Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Centralnotations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform.In this way very elaborated aspects of mathematical statistics can be understoodeasily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating,combination of likelihood and robust M-estimation functions are simple additions/perturbations in A2(Pprior). Weighting observations corresponds to a weightedaddition of the corresponding evidence.Likelihood based statistics for general exponential families turns out to have aparticularly easy interpretation in terms of A2(P). Regular exponential families formfinite dimensional linear subspaces of A2(P) and they correspond to finite dimensionalsubspaces formed by their posterior in the dual information space A2(Pprior).The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P.The discussion of A2(P) valued random variables, such as estimation functionsor likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

Compositional Data in Biomedical Research

Modern methods of compositional data analysis are not well known in biomedical research.Moreover, there appear to be few mathematical and statistical researchersworking on compositional biomedical problems. Like the earth and environmental sciences,biomedicine has many problems in which the relevant scienti c information isencoded in the relative abundance of key species or categories. I introduce three problemsin cancer research in which analysis of compositions plays an important role. Theproblems involve 1) the classi cation of serum proteomic pro les for early detection oflung cancer, 2) inference of the relative amounts of di erent tissue types in a diagnostictumor biopsy, and 3) the subcellular localization of the BRCA1 protein, and it'srole in breast cancer patient prognosis. For each of these problems I outline a partialsolution. However, none of these problems is \solved". I attempt to identify areas inwhich additional statistical development is needed with the hope of encouraging morecompositional data analysts to become involved in biomedical research

A new method for constructing exact tests without making any assumptions

We present a new method for constructing exact distribution-free tests (and confidence intervals) for variables that can generate more than two possible outcomes.This method separates the search for an exact test from the goal to create a non-randomized test. Randomization is used to extend any exact test relating to meansof variables with finitely many outcomes to variables with outcomes belonging to agiven bounded set. Tests in terms of variance and covariance are reduced to testsrelating to means. Randomness is then eliminated in a separate step.This method is used to create confidence intervals for the difference between twomeans (or variances) and tests of stochastic inequality and correlation.

Stepwise multiple testing as formalized data snooping

It is common in econometric applications that several hypothesis tests arecarried out at the same time. The problem then becomes how to decide whichhypotheses to reject, accounting for the multitude of tests. In this paper,we suggest a stepwise multiple testing procedure which asymptoticallycontrols the familywise error rate at a desired level. Compared to relatedsingle-step methods, our procedure is more powerful in the sense that itoften will reject more false hypotheses. In addition, we advocate the useof studentization when it is feasible. Unlike some stepwise methods, ourmethod implicitly captures the joint dependence structure of the teststatistics, which results in increased ability to detect alternativehypotheses. We prove our method asymptotically controls the familywise errorrate under minimal assumptions. We present our methodology in the context ofcomparing several strategies to a common benchmark and deciding whichstrategies actually beat the benchmark. However, our ideas can easily beextended and/or modied to other contexts, such as making inference for theindividual regression coecients in a multiple regression framework. Somesimulation studies show the improvements of our methods over previous proposals. We also provide an application to a set of real data.

A generalization of histogram type estimators

We introduce simple nonparametric density estimators that generalize theclassical histogram and frequency polygon. The new estimators are expressed as linear combination of density functions that are piecewisepolynomials, where the coefficients are optimally chosen in order to minimize the integrated square error of the estimator. We establish the asymptotic behaviour of the proposed estimators, and study theirperformance in a simulation study.

How accurate is the current picture of human genetic variation?

Our understanding of the distribution of worldwide human genomic diversity has greatly increased over recent years thanks to the availability of large data sets derived from short tandem repeats (STRs), insertion deletion polymorphisms (indels) and single nucleotide polymorphisms (SNPs). A concern, however, is that the current picture of worldwide human genomic diversity may be inaccurate because of biases in the selection process of genetic markers (so-called 'ascertainment bias'). To evaluate this problem, we first compared the distribution of genomic diversity between these three types of genetic markers in the populations from the HGDP-CEPH panel for evidence of bias or incongruities. In a second step, using a very relaxed set of criteria to prevent the intrusion of bias, we developed a new set of unbiased STR markers and compared the results against those from available panels. Contrarily to recent claims, our results show that the STR markers suffer from no discernible bias, and can thus be used as a baseline reference for human genetic diversity and population differentiation. The bias on SNPs is moderate compared to that on the set of indels analysed, which we recommend should be avoided for work describing the distribution of human genetic diversity or making inference on human settlement history.

A morphometric study of mechanotransductively induced dermal neovascularization.

BACKGROUND:: Mechanical stretch has been shown to induce vascular remodeling and increase vessel density, but the pathophysiologic mechanisms and the morphologic changes induced by tensile forces to dermal vessels are poorly understood. METHODS:: A custom computer-controlled stretch device was designed and applied to the backs of C57BL/6 mice (n = 38). Dermal and vascular remodeling was studied over a 7-day period. Corrosion casting and three-dimensional scanning electron microscopy and CD31 staining were performed to analyze microvessel morphology. Hypoxia was assessed by immunohistochemistry. Western blot analysis of vascular endothelial growth factor (VEGF) and mRNA expression of VEGF receptors was performed. RESULTS:: Skin stretching was associated with increased angiogenesis as demonstrated by CD31 staining and vessel corrosion casting where intervascular distance and vessel diameter were decreased (p < 0.01). Immediately after stretching, VEGF dimers were increased. Messenger RNA expression of VEGF receptor 1, VEGF receptor 2, neuropilin 1, and neuropilin 2 was increased starting as early as 2 hours after stretching. Highly proliferating epidermal cells induced epidermal hypoxia starting at day 3 (p < 0.01). CONCLUSIONS:: Identification of significant hypoxic cells occurred after identification of neovessels, suggesting an alternative mechanism. Increased expression of angiogenic receptors and stabilization of VEGF dimers may be involved in a mechanotransductive, prehypoxic induction of neovascularization.

Modeling the forensic two-trace problem with Bayesian networks

The forensic two-trace problem is a perplexing inference problem introduced by Evett (J Forensic Sci Soc 27:375-381, 1987). Different possible ways of wording the competing pair of propositions (i.e., one proposition advanced by the prosecution and one proposition advanced by the defence) led to different quantifications of the value of the evidence (Meester and Sjerps in Biometrics 59:727-732, 2003). Here, we re-examine this scenario with the aim of clarifying the interrelationships that exist between the different solutions, and in this way, produce a global vision of the problem. We propose to investigate the different expressions for evaluating the value of the evidence by using a graphical approach, i.e. Bayesian networks, to model the rationale behind each of the proposed solutions and the assumptions made on the unknown parameters in this problem.

Spanning tests in return and stochastic discount factor mean-variance frontiers: A unifying approach

We propose new spanning tests that assess if the initial and additional assets share theeconomically meaningful cost and mean representing portfolios. We prove their asymptoticequivalence to existing tests under local alternatives. We also show that unlike two-step oriterated procedures, single-step methods such as continuously updated GMM yield numericallyidentical overidentifyng restrictions tests, so there is arguably a single spanning test.To prove these results, we extend optimal GMM inference to deal with singularities in thelong run second moment matrix of the influence functions. Finally, we test for spanningusing size and book-to-market sorted US stock portfolios.

The European Social Survey. Methodological aspects

From a scientific point of view, surveys are undoubtedly a valuable tool for the knowledge of the social and political reality. They are widely used in the social sciences research. However, the researcher's task is often disturbed by a series of deficiencies related to some technical aspects that make difficult both the inference and the comparison. The main aim of the present paper is to report and justify the European Social Survey's technical specifications addressed to avoid and/or minimize such deficiencies. The article also gives a characterization of the non-respondents in Spain obtained from the analysis of the 2002 fieldwork data file.

Seeing the Wood through the Trees: The Current State of Higher Systematics in the Strepsirhini.

Strepsirhines comprise 10 living or recently extinct families, ≥50% of extant primate families. Their phylogenetic relationships have been intensively studied, but common topologies have only recently emerged; e.g. all recent reconstructions link the Lepilemuridae and Cheirogaleidae. The position of the indriids, however, remains uncertain, and molecular studies have placed them as the sister to every clade except Daubentonia, the preferred sister group of morphologists. The node subtending Afro-Asian lorisids has been similarly elusive. We probed these phylogenetic inconsistencies using a test data set including 20 strepsirhine taxa and 2 outgroups represented by 3,543 mtDNA base pairs, and 43 selected morphological characters, subjecting the data to maximum parsimony, maximum likelihood and Bayesian inference analyses, and reconstructing topology and node ages jointly from the molecular data using relaxed molecular clock analyses. Our permutations yielded compatible but not identical evolutionary histories, and currently popular techniques seem unable to deal adequately with morphological data. We investigated the influence of morphological characters on tree topologies, and examined the effect of taxon sampling in two experiments: (1) we removed the molecular data only for 5 endangered Malagasy taxa to simulate 'extinction leaving a fossil record'; (2) we removed both the sequence and morphological data for these taxa. Topologies were affected more by the inclusion of morphological data only, indicating that palaeontological studies that involve inserting a partial morphological data set into a combined data matrix of extant species should be interpreted with caution. The gap of approximately 10 million years between the daubentoniid divergence and those of the other Malagasy families deserves more study. The apparently contemporaneous divergence of African and non-daubentoniid Malagasy families 40-30 million years ago may be related to regional plume-induced uplift followed by a global period of cooling and drying. © 2013 S. Karger AG, Basel.

HACEK infective endocarditis: characteristics and outcomes from a large, multi-national cohort.

The HACEK organisms (Haemophilus species, Aggregatibacter species, Cardiobacterium hominis, Eikenella corrodens, and Kingella species) are rare causes of infective endocarditis (IE). The objective of this study is to describe the clinical characteristics and outcomes of patients with HACEK endocarditis (HE) in a large multi-national cohort. Patients hospitalized with definite or possible infective endocarditis by the International Collaboration on Endocarditis Prospective Cohort Study in 64 hospitals from 28 countries were included and characteristics of HE patients compared with IE due to other pathogens. Of 5591 patients enrolled, 77 (1.4%) had HE. HE was associated with a younger age (47 vs. 61 years; p<0.001), a higher prevalence of immunologic/vascular manifestations (32% vs. 20%; p<0.008) and stroke (25% vs. 17% p = 0.05) but a lower prevalence of congestive heart failure (15% vs. 30%; p = 0.004), death in-hospital (4% vs. 18%; p = 0.001) or after 1 year follow-up (6% vs. 20%; p = 0.01) than IE due to other pathogens (n = 5514). On multivariable analysis, stroke was associated with mitral valve vegetations (OR 3.60; CI 1.34-9.65; p<0.01) and younger age (OR 0.62; CI 0.49-0.90; p<0.01). The overall outcome of HE was excellent with the in-hospital mortality (4%) significantly better than for non-HE (18%; p<0.001). Prosthetic valve endocarditis was more common in HE (35%) than non-HE (24%). The outcome of prosthetic valve and native valve HE was excellent whether treated medically or with surgery. Current treatment is very successful for the management of both native valve prosthetic valve HE but further studies are needed to determine why HE has a predilection for younger people and to cause stroke. The small number of patients and observational design limit inferences on treatment strategies. Self selection of study sites limits epidemiological inferences.

Robust response transformations based on optimal prediction

Nonlinear regression problems can often be reduced to linearity by transforming the response variable (e.g., using the Box-Cox family of transformations). The classic estimates of the parameter defining the transformation as well as of the regression coefficients are based on the maximum likelihood criterion, assuming homoscedastic normal errors for the transformed response. These estimates are nonrobust in the presence of outliers and can be inconsistent when the errors are nonnormal or heteroscedastic. This article proposes new robust estimates that are consistent and asymptotically normal for any unimodal and homoscedastic error distribution. For this purpose, a robust version of conditional expectation is introduced for which the prediction mean squared error is replaced with an M scale. This concept is then used to develop a nonparametric criterion to estimate the transformation parameter as well as the regression coefficients. A finite sample estimate of this criterion based on a robust version of smearing is also proposed. Monte Carlo experiments show that the new estimates compare favorably with respect to the available competitors.

What comes to mind

We present a model of intuitive inference, called local thinking, in which anagent combines data received from the external world with information retrieved frommemory to evaluate a hypothesis. In this model, selected and limited recall ofinformation follows a version of the respresentativeness heuristic. The model canaccount for some of the evidence on judgment biases, including conjunction anddisjunction fallacies, but also for several anomalies related to demand for insurance.

Constant coefficient tests for random coefficient regression

Random coefficient regression models have been applied in differentfields and they constitute a unifying setup for many statisticalproblems. The nonparametric study of this model started with Beranand Hall (1992) and it has become a fruitful framework. In thispaper we propose and study statistics for testing a basic hypothesisconcerning this model: the constancy of coefficients. The asymptoticbehavior of the statistics is investigated and bootstrapapproximations are used in order to determine the critical values ofthe test statistics. A simulation study illustrates the performanceof the proposals.