Structuring of Ranked Models

Prognostic procedures can be based on ranked linear models. Ranked regression type models are designed on the basis of feature vectors combined with set of relations defined on selected pairs of these vectors. Feature vectors are composed of numerical results of measurements on particular objects or events. Ranked relations defined on selected pairs of feature vectors represent additional knowledge and can reflect experts' opinion about considered objects. Ranked models have the form of linear transformations of feature vectors on a line which preserve a given set of relations in the best manner possible. Ranked models can be designed through the minimization of a special type of convex and piecewise linear (CPL) criterion functions. Some sets of ranked relations cannot be well represented by one ranked model. Decomposition of global model into a family of local ranked models could improve representation. A procedures of ranked models decomposition is described in this paper.

Regression analysis of ranked segment parameters for optic nerve head classification:a pilot study

Purpose: To determine whether curve-fitting analysis of the ranked segment distributions of topographic optic nerve head (ONH) parameters, derived using the Heidelberg Retina Tomograph (HRT), provide a more effective statistical descriptor to differentiate the normal from the glaucomatous ONH. Methods: The sample comprised of 22 normal control subjects (mean age 66.9 years; S.D. 7.8) and 22 glaucoma patients (mean age 72.1 years; S.D. 6.9) confirmed by reproducible visual field defects on the Humphrey Field Analyser. Three 10°-images of the ONH were obtained using the HRT. The mean topography image was determined and the HRT software was used to calculate the rim volume, rim area to disc area ratio, normalised rim area to disc area ratio and retinal nerve fibre cross-sectional area for each patient at 10°-sectoral intervals. The values were ranked in descending order, and each ranked-segment curve of ordered values was fitted using the least squares method. Results: There was no difference in disc area between the groups. The group mean cup-disc area ratio was significantly lower in the normal group (0.204 ± 0.16) compared with the glaucoma group (0.533 ± 0.083) (p < 0.001). The visual field indices, mean deviation and corrected pattern S.D., were significantly greater (p < 0.001) in the glaucoma group (-9.09 dB ± 3.3 and 7.91 ± 3.4, respectively) compared with the normal group (-0.15 dB ± 0.9 and 0.95 dB ± 0.8, respectively). Univariate linear regression provided the best overall fit to the ranked segment data. The equation parameters of the regression line manually applied to the normalised rim area-disc area and the rim area-disc area ratio data, correctly classified 100% of normal subjects and glaucoma patients. In this study sample, the regression analysis of ranked segment parameters method was more effective than conventional ranked segment analysis, in which glaucoma patients were misclassified in approximately 50% of cases. Further investigation in larger samples will enable the calculation of confidence intervals for normality. These reference standards will then need to be investigated for an independent sample to fully validate the technique. Conclusions: Using a curve-fitting approach to fit ranked segment curves retains information relating to the topographic nature of neural loss. Such methodology appears to overcome some of the deficiencies of conventional ranked segment analysis, and subject to validation in larger scale studies, may potentially be of clinical utility for detecting and monitoring glaucomatous damage. © 2007 The College of Optometrists.

Bayesian Analysis for Nonlinear Regression Model Under Skewed Errors, with Application in Growth Curves

We have considered a Bayesian approach for the nonlinear regression model by replacing the normal distribution on the error term by some skewed distributions, which account for both skewness and heavy tails or skewness alone. The type of data considered in this paper concerns repeated measurements taken in time on a set of individuals. Such multiple observations on the same individual generally produce serially correlated outcomes. Thus, additionally, our model does allow for a correlation between observations made from the same individual. We have illustrated the procedure using a data set to study the growth curves of a clinic measurement of a group of pregnant women from an obstetrics clinic in Santiago, Chile. Parameter estimation and prediction were carried out using appropriate posterior simulation schemes based in Markov Chain Monte Carlo methods. Besides the deviance information criterion (DIC) and the conditional predictive ordinate (CPO), we suggest the use of proper scoring rules based on the posterior predictive distribution for comparing models. For our data set, all these criteria chose the skew-t model as the best model for the errors. These DIC and CPO criteria are also validated, for the model proposed here, through a simulation study. As a conclusion of this study, the DIC criterion is not trustful for this kind of complex model.

Breeding value accuracy estimates for growth traits using random regression and multi-trait models in Nelore cattle

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Variance estimation in ranked set sampling using a concomitant variable

We propose a nonparametric variance estimator when ranked set sampling (RSS) and judgment post stratification (JPS) are applied by measuring a concomitant variable. Our proposed estimator is obtained by conditioning on observed concomitant values and using nonparametric kernel regression.

Logistic regression models for ordinal response: A study of self-efficacy in colorectal cancer screening

The ordinal logistic regression models are used to analyze the dependant variable with multiple outcomes that can be ranked, but have been underutilized. In this study, we describe four logistic regression models for analyzing the ordinal response variable. ^ In this methodological study, the four regression models are proposed. The first model uses the multinomial logistic model. The second is adjacent-category logit model. The third is the proportional odds model and the fourth model is the continuation-ratio model. We illustrate and compare the fit of these models using data from the survey designed by the University of Texas, School of Public Health research project PCCaSO (Promoting Colon Cancer Screening in people 50 and Over), to study the patient’s confidence in the completion colorectal cancer screening (CRCS). ^ The purpose of this study is two fold: first, to provide a synthesized review of models for analyzing data with ordinal response, and second, to evaluate their usefulness in epidemiological research, with particular emphasis on model formulation, interpretation of model coefficients, and their implications. Four ordinal logistic models that are used in this study include (1) Multinomial logistic model, (2) Adjacent-category logistic model [9], (3) Continuation-ratio logistic model [10], (4) Proportional logistic model [11]. We recommend that the analyst performs (1) goodness-of-fit tests, (2) sensitivity analysis by fitting and comparing different models.^

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.

Comparing Near-infrared Conventional Diffuse Reflectance Spectroscopy And Hyperspectral Imaging For Determination Of The Bulk Properties Of Solid Samples By Multivariate Regression: Determination Of Mooney Viscosity And Plasticity Indices Of Natural Rubber.

Conventional reflectance spectroscopy (NIRS) and hyperspectral imaging (HI) in the near-infrared region (1000-2500 nm) are evaluated and compared, using, as the case study, the determination of relevant properties related to the quality of natural rubber. Mooney viscosity (MV) and plasticity indices (PI) (PI0 - original plasticity, PI30 - plasticity after accelerated aging, and PRI - the plasticity retention index after accelerated aging) of rubber were determined using multivariate regression models. Two hundred and eighty six samples of rubber were measured using conventional and hyperspectral near-infrared imaging reflectance instruments in the range of 1000-2500 nm. The sample set was split into regression (n = 191) and external validation (n = 95) sub-sets. Three instruments were employed for data acquisition: a line scanning hyperspectral camera and two conventional FT-NIR spectrometers. Sample heterogeneity was evaluated using hyperspectral images obtained with a resolution of 150 × 150 μm and principal component analysis. The probed sample area (5 cm(2); 24,000 pixels) to achieve representativeness was found to be equivalent to the average of 6 spectra for a 1 cm diameter probing circular window of one FT-NIR instrument. The other spectrophotometer can probe the whole sample in only one measurement. The results show that the rubber properties can be determined with very similar accuracy and precision by Partial Least Square (PLS) regression models regardless of whether HI-NIR or conventional FT-NIR produce the spectral datasets. The best Root Mean Square Errors of Prediction (RMSEPs) of external validation for MV, PI0, PI30, and PRI were 4.3, 1.8, 3.4, and 5.3%, respectively. Though the quantitative results provided by the three instruments can be considered equivalent, the hyperspectral imaging instrument presents a number of advantages, being about 6 times faster than conventional bulk spectrometers, producing robust spectral data by ensuring sample representativeness, and minimizing the effect of the presence of contaminants.

Reproductive behavior, development and eye regression in the cave armored catfish, Ancistrus cryptophthalmus Reis, 1987 (Siluriformes: Loricariidae), breed in laboratory

The troglobitic armored catfish, Ancistrus cryptophthalmus (Loricariidae, Ancistrinae) is known from four caves in the São Domingos karst area, upper rio Tocantins basin, Central Brazil. These populations differ in general body shape and degree of reduction of eyes and of pigmentation. The small Passa Três population (around 1,000 individuals) presents the most reduced eyes, which are not externally visible in adults. A small group of Passa Três catfish, one male and three females, reproduced spontaneously thrice in laboratory, at the end of summertime in 2000, 2003 and 2004. Herein we describe the reproductive behavior during the 2003 event, as well as the early development of the 2003 and 2004 offsprings, with focus on body growth and ontogenetic regression of eyes. The parental care by the male, which includes defense of the rock shelter where the egg clutch is laid, cleaning and oxygenation of eggs, is typical of many loricariids. On the other hand, the slow development, including delayed eye degeneration, low body growth rates and high estimated longevity (15 years or more) are characteristic of precocial, or K-selected, life cycles. In the absence of comparable data for close epigean relatives (Ancistrus spp.), it is not possible to establish whether these features are an autapomorphic specialization of the troglobitic A. cryptophthalmus or a plesiomorphic trait already present in the epigean ancestor, possibly favoring the adoption of the life in the food-poor cave environment. We briefly discuss the current hypotheses on eye regression in troglobitic vertebrates.

Multi-trait and random regression mature weight heritability and breeding value estimates in Nelore cattle

Mature weight breeding values were estimated using a multi-trait animal model (MM) and a random regression animal model (RRM). Data consisted of 82 064 weight records from 8 145 animals, recorded from birth to eight years of age. Weights at standard ages were considered in the MM. All models included contemporary groups as fixed effects, and age of dam (linear and quadratic effects) and animal age as covariates. In the RRM, mean trends were modelled through a cubic regression on orthogonal polynomials of animal age and genetic maternal and direct and maternal permanent environmental effects were also included as random. Legendre polynomials of orders 4, 3, 6 and 3 were used for animal and maternal genetic and permanent environmental effects, respectively, considering five classes of residual variances. Mature weight (five years) direct heritability estimates were 0.35 (MM) and 0.38 (RRM). Rank correlation between sires' breeding values estimated by MM and RRM was 0.82. However, selecting the top 2% (12) or 10% (62) of the young sires based on the MM predicted breeding values, respectively 71% and 80% of the same sires would be selected if RRM estimates were used instead. The RRM modelled the changes in the (co) variances with age adequately and larger breeding value accuracies can be expected using this model.

Residuals for log-Burr XII regression models in survival analysis

In this paper, we compare three residuals to assess departures from the error assumptions as well as to detect outlying observations in log-Burr XII regression models with censored observations. These residuals can also be used for the log-logistic regression model, which is a special case of the log-Burr XII regression model. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and the empirical distribution of each residual is displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended to the modified martingale-type residual in log-Burr XII regression models with censored data.

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.

Log-modified Weibull regression models with censored data: Sensitivity and residual analysis

This paper proposes a regression model considering the modified Weibull distribution. This distribution can be used to model bathtub-shaped failure rate functions. Assuming censored data, we consider maximum likelihood and Jackknife estimators for the parameters of the model. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and we also present some ways to perform global influence. Besides, for different parameter settings, sample sizes and censoring percentages, various simulations are performed and the empirical distribution of the modified deviance residual is displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended for a martingale-type residual in log-modified Weibull regression models with censored data. Finally, we analyze a real data set under log-modified Weibull regression models. A diagnostic analysis and a model checking based on the modified deviance residual are performed to select appropriate models. (c) 2008 Elsevier B.V. All rights reserved.

On estimation and influence diagnostics for zero-inflated negative binomial regression models

The zero-inflated negative binomial model is used to account for overdispersion detected in data that are initially analyzed under the zero-Inflated Poisson model A frequentist analysis a jackknife estimator and a non-parametric bootstrap for parameter estimation of zero-inflated negative binomial regression models are considered In addition an EM-type algorithm is developed for performing maximum likelihood estimation Then the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and some ways to perform global influence analysis are derived In order to study departures from the error assumption as well as the presence of outliers residual analysis based on the standardized Pearson residuals is discussed The relevance of the approach is illustrated with a real data set where It is shown that zero-inflated negative binomial regression models seems to fit the data better than the Poisson counterpart (C) 2010 Elsevier B V All rights reserved

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.