An upper bound on the Bayesian error bars for generalized linear regression

In the Bayesian framework, predictions for a regression problem are expressed in terms of a distribution of output values. The mode of this distribution corresponds to the most probable output, while the uncertainty associated with the predictions can conveniently be expressed in terms of error bars. In this paper we consider the evaluation of error bars in the context of the class of generalized linear regression models. We provide insights into the dependence of the error bars on the location of the data points and we derive an upper bound on the true error bars in terms of the contributions from individual data points which are themselves easily evaluated.

Prediction with Gaussian processes: from linear regression to linear prediction and beyond

The main aim of this paper is to provide a tutorial on regression with Gaussian processes. We start from Bayesian linear regression, and show how by a change of viewpoint one can see this method as a Gaussian process predictor based on priors over functions, rather than on priors over parameters. This leads in to a more general discussion of Gaussian processes in section 4. Section 5 deals with further issues, including hierarchical modelling and the setting of the parameters that control the Gaussian process, the covariance functions for neural network models and the use of Gaussian processes in classification problems.

In vivo analysis of ciliary muscle morphologic changes with accommodation and axial ametropia

Purpose. To use anterior segment optical coherence tomography (AS-OCT) to analyze ciliary muscle morphology and changes with accommodation and axial ametropia. Methods. Fifty prepresbyopic volunteers, aged 19 to 34 years were recruited. High-resolution images were acquired of nasal and temporal ciliary muscles in the relaxed state and at stimulus vergence levels of -4 and -8 D. Objective accommodative responses and axial lengths were also recorded. Two-way, mixed-factor analyses of variance (ANOVAs) were used to assess the changes in ciliary muscle parameters with accommodation and determine whether these changes are dependent on the nasal–temporal aspect or axial length, whereas linear regression analysis was used to analyze the relationship between axial length and ciliary muscle length. Results. The ciliary muscle was longer (r = 0.34, P = 0.02), but not significantly thicker (F = 2.84, P = 0.06), in eyes with greater axial length. With accommodation, the ciliary muscle showed a contractile shortening (F = 42.9. P < 0.001), particularly anteriorly (F = 177.2, P < 0.001), and a thickening of the anterior portion (F= 46.2, P < 0.001). The ciliary muscle was thicker (F = 17.8, P < 0.001) and showed a greater contractile response on the temporal side. Conclusions. The accommodative changes observed support an anterior, as well as centripetal, contractile shift of ciliary muscle mass.

Statnote 19: non-linear regression: fitting an exponential curve to data

Non-linear relationships are common in microbiological research and often necessitate the use of the statistical techniques of non-linear regression or curve fitting. In some circumstances, the investigator may wish to fit an exponential model to the data, i.e., to test the hypothesis that a quantity Y either increases or decays exponentially with increasing X. This type of model is straight forward to fit as taking logarithms of the Y variable linearises the relationship which can then be treated by the methods of linear regression.

Statnote 20: non-linear regression: fitting a general polynomial curve

In some circumstances, there may be no scientific model of the relationship between X and Y that can be specified in advance and indeed the objective of the investigation may be to provide a ‘curve of best fit’ for predictive purposes. In such an example, the fitting of successive polynomials may be the best approach. There are various strategies to decide on the polynomial of best fit depending on the objectives of the investigation.

Mathematical modelling of patient risks in a hospital environment, using multiple regression analysis

The aim of this research work was primarily to examine the relevance of patient parameters, ward structures, procedures and practices, in respect of the potential hazards of wound cross-infection and nasal colonisation with multiple resistant strains of Staphylococcus aureus, which it is thought might provide a useful indication of a patient's general susceptibility to wound infection. Information from a large cross-sectional survey involving 12,000 patients from some 41 hospitals and 375 wards was collected over a five-year period from 1967-72, and its validity checked before any subsequent analysis was carried out. Many environmental factors and procedures which had previously been thought (but never conclusively proved) to have an influence on wound infection or nasal colonisation rates, were assessed, and subsequently dismissed as not being significant, provided that the standard of the current range of practices and procedures is maintained and not allowed to deteriorate. Retrospective analysis revealed that the probability of wound infection was influenced by the patient's age, duration of pre-operative hospitalisation, sex, type of wound, presence and type of drain, number of patients in ward, and other special risk factors, whilst nasal colonisation was found to be influenced by the patient's age, total duration of hospitalisation, sex, antibiotics, proportion of occupied beds in the ward, average distance between bed centres and special risk factors. A multi-variate regression analysis technique was used to develop statistical models, consisting of variable patient and environmental factors which were found to have a significant influence on the risks pertaining to wound infection and nasal colonisation. A relationship between wound infection and nasal colonisation was then established and this led to the development of a more advanced model for predicting wound infections, taking advantage of the additional knowledge of the patient's state of nasal colonisation prior to operation.

Structural brain connectivity and cognitive ability differences: A multivariate distance matrix regression analysis

Neuroimaging research involves analyses of huge amounts of biological data that might or might not be related with cognition. This relationship is usually approached using univariate methods, and, therefore, correction methods are mandatory for reducing false positives. Nevertheless, the probability of false negatives is also increased. Multivariate frameworks have been proposed for helping to alleviate this balance. Here we apply multivariate distance matrix regression for the simultaneous analysis of biological and cognitive data, namely, structural connections among 82 brain regions and several latent factors estimating cognitive performance. We tested whether cognitive differences predict distances among individuals regarding their connectivity pattern. Beginning with 3,321 connections among regions, the 36 edges better predicted by the individuals' cognitive scores were selected. Cognitive scores were related to connectivity distances in both the full (3,321) and reduced (36) connectivity patterns. The selected edges connect regions distributed across the entire brain and the network defined by these edges supports high-order cognitive processes such as (a) (fluid) executive control, (b) (crystallized) recognition, learning, and language processing, and (c) visuospatial processing. This multivariate study suggests that one widespread, but limited number, of regions in the human brain, supports high-level cognitive ability differences. Hum Brain Mapp, 2016. © 2016 Wiley Periodicals, Inc.

On the effect of confounding in linear regression models: an approach based on the theory of quadratic forms

The main topic of this thesis is confounding in linear regression models. It arises when a relationship between an observed process, the covariate, and an outcome process, the response, is influenced by an unmeasured process, the confounder, associated with both. Consequently, the estimators for the regression coefficients of the measured covariates might be severely biased, less efficient and characterized by misleading interpretations. Confounding is an issue when the primary target of the work is the estimation of the regression parameters. The central point of the dissertation is the evaluation of the sampling properties of parameter estimators. This work aims to extend the spatial confounding framework to general structured settings and to understand the behaviour of confounding as a function of the data generating process structure parameters in several scenarios focusing on the joint covariate-confounder structure. In line with the spatial statistics literature, our purpose is to quantify the sampling properties of the regression coefficient estimators and, in turn, to identify the most prominent quantities depending on the generative mechanism impacting confounding. Once the sampling properties of the estimator conditionally on the covariate process are derived as ratios of dependent quadratic forms in Gaussian random variables, we provide an analytic expression of the marginal sampling properties of the estimator using Carlson’s R function. Additionally, we propose a representative quantity for the magnitude of confounding as a proxy of the bias, its first-order Laplace approximation. To conclude, we work under several frameworks considering spatial and temporal data with specific assumptions regarding the covariance and cross-covariance functions used to generate the processes involved. This study allows us to claim that the variability of the confounder-covariate interaction and of the covariate plays the most relevant role in determining the principal marker of the magnitude of confounding.

Exploratory Study Of The Effect Of Lifestyle Counselling On Bone Mineral Density And Body Composition In Users Of The Contraceptive Depot-medroxyprogesterone Acetate.

To compare variations in bone mineral density (BMD) and body composition (BC) in depot-medroxyprogesterone acetate (DMPA) users and nonusers after providing counselling on healthy lifestyle habits. An exploratory study in which women aged 18 to 40 years participated: 29 new DMPA users and 25 new non-hormonal contraceptive users. All participants were advised on healthy lifestyle habits: sun exposure, walking and calcium intake. BMD and BC were assessed at baseline and 12 months later. Statistical analysis included the Mann-Whitney test or Student's t-test followed by multiple linear regression analysis. Compared to the controls, DMPA users had lower BMD at vertebrae L1 and L4 after 12 months of use. They also had a mean increase of 2 kg in total fat mass and an increase of 2.2% in body fat compared to the non-hormonal contraceptive users. BMD loss at L1 was less pronounced in DMPA users with a calcium intake ≥ 1 g/day compared to DMPA users with a lower calcium intake. DMPA use was apparently associated with lower BMD and an increase in fat mass at 12 months of use. Calcium intake ≥ 1 g/day attenuates BMD loss in DMPA users. Counselling on healthy lifestyle habits failed to achieve its aims.

Storage At Low Temperature Differentially Affects The Colour And Carotenoid Composition Of Two Cultivars Of Banana.

Different storage conditions can induce changes in the colour and carotenoid profiles and levels in some fruits. The goal of this work was to evaluate the influence of low temperature storage on the colour and carotenoid synthesis in two banana cultivars: Prata and Nanicão. For this purpose, the carotenoids from the banana pulp were determined by HPLC-DAD-MS/MS, and the colour of the banana skin was determined by a colorimeter method. Ten carotenoids were identified, of which the major carotenoids were all-trans-lutein, all-trans-α-carotene and all-trans-β-carotene in both cultivars. The effect of the low temperatures was subjected to linear regression analysis. In cv. Prata, all-trans-α-carotene and all-trans-β-carotene were significantly affected by low temperature (p<0.01), with negative estimated values (β coefficients) indicating that during cold storage conditions, the concentrations of these carotenoids tended to decrease. In cv. Nanicão, no carotenoid was significantly affected by cold storage (p>0.05). The accumulation of carotenoids in this group may be because the metabolic pathways using these carotenoids were affected by storage at low temperatures. The colour of the fruits was not negatively affected by the low temperatures (p>0.05).

Clinical Influence Of Different Intracanal Medications On Th1-type And Th2-type Cytokine Responses In Apical Periodontitis.

This clinical study assessed the influence of different intracanal medications on Th1-type and Th2-type cytokine responses in apical periodontitis and monitored the levels of bacteria from primarily infection during endodontic procedures. Thirty primarily infected teeth were randomly divided into 3 groups according to the medication selected: chlorhexidine (CHX), 2% CHX gel; Ca(OH)2/SSL, Ca(OH)2 + SSL; and Ca(OH)2/CHX, Ca(OH)2 + 2% CHX gel (all, n = 10). Bacterial sample was collected from root canals, and the interstitial fluid was sampled from lesions. Culture techniques were used to determine bacterial counts (colony-forming units/mL). Th1 (tumor necrosis factor-α, interferon-γ, and interleukin [IL]-2) and Th2 cytokines (IL-4, IL-5, and IL-13) were measured by enzyme-linked immunosorbent assay. All intracanal medication protocols were effective in reducing the bacterial load from root canals (all P < .05) and lowering the levels of Th1-type cytokines in apical lesions (all P < .05), with no differences between them (P > .05). Both Ca(OH)2 treatment protocols significantly increased the levels of Th2-type cytokines (P < .05), with no differences between them (P > .05). Thus, chlorhexidine medication showed the lowest effectiveness in increasing the levels of Th2-type cytokine. After treatment, regardless of the type of medication, the linear regression analysis indicated the down-regulation of Th2-type cytokines by Th1-type cytokines. All intracanal medication protocols were effective in reducing bacterial load and lowering the levels of Th1-type cytokines. Thus, the use of Ca(OH)2 medications contributed to the increase in the Th2-type cytokine response in apical periodontitis.

Patient education in glaucoma: what do patients know about glaucoma?

PURPOSE: To evaluate the knowledge glaucoma patients have about their disease and its treatment. METHODS: One hundred and eighty-three patients were interviewed at the Glaucoma Service of Wills Eye Hospital (Philadelphia, USA, Group 1) and 100 at the Glaucoma Service of University of Campinas (Campinas, Brazil, Group 2). An informal, relaxed atmosphere was created by the interviewer before asking a list of 18 open-ended questions. RESULTS: In Group 1, 44% of the 183 patients did not have an acceptable idea about what glaucoma is, 30% did not know the purpose of the medications they were taking, 47% were not aware of what was an average intraocular pressure, and 45% did not understand why visual fields were examined. In Group 2, 54% gave unsatisfactory answers to the question What is glaucoma?, 54% did not know the purpose of the medications they were taking, 80% were not aware of what was an average intraocular pressure, and 94% did not understand why visual fields were examined (p<0.001). Linear regression analysis demonstrated that level of education was positively correlated to knowledge about glaucoma in both groups (r=0.65, p=0.001). CONCLUSION: This study showed that patients' knowledge about glaucoma varies greatly, and that in an urban, American setting, around one third of the patients have minimal understanding, whereas in an urban setting in Brazil around two thirds of patients were lacking basic information about glaucoma. Innovative and effective methods are needed to correct this situation.