Application of Transformations for Orthogonality

Shabuz, Zillur (2018) Application of Transformations for Orthogonality. PhD thesis, The Open University.

Abstract

In the statistical analysis of multivariate data, principal component analysis is widely used to form orthogonal variables. Realizing the difficulties of interpreting the principal components, Garthwaite et al. (2012) proposed two transformations, each of which yield surrogates of the original variables. Recently, Garthwaite and Koch (2016) proposed a transformation that also produces orthogonal components and can be used to partition the contribution of individual variables to a quadratic form. The aim of this thesis is to discover and explore applications of these transformations. We consider bootstrap methods for forming interval estimates of the contribution of individual variables to a Mahalanobis distance and their percentages. New bootstrap methods are proposed and compared with the percentile, bias-corrected percentile, non-studentized pivotal, and studentized pivotal methods via a large simulation study. The new methods enable use of a broader range of pivotal quantities than with standard pivotal methods, including vector pivotal quantities. Both equal-tailed intervals and shortest intervals are constructed; the latter are particularly attractive when (as here) squared quantities are of interest. Using a transformation to orthogonality, new measures are constructed for evaluating the contribution of individual variables to a regression sum of squares. The transformation yields an orthogonal approximation of the columns of the predictor scores matrix. The new measures are compared with three previously proposed measures through examples, and the properties of the measures are examined. We consider one new procedure and two older procedures for identifying collinear sets. The new procedure is based on transformations that partition variance inflation factors into contributions from individual variables, and they provide detailed information about the collinear sets. The procedures are compared using three examples from published studies that addressed issues of multicollinearity.