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Supplementary Table S6. Normalized expression values from the 90 miRNAs analyzed.

Supplementary Table and Figure Legend. Legends for Supplementary Tables S1-S8 and Supplementary Figures S1-S2.

Better knowledge of the biology of breast cancer has allowed the use of new targeted therapies, leading to improved outcome. High-throughput technologies allow deepening into the molecular architecture of breast cancer, integrating different levels of information, which is important if it helps in making clinical decisions. microRNA (miRNA) and protein expression profiles were obtained from 71 estrogen receptor–positive (ER+) and 25 triple-negative breast cancer (TNBC) samples. RNA and proteins obtained from formalin-fixed, paraffin-embedded tumors were analyzed by RT-qPCR and LC/MS-MS, respectively. We applied probabilistic graphical models representing complex biologic systems as networks, confirming that ER+ and TNBC subtypes are distinct biologic entities. The integration of miRNA and protein expression data unravels molecular processes that can be related to differences in the genesis and clinical evolution of these types of breast cancer. Our results confirm that TNBC has a unique metabolic profile that may be exploited for therapeutic intervention. Cancer Res; 75(11); 2243–53. ©2015 AACR.

Supplementary Figure S2. Hypothesis about how miR449a exerts its regulatory effect over cellular metabolism.

Breast cancer is the most frequent tumor in women and its incidence is increasing. Neoadjuvant chemotherapy has become standard of care as a complement to surgery in locally advanced or poor-prognosis early stage disease. The achievement of a complete response to neoadjuvant chemotherapy correlates with prognosis but it is not possible to predict who will obtain an excellent response. The molecular analysis of the tumor offers a unique opportunity to unveil predictive factors. In this work, gene expression profiling in 279 tumor samples from patients receiving neoadjuvant chemotherapy was performed and probabilistic graphical models were used. This approach enables addressing biological and clinical questions from a Systems Biology perspective, allowing to deal with large gene expression data and their interactions. Tumors presenting complete response to neoadjuvant chemotherapy had a higher activity of immune related functions compared to resistant tumors. Similarly, samples from complete responders presented higher expression of lymphocyte cell lineage markers, immune-activating and immune-suppressive markers, which may correlate with tumor infiltration by lymphocytes (TILs). These results suggest that the patient’s immune system plays a key role in tumor response to neoadjuvant treatment. However, future studies with larger cohorts are necessary to validate these hypotheses.

BACKGROUND Breast cancer is a diverse and heterogeneous disease. The use of neoadjuvant treatments has improved the prognosis of localized breast cancer. However, molecular basis of neoadjuvant treatment response and resistance remains unknown. Clinical data has uncovered the existence of different tumor responses to neoadjudvant chemotherapy, allowing the classification of patients in different groups. Gene expression profile description of the different patient groups provide essential information in the clinical decision making as well as to allow a deeper knowledge of this disease. MATERIALS AND METHODS A breast cancer tumor dataset was obtained from the Gene Expression Omnibus (GSE41998) and from a phase II trial (NCT00455533). 279 tumors from previously untreated women with primary invasive breast adenocarcinoma were included in this study. Whole genome gene expression profiling was performed using Affymetrix GeneChip gene expression microarrays.Differentially expressed genes were chosen selecting 3000 more variable probes among all patients and were used to construct four networks of gene functional interactions, one for all tumors and three for each molecular subtype independently. Functional structure was performed using probabilistic graphical models with local minimum Bayesian Information Criterion. Data analyses were carried out using MeV, BRB Array Tools, R, Cytoscape software suites and DAVID web tools. RESULTS Regardless of tumor molecular subtype, tumors showing a complete response to treatment showed higher "Immune response (MHCII)", "Immune response (chemotaxis)", "Immune response (B cell)“ and "Immune response (Interferon)” nodes activities compared to resistant tumors (stable disease tumors). These differences are also observed when analyzing tumor molecular subgroups (Luminal A, Luminal B and Basal-like) separately. Moreover, complete response tumors, showed significantly higher levels of lymphocytic cell lineage markers (CD4, CD8 and CD20). CONCLUSION This type of approach allows seeing differences at biological process levels rather than at the individual gene level.Tumors that respond to neoadjuvant treatment showed higher “Immune” nodes activity than resistant tumors and these differences were also showed in analyses stratified by molecular subtype. Besides, complete response tumors presented higher values of lymphocyte cell lineage markers which might suggest a greater amount of tumor-infiltrating lymphocytes (TILs). These results can suggest that patients' immune system could play an important role in the response to neoadjuvant chemotherapy treatment. Citation Format: Zamora Auñón P, Zapater-Moros A, Trilla-Fuertes L, Gamez-Pozo A, Prado-Vázquez G, Llorente-Armijo S, Lopez-Vacas R, Main P, Espinosa Arranz E, Fresno-Vara JA. A functional approach to the molecular basis of neoadjuvant treatment response in breast cancer [abstract]. In: Proceedings of the 2017 San Antonio Breast Cancer Symposium; 2017 Dec 5-9; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2018;78(4 Suppl):Abstract nr P6-15-12.

Breast cancer is a heterogeneous disease. In clinical practice, tumors are classified as hormonal receptor positive, Her2 positive and triple negative tumors. In previous works, our group defined a new hormonal receptor positive subgroup, the TN-like subtype, which had a prognosis and a molecular profile more similar to triple negative tumors. In this study, proteomics and Bayesian networks were used to characterize protein relationships in 96 breast tumor samples. Components obtained by these methods had a clear functional structure. The analysis of these components suggested differences in processes such as mitochondrial function or extracellular matrix between breast cancer subtypes, including our new defined subtype TN-like. In addition, one of the components, mainly related with extracellular matrix processes, had prognostic value in this cohort. Functional approaches allow to build hypotheses about regulatory mechanisms and to establish new relationships among proteins in the breast cancer context., PLoS ONE, 15 (6), ISSN:1932-6203

Background Metabolomics has a great potential in the development of new biomarkers in cancer and it has experiment recent technical advances. Methods In this study, metabolomics and gene expression data from 67 localized (stage I to IIIB) breast cancer tumor samples were analyzed, using (1) probabilistic graphical models to define associations using quantitative data without other a priori information; and (2) Flux Balance Analysis and flux activities to characterize differences in metabolic pathways. Results On the one hand, both analyses highlighted the importance of glutamine in breast cancer. Moreover, cell experiments showed that treating breast cancer cells with drugs targeting glutamine metabolism significantly affects cell viability. On the other hand, these computational methods suggested some hypotheses and have demonstrated their utility in the analysis of metabolomics data and in associating metabolomics with patient’s clinical outcome. Conclusions Computational analyses applied to metabolomics data suggested that glutamine metabolism is a relevant process in breast cancer. Cell experiments confirmed this hypothesis. In addition, these computational analyses allow associating metabolomics data with patient prognosis.

Triple-negative breast cancer is a heterogeneous disease characterized by a lack of hormonal receptors and HER2 overexpression. It is the only breast cancer subgroup that does not benefit from targeted therapies, and its prognosis is poor. Several studies have developed specific molecular classifications for triple-negative breast cancer. However, these molecular subtypes have had little impact in the clinical setting. Gene expression data and clinical information from 494 triple-negative breast tumors were obtained from public databases. First, a probabilistic graphical model approach to associate gene expression profiles was performed. Then, sparse k-means was used to establish a new molecular classification. Results were then verified in a second database including 153 triple-negative breast tumors treated with neoadjuvant chemotherapy. Clinical and gene expression data from 494 triple-negative breast tumors were analyzed. Tumors in the dataset were divided into four subgroups (luminal-androgen receptor expressing, basal, claudin-low and claudin-high), using the cancer stem cell hypothesis as reference. These four subgroups were defined and characterized through hierarchical clustering and probabilistic graphical models and compared with previously defined classifications. In addition, two subgroups related to immune activity were defined. This immune activity showed prognostic value in the whole cohort and in the luminal subgroup. The claudin-high subgroup showed poor response to neoadjuvant chemotherapy. Through a novel analytical approach we proved that there are at least two independent sources of biological information: cellular and immune. Thus, we developed two different and overlapping triple-negative breast cancer classifications and showed that the luminal immune-positive subgroup had better prognoses than the luminal immune-negative. Finally, this work paves the way for using the defined classifications as predictive features in the neoadjuvant scenario.

Metabolomics has a great potential in the development of new biomarkers in cancer. In this study, metabolomics and gene expression data from breast cancer tumor samples were analyzed, using (1) probabilistic graphical models to define associations using quantitative data without othera prioriinformation; and (2) Flux Balance Analysis and flux activities to characterize differences in metabolic pathways. On the one hand, both analyses highlighted the importance of glutamine in breast cancer. Moreover, cell experiments showed that treating breast cancer cells with drugs targeting glutamine metabolism significantly affects cell viability. On the other hand, these computational methods suggested some hypotheses and have demonstrated their utility in the analysis of metabolomics data and in associating metabolomics with patient’s clinical outcome.

BackgroundMuscle-invasive bladder tumors are associated with high risk of relapse and metastasis even after neoadjuvant chemotherapy and radical cystectomy. Therefore, further therapeutic options are needed and molecular characterization of the disease may help to identify new targets.ObjectiveThe aim of this work is to characterize muscle-invasive bladder tumors at molecular levels using computational analyses.Design, Settings and ParticipantsThe TCGA cohort of muscle-invasive bladder cancer patients was used to describe these tumors.Outcome Measurements and Statistical AnalysisProbabilistic graphical models, layer analyses based on sparse k-means coupled with Consensus Cluster, and Flux Balance Analysis were applied to characterize muscle-invasive bladder tumors at functional level.ResultsLuminal and Basal groups were identified, and an immune molecular layer with independent value was also described. Luminal tumors had decreased activity in the nodes of epidermis development and extracellular matrix, and increased activity in the node of steroid metabolism leading to a higher expression of androgen receptor.This fact points to androgen receptor as a therapeutic target in this group. Basal tumors were highly proliferative according to Flux Balance Analysis, which make these tumors good candidates for neoadjuvant chemotherapy. Immune-high group had higher expression of immune biomarkers, suggesting that this group may benefit from immune therapy.ConclusionsOur approach, based on layer analyses, established a Luminal group candidate for androgen receptor inhibitor therapy, a proliferative Basal group which seems to be a good candidate for chemotherapy, and an immune-high group candidate for immunotherapy.Patient SummaryMuscle-invasive bladder cancer has a poor prognosis in spite of appropriate therapy. Therefore, it is still necessary to characterize these tumors to propose new therapeutic targets. In this work we used computational analyses to characterize these tumors and propose treatments.

Breast cancer is a heterogeneous disease. In clinical practice, tumors are classified as hormonal receptor positive, Her2 positive and triple negative tumors. In previous works, our group defined a new hormonal receptor positive subgroup, the TN-like subtype, which has a prognosis and a molecular profile more similar to triple negative tumors. In this study, proteomics and Bayesian networks were used to characterize protein relationships in 106 breast tumor samples. Components obtained by these methods had a clear functional structure. The analysis of these components suggested differences in processes such as metastasis or proliferation between breast cancer subtypes, including our new subtype TN-like. In addition, one of the components, mainly related with metastasis, had prognostic value in this cohort. Functional approaches allow to build hypotheses about regulatory mechanisms and to establish new relationships among proteins in the breast cancer context.Author SummaryBreast cancer classification in the clinical practice is defined by three biomarkers (estrogen receptor, progesterone receptor and HER2) into hormone receptor positive, HER2+ and triple negative breast cancer (TNBC). Our group recently described a new ER+ subtype with molecular characteristics and prognosis similar to TNBC. In this study we propose a mathematical method, the Bayesian networks, as a useful tool to study protein interactions and differential biological processes in breast cancer subtypes, characterizing differences in relevant processes such as proliferation or metastasis and associated them with patient prognosis.

Metabolic reprogramming is a hallmark of cancer. We and other authors have previously shown that breast cancer subtypes present metabolism differences. In this study, breast cancer cell lines were treated with metformin and rapamycin. The response was heterogeneous across various breast cancer cells, leading to cell cycle disruption in specific conditions. The molecular effects of these treatments were characterized using sublethal doses, SNP genotyping and mass spectrometry-based proteomics. Protein expression was analyzed using probabilistic graphical models, showing that treatments elicit various responses in some biological processes, providing insights into cell responses to metabolism drugs. Moreover, a flux balance analysis approach using protein expression values was applied, showing that predicted growth rates were comparable with cell viability measurements and suggesting an increase in reactive oxygen species response enzymes due to metformin treatment. In addition, a method to assess flux differences in whole pathways was proposed. Our results show that these various approaches provide complementary information, which can be used to suggest hypotheses about the drugs’ mechanisms of action and the response to drugs that target metabolism.

Background The molecular analysis of melanoma has improved our understanding of the disease. The Cancer Genome Atlas (TCGA) Network proposed the molecular classification of melanoma in three subtypes: keratin-high, immune-high and membrane-low. However, this classification has not translated into therapeutic advances so far. Immunotherapy has contributed to improve survival, yet the mechanisms explaining differences in efficacy have not been elucidated. The aim of this study is to characterize the immune status of melanoma tumors through gene expression, and to analyze if these differences have an impact in the response to immunotherapy. Methods A probabilistic graphical model, followed by successive sparse k-means and consensus cluster analyses, was used to classify melanoma tumor samples from the TCGA cohort. Findings were translated into a cohort of patients treated with anti-PD1 antibodies (GSE78220, Hugo W et al) as a validation dataset. Results A probabilistic graphical model, including the 2,971 more variable genes from 472 melanoma samples from the TCGA dataset was built, and the resulting graph was processed to seek functional structures. Sparse k-means selected 119 genes. Gene ontology analysis showed that these genes were mainly related with immune processes. Immune genes split the population into two groups with different immune status. The so-called immune-high group included 232 patients (49%) and the immune-low group groups 238 patients (51%). The validation dataset GSE78220 provided mRNA expression in melanomas being treated with anti-PD-1 antibodies (28 biopsies belonging to 27 patients). The immune layer was translated to the new cohort by centroid method: 9 patients had immune-low tumors, whereas the remaining 18 had immune-high tumors. Kaplan Meier analysis using the clinical data from the GSE78220 cohort found a favorable response in patients with immune-low tumors (90% long-term survival). Conclusions We found a gene signature related to the tumor immune status that split the TCGA cohort in two groups. When applied to a cohort of patients treated with anti-PD1 antibodies, the group with immune-low tumors had 90% of long-term survival. Legal entity responsible for the study The authors. Funding Has not received any funding. Disclosure All authors have declared no conflicts of interest.

Introduction / Objectives MicroRNAs (miRNAs) constitute a new class of small noncoding RNAs that control post-transcriptionally the expression of gene products, either modulating directly protein translation, or regulating the stability of messenger RNA. There is increasing evidence of the role that miRNAs play in regulating breast cancer gene expression. However, there is little knowledge about the function and targets of miRNAs, and how they regulate complete processes or pathways. The main objective of this study was to unravel biological processes and signaling pathways regulated by miRNAs in breast cancer comparing their expression with the expression of the proteins they regulate. New statistical approaches were conducted in order to associate miRNA and protein quantification results with breast cancer subtype and to evaluate the association of miRNAs and protein expression patterns. Materials and Methods MicroRNA and protein expression were obtained from 79 breast cancer FFPE samples (16 TNBCs and 63 Luminal tumors). RNA was extracted from FFPE samples using RecoverAll (Ambion). MicroRNA expression was analyzed by RT-qPCR using TaqMan Arrays (Applied Biosystems). We selected for subsequent analysis those miRNAs with significant correlation between FF and FFPE samples. Protein extracts from FFPE samples were prepared in 2% SDS buffer using a protocol based on heat-induced antigen retrieval (Gámez-Pozo A et al. Mol Biosyst. 2011; 7: 2368-74). Protein abundance was calculated on the basis of normalized spectral protein intensity (LFQ intensity) using MaxQuant. Probabilistic graphical models are being applied successfully to represent complex biological systems as networks. In this phase of the study, we have chosen an appropriate methodology in the analysis of high dimensional data selecting a forest which minimizes the BIC criterion. This procedure extends the Chow and Liu's approach to the Gaussian case. The software used for implementation is based on the R library gRapHD. Results We measured the expression of 90 miRNAs in 79 breast cancer samples using RT-qPCR. We identified and quantified more than 3000 protein groups. We selected for subsequent analyses more than 1000 quantifiable proteins, defined as those identified at least in 75% of the samples in at least one type of sample with more than two unique peptides. Then, we analyzed the relations between miRNA and protein expressions. We identified miRNAs strongly related with processes considered as hallmarks of cancer, such as cellular adhesion, using probabilistic graphical models. Conclusions The integration of miRNA and protein expression patterns may be useful to describe how miRNAs regulate biological processes and signaling pathways in breast cancer. There is a need of new statistical approaches to evaluate these relations and to obtain meaningful information from such complex and massive data. Citation Information: Cancer Res 2013;73(24 Suppl): Abstract nr P4-07-07.

Breast cancer is a heterogeneous disease comprising a variety of entities with various genetic backgrounds. Estrogen receptor-positive, human epidermal growth factor receptor 2-negative tumors typically have a favorable outcome; however, some patients eventually relapse, which suggests some heterogeneity within this category. In the present study, we used proteomics and miRNA profiling techniques to characterize a set of 102 either estrogen receptor-positive (ER+)/progesterone receptor-positive (PR+) or triple-negative formalin-fixed, paraffin-embedded breast tumors. Protein expression-based probabilistic graphical models and flux balance analyses revealed that some ER+/PR+ samples had a protein expression profile similar to that of triple-negative samples and had a clinical outcome similar to those with triple-negative disease. This probabilistic graphical model-based classification had prognostic value in patients with luminal A breast cancer. This prognostic information was independent of that provided by standard genomic tests for breast cancer, such as MammaPrint, OncoType Dx and the 8-gene Score, The authors would like to acknowledge funding from grants PI12/00444, PI12/01016 and PI15/01310 from the Instituto de Salud Carlos III, Spanish Economy and Competitiveness Ministry, Spain, and co-funded by the FEDER program, “Una forma de hacer Europa”. This study has also been supported by the PRIME-XS project, grant agreement number 262067, funded by the EU’s Seventh Framework Program for Research. AG-P and RL-V are supported by Instituto de Salud Carlos III, and the Spanish Economy and Competitiveness Ministry grants, CA12/00258 and CA12/00264, respectively. . We want to particularly acknowledge the patients in this study for their participation and to the IdiPAZ, I+12 and O+EHUN Biobanks for the generous gifts of clinical samples used in this study. LT-F is supported by the Spanish Economy and Competitiveness Ministry (DI-15-07614). IdiPAZ, I+12 and O+EHUN Biobanks are supported by Instituto de Salud Carlos III, Spanish Economy and Competitiveness Ministry (RD09/0076/00073, RD09/0076/00118 and RD09/0076/00140, respectively) and FarmaIndustria, through the Cooperation Program in Clinical and Translational Research of the Community of Madrid and Basque Autonomous Community

Introduction Melanoma is the most lethal malignancy of the skin. The Cancer Genome Atlas (TCGA) Network proposed a molecular classification of melanoma, consisting in three subtypes: keratin-high, immune-high and membrane-low. However, this classification has not been translated into therapeutic advances yet. The aim of this study is to characterise molecular differences of melanoma at biological and molecular levels and propose a novel molecular classification with clinical impact. Material and methods A novel approach based on the existence of different molecular informative layers was used in this study to establish independent sets of information. A probabilistic graphical model, followed by successive sparse k-means and consensus cluster analyses were used to classify melanoma tumour samples from the TCGA cohort. Then, molecular classification was validated in another public cohort: GSE65904. Results and discussions We established that there were at least two different kinds of molecular information: an immune layer and a histological layer. Immune high and immune low groups were established based on immune layer information. Keratin low-proliferation low, melanogenesis high-membrane low, melanogenesis low-membrane high and keratin high groups were established based on molecular layer information. This suggests that the information related to the immune system is independent from other molecular features and is distributed among the groups established by the TCGA classification. Besides, the immune and histological assignments showed different clinical outcomes in another dataset, identifying two immune groups with prognostic value. Conclusion In this work we proposed a novel analytical approach based on informative molecular layers. In this way, two independent classifications, an immune-based and a histological-based classification, were established. Immune classification showed prognostic value in an independent cohort. Besides, the histological and the immune layers may deserve additional research for define new potential targeted therapies.

The problem of modeling Bayesian networks with continuous nodes deals with discrete approximations and conditional linear Gaussian models. In this article we have considered the possibility of using the exponential power family as conditional probability densities. It will be shown that for some platikurtic conditional distributions\ in this family, conditional regression functions are constant. These results give conditions to avoid compatibility problems when distributions with lighter tails than the normal are used in the description of conditional densities to specify joint densities, like in Bayesian networks.

A Bayesian test for the point null testing problem in the multivariate case is developed. A procedure to get the mixed distribution using the prior density is suggested. For comparisons between the Bayesian and classical approaches, lower bounds on posterior probabilities of the null hypothesis, over some reasonable classes of prior distributions, are computed and compared with the p-value of the classical test. With our procedure, a better approximation is obtained because the p-value is in the range of the Bayesian measures of evidence.

To evaluate the impact of model inaccuracies over the network’s output, after the evidence propagation, in a Gaussian Bayesian network, a sensitivity measure is introduced. This sensitivity measure is the Kullback–Leibler divergence and yields different expressions depending on the type of parameter to be perturbed, i.e. on the inaccurate parameter.In this work, the behavior of this sensitivity measure is studied when model inaccuracies are extreme, i.e. when extreme perturbations of the parameters can exist. Moreover, the sensitivity measure is evaluated for extreme situations of dependence between the main variables of the network and its behavior with extreme inaccuracies. This analysis is performed to find the effect of extreme uncertainty about the initial parameters of the model in a Gaussian Bayesian network and about extreme values of evidence. These ideas and procedures are illustrated with an example.

This article develops a method for computing the sensitivity analysis in a Gaussian Bayesian network. The measure presented is based on the Kullback–Leibler divergence and is useful to evaluate the impact of prior changes over the posterior marginal density of the target variable in the network. We find that some changes do not disturb the posterior marginal density of interest. Finally, we describe a method to compare different sensitivity measures obtained depending on where the inaccuracy was. An example is used to illustrate the concepts and methods presented.

In this paper local sensitivity measures are proposed to evaluate deviations from multivariate normality caused by asymmetry; the model we use to regulate asymmetry is the multivariate skew-normal distribution because it reflects the deviation in a very tractable way. The paper also examines the connection between local sensitivity and Mardia’s and Malkovich-Afifi’s skewness indices. Once the local sensitivity measures have been introduced, we study the effect of local perturbations in asymmetry on the conditional distributions; this issue has important implications because there are many procedures in statistics and other fields where the output depends on the distribution of some variables for known values of the others. The proposed measures use the Kullback-Leibler divergence to evaluate dissimilarities between probability distributions in order to assess deviation from normality on the joint distribution and on the marginal and conditional distributions as well. The results are illustrated with some examples

Better knowledge of the biology of breast cancer has allowed the use of new targeted therapies, leading to improved outcome. High-throughput technologies allow deepening into the molecular architecture of breast cancer, integrating different levels of information, which is important if it helps in making clinical decisions. microRNA (miRNA) and protein expression profiles were obtained from 71 estrogen receptor–positive (ER+) and 25 triple-negative breast cancer (TNBC) samples. RNA and proteins obtained from formalin-fixed, paraffin-embedded tumors were analyzed by RT-qPCR and LC/MS-MS, respectively. We applied probabilistic graphical models representing complex biologic systems as networks, confirming that ER+ and TNBC subtypes are distinct biologic entities. The integration of miRNA and protein expression data unravels molecular processes that can be related to differences in the genesis and clinical evolution of these types of breast cancer. Our results confirm that TNBC has a unique metabolic profile that may be exploited for therapeutic intervention. Cancer Res; 75(11); 2243–53. ©2015 AACR.

The asymptotic behavior in the right tail of the hazard rate function is considered to compare probability distributions. Using this tail ordering, the position of the posterior distribution with respect to the prior and the likelihood distributions is analyzed for a Bayesian location problem, and it is proved that, under rather general conditions, the posterior distribution is equivalent to the lightest-tailed distribution, except when both the likelihood and the prior are very heavy-tailed distributions. The relationship between the posterior distributions based on random samples of sizes n and 1, respectively, is also studied, as well as its dependence on the relative position of the prior distribution and the model for observations in the hazard rate scale.

As an alternative to the multivariate normal distribution we have dealt with a wider class of distributions, including the normal, that considers slightly different tail behavior than the normal tail. This is the multivariate exponential power family of distributions with a kurtosis parameter to give the possible forms of the distributions. To measure distribution deviations the Kullback-Leibler divergence will be used as an asymmetric dissimilarity measure from an information-theoretic basis. Thus, a local quantitative description of the non-normality could be established for joint distributions in this family as well as the impact this perturbation causes in the marginal and conditional distributions.

In this paper the asymptotic relationship between the classical p-value and the infimum (over all unimodal and symmetric distributions) of the posterior probability in the point null hypothesis testing problem is analyzed. It is shown that the ratio between the infimum and the classical p-value has an equivalent asymptotic behavior to the hazard rate of the sample model.

In the problem of testing the point null hypothesis H-0: theta = theta(0) versus H-1: theta not equal theta(0), with a previously given prior density for the parameter theta, we propose the following methodology: to fix an interval of radius epsilon around theta(0) and assign a prior mass, pi(0), to H-0 computed by the density pi(theta) over the interval (theta(0) - epsilon, theta(0) + epsilon), spreading the remainder, 1 - pi(0), over H-1 according to pi(theta). It is shown that for Lindley's paradox, the Normal model with some different priors and Darwin-Fisher's example, this procedure makes the posterior probability of H-0 and the p-value matching better than if the prior mass assigned to H-0 is 0.5.

The Multivariate Exponential Power family is considered for n-dimensional random variables, Z, with a known partition Z (Y,X) of dimensions p and n-p, respectively. An infinitesimal variation of any parameter produces both conditional and marginal distributions perturbations. The aim of our study is to determine the local effect of kurtosis deviations by means of the Kullback-Leibler divergence measure between probability distributions. The additive decomposition of this measure in terms of the conditional and marginal distributions, YǀX and X, has been used to define the relative sensitivity of the conditional distributions family {YǀX = x}. The obtained results show that, for large dimensions, it is nearly p/n.

Se analizan los conceptos de funcion de distribucion propensa, neutra y resistente a producir datos atipicos dependiendo del comportamiento asintotico de la diferencia y la razon de los dos extremos superiores. Posteriormente se caracterizan las primeras definiciones con propiedades de la cola derecha de la funcion de distribucion