Comparing normal score transformations to other transformations

The Normal score transformation (NST) is different from the Box-Cox, arcsine, and log transformations (BAL) in several ways:

• The NST function adapts to each particular dataset, whereas BAL transformations do not (for example, the log transformation function always takes the natural logarithm of the data).
• The goal of the NST is to make the random errors of the whole population (not only the sample) normally distributed. Due to this, it is important that the cumulative distribution of the sample accurately reflects the true cumulative distribution of the whole population (this requires correct sampling of the population and possibly declustering to account for preferential sampling in some locations of the study area). BAL, on the other hand, affects the sample data and can have goals of stabilizing the variance, correcting skewness, or making the distribution closer to normally distributed.
• The NST must occur after detrending the data so that covariance and semivariograms are calculated on residuals after trend correction. In contrast, BAL transformations are used to attempt to remove any relationship between the variance and the trend. Because of this, after the BAL transformation has been applied to the data, you can optionally remove the trend and model spatial autocorrelation. A consequence of this process is that you often get residuals that are approximately normally distributed, but this is not a specific goal of BAL transformations like it is for the NST transformation.