An Evolutionary Topological Theory of Participatory Socioeconomic Development

The epistemological foundation of unity of knowledge is used to formulate a system-model of participatory socioeconomic development. The micro-properties of such a participatory development approach are deeply ethical in nature. In order to bring out the endogenous role of ethics derived from the moral law in reference to the epistemic foundation, and thereby explain their impact on the socioeconomic development experience, the methods of topological space and topological mappings are found to be appropriate for formalizing the complex nature of participatory dynamics. They emerge in the midst of institutional and organic interaction, integration, and evolution. The application of topological methods to participatory socioeconomic development thus opens up a field that is rarely used by mathematical economists. In this sense this article is an original contribution to the area of ethical development theory and its topological explanation and application.     

Definability and Invariance

In his thesis Para uma Teoria Geral dos Homomorfismos (1944), the Portuguese mathematician José Sebastião e Silva constructed an abstract or generalized Galois theory, that is intimately linked to F. Klein’s Erlangen Program and that foreshadows some notions and results of today’s model theory; an analogous theory was independently worked out by M. Krasner in 1938. In this paper, we present a version of the theory making use of tools which were not at Silva’s disposal. At the same time, we tried to keep in mind, so much as possible, the gist of his standpoint.     

Measurement Invariance,Predictive Invariance,and the Duality Paradox

The statistical literature on bias in psychological testing distinguishes at least two forms of bias: measurement bias and predictive bias. Measurement bias concerns group differences in the relationship between a test and the latent variable to be measured. Predictive bias concerns group differences in the relationship between a test and an external criterion. How are these two forms of bias related? For example. if a test is unbiased in the predictive sense, does this fact support the hypothesis that the test is unbiased in the measurement sense? A theorem is given that describes the conditions under which measurement invariance (lack of bias) is consistent with predictive invariance for the linear case. Paradoxically, these two forms of invariance are shown to be inconsistent under realistic conditions. This duality or inconsistency is illustrated in simulated data. The implications of the duality for group differences research are illustrated in real data involving gender and ethnic differences on the SAT. The phenomenon of duality may force a reinterpretation of common empirical findings of test criterion regression slope invariance. and of invariance in test validities. Other implications are discussed.     

Logical Operations and Invariance

I present a notion of invariance under arbitrary surjective mappings for operators on a relational finite type hierarchy generalizing the so-called Tarski–Sher criterion for logicality and I characterize the invariant operators as definable in a fragment of the first-order language. These results are compared with those obtained by Feferman and it is argued that further clarification of the notion of invariance is needed if one wants to use it to characterize logicality.     

Invariance and Scientific Representation

This paper argues for an invariantist view of scientific representation. First, it introduces the deflationary view that sees models in science as no different in essence from symbolic vehicles, which are derivative and adopted pragmatically, as a matter of convention. After analyzing this deflationary view and pointing out its shortcomings, it argues that representations play at least two radically different roles: one purely symbolic and therefore conventional, and the other epistemic. Finally, it argues that although it is correct to say that all particular external vehicles are introduced via some conventions, they get their jobs done because they are invariant with respect to particular conventions.     

Gender Differences in Social Dominance Orientation: Gender Invariance May Be Situation Invariance

Most studies of gender differences in social dominance orientation (SDO) have investigated settings more or less hierarchy-enhancing. The aim of this study was to explore gender differences in SDO (1) within social structures varying in equality-enhancement, i.e., communities differing in political equality between men and women, and (2) settings where equality was maximized and held constant, i.e., democratic, voluntary associations, but varying in gender composition, using survey data from a random sample of 831 Swedish adults (median age?=?47 years). There was a significant interaction effect between gender and membership in voluntary associations dominated by women, with men and women displaying equally low levels of SDO. In all other instances the main effect of gender remained statistically significant.     

Invariance,Symmetry And Rationality

Using recent work by Forster and Sober, I identify circumstances in which appeals to symmetries in physical laws are rational with respect to the aim of predictive accuracy. I then consider a Bayesian account of symmetry, and argue that such an account faces serious problems explaining when and why appeals to symmetry would be rational.     

Topological Essentialism

Philosophical Studies -     

Invariance in the MMPI's component structure

Bernstein and Garbin (1985b) suggested that the Minnesota Multiphasic Personality Inventory's major clinical scales (excluding Scales ? and 5) can be approximated by an oblique three-component structure: (a) Profile Elevation, (b) Test-taking Attitudes, and (c) Optimism-Pessimism, collectively termed the salient weight model. In this study, we found that both this model and the MMPI's principal component structure remain invariant across race, sex, and, as previously noted, context of testing (job applicants vs. inmates in correctional institutions). We further noted that several alternative definitions of Profile Elevation provide equally satisfactory representation of the relations among the scales. This factor invariance is necessary, but not sufficient, for the MMPI to be viewed as unbiased.     

Adult Age Invariance in Sentence Unitization

We examined age differences on a letter detection task that was performed on four-word sentences in order to examine how letter-level and word-level processing is integrated with sentence-level unitization. Sentence-level unitization is defined as the formation of a sentence-level code that delays access to letter-level or word-level codes. There was a consistent word-frequency disadvantage for both age groups in which it took longer to detect letters within high and medium high frequency words than in low frequency words. This finding suggested that word-level and letter-level input channels were involved in a parallel “horse race” to the central processor while simultaneously outputting codes to the sentence-level at the second tier of processing. The present data also revealed that both age groups showed a larger relative increase in reaction time for letter detection for the fourth word position compared to the third word position on syntactically “intact” sentences relative to syntactically “scrambled” sentences. These data indicated that both age groups formed sentence-level codes that made letter-level codes more difficult to access. Finally, older adults' data showed a larger cost than younger adults' data for scrambled sentences than for intact sentences. These results suggest that older adults are more reliant on a syntactical processor that facilitates the parafoveal preview during the reading of syntactically intact sentences. These data suggest that whereas there are age differences in the perceptual processing of letters and words, sentence unitization and syntactical processing remain intact with increasing adult age.     

Reduction Invariance and Prelec's Weighting Functions

Within the framework of separable utility theory, a condition, called reduction invariance, is shown to be equivalent to the 2-parameter family of weighting functions that Prelec (1998) derived from the condition called compound invariance. Reduction invariance, which is a variant on the reduction of compound gambles, is appreciably simpler and more easily testable than compound invariance, and a simpler proof is provided. Both conditions are generalized leading to more general weighting functions that include, as special cases, the families of functions that Prelec called exponential-power and hyperbolic logarithm and that he derived from two other invariance principles. However, of these various families, only Prelec's compound-invariance family includes, as a special case, the power function, which arises from the simplest probabilistic assumption of reduction of compound gambles. Copyright 2001 Academic Press.     

Invariance in Measurement and Prediction Revisited

Borsboom (Psychometrika, 71:425–440, 2006) noted that recent work on measurement invariance (MI) and predictive invariance (PI) has had little impact on the practice of measurement in psychology. To understand this contention, the definitions of MI and PI are reviewed, followed by results on the consistency between the two forms of invariance in the general case. The special parametric cases of factor analysis (strict factorial invariance) and linear regression analyses (strong regression invariance) are then described, along with findings on the inconsistency between the two forms of invariance in this context. Two numerical examples of inconsistency are reviewed in detail. The impact of violations of MI on accuracy of selection is illustrated. Finally, reasons for the slow dissemination of work on invariance are discussed, and the prospects for altering this situation are weighed. This paper is based on the Presidential Address given at the International Meeting of the Psychometric Society in Tokyo, Japan, on July 11, 2007. This research was supported by National Institute of Mental Health grants 1P30 MH 068685-01A1 and RO1 MH64707-01.     

Invariance of the weight parameter in information integration

The relative effect of each informational dimension in an information integration task is a joint function of its weight and the range of values over which it is varied. A method is developed for separating these two factors. Weights obtained by this method were compared across variations of stimulus range. Subjects rated the performance of students on the basis of midterm exam scores and final exam scores. For some subjects, the range of scores on the final exam was twice that on the midterm and the reverse was true for other subjects. An averaging model was shown to describe the results, and weights did not differ for different stimulus ranges. This was true for each of two instructional conditions: one in which a particular weighting strategy was prescribed and one in which there was no prescribed weighting. Students who were instructed to use a 2∶1 weighting were found to assign more than twice as much weight to the final as to the midterm.     

The Objectivity of Values: Invariance without Explanation

This paper develops and motivates minimalism about the objectivity of values: the objectivity of values is no more, and no less, than invariance with respect to possible differences in attitudes. Thus the relation of invariance need not have any particular explanation, or, indeed, any explanation at all, for values to count as fully objective. Values need not be metaphysically real, simply in order to be objective, as according to traditional realist views. But we should not suppose, as some recent writers do, that there is no special issue of objectivity to consider: the issue of objectivity is the issue whether or how values vary with our attitudes.     

Invariance Properties for General Diagnostic Classification Models

In item response theory (IRT), the invariance property states that item parameter estimates are independent of the examinee sample, and examinee ability estimates are independent of the test items. While this property has long been established and understood by the measurement community for IRT models, the same cannot be said for diagnostic classification models (DCMs). DCMs are a newer class of psychometric models that are designed to classify examinees according to levels of categorical latent traits. We examined the invariance property for general DCMs using the log-linear cognitive diagnosis model (LCDM) framework. We conducted a simulation study to examine the degree to which theoretical invariance of LCDM classifications and item parameter estimates can be observed under various sample and test characteristics. Results illustrated that LCDM classifications and item parameter estimates show clear invariance when adequate model data fit is present. To demonstrate the implications of this important property, we conducted additional analyses to show that using pre-calibrated tests to classify examinees provided consistent classifications across calibration samples with varying mastery profile distributions and across tests with varying difficulties.