Statistical methods for identification of golden ratio

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.