Visualizing Type II Error in Normality Tests

Abstract

A Skewed Exponential Power Distribution, with parameters defining kurtosis and skewness, is introduced as a way to visualize Type II error in normality tests. By varying these parameters a mosaic of distributions is built, ranging from double exponential to uniform or from positive to negative exponential; the normal distribution is a particular case located in the center of the mosaic. Using a sequential color scheme, a different color is assigned to each distribution in the mosaic depending on the

Graphical comparison of normality tests for unimodal distribution data

“…Anderson-Darling nortest, Gross and Ligges [18] ad.test(x) D'Agostino-Pearson 2 fBasics, Wuertz et al [20] dagoTest(x) [7] consider the probability density function that is used to create the mosaic distributions. This function is characterized by the mean and variance of the variable considered and also a third parameter, , which is related to kurtosis and varies between 1 (double exponential distribution) and 50 (practically a uniform distribution).…”

“…Each distribution corresponds to the values of and that are indicated. In [7], the R code is included to create mosaics as large as 49x49, although it can easily be changed to obtain larger mosaics.…”

“…Our work proposes a methodology for comparing the power of normality tests to a wide variety of alternative unimodal distributions in a highly visual manner and with a single graph. It is based on the representation suggested by Sánchez-Espigares et al [7], who builds a distribution mosaic in which the kurtosis varies vertically and the skewness horizontally. The mosaic includes distributions such as exponential, Laplace and uniform, with normal occupying the center.…”

Computation of Probability Associated with Anderson–Darling Statistic

“…D'Augostino and Stephens provided different formulas for calculation of p-values associated to the Anderson-Darling statistic (AD), along with a correction for small sample size (AD*) [37]. Their equations are independent of the tested theoretical distribution and highlight the importance of the sample size (Table 3).…”

“…More weight to the tails are given by the Anderson-Darling test compared to Kolmogorov-Smirnov test [36]. The comparisons between different goodness-of-fit tests is frequently conducted by comparing their power [37,38], using or not confidence intervals [39], distribution of p-values [40], or ROC (receiver operating characteristic) analysis [32].…”