I have two independent random samples . The distribution of data in each of the two samples may not be symmetric. Moreover, the asymmetry may not be same in the two data sets of the two corresponding samples. Thus there is necessity of testing of the significance of the difference between the skewness of the two samples. However, it is beyond my knowledge on how to test the significance of this difference. The question thus here is " Does there exist test statistic or testing method for testing of the significance of difference between the skewness of the two samples ? "
Yes there are such tests: > Box, G.E.P. 1953. Non-normality and tests on variances. Biometrika 40: 318-335. >
Cabras S., Castellanos M.E. (2005) Comparing the skewness in two populations using Bayesian inference on skew normal and skew t models, Working Paper, Dep. Stat., Rey Juan Carlos University, http://bayes.escet.urjc.es/publicaciones/wp05-11.pdf.
Joanes D. N., Gill C. A. (1998) Comparing Measures of Sample Skewness and Kurtosis, The Statistician, 47, 183–189.
A randomization test. If the null hypothesis is correct then it makes no difference whether an observation comes from population A or population B. Take all the data and randomly assign each point to population A or B. Calculate the skewness for each population and take the difference. Do this at least 10,000 times, and preferably 50,000+. Sort these values. Take the difference between the skewness of the original data and find that value (or as close as you can get) in the sorted list. What fraction of the values in the list are a large or larger than the observed value? This is the p-value and can be interpreted in the standard way.
One problem that may arise is if there are only a few observations. While the process would work for even 5 or 6 observations in each group, the reliability of calculating skewness on such a small sample is doubtful and the randomization method also starts giving strange answers: With 6 observations there are only 924 possible outcomes. Accurate estimation of an event that occurs only 5% of the time with such few outcomes is doubtful in my opinion. I would like to have 20 observations or more from each population.
@ s. Béatrice Marianne Ewalds-Kvist ,
Thanks for providing the following materials:
Box, G.E.P. 1953. Non-normality and tests on variances. Biometrika 40: 318-335. >
Cabras S., Castellanos M.E. (2005) Comparing the skewness in two populations using Bayesian inference on skew normal and skew t models, Working Paper, Dep. Stat., Rey Juan Carlos University, http://bayes.escet.urjc.es/publicaciones/wp05-11.pdf.
Joanes D. N., Gill C. A. (1998) Comparing Measures of Sample Skewness and Kurtosis, The Statistician, 47, 183–189.
I will go through them and ask again if I face with any doubt / problem.
@ Timothy A Ebert ,
The technique provided by Timothy A Ebert is very interesting. The idea behind the technique is new for me. As I have understood, the technique is logically quite reasonable. It is highly appreciated and recommended.
However, it is not clear to me on how to interpret the results.
The result is a p-value. It is no different in interpretation than running an ANOVA, getting an F-value of 23.45 and finding that the probability of a greater F value is 0.00326. The difference is in the mechanics of how one gets there.
The randomization test is a direct reflection of what we do in standard testing under the frequentist paradigm. A statistical test is an estimate of the probability of finding an observation that is as large or larger than the observed value given that the null-hypothesis is true. If this probability is very small then we decide to reject the null hypothesis.
The description on how to interpret the results of randomized test, provided by Timothy A Ebert , is understandable. Timothy A Ebert is highly appreciated and his answer is recommended.
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