Normal Linear Regression Model Research Articles

Model Selection in Generalized Linear Models

The problem of model selection in regression analysis through the use of forward selection, backward elimination, and stepwise selection has been well explored in the literature. The main assumption in this, of course, is that the data are normally distributed and the main tool used here is either a t test or an F test. However, the properties of these model selection procedures are not well-known. The purpose of this paper is to study the properties of these procedures within generalized linear regression models, considering the normal linear regression model as a special case. The main tool that is being used is the score test. However, the F test and other large sample tests, such as the likelihood ratio and the Wald test, the AIC, and the BIC, are included for the comparison. A systematic study, through simulations, of the properties of this procedure was conducted, in terms of level and power, for symmetric and asymmetric distributions, such as normal, Poisson, and binomial regression models. Extensions for skewed distributions, over-dispersed Poisson (the negative binomial), and over-dispersed binomial (the beta-binomial) regression models, are also given and evaluated. The methods are applied to analyze two health datasets.

Bayes-optimal prediction with frequentist coverage control

This article illustrates how indirect or prior information can be optimally used to construct a prediction region that maintains a target frequentist coverage rate. If the indirect information is accurate, the volume of the prediction region is lower on average than that of other regions with the same coverage rate. Even if the indirect information is inaccurate, the resulting region still maintains the target coverage rate. Such a prediction region can be constructed for models that have a complete sufficient statistic, which includes many widely-used parametric and nonparametric models. Particular examples include a Bayes-optimal conformal prediction procedure that maintains a constant coverage rate across distributions in a nonparametric model, as well as a prediction procedure for the normal linear regression model that can utilize a regularizing prior distribution, yet maintain a frequentist coverage rate that is constant as a function of the model parameters and explanatory variables. No results in this article rely on asymptotic approximations.

The Lifetime Prediction and Insulation Failure Mechanism of XLPE for High-Voltage Cable

The performance parameters and service life of the insulation layer of high-voltage cables are the key to ensure safe cable operation. Through extracting the mechanical parameters of high voltage cable insulation layer under different thermal aging temperatures, a normal linear regression model is established to predict the service life of high voltage cables. Meanwhile, the physicochemical characteristics such as microscopic morphology and molecular structure changes of the cable insulation layer are analyzed, the dielectric properties and breakdown field strength under ac condition are tested and the macroscopic electrical properties of the insulation layer are studied, and the insulation failure mechanism is further analyzed. The experimental results show that the elongation at break retention rate of cross-linked polyethylene (XLPE) increases slowly with the aging time, and then decreases rapidly when it reaches 50% of the initial elongation at break retention rate. The normal linear regression prediction model is established based on the Arrhenius formula, and it can be obtained the life endpoint of XLPE high voltage cable about 65 years at an operating temperature of 70 °C. Further analysis, under the action of heat and oxygen, the crystal zone structure of cable insulation is seriously damaged, and the carbonyl index is significantly increased. Accordingly, when the aging failure point is reached, the dielectric permittivity and dielectric loss of insulation are significantly increased, and the breakdown performance is significantly decreased. The work has an important guiding significance for insulation condition assessment and life prediction of high voltage cable.

Educating Young Consumers about Food Hygiene and Safety with SafeConsume: A Multi-Centre Mixed Methods Evaluation

The SafeConsume educational suite was designed to improve knowledge about undertaking safer food practices and avoiding food-borne illnesses among young people. The resources were designed to support educators and members of the community who teach young people (aged 11–18 years) and include lesson plans and supporting teacher training resources. To assess the efficacy and suitability of the resources, an evaluation of the central lesson, the ‘user journey’, was conducted within four European countries. The mixed-methods evaluation included the following elements: a pre- and post-scenario-based questionnaire, a satisfaction questionnaire, focus groups with students; and interviews with teachers. Data from the scenario-based questionnaires were analysed using a mixed effects normal linear regression model. Qualitative data were thematically analysed, and the main themes were discussed. A total of 171 students and 9 educators took part from schools based in Portugal, Hungary, France and England. The results indicated a significant improvement in students’ knowledge and understanding of appropriate food hygiene practices overall, although this varied among countries. The resources were found to be well-suited to help teachers deliver the lesson, being considered by teachers to be both informative and flexible. Minor alternations were suggested, including alterations to lesson delivery or breaking the lesson into smaller sections, and increasing the lesson’s interactivity.

Impact of administration route on serum progesterone levels in women undergoing artificial endometrial preparation

ObjectiveAre there differences in serum progesterone levels between different routes of exogenous progesterone administration for artificial endometrial preparation? Material and methodsThis prospective, observational, single-centre study included 9 infertile female patients who underwent cycles of artificial endometrial preparation between January and June 2019 with different progesterone formulations (3 cycles in 2 patients; 2 cycles in 2 patients; and 1 cycle in 5 patients). Oestrogen stimulation was followed by vaginal progesterone 400 mg every 12 h (first cycle), subcutaneous progesterone 25 mg every 12 h (second cycle), and intramuscular progesterone 50 mg every 24 h (third cycle). Progesterone therapy was continued for 5 days and daily serum progesterone was recorded. The primary outcome was day 5 serum progesterone. ResultsDay 5 mean ± standard deviation serum progesterone levels after vaginal, subcutaneous, and intramuscular administration were 14.6 ± 5.5, 47.9 ± 22.3, and 60.3 ± 65.5 ng/mL, respectively (p = 0.032 across routes). From day 1 to day 5, the coefficients of variation for serum progesterone were 66% and 75% with the vaginal and subcutaneous routes, respectively, indicating low variability, and 146% with the intramuscular route, indicating high variability. Two linear regression models were conducted: a normal linear regression model, which found no significant effect of administration route on serum progesterone, and a mixed-effects linear regression model, which also showed no statistically significant differences between routes. ConclusionAll routes of progesterone administration showed satisfactory day 5 mean serum progesterone levels, regardless of administration route.

Exact uniformly most powerful postselection confidence distributions

AbstractA conditioning on the event of having selected one model from a set of possibly misspecified normal linear regression models leads to the construction of uniformly optimal conditional confidence distributions. They can be used for valid postselection inference. The constructed conditional confidence distributions are finite sample exact and encompass all information regarding the focus parameter in the selected model. This includes the construction of optimal postselection confidence intervals at all significance levels and uniformly most powerful hypothesis tests.

Rectangular Confidence Regions and Prediction Regions in Multivariate Calibration

The multivariate calibration problem deals with inference concerning an unknown value of a covariate vector based on an observation on a response vector. Two distinct scenarios are considered in the multivariate calibration problem: controlled calibration where the covariates are non-stochastic, and random calibration where the covariates are random. Under controlled calibration, a problem of interest is the computation of a confidence region for the unknown covariate vector. Under random calibration, the problem is that of computing a prediction region for the covariate vector. Assuming the standard multivariate normal linear regression model, rectangular confidence and prediction regions are derived using a parametric bootstrap approach. Numerical results show that the regions accurately maintain the coverage probabilities. The results are illustrated using examples. The regions currently available in the literature are all ellipsoidal, and this work is the first attempt to derive rectangular regions.

Empirical Bayes Model Averaging with Influential Observations: Tuning Zellner’s [formula omitted] Prior for Predictive Robustness

The behavior of Bayesian model averaging (BMA) for the normal linear regression model in the presence of influential observations that contribute to model misfit is investigated. Remedies to attenuate the potential negative impacts of such observations on inference and prediction are proposed. The methodology is motivated by the view that well-behaved residuals and good predictive performance often go hand-in-hand. Focus is placed on regression models that use variants on Zellner's g prior. Studying the impact of various forms of model misfit on BMA predictions in simple situations points to prescriptive guidelines for “tuning” Zellner's g prior to obtain optimal predictions. The tuning of the prior distribution is obtained by considering theoretical properties that should be enjoyed by the optimal fits of the various models in the BMA ensemble. The methodology can be thought of as an “empirical Bayes” approach to modeling, as the data help to inform the specification of the prior in an attempt to attenuate the negative impact of influential cases.

Bayesian Inference of a Non normal Multivariate Partial Linear Regression Model

This research includes the Bayesian estimation of the parameters of the multivariate partial linear regression model when the random error follows the matrix-variate generalized modified Bessel distribution and found the statistical test of the model represented by finding the Bayes factor criterion, the predictive distribution under assumption that the shape parameters are known. The prior distribution about the model parameters is represented by non-informative information, as well as the simulate on the generated data from the model by a suggested way based on different values ​​of the shape parameters, the kernel function used in the generation was a Gaussian kernel function, the bandwidth (Smoothing) parameter was according to the rule of thumb. It found that the posterior marginal probability distribution of the location matrix and the predictive probability distribution is a matrix-t distribution with different parameters, the posterior marginal probability distribution of the scale matrix is proper distribution but it does not belong to the conjugate family, Through the Bayes factor criterion, it was found that the sample that was used in the generation process was drawn from a population that does not belong to the generalized modified Bessel population.

Influence of obstructive sleep apnea on auditory event-related potentials.

To evaluate the influence of obstructive sleep apnea (OSA) on the P300 response of auditory event-related potentials (ERPs) and to correlate the electrophysiological findings with OSA severity. Patients with no OSA and mild, moderate, and severe OSA according to polysomnography (PSG) with normal hearing and no comorbidities were studied. Individuals with a body mass index (BMI) ≥ 40kg/m2, hypertension, diabetes, dyslipidemia, the use of chronic medications, and a risk of hearing loss were excluded. All patients underwent full PSG and auditory ERP measurement using the oddball paradigm with tone burst and speech stimuli. For P300 analysis (latencies and amplitudes), normal multiple linear regression models were adjusted with the groups (No OSA, Mild OSA, Moderate OSA, Severe OSA), age, BMI, and Epworth score as explanatory variables. We studied 54 individuals (47 males) aged 35 ± 8years with a BMI of 28.4 ± 4.3kg/m2. Patients were divided according to the apnea-hypopnea index (AHI) derived from PSG into no OSA (n = 14), mild (n = 16), moderate (n = 12), and severe OSA (n = 12) groups. Patients with severe OSA presented prolonged P300 latencies with tone burst stimuli compared to patients with no OSA and those with mild and moderate OSA. Severe OSA is associated with impairment of the P300 response of auditory ERPs, suggesting a decrease in the processing speed of acoustic information that may be mediated by the level of somnolence.

Semiparametric inference for the scale-mixture of normal partial linear regression model with censored data

In the censored data exploration, the classical linear regression model which assumes normally distributed random errors is perhaps one of the commonly used frameworks. However, practical studies have often criticized the classical linear regression model because of its sensitivity to departure from the normality and partial nonlinearity. This paper proposes to solve these potential issues simultaneously in the context of the partial linear regression model by assuming that the random errors follow a scale-mixture of normal (SMN) family of distributions. The postulated method allows us to model data with great flexibility, accommodating heavy tails and outliers. By implementing the B-spline approximation and using the convenient hierarchical representation of the SMN distributions, a computationally analytical EM-type algorithm is developed for obtaining maximum likelihood (ML) parameter estimates. Various simulation studies are conducted to investigate the finite sample properties, as well as the robustness of the model in dealing with the heavy tails distributed datasets. Real-world data examples are finally analyzed for illustrating the usefulness of the proposed methodology.

Bayesian Reference Analysis for the Generalized Normal Linear Regression Model

This article proposes the use of the Bayesian reference analysis to estimate the parameters of the generalized normal linear regression model. It is shown that the reference prior led to a proper posterior distribution, while the Jeffreys prior returned an improper one. The inferential purposes were obtained via Markov Chain Monte Carlo (MCMC). Furthermore, diagnostic techniques based on the Kullback–Leibler divergence were used. The proposed method was illustrated using artificial data and real data on the height and diameter of Eucalyptus clones from Brazil.

Regression Diagnostics with Predicted Residuals of Linear Model with Improved Singular Value Classification Applied to Forecast the Hydrodynamic Efficiency of Wave Energy Converters

In the preliminary stages of design of the oscillating water column (OWC) type of wave energy converters (WECs), we need a reliable cost- and time-effective method to predict the hydrodynamic efficiency as a function of the design parameters. One of the cheapest approaches is to create a multiple linear regression (MLR) model using an existing data set. The problem with this approach is that the reliability of the MLR predictions depend on the validity of the regression assumptions, which are either rarely tested or tested using sub-optimal procedures. We offer a series of novel methods for assumption diagnostics that we apply in our case study for MLR prediction of the hydrodynamics efficiency of OWC WECs. Namely, we propose: a novel procedure for reliable identification of the zero singular values of a matrix; a modified algorithm for stepwise regression; a modified algorithm to detect heteroskedasticity and identify statistically significant but practically insignificant heteroscedasticity in the original model; a novel test of the validity of the nullity assumption; a modified Jarque–Bera Monte Carlo error normality test. In our case study, the deviations from the assumptions of the classical normal linear regression model were fully diagnosed and dealt with. The newly proposed algorithms based on improved singular value decomposition (SVD) of the design matrix and on predicted residuals were successfully tested with a new family of goodness-of-fit measures. We empirically investigated the correct placement of an elaborate outlier detection procedure in the overall diagnostic sequence. As a result, we constructed a reliable MLR model to predict the hydrodynamic efficiency in the preliminary stages of design. MLR is a useful tool at the preliminary stages of design and can produce highly reliable and time-effective predictions of the OWC WEC performance provided that the constructing and diagnostic procedures are modified to reflect the latest advances in statistics. The main advantage of MLR models compared to other modern black box models is that their assumptions are known and can be tested in practice, which increases the reliability of the model predictions.

Model Averaging and Its Use in Economics

The method of model averaging has become an important tool to deal with model uncertainty, for example in situations where a large amount of different theories exist, as are common in economics. Model averaging is a natural and formal response to model uncertainty in a Bayesian framework, and most of the paper deals with Bayesian model averaging. The important role of the prior assumptions in these Bayesian procedures is highlighted. In addition, frequentist model averaging methods are also discussed. Numerical techniques to implement these methods are explained, and I point the reader to some freely available computational resources. The main focus is on uncertainty regarding the choice of covariates in normal linear regression models, but the paper also covers other, more challenging, settings, with particular emphasis on sampling models commonly used in economics. Applications of model averaging in economics are reviewed and discussed in a wide range of areas including growth economics, production modeling, finance and forecasting macroeconomic quantities. (JEL C11, C15, C20, C52, O47).

A comparison of residual diagnosis tools for diagnosing regression models for count data

BackgroundExamining residuals is a crucial step in statistical analysis to identify the discrepancies between models and data, and assess the overall model goodness-of-fit. In diagnosing normal linear regression models, both Pearson and deviance residuals are often used, which are equivalently and approximately standard normally distributed when the model fits the data adequately. However, when the response vari*able is discrete, these residuals are distributed far from normality and have nearly parallel curves according to the distinct discrete response values, imposing great challenges for visual inspection.MethodsRandomized quantile residuals (RQRs) were proposed in the literature by Dunn and Smyth (1996) to circumvent the problems in traditional residuals. However, this approach has not gained popularity partly due to the lack of investigation of its performance for count regression including zero-inflated models through simulation studies. Therefore, we assessed the normality of the RQRs and compared their performance with traditional residuals for diagnosing count regression models through a series of simulation studies. A real data analysis in health care utilization study for modeling the number of repeated emergency department visits was also presented.ResultsOur results of the simulation studies demonstrated that RQRs have low type I error and great statistical power in comparisons to other residuals for detecting many forms of model misspecification for count regression models (non-linearity in covariate effect, over-dispersion, and zero inflation). Our real data analysis also showed that RQRs are effective in detecting misspecified distributional assumptions for count regression models.ConclusionsOur results for evaluating RQRs in comparison with traditional residuals provide further evidence on its advantages for diagnosing count regression models.

A class of admissible estimators of multiple regression coefficient with an unknown variance

Suppose that we observe , where is an unknown vector with unknown precision τ. Estimating the regression coefficient with known τ has been well studied. However, statistical properties such as admissibility in estimating with unknown τ are not well studied. Han [(2009). Topics in shrinkage estimation and in causal inference (PhD thesis). Warton School, University of Pennsylvania] appears to be the first to consider the problem, developing sufficient conditions for the admissibility of estimating means of multivariate normal distributions with unknown variance. We generalise the sufficient conditions for admissibility and apply these results to the normal linear regression model. 2-level and 3-level hierarchical models with unknown precision τ are investigated when a standard class of hierarchical priors leads to admissible estimators of under the normalised squared error loss. One reason to consider this problem is the importance of admissibility in the hierarchical prior selection, and we expect that our study could be helpful in providing some reference for choosing hierarchical priors.

Bayesian Estimation of Normal Linear Regression Model with Heteroscedasticity Error Structures

Non-constant error variance in Normal Linear Regression Model (NLRM) is an econometric problem generally referred to as heteroscedasticity. Its presence renders statistical inference invalid. Classical approach to its detection, estimation and remediation are widely discussed in the econometric literature. However, estimation of a NLRM using the Bayesian approach when heteroscedasticity problem is present is a major gap in the existing stock of knowledge on this subject. This approach has grown widely in recent times because it combines out-of-sample information with observed data. The study derived Bayesian estimators of the NLRM in the presence of functional forms of heteroscedasticity. Variance was treated as a linear function and as an exponential function of exogenous variables. The estimators are found to be unbiased and consistent and the precision values tend to zero. The estimates obtained from the estimators approximately 95% draws fall within each of the corresponding credible interval. Therefore, the results obtained for the derived Bayesian estimators for different functional forms of heteroscedasticity considered are similar, thus, providing a credible alternative to the existing classical methods which depend solely on the sample information.

Does oil price volatility influence real sector growth? Empirical evidence from Pakistan

The study investigates the short-run and long-run relationship between oil price fluctuation and real sector growth in Pakistan. Four major sectors of the economy (Manufacturing, electricity, transport and communication, and livestock) were analyzed to find any relation. Similar studies can be found in the existing literate, however, the distinguish feature of present study is that it investigates each individual sector’s linkage to oil price changes. Annual time series data of selected sectors ranging from 1976 to 2017 is selected for the study. Classical normal linear regression models under auto regressive distributed lag (ARDL) were employed to study the relationship between economic sectors and oil price fluctuation. Empirical results indicate that changes in oil price adversely affect manufacturing, livestock and electricity sectors in short-run and long-run, while significant positive impact was found on transportation and communication. Consequently, the sectors prone to oil price changes require special attention of policy makers. An expansionary monetary policy can be a short-run solution to reduce the impact of increasing oil price, whereas the government can introduce a policy framework to counter this effect in long-run.

Fast Bayesian variable selection for high dimensional linear models: Marginal solo spike and slab priors

This paper presents a method for fast Bayesian variable selection in the normal linear regression model with high dimensional data. A novel approach is adopted in which an explicit posterior probability for including a covariate is obtained. The method is sequential but not order dependent, one deals with each covariate one by one, and a spike and slab prior is only assigned to the coefficient under investigation. We adopt the well-known spike and slab Gaussian priors with a sample size dependent variance, which achieves strong selection consistency for marginal posterior probabilities even when the number of covariates grows almost exponentially with sample size. Numerical illustrations are presented where it is shown that the new approach provides essentially equivalent results to the standard spike and slab priors, i.e. the same marginal posterior probabilities of the coefficients being nonzero, which are estimated via Gibbs sampling. Hence, we obtain the same results via the direct calculation of $p$ probabilities, compared to a stochastic search over a space of $2^{p}$ elements. Our procedure only requires $p$ probabilities to be calculated, which can be done exactly, hence parallel computation when $p$ is large is feasible.

Exact adaptive confidence intervals for linear regression coefficients

We propose an adaptive confidence interval procedure (CIP) for the coefficients in the normal linear regression model. This procedure has a frequentist coverage rate that is constant as a function of the model parameters, yet provides smaller intervals than the usual interval procedure, on average across regression coefficients. The proposed procedure is obtained by defining a class of CIPs that all have exact $1-\alpha $ frequentist coverage, and then selecting from this class the procedure that minimizes a prior expected interval width. We describe an adaptive approach for estimating the prior distribution from the data, so that the potential risk of a poorly specified prior is reduced. The resulting adaptive confidence intervals maintain exact non-asymptotic $1-\alpha $ coverage if two conditions are met - that the design matrix is full rank (which will be known) and that the errors are normally distributed (which can be checked empirically). No assumptions on the unknown parameters are necessary to maintain exact coverage. Additionally, in a “$p$ growing with $n$” asymptotic scenario, this adaptive FAB procedure is asymptotically Bayes-optimal among $1-\alpha $ frequentist CIPs.