Comparison of Bayesian models for production efficiency

In this paper, we use Markov Chain Monte Carlo (MCMC) methods in order to estimate and compare stochastic production frontier models from a Bayesian perspective. We consider a number of competing models in terms of different production functions and the distribution of the asymmetric error term. All MCMC simulations are done using the package JAGS (Just Another Gibbs Sampler), a clone of the classic BUGS package which works closely with the R package where all the statistical computations and graphics are done.     

A fast algorithm for the coordinate-wise minimum distance estimation

Application of the minimum distance (MD) estimation method to the linear regression model for estimating regression parameters is a difficult and time-consuming process due to the complexity of its distance function, and hence, it is computationally expensive. To deal with the computational cost, this paper proposes a fast algorithm which makes the best use of coordinate-wise minimization technique in order to obtain the MD estimator. R package (KoulMde) based on the proposed algorithm and written in Rcpp is available online.     

Hierarchical spatially varying coefficient and temporal dynamic process models using spTDyn

Bayesian hierarchical spatio-temporal models are becoming increasingly important due to the increasing availability of space-time data in various domains. In this paper we develop a user friendly R package, spTDyn, for spatio-temporal modelling. It can be used to fit models with spatially varying and temporally dynamic coefficients. The former is used for modelling the spatially varying impact of explanatory variables on the response caused by spatial misalignment. This issue can arise when the covariates only vary over time, or when they are measured over a grid and hence do not match the locations of the response point-level data. The latter is to examine the temporally varying impact of explanatory variables in space-time data due, for example, to seasonality or other time-varying effects. The spTDyn package uses Markov chain Monte Carlo sampling written in C, which makes computations highly efficient, and the interface is written in R making these sophisticated modelling techniques easily accessible to statistical analysts. The models and software, and their advantages, are illustrated using temperature and ozone space-time data.     

Generalized Tobit models: diagnostics and application in econometrics

The standard Tobit model is constructed under the assumption of a normal distribution and has been widely applied in econometrics. Atypical/extreme data have a harmful effect on the maximum likelihood estimates of the standard Tobit model parameters. Then, we need to count with diagnostic tools to evaluate the effect of extreme data. If they are detected, we must have available a Tobit model that is robust to this type of data. The family of elliptically contoured distributions has the Laplace, logistic, normal and Student-t cases as some of its members. This family has been largely used for providing generalizations of models based on the normal distribution, with excellent practical results. In particular, because the Student-t distribution has an additional parameter, we can adjust the kurtosis of the data, providing robust estimates against extreme data. We propose a methodology based on a generalization of the standard Tobit model with errors following elliptical distributions. Diagnostics in the Tobit model with elliptical errors are developed. We derive residuals and global/local influence methods considering several perturbation schemes. This is important because different diagnostic methods can detect different atypical data. We implement the proposed methodology in an R package. We illustrate the methodology with real-world econometrical data by using the R package, which shows its potential applications. The Tobit model based on the Student-t distribution with a small quantity of degrees of freedom displays an excellent performance reducing the influence of extreme cases in the maximum likelihood estimates in the application presented. It provides new empirical evidence on the capabilities of the Student-t distribution for accommodation of atypical data.     

Developments in Maximum Likelihood Unit Root Tests

The exact maximum likelihood estimate provides a test statistic for the unit root test that is more powerful than the usual least-squares approach. In this article, a new derivation is given for the asymptotic distribution of this test statistic that is simpler and more direct than the previous method. The response surface regression method is used to obtain a fast algorithm that computes accurate finite-sample critical values. This algorithm is available in the R package mleur that is available on CRAN. The empirical power of the new test is shown to be much better than the usual test not only in the normal case but also for innovations generated from an infinite variance stable distribution as well as for innovations generated from a GARCH(1,1) process.     

Some Results on Fractional Factorial Split-Plot Designs with Multi-Level Factors

Fractional factorial split-plot (FFSP) designs have received much attention in recent years. In this article, the matrix representation for FFSP designs with multi-level factors is first developed, which is an extension of the one proposed by Bingham and Sitter (1999b Bingham , D. , Sitter , R. R. ( 1999b ). Some theoretical results for fractional factorial split-plot designs . Ann. Statist. 27 : 1240 – 1255 . [Google Scholar]) for the two-level case. Based on this representation, periodicity results of maximum resolution and minimum aberration for such designs are derived. Differences between FFSP designs with multi-level factors and those with two-level factors are highlighted.     

Test for Trend With a Multinomial Outcome

ABSTRACT

There is no established procedure for testing for trend with nominal outcomes that would provide both a global hypothesis test and outcome-specific inference. We derive a simple formula for such a test using a weighted sum of Cochran–Armitage test statistics evaluating the trend in each outcome separately. The test is shown to be equivalent to the score test for multinomial logistic regression, however, the new formulation enables the derivation of a sample size formula and multiplicity-adjusted inference for individual outcomes. The proposed methods are implemented in the R package multiCA.     

Visualizing Count Data Regressions Using Rootograms

The rootogram is a graphical tool associated with the work of J. W. Tukey that was originally used for assessing goodness of fit of univariate distributions. Here, we extend the rootogram to regression models and show that this is particularly useful for diagnosing and treating issues such as overdispersion and/or excess zeros in count data models. We also introduce a weighted version of the rootogram that can be applied out of sample or to (weighted) subsets of the data, for example, in finite mixture models. An empirical illustration revisiting a well-known dataset from ethology is included, for which a negative binomial hurdle model is employed. Supplementary materials providing two further illustrations are available online: the first, using data from public health, employs a two-component finite mixture of negative binomial models; the second, using data from finance, involves underdispersion. An R implementation of our tools is available in the R package countreg. It also contains the data and replication code.     

First-order marginalised transition random effects models with probit link function

Marginalised models, also known as marginally specified models, have recently become a popular tool for analysis of discrete longitudinal data. Despite being a novel statistical methodology, these models introduce complex constraint equations and model fitting algorithms. On the other hand, there is a lack of publicly available software to fit these models. In this paper, we propose a three-level marginalised model for analysis of multivariate longitudinal binary outcome. The implicit function theorem is introduced to approximately solve the marginal constraint equations explicitly. probit link enables direct solutions to the convolution equations. Parameters are estimated by maximum likelihood via a Fisher–Scoring algorithm. A simulation study is conducted to examine the finite-sample properties of the estimator. We illustrate the model with an application to the data set from the Iowa Youth and Families Project. The R package pnmtrem is prepared to fit the model.     

A novel Bayesian regression model for counts with an application to health data

Discrete data are collected in many application areas and are often characterised by highly-skewed distributions. An example of this, which is considered in this paper, is the number of visits to a specialist, often taken as a measure of demand in healthcare. A discrete Weibull regression model was recently proposed for regression problems with a discrete response and it was shown to possess desirable properties. In this paper, we propose the first Bayesian implementation of this model. We consider a general parametrization, where both parameters of the discrete Weibull distribution can be conditioned on the predictors, and show theoretically how, under a uniform non-informative prior, the posterior distribution is proper with finite moments. In addition, we consider closely the case of Laplace priors for parameter shrinkage and variable selection. Parameter estimates and their credible intervals can be readily calculated from their full posterior distribution. A simulation study and the analysis of four real datasets of medical records show promises for the wide applicability of this approach to the analysis of count data. The method is implemented in the R package BDWreg.     

Automatic model selection for high-dimensional survival analysis

Many different models for the analysis of high-dimensional survival data have been developed over the past years. While some of the models and implementations come with an internal parameter tuning automatism, others require the user to accurately adjust defaults, which often feels like a guessing game. Exhaustively trying out all model and parameter combinations will quickly become tedious or infeasible in computationally intensive settings, even if parallelization is employed. Therefore, we propose to use modern algorithm configuration techniques, e.g. iterated F-racing, to efficiently move through the model hypothesis space and to simultaneously configure algorithm classes and their respective hyperparameters. In our application we study four lung cancer microarray data sets. For these we configure a predictor based on five survival analysis algorithms in combination with eight feature selection filters. We parallelize the optimization and all comparison experiments with the BatchJobs and BatchExperiments R packages.     

Model selection in quantile regression models

Lasso methods are regularisation and shrinkage methods widely used for subset selection and estimation in regression problems. From a Bayesian perspective, the Lasso-type estimate can be viewed as a Bayesian posterior mode when specifying independent Laplace prior distributions for the coefficients of independent variables [32 T. Park, G. Casella, The Bayesian Lasso, J. Amer. Statist. Assoc. 103 (2008), pp. 681–686. doi: 10.1198/016214508000000337[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. A scale mixture of normal priors can also provide an adaptive regularisation method and represents an alternative model to the Bayesian Lasso-type model. In this paper, we assign a normal prior with mean zero and unknown variance for each quantile coefficient of independent variable. Then, a simple Markov Chain Monte Carlo-based computation technique is developed for quantile regression (QReg) models, including continuous, binary and left-censored outcomes. Based on the proposed prior, we propose a criterion for model selection in QReg models. The proposed criterion can be applied to classical least-squares, classical QReg, classical Tobit QReg and many others. For example, the proposed criterion can be applied to rq(), lm() and crq() which is available in an R package called Brq. Through simulation studies and analysis of a prostate cancer data set, we assess the performance of the proposed methods. The simulation studies and the prostate cancer data set analysis confirm that our methods perform well, compared with other approaches.     

Approximate tolerance intervals for the discrete Poisson–Lindley distribution

The Poisson–Lindley distribution is a compound discrete distribution that can be used as an alternative to other discrete distributions, like the negative binomial. This paper develops approximate one-sided and equal-tailed two-sided tolerance intervals for the Poisson–Lindley distribution. Practical applications of the Poisson–Lindley distribution frequently involve large samples, thus we utilize large-sample Wald confidence intervals in the construction of our tolerance intervals. A coverage study is presented to demonstrate the efficacy of the proposed tolerance intervals. The tolerance intervals are also demonstrated using two real data sets. The R code developed for our discussion is briefly highlighted and included in the tolerance package.     

Analysis of means: a generalized approach using R

Papers on the analysis of means (ANOM) have been circulating in the quality control literature for decades, routinely describing it as a statistical stand-alone concept. Therefore, we clarify that ANOM should rather be regarded as a special case of a much more universal approach known as multiple contrast tests (MCTs). Perceiving ANOM as a grand-mean-type MCT paves the way for implementing it in the open-source software R. We give a brief tutorial on how to exploit R's versatility and introduce the R package ANOM for drawing the familiar decision charts. Beyond that, we illustrate two practical aspects of data analysis with ANOM: firstly, we compare merits and drawbacks of ANOM-type MCTs and ANOVA F-test and assess their respective statistical powers, and secondly, we show that the benefit of using critical values from multivariate t-distributions for ANOM instead of simple Bonferroni quantiles is oftentimes negligible.     

Constructing optimal router bit life sequential experimental designs: New results with a case study

Consider the experiment to improve router bit life as reported in Phadke (1986 Phadke, M. S. 1986. Design optimization case studies. AT &; T Technical Journal 65:51–68.[Crossref], [Web of Science ®] , [Google Scholar]). The goal of the experiment was to increase the life of the router bit before it gets dull, which causes excessive dust formation and consequent expensive cleaning operation to smooth the edges of the boards. A 32-run experimental design was used including seven two-level factors and two four-level factors (cf. Table 1). In this experiment and others, factorial designs with a mixture of two-level and μ( > 2)-level factors may be adopted. Sequential experiments composed of initial experiments and follow-up experiments are widely used to resolve ambiguities involving the aliasing of factorial effects. This article investigates the construction and theoretical properties of optimal designs for sequential experiments with a mixture of α two-level and β μ-level factors for the first time. Constructing optimal design for the router bit life sequential experiment will be discussed for a practical use. From the numerical results, it is found that using a uniform design as the initial experimental design for the router bit life experiment is highly recommended to get an efficient router bit life sequential experimental design. The novelty and significance of the work are evaluated by comparing our results to the existing literature.     

Universally optimal designs under interference models with and without block effects

The concept of neighbor designs was introduced and defined by Rees (1967 Rees, D.H. (1967). Some designs of use in serology. Biometrics 23:779–791.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) along with giving some methods of their construction. Henceforth, many methods of construction of neighbor designs as well as of their generalizations are available in the literature. However, there are only few results on their optimality. Therefore, the purpose of this article is to give an overview of study on this problem. Recent results on optimality of specified neighbor balanced designs under various interference models with block effects are presented and then these results are compared with respective models where block effects are not significant.     

Efficient Mixture Design Fitting Quadratic Surface with Quantile Responses Using First-degree Polynomial

This study considers efficient mixture designs for the approximation of the response surface of a quantile regression model, which is a second degree polynomial, by a first degree polynomial in the proportions of q components. Instead of least squares estimation in the traditional regression analysis, the objective function in quantile regression models is a weighted sum of absolute deviations and the least absolute deviations (LAD) estimation technique should be used (Bassett and Koenker, 1982 Bassett, G., Koenker, R. (1982). An empirical quantile function for linear models with i.i.d. errors. Journal of the American Statistical Association 77:407–415.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]; Koenker and Bassett, 1978 Koenker, R., Bassett, G. (1978). Regression quantiles. Econometrica 46(1):33–50.[Crossref], [Web of Science ®] , [Google Scholar]). Therefore, the standard optimal mixture designs like the D-optimal or A-optimal mixture designs for the least squared estimation are not appropriate. This study explores mixture designs that minimize the bias between the approximated 1st-degree polynomial and a 2nd-degree polynomial response surfaces by the LAD estimation. In contrast to the standard optimal mixture designs for the least squared estimation, the efficient designs might contain elementary centroid design points of degrees higher than two. An example of a portfolio with five assets is given to illustrate the proposed efficient mixture designs in determining the marginal contribution of risks by individual assets in the portfolio.     

Construction of regular 2n41 designs with general minimum lower-order confounding

Mixed-level designs, especially two- and four-level designs, are very useful in practice. In the last two decades, there are quite a few literatures investigating the selection of this kind of optimal designs. Recently, the general minimum lower-order confounding (GMC) criterion (Zhang et al., 2008 Zhang, R.C., Li, P.F., Zhao, S.L., Ai, M.Y. (2008). A general minimum lower order confounding criterion for two-level regular design. Stat. Sin. 18:1689–1705.[Web of Science ®] , [Google Scholar]) gave a new approach for choosing optimal factorials. It is proved that the GMC designs are more powerful than other criteria in the widely practical situations. In this paper, we extend the GMC theory to the mixed-level designs. Under the theory we establish a new criterion for choosing optimal regular two- and four-level designs. Further, a construction method is proposed to obtain all the 2ⁿ4¹ GMC designs with N/4 + 1 ? n + 2 ? 5N/16, where N is the number of runs and n is the number of two-level factors.     

Computer-Generated Change-Over Designs for Correlated Observations

Change-over designs with independently distributed errors in the model have been studied extensively in the literature. Martin and Eccleston (2001 Martin , R. J. , Eccleston , J. A. ( 2001 ). Optimal and near optimal designs for dependent observations . Statist. Applic. 3 : 101 – 116 . [Google Scholar]) gave an algorithm for the generation of efficient change-over designs when the errors are correlated. This article proposes an algorithm for the generation of efficient change-over designs for estimation of direct effects of treatments in the presence of first-order residual effects in the model and when the errors are correlated.     

Construction of some mixed two- and four-level regular designs with GMC criterion

General minimum lower-order confounding (GMC) criterion is to choose optimal designs, which are based on the aliased effect-number pattern (AENP). The AENP and GMC criterion have been developed to form GMC theory. Zhang et al. (2015 Zhang, T.F., Yang, J.F., Li, Z.M., Zhang, R.C. (2015). Construction of regular 2ⁿ4¹ designs with general minimum lower-order confounding. Commun. Stat. - Theory Methods 46:2724–2735.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) introduced GMC 2ⁿ4^m criterion for choosing optimal designs and constructed all GMC 2ⁿ4¹ designs with N/4 + 1 ? n + 2 ? 5N/16. In this article, we analyze the properties of 2ⁿ4¹ designs and construct GMC 2ⁿ4¹ designs with 5N/16 + 1 ? n + 2 < N ? 1, where n and N are, respectively, the numbers of two-level factors and runs. Further, GMC 2ⁿ4¹ designs with 16-run, 32-run are tabulated.