Two-group Case Research Articles

A hybrid Lagrangian–Eulerian model for vector-borne diseases

In this paper, a multi-patch and multi-group vector-borne disease model is proposed to study the effects of host commuting (Lagrangian approach) and/or vector migration (Eulerian approach) on disease spread. We first define the basic reproduction number of the model, R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}_0$$\\end{document}, which completely determines the global dynamics of the model system. Namely, if R0≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}_0 \\le 1 $$\\end{document}, then the disease–free equilibrium is globally asymptotically stable, and if R0>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}_0 > 1 $$\\end{document}, then there exists a unique endemic equilibrium which is globally asymptotically stable. Then, we show that the basic reproduction number has lower and upper bounds which are independent of the host residence times matrix and the vector migration matrix. In particular, nonhomogeneous mixing of hosts and vectors in a homogeneous environment generally increases disease persistence and the basic reproduction number of the model attains its minimum when the distributions of hosts and vectors are proportional. Moreover, R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}_0$$\\end{document} can also be estimated by the basic reproduction numbers of disconnected patches if the environment is homogeneous. The optimal vector control strategy is obtained for a special scenario. In the two-patch and two-group case, we numerically analyze the dependence of the basic reproduction number and the total number of infected people on the host residence times matrix and illustrate the optimal vector control strategy in homogeneous and heterogeneous environments.

A two-group canonical variate analysis biplot for an optimal display of both means and cases

AbstractCanonical variate analysis (CVA) entails a two-sided eigenvalue decomposition. When the number of groups, J, is less than the number of variables, p, at most $$J-1$$ J - 1 eigenvalues are not exactly zero. A CVA biplot is the simultaneous display of the two entities: group means as points and variables as calibrated biplot axes. It follows that with two groups the group means can be exactly represented in a one-dimensional biplot but the individual samples are approximated. We define a criterion to measure the quality of representing the individual samples in a CVA biplot. Then, for the two-group case we propose an additional dimension for constructing an optimal two-dimensional CVA biplot. The proposed novel CVA biplot maintains the exact display of group means and biplot axes, but the individual sample points satisfy the optimality criterion in a unique simultaneous display of group means, calibrated biplot axes for the variables, and within group samples. Although our primary aim is to address two-group CVA, our proposal extends immediately to an optimal three-dimensional biplot when encountering the equally important case of comparing three groups in practice.

Generalised pairwise comparisons for trend: An extension to the win ratio and win odds for dose-response and prognostic variable analysis with arbitrary statements of outcome preference.

The win ratio is a novel approach for handling complex patient outcomes that have seen considerable interest in the medical statistics literature, and operates by considering all-to-all pairwise statements of preference on outcomes. Recent extensions to the method have focused on the two-group case, with few developments made for considering the impact of a well-ordered explanatory variable, which would allow for dose-response analysis or the analysis of links between complex patient outcomes and prognostic variables. Where such methods have been developed, they are semiparametric methods that can only be applied to survival outcomes. In this article, we introduce the generalised pairwise comparison for trend, a modified form of Agresti's generalised odds ratio. This approach is capable of considering arbitrary statements of preference, thus enabling its use across all types of outcome data. We provide a simulation study validating the approach and illustrate it with three clinical applications in stroke research.

A statistical test for detecting discordance in rankings between k groups

ABSTRACTIn several research areas such as psychology, social science, and medicine, studies are conducted in which objects should be ranked by different judges/raters and the concordance of the different rankings is then analyzed. In such studies, it is also frequently of interest to compare the rankings between different groups of judges, e.g. female vs. male judges or judges from different professions. In the two-group case, the two-group concordance test of Schucany & Frawley can be employed for such a comparison. In this article, we propose an extension of this test enabling the comparison of rankings from more than two groups of judges. This test aims to detect disagreement in the average rankings of the objects between k groups with an at least moderate intra-group concordance. We evaluate this test in an extensive simulation study and in an application to data from an aesthetics study. This simulation study shows that the proposed test is able to detect differences between average rankings and performs well even in situations in which the disagreement is comparably small or the intra-group concordance is inhomogeneous.

Simultaneous Sparse Estimation of Canonical Vectors in the p ≫ N Setting

ABSTRACTThis article considers the problem of sparse estimation of canonical vectors in linear discriminant analysis when p ≫ N. Several methods have been proposed in the literature that estimate one canonical vector in the two-group case. However, G − 1 canonical vectors can be considered if the number of groups is G. In the multi-group context, it is common to estimate canonical vectors in a sequential fashion. Moreover, separate prior estimation of the covariance structure is often required. We propose a novel methodology for direct estimation of canonical vectors. In contrast to existing techniques, the proposed method estimates all canonical vectors at once, performs variable selection across all the vectors and comes with theoretical guarantees on the variable selection and classification consistency. First, we highlight the fact that in the N > p setting the canonical vectors can be expressed in a closed form up to an orthogonal transformation. Secondly, we propose an extension of this form to the p ≫ N setting and achieve feature selection by using a group penalty. The resulting optimization problem is convex and can be solved using a block-coordinate descent algorithm. The practical performance of the method is evaluated through simulation studies as well as real data applications. Supplementary materials for this article are available online.

Outcome-adaptive allocation with natural lead-in for three-group trials with binary outcomes

ABSTRACTJust as Bayes extensions of the frequentist optimal allocation design have been developed for the two-group case, we provide a Bayes extension of optimal allocation in the three-group case. We use the optimal allocations derived by Jeon and Hu [Optimal adaptive designs for binary response trials with three treatments. Statist Biopharm Res. 2010;2(3):310–318] and estimate success probabilities for each treatment arm using a Bayes estimator. We also introduce a natural lead-in design that allows adaptation to begin as early in the trial as possible. Simulation studies show that the Bayesian adaptive designs simultaneously increase the power and expected number of successfully treated patients compared to the balanced design. And compared to the standard adaptive design, the natural lead-in design introduced in this study produces a higher expected number of successes whilst preserving power.

Optimal variable selection in multi-group sparse discriminant analysis

This article considers the problem of multi-group classification in the setting where the number of variables $p$ is larger than the number of observations $n$. Several methods have been proposed in the literature that address this problem, however their variable selection performance is either unknown or suboptimal to the results known in the two-group case. In this work we provide sharp conditions for the consistent recovery of relevant variables in the multi-group case using the discriminant analysis proposal of Gaynanova et al. [7]. We achieve the rates of convergence that attain the optimal scaling of the sample size $n$, number of variables $p$ and the sparsity level $s$. These rates are significantly faster than the best known results in the multi-group case. Moreover, they coincide with the minimax optimal rates for the two-group case. We validate our theoretical results with numerical analysis.

Dynamics of social tolerance in the economic interaction model with three groups

The dynamics of social tolerance among three groups of economic agents have been discussed. We show that dynamics of social tolerance between two groups can be greatly influenced by the third group, and a society with three groups exhibits rich behaviours which are very different from the two-group case. In the three-group case, a fully intolerant society may also get total welfare maximization, while a society with higher social costs gets lower total welfare. Cultural integration is vitally important in achieving full tolerance and total welfare maximization. In particular, suitably increase the number of groups, which can reduce the necessary condition of achieving full tolerance, is an effective method if cultural integration is invalid.

Segregation, informativeness and Lorenz dominance

It is possible to partially order cities according to the informativeness of neighborhoods about their ethnic groups. It is also possible to partially order cities with two ethnic groups according to the Lorenz criterion. We show that a segregation order satisfies four well-established segregation principles if and only if it is consistent with the informativeness criterion. We then use this result to show that for the two-group case, the Lorenz and the informativeness criteria are equivalent.

Penalized differential pathway analysis of integrative oncogenomics studies

Through integration of genomic data from multiple sources, we may obtain a more accurate and complete picture of the molecular mechanisms underlying tumorigenesis. We discuss the integration of DNA copy number and mRNA gene expression data from an observational integrative genomics study involving cancer patients. The two molecular levels involved are linked through the central dogma of molecular biology. DNA copy number aberrations abound in the cancer cell. Here we investigate how these aberrations affect gene expression levels within a pathway using observational integrative genomics data of cancer patients. In particular, we aim to identify differential edges between regulatory networks of two groups involving these molecular levels. Motivated by the rate equations, the regulatory mechanism between DNA copy number aberrations and gene expression levels within a pathway is modeled by a simultaneous-equations model, for the one- and two-group case. The latter facilitates the identification of differential interactions between the two groups. Model parameters are estimated by penalized least squares using the lasso (L1) penalty to obtain a sparse pathway topology. Simulations show that the inclusion of DNA copy number data benefits the discovery of gene-gene interactions. In addition, the simulations reveal that cis-effects tend to be over-estimated in a univariate (single gene) analysis. In the application to real data from integrative oncogenomic studies we show that inclusion of prior information on the regulatory network architecture benefits the reproducibility of all edges. Furthermore, analyses of the TP53 and TGFb signaling pathways between ER+ and ER- samples from an integrative genomics breast cancer study identify reproducible differential regulatory patterns that corroborate with existing literature.

Segregation, Informativeness and Lorenz Dominance

It is possible to partially order cities according to the informativeness of neighborhoods about their ethnic groups. It is also possible to partially order cities with two ethnic groups according to them Lorenz criterion. We show that a segregation order satisfies four well-established segregation principles if and only if it is consistent with the informativeness criterion. We then use this result to show that for the two-group case, the Lorenz and the informativeness criteria are equivalent.

Accurate mean comparisons for paired samples with missing data: An application to a smoking-cessation trial

In this paper, we consider mean comparisons for paired samples in which a certain portion of the observations are missing. This type of data commonly arises in medical researches where the outcomes are assessed at two time points after the application of treatments. New methods for statistical inference are proposed by making finiteness correction based on asymptotic expansions of some intuitive statistics. The comparison methods naturally extend to the two-group case after some suitable manipulations. Simulation study is carried out to demonstrate the numerical accuracy of the proposed methods. Data from a smoking-cessation trial are used to illustrate the application of the methods.

Partial least squares discrimination with heterogeneous covariance structures

AbstractBarker and Rayens 1 argued that partial least squares (PLS) is to be preferred over principal components analysis (PCA) when linear discrimination is the goal and dimension reduction is required as a first step. In particular, it is now known that when PLS is used as the dimension reduction tool, information involving Fisher's among‐groups sums‐of‐squares and cross‐products matrix is embedded in the structures extracted to achieve that reduction. Liu and Rayens 2 followed up with a formal proof for the superiority of PLS over PCA in the two‐group case (with respect to formal misclassification probabilities) and pointed to a subclass of oriented PLS (OrPLS) 3 techniques that would always produce a lower misclassification rate than would PLS, for the two‐group problem with like covariance matrices. This paper addresses the situation where these within‐groups covariances are potentially not the same, and hence a single reduction step followed by linear discrimination may not be appropriate. A variety of techniques are compared by way of a large‐scale simulation study, and a version of the ridged subclass studied by Liu and Rayens 2 emerges as the recommended approach from among those studied. An application aimed at distinguishing asymptomatic women at risk for Alzheimer's disease from women not at risk, based on functional magnetic resonance imaging (fMRI) scans, is also presented. Copyright © 2011 John Wiley & Sons, Ltd.

Optimal Nonbipartite Matching and Its Statistical Applications

Matching is a powerful statistical tool in design and analysis. Conventional two-group, or bipartite, matching has been widely used in practice. However, its utility is limited to simpler designs. In contrast, nonbipartite matching is not limited to the two-group case, handling multiparty matching situations. It can be used to find the set of matches that minimize the sum of distances based on a given distance matrix. It brings greater flexibility to the matching design, such as multigroup comparisons. Thanks to improvements in computing power and freely available algorithms to solve nonbipartite problems, the cost in terms of computation time and complexity is low. This article reviews the optimal nonbipartite matching algorithm and its statistical applications, including observational studies with complex designs and an exact distribution-free test comparing two multivariate distributions. We also introduce an R package that performs optimal nonbipartite matching. We present an easily accessible web application to make nonbipartite matching freely available to general researchers.

Inverse Neutron Diffusion Problems in Reactor Design

The parameter estimation problem in neutron diffusion is investigated for a critical reactor. This case is considered as inverse to the classical criticality problem. In this work, a one-dimensional, two-group case is examined in a multiregional core. The flux distributions in a discrete basis, a multiplication factor, region wise fractional powers and group diffusion coefficients of one region are the known data of the inverse problem. It is assumed that no adjoint solution information is available. A non-iterative direct approach is developed to reconstruct nuclear parameters of each region. The numerical results for test cases are given.

PLS and dimension reduction for classification

Barker and Rayens (J Chemometrics 17:166---173, 2003) offered convincing arguments that partial least squares (PLS) is to be preferred over principal components analysis (PCA) when discrimination is the goal and dimension reduction is required, since at least with PLS as the dimension reduction tool, information involving group separation is directly involved in the structure extraction. In this paper the basic results in Barker and Rayens (J Chemometrics 17:166---173, 2003) are reviewed and some of their ideas and comparisons are illustrated on a real data set, something which Barker and Rayens did not do. More importantly, new results are introduced, including a formal proof for the superiority of PLS over PCA in the two-group case, as well as new connections between PLS for discrimination and an extended class of PLS-like techniques known as "oriented PLS" (OrPLS). In the latter case, a particularly simple subclass of OrPLS procedures, when used to achieve the dimension reduction, is shown to always produce a lower misclassification rate than when "ordinary" PLS is used for the same purpose.

Inverse Neutron Diffusion Problems in Reactor Design

The parameter estimation problem in neutron diffusion is investigated for a critical reactor. This case is considered as inverse to the classical criticality problem. In this work, a one-dimensional, two-group case is examined in a multiregional core. The flux distributions in a discrete basis, a multiplication factor, region wise fractional powers and group diffusion coefficients of one region are the known data of the inverse problem. It is assumed that no adjoint solution information is available. A non-iterative direct approach is developed to reconstruct nuclear parameters of each region. The numerical results for test cases are given.

Variable employee productivity in workforce scheduling

This paper considers the problem of developing workforce schedules using groups of employees having different productivity. We show that the existing linear representation of this problem is often inaccurate for high-contact service organizations because it ignores the stochastic nature of customer arrivals. Specifically, the existing representation commonly overestimates the number of less productive employees necessary to deliver a specified, waiting time-based customer service level. We present a new, nonlinear representation of this staffing problem that captures its nonlinear nature and demonstrate its superiority via an extensive set of labor tour scheduling problems for the two-group case.

Robust Confidence Intervals for Effect Size in the Two-Group Case

Robust Confidence Intervals for Effect Size in the Two-Group Case

Linear Discriminant Analysis Versus Logistic Regression: A Comparison of Classification Errors in the Two-Group Case

Classification studies are important for practitioners who need to identify individuals for specialized treatment or intervention. When interventions are irreversible or misclassifications are costly, information about the proficiency of different classification procedures becomes invaluable. This study furnishes information about the relative accuracy of two widely used classification procedures, linear discriminant analysis and logistic regression, under various commonly encountered and interacting conditions. Monte Carlo simulation was used to manipulate four factors under multivariate normality: equality of covariance matrices, degree of group separation, sample size, and prior probabilities. Three criterion measures were employed: total, small-group, and large-group classification error. Interactions of these between factors with two within factors, cut-score and method of classification, were of primary interest.