Growth distribution and transformation
In tree species, the distribution of growth rates among individuals within a species is typically highly right-skewed, and this was the case in the two dominant species, Q. macrocarpa and Q. ellipsoidalis (Fig. 1a). It was thus necessary to transform growth, normalizing the distribution for use in statistical models relating growth to neighborhood density. Log-transformation would be routine but is invalid in this situation due to many growth rates of exactly zero (because of millimeter resolution in dbh). In this situation, one option is to convert zeros to the smallest positive growth rate that could be measured, in this case 1 mm dbh over 5 years. However, such an arbitrary value is a problem. When growth rates are low, the exact choice of a smallest positive has a large impact, because the logarithm varies rapidly at small numbers. In saplings, growth rates are typically < 1 mm y-1, so arbitrarily converting 0 growth to 0.2 vs. 0.1 is a large difference and can affect results.
A better transformation is the square root, or some other similar power, while maintaining any negatives. Define growth g as