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ABSTRACT Scatterplots of biological datasets often have no‐data zones, which suggest constraint or promotion of dependent variables. Although methods exist to estimate boundary lines—that is, to fit lines to the edges of scatters of data points—there are, to our knowledge, none available to assess the significance of the areal extents of no‐data zones. Accordingly, we propose a flexible boundary line definition paired with a permutation test of the magnitude of no‐data zones—rather than testing the shape or slope of the line as current methods do. Our proposed permutation test can be used with any method of defining a boundary line. We demonstrate our approach with empirical datasets, find no‐data zones that methods such as quantile regressions fail to detect, and discuss how our approach can quantify constraint and promotion relationships that are not always apparent with other statistics.

Scatterplots of biological datasets often have no-data zones, which suggest constraint or promotion of dependent variables. Although methods exist to estimate boundary lines-that is, to fit lines to the edges of scatters of data points-there are, to our knowledge, none available to assess the significance of the areal extents of no-data zones. Accordingly, we propose a flexible boundary line definition paired with a permutation test of the magnitude of no-data zones-rather than testing the shape or slope of the line as current methods do. Our proposed permutation test can be used with any method of defining a boundary line. We demonstrate our approach with empirical datasets, find no-data zones that methods such as quantile regressions fail to detect, and discuss how our approach can quantify constraint and promotion relationships that are not always apparent with other statistics., Competing Interests: The authors declare no conflicts of interest., (© 2024 The Author(s). Ecology and Evolution published by John Wiley & Sons Ltd.)

Statistical analyses of directional or angular data have applications in a variety of fields, such as geology, meteorology and bioinformatics. There is substantial literature on descriptive and inferential techniques for univariate angular data, with the bivariate (or more generally, multivariate) cases receiving more attention in recent years. More specifically, the bivariate wrapped normal, von Mises sine and von Mises cosine distributions, and mixtures thereof, have been proposed for practical use. However, there is a lack of software implementing these distributions and the associated inferential techniques. In this article, we introduce BAMBI, an R package for analyzing bivariate (and univariate) angular data. We implement random data generation, density evaluation, and computation of theoretical summary measures (variances and correlation coefficients) for the three aforementioned bivariate angular distributions, as well as two univariate angular distributions: the univariate wrapped normal and the univariate von Mises distribution. The major contribution of BAMBI to statistical computing is in providing Bayesian methods for modeling angular data using finite mixtures of these distributions. We also provide functions for visual and numerical diagnostics and Bayesian inference for the fitted models. In this article, we first provide a brief review of the distributions and techniques used in BAMBI, then describe the capabilities of the package, and finally conclude with demonstrations of mixture model fitting using BAMBI on the two real data sets included in the package, one univariate and one bivariate.

We develop an empirical likelihood (EL) approach to test independence of two univariate random variables X and Y versus the alternative that X and Y are strictly positive quadrant dependent (PQD). Establishing this type of ordering between X and Y is of interest in many applications, including finance, insurance, engineering, and other areas. Adopting the framework in Einmahl and McKeague, we create a distribution-free test statistic that integrates a localized EL ratio test statistic with respect to the empirical joint distribution of X and Y. When compared to well-known existing tests and distance-based tests we develop by using copula functions, simulation results show the EL testing procedure performs well in a variety of scenarios when X and Y are strictly PQD. We use three datasets for illustration and provide an online R resource practitioners can use to implement the methods in this article. for this article are available online. [ABSTRACT FROM AUTHOR]

For technical applications the dependability generally measures availability, reliability, maintainability, and connected activities like costs. The common effect of two or many properties on the same objects can be estimated with multivariate distributions using copula structures.

Through the lenses of statistical investigations and cognitive demands, we examined bivariate data tasks offered in US high school mathematics textbook series—a popular representative of three curriculum types: traditional, integrated, and hybrid. We developed a framework grounded in literature of association topics for the inclusion and exclusion of tasks. Using the Guidelines for Assessment and Instruction of Statistics Education (GAISE) framework, textbook tasks were coded for four investigation components (formulate questions, collect data, analyze data, and interpret results) and levels of statistical sophistication, as well as levels of cognitive demand as suggested by the Mathematical Complexity framework. Across the three series 582 statistical association tasks, all components of statistical investigation were evident with different levels of treatment: (a) all questions for statistical investigations were provided by textbook authors; (b) tasks rarely afforded student opportunities to collect data; and (c) nearly all of the tasks required students to analyze data and most required them to interpret results. Tasks in the integrated series were more numerous (n = 246) and required higher levels of mathematical complexity and statistical sophistication than tasks in the traditional and hybrid series. The vast majority of tasks were coded at the GAISE Level B for analyze data and interpret results and moderate level for mathematical complexity. Further analyses show the concordance between the developmental levels for statistical sophistication and mathematical complexity. Suggestions for curriculum development, content analysis, and future research are provided. [ABSTRACT FROM AUTHOR]

In recent years, the use of copulas has grown rapidly, especially in survival analysis. In this paper, we introduce a bivariate modified Weibull distribu-tion derived from the Farlie{Gumbel{Morgenstern (FGM), a copula function commonly used to model very weak linear dependences. Considering the presence of non censored data and censored data, an extensive simulation study was developed to check the performance of the maximum likelihood method in estimating the parameters of the proposed model. Maximum like-lihood and Bayesian approaches for the estimation of the model parameters are presented. In the Bayesian analysis, the posterior distributions of the pa-rameters are estimated using Markov chain Monte Carlo (MCMC) method-ology. An example, considering a real data set, is introduced to illustrate the proposed methodology. [ABSTRACT FROM AUTHOR]

For technical applications the dependability generally measures availability, reliability, maintainability, and connected activities like costs. The common effect of two or many properties on the same objects can be estimated with multivariate distributions using copula structures. [ABSTRACT FROM AUTHOR]

This system paper presents the Topology ToolKit (TTK), a software platform designed for the topological analysis of scalar data in scientific visualization. While topological data analysis has gained in popularity over the last two decades, it has not yet been widely adopted as a standard data analysis tool for end users or developers. TTK aims at addressing this problem by providing a unified, generic, efficient, and robust implementation of key algorithms for the topological analysis of scalar data, including: critical points, integral lines, persistence diagrams, persistence curves, merge trees, contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots, Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due to a tight integration with ParaView. It is also easily accessible to developers through a variety of bindings (Python, VTK/C++) for fast prototyping or through direct, dependency-free, C++, to ease integration into pre-existing complex systems. While developing TTK, we faced several algorithmic and software engineering challenges, which we document in this paper. In particular, we present an algorithm for the construction of a discrete gradient that complies to the critical points extracted in the piecewise-linear setting. This algorithm guarantees a combinatorial consistency across the topological abstractions supported by TTK, and importantly, a unified implementation of topological data simplification for multi-scale exploration and analysis. We also present a cached triangulation data structure, that supports time efficient and generic traversals, which self-adjusts its memory usage on demand for input simplicial meshes and which implicitly emulates a triangulation for regular grids with no memory overhead. Finally, we describe an original software architecture, which guarantees memory efficient and direct accesses to TTK features, while still allowing for researchers powerful and easy bindings and extensions. TTK is open source (BSD license) and its code. online documentation and video tutorials are available on TTK's website [108]. [ABSTRACT FROM PUBLISHER]

Isosurfaces are fundamental geometrical objects for the analysis and visualization of volumetric scalar fields. Recent work has generalized them to bivariate volumetric fields with fiber surfaces, the pre-image of polygons in range space. However, the existing algorithm for their computation is approximate, and is limited to closed polygons. Moreover, its runtime performance does not allow instantaneous updates of the fiber surfaces upon user edits of the polygons. Overall, these limitations prevent a reliable and interactive exploration of the space of fiber surfaces. This paper introduces the first algorithm for the exact computation of fiber surfaces in tetrahedral meshes. It assumes no restriction on the topology of the input polygon, handles degenerate cases and better captures sharp features induced by polygon bends. The algorithm also allows visualization of individual fibers on the output surface, better illustrating their relationship with data features in range space. To enable truly interactive exploration sessions, we further improve the runtime performance of this algorithm. In particular, we show that it is trivially parallelizable and that it scales nearly linearly with the number of cores. Further, we study acceleration data-structures both in geometrical domain and range space and we show how to generalize interval trees used in isosurface extraction to fiber surface extraction. Experiments demonstrate the superiority of our algorithm over previous work, both in terms of accuracy and running time, with up to two orders of magnitude speedups. This improvement enables interactive edits of range polygons with instantaneous updates of the fiber surface for exploration purpose. A VTK-based reference implementation is provided as additional material to reproduce our results. [ABSTRACT FROM AUTHOR]

This paper extends the analysis of the bivariate Seemingly Unrelated Regression (SUN) Tobit model by modeling its nonlinear dependence structure through the Clayton copula. The ability to capture/model the lower tail dependence of the SUN Tobit model where some data are censored (generally, left-censored at zero) is an useful feature of the Clayton copula. We propose a modified version of the (classical) Inference Function for Margins (IFS) method by Joe and XP [H. Joe and J.J. XP,The estimation method of inference functions for margins for multivariate models, Tech. Rep. 166, Department of Statistics, University of British Columbia, 1996], which we refer to as Modified Inference Function for Margins (MIFF) method, to obtain the (point) estimates of the marginal and Clayton copula parameters. More specifically, we employ the (frequenting) data augmentation technique at the second stage of the IFS method (the first stage of the MIFF method is equivalent to the first stage of the IFS method) to generate the censored observations and then estimate the Clayton copula parameter. This process (data augmentation and copula parameter estimation) is repeated until convergence. Such modification at the second stage of the usual estimation method is justified in order to obtain continuous marginal distributions, which ensures the uniqueness of the resulting Clayton copula, as stated by Solar's [A. Solar,Fonctions de répartition à n dimensions et leurs marges, Publ. de l'Institut de Statistique de l'Université de Paris 8 (1959), pp. 229–231] theorem; and also to provide an unbiased estimate of the association parameter (the IFS method provides a biased estimate of the Clayton copula parameter in the presence of censored observations in both margins). Since the usual asymptotic approach, that is the computation of the asymptotic covariance matrix of the parameter estimates, is troublesome in this case, we also propose the use of resampling procedures (bootstrap methods, such as standard normal and percentile, by Efron and Tibshirani [B. Efron and R.J. Tibshirani,An Introduction to the Bootstrap, Chapman & Hall, New York, 1993] to obtain confidence intervals for the model parameters. [ABSTRACT FROM PUBLISHER]

This work is concerned with understanding common population level effects of stroke on motor control while accounting for possible subject level idiosyncratic effects. Upper extremity motor control for each subject is assessed through repeated planar reaching motions from a central point to eight prespecified targets arranged on a circle. We observe the kinematic data for hand position as a bivariate function of time for each reach. Our goal is to estimate the bivariate function-on-scalar regression with subject level random functional effects while accounting for potential correlation in residual curves; covariates of interest are severity of motor impairment and target number. We express fixed effects and random effects by using penalized splines, and we allow for residual correlation by using a Wishart prior distribution. Parameters are jointly estimated in a Bayesian framework, and we implement a computationally efficient approximation algorithm using variational Bayes methods. Simulations indicate that the method proposed yields accurate estimation and inference, and application results suggest that the effect of stroke on motor control has a systematic component observed across subjects. [ABSTRACT FROM AUTHOR]

This paper proposes univariate and bivariate density estimation techniques whereby averaged shifted histograms are smoothed by means of polynomials. In the univariate case, the density estimate is obtained as a moment-based polynomial approximation to an averaged shifted histogram. The polynomials are expressed as linear combinations of Legendre polynomials. In the bivariate case, the product of polynomial approximations to the marginal density functions is adjusted by making use of a bivariate polynomial whose coefficients are such that the joint moments of the resulting distribution agree with the sample moments up to a certain order. As well, it is explained that this approach actually gives rise to copula density functions. Alternatively, a density estimate may be obtained by smoothing a bivariate averaged shifted histogram with a least-squares approximating polynomial. Several illustrative examples are presented. [ABSTRACT FROM AUTHOR]

This article considers the estimation for bivariate distribution function (d.f.) $$F_0(t, z)$$ of survival time $$T$$ and covariate variable $$Z$$ based on bivariate data where $$T$$ is subject to right censoring. We derive the empirical likelihood-based bivariate nonparametric maximum likelihood estimator $$\hat{F}_n(t,z)$$ for $$F_0(t,z)$$ , which has an explicit expression and is unique in the sense of empirical likelihood. Other nice features of $$\hat{F}_n(t,z)$$ include that it has only nonnegative probability masses, thus it is monotone in bivariate sense. We show that under $$\hat{F}_n(t,z)$$ , the conditional d.f. of $$T$$ given $$Z$$ is of the same form as the Kaplan-Meier estimator for the univariate case, and that the marginal d.f. $$\hat{F}_n(\infty ,z)$$ coincides with the empirical d.f. of the covariate sample. We also show that when there is no censoring, $$\hat{F}_n(t,z)$$ coincides with the bivariate empirical d.f. For discrete covariate $$Z$$ , the strong consistency and weak convergence of $$\hat{F}_n(t,z)$$ are established. Some simulation results are presented. [ABSTRACT FROM AUTHOR]

This paper uses diachronic corpus data to visualize language change in a dynamic fashion. Bivariate and multivariate data sets form the input for so-called motion charts, i.e. series of diachronically ordered scatterplots that can be viewed in sequence. Based on data from COHA (Davies 2010), two case studies illustrate recent changes in American English. The first study visualizes change in a diachronic analysis of ambicategorical nouns and verbs such as hope or drink; the second study shows structural change in the behavior of complement-taking predicates such as expect or remember. Whereas motion charts are typically used to represent bivariate data sets, it is argued here that they are also useful for the analysis of multivariate data over time. The present paper submits multivariate diachronic data to a multi-dimensional scaling analysis. Viewing the resulting data points in separate time slices offers a holistic and intuitive representation of complex linguistic change. [ABSTRACT FROM AUTHOR]

In this article we construct a kernel estimate of the probability function from bivariate data when a component is subject to random left-truncation. We establish consistency and asymptotic normality of the proposed estimator using a strong approximation result. Simulation studies show that the proposed procedure gives a good estimate of the true density function even when the sample size is moderate. [ABSTRACT FROM AUTHOR]

Hougaard's (1986) bivariate Weibull distribution with positive stable frailties is applied to matched pairs survival data when either or both components of the pair may be censored and covariate vectors may be of arbitrary fixed length. When there is no censoring, we quantify the corresponding gain in Fisher information over a fixed-effects analysis. With the appropriate parameterization, the results take a simple algebraic form. An alternative marginal ("independence working model") approach to estimation is also considered. This method ignores the correlation between the two survival times in the derivation of the estimator, but provides a valid estimate of standard error. It is shown that when both the correlation between the two survival times is high, and the ratio of the within-pair variability to the between-pair variability of the covariates is high, the fixed-effects analysis captures most of the information about the regression coefficient but the independence working model does badly. When the correlation is low, and/or most of the variability of the covariates occurs between pairs, the reverse is true. The random effects model is applied to data on skin grafts, and on loss of visual acuity among diabetics. In conclusion some extensions of the methods are indicated and they are placed in a wider context of Generalized Estimation Equation methodology. [ABSTRACT FROM AUTHOR]

This article describes a new, biologically motivated model for bivariate data containing a trend line. The model is characterized by a mixture in which the component means lie on a line. Consistent methods for estimating the parameters of the mixture line are derived, as are procedures for obtaining a confidence interval for the slope. The methods are proved valid within a large class of models for bivariate data, including an errors-invariables model, and are illustrated with data on Canadian herring. This work provides one possible objective resolution of a (sometimes heated) debate in allometry and elsewhere on how to define and estimate a linear trend through bivariate data from a natural population. [ABSTRACT FROM AUTHOR]

We develop an empirical likelihood approach to test independence of two univariate random variables X and Y versus the alternative that X and Y are strictly positive quadrant dependent (PQD). Establishing this type of ordering between X and Y is of interest in many applications, including finance, insurance, engineering, and other areas. Adopting the framework in Einmahl and McKeague (2003, Bernoulli ), we create a distribution-free test statistic that integrates a localized empirical likelihood ratio test statistic with respect to the empirical joint distribution of X and Y . When compared to well known existing tests and distance-based tests we develop by using copula functions, simulation results show the EL testing procedure performs well in a variety of scenarios when X and Y are strictly PQD. We use three data sets for illustration and provide an online R resource practitioners can use to implement the methods in this article.

Mortality rates typically vary smoothly over age and time. Exceptions occur owing to events such as wars and epidemics, which create ridges in the age‐period surface of mortality rates in a particular year or for cohorts born in a particular year. We propose a new practical method for modelling the age‐period surface of mortality rates. Our approach uses L1 regularization with bivariate smoothing and allows for age‐varying period and cohort ridges in the otherwise smooth surface. Cross‐validation on data from many countries and from simulations demonstrates that our approach is superior to existing approaches in estimating the "true" age‐period mortality surface. It also provides greater insight into the underlying mortality dynamics, informing mortality modelling, analysis and forecasting. Although designed for the modelling of mortality rates, our method can also be applied to any bivariate data with occasional ridges and extends the statistical literature on quantile smoothing. © 2018 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The role of scientific research is not limited to the description and analysis of single phenomena occurring independently one from each other (univariate analysis). Even though univariate analysis has a pivotal role in statistical analysis, and is useful to find errors inside datasets, to familiarize with and to aggregate data, to describe and to gather basic information on simple phenomena, it has a limited cognitive impact. Therefore, research also and mostly focuses on the relationship that single phenomena may have with each other. More specifically, bivariate analysis explores how the dependent ("outcome") variable depends or is explained by the independent ("explanatory") variable (asymmetrical analysis), or it explores the association between two variables without any cause and effect relationship (symmetrical analysis). In this paper we will introduce the concept of "causation", dependent ("outcome") and independent ("explanatory") variable. Also, some statistical techniques used for the analysis of the relationship between the two variables will be presented, based on the type of variable (categorical or continuous)., Competing Interests: Conflicts of Interest: The authors have no conflicts of interest to declare.

This work is concerned with understanding common population-level effects of stroke on motor control while accounting for possible subject-level idiosyncratic effects. Upper extremity motor control for each subject is assessed through repeated planar reaching motions from a central point to eight pre-specified targets arranged on a circle. We observe the kinematic data for hand position as a bivariate function of time for each reach. Our goal is to estimate the bivariate function-on-scalar regression with subject-level random functional effects while accounting for potential correlation in residual curves; covariates of interest are severity of motor impairment and target number. We express fixed effects and random effects using penalized splines, and allow for residual correlation using a Wishart prior distribution. Parameters are jointly estimated in a Bayesian framework, and we implement a computationally efficient approximation algorithm using variational Bayes. Simulations indicate that the proposed method yields accurate estimation and inference, and application results suggest that the effect of stroke on motor control has a systematic component observed across subjects.

A proposed scatterplot design combines two other designs. One, proposed by Edward Tufte, uses the range-frame; the other, advised by William Cleveland, uses the four-scale-line rectangle. The hybrid design aims to present bivariate data with more information than classic scatterplots or Tufte's and Cleveland's designs. It works well for both single and multiple scatterplots, in the latter case offering efficient comparison of the datasets in the plots. [ABSTRACT FROM PUBLISHER]