Some properties of two measures of multivariate association

Two kinds of measures of multivariate association, based on Wilks' and the Bartlett-Nanda-Pillai trace criterionV, respectively, are compared in terms of properties of the univariateR ² which they generalize. A unified set of derivations of the properties is provided which are self-contained and not restricted to decompositions in canonical variates. One conclusion is that asymmetric index based on allows generalization of the multiplicative decomposition ofR ² in terms of squared partial correlations, but not the additive decomposition in terms of squared semipartial correlations, while the reverse is true for anasymmetric index based onV.We are indebted to Jos M. F. ten Berge for some fruitful discussions.     

Ezekiel’s classic estimator of the population squared multiple correlation coefficient: Monte Carlo-based extensions and refinements

Abstract

Ezekiel’s adjusted R² is widely used in linear regression analysis. The present study examined the statistical properties of Ezekiel’s measure through a series of Monte Carlo simulations. Specifically, we examined the bias and root mean squared error (RMSE) of Ezekiel’s adjusted R² relative to (a) the sample R² statistic, and (b) the sample R² minus the expected value of R². Simulation design factors consisted of sample sizes (N?=?50, 100, 200, 400), number of predictors (2, 3, 4, 5, 6), and population squared multiple correlations (ρ² = 0, .10, .25, .40, .60). Factorially crossing these design factors resulted in 100 simulation conditions. All populations were normal/Gaussian, and for each condition, we drew 10,000 Monte Carlo samples. Regarding systematic variation (bias), results indicated that with few exceptions, Ezekiel’s adjusted R² demonstrated the lowest bias. Regarding unsystematic variation (RMSE), the performance of Ezekiel’s measure was comparable to the other statistics, suggesting that the bias-variance tradeoff is minimal for Ezekiel’s adjusted R². Additional findings indicated that sample size-to-predictor ratios of 66.67 and greater were associated with low bias and that ratios of this magnitude were accompanied by large sample sizes (N?=?200 and 400), thus suggesting that researchers using Ezekiel’s adjusted R² should aim for sample sizes of 200 or greater in order to minimize bias when estimating the population squared multiple correlation coefficient. Overall, these findings indicate that Ezekiel’s adjusted R² has desirable properties and, in addition, these findings bring needed clarity to the statistical literature on Ezekiel’s classic estimator.     

A new allotropic structure of silver nanocrystals nucleated and grown over planar polymer molecules

Polymer molecules of polyvinyl alcohol (PVA) serve as a good Ag⁺?→?Ag⁰ reducer as well as a template to produce Ag⁰ nanocrystals of anisotropic shapes. Thin Ag–PVA films (50–100?µm thick) were prepared by carrying out the reaction in aqueous PVA (under heating conditions) and then spin casting the sample in this shape. A topotactic Ag⁺?→?Ag⁰ reaction and in-situ Ag⁰ growth occur in support over the PVA molecules (as per the local structure and Ag⁰–PVA interactions) of elongated surfaces. The Ag⁰ state is confirmed with 3d _5/2 and 3d _3/2 XPS signals of 368.37 and 374.13?eV binding energies, respectively. As analyzed with X-ray diffraction, it is a new allotrope with an orthorhombic structure, lattice parameters a?=?0.8535, b?=?0.8410?nm and c?=?0.4119?nm, and density ρ?=?9.69?g?cm^?3. On heating in air, the well-known cubic structure develops, a?=?0.4082?nm (ρ?=?10.53?g?cm^?3) after 150°C, whereas a?=?0.4071?nm (ρ?= 10.61?g?cm^?3) after 300°C for 2?h. According to proposed models, it is a vacancy-stabilized allotrope, which evolves from multiply twinned particles.     

A quasi-nonmetric method for multidimensional scaling VIA an extended euclidean model

An Extended Two-Way Euclidean Multidimensional Scaling (MDS) model which assumes both common and specific dimensions is described and contrasted with the standard (Two-Way) MDS model. In this Extended Two-Way Euclidean model then stimuli (or other objects) are assumed to be characterized by coordinates onR common dimensions. In addition each stimulus is assumed to have a dimension (or dimensions) specific to it alone. The overall distance between objecti and objectj then is defined as the square root of the ordinary squared Euclidean distance plus terms denoting the specificity of each object. The specificity,s _j , can be thought of as the sum of squares of coordinates on those dimensions specific to objecti, all of which have nonzero coordinatesonly for objecti. (In practice, we may think of there being just one such specific dimension for each object, as this situation is mathematically indistinguishable from the case in which there are more than one.)We further assume that _ij =F(d _ij ) +e _ij where _ij is the proximity value (e.g., similarity or dissimilarity) of objectsi andj,d _ij is the extended Euclidean distance defined above, whilee _ij is an error term assumed i.i.d.N(0, ²).F is assumed either a linear function (in the metric case) or a monotone spline of specified form (in the quasi-nonmetric case). A numerical procedure alternating a modified Newton-Raphson algorithm with an algorithm for fitting an optimal monotone spline (or linear function) is used to secure maximum likelihood estimates of the paramstatistics) can be used to test hypotheses about the number of common dimensions, and/or the existence of specific (in addition toR common) dimensions.This approach is illustrated with applications to both artificial data and real data on judged similarity of nations.     

Investigations of the optical spectra and local structure for nickel(II) ions in XF2 (X = Mg,Zn) single crystals

A complete diagonalization energy matrix method (CDM) is proposed for 3d ⁸ ions in orthorhombic site (D _{2 h} ) symmetry. As an application, the optical spectra of XF₂ : Ni²⁺ (X = Mg, Zn) is well explained on the basis of both the CDM and the semi-self-consistent field d-orbital (semi-SCF d-orbital) theory. In addition, by establishing the relationship between the optical spectra and local structure, it has been possible to calculate the crystal structure parameters for XF₂:Ni²⁺ from optical measurements on Ni²⁺ ions. The theoretical results are in good agreement with experimental findings.     

Reliability for the law of comparative judgment

A variance-components analysis is presented for paired comparisons in terms of three components:s, the scale value of the stimuli;d, a deviation from the linear model specified by the law of comparative judgment; andb, a binomial error component. Estimates are given for each of the three variances, _s ² , _d ² , and _b ² . Several coefficients, analogous to reliability coefficients, based on these three variances are indicated. The techniques are illustrated in a replicated comparison of handwriting specimens.This research was jointly supported in part by Princeton University, the Office of Naval Research under contract Nonr-1858(15), and the National Science Foundation under grant NSF G-642, and in part by Educational Testing Service. Reproduction in whole or in part is permitted for any purpose of the United States Government.Thanks are due to Ledyard Tucker and Frederic Lord for valuable suggestions on the development presented here.     

Axiomatizing logics closely related to varieties

Let V be a s.f.b. (strongly finitely based, see below) variety of algebras. The central result is Theorem 2 saying that the logic defined by all matrices (A, d) with d A V is finitely based iff the A V have 1^st order definable cosets for their congruences. Theorem 3 states a similar axiomatization criterion for the logic determined by all matrices (A, ^A), A V, a term which is constant in V. Applications are given in a series of examples.     

Internal Cue Theory: Calibration and Resolution of Confidence in General Knowledge

A theory is proposed in which beliefs in the form of internal cue validities mediate the processing of ecological cue validities in the assessment of confidence. The conditions necessary for perfect calibration are specified: (a) correspondence between ecological and internal validity, (b) perfect translation of internal validity into a confidence assessment, and (c) consistent utilization of cues. Process errors are then added to these conditions to investigate how calibration is affected by error variance of confidence assessments. To accomplish this, the calibration score (C) is decomposed into three additive parts: D² = bias, i.e., the squared difference between mean confidence and proportion correct; R² = resolution, i.e., the squared difference between the standard deviations of confidence and proportion correct; L = linearity, i.e., how closely the calibration curve follows a linear function. In the equation C = D² + R² + L, R² (resolution) reflects the subject′s ability to discriminate cue validities. Selection of items is a critical factor in studies of confidence. Informal selection with a tendency to avoid easy items results in overconfidence. Internal cue theory predicts both that overconfidence should disappear (in accordance with previous research) and that resolution should improve when item selection is made representative of the natural environment. Both predictions are confirmed by data from published studies on confidence in general knowledge. It is noteworthy that resolution is still poor and accounts for the major portion of miscalibration under representative item selection.     

A note on moran's measure of multiple rank correlation

A measure of multiple rank correlation,T _y.12 ², is proposed for the situation with no tied observations in the variables. The measure is a weighted average of two squared Kendall taus. It is shown thatT _y.12 ² is equivalent to a statistic previously proposed by Moran and thus a new interpretation is given to Moran's statistic.The author wishes to thank Nancy Anderson, Willard Larkin, and Kent Norman for their helpful comments.     

The asymptotic distribution of commonality components

Commonality components have been defined as a method of partitioning squared multiple correlations. In this paper, the asymptotic joint distribution of all 2 ^k – 1 squared multiple correlations is derived. The asymptotic joint distribution of linear combinations of squared multiple correlations is obtained as a corollary. In particular, the asymptotic joint distribution of commonality components are derived as a special case. Simultaneous and nonsimultaneous asymptotic confidence intervals for commonality components can be obtained from this distribution.This work was supported in part by the Spencer Foundation and the National Science Foundation.The authors are grateful to Bryna Siegel-Gorlick for her help in obtaining the data used in Example 4.3, and to the referees for their comments and suggestions.     

Comments on “the measurement of factorial indeterminacy”

Guttman's index of indeterminacy (2² – 1) measures the potential amount of uncertainty in picking the right alternative interpretation for a factor. When alternative solutions for a factor are equally likely to be correct, then the squared multiple correlation ² for predicting the factor from the observed variables is the average correlation _AB between independently selected alternative solutionsA andB, while var ( _AB )=(1 – ²)²/s, wheres is the dimensionality of the space in which unpredicted components of alternative solutions are to be found. When alternative solutions for the factor are not equally likely to be chosen, ² is the lower bound for E( _AB ); however, E( _AB ) need not be a modal value in the distribution of _AB . Guttman's index and E( _AB ) measure different aspects of the same indeterminacy problem.     

Intermediate logics with the same disjunctionless fragment as intuitionistic logic

Given an intermediate prepositional logic L, denote by L ^–d its disjuctionless fragment. We introduce an infinite sequence {J _n}_n1 of propositional formulas, and prove:(1)For any L: L ^–d =I ^–d (I=intuitionistic logic) if and only if J _n L for every n 1.Since it turns out that L{J _{n} n}1 = Ø for any L having the disjunction property, we obtain as a corollary that L ^–d = I ^–d for every L with d.p. (cf. open problem 7.19 of [5]). Algebraic semantic is used in the proof of the if part of (1). In the last section of the paper we provide a characterization in Kripke's semantic for the logics J _n =I+ +J _n (n 1).     

Sex differences in achievement self-attributions: An effect-size analysis

An effect-size analysis of the findings on sex differences in the use of achievement self-attributions was performed to determine if there were relationships between these two variables that accounted for more than 5% of the variance. The effect-size index used was an estimate of omega squared ( ² ). Two kinds of sex difference effects were examined: (a) the main effect for sex and (b) the simple sex difference effects for success and failure, respectively. With the exception of luck attributions for success, whose ² was .01, all ² s were less than .01. It was concluded that the studies surveyed provided no evidence of the existence of consequential relationships between sex, achievement, and self-attributions.     

Comparing the squared multiple correlation coefficients of non‐nested models: An examination of confidence intervals and hypothesis testing

The performance of the asymptotic method for comparing the squared multiple correlations of non‐nested models was investigated. Specifically, the increase in a given regression model's R² when one predictor is added was compared to the increase in the same model's R² when another predictor is added. This comparison can be used to determine predictor importance and is the basis for procedures such as Dominance Analysis. Results indicate that the asymptotic procedure provides the expected coverage rates for sample sizes of 200 or more, but in many cases much higher sample sizes are required to achieve adequate power. Guidelines and computations are provided for the determination of adequate sample sizes for hypothesis testing.     

A study of the spin-Hamiltonian parameters for Cr5+ at the rhombically-distorted tetrahedral P5+ site of Ca2PO4Cl crystal using a two-mechanism model

The complete high-order perturbation formulae of spin-Hamiltonian (SH) parameters (g factors g_i and hyperfine structure constants A_i , where i = x, y, z) containing contributions from both the crystal-field (CF) and charge-transfer (CT) mechanisms (the latter mechanism is neglected in the widely-used CF theory) are established for d¹ ions in rhombic tetrahedra. From these formulae, the SH parameters of Cr⁵⁺ ion at the rhombically-distorted tetrahedral P⁵⁺ site of Ca₂PO₄Cl crystal are calculated. The CF and CT energy levels used in the calculation are obtained from the optical spectra of the studied Ca₂PO₄Cl : Cr⁵⁺ crystal. The calculated results showed reasonable agreement with the experimental values. The signs of the hyperfine structure constants A_i and the relative importance of the CT mechanism to SH parameters are acquired from the calculations.     

Efficiency of multiple-choice tests as a function of spread of item difficulties

The validity of a univocal multiple-choice test is determined for varying distributions of item difficulty and varying degrees of item precision. Validity is a function of _d ² + _v ² , where _d measures item unreliability and _v measures the spread of item difficulties. When this variance is very small, validity is high for one optimum cutting score, but the test gives relatively little valid information for other cutting scores. As this variance increases, eta increases up to a certain point, and then begins to decrease. Screening validity at the optimum cutting score declines as this variance increases, but the test becomes much more flexible, maintaining the same validity for a wide range of cutting scores. For items of the type ordinarily used in psychological tests, the test with uniform item difficulty gives greater over-all validity, and superior validity for most cutting scores, compared to a test with a range of item difficulties. When a multiple-choice test is intended to reject the poorestF per cent of the men tested, items should on the average be located at or above the threshold for men whose true ability is at theFth percentile.This research was performed under contract Nop 536 with the Bureau of Naval Personnel, and received additional support from the Bureau of Research and Service, College of Education, University of Illinois.     

Finite structural axiomatization of every finite-valued propositional calculus

In [2] A. Wroski proved that there is a strongly finite consequence C which is not finitely based i.e. for every consequence C ⁺ determined by a finite set of standard rules C C ⁺. In this paper it will be proved that for every strongly finite consequence C there is a consequence C ⁺ determined by a finite set of structural rules such that C(Ø)=C ⁺(Ø) and = (where , are consequences obtained by adding to the rules of C, C ⁺ respectively the rule of substitution). Moreover it will be shown that under certain assumptions C=C ⁺.     

Extensions of the significance test for one-parameter signal detection hypotheses

The basic models of signal detection theory involve the parametric measure,d, generally interpreted as a detectability index. Given two observers, one might wish to know whether their detectability indices are equal or unequal. Gourevitch and Galanter (1967) proposed a large sample statistical test that could be used to test the hypothesis of equald values. In this paper, their large two sample test is extended to aK-sample detection test. If the null hypothesisd ₁=d ₂=...=d _K is rejected, one can employ the post hoc confidence interval procedure described in this paper to locate possible statistically significant sources of variance and differences. In addition, it is shown how one can use the Gourevitch and Galanter statistics to testd=0 for a single individual.This paper was written while the author was associated with the Institute of Human Learning at the University of California at Berkeley.     

Limited‐information goodness‐of‐fit testing of item response theory models for sparse 2P tables

Bartholomew and Leung proposed a limited‐information goodness‐of‐fit test statistic (Y) for models fitted to sparse 2^P contingency tables. The null distribution of Y was approximated using a chi‐squared distribution by matching moments. The moments were derived under the assumption that the model parameters were known in advance and it was conjectured that the approximation would also be appropriate when the parameters were to be estimated. Using maximum likelihood estimation of the two‐parameter logistic item response theory model, we show that the effect of parameter estimation on the distribution of Y is too large to be ignored. Consequently, we derive the asymptotic moments of Y for maximum likelihood estimation. We show using a simulation study that when the null distribution of Y is approximated using moments that take into account the effect of estimation, Y becomes a very useful statistic to assess the overall goodness of fit of models fitted to sparse 2^P tables.