Abstract
Several measures of skewness and kurtosis were proposed by Hogg (1974) in order to reduce the bias of conventional estimators when the distribution is non-normal. Here we conducted a Monte Carlo simulation study to compare the performance of conventional and Hogg’s estimators, considering the most frequent continuous distributions used in health, education, and social sciences (gamma, lognormal and exponential distributions). In order to determine the bias, precision and accuracy of the skewness and kurtosis estimators for each distribution we calculated the relative bias, the coefficient of variation, and the scaled root mean square error. The effect of sample size on the estimators is also analyzed. In addition, a SAS program for calculating both conventional and Hogg’s estimators is presented. The results indicated that for the non-normal distributions investigated, the estimators of skewness and kurtosis which best reflect the shape of the distribution are Hogg’s estimators. It should also be noted that Hogg’s estimators are not as affected by sample size as are conventional estimators.
Highlights
In the health and social sciences, outcome variables usually show values of skewness and kurtosis that clearly deviate from the normal distribution [1,2]
We constructed macros that enabled pseudorandom variables to be generated for the different distributions, and which could calculate kurtosis and skewness both with the conventional estimators that SAS uses by default (Kr1 and Sk1), and with those of Hogg (Kr2, Kr3, and Sk2)
Comparison of conventional estimators of kurtosis and skewness with those proposed by Hogg reveals, for the gamma, lognormal and exponential distributions, the following: (a) all the estimators have negative values of relative bias (RB); in other words, they are biased negatively; (b) the
Summary
In the health and social sciences, outcome variables usually show values of skewness and kurtosis that clearly deviate from the normal distribution [1,2]. Micceri [2] analyzed the distributional characteristics of over 400 large-sample achievement and psychometric measures and found several classes of deviation from the normal distribution in addition to skewness and kurtosis, leading him to conclude that normal distributions are uncommon with real data. This was further illustrated in the study by Blanca et al [1], who analyzed 693 distributions corresponding to cognitive ability and other psychological variables derived from 130 different populations. Further examples cited by Arnau et al [5]
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