Numerical solutions of elliptic inverse problems via the equation error method

To estimate a parameter in an elliptic boundary value problem, the method of equation error chooses the value that minimizes the error in the PDE and boundary condition (the solution of the BVP having been replaced by a measurement). The estimated parameter converges to the exact value as the measured data converge to the exact value, provided Tikhonov regularization is used to control the instability inherent in the problem. The error in the estimated solution can be bounded in an appropriate quotient norm; estimates can be derived for both the underlying (infinite-dimensional) problem and a finite-element discretization that can be implemented in a practical algorithm. Numerical experiments demonstrate the efficacy and limitations of the method.

On Graphically Checking Goodness-of-fit of Binary Logistic Regression Models

OBJECTIVES: This paper is concerned with checking goodness-of-fit of binary logistic regression models. For the practitioners of data analysis, the broad classes of procedures for checking goodness-of-fit available in the literature are described. The challenges of model checking in the context of binary logistic regression are reviewed. As a viable solution, a simple graphical procedure for checking goodness-of-fit is proposed. METHODS: The graphical procedure proposed relies on pieces of information available from any logistic analysis; the focus is on combining and presenting these in an informative way. RESULTS: The information gained using this approach is presented with three examples. In the discussion, the proposed method is put into context and compared with other graphical procedures for checking goodness-of-fit of binary logistic models available in the literature. CONCLUSION: A simple graphical method can significantly improve the understanding of any logistic regression analysis and help to prevent faulty conclusions.

Prediction of 1-year mortality in patients with acute coronary syndromes undergoing percutaneous coronary intervention: validation of the logistic clinical SYNTAX (Synergy Between Percutaneous Coronary Interventions With Taxus and Cardiac Surgery) score

OBJECTIVES This study sought to validate the Logistic Clinical SYNTAX (Synergy Between Percutaneous Coronary Intervention With Taxus and Cardiac Surgery) score in patients with non-ST-segment elevation acute coronary syndromes (ACS), in order to further legitimize its clinical application. BACKGROUND The Logistic Clinical SYNTAX score allows for an individualized prediction of 1-year mortality in patients undergoing contemporary percutaneous coronary intervention. It is composed of a "Core" Model (anatomical SYNTAX score, age, creatinine clearance, and left ventricular ejection fraction), and "Extended" Model (composed of an additional 6 clinical variables), and has previously been cross validated in 7 contemporary stent trials (>6,000 patients). METHODS One-year all-cause death was analyzed in 2,627 patients undergoing percutaneous coronary intervention from the ACUITY (Acute Catheterization and Urgent Intervention Triage Strategy) trial. Mortality predictions from the Core and Extended Models were studied with respect to discrimination, that is, separation of those with and without 1-year all-cause death (assessed by the concordance [C] statistic), and calibration, that is, agreement between observed and predicted outcomes (assessed with validation plots). Decision curve analyses, which weight the harms (false positives) against benefits (true positives) of using a risk score to make mortality predictions, were undertaken to assess clinical usefulness. RESULTS In the ACUITY trial, the median SYNTAX score was 9.0 (interquartile range 5.0 to 16.0); approximately 40% of patients had 3-vessel disease, 29% diabetes, and 85% underwent drug-eluting stent implantation. Validation plots confirmed agreement between observed and predicted mortality. The Core and Extended Models demonstrated substantial improvements in the discriminative ability for 1-year all-cause death compared with the anatomical SYNTAX score in isolation (C-statistics: SYNTAX score: 0.64, 95% confidence interval [CI]: 0.56 to 0.71; Core Model: 0.74, 95% CI: 0.66 to 0.79; Extended Model: 0.77, 95% CI: 0.70 to 0.83). Decision curve analyses confirmed the increasing ability to correctly identify patients who would die at 1 year with the Extended Model versus the Core Model versus the anatomical SYNTAX score, over a wide range of thresholds for mortality risk predictions. CONCLUSIONS Compared to the anatomical SYNTAX score alone, the Core and Extended Models of the Logistic Clinical SYNTAX score more accurately predicted individual 1-year mortality in patients presenting with non-ST-segment elevation acute coronary syndromes undergoing percutaneous coronary intervention. These findings support the clinical application of the Logistic Clinical SYNTAX score.

The Relationship Between Sociosexuality and Aspects of Body Image in Men and Women: A Structural Equation Modeling Approach

The present study investigated the association between individual differences in sociosexual orientation and four aspects of body image in 156 male and 136 female students. While men were characterized by a less restricted sociosexual orientation, higher self-perceived physical attractiveness, and more pronounced self-rated physical assertiveness, women placed more emphasis on accentuation of body presentation. Structural equation modeling revealed significant positive relationships between sociosexual attitudes and physical attractiveness and accentuation of body presentation as well as between sociosexual behavior and physical attractiveness for the total sample. When introducing sex as a grouping variable, the attitudinal and behavioral components of sociosexuality were reliably related to both physical attractiveness and accentuation of body presentation as two aspects of body image in men, but not in women. Furthermore, our findings suggest that accentuation of body presentation represents a goal-directed behavior in men to increase the likelihood of having uncommitted sex but serves additional functions widely unrelated to unrestrictive sociosexual behavior in women.

A case of distinct difference between the measured blood Ethanol concentration and the concentration estimated by Widmark's equation

In the last century, several mathematical models have been developed to calculate blood ethanol concentrations (BAC) from the amount of ingested ethanol and vice versa. The most common one in the field of forensic sciences is Widmark's equation. A drinking experiment with 10 voluntary test persons was performed with a target BAC of 1.2 g/kg estimated using Widmark's equation as well as Watson's factor. The ethanol concentrations in the blood were measured using headspace gas chromatography/flame ionization and additionally with an alcohol Dehydrogenase (ADH)-based method. In a healthy 75-year-old man a distinct discrepancy between the intended and the determined blood ethanol concentration was observed. A blood ethanol concentration of 1.83 g/kg was measured and the man showed signs of intoxication. A possible explanation for the discrepancy is a reduction of the total body water content in older people. The incident showed that caution is advised when using the different mathematical models in aged people. When estimating ethanol concentrations, caution is recommended with calculated results due to potential discrepancies between mathematical models and biological systems

Effects of two traits of the ecological state equation on our understanding of species coexistence and ecosystem services

Species coexistence has been a fundamental issue to understand ecosystem functioning since the beginnings of ecology as a science. The search of a reliable and all-encompassing explanation for this issue has become a complex goal with several apparently opposing trends. On the other side, seemingly unconnected with species coexistence, an ecological state equation based on the inverse correlation between an indicator of dispersal that fits gamma distribution and species diversity has been recently developed. This article explores two factors, whose effects are inconspicuous in such an equation at the first sight, that are used to develop an alternative general theoretical background in order to provide a better understanding of species coexistence. Our main outcomes are: (i) the fit of dispersal and diversity values to gamma distribution is an important factor that promotes species coexistence mainly due to the right-skewed character of gamma distribution; (ii) the opposite correlation between species diversity and dispersal implies that any increase of diversity is equivalent to a route of “ecological cooling” whose maximum limit should be constrained by the influence of the third law of thermodynamics; this is in agreement with the well-known asymptotic trend of diversity values in space and time; (iii) there are plausible empirical and theoretical ways to apply physical principles to explain important ecological processes; (iv) the gap between theoretical and empirical ecology in those cases where species diversity is paradoxically high could be narrowed by a wave model of species coexistence based on the concurrency of local equilibrium states. In such a model, competitive exclusion has a limited but indispensable role in harmonious coexistence with functional redundancy. We analyze several literature references as well as ecological and evolutionary examples that support our approach, reinforcing the meaning equivalence between important physical and ecological principles.

Predicting childhood sexual or physical abuse: a logistic regression geo-mapping approach to prevention.

This study investigates the degree to which gender, ethnicity, relationship to perpetrator, and geomapped socio-economic factors significantly predict the incidence of childhood sexual abuse, physical abuse and non- abuse. These variables are then linked to geographic identifiers using geographic information system (GIS) technology to develop a geo-mapping framework for child sexual and physical abuse prevention.

Development of a Bayesian Joint Logistic Model to Better Study the Association between Haplotypes and Disease

In 2011, there will be an estimated 1,596,670 new cancer cases and 571,950 cancer-related deaths in the US. With the ever-increasing applications of cancer genetics in epidemiology, there is great potential to identify genetic risk factors that would help identify individuals with increased genetic susceptibility to cancer, which could be used to develop interventions or targeted therapies that could hopefully reduce cancer risk and mortality. In this dissertation, I propose to develop a new statistical method to evaluate the role of haplotypes in cancer susceptibility and development. This model will be flexible enough to handle not only haplotypes of any size, but also a variety of covariates. I will then apply this method to three cancer-related data sets (Hodgkin Disease, Glioma, and Lung Cancer). I hypothesize that there is substantial improvement in the estimation of association between haplotypes and disease, with the use of a Bayesian mathematical method to infer haplotypes that uses prior information from known genetics sources. Analysis based on haplotypes using information from publically available genetic sources generally show increased odds ratios and smaller p-values in both the Hodgkin, Glioma, and Lung data sets. For instance, the Bayesian Joint Logistic Model (BJLM) inferred haplotype TC had a substantially higher estimated effect size (OR=12.16, 95% CI = 2.47-90.1 vs. 9.24, 95% CI = 1.81-47.2) and more significant p-value (0.00044 vs. 0.008) for Hodgkin Disease compared to a traditional logistic regression approach. Also, the effect sizes of haplotypes modeled with recessive genetic effects were higher (and had more significant p-values) when analyzed with the BJLM. Full genetic models with haplotype information developed with the BJLM resulted in significantly higher discriminatory power and a significantly higher Net Reclassification Index compared to those developed with haplo.stats for lung cancer. Future analysis for this work could be to incorporate the 1000 Genomes project, which offers a larger selection of SNPs can be incorporated into the information from known genetic sources as well. Other future analysis include testing non-binary outcomes, like the levels of biomarkers that are present in lung cancer (NNK), and extending this analysis to full GWAS studies.

Structural equation modeling of the medical outcome study: Short Form 36 (SF-36)

The factorial validity of the SF-36 was evaluated using confirmatory factor analysis (CFA) methods, structural equation modeling (SEM), and multigroup structural equation modeling (MSEM). First, the measurement and structural model of the hypothesized SF-36 was explicated. Second, the model was tested for the validity of a second-order factorial structure, upon evidence of model misfit, determined the best-fitting model, and tested the validity of the best-fitting model on a second random sample from the same population. Third, the best-fitting model was tested for invariance of the factorial structure across race, age, and educational subgroups using MSEM.^ The findings support the second-order factorial structure of the SF-36 as proposed by Ware and Sherbourne (1992). However, the results suggest that: (a) Mental Health and Physical Health covary; (b) general mental health cross-loads onto Physical Health; (c) general health perception loads onto Mental Health instead of Physical Health; (d) many of the error terms are correlated; and (e) the physical function scale is not reliable across these two samples. This hierarchical factor pattern was replicated across both samples of health care workers, suggesting that the post hoc model fitting was not data specific. Subgroup analysis suggests that the physical function scale is not reliable across the "age" or "education" subgroups and that the general mental health scale path from Mental Health is not reliable across the "white/nonwhite" or "education" subgroups.^ The importance of this study is in the use of SEM and MSEM in evaluating sample data from the use of the SF-36. These methods are uniquely suited to the analysis of latent variable structures and are widely used in other fields. The use of latent variable models for self reported outcome measures has become widespread, and should now be applied to medical outcomes research. Invariance testing is superior to mean scores or summary scores when evaluating differences between groups. From a practical, as well as, psychometric perspective, it seems imperative that construct validity research related to the SF-36 establish whether this same hierarchical structure and invariance holds for other populations.^ This project is presented as three articles to be submitted for publication. ^

Predicting 3-year mortality after percutaneous coronary intervention: updated logistic clinical SYNTAX score based on patient-level data from 7 contemporary stent trials

OBJECTIVES This study aimed to update the Logistic Clinical SYNTAX score to predict 3-year survival after percutaneous coronary intervention (PCI) and compare the performance with the SYNTAX score alone. BACKGROUND The SYNTAX score is a well-established angiographic tool to predict long-term outcomes after PCI. The Logistic Clinical SYNTAX score, developed by combining clinical variables with the anatomic SYNTAX score, has been shown to perform better than the SYNTAX score alone in predicting 1-year outcomes after PCI. However, the ability of this score to predict long-term survival is unknown. METHODS Patient-level data (N = 6,304, 399 deaths within 3 years) from 7 contemporary PCI trials were analyzed. We revised the overall risk and the predictor effects in the core model (SYNTAX score, age, creatinine clearance, and left ventricular ejection fraction) using Cox regression analysis to predict mortality at 3 years. We also updated the extended model by combining the core model with additional independent predictors of 3-year mortality (i.e., diabetes mellitus, peripheral vascular disease, and body mass index). RESULTS The revised Logistic Clinical SYNTAX models showed better discriminative ability than the anatomic SYNTAX score for the prediction of 3-year mortality after PCI (c-index: SYNTAX score, 0.61; core model, 0.71; and extended model, 0.73 in a cross-validation procedure). The extended model in particular performed better in differentiating low- and intermediate-risk groups. CONCLUSIONS Risk scores combining clinical characteristics with the anatomic SYNTAX score substantially better predict 3-year mortality than the SYNTAX score alone and should be used for long-term risk stratification of patients undergoing PCI.

Solution of the relativistic Schrödinger equation for the δ ′ -Function potential in one dimension using cutoff regularization

We study the relativistic version of the Schrödinger equation for a point particle in one dimension with the potential of the first derivative of the delta function. The momentum cutoff regularization is used to study the bound state and scattering states. The initial calculations show that the reciprocal of the bare coupling constant is ultraviolet divergent, and the resultant expression cannot be renormalized in the usual sense, where the divergent terms can just be omitted. Therefore, a general procedure has been developed to derive different physical properties of the system. The procedure is used first in the nonrelativistic case for the purpose of clarification and comparisons. For the relativistic case, the results show that this system behaves exactly like the delta function potential, which means that this system also shares features with quantum filed theories, like being asymptotically free. In addition, in the massless limit, it undergoes dimensional transmutation, and it possesses an infrared conformal fixed point. The comparison of the solution with the relativistic delta function potential solution shows evidence of universality.