Effect of various transformations on the analysis of percentage data

Proper analysis of transformed data arrays (such as percentages) requires paying special attention to the effects of the transformation process itself. Effects of several commonly used transformations (including percentage formation, row and column normalization, and the square root transformation) have been examined with emphasis placed on changes in the statistical and geometrical properties of column vectors that accompany the application of the transformation. Even though many transformations, including taking the square root, open up the percentage array, this does not allow one to ignore the fact that percentage formation may have considerably modified the statistical and geometrical properties of the columns of the matrix. In preparing to analyze percentages one should give serious consideration to using the row normalized form of the data matrix. The individual elements in such a matrix are the direction cosines of the vector in M-dimensional space, the row vectors are of unit length, and the row normalized matrix computed from the closed array is equal to the row normalized, open matrix that is unobservable. Application of a column transformation (such as range restriction and proportion of the maximum) destroys the equality of the open and percentage row normalized matrices. Despite repeated claims to the contrary, one can not deduce the statistical and geometrical properties of the open matrix given only the statistical and geometrical properties of the closed matrix.