Methods of Estimation in Multiple Linear Regression: Application to Clinical Data

Nesse artigo, tem-se o interesse em avaliar diferentes estratégias de estimação de parâmetros para um modelo de regressão linear múltipla. Para a estimação dos parâmetros do modelo foram utilizados dados de um ensaio clínico em que o interesse foi verificar se o ensaio mecânico da propriedade de força máxima (EM-FM) está associada com a massa femoral, com o diâmetro femoral e com o grupo experimental de ratas ovariectomizadas da raça Rattus norvegicus albinus, variedade Wistar. Para a estimação dos parâmetros do modelo serão comparadas três metodologias: a metodologia clássica, baseada no método dos mínimos quadrados; a metodologia Bayesiana, baseada no teorema de Bayes; e o método Bootstrap, baseado em processos de reamostragem.

Methods to verify parameter equality in nonlinear regression models

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Data methods in optometry. Part 6: fitting a regression line to data

1. Fitting a linear regression to data provides much more information about the relationship between two variables than a simple correlation test. A goodness of fit test of the line should always be carried out. Hence, r squared estimates the strength of the relationship between Y and X, ANOVA whether a statistically significant line is present, and the ‘t’ test whether the slope of the line is significantly different from zero. 2. Always check whether the data collected fit the assumptions for regression analysis and, if not, whether a transformation of the Y and/or X variables is necessary. 3. If the regression line is to be used for prediction, it is important to determine whether the prediction involves an individual y value or a mean. Care should be taken if predictions are made close to the extremities of the data and are subject to considerable error if x falls beyond the range of the data. Multiple predictions require correction of the P values. 3. If several individual regression lines have been calculated from a number of similar sets of data, consider whether they should be combined to form a single regression line. 4. If the data exhibit a degree of curvature, then fitting a higher-order polynomial curve may provide a better fit than a straight line. In this case, a test of whether the data depart significantly from a linear regression should be carried out.

Data analysis methods in optometry Part 7: multiple linear regression

Multiple regression analysis is a complex statistical method with many potential uses. It has also become one of the most abused of all statistical procedures since anyone with a data base and suitable software can carry it out. An investigator should always have a clear hypothesis in mind before carrying out such a procedure and knowledge of the limitations of each aspect of the analysis. In addition, multiple regression is probably best used in an exploratory context, identifying variables that might profitably be examined by more detailed studies. Where there are many variables potentially influencing Y, they are likely to be intercorrelated and to account for relatively small amounts of the variance. Any analysis in which R squared is less than 50% should be suspect as probably not indicating the presence of significant variables. A further problem relates to sample size. It is often stated that the number of subjects or patients must be at least 5-10 times the number of variables included in the study.5 This advice should be taken only as a rough guide but it does indicate that the variables included should be selected with great care as inclusion of an obviously unimportant variable may have a significant impact on the sample size required.

Data methods in optometry. Part 10: non-linear regression analysis

1. The techniques associated with regression, whether linear or non-linear, are some of the most useful statistical procedures that can be applied in clinical studies in optometry. 2. In some cases, there may be no scientific model of the relationship between X and Y that can be specified in advance and the objective may be to provide a ‘curve of best fit’ for predictive purposes. In such cases, the fitting of a general polynomial type curve may be the best approach. 3. An investigator may have a specific model in mind that relates Y to X and the data may provide a test of this hypothesis. Some of these curves can be reduced to a linear regression by transformation, e.g., the exponential and negative exponential decay curves. 4. In some circumstances, e.g., the asymptotic curve or logistic growth law, a more complex process of curve fitting involving non-linear estimation will be required.

Probabilistic methods for seasonal forecasting in a changing climate: Cox-type regression models.

For climate risk management, cumulative distribution functions (CDFs) are an important source of information. They are ideally suited to compare probabilistic forecasts of primary (e.g. rainfall) or secondary data (e.g. crop yields). Summarised as CDFs, such forecasts allow an easy quantitative assessment of possible, alternative actions. Although the degree of uncertainty associated with CDF estimation could influence decisions, such information is rarely provided. Hence, we propose Cox-type regression models (CRMs) as a statistical framework for making inferences on CDFs in climate science. CRMs were designed for modelling probability distributions rather than just mean or median values. This makes the approach appealing for risk assessments where probabilities of extremes are often more informative than central tendency measures. CRMs are semi-parametric approaches originally designed for modelling risks arising from time-to-event data. Here we extend this original concept beyond time-dependent measures to other variables of interest. We also provide tools for estimating CDFs and surrounding uncertainty envelopes from empirical data. These statistical techniques intrinsically account for non-stationarities in time series that might be the result of climate change. This feature makes CRMs attractive candidates to investigate the feasibility of developing rigorous global circulation model (GCM)-CRM interfaces for provision of user-relevant forecasts. To demonstrate the applicability of CRMs, we present two examples for El Ni ? no/Southern Oscillation (ENSO)-based forecasts: the onset date of the wet season (Cairns, Australia) and total wet season rainfall (Quixeramobim, Brazil). This study emphasises the methodological aspects of CRMs rather than discussing merits or limitations of the ENSO-based predictors.

Petroleomics By Electrospray Ionization Ft-icr Mass Spectrometry Coupled To Partial Least Squares With Variable Selection Methods: Prediction Of The Total Acid Number Of Crude Oils.

Negative-ion mode electrospray ionization, ESI(-), with Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR MS) was coupled to a Partial Least Squares (PLS) regression and variable selection methods to estimate the total acid number (TAN) of Brazilian crude oil samples. Generally, ESI(-)-FT-ICR mass spectra present a power of resolution of ca. 500,000 and a mass accuracy less than 1 ppm, producing a data matrix containing over 5700 variables per sample. These variables correspond to heteroatom-containing species detected as deprotonated molecules, [M - H](-) ions, which are identified primarily as naphthenic acids, phenols and carbazole analog species. The TAN values for all samples ranged from 0.06 to 3.61 mg of KOH g(-1). To facilitate the spectral interpretation, three methods of variable selection were studied: variable importance in the projection (VIP), interval partial least squares (iPLS) and elimination of uninformative variables (UVE). The UVE method seems to be more appropriate for selecting important variables, reducing the dimension of the variables to 183 and producing a root mean square error of prediction of 0.32 mg of KOH g(-1). By reducing the size of the data, it was possible to relate the selected variables with their corresponding molecular formulas, thus identifying the main chemical species responsible for the TAN values.

Censored Linear Regression Models For Irregularly Observed Longitudinal Data Using The Multivariate-t Distribution.

In acquired immunodeficiency syndrome (AIDS) studies it is quite common to observe viral load measurements collected irregularly over time. Moreover, these measurements can be subjected to some upper and/or lower detection limits depending on the quantification assays. A complication arises when these continuous repeated measures have a heavy-tailed behavior. For such data structures, we propose a robust structure for a censored linear model based on the multivariate Student's t-distribution. To compensate for the autocorrelation existing among irregularly observed measures, a damped exponential correlation structure is employed. An efficient expectation maximization type algorithm is developed for computing the maximum likelihood estimates, obtaining as a by-product the standard errors of the fixed effects and the log-likelihood function. The proposed algorithm uses closed-form expressions at the E-step that rely on formulas for the mean and variance of a truncated multivariate Student's t-distribution. The methodology is illustrated through an application to an Human Immunodeficiency Virus-AIDS (HIV-AIDS) study and several simulation studies.

Regression models for grouped survival data: Estimation and sensitivity analysis

In this study, regression models are evaluated for grouped survival data when the effect of censoring time is considered in the model and the regression structure is modeled through four link functions. The methodology for grouped survival data is based on life tables, and the times are grouped in k intervals so that ties are eliminated. Thus, the data modeling is performed by considering the discrete models of lifetime regression. The model parameters are estimated by using the maximum likelihood and jackknife methods. To detect influential observations in the proposed models, diagnostic measures based on case deletion, which are denominated global influence, and influence measures based on small perturbations in the data or in the model, referred to as local influence, are used. In addition to those measures, the local influence and the total influential estimate are also employed. Various simulation studies are performed and compared to the performance of the four link functions of the regression models for grouped survival data for different parameter settings, sample sizes and numbers of intervals. Finally, a data set is analyzed by using the proposed regression models. (C) 2010 Elsevier B.V. All rights reserved.

Quantification of chlorpheniramine maleate enantiomers by ultraviolet spectroscopy and chemometric methods

Chlorpheniramine maleate (CLOR) enantiomers were quantified by ultraviolet spectroscopy and partial least squares regression. The CLOR enantiomers were prepared as inclusion complexes with beta-cyclodextrin and 1-butanol with mole fractions in the range from 50 to 100%. For the multivariate calibration the outliers were detected and excluded and variable selection was performed by interval partial least squares and a genetic algorithm. Figures of merit showed results for accuracy of 3.63 and 2.83% (S)-CLOR for root mean square errors of calibration and prediction, respectively. The ellipse confidence region included the point for the intercept and the slope of 1 and 0, respectively. Precision and analytical sensitivity were 0.57 and 0.50% (S)-CLOR, respectively. The sensitivity, selectivity, adjustment, and signal-to-noise ratio were also determined. The model was validated by a paired t test with the results obtained by high-performance liquid chromatography proposed by the European pharmacopoeia and circular dichroism spectroscopy. The results showed there was no significant difference between the methods at the 95% confidence level, indicating that the proposed method can be used as an alternative to standard procedures for chiral analysis.

Exploratory Approach to the use of Techniques of Regression Analysis and Pooled Analysis in Studies of Consumer Behavior

This paper is part of a large study to assess the adequacy of the use of multivariate statistical techniques in theses and dissertations of some higher education institutions in the area of marketing with theme of consumer behavior from 1997 to 2006. The regression and conjoint analysis are focused on in this paper, two techniques with great potential of use in marketing studies. The objective of this study was to analyze whether the employement of these techniques suits the needs of the research problem presented in as well as to evaluate the level of success in meeting their premisses. Overall, the results suggest the need for more involvement of researchers in the verification of all the theoretical precepts of application of the techniques classified in the category of investigation of dependence among variables.

Validated spectrophotometric and spectrofluorimetric methods for determination of chloroaluminum phthalocyanine in nanocarriers

UV-VIS-Spectrophotometric and spectrofluorimetric methods have been developed and validated allowing the quantification of chloroaluminum phthalocyanine (CIAIPc) in nanocarriers. In order to validate the methods, the linearity, limit of detection (LOD), limit of quantification (LOQ), precision, accuracy, and selectivity were examined according to USP 30 and ICH guidelines. Linearities range were found between 0.50-3.00 mu g.mL(-1) (Y=0.3829 X [CIAIPc, mu g.mL(-1)] + 0.0126; r=0.9992) for spectrophotometry, and 0.05-1.00 mu g.mL(-1) (Y=2.24 x 10(6) X [CIAIPc, mu g.L(-1)] + 9.74 x 10(4); r=0.9978) for spectrofluorimetry. In addition, ANOVA and Lack-of-fit tests demonstrated that the regression equations were statistically significant (p<0.05), and the resulting linear model is fully adequate for both analytical methods. The LOD values were 0.09 and 0.01 mu g.mL(-1), while the LOCI were 0.27 and 0.04 mu g.mL(-1) for spectrophotometric and spectrofluorimetric methods, respectively. Repeatability and intermediate precision for proposed methods showed relative standard deviation (RSD) between 0.58% to 4.80%. The percent recovery ranged from 98.9% to 102.7% for spectrophotometric analyses and from 94.2% to 101.2% for spectrofluorimetry. No interferences from common excipients were detected and both methods were considered specific. Therefore, the methods are accurate, precise, specific, and reproducible and hence can be applied for quantification of CIAIPc in nanoemulsions (NE) and nanocapsules (NC).

A comparison of computer-based methods for the determination of onset of muscle contraction using electromyography

Little consensus exists in the literature regarding methods for determination of the onset of electromyographic (EMG) activity. The aim of this study was to compare the relative accuracy of a range of computer-based techniques with respect to EMG onset determined visually by an experienced examiner. Twenty-seven methods were compared which varied in terms of EMG processing (low pass filtering at 10, 50 and 500 Hz), threshold value (1, 2 and 3 SD beyond mean of baseline activity) and the number of samples for which the mean must exceed the defined threshold (20, 50 and 100 ms). Three hundred randomly selected trials of a postural task were evaluated using each technique. The visual determination of EMG onset was found to be highly repeatable between days. Linear regression equations were calculated for the values selected by each computer method which indicated that the onset values selected by the majority of the parameter combinations deviated significantly from the visually derived onset values. Several methods accurately selected the time of onset of EMG activity and are recommended for future use. Copyright (C) 1996 Elsevier Science Ireland Ltd.

Methods for exploring the morpho-functional relations of the aortic depressor nerve in experimental diabetes

The present study investigated morpho-functional relations of the aortic depressor nerve (ADN) 5, 15 and 120 days after the onset of streptozotocin-induced diabetes in rats. Time control animals received vehicle. Under pentobarbital anesthesia, ADN activity was recorded simultaneously with arterial pressure. After the recordings, nerves were prepared for light microscopy study and morphometry. ADN function was accessed by means of pressure-nerve activity curve (fitted by sigmoidal regression) and cross-spectral analysis between mean arterial pressure (MAP) and ADN activity. The relation between morphological (myelinated fibers number and density, total myelin area, total fiber area and percentage of occupancy) and functional (gain, signal/noise relation, frequency) parameters were accessed by linear regression analysis and correlation coefficient calculations. Functional parameters obtained by means of the sigmoidal regression curve as well as by cross-spectral analysis were similar in diabetic and control rats. Morphometric parameters of the ADN were similar between groups 5 days after the onset of diabetes. Average myelin area and myelinated fiber area were significantly smaller on diabetic rats 15 and 120 days after the onset of diabetes, being the myelinated fiber and respective axons area and diameter also smaller on 120 days group. Nevertheless, G ratio (ratio between axon and fiber diameter) was nearly 0.6 and not different between groups or experimental times. No significant relationship between morphological and functional parameters was detected in all experimental groups. The present study suggests that ADN diabetic neuropathy was time-dependent, with damage to myelinated fibers to be the primary event, not evidenced by physiological methods. (C) 2010 Elsevier B.V. All rights reserved.

Proximal arterial vasoconstriction precedes regression of the hyaloid vasculature

Purpose: To determine whether constriction of proximal arterial vessels precedes involution of the distal hyaloid vasculature in the mouse, under normal conditions, and whether this vasoconstriction is less pronounced when the distal hyaloid network persists, as it does in oxygen-induced retinopathy (OIR). Methods: Photomicrographs of the vasa hyaloidea propria were analysed from pre-term pups (1-2 days prior to birth), and on Days 1-11 post-birth. The OIR model involved exposing pups to similar to 90% O-2 from D1-5, followed by return to ambient air. At sampling times pups were anaesthetised and perfused with india ink. Retinal flatmounts were also incubated with FITC-lectin (BS-1, G. simplicifolia,); this labels all vessels, allowing identification of vessels not patent to the perfusate. Results: Mean diameter of proximal hyaloid vessels in preterm pups was 25.44 +/- 1.98 mum; +/-1 SEM). Within 3-12 hrs of birth, significant vasoconstriction was evident (diameter:12.45 +/- 0.88 mum), and normal hyaloid regression subsequently occurred. Similar vasoconstriction occurred in the O-2-treated group, but this was reversed upon return to room air, with significant dilation of proximal vessels by D7 (diameter: 31.75 +/- 11.99 mum) and distal hyaloid vessels subsequently became enlarged and tortuous. Conclusions: Under normal conditions, vasoconstriction of proximal hyaloid vessels occurs at birth, preceding attenuation of distal hyaloid vessels. Vasoconstriction also occurs in O-2-treated pups during treatment, but upon return to room air, the remaining hyaloid vessels dilate proximally, and the distal vessels become dilated and tortuous. These observations support the contention that regression of the hyaloid network is dependent, in the first instance, on proximal arterial vasoconstriction.