Several biological systems such as the biomechanics of human heart, locomotion, and phyllotaxis of plants present a harmonic behavior because their fractal structure are associated to the golden ratio. The golden ratio (Φ = 1.618033988749…), also known as Phi, golden mean, golden section or divine proportion, is an irrational constant found in various forms in nature and recently has been used in many health areas. However, there is no literature on a specific statistical test to identify the golden ratio structures. To validate the results from each survey, it is necessary that statistical techniques be correctly selected and implemented, and the absence of a test to identify the golden ratio may undermines the scientific papers which have this goal. Since the golden number is a ratio, some tests have been wrongly applied in its identification. The objective of this paper is to present and to evaluate methods for identification of golden ratio. Four tests were evaluated: t-Student with ratio statistic (TR), with delta statistic (TΔ), with difference statistic (TED), and Wilcoxon test with statistic difference (WD). Data simulating different samples sizes (n = 2-200) and variability scenarios were used. The tests were assessed regarding type I error rate and power. For TΔ, type I error rate increased along with sample size and variability, achieving 50% in the scenario of relative standard deviation of 12.5% and 20.0% for line segments of lengths a and b, and sample size equal 200. This test also showed lower power when compared to the others in all scenarios. Similarly, for TR, the type I error rate was sensitive to the increasing in sample size, varying from 5 to 60%. On the other hand, WD and TED were associated to low type I error rates (around 5%) and high power (6.1% for sample size equal 2-100% for sample size equal 200). The TΔ and TR were inadequate to identify the golden ratio, since they did not controlled the type I error rate and/or presented low power, leading to possible erroneous conclusions. Therefore WD and TED, both with statistical of difference, appeared as the most appropriate methods to test golden ratio structures.

译文

几种生物系统,例如人类心脏的生物力学,植物的运动和叶序形成了调和行为,因为它们的分形结构与黄金比例有关。黄金比例 (Φ   =   1.618033988749…),也称为Phi,黄金中庸,黄金分割或神圣比例,是自然界中各种形式的非理性常数,最近已在许多健康领域使用。然而,没有关于确定黄金比例结构的特定统计检验的文献。为了验证每次调查的结果,有必要正确选择和实施统计技术,而缺乏确定黄金比例的测试可能会破坏具有此目标的科学论文。由于黄金数字是一个比率,因此在识别时错误地应用了一些测试。本文的目的是介绍和评估黄金分割率的识别方法。评估了四个测试: 具有比率统计 (TR) 的t学生,具有增量统计 (t Δ),具有差异统计 (TED) 和具有统计差异 (WD) 的Wilcoxon检验。使用了模拟不同样本大小 (n   =   2-200) 和可变性场景的数据。对测试进行了I型错误率和功率评估。对于t Δ,I型错误率随着样本大小和变异性而增加,在长度为a和b的线段的12.5% 和20.0% 的相对标准偏差的情况下实现了50%,并且样本大小200相等。与所有情况下的其他测试相比,该测试还显示出较低的功率。同样,对于TR,I型错误率对样本量的增加敏感,从5到60% 不等。另一方面,WD和TED与低I型错误率 (约5%) 和高功率 (样本大小等于2-100%,样本大小等于200的6.1%) 相关。T Δ 和TR不足以识别黄金比例,因为它们没有控制I型错误率和/或呈现低功率,从而导致可能的错误结论。因此,WD和TED都具有统计学差异,是测试黄金比例结构的最合适方法。

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