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Analysis of distance for structured multivariate data and extensions to multivariate analysis of variance

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  • J. C. Gower
  • W. J. Krzanowski

Abstract

Many data sets in practice fit a multivariate analysis of variance (MANOVA) structure but are not consonant with MANOVA assumptions. One particular such data set from economics is described. This set has a 24 factorial design with eight variables measured on each individual, but the application of MANOVA seems inadvisable given the highly skewed nature of the data. To establish a basis for analysis, we examine the structure of distance matrices in the presence of a priori grouping of units and show how the total squared distance among the units of a multivariate data set can be partitioned according to the factors of an external classification. The partitioning is exactly analogous to that in the univariate analysis of variance. It therefore provides a framework for the analysis of any data set whose structure conforms to that of MANOVA, but which for various reasons cannot be analysed by this technique. Descriptive aspects of the technique are considered in detail, and inferential questions are tackled via randomization tests. This approach provides a satisfactory analysis of the economics data.

Suggested Citation

  • J. C. Gower & W. J. Krzanowski, 1999. "Analysis of distance for structured multivariate data and extensions to multivariate analysis of variance," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 48(4), pages 505-519.
    • Handle: RePEc:bla:jorssc:v:48:y:1999:i:4:p:505-519
    DOI: 10.1111/1467-9876.00168
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    1. repec:kap:iaecre:v:15:y:2009:i:2:p:186-206 is not listed on IDEAS
    2. Jun Li & Jifei Ban & Louis S. Santiago, 2011. "Nonparametric Tests for Homogeneity of Species Assemblages: A Data Depth Approach," Biometrics, The International Biometric Society, vol. 67(4), pages 1481-1488, December.
    3. Rathke, Alex A.T. & Rezende, Amaury J. & Watrin, Christoph, 2020. "Classification of transfer pricing systems across countries," International Economics, Elsevier, vol. 164(C), pages 151-167.
    4. Matthias Studer & Gilbert Ritschard & Alexis Gabadinho & Nicolas S. Müller, 2011. "Discrepancy Analysis of State Sequences," Sociological Methods & Research, , vol. 40(3), pages 471-510, August.
    5. W. J. Krzanowski, 2004. "Biplots for Multifactorial Analysis of Distance," Biometrics, The International Biometric Society, vol. 60(2), pages 517-524, June.
    6. Philip T. Reiss & M. Henry H. Stevens & Zarrar Shehzad & Eva Petkova & Michael P. Milham, 2010. "On Distance-Based Permutation Tests for Between-Group Comparisons," Biometrics, The International Biometric Society, vol. 66(2), pages 636-643, June.
    7. Marti J. Anderson, 2006. "Distance-Based Tests for Homogeneity of Multivariate Dispersions," Biometrics, The International Biometric Society, vol. 62(1), pages 245-253, March.
    8. Nickolay T. Trendafilov & Tsegay Gebrehiwot Gebru, 2016. "Recipes for sparse LDA of horizontal data," METRON, Springer;Sapienza Università di Roma, vol. 74(2), pages 207-221, August.
    9. Łukasz Smaga & Jin‐Ting Zhang, 2020. "Linear hypothesis testing for weighted functional data with applications," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(2), pages 493-515, June.
    10. Vsevolozhskaya, O.A. & Greenwood, M.C. & Bellante, G.J. & Powell, S.L. & Lawrence, R.L. & Repasky, K.S., 2013. "Combining functions and the closure principle for performing follow-up tests in functional analysis of variance," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 175-184.
    11. W. J. Krzanowski, 2006. "Sensitivity in Metric Scaling and Analysis of Distance," Biometrics, The International Biometric Society, vol. 62(1), pages 239-244, March.
    12. Irène Gijbels & Marek Omelka, 2013. "Testing for Homogeneity of Multivariate Dispersions Using Dissimilarity Measures," Biometrics, The International Biometric Society, vol. 69(1), pages 137-145, March.
    13. J. Fernando Vera & Rodrigo Macías, 2017. "Variance-Based Cluster Selection Criteria in a K-Means Framework for One-Mode Dissimilarity Data," Psychometrika, Springer;The Psychometric Society, vol. 82(2), pages 275-294, June.

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