The source R code to compute p-value of CCOT.

Testing whether data are from a normal distribution is a traditional problem and is of great concern for data analyses. The normality is the premise of many statistical methods, such as t-test, Hotelling T2 test and ANOVA. There are numerous tests in the literature and the commonly used ones are Anderson-Darling test, Shapiro-Wilk test and Jarque-Bera test. Each test has its own advantageous points since they are developed for specific patterns and there is no method that consistently performs optimally in all situations. Since the data distribution of practical problems can be complex and diverse, we propose a Cauchy Combination Omnibus Test (CCOT) that is robust and valid in most data cases. We also give some theoretical results to analyze the good properties of CCOT. Two obvious advantages of CCOT are that not only does CCOT have a display expression for calculating statistical significance, but extensive simulation results show its robustness regardless of the shape of distribution the data comes from. Applications to South African Heart Disease and Neonatal Hearing Impairment data further illustrate its practicability.</div

The p-values of normality tests for South African heart disease data.

The p-values of normality tests for South African heart disease data.</p

The empirical power for different sample sizes under remaining 8 common non-normal distributions at nominal significance level of 0.05 based on 10,000 replications.

The empirical power for different sample sizes under remaining 8 common non-normal distributions at nominal significance level of 0.05 based on 10,000 replications.</p

The density function curves for Scenarios (i)—(v) under BN model.

The solid and dashed lines represent the density function curves of standardized BN samples and standard normally distributed samples, respectively. The values of sk and ku below these subfigures are the skewness and kurtosis. (PDF)</p

The density function curves, skewness and kurtosis of 12 common non-normal distributions.

The solid and dashed lines represent the density curves of 12 common non-normal distributed data and standard normally distributed data, respectively. (PDF)</p

The Q-Q plots of patient group and health group for biomarkers TEOAE and ABR.

The Q-Q plots of patient group and health group for biomarkers TEOAE and ABR.</p

The p-values of normality tests for neonatal hearing impairment female data.

The p-values of normality tests for neonatal hearing impairment female data.</p

The empirical type I error rates of eight tests for different sample sizes under nominal significance level 0.05 based on 10,000 replications.

The empirical type I error rates of eight tests for different sample sizes under nominal significance level 0.05 based on 10,000 replications.</p

The empirical power for different sample sizes under nominal significance level 0.05 based on 10,000 replications.

The empirical power for different sample sizes under nominal significance level 0.05 based on 10,000 replications.</p

The empirical power for scenarios (i)—(v) under nominal significance level 0.05 based on 10,000 replications.

The empirical power for scenarios (i)—(v) under nominal significance level 0.05 based on 10,000 replications.</p