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Power transformations in correspondence analysis

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  • Greenacre, Michael

Abstract

Power transformations of positive data tables, prior to applying the correspondence analysis algorithm, are shown to open up a family of methods with direct connections to the analysis of log-ratios. Two variations of this idea are illustrated. The first approach is simply to power transform the original data and perform a correspondence analysis -- this method is shown to converge to unweighted log-ratio analysis as the power parameter tends to zero. The second approach is to apply the power transformation to the contingency ratios, that is, the values in the table relative to expected values based on the marginals -- this method converges to weighted log-ratio analysis, or the spectral map. Two applications are described: first, a matrix of population genetic data which is inherently two-dimensional, and second, a larger cross-tabulation with higher dimensionality, from a linguistic analysis of several books.

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  • Greenacre, Michael, 2009. "Power transformations in correspondence analysis," Computational Statistics & Data Analysis, Elsevier, vol. 53(8), pages 3107-3116, June.
    • Handle: RePEc:eee:csdana:v:53:y:2009:i:8:p:3107-3116
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    References listed on IDEAS

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    1. Michael Greenacre, 2008. "Correspondence analysis of raw data," Economics Working Papers 1112, Department of Economics and Business, Universitat Pompeu Fabra, revised Jul 2009.
    2. John Aitchison & Michael Greenacre, 2002. "Biplots of compositional data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 51(4), pages 375-392, October.
    3. Nenadic, Oleg & Greenacre, Michael, 2007. "Correspondence Analysis in R, with Two- and Three-dimensional Graphics: The ca Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 20(i03).
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    1. Blasius, J. & Greenacre, M. & Groenen, P.J.F. & van de Velden, M., 2009. "Special issue on correspondence analysis and related methods," Computational Statistics & Data Analysis, Elsevier, vol. 53(8), pages 3103-3106, June.
    2. Lombardo, Rosaria & Camminatiello, Ida & D'Ambra, Antonello & Beh, Eric J., 2021. "Assessing the Italian tax courts system by weighted three-way log-ratio analysis," Socio-Economic Planning Sciences, Elsevier, vol. 73(C).
    3. Tsagris, Michail & Preston, Simon & T.A. Wood, Andrew, 2016. "Improved classi cation for compositional data using the $\alpha$-transformation," MPRA Paper 67657, University Library of Munich, Germany.
    4. Michael Greenacre, 2023. "The chi-square standardization, combined with Box-Cox transformation, is a valid alternative to transforming to logratios in compositional data analysis," Economics Working Papers 1857, Department of Economics and Business, Universitat Pompeu Fabra.
    5. Michail Tsagris & Simon Preston & Andrew T. A. Wood, 2016. "Improved Classification for Compositional Data Using the α-transformation," Journal of Classification, Springer;The Classification Society, vol. 33(2), pages 243-261, July.
    6. Tsai, Arthur C. & Liou, Michelle & Simak, Maria & Cheng, Philip E., 2017. "On hyperbolic transformations to normality," Computational Statistics & Data Analysis, Elsevier, vol. 115(C), pages 250-266.
    7. J. L. Scealy & Patrice de Caritat & Eric C. Grunsky & Michail T. Tsagris & A. H. Welsh, 2015. "Robust Principal Component Analysis for Power Transformed Compositional Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 136-148, March.
    8. Antonello D’Ambra & Anna Crisci & Luigi D’Ambra, 2017. "Weighted log ratio analysis by means of Poisson factor models: a case study to evaluate the quality of the public services offered to the citizens," Quality & Quantity: International Journal of Methodology, Springer, vol. 51(2), pages 629-639, March.
    9. Ida Camminatiello & Antonello D’Ambra & Luigi D’Ambra, 2022. "The association in two-way ordinal contingency tables through global odds ratios," METRON, Springer;Sapienza Università di Roma, vol. 80(1), pages 9-22, April.
    10. Michael Greenacre & Paul Lewi, 2009. "Distributional Equivalence and Subcompositional Coherence in the Analysis of Compositional Data, Contingency Tables and Ratio-Scale Measurements," Journal of Classification, Springer;The Classification Society, vol. 26(1), pages 29-54, April.

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    JEL classification:

    • C19 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Other
    • C88 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Other Computer Software

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