An Introduction to Kristof’s Theorem for Solving Least-Square Optimization Problems Without Calculus

Kristof’s Theorem (Kristof, 1970 Kristof, W. (1970). A theorem on the trace of certain matrix products and some applications. Journal of Mathematical Psychology, 7(3), 515–530. https://doi.org/10.1016/0022-2496(70)90037-4[Crossref], [Web of Science ®] , [Google Scholar]) describes a matrix trace inequality that can be used to solve a wide-class of least-square optimization problems without calculus. Considering its generality, it is surprising that Kristof’s Theorem is rarely used in statistics and psychometric applications. The underutilization of this method likely stems, in part, from the mathematical complexity of Kristof’s (1964 Kristof, W. (1964). Die beste orthogonale Transformation zur gegenseitigen Uberfiihrung zweier Faktorenmatrizen. Diagnostica, 10, 87–90. [Google Scholar], 1970 Kristof, W. (1970). A theorem on the trace of certain matrix products and some applications. Journal of Mathematical Psychology, 7(3), 515–530. https://doi.org/10.1016/0022-2496(70)90037-4[Crossref], [Web of Science ®] , [Google Scholar]) writings. In this article, I describe the underlying logic of Kristof’s Theorem in simple terms by reviewing four key mathematical ideas that are used in the theorem’s proof. I then show how Kristof’s Theorem can be used to provide novel derivations to two cognate models from statistics and psychometrics. This tutorial includes a glossary of technical terms and an online supplement with R (R Core Team, 2017 R Core Team (2017). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. Retrieved from https://www.R-project.org/ [Google Scholar]) code to perform the calculations described in the text.     

Psychobiology of altered states of consciousness

The article reviews the current knowledge regarding altered states of consciousness (ASC) (a) occurring spontaneously, (b) evoked by physical and physiological stimulation, (c) induced by psychological means, and (d) caused by diseases. The emphasis is laid on psychological and neurobiological approaches. The phenomenological analysis of the multiple ASC resulted in 4 dimensions by which they can be characterized: activation, awareness span, self-awareness, and sensory dynamics. The neurophysiological approach revealed that the different states of consciousness are mainly brought about by a compromised brain structure, transient changes in brain dynamics (disconnectivity), and neurochemical and metabolic processes. Besides these severe alterations, environmental stimuli, mental practices, and techniques of self-control can also temporarily alter brain functioning and conscious experience.     

Lexical selection in bilingual speech production does not involve language suppression

The "hard problem" in bilingual lexical access arises when translation-equivalent lexical representations are activated to roughly equal levels and, thus, compete equally for lexical selection. The language suppression hypothesis (D. W. Green, 1998) solves this hard problem through the suppression of lexical representations in the nontarget language. Following from this proposal is the prediction that lexical selection should take longer on a language switch trial because the to-be-selected representation was just suppressed on the previous trial. Inconsistent with this prediction, participants took no longer to name pictures in their dominant language on language switch trials than they did on nonswitch trials. These findings indicate that nontarget lexical representations are not suppressed. The authors suggest that these results undermine the viability of the language suppression hypothesis as a possible solution to the hard problem in bilingual lexical access.     

Measuring motivations for intercollegiate athlete alcohol use: a confirmatory factor analysis of the drinking motives measure

This study examined the psychometric properties of the Drinking Motives Measure (DMM) on a sample of 227 collegiate athletes. Confirmatory factor analyses indicated that the 4-factor structure of the DMM provided a better fit than either 2- or 1-factor models, but the overall fit of the 4-factor model was moderate at best. A revised 3-factor model consistent with prior research (M. L. Cooper, M. Russell, J. B. Skinner, & M. Windle, 1992) provided the best fit. Hierarchical multiple regression analyses indicated that the 3 DMM factors included in the revised model accounted for 17%-21% of the unique variance on alcohol consumption variables. Results provide preliminary evidence supporting the internal consistency, construct validity, and convergent validity of the revised 3-factor DMM with collegiate athletes.     

PROSPECTS FOR THE FIELD OF SCIENCE AND RELIGION: AN OCTOPUS VIEW

The organic unity between the head and the vital arms of the octopus is proposed as a metaphor for science and religion as an academic field. While the specific object of the field is to pursue second‐order reflections on existing and possible relations between sciences and religions, it is argued that several aspects of realism and normativity are constitutive to the field. The vital arms of the field are related to engagements with distinctive scientific theories, specialized philosophy of science, representative theological proposals, and the input from the study of world religions.     

Why envy outperforms admiration

Four studies tested the hypothesis that the emotion of benign envy, but not the emotions of admiration or malicious envy, motivates people to improve themselves. Studies 1 to 3 found that only benign envy was related to the motivation to study more (Study 1) and to actual performance on the Remote Associates Task (which measures intelligence and creativity; Studies 2 and 3). Study 4 found that an upward social comparison triggered benign envy and subsequent better performance only when people thought self-improvement was attainable. When participants thought self-improvement was hard, an upward social comparison led to more admiration and no motivation to do better. Implications of these findings for theories of social emotions such as envy, social comparisons, and for understanding the influence of role models are discussed.