PSPLINE: Stata module providing a penalized spline scatterplot smoother based on linear mixed model technology

Pspline uses xtmixed to fit a penalized spline regression and plots the smoothed function. Additional covariates can be specified to adjust the smooth and plot partial residuals.

Finite mixture regression model with random effects: application to neonatal hospital length of stay

A two-component mixture regression model that allows simultaneously for heterogeneity and dependency among observations is proposed. By specifying random effects explicitly in the linear predictor of the mixture probability and the mixture components, parameter estimation is achieved by maximising the corresponding best linear unbiased prediction type log-likelihood. Approximate residual maximum likelihood estimates are obtained via an EM algorithm in the manner of generalised linear mixed model (GLMM). The method can be extended to a g-component mixture regression model with the component density from the exponential family, leading to the development of the class of finite mixture GLMM. For illustration, the method is applied to analyse neonatal length of stay (LOS). It is shown that identification of pertinent factors that influence hospital LOS can provide important information for health care planning and resource allocation. (C) 2002 Elsevier Science B.V. All rights reserved.

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.

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.

Dimension Reduction of the Explanatory Variables in Multiple Linear Regression

2002 Mathematics Subject Classification: 62J05, 62G35.

Analysing Conjoint Analysis Data by a Random Coefficient Regression Model

2000 Mathematics Subject Classification: 62J12, 62K15, 91B42, 62H99.

Classification of Texts' Authorship Using a Regression Model on Compressed Data

2010 Mathematics Subject Classification: 68T50,62H30,62J05.

Seemingly unrelated linear regression for contaminated data based on Gaussian mixtures

In this thesis, new classes of models for multivariate linear regression defined by finite mixtures of seemingly unrelated contaminated normal regression models and seemingly unrelated contaminated normal cluster-weighted models are illustrated. The main difference between such families is that the covariates are treated as fixed in the former class of models and as random in the latter. Thus, in cluster-weighted models the assignment of the data points to the unknown groups of observations depends also by the covariates. These classes provide an extension to mixture-based regression analysis for modelling multivariate and correlated responses in the presence of mild outliers that allows to specify a different vector of regressors for the prediction of each response. Expectation-conditional maximisation algorithms for the calculation of the maximum likelihood estimate of the model parameters have been derived. As the number of free parameters incresases quadratically with the number of responses and the covariates, analyses based on the proposed models can become unfeasible in practical applications. These problems have been overcome by introducing constraints on the elements of the covariance matrices according to an approach based on the eigen-decomposition of the covariance matrices. The performances of the new models have been studied by simulations and using real datasets in comparison with other models. In order to gain additional flexibility, mixtures of seemingly unrelated contaminated normal regressions models have also been specified so as to allow mixing proportions to be expressed as functions of concomitant covariates. An illustration of the new models with concomitant variables and a study on housing tension in the municipalities of the Emilia-Romagna region based on different types of multivariate linear regression models have been performed.

Bayesian Linear Regression for estimation of the Fractal Dimension of the Cerebral Cortex

The cerebral cortex presents self-similarity in a proper interval of spatial scales, a property typical of natural objects exhibiting fractal geometry. Its complexity therefore can be characterized by the value of its fractal dimension (FD). In the computation of this metric, it has usually been employed a frequentist approach to probability, with point estimator methods yielding only the optimal values of the FD. In our study, we aimed at retrieving a more complete evaluation of the FD by utilizing a Bayesian model for the linear regression analysis of the box-counting algorithm. We used T1-weighted MRI data of 86 healthy subjects (age 44.2 ± 17.1 years, mean ± standard deviation, 48% males) in order to gain insights into the confidence of our measure and investigate the relationship between mean Bayesian FD and age. Our approach yielded a stronger and significant (P < .001) correlation between mean Bayesian FD and age as compared to the previous implementation. Thus, our results make us suppose that the Bayesian FD is a more truthful estimation for the fractal dimension of the cerebral cortex compared to the frequentist FD.

A log-extended Weibull regression model

A bathtub-shaped failure rate function is very useful in survival analysis and reliability studies. The well-known lifetime distributions do not have this property. For the first time, we propose a location-scale regression model based on the logarithm of an extended Weibull distribution which has the ability to deal with bathtub-shaped failure rate functions. We use the method of maximum likelihood to estimate the model parameters and some inferential procedures are presented. We reanalyze a real data set under the new model and the log-modified Weibull regression model. We perform a model check based on martingale-type residuals and generated envelopes and the statistics AIC and BIC to select appropriate models. (C) 2009 Elsevier B.V. All rights reserved.

Quantification of endocrine disruptors and pesticides in water by gas chromatography–tandem mass spectrometry. Method validation using weighted linear regression schemes

Amulti-residue methodology based on a solid phase extraction followed by gas chromatography–tandem mass spectrometry was developed for trace analysis of 32 compounds in water matrices, including estrogens and several pesticides from different chemical families, some of them with endocrine disrupting properties. Matrix standard calibration solutions were prepared by adding known amounts of the analytes to a residue-free sample to compensate matrix-induced chromatographic response enhancement observed for certain pesticides. Validation was done mainly according to the International Conference on Harmonisation recommendations, as well as some European and American validation guidelines with specifications for pesticides analysis and/or GC–MS methodology. As the assumption of homoscedasticity was not met for analytical data, weighted least squares linear regression procedure was applied as a simple and effective way to counteract the greater influence of the greater concentrations on the fitted regression line, improving accuracy at the lower end of the calibration curve. The method was considered validated for 31 compounds after consistent evaluation of the key analytical parameters: specificity, linearity, limit of detection and quantification, range, precision, accuracy, extraction efficiency, stability and robustness.

Association between LDL-C, Non HDL-C, and Apolipoprotein B Levels with Coronary Plaque Regression

Background:Previous reports have inferred a linear relationship between LDL-C and changes in coronary plaque volume (CPV) measured by intravascular ultrasound. However, these publications included a small number of studies and did not explore other lipid markers.Objective:To assess the association between changes in lipid markers and regression of CPV using published data.Methods:We collected data from the control, placebo and intervention arms in studies that compared the effect of lipidlowering treatments on CPV, and from the placebo and control arms in studies that tested drugs that did not affect lipids. Baseline and final measurements of plaque volume, expressed in mm3, were extracted and the percentage changes after the interventions were calculated. Performing three linear regression analyses, we assessed the relationship between percentage and absolute changes in lipid markers and percentage variations in CPV.Results:Twenty-seven studies were selected. Correlations between percentage changes in LDL-C, non-HDL-C, and apolipoprotein B (ApoB) and percentage changes in CPV were moderate (r = 0.48, r = 0.47, and r = 0.44, respectively). Correlations between absolute differences in LDL-C, non‑HDL-C, and ApoB with percentage differences in CPV were stronger (r = 0.57, r = 0.52, and r = 0.79). The linear regression model showed a statistically significant association between a reduction in lipid markers and regression of plaque volume.Conclusion:A significant association between changes in different atherogenic particles and regression of CPV was observed. The absolute reduction in ApoB showed the strongest correlation with coronary plaque regression.

Bayesian Model Averaging in the Instrumental Variable Regression Model

This paper considers the instrumental variable regression model when there is uncertainty about the set of instruments, exogeneity restrictions, the validity of identifying restrictions and the set of exogenous regressors. This uncertainty can result in a huge number of models. To avoid statistical problems associated with standard model selection procedures, we develop a reversible jump Markov chain Monte Carlo algorithm that allows us to do Bayesian model averaging. The algorithm is very exible and can be easily adapted to analyze any of the di¤erent priors that have been proposed in the Bayesian instrumental variables literature. We show how to calculate the probability of any relevant restriction (e.g. the posterior probability that over-identifying restrictions hold) and discuss diagnostic checking using the posterior distribution of discrepancy vectors. We illustrate our methods in a returns-to-schooling application.