Projection error term in Gower's interpolation

New points can be superimposed on a Euclidean configuration obtained as a result of a metric multidimensional scaling at coordinates given by Gower's interpolation formula. The procedure amounts to discarding a, possibly nonnull, coordinate along an additional dimension. We derive an analytical formula for this projection error term and, for real data problems, we describe a statistical method for testing its significance, as a cautionary device prior to further distance-based predictions.     

Goodness-of-fit tests based on maximum correlations and their orthogonal decompositions

Summary. We propose a goodness-of-fit statistic Q _n based on the Hoeffding maximum correlation for testing uniformity and we show its relationship to Gini's mean difference. We compute exact and asymptotic critical values and study the power of the test proposed against a representative set of alternatives.     

Socio-economic Indexes in Surveys for Comparisons between Countries

The study of socio-economic inequalities from across-national perspective has been hampered by the lack of adequate common indices of socio-economic status that can be used in a self-report survey instrument. This paper examines the construction and the properties of global social indexes in general, and of the Family Affluence Scale (henceforth FAS) in particular. The paper proposes a new strategy for making comparisons of the global index with stratified data, building a revised FAS based on Adapted Canonical Variate Analysis (henceforth ACVA). This alternative strategy for constructing a global index is available in standard software, and the new proposal for stratified data only requires a simple program, which is justified, explained and provided in the text. Data come from the 1998 Health Behaviour in School-Aged Children (HBSC), a WHO Cross-National Study using cluster sampling of schoolchildren from five countries: Denmark, Latvia, Portugal, Scotland and the USA. The results reveal that in every country we would have had a completely different evaluation of the three indicators of Family Affluence if we had used either linear or nonlinear approaches to compute the global indexes. Moreover, Family Affluence comparisons among countries shows that the relative contribution of the three indicators to the overall FAS, changes from country to country. We conclude that separate indicators of Family Affluence are not equally relevant in each country and, as a consequence, do not contribute equally to the global index. For cross-cultural studies, the strategy for constructing an index should be country specific. The methodological developments presented in the paper open up opportunities to study socio-economic patterning of health among young people in the developed world, since self completed surveys can now employ a common measure of family material wealth. The findings show that the RFAS (Revised FAS) is a useful index of socio-economic status for use in national and cross-national surveys of adolescent health and health behaviour. The new strategy for weighting observed indicators in the index gives it enhanced power to detect in equalities.     

Karhunen–Loève Basis in Goodness-of-Fit Tests Decomposition: An Evaluation

In a previous article (Grané and Fortiana, 2006 Grané , A. , Fortiana , J. ( 2006 ). An adaptive goodness-of-fit test . Commun. Statist. A. Theor. Meth 35 ( 6 ): 1141 – 1155 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), we studied a flexible class of goodness-of-fit tests associated with an orthogonal sequence, the Karhunen–Loève decomposition of a stochastic process derived from the null hypothesis. Generally speaking, these tests outperform Kolmogorov–Smirnov and Cramér–von Mises, but we registered several exceptions. In this work we investigate the cause of these anomalies and, more precisely, whether and when such poor behavior may be attributed to the orthogonal sequence itself, by replacing it with the Legendre polynomials, a commonly used basis for smooth tests. We find an easily computable formula for the Bahadur asymptotic relative efficiency, a helpful quantity in choosing an adequate basis.     

Selection of Predictors in Distance-Based Regression

Distance-based regression is a prediction method consisting of two steps: from distances between observations we obtain latent variables which, in turn, are the regressors in an ordinary least squares linear model. Distances are computed from actually observed predictors by means of a suitable dissimilarity function. Being generally nonlinearly related with the response, their selection by the usual F tests is unavailable. In this article, we propose a solution to this predictor selection problem by defining generalized test statistics and adapting a nonparametric bootstrap method to estimate their p-values. We include a numerical example with automobile insurance data.     

An Adaptive Goodness-of-Fit Test

In Fortiana and Grané (2003 Fortiana , J. , Grané , A. ( 2003 ). Goodness-of-fit tests based on maximum correlations and their orthogonal decompositions . J. Roy. Statist. Soc. B 65 ( 1 ): 115 – 126 . [CSA] [CROSSREF] [Crossref] , [Google Scholar]), we introduced the statistic Q _n , based on Hoeffding's maximum correlation, as a general-purpose goodness-of-fit test of uniformity. It admits an expansion along a countable set of orthogonal axes, originating a sequence of statistics. Linear combinations of a given number p of terms in this sequence have easy-to-compute probability distributions, either the exact ones for a finite sample or their normal asymptotic approximations for a large sample. In this article we develop an algorithm for tailoring a statistic within this class of linear combinations to test uniformity with optimal power against a specific alternative or family of alternatives.     

Interaction Terms in Distance-Based Regression

We propose a method of including polynomial and interaction terms in Distance-Based Regression (Cuadras and Arenas, 1990 Cuadras , C. M. , Arenas , C. ( 1990 ). A distance based regression model for prediction with mixed data . Commun. Statist. A Theor. Meth. 19 : 2261 – 2279 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), relying on properties of a semi-Hadamard or Khatri-Rao product of matrices. We demonstrate its application to real data examples.