Downloadable articles

Abstract: Joint modelling has gained attention in longitudinal studies incorporating biomarkers and survival data. In the context of chronic diseases, patient evolution is often tracked through multiple assessments, with patient-reported outcomes playing a crucial role. The Beta-Binomial distribution is suggested as a suitable model for these longitudinal variables. However, its integration into joint modelling remains unexplored. This study introduces an estimation procedure for analyzing longitudinal patient-reported outcomes and survival data together. We compare different estimation approaches through simulation experiments, including the proposed model. Furthermore, the methodologies are applied to real data from a follow-up study on chronic obstructive pulmonary disease patients

Keywords: joint modelling, Beta-Binomial regression, patient-reported outcomes, survival analysis

Pages: 155–182

DOI:10.57645/ 20.8080.02.17

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Abstract: In many biomedical studies, recurrent or consecutive events may occur during the follow up of the individuals. This situation can be found, for example, in transplant studies, where there are two consecutive events which give rise to two times of interest subject to a common random right-censoring time, the first one being the elapsed time from acceptance into the transplantation program to transplant, and the second one the time from transplant to death. In this work, we incorporate the information of a continuous covariate into the bivariate distribution of the two gap times of interest and propose a non-parametric method to cope with it. We prove the asymptotic properties of the proposed method and carry out a simulation study to see the performance of this approach. Additionally, we illustrate its use with Stanford heart transplant data and colon cancer data.

Keywords: bivariate distribution, copula function, covariate, serial dependence, random censoring, kernel estimation

Pages: 183–208

DOI:10.57645/ 20.8080.02.18

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Abstract: Multistate models are well developed for continuous and discrete times under a first order Markov assumption. Motivated by a cohort of COVID-19 patients, a multistate model was designed based on 14 transitions among 7 states of a patient. Since a preliminary analysis showed that the first-order Markov condition was not met for some transitions, we have developed a second-order Markov model where the future evolution not only depends on the state at the current time but also on the state at the preceding time. Under a discrete time analysis, assuming homogeneity and that past information is restricted to two consecutive times, we expanded the transition probability matrix and proposed an extension of the Chapman-Kolmogorov equations. We propose two estimators for the second-order transition probabilities and illustrate them within the cohort of COVID-19 patients.

Keywords: multistate models, non-Markov, COVID-19

Pages: 209–234

DOI:10.57645/ 20.8080.02.19

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Abstract: Prediction of a traffic accident cost is one of the major problems in motor insurance. To identify the factors that influence costs is one of the main challenges of actuarial modelling. Telematics data about individual driving patterns could help calculating the expected claim severity in motor insurance. We propose using single-index models to assess the marginal effects of covariates on the claim severity conditional distribution. Thus, drivers with a claim cost distribution that has a long tail can be identified. These are risky drivers, who should pay a higher insurance premium and for whom preventative actions can be designed. A new kernel approach to estimate the covariance matrix of coefficients’ estimator is outlined. Its statistical properties are described and an application to an innovative data set containing information on driving styles is presented. The method provides good results when the response variable is skewed.

Keywords: covariance matrix of estimator, kernel estimator, marginal effects, telematics covariates, right-skewed cost variable

Pages: 235–258

DOI:10.57645/ 20.8080.02.20

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Abstract: Gaussian random fields with Matérn covariance functions are popular models in spatial statistics and machine learning. In this work, we develop a spatio-temporal extension of the Gaussian Matérn fields formulated as solutions to a stochastic partial differential equation. The spatially stationary subset of the models have marginal spatial Matérn covariances, and the model also extends to Whittle-Matérn fields on curved manifolds, and to more general non-stationary fields. In addition to the parameters of the spatial dependence (variance, smoothness, and practical correlation range) it additionally has parameters controlling the practical correlation range in time, the smoothness in time, and the type of non-separability of the spatio-temporal covariance. Through the separability parameter, the model also allows for separable covariance functions. We provide a sparse representation based on a finite element approximation, that is well suited for statistical inference and which is implemented in the R-INLA software. The flexibility of the model is illustrated in an application to spatio-temporal modeling of global temperature data.

Keywords: Stochastic partial differential equations, diffusion, Gaussian fields, non-separable space-time models, INLA, finite element methods

Pages: 3–66

DOI:10.57645/20.8080.02.13

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Abstract: In complex survey data, each sampled observation has assigned a sampling weight, indicating the number of units that it represents in the population. Whether sampling weights should or not be considered in the estimation process of model parameters is a question that still continues to generate much discussion among researchers in different fields. We aim to contribute to this debate by means of a real data based simulation study in the framework of logistic regression models. In order to study their performance, three methods have been considered for estimating the coefficients of the logistic regression model: a) the unweighted model, b) the weighted model, and c) the unweighted mixed model. The results suggest the use of the weighted logistic regression model is superior, showing the importance of using sampling weights in the estimation of the model parameters.

Keywords: complex survey data, sampling weights, logistic regression, estimation of model parameters, real data based simulation study

Pages: 67–92

DOI:10.57645/20.8080.02.14

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Abstract: In this paper we review some methods proposed in the literature for combining a nonprobability and a probability sample with the purpose of obtaining an estimator with a smaller bias and standard error than the estimators that can be obtained using only the probability sample. We propose a new methodology based on the kernel weighting method. We discuss the properties of the new estimator when there is only selection bias and when there are both coverage and selection biases. We perform an extensive simulation study to better understand the behaviour of the proposed estimator.

Keywords: Kernel weighting, survey sampling, non-probability sample, coverage bias, selection bias

Pages: 93–124

DOI:10.57645/20.8080.02.15

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Abstract: Household composition reveals vital aspects of the socioeconomic situation and major changes in developed countries for decision-making and mapping the distribution of single-person households is highly relevant and useful. Driven by the Spanish Household Budget Survey data, we propose a new statistical methodology for small area estimation of proportions and total counts of single-person households. Estimation domains are defined as crosses of province, sex and age group of the main breadwinner of the household. Predictors are based on area-level zero-inflated Poisson mixed models. Model parameters are estimated by maximum likelihood and mean squared errors by parametric bootstrap. Several simulation experiments are carried out to empirically investigate the properties of these estimators and predictors. Finally, the paper concludes with an application to real data from 2016.

Keywords: Small area estimation, zero-inflated Poisson mixed model, area-level data, Household Budget Survey, single-person household

Pages: 125–152

DOI:10.57645/20.8080.02.16

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Abstract: The year 2022 marked 40 years since Aitchison published the article “The statistical analysis of compositional data”. It is considered to be the foundation of contemporary compositional data analysis. It is time to review what has been accomplished in the feld and what needs to be addressed. Astonishingly enough, many aspects seen as challenging in 1982 continue to lead to fruitful scholarly work. We commence with a bibliometric study and continue with some hot topics such as multi-way compositions, compositional regression models, dealing with zero values, non-logratio transformations, new application felds, and a number of current loose ends. Finally, a tentative future research agenda is outlined.

Keywords: Compositional data (CoDa), logratios, Aitchison geometry, multi-way compositions, zero replacement, compositional regression

Pages: 207–228

DOI: 10.57645/20.8080.02.6

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Abstract: Research in compositional data analysis was motivated by spurious (Pearson) correlation. Spurious results are due to semantic incoherence, but the question of ways to relate parts in a statistically consistent way remains open. To solve this problem, we first define a coherent system of functions with respect to a subcomposition and analyze the space of parts. This leads to understanding why measures like covariance and correlation depend on the subcomposition considered, while measures like the distance between parts are independent of the same. It allows the definition of a novel index of proportionality between parts.

Keywords: Compositional data analysis, Aitchison geometry, simplex, compositional parts, proportionality, dominance, correlation

Pages: 229–244

DOI: 10.57645/20.8080.02.7

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Abstract: We propose an estimation procedure for covariation in wide compositional data sets. For compositions, widely-used logratio variables are interdependent due to a common reference. Logratio uncorrelated compositions are linearly independent before the unitsum constraint is imposed. We show how they are used to construct bespoke shrinkage targets for logratio covariance matrices and test a simple procedure for partial correlation estimates on both a simulated and a single-cell gene expression data set. For the underlying counts, different zero imputations are evaluated. The partial correlation induced by the closure is derived analytically. Data and code are available from GitHub.

Keywords: Compositional covariance structure, logratio analysis, partial correlation, James-Stein shrinkage

Pages: 245–268

DOI: 10.57645/20.8080.02.8

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Abstract: Compositional data, multivariate observations carrying relative information, are popularly expressed in additive logratio coordinates which are easily interpretable as they use one of the components as ratioing part to produce pairwise logratios. These coordinates are however oblique and they lead to issues when applying multivariate methods on them, including widely-used techniques such as principal component analysis and linear regression. In this paper we propose a way to redefine alr coordinates with respect to an orthonormal system and we also extend the idea to the case of compositional tables. The new approach is demonstrated in an application to movement behavior data.

Keywords: Compositional data, compositional tables, regression, principal component analysis

Pages: 269–294

DOI: 10.57645/20.8080.02.9

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Abstract: The process of supervised classification when the data set consists of probability density functions is studied. Due to the relative information contained in densities, it is necessary to convert the functional data analysis methods into an appropriate framework, here represented by the Bayes spaces. This work develops Bayes space counterparts to a set of commonly used functional methods with a focus on classification. Hereby, a clear guideline is provided on how some classification approaches can be adapted for the case of densities. Comparison of the methods is based on simulation studies and real-world applications, reflecting their respective strengths and weaknesses.

Keywords: Probability density functions, Bayes spaces, classification, functional data analysis

Pages: 295–322

DOI: 10.57645/20.8080.02.10

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Abstract: Many of the most popular statistical techniques incorporate optimisation problems in their inner workings. A convex optimisation problem is defined as the problem of minimising a convex function over a convex set. When traditional methods are applied to compositional data, misleading and incoherent results could be obtained. In this paper, we fill a gap in the specialised literature by introducing and rigorously defining novel concepts of convex optimisation for compositional data according to the Aitchison geometry. Convex sets and convex functions on the simplex are defined and illustrated.

Keywords: Compositional data, logratio, simplex, proportion, function, convexity, optimisation

Pages: 323–344

DOI: 10.57645/20.8080.02.11

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Abstract: In addition to approaches based on a number of physical and chemical analyses, statistical methods have been commonly used for determining the modes of occurrence of elements in coal. The Pearson correlation coefficient of element concentrations vs. ash yields is the simplest method that has been widely used. Concentrations of elements in coal are usually reported on two bases: whole-coal and ash bases. Coal compositional data on whole-coal basis can be converted back to ash basis. However, in many cases, the correlation between corresponding pairs of elements in coal is inconsistent when reported on whole-coal versus ash bases. Therefore, traditional statistical methods, such as correlation analysis, based on whole-coal and ash bases can sometimes lead to misleading or confusing results. Previous investigations have suggested using logratio variance or related parameters (i.e., stability) to examine these data, as they provide consistent results regardless of the sample basis. However, logratio variance based approaches are unable to distinguish the inverse relationships between parts. To provide more clarity on the relationships between parts, weighted symmetric pivot coordinates are used to analyze the correlation between elements in coal on whole-coal basis and ash basis. To illustrate this approach, 106 late Paleozoic coal samples from the Datanhao and Adaohai coal mines, Daqingshan Coalfeld, northern China, are used for performance evaluation. Experimental results show that the weight symmetric pivots method is more effective than the stability method in predicting the modes of occurrence of elements in coal for these samples, providing deeper insight than logratio variance based approaches.

Keywords: whole-coal basis, ash basis, correlation, WSPC method

Pages: 345–362

DOI: 10.57645/20.8080.02.12

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Abstract: Rapid advancements in new technologies, especially information and communication technologies (ICT), have significantly increased the number of sensors that capture data, namely those embedded in mobile devices. This wealth of data has garnered particular interest in analyzing transport systems, with some researchers arguing that the data alone are sufficient enough to render transport models unnecessary. However, this paper takes a contrary position and holds that models and data are not mutually exclusive but rather depend upon each other. Transport models are built upon established families of optimization and simulation approaches, and their development aligns with the scientific principles of operations research, which involves acquiring knowledge to derive modeling hypotheses. We provide an overview of these modeling principles and their application to transport systems, presenting numerous models that vary according to study objectives and corresponding modeling hypotheses. The data required for building, calibrating, and validating selected models are discussed, along with examples of using data analytics techniques to collect and handle the data supplied by ICT applications. The paper concludes with some comments on current and future trends.

Keywords: Optimization, Simulation, Data Analytics, Traffic Assignment, Traffic Simulation

Pages: 3–80

DOI:10.57645/20.8080.02.1

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Abstract: More than ever, healthcare systems can use data, predictive models, and intelligent algorithms to optimize their operations and the service they provide. This paper reviews the existing literature regarding the use of data science/analytics methods and artificial intelligence algorithms in healthcare. The paper also discusses how healthcare organizations can benefit from these tools to efficiently deal with a myriad of new possibilities and strategies. Examples of real applications are discussed to illustrate the potential of these methods. Finally, the paper highlights the main challenges regarding the use of these methods in healthcare, as well as some open research lines.

Keywords: e-health, data science, analytics, artificial intelligence, machine learning.

Pages: 81–128

DOI:10.57645/20.8080.02.2

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Abstract: Data envelopment analysis is not typically used for bankruptcy prediction. However, this paper shows that a correctly set up a model for this approach can be very useful in that context. A superefficiency model was applied to classify bankrupt and actively manufactured companies in the European Union. To select an appropriate threshold, the Youden index and the distance from the corner were used in addition to the total accuracy. The results indicate that selecting a suitable threshold improves specificity visibly with only a small reduction in the total accuracy. The thresholds of the best models appear to be robust enough for predictions in different time and economic sectors.

Keywords: Bankruptcy prediction, data envelopment analysis, ROC curve, threshold optimization, validation

Pages: 129–150

DOI:10.57645/20.8080.02.3

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Abstract: This paper assesses the two current major alternatives for ecological inference, based on a multinomial-Dirichlet Bayesian model and on mathematical programming. Their performance is evaluated in a database made up of almost 2000 real datasets for which the actual cross-distributions are known. The analysis reveals both approaches as complementarity, each one of them performing better in a different area of the simplex space, although with Bayesian solutions deteriorating when the amount of information is scarce. After offering some guidelines regarding the appropriate contexts for employing each one of the algorithms, we conclude with some ideas for exploiting their complementarities.

Keywords: Ecological inference; Voter transitions; US voting rights; two-way contingency tables; ei.MD.bayes; lphom; R-packages

Pages: 151–186

DOI:10.57645/20.8080.02.4

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Abstract: In the presence of a continuous response test/biomarker, it is often necessary to identify a cut-off point value to aid binary classification between diseased and non-diseased subjects. The symmetry-point approach which maximizes simultaneously both types of correct classification is one way to determine an optimal cut-off point. In this article, we study methods for constructing confidence intervals independently for the symmetry point and its corresponding sensitivity, as well as respective joint nonparametric confidence regions. We illustrate using data on the generation of antibodies elicited two weeks post-injection after the second dose of the Pfizer/BioNTech vaccine in adult healthcare workers.

Keywords: Empirical likelihood function, Empirical chi-square function, Box-Cox transformation, Confidence regions, Sensitivity, Specificity

Pages: 187–204

DOI:10.57645/20.8080.02.5

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Abstract: In this work we study the problem of modelling and forecasting the dynamics of the influenza virus in Brazil at a given month, from data on reported cases and genetic diversity collected from previous months, in other locations. Granger causality is employed as a tool to assess possible predictive relationships between covariates. For modelling and forecasting purposes, a time series regression approach is applied considering lagged information regarding reported cases and genetic diversity in other regions. Three different models are analysed, including stepwise time series regression and LASSO.

Keywords: Flu, time series regression, variable selection, genetic diversity, Granger causality

Pages: 161–188

DOI:10.2436/20.8080.02.122

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Abstract: The Split-Then-Combine approach has previously been used to generate the weights of forecasts in a combination in the Euclidean space. This paper extends this approach to combine forecasts inside the simplex space, the sample space of positive weights adding up to one. As it turns out, the simplicial statistic given by the sample centre compares favourably against the fixed-weight, average forecast. Besides, we also develop a Combination-After-Selection method to get rid of redundant forecasters. We apply these approaches to make out-of-sample one-step ahead combinations and subcombinations of forecasts for several economic variables. This methodology is particularly useful when the sample size is smaller than the number of forecasts, a case where other methods (e.g., ordinary least squares or principal component analysis) are not applicable.

Keywords: Aitchison geometry, Combination-After-Selection, Dimensionality problem, Simplex, Split-Then-Combine

Pages: 189–216

DOI:10.2436/20.8080.02.123

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Abstract: This paper recasts the problem of missing values in the covariates of a regression model as a latent Gaussian Markov random field (GMRF) model in a fully Bayesian framework. The proposed approach is based on the definition of the covariate imputation sub-model as a latent effect with a GMRF structure. This formulation works for continuous covariates but for categorical covariates a typical multiple imputation approach is employed. Both techniques can be easily combined for the case in which continuous and categorical variables have missing values. The resulting Bayesian hierarchical model naturally fts within the integrated nested Laplace approximation (INLA) framework, which is used for model fitting. Hence, this work fills an important gap in the INLA methodology as it allows to treat models with missing values in the covariates. As in any other fully Bayesian framework, by relying on INLA for model fitting it is possible to formulate a joint model for the data, the imputed covariates and their missingness mechanism. In this way, it is possible to tackle the more general problem of assessing the missingness mechanism by conducting a sensitivity analysis on the different alternatives to model the non-observed covariates. Finally, the proposed approach is illustrated in two examples on modeling health risk factors and disease mapping.

Keywords: Imputation, missing values, GMRF, INLA, sensitivity analysis

Pages: 217–244

DOI:10.2436/20.8080.02.124

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Abstract: To generate a circular distribution, we use the alternate-wrapping technique (unlike the usual wrapping), by wrapping in the alternate directions, after each single-wrapping. The resulting distribution is called alternate-wrapped distribution. Some general properties and distinctions between the two wrapping schemes are indicated. As an illustration, alternate-wrapped-exponential distribution and alternate-wrapped-normal distribution are considered. The moment and maximum likelihood estimator of the parameters of alternative-wrapped-exponential distribution are obtained and their performance is evaluated using simulation. Maximum likelihood estimators are obtained for the parameters of the alternate-wrapped-normal distribution and simulation study is conducted, and this distribution is used to analyse a real-life data set and is compared with the wrapped normal distribution.

Keywords: Akaike information criterion, Bayesian information criterion, circular distribution, exponential distribution, trigonometric moments, wrapped normal distribution

Pages: 245–262

DOI:10.2436/20.8080.02.125

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Abstract: The Hawkes process is a self-exciting Poisson point process, characterised by a conditional intensity function. Since its introduction fifty years ago, it has been the subject of numerous research directions and continues to inspire new methodological and theoretical developments as well as new applications. This paper marks half a century of interest in Hawkes processes by presenting a snapshot of four state-of-the-art research directions, categorised as frequentist and Bayesian methods, other modelling approaches and notable theoretical developments. A particular focus is on nonparametric approaches, with advances in kernel estimation and computational efficiencies. A survey of real world applications is provided to illustrate the breadth of application of this remarkable approach.

Keywords: Hawkes process, point process, nonparametric

Pages: 3–38

DOI:10.2436/20.8080.02.116

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Abstract: At least one unusual event appears in some count datasets. It will lead to a more concentrated (or dispersed) distribution than the Poisson, gamma, Weibull, Conway-Maxwell-Poisson (CMP), and Faddy (1997) models can accommodate. These well-known count models are based on the monotonic rates of interarrival times between successive events. Under the assumption of non-monotonic rates and independent exponential interarrival times, a new class of parametric models for unusual-event (UE) count data is proposed. These models are applied to two empirical applications, the number of births and the number of bids, and yield considerably better results to the above well-known count models.

Keywords: Poisson count model, Gamma count model, Weibull count model, Conway-Maxwell-Poisson count model, Faddy count model

Pages: 39–66

DOI:10.2436/20.8080.02.117

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Abstract: In this paper, we consider the problem of estimating the finite population cumulative distribution function (CDF) in a complex survey sampling, which includes two-stage and three-stage cluster sampling schemes with and without stratification. We propose two new families of CDF estimators using supplementary information on a single auxiliary variable. Explicit mathematical expressions of the biases and mean squared errors of the proposed CDF estimators are developed under the first order of the approximation. Real datasets are also considered to support the proposed theory.

Keywords: Ratio estimator, exponential ratio estimator, auxiliary information, stratification, two-stage and three-stage cluster sampling, relative efficiencies, bias, mean-squared error

Pages: 67–94

DOI:10.2436/20.8080.02.118

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Abstract: In this article we consider an extension of the penalized splines approach in the context of censored semiparametric modelling using Kaplan-Meier weights to take into account the effect of censorship. We proposed an estimation method and develop statistical inferences in the model. Using various simulation studies we show that the performance of the method is quite satisfactory. A real data set is used to illustrate that the proposed method is comparable to parametric approaches when assuming a probability distribution of the response variable and/or the functional form. However, our proposal does not need these assumptions since it avoids model specification problems

Keywords: Censored data, Kaplan-Meier weights, P-splines, semiparametric models, survival analysis

Pages: 95– 114

DOI:10.2436/20.8080.02.119

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Abstract: Precision medicine allows the extraction of information from complex datasets to facilitate clinical decision-making at the individual level. Topological Data Analysis (TDA) offers promising tools that complement current analytical methods in precision medicine studies. We introduce the fundamental concepts of the TDA corpus (the simplicial complex, the Mapper graph, the persistence diagram and persistence landscape). We show how these can be used to enhance the prediction of clinical outcomes and to identify novel subpopulations of interest, particularly applied to understand remission of depression in data from the GENDEP clinical trial.

Keywords: Precision medicine, data shape, topology, topological data analysis, persistence diagram, Mapper, persistence landscapes, machine learning

Pages: 115– 136

DOI:10.2436/20.8080.02.120

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Abstract: In the context of logistic regression models, a cut-off point is usually selected to dichotomize the estimated predicted probabilities based on the model. The techniques proposed to estimate optimal cut-off points in the literature, are commonly developed to be applied in simple random samples and their applicability to complex sampling designs could be limited. Therefore, in this work we propose a methodology to incorporate sampling weights in the estimation process of the optimal cut-off points, and we evaluate its performance using a real data-based simulation study. The results suggest the convenience of considering sampling weights for estimating optimal cut-off points.

Keywords: Optimal cut-off points, complex survey data, sampling weights

Pages: 137– 158

DOI:10.2436/20.8080.02.121

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Abstract: In this paper, a general nonparametric estimator of the probability of default is proposed and studied. It is derived from an estimator of the conditional survival function for censored data obtained with a double smoothing, on the covariate and on the variable of interest. An empirical study, based on modified real data, illustrates its practical application and a simulation study shows the performance of the proposed estimator and compares its behaviour with smoothed estimators only in the covariate. Asymptotic expressions for the bias and the variance of the probability of default estimator are found and asymptotic normality is proved.

Keywords: Censored data, kernel method, probability of default, risk analysis, survival analysis

Pages: 93–120

DOI:10.2436/20.8080.02.111

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Abstract: Due to the large amount of accidents negatively affecting the wellbeing of the survivors and their families, a substantial amount of research is conducted to determine the causes of road accidents. This type of data come in the form of non-negative integers and may be modelled using the Poisson regression model. Unfortunately, the commonly used maximum likelihood estimator is unstable when the explanatory variables of the Poisson regression model are highly correlated. Therefore, this paper proposes a new almost unbiased estimator which reduces the instability of the maximum likelihood estimator and at the same time produce smaller mean squared error. We study the statistical properties of the proposed estimator and a simulation study has been conducted to compare the performance of the estimators in the smaller mean squared error sense. Finally, Swedish traffic fatality data are analyzed to show the benefit of the proposed method.

Keywords: Applied traffic modeling, Maximum likelihood estimator, mean squared error matrix, Poisson regression, simulation study, traffic fatality

Pages: 121–142

DOI:10.2436/20.8080.02.112

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Abstract: Laparoscopy is an operation carried out in the abdomen through small incisions with visual control by a camera. This technique needs the abdomen to be insufflated with carbon dioxide to obtain a working space for surgical instruments’ manipulation. Identifying the critical point at which insufflation should be limited is crucial to maximizing surgical working space and minimizing injurious effects. A Bayesian nonlinear growth mixed-effects model for the relationship between the insufflation pressure and the intra–abdominal volume generated is discussed as well as its plausibility to represent the data.

Keywords: Intra-abdominal pressure, logistic growth function, Markov chain, Monte Carlo methods, random effects

Pages: 143–162

DOI:10.2436/20.8080.02.113

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Abstract: Bilinear regression models involving a nonlinear interaction term are applied in many fields (e.g., Goodman’s RC model, Lee-Carter mortality model or CAPM financial model). In many of these contexts data often exhibit extreme values. We propose the use of bilinear models to estimate the median of the conditional distribution in the presence of extreme values. The aim of this paper is to provide alternative methods to estimate median bilinear models. A calibration strategy based on an iterative estimation process of a sequence of median linear regression is developed. Mean and median bilinear models are compared in two applications with extreme observations. The first application deals with simulated data. The second application refers to Spanish mortality data involving years with atypical high mortality (Spanish flu, civil war and HIV/AIDS). The performance of the median bilinear model was superior to that of the mean bilinear model. Median bilinear models may be a good alternative to mean bilinear models in the presence of extreme values when the centre of the conditional distribution is of interest.

Keywords: Outliers, quantile regression, single factor models, nonlinear, multiplicative

Pages: 163–180

DOI:10.2436/20.8080.02.114

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Abstract: In this paper we introduce an extension of the power Maxwell distribution. We also discuss a reparametrized version of this model applied to quantile regression. Some properties of the model and estimation based on the maximum likelihood estimation method are studied. We also present a simulation study to assess the performance of estimators in such finite samples, and two applications to real data sets to illustrate the model. .

Keywords: Maxwell distribution, exponentiated distributions, maximum likelihood, quantile regression

Pages: 181–200

DOI:10.2436/20.8080.02.115

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Abstract: This article is a tribute to the contributions of Florence Nightingale to statistics and statistical graphics on her bicentennial. We start with her most famous “rose” diagram and describe how she came to this graphic, designed to influence medical practice in the British army. But this study takes us backward in time to consider where and when the ideas of radial diagrams arose, why they were useful, and why we call these her “radiant diagrams.”

Keywords: Data visualization, polar area diagram, radial diagram, nursing, sanitation

Pages: 3–18

DOI: 10.2436/20.8080.02.106

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Abstract: We construct credible intervals to estimate the mean organism (zooplankton and phytoplankton) concentration in ballast water via a decision-theoretic approach. To obtain the required optimal sample size, we use a total cost minimization criterion defined as the sum of the sampling cost and the Bayes risk either under a Poisson or a negative binomial model for organism counts, both with a gamma prior distribution. Such credible intervals may be employed to verify whether the ballast water discharged from a ship is in compliance with international standards. We also conduct a simulation study to evaluate the credible interval lengths associated with the proposed optimal sample sizes.

Keywords: Optimal sample size, Bayes risk, Poisson distribution, negative binomial distribution

Pages: 19–32

DOI: 10.2436/20.8080.02.107

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Abstract: Dating is a key element for archaeologists. We propose a Bayesian approach to provide chronology to sites that have neither radiocarbon dating nor clear stratigraphy and whose only information comes from lithic arrowheads. This classifier is based on the Dirichlet-multinomial inferential process and posterior predictive distributions. The procedure is applied to predict the period of a set of undated sites located in the east of the Iberian Peninsula during the 4th and 3rd millennium cal BC.

Keywords: Bifacial flint arrowheads, chronological model, Dirichlet-multinomial process, posterior predictive distribution, radiocarbon dating

Pages: 33–46

DOI: 10.2436/20.8080.02.108

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Abstract: In regression, the quality of estimators is known to be very sensitive to the presence of spurious variables and outliers. Unfortunately, this is a frequent situation when dealing with real data. To handle outlier proneness and achieve variable selection, we propose a robust method performing the outright rejection of discordant observations together with the selection of relevant variables. A natural way to define the corresponding optimization problem is to use the ℓ0 norm and recast it as a mixed integer optimization problem. To retrieve this global solution more efficiently, we suggest the use of additional constraints as well as a clever initialization. To this end, an efficient and scalable non-convex proximal alternate algorithm is introduced. An empirical comparison between the ℓ0 norm approach and its ℓ1 relaxation is presented as well. Results on both synthetic and real data sets provided that the mixed integer programming approach and its discrete first order warm start provide high quality solutions.

Keywords: Robust optimization, statistical learning, linear regression, variable selection, outlier detection, mixed integer programming

Pages: 47–66

DOI:10.2436/20.8080.02.109

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Abstract: This paper proposes a generalized framework to analyze spatial count data under a unilateral regular lattice structure based on thinning type models. We start from the simple spatial integer-valued auto-regressive model of order 1. We extend this model in certain directions. First, we consider various distributions as choices for the innovation distribution to allow for additional overdispersion. Second, we allow for use of covariate information, leading to a non-stationary model. Finally, we derive and use other models related to this simple one by considering simplification on the existing model. Inference is based on conditional maximum likelihood approach. We provide simulation results under different scenarios to understand the behaviour of the conditional maximum likelihood. A real data application is also provided. Remarks on how the results extend to other families of models are also given.

Keywords: Unilateral, spatial, regular, lattice, thinning

Pages: 67–90

DOI: 10.2436/20.8080.02.110

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Abstract: In order to apply group sequential methods for interim analysis for early stopping in clinical trials, the joint distribution of test statistics over time has to be known. Often the distribution is multivariate normal or asymptotically so, and an application of group sequential methods requires multivariate integration to determine the group sequential boundaries. However, if the increments between successive test statistics are independent, the multivariate integration reduces to a univariate integration involving simple recursion based on convolution. This allows application of standard group sequential methods. In this paper we review group sequential methods and the development that established independent increments in test statistics for the primary outcomes of longitudinal or failure time data.

Keywords: Failure time data, interim analysis, longitudinal data, clinical trials, repeated significance tests, sequential methods

Pages: 223–264

DOI: 10.2436/20.8080.02.101

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Abstract: A new discrete two-parameter distribution is introduced by discretizing a generalized half-normal distribution. The model is useful for fitting overdispersed as well as underdispersed data. The failure function can be decreasing, bathtub shaped or increasing. A reparameterization of the distribution is introduced for use in a regression model based on the median. The behaviour of the maximum likelihood estimates is studied numerically, showing good performance in finite samples. Three real data set applications reveal that the new model can provide a better explanation than some other competitors.

Keywords: Discretizing, generalized half-normal distribution, failure function, health, quantile regression, stochastic order

Pages: 265–284

DOI: 10.2436/20.8080.02.102

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Abstract: A major operational task in city logistics is related to waste collection. Due to large problem sizes and numerous constraints, the optimization of real-life waste collection problems on a daily basis requires the use of metaheuristic solving frameworks to generate near-optimal collection routes in low computation times. This paper presents a simheuristic algorithm for the time-dependent waste collection problem with stochastic travel times. By combining Monte Carlo simulation with a biased randomized iterated local search metaheuristic, time-varying and stochastic travel speeds between different network nodes are accounted for. The algorithm is tested using real instances in a medium-sized city in Spain.

Keywords: Waste collection management, vehicle routing problem, stochastic optimization, simheuristics, biased randomization, case study

Pages: 285–310

DOI: 10.2436/20.8080.02.103

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Abstract: Many decision-making processes in our society involve NP-hard optimization problems. The largescale, dynamism, and uncertainty of these problems constrain the potential use of stand-alone optimization methods. The same applies for isolated simulation models, which do not have the potential to find optimal solutions in a combinatorial environment. This paper discusses the utilization of modelling and solving approaches based on the integration of simulation with metaheuristics. These ‘simheuristic’ algorithms, which constitute a natural extension of both metaheuristics and simulation techniques, should be used as a ‘first-resort’ method when addressing large-scale and NP-hard optimization problems under uncertainty –which is a frequent case in real-life applications. We outline the benefits and limitations of simheuristic algorithms, provide numerical experiments that validate our arguments, review some recent publications, and outline the best practices to consider during their design and implementation stages.

Keywords: Simulation, metaheuristics, combinatorial optimization, simheuristics

Pages: 311–334

DOI: 10.2436/20.8080.02.104

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Abstract: Mixed Poisson models are most relevant to the analysis of longitudinal count data in various disciplines. A conventional specification of such models relies on the normality of unobserved heterogeneity effects. In practice, such an assumption may be invalid, and non-normal cases are appealing. In this paper, we propose a modelling strategy by allowing the vector of effects to follow the multivariate skew-normal distribution. It can produce dependence between the correlated longitudinal counts by imposing several structures of mixing priors. In a Bayesian setting, the estimation process proceeds by sampling variants from the posterior distributions. We highlight the usefulness of our approach by conducting a simulation study and analysing two real-life data sets taken from the German Socioeconomic Panel and the US Centers for Disease Control and Prevention. By a comparative study, we indicate that the new approach can produce more reliable results compared to traditional mixed models to fit correlated count data.

Keywords: Bayesian computation, correlated random effects, hierarchical representation, longitudinal data, multivariate skew-normal distribution, over-dispersion

Pages: 335–356

DOI: 10.2436/20.8080.02.105

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Abstract: Average incomes and poverty proportions are additive parameters obtained as averages of a given function of an income variable. As the variable income has an asymmetric distribution, it is not properly modelled via normal distributions. When dealing with this type of variable, a first option is to apply transformations that approximate normality. A second option is to use nonsymmetric distributions from the exponential family. This paper proposes unit-level generalized linear mixed models for modelling asymmetric positive variables and for deriving three types of predictors of small area additive parameters, called empirical best, marginal and plug-in. The parameters of the introduced model are estimated by applying the maximum likelihood method to the Laplace approximation of the likelihood. The mean squared errors of the predictors are estimated by parametric bootstrap. The introduced methodology is applied and illustrated under unit-level gamma mixed models. Some simulation experiments are carried out to study the behaviour of the fitting algorithm, the small area predictors and the bootstrap estimator of the mean squared errors. By using data of the Spanish living condition survey of 2013, an application to the estimation of average incomes and poverty proportions in counties of the region of Valencia is given.

Keywords: Average income, poverty proportion, generalized linear mixed models, empirical best predictor, mean squared error, bootstrap

Pages: 3–38

DOI: 10.2436/20.8080.02.93

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Abstract: Archetypal analysis is an exploratory tool that explains a set of observations as mixtures of pure (extreme) patterns. If the patterns are actual observations of the sample, we refer to them as archetypoids. For the first time, we propose to use archetypoid analysis for binary observations. This tool can contribute to the understanding of a binary data set, as in the multivariate case. We illustrate the advantages of the proposed methodology in a simulation study and two applications, one exploring objects (rows) and the other exploring items (columns). One is related to determining student skill set profiles and the other to describing item response functions.

Keywords: Dichotomous item test, archetypal analysis, functional data analysis, item response theory, skill profile

Pages: 39–66

DOI: 10.2436/20.8080.02.94

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Abstract: One of the main challenges researchers face is to identify the most relevant features in a prediction model. As a consequence, many regularized methods seeking sparsity have flourished. Although sparse, their solutions may not be interpretable in the presence of spurious coefficients and correlated features. In this paper we aim to enhance interpretability in linear regression in presence of multicollinearity by: (i) forcing the sign of the estimated coefficients to be consistent with the sign of the correlations between predictors, and (ii) avoiding spurious coefficients so that only significant features are represented in the model. This will be addressed by modelling constraints and adding them to an optimization problem expressing some estimation procedure such as ordinary least squares or the lasso. The so-obtained constrained regression models will become Mixed Integer Quadratic Problems. The numerical experiments carried out on real and simulated datasets show that tightening the search space of some standard linear regression models by adding the constraints modelling (i) and/or (ii) help to improve the sparsity and interpretability of the solutions with competitive predictive quality.

Keywords: Linear regression, Multicollinearity, Sparsity, Cardinality constraint, Mixed Integer Non Linear Programming

Pages: 67–98

DOI: 10.2436/20.8080.02.95

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Abstract: The logratio-normal-multinomial distribution is a count data model resulting from compounding a multinomial distribution for the counts with a multivariate logratio-normal distribution for the multinomial event probabilities. However, the logratio-normal-multinomial probability mass function does not admit a closed form expression and, consequently, numerical approximation is required for parameter estimation. In this work, different estimation approaches are introduced and evaluated. We concluded that estimation based on a quasi-Monte Carlo Expectation-Maximisation algorithm provides the best overall results. Building on this, the performances of the Dirichlet-multinomial and logratio-normal-multinomial models are compared through a number of examples using simulated and real count data.

Keywords: Count data, Compound probability distribution, Dirichlet Multinomial, Logratio coordinates, Monte Carlo method, Simplex

Pages: 99–126

DOI: 10.2436/20.8080.02.96

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Abstract: In this paper, we obtain the Bartlett factor for the likelihood ratio statistic and the Bartlett-type correction factor for the score and gradient test in censored data from a Weibull distribution. The expressions derived are simple, we only have to define a few matrices. We conduct an extensive Monte Carlo study to evaluate the performance of the corrected tests in small sample sizes and we show how they improve the original versions. Finally, we apply the results to a real data set with a small sample size illustrating that conclusions about the regressors could be different if corrections were not applied to the three mentioned classical statistics for the hypothesis test.

Keywords: Bartlett correction, censored data, Weibull distribution, chi-squared distribution, maximum likelihood estimates, type I and II censoring

Pages: 127–140

DOI: 10.2436/20.8080.02.97

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Abstract: The introduction of Electric Vehicles (EVs) in modern fleets facilitates green road transportation. However, the driving ranges of EVs are limited by the duration of their batteries, which arise new operational challenges. Hybrid fleets of gas and EVs might be heterogeneous both in loading capacities as well as in driving-range capabilities,whichmakes the design of efficient routing plans a difficult task. In this paper, we propose a newMulti-Round IteratedGreedy (MRIG) metaheuristic to solve the Heterogeneous Vehicle Routing Problem with Multiple Driving ranges and loading capacities (HeVRPMD). MRIG uses a successive approximations method to offer the decision maker a set of alternative fleet configurations,with different distance-based costs and green levels. The numerical experiments show that MRIG is able to outperform previous works dealing with the homogeneous version of the problem, which assumes the same loading capacity for all vehicles in the fleet. The numerical experiments also confirm that the proposed MRIG approach extends previous works by solving a more realistic HeVRPMD and provides the decision-maker with fleets with higher green levels.

Keywords: Vehicle Routing Problem, Electric Vehicles, Heterogeneous Fleet, Multiple Driving Ranges. Iterated Greedy heuristic, Successive Approximations Method

Pages: 141–170

DOI: 10.2436/20.8080.02.98

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Abstract: An important problem in Statistics is the study of longitudinal data taking into account the effect of other explanatory variables, such as treatments and time and, simultaneously, the incorporation into the model of the time dependence between observations on the same individual. The latter is specially relevant in the case of nonstationary correlations, and nonconstant variances for the different time point at which measurements are taken. Antedependence models constitute a well known commonly used set of models that can accommodate this behaviour. These covariance models can include too many parameters and estimation can be a complicated optimization problem requiring the use of complex algorithms and programming. In this paper, a new Bayesian approach to analyse longitudinal data within the context of antedependence models is proposed. This innovative approach takes into account the possibility of having nonstationary correlations and variances, and proposes a robust and computationally efficient estimation method for this type of data. We consider the joint modelling of the mean and covariance structures for the general antedependence model, estimating their parameters in a longitudinal data context. Our Bayesian approach is based on a generalization of the Gibbs sampling and Metropolis-Hastings by blocks algorithm, properly adapted to the antedependence models longitudinal data settings. Finally, we illustrate the proposed methodology by analysing several examples where antedependence models have been shown to be useful: the small mice, the speech recognition and the race data sets.

Keywords: Antedependence models, Bayesian methods, Gibbs sampling, Mean-covariance modelling, Nonstationary correlation

Pages: 171–200

DOI: 10.2436/20.8080.02.99

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Abstract: Compositional data analysis is concerned with the relative importance of positive variables, expressed through their log-ratios. The literature has proposed a range of manners to compute log-ratios, some of whose interrelationships have never been reported when used as explanatory variables in regression models. This article shows their similarities and differences in interpretation based on the notion that one log-ratio has to be interpreted keeping all others constant. The article shows that centred, additive, pivot, balance and pairwise log-ratios lead to simple reparametrizations of the same model which can be combined to provide useful tests and comparable effect size estimates.

Keywords: Compositional regression models, CoDa, composition as explanatory, centred log-ratios, pivot coordinates, pairwise log-ratios, additive log-ratios, effect size

Pages: 201–220

DOI: 10.2436/20.8080.02.100

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Abstract: The inclusion of stakeholder behaviour in Operations Research / Industrial Engineering (OR/IE) models has gained much attention in recent years. Behavioural and cognitive traits of people and groups have been integrated in simulation models (mainly through agent-based approaches) as well as in optimization algorithms. However, especially the influence of relations between different actors in human networks is a broad and interdisciplinary topic that has not yet been fully investigated. This paper analyses, from an OR/IE point of view, the existing literature on behaviour-related factors in human networks. This review covers different application fields, including: supply chain management, public policies in emergency situations, and Internet-based human networks. The review reveals that the methodological approach of choice (either simulation or optimization) is highly dependent on the application area. However, an integrated approach combining simulation and optimization is rarely used. Thus, the paper proposes the hybridization of simulation with optimization as one of the best strategies to incorporate human behaviour in human networks and the resulting uncertainty, randomness, and dynamism in related OR/IE models.

Keywords: Modelling human behaviour, human networks, simulation, optimization, simheuristics

Pages: 193–222

DOI: 10.2436/20.8080.02.85

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Abstract: We propose a new type of risk measure for non-negative random variables that focuses on the tail of the distribution. The measure is inspired in general parametric distributions that are well-known in the statistical analysis of the size of income. We derive simple expressions for the conditional moments of these distributions, and we show that they are suitable for analysis of tail risk. The proposed method can easily be implemented in practice because it provides a simple one-step way to compute value-at-risk and tail value-at-risk. We show an illustration with currency exchange data. The data and implementation are open access for reproducibility.

Keywords: Moments, multi-period risk assessment, value-at-risk

Pages: 223–236

DOI: 10.2436/20.8080.02.86

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Abstract: False discovery rate (FDR) control is important in multiple testing scenarios that are common in neuroimaging experiments, and p-values from such experiments may often arise from some discretely supported distribution or may be grouped in some way. Two situations that may lead to discretely supported distributions are when the p-values arise from Monte Carlo or permutation tests are used. Grouped p-values may occur when p-values are quantized for storage. In the neuroimaging context, grouped p-values may occur when data are stored in an integer-encoded form. We present a method for FDR control that is applicable in cases where only p-values are available for inference, and when those p-values are discretely supported or grouped. We assess our method via a comprehensive set of simulation scenarios and find that our method can outperform commonly used FDR control schemes in various cases. An implementation to a mouse imaging data set is used as an example to demonstrate the applicability of our approach.

Keywords: Censored data, data quantization, discrete support, empirical-Bayes, false discovery rate control, grouped data, incompletely observed data, mixture model

Pages: 237–258

DOI: 10.2436/20.8080.02.87

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Abstract: Interval-grouped data appear when the observations are not obtained in continuous time, but monitored in periodical time instants. In this framework, a nonparametric kernel distribution estimator is proposed and studied. The asymptotic bias, variance and mean integrated squared error of the new approach are derived. From the asymptotic mean integrated squared error, a plug-in bandwidth is proposed. Additionally, a bootstrap selector to be used in this context is designed. Through a comprehensive simulation study, the behaviour of the estimator and the bandwidth selectors considering different scenarios of data grouping is shown. The performance of the different approaches is also illustrated with a real grouped emergence data set of Avena sterilis (wild oat).

Keywords: Bootstrap bandwidth, cumulative distribution function estimator, interval data, plug-in bandwidth

Pages: 259–288

DOI: 10.2436/20.8080.02.88

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Abstract: It is well known that outliers can affect both the estimation of parameters and volatilities when fitting a univariate GARCH-type model. Similar biases and impacts are expected to be found on correlation dynamics in the context of multivariate time series. We study the impact of outliers on the estimation of correlations when fitting multivariate GARCH models and propose a general detection algorithm based on wavelets, that can be applied to a large class of multivariate volatility models. Its effectiveness is evaluated through a Monte Carlo study before it is applied to real data. The method is both effective and reliable, since it detects very few false outliers.

Keywords: Correlations, multivariate GARCH models, outliers, wavelets.

Pages: 289–316

DOI: 10.2436/20.8080.02.89

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Abstract: We propose a class of goodness–of–fit test procedures for arbitrary parametric families of circular distributions with unknown parameters. The tests make use of the specific form of the characteristic function of the family being tested, and are shown to be consistent. We derive the asymptotic null distribution and suggest that the new method be implemented using a bootstrap resampling technique that approximates this distribution consistently. As an illustration, we then specialize this method to testing whether a given data set is from the von Mises distribution, a model that is commonly used and for which considerable theory has been developed. An extensive Monte Carlo study is carried out to compare the new tests with other existing omnibus tests for this model. An application involving five real data sets is provided in order to illustrate the new procedure.

Keywords: Goodness-of-fit, Circular data, Empirical characteristic function, Maximum likelihood estimation, von Mises distribution.

Pages: 317–336

DOI: 10.2436/20.8080.02.90

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Abstract: Public services, such as higher education, medical services, libraries or public administration offices, provide services to their customers. To obtain opinion-satisfaction indices of customers, it would be necessary to survey all the customers of the service (census), which is impossible. What is possible is to estimate the indices by surveying a random customer sample. The efficiency obtained with the classic data envelopment analysis models, considering the opinion indices of the customers of the public service as output data estimated with a user sample, will be an estimation of the obtained efficiency if the census is available. This paper proposes a bootstrap methodology to build a confidence region to simultaneously estimate the population data envelopment analysis efficiency score vector of a set of public service-producing units, with a fixed confidence level and using deterministic input data and estimated customer opinion indices as output data. The usefulness of the result is illustrated by describing a case study comparing the efficiency of libraries.

Keywords: Data envelopment analysis, sampling survey research, public sector, bootstrap, simultaneous confidence region.

Pages: 337–354

DOI: 10.2436/20.8080.02.91

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Abstract: In this article, we consider two univariate random environment integer-valued autoregressive processes driven by the same hidden process. A model of this kind is capable of describing two correlated non-stationary counting time series using its marginal variable parameter values. The properties of the model are presented. Some parameter estimators are described and implemented on the simulated time series. The introduction of this bivariate integer-valued autoregressive model with a random environment is justified at the end of the paper, where its real-life data-fitting performance was checked and compared to some other appropriate models. The forecasting properties of the model are tested on a few data sets, and forecasting errors are discussed through the residual analysis of the components that comprise the model.

Keywords: INAR, negative binomial thinning, random states, time series of count, non-stationary process.

Pages: 355–384

DOI: 10.2436/20.8080.02.92

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Abstract: Green transportation is becoming relevant in the context of smart cities, where the use of electric vehicles represents a promising strategy to support sustainability policies. However the use of electric vehicles shows some drawbacks as well, such as their limited driving-range capacity. This paper analyses a realistic vehicle routing problem in which both driving-range constraints and stochastic travel times are considered. Thus, the main goal is to minimize the expected time-based cost required to complete the freight distribution plan. In order to design reliable Routing plans, a simheuristic algorithm is proposed. It combines Monte Carlo simulation with a multi-start metaheuristic, which also employs biased-randomization techniques. By including simulation, simheuristics extend the capabilities of metaheuristics to deal with stochastic problems. A series of computational experiments are performed to test our solving approach as well as to analyse the effect of uncertainty on the routing plans.

Keywords: Vehicle routing problem, electric vehicles, green transport and logistics, smart cities, simheuristics, biased-randomized heuristics

Pages: 3–24

DOI: 10.2436/20.8080.02.77

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Abstract: We introduce new consistent and scale-free goodness-of-fit tests for the exponential distribution based on the Puri-Rubin characterization. For the construction of test statistics we employ weighted L² distance between V-empirical Laplace transforms of random variables that appear in the characterization. We derive the asymptotic behaviour under the null hypothesis as well as under fixed alternatives. We compare our tests, in terms of the Bahadur efficiency, to the likelihood ratio test, as well as some recent characterization based goodness-of-fit tests for the exponential distribution. We also compare the power of our tests to the power of some recent and classical exponentiality tests. According to both criteria, our tests are shown to be strong and outperform most of their competitors.

Keywords: Goodness-of-fit, exponential distribution, Laplace transform, Bahadur efficiency, V-statistics with estimated parameters

Pages: 25–50

DOI: 10.2436/20.8080.02.78

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Abstract: In this paper we propose a Bayesian hierarchical spatio-temporal model for the joint analysis of multiple diseases which includes specific and shared spatial and temporal effects. Dependence on shared terms is controlled by disease-specific weights so that their posterior distribution can be used to identify diseases with similar spatial and temporal patterns. The model proposed here has been used to study three different causes of death (oral cavity, esophagus and stomach cancer) in Spain at the province level. Shared and specific spatial and temporal effects have been estimated and mapped in order to study similarities and differences among these causes. Furthermore, estimates using Markov chain Monte Carlo and the integrated nested Laplace approximation are compared.

Keywords: Bayesian modelling, Joint modelling, Multivariate disease mapping, Shared components. Spatio-temporal epidemiology

Pages: 51–74

DOI: 10.2436/20.8080.02.79

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Abstract: Road freight transportation includes various internal and external costs that need to be accounted for in the construction of efficient routing plans. Typically, the resulting optimization problem is formulated as a vehicle routing problem in any of its variants. While the traditional focus of the vehicle routing problem was the minimization of internal routing costs such as travel distance or duration, numerous approaches to include external factors related to environmental routing aspects have been recently discussed in the literature. However, internal and external routing costs are often treated as competing objectives. This paper discusses the internalization of external routing costs through the consideration of green taxes and green tolls. Numeric experiments with a biased-randomization savings algorithm, show benefits of combining internal and external costs in delivery route planning.

Keywords: Vehicle routing problem, biased randomization, green logistics, negative road externalities, internalization

Pages: 75–94

DOI: 10.2436/20.8080.02.80

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Abstract: In the last decades, a lot of research papers applying statistical methods for analysing sports data have been published. Football, also called soccer, is one of the most popular sports all over the world organised in national championships in a round robin format in which the team reaching the most points at the end of the tournament wins the competition. The aim of this work is to develop a suitable probability model for studying the points achieved by a team in a football match. For this purpose, we built a discrete probability distribution taking values, zero for losing, one for a draw and three for a victory. We test its performance using data from the Spanish Football League (First division) during the 2013-14 season. Furthermore, the model provides an attractive framework for predicting points and incorporating covariates in order to study the factors affecting the points achieved by the teams.

Keywords: Covariate, football data, forecasting, regression, sport statistics, truncated distribution, weighted distribution

Pages: 95–112

DOI: 10.2436/20.8080.02.81

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Abstract: Pearson’s chi-square test is widely employed in social and health sciences to analyse categorical data and contingency tables. For the test to be valid, the sample size must be large enough to provide a minimum number of expected elements per category. This paper develops functions for regrouping strata automatically, thus enabling the goodness-of-fit test to be performed within an iterative procedure. The usefulness and performance of these functions is illustrated by means of a simulation study and the application to different datasets. Finally, the iterative use of the functions is applied to the Continuous Sample of Working Lives, a dataset that has been used in a considerable number of studies, especially on labour economics and the Spanish public pension system.

Keywords: Goodness-of-fit chi-square test, statistical software, Visual Basic for Applications, Mathematica, Continuous Sample of Working Lives

Pages: 113–142

DOI: 10.2436/20.8080.02.83

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Abstract: When the same data are used to fit a model and estimate its predictive performance, this estimate may be optimistic, and its correction is required. The aim of this work is to compare the behaviour of different methods proposed in the literature when correcting for the optimism of the estimated area under the receiver operating characteristic curve in logistic regression models. A simulation study (where the theoretical model is known) is conducted considering different number of covariates, sample size, prevalence and correlation among covariates. The results suggest the use of k-fold cross-validation with replication and bootstrap.

Keywords: Prediction models, logistic regression, area under the receiver operating characteristic curve, validation, bootstrap

Pages: 145–162

DOI: 10.2436/20.8080.02.82

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Abstract: The problem of finding optimal exact designs is more challenging than that of approximate optimal designs. In the present paper, we develop two efficient algorithms to numerically construct exact designs for mixture experiments. The first is a novel approach to the well-known multiplicative algorithm based on sets of permutation points, while the second uses genetic algorithms. Using (i) linear and non-linear models, (ii) D- and I-optimality criteria, and (iii) constraints on the ingredients, both approaches are explored through several practical problems arising in the chemical, pharmaceutical and oil industry.

Keywords: Optimal experimental design, D-optimality, I-optimality, mixture experiments, multiplicative algorithm, genetic algorithm, exact designs

Pages: 163–190

DOI: 10.2436/20.8080.02.84

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Abstract: The discrete case of Bayes’ formula is considered the paradigm of information acquisition. Prior and posterior probability functions, as well as likelihood functions, called evidence functions, are compositions following the Aitchison geometry of the simplex, and have thus vector character. Bayes’ formula becomes a vector addition. The Aitchison norm of an evidence function is introduced as a scalar measurement of information. A fictitious fire scenario serves as illustration. Two different inspections of affected houses are considered. Two questions are addressed: (a) which is the information provided by the outcomes of inspections, and (b) which is the most informative inspection.

Keywords: Evidence function, Bayes’ formula, Aitchison geometry, compositions, orthonormal basis, simplex, scalar information

Pages: 101–124

DOI: 10.2436/20.8080.02.71

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Abstract: Nearest neighbour methods are employed for drawing inferences about spatial patterns of points from two or more classes. We introduce a new pattern called correspondence which is motivated by (spatial) niche/habitat specificity and segregation, and define an associated contingency table called a correspondence contingency table, and examine the relation of correspondence with the motivating patterns (namely, segregation and niche specificity). We propose tests based on the correspondence contingency table for testing self and mixed correspondence and determine the appropriate null hypotheses and the underlying conditions appropriate for these tests. We compare finite sample performance of the tests in terms of empirical size and power by extensive Monte Carlo simulations and illustrate the methods on two artificial data sets and one real-life ecological data set.

Keywords: Association, complete spatial randomness, habitat/niche specificity, independence, random labelling, segregation

Pages: 125–158

DOI: 10.2436/20.8080.02.72

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Abstract: One of the main sources of inaccuracy in modern survey techniques, such as online and smartphone surveys, is the absence of an adequate sampling frame that could provide a probabilistic sampling. This kind of data collection leads to the presence of high amounts of bias in final estimates of the survey, specially if the estimated variables (also known as target variables) have some influence on the decision of the respondent to participate in the survey. Various correction techniques, such as calibration and propensity score adjustment or PSA, can be applied to remove the bias. This study attempts to analyse the efficiency of correction techniques in multiple situations, applying a combination of propensity score adjustment and calibration on both types of variables (correlated and not correlated with the missing data mechanism) and testing the use of a reference survey to get the population totals for calibration variables. The study was performed using a simulation of a fictitious population of potential voters and a real volunteer survey aimed to a population for which a complete census was available. Results showed that PSA combined with calibration results in a bias removal considerably larger when compared with calibration with no prior adjustment. Results also showed that using population totals from the estimates of a reference survey instead of the available population data does not make a difference in estimates accuracy, although it can contribute to slightly increment the variance of the estimator.

Keywords: Online surveys, Smartphone surveys, propensity score adjustment, calibration, simulation

Pages: 159–182

DOI: 10.2436/20.8080.02.73

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Abstract: Surveys applying quota sampling in their final step are widely used in opinion and market research all over the world. This is also the case in Spain, where the surveys carried out by CIS (a public institution for sociological research supported by the government) have become a point of reference. The rules used by CIS to select individuals within quotas, however, could be improved as they lead to biases in age distributions. Analysing more than 545,000 responses collected in the 220 monthly barometers conducted between 1997 and 2016 by CIS, we compare the empirical distributions of the barometers with the expected distributions from the sample design and/or target populations. Among other results, we find, as a consequence of the rules used, significant overrepresentations in the observed proportions of respondents with ages equal to the minimum and maximum of each quota (age and gender group). Furthermore, in line with previous literature, we also note a significant overrepresentation of ages ending in zero. After offering simple solutions to avoid all these biases, we discuss some of their consequences for modelling and inference and about limitations and potentialities of CIS data

Keywords: Centre for Sociological Research, quota sampling, fieldwork rules, age and gender groups, inter-quota distributions, intra-quota distributions

Pages: 183–206

DOI: 10.2436/20.8080.02.74

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Abstract: Authentication of extra virgin olive oil requires fast and cost-effective analytical procedures, such as near infrared spectroscopy. Multivariate analysis and chemometrics have been successfully applied in several papers to gather qualitative and quantitative information of extra virgin olive oils from near infrared spectra. Moreover, there are many examples in the literature analysing the effect of agro-climatic conditions on food content, in general, and in olive oil components, in particular. But the majority of these studies considered a factor, a non-numerical variable, containing this meteorological information. The present work uses all the agro-climatic data with the aim of highlighting the linear relationships between them and the near infrared spectra. The study begins with a graphical motivation, continues with a bivariate analysis and, finally, applies redundancy analysis to extend and confirm the previous conclusions.

Keywords: Extra virgin olive oil, infrared spectroscopy, agro-climatic data, linear correlations, redundancy analysis

Pages: 209–236

DOI: 10.2436/20.8080.02.75

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Abstract: Two new implementations for fitting Poisson excess relative risk methods are proposed for assumed simple models. This allows for estimation of the excess relative risk associated with a unique exposure, where the background risk is modelled by a unique categorical variable, for example gender or attained age levels. Additionally, it is shown how to fit general Poisson linear relative risk models in R. Both simple methods and the R fitting are illustrated in three examples. The first two examples are from the radiation epidemiology literature. Data in the third example are randomly generated with the purpose of sharing it jointly with the R scripts.

Keywords: Radiation epidemiology, Poisson non-linear regression, improper priors, R programming

Pages: 237–252

DOI: 10.2436/20.8080.02.76

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Abstract: In this paper, it is proposed a Bayesian analysis of a time series in the presence of a random change-point and autoregressive terms. The development of this model was motivated by a data set related to the monthly number of asthma medications dispensed by the public health services of Ribeirão Preto, Southeast Brazil, from 1999 to 2011. A pronounced increase trend has been observed from 1999 to a specific change-point, with a posterior decrease until the end of the series. In order to obtain estimates for the parameters of interest, a Bayesian Markov Chain Monte Carlo (MCMC) simulation procedure using the Gibbs sampler algorithm was developed. The Bayesian model with autoregressive terms of order 1 fits well to the data, allowing to estimate the change-point at July 2007, and probably reflecting the results of the new health policies and previously adopted programs directed toward patients with asthma. The results imply that the present model is useful to analyse the monthly number of dispensed asthma medications and it can be used to describe a broad range of epidemiological time series data where a change-point is present.

Keywords: Time series, regression models, Bayesian methods, change-point model, epidemiological data

Pages: 3–26

DOI: 10.2436/20.8080.02.66

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Abstract: In this paper we study the complexity of a functional data set drawn from particular processes by means of a two-step approach. The first step considers a new graphical tool for assessing to which family the data belong: the main aim is to detect whether a sample comes from a monomial or an exponential family. This first tool is based on a nonparametric kNN estimation of small ball probability. Once the family is specified, the second step consists in evaluating the extent of complexity by estimating some specific indexes related to the assigned family. It turns out that the developed methodology is fully free from assumptions on model, distribution as well as dominating measure. Computational issues are carried out by means of simulations and finally the method is applied to analyse some financial real curves dataset.

Keywords: Small ball probability, log-Volugram, random processes, complexity class, complexity index, knn estimation, functional data analysis

Pages: 27–44

DOI: 10.2436/20.8080.02.67

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Abstract: Suppose the regression vector-parameter is subjected to lie in a subspace hypothesis in a linear regression model. In situations where the use of least absolute and shrinkage selection operator (LASSO) is desired, we propose a restricted LASSO estimator. To improve its performance, LASSO-type shrinkage estimators are also developed and their asymptotic performance is studied. For numerical analysis, we used relative efficiency and mean prediction error to compare the estimators which resulted in the shrinkage estimators to have better performance compared to the LASSO.

Keywords: Double shrinking, LASSO, preliminary test LASSO, restricted LASSO, Stein-type shrinkage LASSO

Pages: 45–58

DOI: 10.2436/20.8080.02.68

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Abstract: Heteroscedasticity produces a lack of type I error control in Student’s t test for difference between means. Pretesting for it (e.g., by means of Levene’s test) should be avoided as this also induces type I error. These pretests are inadequate for their objective: not rejecting the null hypotheses is not a proof of homoscedasticity; and rejecting it may simply suggest an irrelevant heteroscedasticity. We propose a method to establish irrelevance limits for the ratio of variances. In conjunction with a test for dispersion equivalence, this appears to be a more affordable pretesting strategy.

Keywords: Homoscedasticity, equivalence test, indifference zone, pretest, Student’s t test

Pages: 59–72

DOI: 10.2436/20.8080.02.69

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Abstract: Usual assumptions on the statistical properties of daily net cash flows include normality, absence of correlation and stationarity. We provide a comprehensive study based on a real-world cash flow data set showing that: (i) the usual assumption of normality, absence of correlation and stationarity hardly appear; (ii) non-linearity is often relevant for forecasting; and (iii) typical data transformations have little impact on linearity and normality. This evidence may lead to consider a more data-driven approach such as time-series forecasting in an attempt to provide cash managers with expert systems in cash management.

Keywords: Statistics, forecasting, cash flow, non-linearity, time-series

Pages: 73–98

DOI: 10.2436/20.8080.02.70

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Abstract: Molenberghs, Verbeke, and Demétrio (2007) and Molenberghs et al. (2010) proposed a general framework to model hierarchical data subject to within-unit correlation and/or overdispersion. The framework extends classical overdispersion models as well as generalized linear mixed models. Subsequent work has examined various aspects that lead to the formulation of several extensions. A unified treatment of the model framework and key extensions is provided. Particular extensions discussed are: explicit calculation of correlation and other moment-based functions, joint modelling of several hierarchical sequences, versions with direct marginally interpretable parameters, zero-inflation in the count case, and influence diagnostics. The basic models and several extensions are illustrated using a set of key examples, one per data type (count, binary, multinomial, ordinal, and time-to-event).

Keywords: Conjugacy, frailty, joint modelling, marginalized multilevel model, mixed model, overdispersion, underdispersion, variance component, zero-inflation.

Pages: 191–254

DOI: 10.2436/20.8080.02.58

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Abstract: We study the factors which may affect students’ marks in two modules, mathematics and statistics, taught consecutively in the first year of a Business Administration Studies degree course. For this purpose, we introduce a suitable bivariate regression model in which the dependent variables have bounded support and the marginal means are functions of explanatory variables. The marginal probability density functions have a classical beta distribution. Simulation experiments were performed to observe the behaviour of the maximum likelihood estimators. Comparisons with univariate beta regression models show the proposed bivariate regression model to be superior.

Keywords: Beta distribution, bivariate beta distribution, conditional distributions, covariate, marginal distributions, regression, mathematics, statistics, business studies.

Pages: 255–276

DOI: 10.2436/20.8080.02.59

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Abstract: A methodological approach for modelling the spatial distribution of bioclimatic indices is proposed in this paper. The value of the bioclimatic index is modelled with a hierarchical Bayesian model that incorporates both structured and unstructured random effects. Selection of prior distributions is also discussed in order to better incorporate any possible prior knowledge about the parameters that could refer to the particular characteristics of bioclimatic indices. MCMC methods and distributed programming are used to obtain an approximation of the posterior distribution of the parameters and also the posterior predictive distribution of the indices. One main outcome of the proposal is the spatial bioclimatic probability distribution of each bioclimatic index, which allows researchers to obtain the probability of each location belonging to different bioclimates. The methodology is evaluated on two indices in the Island of Cyprus.

Keywords: Bioclimatology, geostatistics, parallel computation, spatial prediction.

Pages: 277–296

DOI: 10.2436/20.8080.02.60

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Abstract: Cutaneous melanoma is thought to be triggered by intense, occasional exposure to ultraviolet radiation, either from the sun or tanning beds, especially in people who are genetically predisposed to the disease. When skin cells are damaged by ultraviolet light in this way, often showing up as a sunburn, they are more prone to genetic defects that cause them to rapidly multiply and form potentially fatal (malignant) tumors. Melanoma originates in a type of skin cell called a melanocyte, such cells help produce the pigments of our skin, hair, and eyes. We propose a new cure rate survival regression model for predicting cutaneous melanoma. We assume that the unknown number of competing causes that can influence the survival time is governed by a power series Distribution and that the time until the tumor cells are activated follows the Pareto IV distribution. The parameter estimation is based on the EM algorithm which for this model can be implemented in a simple way in computational terms. Simulation studies are presented, showing the good performance of the proposed estimation procedure. Finally, two real applications related to a cutaneous melanoma and melanoma data sets are presented.

Keywords: Competing risks, cure rate models, EM algorithm, Pareto IV distribution, power series distribution.

Pages: 297–318

DOI: 10.2436/20.8080.02.61

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Abstract: This paper has two complementary objectives: on the one hand, it introduces the EURO method for the estimation of (regional) Social Accounting Matrices. This method is widely used by Eurostat for the estimation of missing national Supply, Use and Input-output tables but it has not been used before within the context of social accounting matrices or of regional statistics and/or regional impact analyses. On the other hand, this work discusses the possibility of producing non-survey based regional Social Accounting Matrices that may eventually allow the user to carry out impact analyses such as those of rural development policies, among others. The analysis is carried out for 12 selected European regions based on clusters.

Keywords: Social accounting matrices, rural development, European regions, impact analysis.

Pages: 319–346

DOI: 10.2436/20.8080.02.62

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Abstract: Aging societies have given rise to important challenges in the field of health insurance. Elderly policyholders need to be provided with fair premiums based on their individual health status, whereas insurance companies want to plan for the potential costs of tackling lifetimes above mean expectations. In this article, we focus on a large cohort of policyholders in Barcelona (Spain), aged 65 years and over. A shared-parameter joint model is proposed to analyse the relationship between annual demand for emergency claims and time until death outcomes, which are subject to left truncation. We compare different functional forms of the association between both processes, and, furthermore, we illustrate how the fitted model provides time-dynamic predictions of survival probabilities. The parameter estimation is performed under the Bayesian framework using Markov chain Monte Carlo methods.

Keywords: Joint models, panel count data, left truncation, Bayesian framework, health insurance.

Pages:347–372

DOI: 10.2436/20.8080.02.63

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Abstract: Catastrophe bonds are financial instruments designed to transfer risk of monetary losses arising from earthquakes, hurricanes, or floods to the capital markets. The insurance and reinsurance industry, governments, and private entities employ them frequently to obtain coverage. Parametric catastrophe bonds base their payments on physical features. For instance, given parameters such as magnitude of the earthquake and the location of its epicentre, the bond may pay a fixed amount or not pay at all. This paper reviews statistical and machine learning techniques for designing trigger mechanisms and includes a computational experiment. Several lines of future research are discussed.

Keywords: Catastrophe bonds, risk of natural hazards, classification techniques, earthquakes, insurance.

Pages: 373–392

DOI: 10.2436/20.8080.02.64

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Abstract: Since its appearance in the 1990s, horizontal collaboration (HC) practices have revealed themselves as catalyzers for optimizing the distribution of goods in freight transport logistics. After introducing the main concepts related to HC, this paper offers a literature review on the topic and provides a classification of best practices in HC. Then, the paper analyses the main benefits and optimization challenges associated with the use of HC at the strategic, tactical, and operational levels. Emerging trends such as the concept of ‘green’ or environmentally-friendly HC in freight transport logistics are also introduced. Finally, the paper discusses the need of using hybrid optimization methods, such as simheuristics and learnheuristics, in solving some of the previously identified challenges in real-life scenarios dominated by uncertainty and dynamic conditions.

Keywords: Horizontal collaboration, freight transport, sustainable logistics, supply chain management, combinatorial optimization.

Pages: 393–414