaussian E[(X−X¯)4]=3E[(X−X¯)2]2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8tuQ8FMI8Gi=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciGacaGaaeqabaqadeqadaaakeaaieaacaWFfbGaai4waiaacIcacaWFybGaeyOeI0Iab8hwayaaraGaaiykamaaCaaaleqabaGaaGinaaaakiaac2facqGH9aqpcaaIZaGaa8xraiaacUfacaGGOaGaa8hwaiabgkHiTiqa=HfagaqeaiaacMcadaahaaWcbeqaaiaaikdaaaGccaGGDbWaaWbaaSqabeaacaaIYaaaaaaa@44F6@, so that K(X) = 0. Most nongaussian distributions necessarily have either K > 0, in which case they are called supergaussian or leptokurtic, or K < 0, in which case they are called subgaussian or platykurtic. Specifically, a mixture of two approximately equal mass gaussians must have negative kurtosis because the two modes on either side of the center of mass effectively flatten out the distribution. To see this, consider a gene whose expression profile is described by a mixture of two gaussians. Then, the kurtosis, K, is a function of two parameters (we assume for simplicity that the gaussians are of equal variance σ², although this assumption is not needed for the result below); the effect size of the gene, as defined by the effective separation e between the two gaussians (e = μ/σ, where μ is the separation), and the ratio of cluster weights (π₁, π₂), that is R = π₁/(1 - π₁). Specifically, a short computation reveals that

K ( e , R ) = e 4 R ( R − a ) ( R − b ) ( 1 + R ) 4 ( 1 + R ( 1 + R ) 2 e 2 ) 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8tuQ8FMI8Gi=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciGacaGaaeqabaqadeqadaaakeaacaWGlbGaaiikaiaadwgacaGGSaGaamOuaiaacMcacqGH9aqpcaWGLbWaaWbaaSqabeaacaaI0aaaaOWaaSaaaeaacaWGsbGaaiikaiaadkfacqGHsislcaWGHbGaaiykaiaacIcacaWGsbGaeyOeI0IaamOyaiaacMcaaeaacaGGOaGaaGymaiabgUcaRiaadkfacaGGPaWaaWbaaSqabeaacaaI0aaaaOGaaiikaiaaigdacqGHRaWkdaWcaaqaaiaadkfaaeaacaGGOaGaaGymaiabgUcaRiaadkfacaGGPaWaaWbaaSqabeaacaaIYaaaaaaakiaadwgadaahaaWcbeqaaiaaikdaaaGccaGGPaWaaWbaaSqabeaacaaIYaaaaaaaaaa@53BF@

where a and b are the quadratic roots 2 ± √3. Thus, for e ≠ 0, the kurtosis is negative if and only if (2 - √3) < R < (2 + √3). This in turn requires the smallest cluster weight, π_min, to be in the range (approximately) of 0.22 <π_min < 0.5. It follows that for the case of approximately equal weights, where R ≅ 1 (π_min ≅ 0.5), the kurtosis is always negative and in the limit of large cluster separations (when e >> 1) the kurtosis decreases monotonically, asymptotically approaching the lower bound -2. Thus, kurtosis provides a useful measure for ranking genes based on how platykurtic their profiles are.

Given a gene's expression profile x = (x₁, ..., x_n), an unbiased estimate for the kurtosis 50 is as follows:

K ( x ) ≙ n ( n + 1 ) ∑ i = 1 n ( x i − x ¯ ) 4 ( n − 1 ) ( n − 2 ) ( n − 3 ) σ 4 − 3 ( n − 1 ) 2 ( n − 2 ) ( n − 3 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8tuQ8FMI8Gi=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@65B9@

where x¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8tuQ8FMI8Gi=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciGacaGaaeqabaqadeqadaaakeaaceWG4bGbaebaaaa@3442@ and σ are the mean and standard deviation estimates of the profile. A standard error estimate of K was obtained by performing 10,000 random simulations, with n = 186 (number of ER^- samples), which showed that the standard error estimate, 0.36, was essentially identical to the theoretical estimate, √(24/n) 50.

Two notes with the feature selection step are in order. First, the kurtosis threshold used to select features depends on how large the smallest subgroup must be. Generally, given the effective separation values that are typical for differential gene expression, we find that a zero kurtosis threshold (as used in this report) generally picks out subgroups within the individual gene expression profiles that are at least as large as 30% of the total sample size 17. Second, in principle, genes defining major subclasses could be found using a clustering step to infer two clusters (PAC) and setting a lower bound threshold (for instance, 30%) on the size of the smallest cluster. However, this approach is computationally more expensive, because PAC attempts to estimate the optimal number of clusters in the profile. However, this model selection step is a necessary one to ensure that profiles for which there is no objective evidence of bimodality are excluded (see below).

Having selected the genes defining the largest subclasses, we next apply PAC to each of these genes to remove those for which there is no evidence of bimodality (gaussian profiles that spuriously have negative kurtosis values). Specifically, given a gene's expression profile x = (x₁, ..., x_n), we model this as a random sample of a univariate random variable X, whose density function is possibly a mixture of Gaussians:

p ( x i | θ ) = ∑ k = 1 C M π k G ( x i | μ k , σ k ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8tuQ8FMI8Gi=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciGacaGaaeqabaqadeqadaaakeaacaWGWbGaaiikaiaadIhadaWgaaWcbaGaamyAaaqabaGccaGG8bGaeqiUdeNaaiykaiabg2da9maaqahabaGaeqiWda3aaSbaaSqaaiaadUgaaeqaaOGaam4raiaacIcacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaaiiFaiabeY7aTnaaBaaaleaacaWGRbaabeaakiaacYcacqaHdpWCdaWgaaWcbaGaam4AaaqabaGccaGGPaaaleaacaWGRbGaeyypa0JaaGymaaqaaiaadoeadaWgaaadbaGaamytaaqabaaaniabggHiLdaaaa@50B0@

Where π_k are the weights of the components, (μ_k, σ_k) are the mean and standard deviation of the univariate gaussian k, and θ denotes the set of all parameters. In the above, C_M denotes the maximum number of clusters that can be inferred, which in our application we set to 2. The optimal number of clusters, C, can be inferred using one of various approaches. One possibility is to use the EM algorithm to learn the parameters for the two different models C = 1 and C = 2, and perform model selection using the Bayesian Information Criterion (BIC) score 5152. Alternatively, the optimal number of clusters, C, can be inferred using a lower bound on the model evidence, as provided by a variational Bayesian (VB) approach 395354. The results we report here were obtained using the VB algorithm for model selection. Thus, genes for which C = 1 were excluded from further analysis. Finally, association with the phenotype (here prognosis) was determined using Fisher's exact test to test whether poor outcome events were unevenly distributed across the two clusters.

All analyses were performed using the R statistical programming language 55. The following add-on packages were used: vabayelMix for the PACK implementation, survival for the Cox regression models, qvalue for FDR estimation, and cluster for the partitioning around medoids (pam) clustering algorithm.

The classification of the samples in the NKI2, EMC, and NCH cohorts into the intrinsic subtypes was performed using the single sample predictor (SSP) 23 and was done for each cohort separately because this guaranteed a larger number of overlapping genes. In the SSP, samples were assigned the intrinsic subtype for which the corresponding Spearman rank correlation between the sample and SSP centroid was maximal 23.

From the hierarchical clustering with Pearson correlation metric and complete linkage diagram (Figure 3a) we constructed mean centroids for each of the five subclasses. Classification of external samples to these centroids was performed using the nearest centroid criterion. Because these centroids were defined over ER^- samples only, external samples (which may not be ER^-) may not show strong correlation to any of these centroids. We thus validated, through 10,000 Monte Carlo (MC) randomisations, that samples with a maximal pearson coefficient larger than 0.25 were significantly correlated with the corresponding centroid (P < 0.0001). Samples with maximal correlation coefficients smaller than 0.25 were deemed to be unclassifiable.

The basal marker used in Figure 3 was derived by first mapping ten validated basal markers (CRYAB, ANXA8, LAMC2, LAMB3, ITGA6, KRT17, KRT15, KRT13, KRT6B, and KRT5) 27 onto the integrated data set of 5,007 genes. For each of these markers samples were ranked in order of decreasing expression or 'basalness'. For each sample in the integrated cohort an average rank was then computed over the ten basal markers. The average ranks were then rescaled onto the unit interval (0,1), with '1' indicating highest expression for basal markers. The marker for the ERBB2 subtype was obtained in an analogous manner using three genes in the ERBB2 amplicon (ERBB2, GRB7, and STARD3).

For the samples from our NCH cohort 9 we used the following scoring method. Lymphocytic infiltration (LI) was assessed in whole tumour sections from frozen sections stained with hematoxylin and eosin. The intensity of lymphocytic infiltrate was first graded semi-quantitatively as minimal or mild (1), moderate (2), and marked (3). The LI scores were then dichotomized (we considered mild and moderate as low LI and marked as high LI) to make them comparable with the binary LI scores used by van 't Veer and coworkers 3.