Mosaic normality test

Abstract

A procedure is proposed here for jointly visualizing the compatibility of a sample with a family of Skewed Exponential Power Distributions, of which the distributions known as Normal, Exponential, Laplace and Uniform are particular cases. The procedure involves constructing a mosaic that contains these distributions in such a way that the asymmetry varies from left to right and the kurtosis varies from top to bottom. The null hypothesis that the sample belongs to each of the mosaic distributions is tested, with the corresponding box for each distribution being shaded in a gray scale according to the pvalue obtained. The location and shape of the shaded area that appears on the mosaic facilitates not only identifying which distributions are compatible with the sample, but also assessing the power of the test performed. All the parameters that define the distribution are considered to be known. The problem remains open for expanding the procedure to cases in which these parameters are not known but must be estimated from the sample. As additional material, a code written in R that carries out the proposed test is included.

Normality assessment, few paradigms and use cases

“…When we have a large data sample, the central limit theorem states that the violation of normality is not a major problem, although, to obtain some significant conclusions it is advisable to assume the normality of the data regardless of sample size (19,27). For example, to verify the normality of the data, the Shapiro-Wilk test is recommended for small data samples (n <50), although it can be applied just as well for large sample sizes (12,27).…”

“…When we have a large data sample, the central limit theorem states that the violation of normality is not a major problem, although, to obtain some significant conclusions it is advisable to assume the normality of the data regardless of sample size (19,27). For example, to verify the normality of the data, the Shapiro-Wilk test is recommended for small data samples (n <50), although it can be applied just as well for large sample sizes (12,27). Unfortunately, this test, like the others studied, can misidentify the distributions for data sets less than to 30, this test should be supplemented with graphical methods to have a confirmation of the test result (Figure 3).…”

“…In our opinion, it is recommended that the analytical values be supplemented with a verification of the graphical representation of the data -especially for small data sets (fewer than 30 measurements) -which allows the identification of abnormal values that may influence the test result (27). We must not forget that the graphical and semigraphic methods (CDF, QQ plot), although present in most statistical programs, require a certain degree of user experience in using these methods (1).…”