Transformation using (x + 0.5) to stabilize the variance of populations

摘要

Transformation is required to achieve homoscedasticity when we perform ANOVA to test the effect of factors on population abundance. The effectiveness of transformations decreases when the data contain zeros. Especially, the logarithmic transformation or the Box-Cox transformation is not applicable in such a case. For the logarithmic transformation, 1 is traditionally added to avoid such problems. However, there is no concrete foundation as to why 1 is added rather than other constants, such as 0.5 or 2, although the result of ANOVA is much influenced by the added constant. In this paper, I suggest that 0.5 is preferable to 1 as an added constant, because a discrete distribution defined in {0, 1, 2,... } is approximately described by a corresponding continuous distribution defined in (0, infinity) if we add 0.5. Numerical investigation confirms this prediction.