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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.

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.

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.