On the asymptotic distribution of Pearson’s X 2 in cross-validation samples

In categorical data analysis, two-sample cross-validation is used not only for model selection but also to obtain a realistic impression of the overall predictive effectiveness of the model. The latter is of particular importance in the case of highly parametrized models capable of capturing every idiosyncracy of the calibrating sample. We show that for maximum likelihood estimators or other asymptotically efficient estimators Pearson's X ² is not asymptotically chi-square in the two-sample cross-validation framework due to extra variability induced by using different samples for estimation and goodness-of-fit testing. We propose an alternative test statistic, X _xval ², obtained as a modification of X ² which is asymptotically chi-square with C−1 degrees of freedom in cross-validation samples. Stochastically, X _xval ² ≤ X ². Furthermore, the use of X ² instead of X _xval ² with a χ _C −1² reference distribution may provide an unduly poor impression of fit of the model in the cross-validation sample. This paper is dedicated to the memory of Michael V. Levine. Requests for reprints should be sent to Albert Maydeu-Olivares, Faculty of Psychology, University of Barcelona, P. Valle de Hebrón, 171, 0835 Barcelona, Spain.     

Limited Information Goodness-of-fit Testing in Multidimensional Contingency Tables

We introduce a family of goodness-of-fit statistics for testing composite null hypotheses in multidimensional contingency tables. These statistics are quadratic forms in marginal residuals up to order r. They are asymptotically chi-square under the null hypothesis when parameters are estimated using any asymptotically normal consistent estimator. For a widely used item response model, when r is small and multidimensional tables are sparse, the proposed statistics have accurate empirical Type I errors, unlike Pearson's X ². For this model in nonsparse situations, the proposed statistics are also more powerful than X ². In addition, the proposed statistics are asymptotically chi-square when applied to subtables, and can be used for a piecewise goodness-of-fit assessment to determine the source of misfit in poorly fitting models. This research has been supported by the Department of Universities, Research, and Information Society (DURSI) of the Catalan Government, by grant BSO2003-08507 of the Spanish Ministry of Science and Technology, and an NSERC Canada grant. We are grateful to the referees for comments leading to improvements.     

Critical Notice: Paul Russell's The Riddle of Hume's Treatise: Skepticism,Naturalism, and Irreligion (OUP 2008)

In The Riddle of Hume's Treatise: Skepticism, Naturalism, and Irreligion (2008), Paul Russell makes a strong case for the claim that “The primary aim of Hume's series of skeptical arguments, as developed and distributed throughout the Treatise, is to discredit the doctrines and dogmas of Christian philosophy and theology with a view toward redirecting our philosophical investigations to areas of ‘common life,’ with the particular aim of advancing ‘the science of man’” (2008, 290). Understanding Hume in this way, according to Russell, sheds light on the “ultimate riddle” of the Treatise: “is it possible to reconcile Hume's (extreme) skeptical principles and conclusions with his aim to advance the ‘science of man’” (2008, 3)? Or does Hume's skepticism undermine his “secular, scientific account of the foundations of moral life in human nature” (290)? Russell's controversial thesis is that “the irreligious nature of Hume's fundamental intentions in the Treatise” is essential to solving the riddle (11). Russell makes a compelling case for Hume's irreligion as well as his atheism. Contrary to this interpretation I argue that Hume is an irreligious theist and not an atheist.     

Counseling to End an Epidemic: Revisiting the Ethics of HIV/AIDS

In the decades since HIV/AIDS was first identified, medical discoveries have advanced prevention and treatment, and shifting demographics have changed the face of the illness. Despite these changes, the counseling literature on HIV/AIDS and its ethical implications for counselors have remained limited and stagnant. Whereas past discussions have focused primarily on issues of confidentiality, this article addresses multiple ethical implications of HIV/AIDS to reflect current knowledge about the illness and its effects on clients.