The Impact of Fallible Item Parameter Estimates on Latent Trait Recovery

In this paper we propose an upward correction to the standard error (SE) estimation of [^(q)]_ML\hat{\theta}_{\mathrm{ML}} , the maximum likelihood (ML) estimate of the latent trait in item response theory (IRT). More specifically, the upward correction is provided for the SE of [^(q)]_ML\hat{\theta}_{\mathrm{ML}} when item parameter estimates obtained from an independent pretest sample are used in IRT scoring. When item parameter estimates are employed, the resulting latent trait estimate is called pseudo maximum likelihood (PML) estimate. Traditionally, the SE of [^(q)]_ML\hat{\theta}_{\mathrm{ML}} is obtained on the basis of test information only, as if the item parameters are known. The upward correction takes into account the error that is carried over from the estimation of item parameters, in addition to the error in latent trait recovery itself. Our simulation study shows that both types of SE estimates are very good when θ is in the middle range of the latent trait distribution, but the upward-corrected SEs are more accurate than the traditional ones when θ takes more extreme values.     

Theories with the Independence Property

A first-order theory T{{\mathcal T}} has the Independence Property provided T   \vdash (Q)(FT F₁ ú. . .úF_n){{{\mathcal T} \, \, \vdash (Q)(\Phi \Rightarrow {\Phi_1} \vee.\,.\,.\vee {\Phi_n})}} implies T   \vdash (Q)(FT F_i){{{\mathcal T} \, \, \vdash (Q)(\Phi \Rightarrow {\Phi_i})}} for some i whenever F,F₁, . . . ,F_n{{\Phi,\Phi_1,\,.\,.\,.\,,\Phi_n}} are formulae of a suitable type and (Q) is any quantifier sequence. Variants of this property have been noticed for some time in logic programming and in linear programming.     

A scaled difference chi-square test statistic for moment structure analysis

A family of scaling corrections aimed to improve the chi-square approximation of goodness-of-fit test statistics in small samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, Satorra-Bentler's (SB) scaling corrections are available in standard computer software. Often, however, the interest is not on the overall fit of a model, but on a test of the restrictions that a null model sayM ₀ implies on a less restricted oneM ₁. IfT ₀ andT ₁ denote the goodness-of-fit test statistics associated toM ₀ andM ₁, respectively, then typically the differenceT _d =T ₀−T ₁ is used as a chi-square test statistic with degrees of freedom equal to the difference on the number of independent parameters estimated under the modelsM ₀ andM ₁. As in the case of the goodness-of-fit test, it is of interest to scale the statisticT _d in order to improve its chi-square approximation in realistic, that is, nonasymptotic and nonormal, applications. In a recent paper, Satorra (2000) shows that the difference between two SB scaled test statistics for overall model fit does not yield the correct SB scaled difference test statistic. Satorra developed an expression that permits scaling the difference test statistic, but his formula has some practical limitations, since it requires heavy computations that are not available in standard computer software. The purpose of the present paper is to provide an easy way to compute the scaled difference chi-square statistic from the scaled goodness-of-fit test statistics of modelsM ₀ andM ₁. A Monte Carlo study is provided to illustrate the performance of the competing statistics. This research was supported by the Spanish grants PB96-0300 and BEC2000-0983, and USPHS grants DA00017 and DA01070.     

A 2-categorial Generalization of the Concept of Institution

After defining, for each many-sorted signature Σ = (S, Σ), the category Ter(Σ), of generalized terms for Σ (which is the dual of the Kleisli category for \mathbb T_S{\mathbb {T}_{\bf \Sigma}}, the monad in Set ^S determined by the adjunction T_S \dashv G_S{{\bf T}_{\bf \Sigma} \dashv {\rm G}_{\bf \Sigma}} from Set ^S to Alg(Σ), the category of Σ-algebras), we assign, to a signature morphism d from Σ to Λ, the functor d_à{{\bf d}_\diamond} from Ter(Σ) to Ter(Λ). Once defined the mappings that assign, respectively, to a many-sorted signature the corresponding category of generalized terms and to a signature morphism the functor between the associated categories of generalized terms, we state that both mappings are actually the components of a pseudo-functor Ter from Sig to the 2-category Cat. Next we prove that there is a functor Tr^Σ, of realization of generalized terms as term operations, from Alg(Σ) × Ter(Σ) to Set, that simultaneously formalizes the procedure of realization of generalized terms and its naturalness (by taking into account the variation of the algebras through the homomorphisms between them). We remark that from this fact we will get the invariance of the relation of satisfaction under signature change. Moreover, we prove that, for each signature morphism d from Σ to Λ, there exists a natural isomorphism θ ^d from the functor Tr^L °(Id ×d_à){{{\rm Tr}^{\bf {\bf \Lambda}} \circ ({\rm Id} \times {\bf d}_\diamond)}} to the functor Tr^S °(d^* ×Id){{\rm Tr}^{\bf \Sigma} \circ ({\bf d}^* \times {\rm Id})}, both from the category Alg(Λ) × Ter(Σ) to the category Set, where d* is the value at d of the arrow mapping of a contravariant functor Alg from Sig to Cat, that shows the invariant character of the procedure of realization of generalized terms under signature change. Finally, we construct the many-sorted term institution by combining adequately the above components (and, in a derived way, the many-sorted specification institution), but for a strict generalization of the standard notion of institution.     

The Power of Belnap: Sequent Systems for <Emphasis Type="BoldItalic">SIXTEEN</Emphasis><Subscript><Emphasis Type="Bold">3</Emphasis></Subscript>

The trilattice SIXTEEN₃\textit{SIXTEEN}_3 is a natural generalization of the well-known bilattice FOUR₂\textit{FOUR}_2. Cut-free, sound and complete sequent calculi for truth entailment and falsity entailment in SIXTEEN₃\textit{SIXTEEN}_3 are presented.     

The Geometry of Enhancement in Multiple Regression

In linear multiple regression, “enhancement” is said to occur when R ²=b′r>r′r, where b is a p×1 vector of standardized regression coefficients and r is a p×1 vector of correlations between a criterion y and a set of standardized regressors, x. When p=1 then b≡r and enhancement cannot occur. When p=2, for all full-rank R _xx≠I, R _xx=E[xx′]=V Λ V′ (where V Λ V′ denotes the eigen decomposition of R _xx; λ ₁>λ ₂), the set B₁:={b_i:R²=b_i￠r_i=r_i￠r_i;0 < R² ￡ 1}\boldsymbol{B}_{1}:=\{\boldsymbol{b}_{i}:R^{2}=\boldsymbol{b}_{i}'\boldsymbol{r}_{i}=\boldsymbol{r}_{i}'\boldsymbol{r}_{i};0R² ￡ 1;R²l_p ￡ r_i￠r_i < R²}0p≥3 (and λ ₁>λ ₂>⋯>λ _p ), both sets contain an uncountably infinite number of vectors. Geometrical arguments demonstrate that B ₁ occurs at the intersection of two hyper-ellipsoids in ℝ ^p . Equations are provided for populating the sets B ₁ and B ₂ and for demonstrating that maximum enhancement occurs when b is collinear with the eigenvector that is associated with λ _p (the smallest eigenvalue of the predictor correlation matrix). These equations are used to illustrate the logic and the underlying geometry of enhancement in population, multiple-regression models. R code for simulating population regression models that exhibit enhancement of any degree and any number of predictors is included in Appendices A and B.     

Empirically Corrected Rescaled Statistics for SEM with Small N and Large p

Survey data often contain many variables. Structural equation modeling (SEM) is commonly used in analyzing such data. With typical nonnormally distributed data in practice, a rescaled statistic T_rml proposed by Satorra and Bentler was recommended in the literature of SEM. However, T_rml has been shown to be problematic when the sample size N is small and/or the number of variables p is large. There does not exist a reliable test statistic for SEM with small N or large p, especially with nonnormally distributed data. Following the principle of Bartlett correction, this article develops empirical corrections to T_rml so that the mean of the empirically corrected statistics approximately equals the degrees of freedom of the nominal chi-square distribution. Results show that empirically corrected statistics control type I errors reasonably well even when N is smaller than 2p, where T_rml may reject the correct model 100% even for normally distributed data. The application of the empirically corrected statistics is illustrated via a real data example.     

Correcting Too Much or Too Little? The Performance of Three Chi-Square Corrections

This simulation study investigates the performance of three test statistics, T₁, T₂, and T₃, used to evaluate structural equation model fit under non normal data conditions. T₁ is the well-known mean-adjusted statistic of Satorra and Bentler. T₂ is the mean-and-variance adjusted statistic of Sattertwaithe type where the degrees of freedom is manipulated. T₃ is a recently proposed version of T₂ that does not manipulate degrees of freedom. Discrepancies between these statistics and their nominal chi-square distribution in terms of errors of Type I and Type II are investigated. All statistics are shown to be sensitive to increasing kurtosis in the data, with Type I error rates often far off the nominal level. Under excess kurtosis true models are generally over-rejected by T₁ and under-rejected by T₂ and T₃, which have similar performance in all conditions. Under misspecification there is a loss of power with increasing kurtosis, especially for T₂ and T₃. The coefficient of variation of the nonzero eigenvalues of a certain matrix is shown to be a reliable indicator for the adequacy of these statistics.     

Types of \mathsf{I} -Free Hereditary Right Maximal Terms

The implicational fragment of the relevance logic “ticket entailment” is closely related to the so-called hereditary right maximal terms. I prove that the terms that need to be considered as inhabitants of the types which are theorems of T_→ are in normal form and built in all but one case from and only. As a tool in the proof ordered term rewriting systems are introduced. Based on the main theorem I define FIT_→ – a Fitch-style calculus (related to FT_→) for the implicational fragment of ticket entailment.     

A first-principles study of the formation of θ′′ phase in Al–Cu alloys

The θ′′-Al₃Cu phase plays an important role in the precipitation process of Al–Cu alloys. This phase has a sandwich structure—every two {200}_Cu layers are separated by three {200}_Al layers. To analyse the formation mechanism of this structure, the elastic strain energy of the {200}_Cu and {200}_Al layers, and the chemical bonding energy that reflects the interaction between the electrons in Cu and neighbouring Al atoms are calculated and analysed by first-principles calculations, projected density of states and Bader analysis. Our computation results reveal that this sandwich structure is energetically preferred in the competition of elastic strain and chemical bonding energies. To minimise the elastic strain energy of {200}_Al and {200}_Cu layers, the {200}_Cu layers prefer being apart from each other, whereas the chemical bonding energy favours the opposite arrangement because the intermetallic bond between Al and Cu atoms may form through p-d hybridization.     

Finite basis problems and results for quasivarieties

Let be a finite collection of finite algebras of finite signature such that SP( ) has meet semi-distributive congruence lattices. We prove that there exists a finite collection ₁ of finite algebras of the same signature, , such that SP( ₁) is finitely axiomatizable.We show also that if , then SP( ₁) is finitely axiomatizable. We offer new proofs of two important finite basis theorems of D. Pigozzi and R. Willard. Our actual results are somewhat more general than this abstract indicates.While working on this paper, the first author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA) grant no. T37877 and the second author was supported by the US National Science Foundation grant no. DMS-0245622.Special issue of Studia Logica: Algebraic Theory of Quasivarieties Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko     

On the dynamics of institutional agreements

In this paper we investigate a logic for modelling individual and collective acceptances that is called acceptance logic. The logic has formulae of the form A_G:x j{\rm A}_{G:x} \varphi reading ‘if the agents in the set of agents G identify themselves with institution x then they together accept that j{\varphi} ’. We extend acceptance logic by two kinds of dynamic modal operators. The first kind are public announcements of the form x!y{x!\psi}, meaning that the agents learn that y{\psi} is the case in context x. Formulae of the form [x!y]j{[x!\psi]\varphi} mean that j{\varphi} is the case after every possible occurrence of the event x!ψ. Semantically, public announcements diminish the space of possible worlds accepted by agents and sets of agents. The announcement of ψ in context x makes all \lnoty{\lnot\psi} -worlds inaccessible to the agents in such context. In this logic, if the set of accessible worlds of G in context x is empty, then the agents in G are not functioning as members of x, they do not identify themselves with x. In such a situation the agents in G may have the possibility to join x. To model this we introduce here a second kind of dynamic modal operator of acceptance shifting of the form G:x-y{G:x\uparrow\psi}. The latter means that the agents in G shift (change) their acceptances in order to accept ψ in context x. Semantically, they make ψ-worlds accessible to G in the context x, which means that, after such operation, G is functioning as member of x (unless there are no ψ-worlds). We show that the resulting logic has a complete axiomatization in terms of reduction axioms for both dynamic operators. In the paper we also show how the logic of acceptance and its dynamic extension can be used to model some interesting aspects of judgement aggregation. In particular, we apply our logic of acceptance to a classical scenario in judgment aggregation, the so-called ‘doctrinal paradox’ or ‘discursive dilemma’ (Pettit, Philosophical Issues 11:268–299, 2001; Kornhauser and Sager, Yale Law Journal 96:82–117, 1986).     

用于处理不努力作答的标准化残差系列方法和混合多层模型法的比较

文章采用模拟研究, 分别在混合多层模型假设满足和违背的情境下, 比较了混合多层模型方法与标准化残差系列方法在识别不努力作答和参数估计方面的表现。结果显示：(1)不存在不努力作答或其严重性低时, 各方法表现接近; (2)不努力作答严重性高时, 固定参数迭代标准化残差法普遍更优, 混合多层模型法仅在假设满足且两种作答反应时差异大的条件下表现较好。建议实际应用中优先选择固定参数迭代标准化残差法。     

Variation with density of the transport properties of YBaCuO superconductors

Abstract

The dependence of the normal-state resistivity, resistive superconducting transition (T _c, ΔT _c), and of the upper critical-field slope (dH _c2/dT|T=T_c) on density has been investigated for several YBa₂Cu₃O_9-x(x?2.1) samples. The resistivity decreases rapidly with increasing density, whereas T _c and ΔT _c are rather insensitive to a change in density. dH _c2/dT|T_c depends sensitively on the preparation conditions. The implications of these results both for the evaluation of the parameters relevant to the understanding of the nature of superconductivity and for the technological applications of granular superconductors are briefly discussed.     

Dynamic Topological Logic Interpreted over Minimal Systems

Dynamic Topological Logic (DTL\mathcal{DTL}) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems, which are pairs consisting of a topological space X and a continuous function f : X → X. The function f is seen as a change in one unit of time; within DTL\mathcal{DTL} one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior of f is particularly interesting is that of minimal systems; these are dynamic topological systems which admit no proper, closed, f-invariant subsystems. In such systems the orbit of every point is dense, which within DTL\mathcal{DTL} translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while DTL\mathcal{DTL}s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that DTL\mathcal{DTL} interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of DTL\mathcal{DTL} which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic.     

The order–dsorder transition at twin boundaries in Cu3Au as studied by TEM

Abstract

The order-disorder phase transition at ∑ = 3{111}- and {211}-type twin boundaries has been studied in the L1²-ordered alloy Cu³Au employing in situ heating in a transmission electron microscope. Evidence is presented for an order-disorder phase transition occurring in these boundaries prior to the bulk transition. The temperature difference ΔT between the transition temperature of both boundary types and the bulk is estimated as 0.5K <ΔT<2K. No difference in T _c for the twin boundaries can be established as yet. The nature of the order-disorder transition in both twin boundaries is presumably a second-order phase transition.     

A Method of Generating Modal Logics Defining Jaśkowski’s Discussive Logic D<Subscript>2</Subscript>

Ja?kowski’s discussive logic D₂ was formulated with the help of the modal logic S5 as follows (see [7, 8]): \({A \in {D_{2}}}\) iff \({\ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\rm S}5}\), where (–)^? is a translation of discussive formulae from For^d into the modal language. We say that a modal logic L defines D₂ iff \({{\rm D}_{2} = \{A \in {\rm For^{\rm d}} : \ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\it L}\}}\). In [14] and [10] were respectively presented the weakest normal and the weakest regular logic which (?): have the same theses beginning with ‘\({\diamond}\)’ as S5. Of course, all logics fulfilling the above condition, define D₂. In [10] it was prowed that in the cases of logics closed under congruence the following holds: defining D₂ is equivalent to having the property (?). In this paper we show that this equivalence holds also for all modal logics which are closed under replacement of tautological equivalents (rte-logics).We give a general method which, for any class of modal logics determined by a set of joint axioms and rules, generates in the given class the weakest logic having the property (?). Thus, for the class of all modal logics we obtain the weakest modal logic which owns this property. On the other hand, applying the method to various classes of modal logics: rte-logics, congruential, monotonic, regular and normal, we obtain the weakest in a given class logic defining D₂.     

A Note on Neat Reducts

SC, CA, QA and QEA denote the class of Pinter’s substitution algebras, Tarski’s cylindric algebras, Halmos’ quasi-polyadic and quasi-polyadic equality algebras, respectively. Let . and . We show that the class of n dimensional neat reducts of algebras in K _m is not elementary. This solves a problem in [2]. Also our result generalizes results proved in [1] and [2]. Presented by Robert Goldblatt