Robust functional principal component analysis via a functional pairwise spatial sign operator

Functional principal component analysis (FPCA) has been widely used to capture major modes of variation and reduce dimensions in functional data analysis. However, standard FPCA based on the sample covariance estimator does not work well if the data exhibits heavy-tailedness or outliers. To address this challenge, a new robust FPCA approach based on a functional pairwise spatial sign (PASS) operator, termed PASS FPCA, is introduced. We propose robust estimation procedures for eigenfunctions and eigenvalues. Theoretical properties of the PASS operator are established, showing that it adopts the same eigenfunctions as the standard covariance operator and also allows recovering ratios between eigenvalues. We also extend the proposed procedure to handle functional data measured with noise. Compared to existing robust FPCA approaches, the proposed PASS FPCA requires weaker distributional assumptions to conserve the eigenspace of the covariance function. Specifically, existing work are often built upon a class of functional elliptical distributions, which requires inherently symmetry. In contrast, we introduce a class of distributions called the weakly functional coordinate symmetry (weakly FCS), which allows for severe asymmetry and is much more flexible than the functional elliptical distribution family. The robustness of the PASS FPCA is demonstrated via extensive simulation studies, especially its advantages in scenarios with nonelliptical distributions. The proposed method was motivated by and applied to analysis of accelerometry data from the Objective Physical Activity and Cardiovascular Health Study, a large-scale epidemiological study to investigate the relationship between objectively measured physical activity and cardiovascular health among older women.     

Does genetic counseling have any impact on management of breast cancer risk?

Despite there being an increasing literature on the impact of cancer genetic counseling on risk perception and mental health, there is a lack of data describing impact on risk management. Genetic counseling and testing for cancer predisposition genes aims to improve the future health of those at high risk through appropriate surveillance and screening. However, management of breast cancer risk in women with a family history of this disease is an area of controversy. Counseling services may recommend specific risk management options to women, who then rely on their local screening service to make provision. This study investigated the impact of genetic counseling on management of breast cancer risk in women attending Cancer Family Clinics. A total of 293 women attending four genetic clinics were enrolled. Rates of breast self-examination, clinical breast examination, mammography, biopsy, detected cancers, and other screenings were documented. Participants' perceived benefits and barriers to mammography were assessed along with cancer worry. Results show that rates of mammography, clinical breast examination, and breast self-examination were increased following clinic attendance (p < 0.001). Women in the under 35 age-group had limited access to screening. Rates for biopsy and detected cancers were low. Women reported positive attitudes to mammography, with few reported barriers. Contrary to previous studies, there was no evidence that anxiety about breast cancer impedes uptake of health surveillance methods. Genetic counseling had a positive impact on management of breast cancer risk. Whether this translates into future health gains remains to be established.     

On the application of (topographic) independent and tree-dependent component analysis for the examination of DCE-MRI data

In this contribution we investigate the applicability of different methods from the field of independent component analysis (ICA) for the examination of dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) data from breast cancer research. DCE-MRI has evolved in recent years as a powerful complement to X-ray based mammography for breast cancer diagnosis and monitoring. In DCE-MRI the time related development of the signal intensity after the administration of a contrast agent can provide valuable information about tissue states and characteristics. To this end, techniques related to ICA, offer promising options for data integration and feature extraction at voxel level. In order to evaluate the applicability of ICA, topographic ICA and tree-dependent component analysis (TCA), these methods are applied to twelve clinical cases from breast cancer research with a histopathologically confirmed diagnosis. For ICA these experiments are complemented by a reliability analysis of the estimated components. The outcome of all algorithms is quantitatively evaluated by means of receiver operating characteristics (ROC) statistics whereas the results for specific data sets are discussed exemplarily in terms of reification, score-plots and score images.     

Machine Learning Based Computer Aided Diagnosis of Breast Cancer Utilizing Anthropometric and Clinical Features

Breast cancer is one of the most prevalent types of cancers in females, which has become rampant all over the world in recent years. The survival rate of breast cancer patients degrades considerably for patients diagnosed at an advanced stage compared to those diagnosed at an early stage. The objective of this study is two folds. The first one is to find the most relevant biomarkers of breast cancer, which can be attained from regular blood analysis and anthropometric measurements. The other one is to improve the performance of current computer-aided diagnosis (CAD) system of early breast cancer detection. This study utilized a recent data set containing nine anthropometric and clinical attributes. In our methodology, first, we performed multicollinearity analysis and ranked the features based on the weighted average score obtained from four filter-based feature evaluation methods such as F-score, information gain, chi-square statistic, and Minimum Redundancy Maximum Relevance. Next, to improve the separability of the target classes, we scaled and weighted the dataset using min-max normalization and similarity-based attribute weighting by the k-means clustering algorithm, respectively. Finally, we trained standard machine learning (ML) models and evaluated the performance metrics by 10-fold cross-validation method. Our support vector machine (SVM) model with radial basis function (RBF) kernel appeared to be the most successful classifier by utilizing six features, namely, Body Mass Index (BMI), Age, Glucose, MCP-1, Resistin, and Insulin. The obtained classification accuracy, sensitivity, and specificity are 93.9% (95% CI: 93.2–94.6%), 95.1% (95% CI: 94.4–95.8%), and 94.0% (95% CI: 93.3–94.7%), respectively; these performance metrics outperformed state-of-the-art methods reported in the literature. The developed model could potentially assist the medical experts for the early diagnosis of breast cancer by employing a set of attributes that can be easily obtained from regular blood analysis and anthropometric measurements.     

What Is the Optimal Threshold at Which to Recommend Breast Biopsy?

Background

A 2% threshold, traditionally used as a level above which breast biopsy recommended, has been generalized to all patients from several specific situations analyzed in the literature. We use a sequential decision analytic model considering clinical and mammography features to determine the optimal general threshold for image guided breast biopsy and the sensitivity of this threshold to variation of these features.

Methodology/Principal Findings

We built a decision analytical model called a Markov Decision Process (MDP) model, which determines the optimal threshold of breast cancer risk to perform breast biopsy in order to maximize a patient’s total quality-adjusted life years (QALYs). The optimal biopsy threshold is determined based on a patient’s probability of breast cancer estimated by a logistic regression model (LRM) which uses demographic risk factors (age, family history, and hormone use) and mammographic findings (described using the established lexicon–BI-RADS). We estimate the MDP model''s parameters using SEER data (prevalence of invasive vs. in situ disease, stage at diagnosis, and survival), US life tables (all cause mortality), and the medical literature (biopsy disutility and treatment efficacy) to determine the optimal “base case” risk threshold for breast biopsy and perform sensitivity analysis. The base case MDP model reveals that 2% is the optimal threshold for breast biopsy for patients between 42 and 75 however the thresholds below age 42 is lower (1%) and above age 75 is higher (range of 3–5%). Our sensitivity analysis reveals that the optimal biopsy threshold varies most notably with changes in age and disutility of biopsy.

Conclusions/Significance

Our MDP model validates the 2% threshold currently used for biopsy but shows this optimal threshold varies substantially with patient age and biopsy disutility.     

Analysis of 5382insC (BRCA1) and 6174delT (BRCA2) mutations in 382 healthy Chilean women with a family history of breast cancer

Breast cancer is the most common malignancy among women. Chilean studies reveal that this cancer presents the third highest mortality rate. A family history of breast cancer is one of the major risk factors for the development of this disease. BRCA1 and BRCA2 are the two main hereditary breast cancer susceptibility genes, and mutations in these genes are related to inherited breast cancer. In specific populations only some mutations have been found to be associated with susceptibility. The purpose of this study was to establish the frequency of 5382insC (BRCA1) and 6174delT (BRCA2) germline mutations in 382 healthy Chilean women with at least two relatives affected with breast cancer and in probands and their relatives from 8 high risk families for breast cancer, using mismatch PCR assay. The results obtained showed that 5382insC and 6174delT mutations were not found in either of the groups studied. The ethnic origin of the contemporary Chilean population and the data reported in the literature suggest that these mutations may be absent or have a very low frequency in this population.. This genetic study is part of a breast cancer screening program that also includes annual mammography and clinical breast examination over a five-year period. Strategies to reduce morbidity and mortality associated with breast cancer lie in early detection in women with genetic risk.     

全数字化X线摄影联合MR 动态增强检查诊断早期乳腺癌的 临床价值研究

摘要目的：分析早期乳腺癌的全数字X 线摄影与MRI影像学表现,评价全数字X 线摄影联合MRI 检查在早期乳腺癌诊断中的 临床价值。方法：回顾性分析2009 年10 月至2012 年5月在我院经穿刺或手术病理证实为早期乳腺癌的42例患者的临床资料, 术前均行数字X线及动态增强MR 检查,比较两种方法单独使用和联合使用的诊断乳腺癌的准确率。结果：全数字化X 线摄片 诊断早期乳腺癌的准确率为69.0%(29/42),动态增强MR 检查为95.2%(40/42),两者比较差异有统计学意义(P<0.05);两者联合使 用诊断早期乳腺癌的准确率为97.6%(41/42)。结论：动态增强MR 检查对早期乳腺癌的诊断价值明显优于全数字X线摄影,但后 者对微小钙化显示较好,两者联合可提高诊断正确率,尤其对多腺体型和致密型乳腺的早期乳腺癌的检出具有重要的价值。     

The role of various modalities in breast imaging

Background: Breast cancer is the most common type of cancer in women worldwide. Mammography is considered the "gold standard" in the evaluation of the breast from an imaging perspective. Apart from mammography, ultrasound examination and magnetic resonance imaging are being offered as adjuncts to the preoperative workup. Recently, other new modalities like positron emission tomography, 99mTc-sestamibi scintimammography, and electrical impedance tomography (EIT) are also being offered. However, there is still controversy over the most appropriate use of these new modalities. Based on the literature, this review evaluates the role of various modalities used in the screening and diagnosis of breast cancer. Methods and Results: Based on relevant literatures this article gives an overview of the old and new modalities used in the field of breast imaging. A narrative literature review of all the relevant papers known to the authors was conducted. The search of literatures was done using pubmed and ovid search engines. Additional references were found through bibliography reviews of relevant articles. It was clear that though various new technics and methods have emerged, none have substituted mammography and it is still the only proven screening method for the breast as of date. Conclusion: From the literature it is clear that apropos modern radiology's impact on diagnosis, staging and patient follow-up, only one imaging technique has had a significant impact on screening asymptomatic individuals for cancer i.e.; low-dose mammography. Mammography is the only screening test proven in breast imaging. Positron emission tomography (PET) also plays an important role in staging breast cancer and monitoring treatment response. As imaging techniques improve, the role of imaging will continue to evolve with the goal remaining a decrease in breast cancer morbidity and mortality. Progress in the development and commercialisation of EIT breast imaging system will definitely help to promote other systems and applications based on the EIT and similar visualization methods. Breast ultrasound and breast magnetic resonance imaging (MRI) are frequently used adjuncts to mammography in today's clinical practice and these techniques enhance the radiologist's ability to detect cancer and assess disease extent, which is crucial in treatment planning and staging.     

Robust Classification of Functional and Quantitative Image Data Using Functional Mixed Models

Summary This article introduces new methods for performing classification of complex, high‐dimensional functional data using the functional mixed model (FMM) framework. The FMM relates a functional response to a set of predictors through functional fixed and random effects, which allows it to account for various factors and between‐function correlations. The methods include training and prediction steps. In the training steps we train the FMM model by treating class designation as one of the fixed effects, and in the prediction steps we classify the new objects using posterior predictive probabilities of class. Through a Bayesian scheme, we are able to adjust for factors affecting both the functions and the class designations. While the methods can be used in any FMM framework, we provide details for two specific Bayesian approaches: the Gaussian, wavelet‐based FMM (G‐WFMM) and the robust, wavelet‐based FMM (R‐WFMM). Both methods perform modeling in the wavelet space, which yields parsimonious representations for the functions, and can naturally adapt to local features and complex nonstationarities in the functions. The R‐WFMM allows potentially heavier tails for features of the functions indexed by particular wavelet coefficients, leading to a down‐weighting of outliers that makes the method robust to outlying functions or regions of functions. The models are applied to a pancreatic cancer mass spectroscopy data set and compared with other recently developed functional classification methods.     

Imaging Breast Density: Established and Emerging Modalities

Mammographic density has been proven as an independent risk factor for breast cancer. Women with dense breast tissue visible on a mammogram have a much higher cancer risk than women with little density. A great research effort has been devoted to incorporate breast density into risk prediction models to better estimate each individual’s cancer risk. In recent years, the passage of breast density notification legislation in many states in USA requires that every mammography report should provide information regarding the patient’s breast density. Accurate definition and measurement of breast density are thus important, which may allow all the potential clinical applications of breast density to be implemented. Because the two-dimensional mammography-based measurement is subject to tissue overlapping and thus not able to provide volumetric information, there is an urgent need to develop reliable quantitative measurements of breast density. Various new imaging technologies are being developed. Among these new modalities, volumetric mammographic density methods and three-dimensional magnetic resonance imaging are the most well studied. Besides, emerging modalities, including different x-ray–based, optical imaging, and ultrasound-based methods, have also been investigated. All these modalities may either overcome some fundamental problems related to mammographic density or provide additional density and/or compositional information. The present review article aimed to summarize the current established and emerging imaging techniques for the measurement of breast density and the evidence of the clinical use of these density methods from the literature.     

Toward the precision breast cancer survival prediction utilizing combined whole genome-wide expression and somatic mutation analysis

Background

Breast cancer is the most common type of invasive cancer in woman. It accounts for approximately 18% of all cancer deaths worldwide. It is well known that somatic mutation plays an essential role in cancer development. Hence, we propose that a prognostic prediction model that integrates somatic mutations with gene expression can improve survival prediction for cancer patients and also be able to reveal the genetic mutations associated with survival.

Method

Differential expression analysis was used to identify breast cancer related genes. Genetic algorithm (GA) and univariate Cox regression analysis were applied to filter out survival related genes. DAVID was used for enrichment analysis on somatic mutated gene set. The performance of survival predictors were assessed by Cox regression model and concordance index(C-index).

Results

We investigated the genome-wide gene expression profile and somatic mutations of 1091 breast invasive carcinoma cases from The Cancer Genome Atlas (TCGA). We identified 118 genes with high hazard ratios as breast cancer survival risk gene candidates (log rank p?<? 0.0001 and c-index?=?0.636). Multiple breast cancer survival related genes were found in this gene set, including FOXR2, FOXD1, MTNR1B and SDC1. Further genetic algorithm (GA) revealed an optimal gene set consisted of 88 genes with higher c-index (log rank p?<? 0.0001 and c-index?=?0.656). We validated this gene set on an independent breast cancer data set and achieved a similar performance (log rank p?<? 0.0001 and c-index?=?0.614). Moreover, we revealed 25 functional annotations, 15 gene ontology terms and 14 pathways that were significantly enriched in the genes that showed distinct mutation patterns in the different survival risk groups. These functional gene sets were used as new features for the survival prediction model. In particular, our results suggested that the Fanconi anemia pathway had an important role in breast cancer prognosis.

Conclusions

Our study indicated that the expression levels of the gene signatures remain the effective indicators for breast cancer survival prediction. Combining the gene expression information with other types of features derived from somatic mutations can further improve the performance of survival prediction. The pathways that were associated with survival risk suggested by our study can be further investigated for improving cancer patient survival.

     

Trends in breast cancer mortality in Sweden before and after implementation of mammography screening

Background

Incidence-based mortality modelling comparing the risk of breast cancer death in screened and unscreened women in nine Swedish counties has suggested a 39% risk reduction in women 40 to 69 years old after introduction of mammography screening in the 1980s and 1990s.

Objective

We evaluated changes in breast cancer mortality in the same nine Swedish counties using a model approach based on official Swedish breast cancer mortality statistics, robust to effects of over-diagnosis and treatment changes. Using mortality data from the NordCan database from 1974 until 2003, we estimated the change in breast cancer mortality before and after introduction of mammography screening in at least the 13 years that followed screening start.

Results

Breast mortality decreased by 16% (95% CI: 9 to 22%) in women 40 to 69, and by 11% (95% CI: 2 to 20%) in women 40 to 79 years of age.

Discussion

Without individual data it is impossible to completely separate the effects of improved treatment and health service organisation from that of screening, which would bias our results in favour of screening. There will also be some contamination of post-screening mortality from breast cancer diagnosed prior to screening, beyond our attempts to adjust for delayed benefit. This would bias against screening. However, our estimates from publicly available data suggest considerably lower benefits than estimates based on comparison of screened versus non-screened women.     

Testing the Independence of Two Diagnostic Tests

Consider two diagnostic procedures having binary outcomes. If one of the tests results in a positive finding, a more definitive diagnostic procedure will be administered to establish the presence or absence of a disease. The use of both tests will improve the overall screening sensitivity when the two tests are independent, compared with employing two tests that are positively correlated. We estimate the correlation coefficient of the two tests and derive statistical methods for testing the independence of the two diagnostic procedures conditional on disease status. The statistical tests are used to investigate the independence of mammography and clinical breast exams aimed at establishing the benefit of early detection of breast cancer. The data used in the analysis are obtained from periodic screening examinations of three randomized clinical trials of breast cancer screening. Analysis of each of these trials confirms the independence of the clinical breast and mammography examinations. Based on these three large clinical trials, we conclude that a clinical breast exam considerably increases the overall sensitivity relative to screening with mammography alone and should be routinely included in early breast cancer detection programs.     

Family-based association studies for next-generation sequencing

An individual's disease risk is determined by the compounded action of both common variants, inherited from remote ancestors, that segregated within the population and rare variants, inherited from recent ancestors, that segregated mainly within pedigrees. Next-generation sequencing (NGS) technologies generate high-dimensional data that allow a nearly complete evaluation of genetic variation. Despite their promise, NGS technologies also suffer from remarkable limitations: high error rates, enrichment of rare variants, and a large proportion of missing values, as well as the fact that most current analytical methods are designed for population-based association studies. To meet the analytical challenges raised by NGS, we propose a general framework for sequence-based association studies that can use various types of family and unrelated-individual data sampled from any population structure and a universal procedure that can transform any population-based association test statistic for use in family-based association tests. We develop family-based functional principal-component analysis (FPCA) with or without smoothing, a generalized T(2), combined multivariate and collapsing (CMC) method, and single-marker association test statistics. Through intensive simulations, we demonstrate that the family-based smoothed FPCA (SFPCA) has the correct type I error rates and much more power to detect association of (1) common variants, (2) rare variants, (3) both common and rare variants, and (4) variants with opposite directions of effect from other population-based or family-based association analysis methods. The proposed statistics are applied to two data sets with pedigree structures. The results show that the smoothed FPCA has a much smaller p value than other statistics.     

全数字化X线摄影联合MR动态增强检查诊断早期乳腺癌的临床价值研究

目的：分析早期乳腺癌的全数字X线摄影与MRI影像学表现,评价全数字X线摄影联合MRI检查在早期乳腺癌诊断中的临床价值。方法：回顾性分析2009年10月至2012年5月在我院经穿刺或手术病理证实为早期乳腺癌的42例患者的临床资料,术前均行数字X线及动态增强MR检查,比较两种方法单独使用和联合使用的诊断乳腺癌的准确率。结果：全数字化X线摄片诊断早期乳腺癌的准确率为69．0％（29／42）,动态增强MR检查为95．2％（40／42）,两者比较差异有统计学意义（P〈0．05）;两者联合使用诊断早期乳腺癌的准确率为97．6％（41／42）。结论：动态增强MR检查对早期乳腺癌的诊断价值明显优于全数字X线摄影,但后者对微小钙化显示较好,两者联合可提高诊断正确率,尤其对多腺体型和致密型乳腺的早期乳腺癌的检出具有重要的价值。     

Management of women at increased risk for breast cancer: preliminary results from a new program

OBJECTIVE: To examine the characteristics of malignant tumours that develop in women undergoing surveillance for increased risk for breast cancer and to identify presentation patterns in order to determine the respective roles of mammography, clinical breast examination (CBE) and breast self-examination (BSE). SETTING: Breast Diagnostic Clinic and Familial Breast Cancer Clinic at Toronto-Sunnybrook Regional Cancer Centre. PARTICIPANTS: A total of 1044 women evaluated for breast cancer risk from Oct. 1, 1990, to Dec. 31, 1996, of whom 381 were categorized as being at high risk, 204 as being at moderate risk, 401 as being at slightly increased risk and 58 as being at no appreciably increased risk. PROGRAM COMPONENTS: Comprehensive review and discussion of risk factors, clinical assessment, surveillance recommendations that include mammography, CBE and BSE, genetics consultation (Familial Breast Cancer Clinic) and psychosocial support. Data are captured prospectively, updated at each visit and audited every 3 to 6 months. PROGRAM OUTCOMES: During the study period breast cancer was diagnosed in 24 patients, 12 in the high-risk group, 4 in the moderate-risk group and 8 in the group at slightly increased risk. The mean age at diagnosis was 47 (range 32 to 82) years. Ten cases of cancer were diagnosed during surveillance (incident cancer), 5 in women under age 50. The mean length of time from initial assessment to diagnosis was 28.6 (range 12 to 51) months. Of the 24 women, 17 reported a family history of breast cancer. The mean age at diagnosis in this cohort was 45.5 years, and the diagnosis was made under age 50 in 10 patients (59%). The mean earliest age at which breast cancer was diagnosed in a family member was 42.5 years. CONCLUSIONS: These preliminary results suggest that surveillance of women at increased risk for breast cancer may be useful in detecting disease at an early stage. The regular performance of mammography, CBE and BSE appears necessary to achieve these results.     

A critical evaluation of network and pathway-based classifiers for outcome prediction in breast cancer

Recently, several classifiers that combine primary tumor data, like gene expression data, and secondary data sources, such as protein-protein interaction networks, have been proposed for predicting outcome in breast cancer. In these approaches, new composite features are typically constructed by aggregating the expression levels of several genes. The secondary data sources are employed to guide this aggregation. Although many studies claim that these approaches improve classification performance over single genes classifiers, the gain in performance is difficult to assess. This stems mainly from the fact that different breast cancer data sets and validation procedures are employed to assess the performance. Here we address these issues by employing a large cohort of six breast cancer data sets as benchmark set and by performing an unbiased evaluation of the classification accuracies of the different approaches. Contrary to previous claims, we find that composite feature classifiers do not outperform simple single genes classifiers. We investigate the effect of (1) the number of selected features; (2) the specific gene set from which features are selected; (3) the size of the training set and (4) the heterogeneity of the data set on the performance of composite feature and single genes classifiers. Strikingly, we find that randomization of secondary data sources, which destroys all biological information in these sources, does not result in a deterioration in performance of composite feature classifiers. Finally, we show that when a proper correction for gene set size is performed, the stability of single genes sets is similar to the stability of composite feature sets. Based on these results there is currently no reason to prefer prognostic classifiers based on composite features over single genes classifiers for predicting outcome in breast cancer.     

The possible role of lipid peroxidation in breast cancer risk

Breast cancer remains the commonest cause of death from cancer in women in most of the Western world. There is considerable evidence that breast cancer risk is influenced by environmental factors and can therefore potentially be modified. In this paper we describe evidence suggesting a relationship of lipid peroxidation to breast cancer risk, and propose that the method used to generate this information might usefully be applied to other disease states, and make some suggestions for further work. We have compared the urinary excretion of the mutagen malonaldehyde (MDA) in premenopausal women at different risks for breast cancer as determined by the appearance of the breast parenchyma on mammography. MDA was measured in 24-h urine samples from both groups and excretion in 30 women with mammographic dysplasia (high risk) was found to be approximately double that of 16 women without these radiological changes (p less than 0.02). These results suggest that mammographic dysplasia may be associated with lipid peroxidation. Further study of environmental factors associated with states that precede the development of breast and other cancers may lead to the identification of factors that can be modified and that may prevent the development of malignant disease.     

Inference of patient‐specific subpathway activities reveals a functional signature associated with the prognosis of patients with breast cancer

Breast cancer is one of the most deadly forms of cancer in women worldwide. Better prediction of breast cancer prognosis is essential for more personalized treatment. In this study, we aimed to infer patient‐specific subpathway activities to reveal a functional signature associated with the prognosis of patients with breast cancer. We integrated pathway structure with gene expression data to construct patient‐specific subpathway activity profiles using a greedy search algorithm. A four‐subpathway prognostic signature was developed in the training set using a random forest supervised classification algorithm and a prognostic score model with the activity profiles. According to the signature, patients were classified into high‐risk and low‐risk groups with significantly different overall survival in the training set (median survival of 65 vs 106 months, P = 1.82e‐13) and test set (median survival of 75 vs 101 months, P = 4.17e‐5). Our signature was then applied to five independent breast cancer data sets and showed similar prognostic values, confirming the accuracy and robustness of the subpathway signature. Stratified analysis suggested that the four‐subpathway signature had prognostic value within subtypes of breast cancer. Our results suggest that the four‐subpathway signature may be a useful biomarker for breast cancer prognosis.