Profile transformations for reciprocal averaging and singular value decomposition

Power transformations of count data, including cell frequencies of a contingency table, have been well understood for nearly 100 years, with much of the attention focused on the square root transformation. Over the past 15 years, this topic has been the focus of some new insights into areas of correspondence analysis where two forms of power transformation have been discussed. One type considers the impact of raising the joint proportions of the cell frequencies of a table to a known power while the other examines the power transformation of the relative distribution of the cell frequencies. While the foundations of the graphical features of correspondence analysis rest with the numerical algorithms like reciprocal averaging, and other analogous techniques, discussions of the role of power transformations in reciprocal averaging have not been described. Therefore, this paper examines this link where a power transformation is applied to the cell frequencies of a two-way contingency table. In doing so, we show that reciprocal averaging can be performed under such a transformation to obtain row and column scores that provide the maximum association between the variables and the greatest discrimination between the categories. Finally, we discuss the connection between performing reciprocal averaging and singular value decomposition under this type of power transformation. The R function, powerRA.exe is included in the Appendix and performs reciprocal averaging of a power transformation of the cell frequencies of a two-way contingency table.