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Influence Functions for Dimension Reduction Methods: An Example Influence Study of Principal Hessian Direction Analysis

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  • LUKE A. PRENDERGAST
  • JODIE A. SMITH

Abstract

. The first goal of this article is to consider influence analysis of principal Hessian directions (pHd) and highlight how such an analysis can provide valuable insight into its behaviour. Such insight includes reasons as to why pHd can sometimes return informative results when it is not expected to do so, and why many prefer a residuals‐based pHd method over its response‐based counterpart. The secondary goal of this article is to introduce a new influence measure applicable to many dimension reduction methods based on average squared canonical correlations. A general form of this measure is also given, allowing for application to dimension reduction methods other than pHd. A sample version of the measure is considered, with respect to pHd, with two example data sets.

Suggested Citation

  • Luke A. Prendergast & Jodie A. Smith, 2010. "Influence Functions for Dimension Reduction Methods: An Example Influence Study of Principal Hessian Direction Analysis," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(4), pages 588-611, December.
    • Handle: RePEc:bla:scjsta:v:37:y:2010:i:4:p:588-611
    DOI: 10.1111/j.1467-9469.2009.00666.x
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    References listed on IDEAS

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    1. Luke A. Prendergast, 2007. "Implications of influence function analysis for sliced inverse regression and sliced average variance estimation," Biometrika, Biometrika Trust, vol. 94(3), pages 585-601.
    2. L. A. Prendergast, 2005. "Influence Functions for Sliced Inverse Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(3), pages 385-404, September.
    3. Lexin Li & R. Dennis Cook & Christopher J. Nachtsheim, 2005. "Model‐free variable selection," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 285-299, April.
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    1. Heng-Hui Lue, 2015. "An Inverse-regression Method of Dependent Variable Transformation for Dimension Reduction with Non-linear Confounding," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(3), pages 760-774, September.
    2. Prendergast, Luke A. & Smith, Jodie A., 2022. "Influence functions for linear discriminant analysis: Sensitivity analysis and efficient influence diagnostics," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    3. Luke A. Prendergast & Simon J. Sheather, 2013. "On Sensitivity of Inverse Response Plot Estimation and the Benefits of a Robust Estimation Approach," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(2), pages 219-237, June.

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