On the explanatory power of principal components

We show that if we have an orthogonal base ($u_1,\ldots,u_p$) in a $p$-dimensional vector space, and select $p+1$ vectors $v_1,\ldots, v_p$ and $w$ such that the vectors traverse the origin, then the probability of $w$ being to closer to all the vectors in the base than to $v_1,\ldots, v_p$ is at least 1/2 and converges as $p$ increases to infinity to a normal distribution on the interval [-1,1]; i.e., $\Phi(1)-\Phi(-1)\approx0.6826$. This result has relevant consequences for Principal Components Analysis in the context of regression and other learning settings, if we take the orthogonal base as the direction of the principal components.Comment: 10 pages, 3 figure

Cross-validation and Peeling Strategies for Survival Bump Hunting using Recursive Peeling Methods

We introduce a framework to build a survival/risk bump hunting model with a censored time-to-event response. Our Survival Bump Hunting (SBH) method is based on a recursive peeling procedure that uses a specific survival peeling criterion derived from non/semi-parametric statistics such as the hazards-ratio, the log-rank test or the Nelson-Aalen estimator. To optimize the tuning parameter of the model and validate it, we introduce an objective function based on survival or prediction-error statistics, such as the log-rank test and the concordance error rate. We also describe two alternative cross-validation techniques adapted to the joint task of decision-rule making by recursive peeling and survival estimation. Numerical analyses show the importance of replicated cross-validation and the differences between criteria and techniques in both low and high-dimensional settings. Although several non-parametric survival models exist, none addresses the problem of directly identifying local extrema. We show how SBH efficiently estimates extreme survival/risk subgroups unlike other models. This provides an insight into the behavior of commonly used models and suggests alternatives to be adopted in practice. Finally, our SBH framework was applied to a clinical dataset. In it, we identified subsets of patients characterized by clinical and demographic covariates with a distinct extreme survival outcome, for which tailored medical interventions could be made. An R package `PRIMsrc` is available on CRAN and GitHub.Comment: Keywords: Exploratory Survival/Risk Analysis, Survival/Risk Estimation & Prediction, Non-Parametric Method, Cross-Validation, Bump Hunting, Rule-Induction Metho

Unsupervised Bump Hunting Using Principal Components

Principal Components Analysis is a widely used technique for dimension reduction and characterization of variability in multivariate populations. Our interest lies in studying when and why the rotation to principal components can be used effectively within a response-predictor set relationship in the context of mode hunting. Specifically focusing on the Patient Rule Induction Method (PRIM), we first develop a fast version of this algorithm (fastPRIM) under normality which facilitates the theoretical studies to follow. Using basic geometrical arguments, we then demonstrate how the PC rotation of the predictor space alone can in fact generate improved mode estimators. Simulation results are used to illustrate our findings.Comment: 24 pages, 9 figure

Metabolomics of ApcMin/+ mice genetically susceptible to intestinal cancer

BACKGROUND: To determine how diets high in saturated fat could increase polyp formation in the mouse model of intestinal neoplasia, Apc( Min/+ ), we conducted large-scale metabolome analysis and association study of colon and small intestine polyp formation from plasma and liver samples of Apc( Min/+ ) vs. wild-type littermates, kept on low vs. high-fat diet. Label-free mass spectrometry was used to quantify untargeted plasma and acyl-CoA liver compounds, respectively. Differences in contrasts of interest were analyzed statistically by unsupervised and supervised modeling approaches, namely Principal Component Analysis and Linear Model of analysis of variance. Correlation between plasma metabolite concentrations and polyp numbers was analyzed with a zero-inflated Generalized Linear Model. RESULTS: Plasma metabolome in parallel to promotion of tumor development comprises a clearly distinct profile in Apc( Min/+ ) mice vs. wild type littermates, which is further altered by high-fat diet. Further, functional metabolomics pathway and network analyses in Apc( Min/+ ) mice on high-fat diet revealed associations between polyp formation and plasma metabolic compounds including those involved in amino-acids metabolism as well as nicotinamide and hippuric acid metabolic pathways. Finally, we also show changes in liver acyl-CoA profiles, which may result from a combination of Apc( Min/+ )-mediated tumor progression and high fat diet. The biological significance of these findings is discussed in the context of intestinal cancer progression. CONCLUSIONS: These studies show that high-throughput metabolomics combined with appropriate statistical modeling and large scale functional approaches can be used to monitor and infer changes and interactions in the metabolome and genome of the host under controlled experimental conditions. Further these studies demonstrate the impact of diet on metabolic pathways and its relation to intestinal cancer progression. Based on our results, metabolic signatures and metabolic pathways of polyposis and intestinal carcinoma have been identified, which may serve as useful targets for the development of therapeutic interventions

Urinary Protein Profiles in a Rat Model for Diabetic Complications

Diabetes mellitus is estimated to affect ∼24 million people in the United States and more than 150 million people worldwide. There are numerous end organ complications of diabetes, the onset of which can be delayed by early diagnosis and treatment. Although assays for diabetes are well founded, tests for its complications lack sufficient specificity and sensitivity to adequately guide these treatment options. In our study, we employed a streptozotocin- induced rat model of diabetes to determine changes in urinary protein profiles that occur during the initial response to the attendant hyperglycemia (e.g. the first two months) with the goal of developing a reliable and reproducible method of analyzing multiple urine samples as well as providing clues to early markers of disease progression. After filtration and buffer exchange, urinary proteins were digested with a specific protease, and the relative amounts of several thousand peptides were compared across rat urine samples representing various times after administration of drug or sham control. Extensive data analysis, including imputation of missing values and normalization of all data was followed by ANOVA analysis to discover peptides that were significantly changing as a function of time, treatment and interaction of the two variables. The data demonstrated significant differences in protein abundance in urine before observable pathophysiological changes occur in this animal model and as function of the measured variables. These included decreases in relative abundance of major urinary protein precursor and increases in pro-alpha collagen, the expression of which is known to be regulated by circulating levels of insulin and/or glucose. Peptides from these proteins represent potential biomarkers, which can be used to stage urogenital complications from diabetes. The expression changes of a pro-alpha 1 collagen peptide was also confirmed via selected reaction monitoring

Metabolomics of ApcMin/+\u3c/sup\u3e Mice Genetically Susceptible to Intestinal Cancer

Background: To determine how diets high in saturated fat could increase polyp formation in the mouse model of intestinal neoplasia, ApcMin/+, we conducted large-scale metabolome analysis and association study of colon and small intestine polyp formation from plasma and liver samples of ApcMin/+ vs. wild-type littermates, kept on low vs. high-fat diet. Label-free mass spectrometry was used to quantify untargeted plasma and acyl-CoA liver compounds, respectively. Differences in contrasts of interest were analyzed statistically by unsupervised and supervised modeling approaches, namely Principal Component Analysis and Linear Model of analysis of variance. Correlation between plasma metabolite concentrations and polyp numbers was analyzed with a zero-inflated Generalized Linear Model.Results: Plasma metabolome in parallel to promotion of tumor development comprises a clearly distinct profile in ApcMin/+ mice vs. wild type littermates, which is further altered by high-fat diet. Further, functional metabolomics pathway and network analyses in ApcMin/+ mice on high-fat diet revealed associations between polyp formation and plasma metabolic compounds including those involved in amino-acids metabolism as well as nicotinamide and hippuric acid metabolic pathways. Finally, we also show changes in liver acyl-CoA profiles, which may result from a combination of ApcMin/+-mediated tumor progression and high fat diet. The biological significance of these findings is discussed in the context of intestinal cancer progression.Conclusions: These studies show that high-throughput metabolomics combined with appropriate statistical modeling and large scale functional approaches can be used to monitor and infer changes and interactions in the metabolome and genome of the host under controlled experimental conditions. Further these studies demonstrate the impact of diet on metabolic pathways and its relation to intestinal cancer progression. Based on our results, metabolic signatures and metabolic pathways of polyposis and intestinal carcinoma have been identified, which may serve as useful targets for the development of therapeutic interventions. © 2014 Dazard et al.; licensee BioMed Central Ltd

Studying genetic determinants of natural variation in human gene expression using Bayesian ANOVA

Standard genetic mapping techniques scan chromosomal segments for location of genetic linkage and association signals. The majority of these methods consider only correlations at single markers and/or phenotypes with explicit detailing of the genetic structure. These methods tend to be limited by their inability to consider the effect of large numbers of model variables jointly. In contrast, we propose a Bayesian analysis of variance (ANOVA) method to categorize individuals based on similarity of multidimensional profiles and attempt to analyze all variables simultaneously. Using Problem 1 of the Genetic Analysis Workshop 15 data set, we demonstrate the method's utility for joint analysis of gene expression levels and single-nucleotide polymorphism genotypes. We show that the method extracts similar information to that of previous genetic mapping analyses, and suggest extensions of the method for mining unique information not previously found

Metabolomic Analysis of Liver Tissue from the VX2 Rabbit Model of Secondary Liver Tumors

Purpose. The incidence of liver neoplasms is rising in USA. The purpose of this study was to determine metabolic profiles of liver tissue during early cancer development. Methods. We used the rabbitVX2 model of liver tumors (LT) and a control group consisting of sham animals implanted with Gelfoam into their livers (LG). After two weeks from implantation, liver tissue from lobes with and without tumor was obtained from experimental animals (LT+/LT−) as well as liver tissue from controls (LG+/LG−). Peaks obtained by Gas Chromatography-Mass Spectrometry were subjected to identification. 56 metabolites were identified and their profiles compared between groups using principal component analysis (PCA) and a mixed-effect two-way ANOVA model. Results. Animals recovered from surgery uneventfully. Analyses identified a metabolite profile that significantly differs in experimental conditions after controlling the False Discovery Rate (FDR). 16 metabolites concentrations differed significantly when comparing samples from (LT+/LT−) to samples from (LG+/LG−) livers. A significant difference was also shown in 20 metabolites when comparing samples from (LT+) liver lobes to samples from (LT−) liver lobes. Conclusion. Normal liver tissue harboring malignancy had a distinct metabolic signature. The role of metabolic profiles on liver biopsies for the detection of early liver cancer remains to be determined