On the Escaig obstacle hypothesis for cross-slip in face-centered-cubic materials

There is a significant body of literature wherein a linear approximation of Escaig's model is used to justify the large experimentally measured activation-volumes for cross-slip in face-centered-cubic copper. Here, by examining the error between the linear approximation and the original theory, we show that this explanation is not satisfactory. The calculated value for activation volume in copper, using Escaig's original equations, yields ～60b ³ (b?=?Burgers vector) while the linear approximation yields 200b ³, the latter result fortuitously matching the experimental values.     

Approximating the conditional distribution of person FIT indexes for checking the rasch model

We present a simple approximation to the conditional distribution of goodness-of-fit statistics for the Rasch model, assuming that the item difficulties are known. The approximation is easily programmed, and gives relatively accurate assessments of conditionalp-values for tests of length 10 or more. A few generalizations are discussed.     

An approximation method for obtaining a maximized multiple criterion

A new approximation method for obtaining a maximized multiple criterion, based on the formula of Edgerton and Kolbe (2), is presented. By applications to examples from the literature (1, 2), the new method is evaluated in comparison with the Horst approximation (1), a suggested revision of the Horst procedure, and the more exact but more laborious iterative method for the principal axis solution of Hotelling.     

Single-particle excitations in a two-dimensional strong-coupling superconductor

We present calculations of the single-particle excitation spectrum for a twodimensional strong-coupling superconductor in a conserving approximation. Spectral weight at low frequency is substantially reduced as the superconducting transition is approached from the normal state. The suppression of low-energy excitations is a consequence of Cooper-pair fluctuations that are described self-consistently in the fluctuation exchange approximation. The static and uniform electromagnetic response provides a measure of the super fluid density and a fully self-consistent indication of the superconducting transition temperature.     

Generalized Rough Sets (Preclusivity Fuzzy-Intuitionistic (BZ) Lattices)

The standard Pawlak approach to rough set theory, as an approximation space consisting of a universe U and an equivalence (indiscernibility) relation R U x U, can be equivalently described by the induced preclusivity ("discernibility") relation U x U \ R, which is irreflexive and symmetric.We generalize the notion of approximation space as a pair consisting of a universe U and a discernibility or preclusivity (irreflexive and symmetric) relation, not necessarily induced from an equivalence relation. In this case the "elementary" sets are not mutually disjoint, but all the theory of generalized rough sets can be developed in analogy with the standard Pawlak approach. On the power set of the universe, the algebraic structure of the quasi fuzzy-intuitionistic "classical" (BZ) lattice is introduced and the sets of all "closed" and of all "open" definable sets with the associated complete (in general nondistributive) ortholattice structures are singled out.The rough approximation of any fixed subset of the universe is the pair consisting of the best "open" approximation from the bottom and the best "closed" approximation from the top. The properties of this generalized rough approximation mapping are studied in the context of quasi-BZ lattice structures of "closed-open" ordered pairs (the "algebraic logic" of generalized rough set theory), comparing the results with the standard Pawlak approach. A particular weak form of rough representation is also studied.     

Sequential Down Syndrome Screening: The Importance of First and Second Trimester Test Correlations when Calculating Risk

Down syndrome screening can be particularly effective when both first and second trimester tests are performed. However, the counseling of women who have received sequential first and second trimester screening can be problematic. We evaluated an approximation where the post-test risk from the first trimester screening is used as the new a priori risk for the second trimester screening. The approximation disregards between-trimester test correlations. The Down syndrome detection rate based on the approximation (90.2%) would be close to that obtained when all correlations were considered (90.8%) but the false positive rate would be 26% higher (3.9% versus 3.1%, respectively). For any particular woman, the use of the approximation could result in highly inaccurate risks. We conclude that the correlations that exist between first and second trimester screening tests preclude the use of second trimester risks derived from the direct product of separate first and second trimester screening. Counseling issues in the delivery of sequential screening are discussed.     

On approximate confidence intervals for measures of concordance

The use ofU-statistics based on rank correlation coefficients in estimating the strength of concordance among a group of rankers is examined for cases where the null hypothesis of random rankings is not tenable. The studentizedU-statistics is asymptotically distribution-free, and the Student-t approximation is used for small and moderate sized samples. An approximate confidence interval is constructed for the strength of concordance. Monte Carlo results indicate that the Student-t approximation can be improved by estimating the degrees of freedom.Research partially supported on ONR Contract N00014-82-K-0207.     

Agreement between observers when the categories are not specified in advance

To test the agreement between two observers who categorize a number of objects when the categories have not been specified in advance, Brennan & Light (1974) developed a statistic A′ and suggested a normal approximation for its distribution. In this paper it is shown that this approximation is inadequate particularly when one, or both, of the observers place a fairly equal number of objects in all of their categories. A chi-squared approximation to the distribution of A′ is developed and is shown to work well in a variety of situations. The relative powers of A′ and the ordinary X² test for association are dependent on the type of ‘agreement between the observers’ that is assumed. However a simulation for a fairly general type of agreement indicates that the X² test is more powerful. As the X² test is also much easier to apply, it would seem preferable in most situations.     

Generating interior search directions for multiobjective linear programming

A new multiobjective linear programming (MOLP) algorithm is presented. The algorithm uses a variant of Karmarkar's interior-point algorithm known as the affine-scaling primal algorithm. Using this single-objective algorithm, interior search directions are generated and used to provide an approximation to the gradient of the (implicitly known) utility function. The approximation is guided by assessing locally relevant preference information for the various interior directions through interaction with a decision maker (DM). The resulting algorithm is an interactive approach that makes its progress towards the solution through the interior of the constraints polytope.     

Factors Affecting the Adequacy and Preferability of Semiparametric Groups-Based Approximations of Continuous Growth Trajectories

Psychologists have long been interested in characterizing individual differences in change over time. It is often plausible to assume that the distribution of these individual differences is continuous in nature, yet theory is seldom so specific as to designate its parametric form (e.g., normal). Semiparametric groups-based trajectory models (SPGMs) were thus developed to provide a discrete approximation for continuously distributed growth of unknown form. Previous research has demonstrated the adequacy of the approximation provided by SPGM but only under relatively narrow, theoretically optimal conditions. Under alternative conditions, which may be more common in practice (e.g., higher dimension random effects, smaller sample sizes), this study shows that approximation adequacy can suffer. Furthermore, this study also evaluates whether SPGM's discrete approximation is preferable to a parametric trajectory model that assumes normally distributed random effects when in fact the distribution is modestly nonnormal. The answer is shown to depend on distributional characteristics of both repeated measures (binary or continuous) and random effects (bimodal or skewed). Implications for practice are discussed in light of empirical examples on externalizing behavior.     

Estimation of density of charged defects in amorphous chalcogenides from a.c. conductivity: Random-walk approach for bipolarons based on correlated barrier hopping

Abstract

Although the pair approximation (PA) based on correlated bamer hopping (CBH) is now fairly well established to explain the a.c. conductivity of amorphous chalcogenides, the density of charged states deduced from PA is large compared with those estimated from other studies, namely light-induced electron spin resonance and drift mobility. We have used the continuous-time random-walk approximation based on CBH to estimate the density of charged defects and have applied it to experimental data for amorphous As₂Se₃, and a density of charged defects consistent with other measurements has been estimated.     

Some exact results for random walk models with applications

This article presents a random walk model that can be analyzed without recourse to Wald's (1947) approximation, which neglects the excess over the absorbing barriers. Hence, the model yields exact predictions for the absorption probabilities and all mean conditional absorption times. We derive these predictions in some detail and fit them to the extensive data of an identification experiment published by Green et al. (1983). The fit of the model seems satisfactory. The relationship of the model to existing classes of random walk models (SPRT and SSR; see Luce, 1986) is discussed; for certain combinations of its parameters, the model belongs either to the SPRT or to the SSR class, or to both. We stress the theoretical significance of the knowledge of exact results for the evaluation of Wald's approximation and general properties of the several models proposed derived from this approximation.     

The relationship between Fisher's exact test and Pearson's chi-square test: A bayesian perspective

It is demonstrated in this paper that two major tests for 2 × 2 talbes are highly related from a Bayesian perspective. Although it is well-known that Fisher's exact and Pearson's chi-square tests are asymptotically equivalent, the present analysis shows that a formal similarity also exists in small samples. The key assumption that leads to the resemblance is the presence of a continuous parameter measuring association. In particular, it is shown that Pearson's probability can be obtained by integrating a two-moment approximation to the posterior distribution of the log-odds ratio. Furthermore, Pearson's chi-square test gave an excellent approximation to the actual Bayes probability in all 2×2 tables examined, except for those with extremely disproportionate marginal frequencies.     

Estimation Methods for Mixed Logistic Models with Few Clusters

For mixed models generally, it is well known that modeling data with few clusters will result in biased estimates, particularly of the variance components and fixed effect standard errors. In linear mixed models, small sample bias is typically addressed through restricted maximum likelihood estimation (REML) and a Kenward-Roger correction. Yet with binary outcomes, there is no direct analog of either procedure. With a larger number of clusters, estimation methods for binary outcomes that approximate the likelihood to circumvent the lack of a closed form solution such as adaptive Gaussian quadrature and the Laplace approximation have been shown to yield less-biased estimates than linearization estimation methods that instead linearly approximate the model. However, adaptive Gaussian quadrature and the Laplace approximation are approximating the full likelihood rather than the restricted likelihood; the full likelihood is known to yield biased estimates with few clusters. On the other hand, linearization methods linearly approximate the model, which allows for restricted maximum likelihood and the Kenward-Roger correction to be applied. Thus, the following question arises: Which is preferable, a better approximation of a biased function or a worse approximation of an unbiased function? We address this question with a simulation and an illustrative empirical analysis.     

Approximate uniqueness estimates for singular correlation matrices

The residual variance (one minus the squared multiple correlation) is often used as an approximation to the uniqueness in factor analysis. An upper bound approximation to the residual variance is given for the case when the correlation matrix is singular. The approximation is computationally simpler than the exact solution, especially since it can be applied routinely without prior knowledge as to the singularity or nonsingularity of the correlation matrix.     

On the chi‐square approximation to the exact distribution of goodness‐of‐fit statistics in multinomial models with composite hypotheses

Multinomial models are increasingly being used in psychology, and this use always requires estimating model parameters and testing goodness of fit with a composite null hypothesis. Goodness of fit is customarily tested with recourse to the asymptotic approximation to the distribution of the statistics. An assessment of the quality of this approximation requires a comparison with the exact distribution, but how to compute this exact distribution when parameters are estimated from the data appears never to have been defined precisely. The main goal of this paper is to compare two different approaches to defining this exact distribution. One of the approaches uses the marginal distribution and is, therefore, independent of the data; the other approach uses the conditional distribution of the statistics given the estimated parameters and, therefore, is data—dependent. We carried out a thorough study involving various parameter estimation methods and goodness‐of‐fit statistics, all of them members of the general class of power‐divergence measures. Included in the study were multinomial models with three to five cells and up to three parameters. Our results indicate that the asymptotic distribution is rarely a good approximation to the exact marginal distribution of the statistics, whereas it is a good approximation to the exact conditional distribution only when the vector of expected frequencies is interior to the sample space of the multinomial distribution.     

Approximation of Value Efficient Solutions

We introduce an approximation set to the value efficient set in multiobjective problems under partial information on the decision maker's preferences modelled by a vector value function. We show monotonicity and convergence properties based on increasingly precise vector value functions with two components, which improve the approximation and might be a support to possible solution methods. © 1997 John Wiley & Sons, Ltd.     

A note on the expected value of the Rand index

Two expectations of the adjusted Rand index (ARI) are compared. It is shown that the expectation derived by Morey and Agresti (1984, Educational and Psychological Measurement, 44, 33) under the multinomial distribution to approximate the exact expectation from the hypergeometric distribution (Hubert & Arabie, 1985, Journal of Classification, 2, 193) provides a poor approximation, and, in some cases, the difference between the two expectations can increase with the sample size. Proofs concerning the minimum and maximum difference between the two expectations are provided, and it is shown through simulation that the ARI can differ significantly depending on which expectation is used. Furthermore, when compared in a hypothesis testing framework, multinomial approximation overly favours the null hypothesis.     

Likelihood-ratio decision strategy for independent observations in thesame-different task: An approximation to the detection-theoretic model

An optimal decision strategy for deciding whether two things are the same or different is to adopt a likelihood-ratio criterion. The parametric equations for the receiver operating characteristic (ROC) based on the likelihood-ratio strategy when observations are independent are complicated; they require the numerical evaluation of a double integral. An approximation to the parametric equations for the likelihood-ratio strategy was developed. This approximation takes the form of a pair of equations that describe ROCs virtually indistinguishable from those of the full model.