187-2007: SAS® Macros for Assessing the Symmetry of a Data Set

Skewness indicates a lack of symmetry in a distribution. The coefficient of skewness is the commonly used measure to identify a lack of symmetry in the underlying data, although graphical procedures can also be effective. We have developed two comprehensive and efficient SAS ® macros for computing the various skewness measures and the appropriate power transformation, if one exists, to make an asymmetric distribution symmetric. We present two SAS ® macros, %Symm and %Symmchk, which primarily utilize the UNIVARIATE procedure in SAS ® and a few other built-in macro calls. Our SAS ® macros provide a menu of options to perform all computations efficiently and suggest an appropriate power transformation, if one exists, to make the data symmetric. This latter part is noteworthy in that we not only address the problem i.e., identify whether a given data set has an underlying symmetric distribution, but also provide a solution. Real data from clinical studies will be used to demonstrate the methods and its application using the SAS ® macros. INTRODUCTION The first step in any statistical analysis includes summarizing the characteristics of the underlying data. All standard statistical packages routinely provide the summary statistics information, and this often includes a sample skewness score, which is a measure of symmetry. Symmetry is a rather complex property of probability distributions and it is difficult to identify deviations from it in a small number of observations. Broadly speaking, a dataset or a distribution is said to be symmetric if it looks same to the right and left of the center point. One of the numerous reasons for checking symmetry in a given data set is because many statistical tests rely strongly on the assumption of normality, which in turn relies on symmetry. Thus, a skewness measure can provide valuable information on issues such as data transformation, outlier detection, distribution fitting etc. so as to ensure that an appropriate analysis procedure (parametric versus non-parametric) is employed. In this paper, we present a novel approach to assessing symmetry, which is simple and computationally less complex. This approach rests on the inherent characteristic of a symmetric distribution: the corresponding upper and lower percentiles are equidistant from its median. 1 We also introduce the other commonly used skewness measures namely the traditional coefficient of skewness index, skewness index based on the L-moments discussed by Hosking 2 , the Quartile and the Octile skewness coefficients proposed by Hinkley 3 , and the symmetry test developed by Randles et al. 4 We discuss the pros and cons of the different approaches, and introduce our comprehensive SAS ® macro that performs these computations efficiently in addition to providing a possible power law transformation to make the variable symmetric where appropriate. 1,5 Two slightly different versions of a macro are available for checking for symmetry. METHODS Let x(1), x(2), ..., x(n) be the ordered random sample of size n from a distribution of the random variable X with mean μ and variance σ 2 . 1) The coefficient of skewness: S1 = ( ) n r i 3 i=1 r 3/2 2 (x -x) m , where m = n m ∑ . For symmetrical distributions, S1 has expectation 0, i.e., when the data is symmetric, the sample skewness coefficient is near zero. If S1 > 0, then the distribution is asymmetric with a positive skew and if S1 < 0, then the distribution is asymmetric with a negative skew. 2) Skewness index based on the L-moments 2 : S2 = 3 2 l l , where: 2 2 l = 2w x , 3 3 2 l = 6w 6w + x , and