Normal Regression Model Research Articles

Mixtures of multivariate contaminated normal regression models

Mixtures of regression models (MRMs) are widely used to investigate the relationship between variables coming from several unknown latent homogeneous groups. Usually, the conditional distribution of the response in each mixture component is assumed to be (multivariate) normal (MN-MRM). To robustify the approach with respect to possible elliptical heavy-tailed departures from normality, due to the presence of mild outliers, the multivariate contaminated normal MRM is here introduced. In addition to the parameters of the MN-MRM, each mixture component has a parameter controlling the proportion of outliers and one specifying the degree of contamination with respect to the response variable(s). Crucially, these parameters do not have to be specified a priori, adding flexibility to our approach. Furthermore, once the model is estimated and the observations are assigned to the groups, a finer intra-group classification in typical points and (mild) outliers, can be directly obtained. Identifiability conditions are provided, an expectation-conditional maximization algorithm is outlined for parameter estimation, and various implementation and operational issues are discussed. Properties of the estimators of the regression coefficients are evaluated through Monte Carlo experiments and compared with other procedures. The performance of this novel family of models is also illustrated on artificial and real data, with particular emphasis to the application in allometric studies.

The Kumaraswamy normal linear regression model with applications

ABSTRACTFor any continuous baseline G distribution, Cordeiro and Castro pioneered the Kumaraswamy-G family of distributions with two extra positive parameters, which generalizes both Lehmann types I and II classes. We study some mathematical properties of the Kumaraswamy-normal (KwN) distribution including ordinary and incomplete moments, mean deviations, quantile and generating functions, probability weighted moments, and two entropy measures. We propose a new linear regression model based on the KwN distribution, which extends the normal linear regression model. We obtain the maximum likelihood estimates of the model parameters and provide some diagnostic measures such as global influence, local influence, and residuals. We illustrate the potentiality of the introduced models by means of two applications to real datasets.

Circular interpretation of regression coefficients.

The interpretation of the effect of predictors in projected normal regression models is not straight‐forward. The main aim of this paper is to make this interpretation easier such that these models can be employed more readily by social scientific researchers. We introduce three new measures: the slope at the inflection point (b c), average slope (AS) and slope at mean (SAM) that help us assess the marginal effect of a predictor in a Bayesian projected normal regression model. The SAM or AS are preferably used in situations where the data for a specific predictor do not lie close to the inflection point of a circular regression curve. In this case b c is an unstable and extrapolated effect. In addition, we outline how the projected normal regression model allows us to distinguish between an effect on the mean and spread of a circular outcome variable. We call these types of effects location and accuracy effects, respectively. The performance of the three new measures and of the methods to distinguish between location and accuracy effects is investigated in a simulation study. We conclude that the new measures and methods to distinguish between accuracy and location effects work well in situations with a clear location effect. In situations where the location effect is not clearly distinguishable from an accuracy effect not all measures work equally well and we recommend the use of the SAM.

Statistical disclosure control via sufficiency under the multiple linear regression model

In this article we show, under the normal multiple linear regression model, how synthetic data can be generated using the principle of sufficiency. An advantage of this approach is that if the regression model assumed by the synthetic data producer is correctly specified, then the synthetic data have the same joint distribution as the original data, and therefore one can use standard regression methodology and software to analyze the synthetic data. If the same regression model used to generate the synthetic data is also used for data analysis, and the data are analyzed using standard regression methodology, then the synthetic data yield identical inference to that of the original data. We also study the effects of overfitting or under-fitting the linear regression model. We show that even if the data producer overspecifies the regression model when creating the synthetic data, the synthetic data will still have the same distribution as the original data, and hence valid inference can be obtained. However, if the data producer underspecifies the linear regression model, then one cannot expect to obtain valid inference from the synthetic data. The disclosure risk of the proposed method relative to a standard synthetic data method is also examined.

A unified censored normal regression model for qPCR differential gene expression analysis.

Reverse transcription quantitative polymerase chain reaction (RT-qPCR) is considered as the gold standard for accurate, sensitive, and fast measurement of gene expression. Prior to downstream statistical analysis, RT-qPCR fluorescence amplification curves are summarized into one single value, the quantification cycle (Cq). When RT-qPCR does not reach the limit of detection, the Cq is labeled as “undetermined”. Current state of the art qPCR data analysis pipelines acknowledge the importance of normalization for removing non-biological sample to sample variation in the Cq values. However, their strategies for handling undetermined Cq values are very ad hoc. We show that popular methods for handling undetermined values can have a severe impact on the downstream differential expression analysis. They introduce a considerable bias and suffer from a lower precision. We propose a novel method that unites preprocessing and differential expression analysis in a single statistical model that provides a rigorous way for handling undetermined Cq values. We compare our method with existing approaches in a simulation study and on published microRNA and mRNA gene expression datasets. We show that our method outperforms traditional RT-qPCR differential expression analysis pipelines in the presence of undetermined values, both in terms of accuracy and precision.

The performance of model averaged tail area confidence intervals

ABSTRACTWe investigate the exact coverage and expected length properties of the model averaged tail area (MATA) confidence interval proposed by Turek and Fletcher, CSDA, 2012, in the context of two nested, normal linear regression models. The simpler model is obtained by applying a single linear constraint on the regression parameter vector of the full model. For given length of response vector and nominal coverage of the MATA confidence interval, we consider all possible models of this type and all possible true parameter values, together with a wide class of design matrices and parameters of interest. Our results show that, while not ideal, MATA confidence intervals perform surprisingly well in our regression scenario, provided that we use the minimum weight within the class of weights that we consider on the simpler model.

Improved heteroskedasticity likelihood ratio tests in symmetric nonlinear regression models

In this paper we address the issue of testing inference of the dispersion parameter in heteroscedastic symmetric nonlinear regression models considering small samples. We derive Bartlett corrections to improve the likelihood ratio as well modified profile likelihood ratio tests. Our results extend some of those obtained in Cordeiro (J Stat Comput Simul 74:609–620, 2004) and Ferrari et al. (J Stat Plan Inference 124:423–437, 2004), who consider a symmetric nonlinear regression model and normal linear regression model, respectively. We also present the bootstrap and bootstrap Bartlett corrected likelihood ratio tests. Monte Carlo simulations are carried out to compare the finite sample performances of the three corrected tests and their uncorrected versions. The numerical evidence shows that the corrected modified profile likelihood ratio test, the bootstrap and bootstrap Bartlett corrected likelihood ratio test perform better than the other ones. We also present an empirical application.

A semiparametric mixture regression model for longitudinal data

A normal semiparametric mixture regression model is proposed for longitudinal data. The proposed model contains one smooth term and a set of possible linear predictors. Model terms are estimated using the penalized likelihood method with the EM algorithm. A computationally feasible alternative method that provides an approximate solution is also introduced. Simulation experiments and a real data example are used to illustrate the methods.

Fixed-width confidence interval for covariate-adjusted response-adaptive designs

In this paper, we obtain fixed-width confidence interval for covariate-adjusted response-adaptive designs. Specifically, we consider logistic regression model and the normal regression model for binary and continuous responses, respectively, both in the situations for presence and absence of treatment–covariate interactions. Simulation study and real-data analysis are carried out.

Censored Regression Modeling To Predict Virus Inactivation in Wastewaters.

Among the many uncertainties presented by poorly studied pathogens is possible transmission via human fecal material or wastewaters. Such worries were a documented concern during the 2013 Ebola outbreak in West Africa. Using published experimental data on virus inactivation rates in wastewater and similar matrices, we extracted data to construct a model predicting the T90 (1 × log10 inactivation measured in seconds) of a virus. Extracted data were as follows: RNA or DNA genome, enveloped or not, primary transmission pathway, temperature, pH, light levels, and matrix. From the primary details, we further determined matrix level of contamination, genus, and taxonomic family. Prior to model construction, three records were separated for verification. A censored normal regression model provided the best fit model, which predicted T90 from DNA or RNA structure, enveloped status, whether primary transmission pathway was fecal-oral, temperature, and whether contamination was low, medium, or high. Model residuals and predicted values were evaluated against observed values. Mean values of model predictions were compared to independent data and considering 95% confidence ranges (which could be quite large). A relatively simple model can predict virus inactivation rates from virus and matrix attributes, providing valuable input when formulating risk management strategies for little studied pathogens.

MicroRNA profiling in clear cell renal cell carcinoma tissues potentially links tumorigenesis and recurrence with obesity

Background:Twenty to 40% localised RCC patients may experience recurrence after curative surgery. Limited miRNA predictors have been identified for ccRCC recurrence.Methods:Through a multi-phase study design, we analysed miRNAs in tissues obtained from 203 ccRCC patients. Paired t-test was used for tumour–normal comparisons and Cox regression model was performed to compute hazard ratios (HRs) and corresponding 95% CIs.Results:A 17-miRNA signature was identified that can concordantly classify >95% of tumour/adjacent normal samples. Significant enrichment was found as 6 out of 17 miRNAs were associated with obesity (binomial probability=0.001). Decreased levels of miR-204-5p and miR-139-5p were each associated with an approximately three-fold increased risk of recurrence (P<0.01). Risk score was generated based on expressions of miR-204-5p and miR-139-5p, and the trend test was significant in both discovery and validation sets (Pfor trend<0.05). Striking MST reduction was observed for patients with a high-risk score (high vs low: discovery, 9.4 vs >97.7 months; validation, 20.8 vs >70.3 months). Expressions of miR-204-5p were also associated with body mass index (β=5.64, P<0.001). Significant inverse correlations were observed and validated between miR-204-5p and 13 obesity-related genes (r<0, P<0.01).Conclusions:We identified 17 miRNAs dysregulated in ccRCC tissues and showed that low expressions of miR-204-5p and miR-139-5p were associated with the higher risk of recurrence. The link between miR-204-5p and ccRCC recurrence may be partially mediated by regulating the expression of targeted obesity-related genes.

Orthogonality of the mean and error distribution in generalized linear models

ABSTRACTWe show that the mean-model parameter is always orthogonal to the error distribution in generalized linear models. Thus, the maximum likelihood estimator of the mean-model parameter will be asymptotically efficient regardless of whether the error distribution is known completely, known up to a finite vector of parameters, or left completely unspecified, in which case the likelihood is taken to be an appropriate semiparametric likelihood. Moreover, the maximum likelihood estimator of the mean-model parameter will be asymptotically independent of the maximum likelihood estimator of the error distribution. This generalizes some well-known results for the special cases of normal, gamma, and multinomial regression models, and, perhaps more interestingly, suggests that asymptotically efficient estimation and inferences can always be obtained if the error distribution is non parametrically estimated along with the mean. In contrast, estimation and inferences using misspecified error distributions or variance functions are generally not efficient.

Consistency of Bayes factor for nonnested model selection when the model dimension grows

Zellner's $g$-prior is a popular prior choice for the model selection problems in the context of normal regression models. Wang and Sun [J. Statist. Plann. Inference 147 (2014) 95-105] recently adopt this prior and put a special hyper-prior for $g$, which results in a closed-form expression of Bayes factor for nested linear model comparisons. They have shown that under very general conditions, the Bayes factor is consistent when two competing models are of order $O(n^{\tau})$ for $\tau <1$ and for $\tau=1$ is almost consistent except a small inconsistency region around the null hypothesis. In this paper, we study Bayes factor consistency for nonnested linear models with a growing number of parameters. Some of the proposed results generalize the ones of the Bayes factor for the case of nested linear models. Specifically, we compare the asymptotic behaviors between the proposed Bayes factor and the intrinsic Bayes factor in the literature.

Nonlinear regression models under skew scale mixtures of normal distributions

Normal nonlinear regression models are applied in some areas of the sciences and engineering to explain or describe the phenomena under study. However, it is well known that several phenomena are not always represented by the normal model due to lack of symmetry or the presence of heavy- and light-tailed distributions related to the normal law in the data. This paper proposes an extension of nonlinear regression models using the skew-scale mixtures of normal (SSMN) distributions proposed by Ferreira et al. (2011). This class of models provides a useful generalization of the symmetrical nonlinear regression models since the random term distributions cover both asymmetric and heavy-tailed distributions, such as the skew-t-normal, skew-slash and skew-contaminated normal, among others. An expectation–maximization (EM) algorithm for maximum likelihood (ML) estimates is presented and the observed information matrix is derived analytically. Some simulation studies are presented to examine the performance of the proposed methods, with relation to robustness and asymptotic properties of the ML estimates. Finally, an illustration of the method is presented considering a dataset previously analyzed under normal and skew-normal (SN) nonlinear regression models. The main conclusion is that the ML estimates from the heavy tails SSMN nonlinear models are more robust against outlying observations compared to the corresponding SN estimates.

A skew–normal mixture of joint location, scale and skewness models

Normal mixture regression models are one of the most important statistical data analysis tools in a heterogeneous population. When the data set under consideration involves asymmetric outcomes, in the last two decades, the skew normal distribution has been shown beneficial in dealing with asymmetric data in various theoretic and applied problems. In this paper, we propose and study a novel class of models: a skew–normal mixture of joint location, scale and skewness models to analyze the heteroscedastic skew–normal data coming from a heterogeneous population. The issues of maximum likelihood estimation are addressed. In particular, an Expectation–Maximization (EM) algorithm for estimating the model parameters is developed. Properties of the estimators of the regression coefficients are evaluated through Monte Carlo experiments. Results from the analysis of a real data set from the Body Mass Index (BMI) data are presented.

Predicting chlorophyll-a using Landsat 8 OLI sensor data and the non-linear RANSAC method – a case study of Nakdong River, South Korea

ABSTRACTTo predict the degree of chlorophyll-a (chl-a) concentration as a water quality indicator, we propose a novel method that uses data collected by satellite remote sensors and a non-linear random sample consensus (NL-RANSAC) algorithm. In this study, multispectral images obtained from the Landsat OLI sensor and in situ sampling data obtained from the Nakdong River in South Korea are used for estimating a regression model. Because the RANSAC algorithm based on linear models is not appropriate for modelling non-linear data distribution, such as chl-a concentration and influential outlier data, in this article an NL-RANSAC model represented by an explicit polynomial curve is proposed instead of the RANSAC. The results of experiments in which NL-RANSAC-based regressions and normal regression models were compared using a calibration and a validation set clearly show that the proposed second-order NL-RANSAC is a good choice for estimating chl-a concentration, because it shows an average 0.3987 higher assessment than normal regression models in validation set. In addition, we prove that the second-order NL-RANSAC model is the most appropriate regression model for estimating chl-a concentration by using Landsat 8 OLI sensor imagery in the midstream of the Nakdong River.

Objective Bayesian hypothesis testing in regression models with first-order autoregressive residuals

ABSTRACTThis article considers the objective Bayesian testing in the normal regression models with first-order autoregressive residuals. We propose some solutions based on a Bayesian model selection procedure to this problem where no subjective input is considered. We construct the proper priors for testing the autocorrelation coefficient based on measures of divergence between competing models, which is called the divergence-based (DB) priors and then propose the objective Bayesian decision-theoretic rule, which is called the Bayesian reference criterion (BRC). Finally, we derive the intrinsic test statistic for testing the autocorrelation coefficient. The behavior of the Bayes factor-based DB priors is examined by comparing with the BRC in a simulation study and an example.

The Mean-shift Outlier Model under Skew Normal Distribution

Asymmetric models have been extensively studied in recent years, in situations where the normality assumption is not satisfied due to lack of symmetry of the data. Techniques for assessing the quality of fit and diagnostic analysis are important for model validation. This paper presents a study of the mean-shift method for detecting outliers in asymmetric normal regression models. Analytical solutions for the estimators of the parameters are obtained using the algorithm. Simulation studies and application to real data are presented, showing the efficiency of the method in detecting outliers.

A non-iterative Bayesian sampling algorithm for censored Student-t linear regression models

ABSTRACTIn this paper, we consider an effective Bayesian inference for censored Student-t linear regression model, which is a robust alternative to the usual censored Normal linear regression model. Based on the mixture representation of the Student-t distribution, we propose a non-iterative Bayesian sampling procedure to obtain independently and identically distributed samples approximately from the observed posterior distributions, which is different from the iterative Markov Chain Monte Carlo algorithm. We conduct model selection and influential analysis using the posterior samples to choose the best fitted model and to detect latent outliers. We illustrate the performance of the procedure through simulation studies, and finally, we apply the procedure to two real data sets, one is the insulation life data with right censoring and the other is the wage rates data with left censoring, and we get some interesting results.

New Flexible Regression Models Generated by Gamma Random Variables with Censored Data

We propose and study a new log-gamma Weibull regression model. We obtain explicit expressions for the raw and incomplete moments, quantile and generating functions and mean deviations of the log-gamma Weibull distribution. We demonstrate that the new regression model can be applied to censored data since it represents a parametric family of models which includes as sub-models several widely-known regression models and therefore can be used more effectively in the analysis of survival data. We obtain the maximum likelihood estimates of the model parameters by considering censored data and evaluate local influence on the estimates of the parameters by taking different perturbation schemes. Some global-influence measurements are also investigated. Further, for different parameter settings, sample sizes and censoring percentages, various simulations are performed. In addition, the empirical distribution of some modified residuals are displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be extended to a modified deviance residual in the proposed regression model applied to censored data. We demonstrate that our extended regression model is very useful to the analysis of real data and may give more realistic fits than other special regression models.