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