Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators

In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator $\mathbf{P}$ consisting of finitely or countably many distributional operators $P_n$, which are defined on the dual space of the Schwartz space. The types of operators we consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain appropriate full-space Green function $G$ with respect to $L:=\mathbf{P}^{\ast T}\mathbf{P}$ now becomes a conditionally positive definite function. In order to support this claim we ensure that the distributional adjoint operator $\mathbf{P}^{\ast}$ of $\mathbf{P}$ is well-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function $G$ can be isometrically embedded into or even be isometrically equivalent to a generalized Sobolev space. As an application, we take linear combinations of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant $s_{f,X}$ to data values sampled from an unknown generalized Sobolev function $f$ at data sites located in some set $X \subset \mathbb{R}^d$. We provide several examples, such as Mat\'ern kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert spaces of well-known reproducing kernels are isometrically equivalent to a generalized Sobolev space. These examples further illustrate how we can rescale the Sobolev spaces by the vector distributional operator $\mathbf{P}$. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the "best" kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D. thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}

Assessing quality of life in a clinical study on heart rehabilitation patients: how well do value sets based on given or experienced health states reflect patients' valuations?

Background: Quality of life as an endpoint in a clinical study may be sensitive to the value set used to derive a single score. Focusing on patients' actual valuations in a clinical study, we compare different value sets for the EQ-5D-3L and assess how well they reproduce patients' reported results. Methods: A clinical study comparing inpatient (n = 98) and outpatient (n = 47) rehabilitation of patients after an acute coronary event is re-analyzed. Value sets include: 1. Given health states and time-trade-off valuation (GHS-TTO) rendering economic utilities;2. Experienced health states and valuation by visual analog scale (EHS-VAS). Valuations are compared with patient-reported VAS rating. Accuracy is assessed by mean absolute error (MAE) and by Pearson's correlation.. External validity is tested by correlation with established MacNew global scores. Drivers of differences between value sets and VAS are analyzed using repeated measures regression. Results: EHS-VAS had smaller MAEs and higher. in all patients and in the inpatient group, and correlated best with MacNew global score. Quality-adjusted survival was more accurately reflected by EHS-VAS. Younger, better educated patients reported lower VAS at admission than the EHS-based value set. EHS-based estimates were mostly able to reproduce patient-reported valuation. Economic utility measurement is conceptually different, produced results less strongly related to patients' reports, and resulted in about 20 % longer quality-adjusted survival. Conclusion: Decision makers should take into account the impact of choosing value sets on effectiveness results. For transferring the results of heart rehabilitation patients from another country or from another valuation method, the EHS-based value set offers a promising estimation option for those decision makers who prioritize patient-reported valuation. Yet, EHS-based estimates may not fully reflect patient-reported VAS in all situations

Smoking and health-related quality of life in English general population: Implications for economic evaluations

Copyright @ 2012 Vogl et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.This article has been made available through the Brunel Open Access Publishing Fund.Background: Little is known as to how health-related quality of life (HRQoL) when measured by generic instruments such as EQ-5D differ across smokers, ex-smokers and never-smokers in the general population; whether the overall pattern of this difference remain consistent in each domain of HRQoL; and what implications this variation, if any, would have for economic evaluations of tobacco control interventions. Methods: Using the 2006 round of Health Survey for England data (n = 13,241), this paper aims to examine the impact of smoking status on health-related quality of life in English population. Depending upon the nature of the EQ-5D data (i.e. tariff or domains), linear or logistic regression models were fitted to control for biology, clinical conditions, socio-economic background and lifestyle factors that an individual may have regardless of their smoking status. Age- and gender-specific predicted values according to smoking status are offered as the potential 'utility' values to be used in future economic evaluation models. Results: The observed difference of 0.1100 in EQ-5D scores between never-smokers (0.8839) and heavy-smokers (0.7739) reduced to 0.0516 after adjusting for biological, clinical, lifestyle and socioeconomic conditions. Heavy-smokers, when compared with never-smokers, were significantly more likely to report some/severe problems in all five domains - mobility (67%), self-care (70%), usual activity (42%), pain/discomfort (46%) and anxiety/depression (86%) -. 'Utility' values by age and gender for each category of smoking are provided to be used in the future economic evaluations. Conclusion: Smoking is significantly and negatively associated with health-related quality of life in English general population and the magnitude of this association is determined by the number of cigarettes smoked. The varying degree of this association, captured through instruments such as EQ-5D, may need to be fed into the design of future economic evaluations where the intervention being evaluated affects (e.g. tobacco control) or is affected (e.g. treatment for lung cancer) by individual's (or patients') smoking status

Adiposity has differing associations with incident coronary heart disease and mortality in the Scottish population: cross-sectional surveys with follow-up

Objective: Investigation of the association of excess adiposity with three different outcomes: all-cause mortality, coronary heart disease (CHD) mortality and incident CHD. Design: Cross-sectional surveys linked to hospital admissions and death records. Subjects: 19 329 adults (aged 18–86 years) from a representative sample of the Scottish population. Measurements: Gender-stratified Cox proportional hazards models were used to estimate hazard ratios (HRs) for all-cause mortality, CHD mortality and incident CHD. Separate models incorporating the anthropometric measurements body mass index (BMI), waist circumference (WC) or waist–hip ratio (WHR) were created adjusted for age, year of survey, smoking status and alcohol consumption. Results: For both genders, BMI-defined obesity (greater than or equal to30 kg m−2) was not associated with either an increased risk of all-cause mortality or CHD mortality. However, there was an increased risk of incident CHD among the obese men (hazard ratio (HR)=1.78; 95% confidence interval=1.37–2.31) and obese women (HR=1.93; 95% confidence interval=1.44–2.59). There was a similar pattern for WC with regard to the three outcomes; for incident CHD, the HR=1.70 (1.35–2.14) for men and 1.71 (1.28–2.29) for women in the highest WC category (men greater than or equal to102 cm, women greater than or equal to88 cm), synonymous with abdominal obesity. For men, the highest category of WHR (greater than or equal to1.0) was associated with an increased risk of all-cause mortality (1.29; 1.04–1.60) and incident CHD (1.55; 1.19–2.01). Among women with a high WHR (greater than or equal to0.85) there was an increased risk of all outcomes: all-cause mortality (1.56; 1.26–1.94), CHD mortality (2.49; 1.36–4.56) and incident CHD (1.76; 1.31–2.38). Conclusions: In this study excess adiposity was associated with an increased risk of incident CHD but not necessarily death. One possibility is that modern medical intervention has contributed to improved survival of first CHD events. The future health burden of increased obesity levels may manifest as an increase in the prevalence of individuals living with CHD and its consequences

Variation of health-related quality of life assessed by caregivers and patients affected by severe childhood infections.

BACKGROUND: The agreement between self-reported and proxy measures of health status in ill children is not well established. This study aimed to quantify the variation in health-related quality of life (HRQOL) derived from young patients and their carers using different instruments. METHODS: A hospital-based cross-sectional survey was conducted between August 2010 and March 2011. Children with meningitis, bacteremia, pneumonia, acute otitis media, hearing loss, chronic lung disease, epilepsy, mild mental retardation, severe mental retardation, and mental retardation combined with epilepsy, aged between five to 14 years in seven tertiary hospitals were selected for participation in this study. The Health Utilities Index Mark 2 (HUI2), and Mark 3 (HUI3), and the EuroQoL Descriptive System (EQ-5D) and Visual Analogue Scale (EQ-VAS) were applied to both paediatric patients (self-assessment) and caregivers (proxy-assessment). RESULTS: The EQ-5D scores were lowest for acute conditions such as meningitis, bacteremia, and pneumonia, whereas the HUI3 scores were lowest for most chronic conditions such as hearing loss and severe mental retardation. Comparing patient and proxy scores (n = 74), the EQ-5D exhibited high correlation (r = 0.77) while in the HUI2 and HUI3 patient and caregiver scores were moderately correlated (r = 0.58 and 0.67 respectively). The mean difference between self and proxy-assessment using the HUI2, HUI3, EQ-5D and EQ-VAS scores were 0.03, 0.05, -0.03 and -0.02, respectively. In hearing-impaired and chronic lung patients the self-rated HRQOL differed significantly from their caregivers. CONCLUSIONS: The use of caregivers as proxies for measuring HRQOL in young patients affected by pneumococcal infection and its sequelae should be employed with caution. Given the high correlation between instruments, each of the HRQOL instruments appears acceptable apart from the EQ-VAS which exhibited low correlation with the others

CARDIOVASCULAR MEDICINE C reactive protein concentration and recurrence of atrial fibrillation after electrical cardioversion

Background: To test the hypothesis that a high C reactive protein (CRP) concentration would predict recurrence of atrial fibrillation (AF) after cardioversion in patients taking antiarrhythmic drugs. Methods: 111 patients who underwent direct current cardioversion for symptomatic AF were enrolled. Blood was drawn for CRP determination before cardioversion on the same day. All patients were taking antiarrhythmic drugs before and after electrical cardioversion. Results: After a mean follow up of 76 days, 75 patients had recurrence of AF. In univariate analysis, the median CRP concentration was significantly higher in patients with AF recurrence (3.95 mg/l v 1.81 mg/l, p = 0.002). Among the 55 patients with CRP in the upper 50th centile, 44 (80%) experienced recurrence of AF over a total follow up of 8.98 patient years, whereas among the 56 patients with CRP in the lower 50th centile, 31 (55%) experienced recurrence of AF over a total follow up of 14.3 patient years (p , 0.001). The adjusted hazard ratio comparing the upper 50th centile of CRP with the lower 50th centile of CRP was 2.0 (95% confidence interval 1.2 to 3.2, p = 0.007). Conclusions: CRP is independently associated with recurrence of AF after electrical cardioversion among patients taking antiarrhythmic drugs. These results suggest that inflammation may have a role in the pathogenesis of AF resistant to antiarrhythmic drugs

Patient-rated health status predicts prognosis following percutaneous coronary intervention with drug-eluting stenting

Purpose: In patients treated with percutaneous coronary intervention (PCI) with the paclitaxel-eluting stent, we examined whether patient-rated health status predicts adverse clinical events. Methods: Consecutive PCI patients treated with drug-eluting stenting (N = 870; 72.2% men; mean age = 62.6 ± 11.5) completed the EQ-5D post-PCI. The EQ-5D levels were dichotomized into 'no problems' (level 1) versus 'problems' (levels 2, 3); the visual analogue scale (VAS) was dichotomized using the 25th percentile (cut-off ≤60) indicating poor health status. Patients were followed up for 1-year clinical events (death or non-fatal myocardial infarction (MI)). Results: There were 53 deaths/MIs at follow-up. The EQ-5D health status dimensions mobility (HR:2.23; 95% CI:1.25-3.97), self-care (HR:3.09; 95% CI:1.54-6.20), and self-reported health status as measured with the EQ-VAS (HR:2.94; 95% CI:1.65-5.25) were independent predictors of death/MI and added to the predictive value of a model comprised of demographic and clinical characteristics. The EQ-5D dimensions usual activities, pain/discomfort, and anxiety/depression were not associated with adverse clinical events in adjusted analysis. Conclusions: Patient-rated health status predicted adverse clinical events at 1-year follow-up in PCI patients treated with drug-eluting stenting, with the risk being more than 2-fold indepe

Chronic non-specific low back pain - sub-groups or a single mechanism?

Copyright 2008 Wand and O'Connell; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Background: Low back pain is a substantial health problem and has subsequently attracted a considerable amount of research. Clinical trials evaluating the efficacy of a variety of interventions for chronic non-specific low back pain indicate limited effectiveness for most commonly applied interventions and approaches. Discussion: Many clinicians challenge the results of clinical trials as they feel that this lack of effectiveness is at odds with their clinical experience of managing patients with back pain. A common explanation for this discrepancy is the perceived heterogeneity of patients with chronic non-specific low back pain. It is felt that the effects of treatment may be diluted by the application of a single intervention to a complex, heterogeneous group with diverse treatment needs. This argument presupposes that current treatment is effective when applied to the correct patient. An alternative perspective is that the clinical trials are correct and current treatments have limited efficacy. Preoccupation with sub-grouping may stifle engagement with this view and it is important that the sub-grouping paradigm is closely examined. This paper argues that there are numerous problems with the sub-grouping approach and that it may not be an important reason for the disappointing results of clinical trials. We propose instead that current treatment may be ineffective because it has been misdirected. Recent evidence that demonstrates changes within the brain in chronic low back pain sufferers raises the possibility that persistent back pain may be a problem of cortical reorganisation and degeneration. This perspective offers interesting insights into the chronic low back pain experience and suggests alternative models of intervention. Summary: The disappointing results of clinical research are commonly explained by the failure of researchers to adequately attend to sub-grouping of the chronic non-specific low back pain population. Alternatively, current approaches may be ineffective and clinicians and researchers may need to radically rethink the nature of the problem and how it should best be managed