Vector Analytics, LLC, Wilmington, DE, United States of America.
PLoS One. 2020 Dec 28;15(12):e0244517. doi: 10.1371/journal.pone.0244517. eCollection 2020.
We consider the problem of constructing a complete set of parameters that account for all of the degrees of freedom for point-biserial variation. We devise an algorithm where sort as an intrinsic property of both numbers and labels, is used to generate the parameters. Algebraically, point-biserial variation is represented by a Cartesian product of statistical parameters for two sets of [Formula: see text] data, and the difference between mean values (δ) corresponds to the representation of variation in the center of mass coordinates, (δ, μ). The existence of alternative effect size measures is explained by the fact that mathematical considerations alone do not specify a preferred coordinate system for the representation of point-biserial variation. We develop a novel algorithm for estimating the nonoverlap proportion (ρpb) of two sets of [Formula: see text] data. ρpb is obtained by sorting the labeled [Formula: see text] data and analyzing the induced order in the categorical data using a diagonally symmetric 2 × 2 contingency table. We examine the correspondence between ρpb and point-biserial correlation (rpb) for uniform and normal distributions. We identify the [Formula: see text], [Formula: see text], and [Formula: see text] representations for Pearson product-moment correlation, Cohen's d, and rpb. We compare the performance of rpb versus ρpb and the sample size proportion corrected correlation (rpbd), confirm that invariance with respect to the sample size proportion is important in the formulation of the effect size, and give an example where three parameters (rpbd, μ, ρpb) are needed to distinguish different forms of point-biserial variation in CART regression tree analysis. We discuss the importance of providing an assessment of cost-benefit trade-offs between relevant system parameters because 'substantive significance' is specified by mapping functional or engineering requirements into the effect size coordinates. Distributions and confidence intervals for the statistical parameters are obtained using Monte Carlo methods.
我们考虑构建一个完整的参数集，以解释点双列变异的所有自由度。我们设计了一种算法，其中排序作为数字和标签的固有属性，用于生成参数。从代数角度看，点双列变异由两组[公式：见文本]数据的统计参数的笛卡尔积表示，均值差（δ）对应于质心坐标（δ，μ）的变化表示。替代效应量测的存在可以解释为数学考虑本身并不能为点双列变异的表示指定一个首选的坐标系。我们开发了一种新的算法来估计两组[公式：见文本]数据的非重叠比例（ρpb）。ρpb 通过对标记的[公式：见文本]数据进行排序，并使用对角对称的 2×2 列联表分析分类数据中的诱导顺序来获得。我们检查了 ρpb 与点双列相关系数（rpb）在均匀分布和正态分布下的对应关系。我们确定了皮尔逊积差相关系数（rpb）、科恩氏 d 和 rpb 的[公式：见文本]、[公式：见文本]和[公式：见文本]表示。我们比较了 rpb 与 ρpb 以及样本大小比例校正相关系数（rpbd）的性能，确认了样本大小比例不变性在效应量的制定中很重要，并给出了一个例子，其中在 CART 回归树分析中需要三个参数（rpbd、μ、ρpb）来区分不同形式的点双列变异。我们讨论了提供对相关系统参数的成本效益权衡评估的重要性，因为“实质性意义”是通过将功能或工程要求映射到效应量坐标来指定的。使用蒙特卡罗方法获得统计参数的分布和置信区间。
