This Demonstration shows the effect of a power data transformation, , on data, , from simulated samples of size or from normal, exponential, lognormal, inverse Gaussian, or Weibull distributions for . In practice, a suitable power transformation can be selected by examining the effect of the transformation using a boxandwhisker plot. The simplest power transformation which makes the data approximately symmetric is selected. With actual data, often corresponding to reciprocal, log, square root, or no transformation. Two skewness statistics—the usual Pearson skewness, , and the Bowley skewness, —are displayed for comparison with the plot. Another method for choosing treats as a parameter and makes the assumption that for some value of , the data is normally distributed. Under this assumption, the likelihood function may be obtained and it may be numerically maximized to obtain the maximum likelihood estimate for , . A range of plausible values for is given by all for which , where . Try experimenting with different sample sizes and different distributions. In actual applications, real data (not simulated data) would be used. Using a suitable power transformation often simplifies the statistical analysis.
Data transformations such as squareroot and logs are often used in statistics to improve the model assumptions. See [1] for examples, with actual data, of the use of boxandwhisker plots to choose a transformation. Using Mathematica's builtin functions Manipulate and BoxWhisker with the family of power transformations provides a simple and effective method for choosing a suitable transformation with real data. For comparison and for pedagogical purposes, we have included skewness and maximum likelihood methods for choosing . [2] discusses the use of maximum likelihood estimation for in the family of power transformations, . [3] discusses choosing a power transformation by minimizing absolute skewness. The robust skewness statistic computed using QuartileSkewness is sometimes called Bowley skewness. The use of the relative likelihood function for statistical inference is discussed in the books [4] and [5]. [1] W. S. Cleveland, Visualizing Data, Summit, NJ: Hobart Press, 1993. [2] G. E. P. Box and D. R. Cox, "An Analysis of Transformations," Journal of the Royal Statistical Society B, 26(2), 1964 pp. 211–252. [3] D. V. Hinkley, "On Power Transformations to Symmetry,” Biometrika, 62, 1975 pp. 101–111. [4] A. Azzalini, Statistical Inference, Boca Raton, FL: Chapman & Hall/CRC, 1996. [5] D. A. Sprott, Statistical Inference in Science, New York: Springer, 2000.
