Testes de hipóteses frequentistas e bayesianos para razão áurea via simulação Monte Carlo

Abstract

The golden ratio is an irrational constant that has been investigated by researchers from various fields of knowledge. Present in nature and widely used in paintings, sculptures and buildings, the golden ratio is related to the perfect beauty and the ideal proportion. Currently research has associated properties such as proper functioning, efficiency and stability at biological structures that have golden ratio. In order to study the golden ratio, researchers have used different methodologies, such as, the Student’s test, non-parametric tests and, in many cases, subjective criteria. There is no, in the literature, a specific test to infer about the mean ratio of two populations. Many researchers, to infer if the rate of certain segments are, on average, equal to the number of gold , use the of sample rates average to estimate the mean of population rates and use the Student’s test. The ratio of two independent random variables standard normal is a random variable that follows a Cauchy distribution. However, when the random variables are dependent, as is the case where it is desired infer about the golden ratio, the ratio probability distribution is not known. The infer process based on assumptions that are not satisfied can lead to unreliable results. One goal of this study was to evaluate the viability of the Student’s test with different statistics, some of them, already used by researchers, and a proposal one to infer the golden ratio. We evaluated other methodologies proposed: the nonparametric Wilcoxon test and the Bayesian test with non-informative priori. The evaluation of the methodologies of parametric tests and nonparametric occurred by quantifying and comparing of the tests type I errors rate and power in different situations variability and sample sizes, via Monte Carlo simulation. To evaluate the Bayesian test, it was quantified rejection rates of H0 in the simulations when the samples were generated under H0 and under H1. As an application, it was obtained data refers the measuring of the arms of nine pairs of the chromosomes of the cell Cerbiatta genotype of Lactuca sativa L., the lettuce and the tests were compared when applied in this sample. All simulations and comparisons of tests were performed using the statistical software R. The Student’s test with the statistics used in literature was liberal or had power rates less than 95% in most scenarios and therefore is not recommended. The Bayesian test with Jeffreys’s priori was equivalent to the Student’s test with the statistical proposal, which in turn performed better on the control of Type I error rates, but showed lower power rates less than 95 % for small samples, especially when the variability of the two segments is large. It was verified that there is the presence of golden ratio in the rate between long and short arms in the fifth pair of chromosome.