A semi-orthogonal dependent factor solution

The paper presents a general framework for the dependent factor method in which judgmental as well as analytic criteria may be employed.The procedure involves a semi-orthogonal transformation of an oblique solution comprising a number of reference factors and a number of experimental factors (composites). It determines analytically the residual factors in the experimental field keeping the reference field constant. The method is shown to be a generalization of the multiple factor procedure [Thurstone, 1947] in so far as it depends on the use of generalized inverse in derivingpartial structure from total pattern. It is also shown to provide an example of the previously empty category (Case III) of the Harris-Kaiser generalization [1964]. A convenient computational procedure is provided. It is based on an extension of Aitken's [1937] method of pivotal condensation of a triple-product matrix to the evaluation of a matrix of the formH — VA ^–1 U' (for a nonsingularA).     

A note concerning the creager—Valentine paper

TheU _i statistic, discussed by Creager and Valentine in a recent paper, is defined as a standard error of estimate for the regression of a composite, consisting ofn components, on then–1 components, exclusive of thejth component. It is shown that this standard error equals the standard error of estimate for the prediction of thejth component from the othern–1 components. This equality is used to suggest an efficient procedure to compute all primary uniquenessesU _i .     

Maximization of sums of quotients of quadratic forms and some generalizations

Monotonically convergent algorithms are described for maximizing six (constrained) functions of vectors x, or matricesX with columns x₁, ..., x _r . These functions are h₁(x)= _k (xA _kx)(xC _kx)^–1, H₁(X)= _k tr (XA _k X)(XC _k X)^–1, h₁(X)= _{k l} (x _l A _kx _l ) (x _l C _kx _l )^–1 withX constrained to be columnwise orthonormal, h₂(x)= _k (xA _kx)²(xC _kx)^–1 subject to xx=1, H₂(X)= _k tr(XA _kX)(XA_kX)(XC_kX)^–1 subject toXX=I, and h₂(X)= _{k l} (x _l A _kx _l )² (x _l C _kX _l )^–1 subject toXX=I. In these functions the matricesC _k are assumed to be positive definite. The matricesA _k can be arbitrary square matrices. The general formulation of the functions and the algorithms allows for application of the algorithms in various problems that arise in multivariate analysis. Several applications of the general algorithms are given. Specifically, algorithms are given for reciprocal principal components analysis, binormamin rotation, generalized discriminant analysis, variants of generalized principal components analysis, simple structure rotation for one of the latter variants, and set component analysis. For most of these methods the algorithms appear to be new, for the others the existing algorithms turn out to be special cases of the newly derived general algorithms.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author. The author is obliged to Jos ten Berge for stimulating this research and for helpful comments on an earlier version of this paper.     

Henkin Quantifiers and the Definability of Truth

Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) studied basic model-theoretical properties of an extension L _* ¹(H) of ordinary first-order languages in which every sentence is a first-order sentence prefixed with a Henkin quantifier. In this paper we consider a generalization of Walkoe's languages: we close L _* ¹(H) with respect to Boolean operations, and obtain the language L ¹(H). At the next level, we consider an extension L _* ²(H) of L ¹(H) in which every sentence is an L ¹(H)-sentence prefixed with a Henkin quantifier. We repeat this construction to infinity. Using the (un)-definability of truth – in – N for these languages, we show that this hierarchy does not collapse. In addition, we compare some of the present results to the ones obtained by Kripke (1975), McGee (1991), and Hintikka (1996).     

Finite condensations of recursive linear orders

The complexity of aII ₄ set of natural numbers is encoded into a linear order to show that the finite condensation of a recursive linear order can beII ₂–II ₁. A priority argument establishes the same result, and is extended to a complete classification of finite condensations iterated finitely many times.     

Repeated measurements analysis of variance when the correlations have a certain pattern

Consider a repeated measurements study in which a single observation is taken at each oft(t > 2) equally spaced time points for each ofn subjects. This paper uses a result of Box to find the appropriate multiplicative correction term for the degrees of freedom in the analysis of variance table for the case when there is equal variability per time point, and when the correlation between observationsu time units apart is equal to ^u . For large values oft, the multiplicative correction has a lower bound approximately equal to 2.5/(t – 1).Dr. Fleiss is also with the Biometrics Research Unit of the New York State Psychiatric Institute. This work was supported in part by grant DE 04068 from the National Institute of Dental Research.     

A linear conservative extension of Zermelo-Fraenkel set theory

In this paper, we develop the system LZF of set theory with the unrestricted comprehension in full linear logic and show that LZF is a conservative extension of ZF^– i.e., the Zermelo-Fraenkel set theory without the axiom of regularity. We formulate LZF as a sequent calculus with abstraction terms and prove the partial cut-elimination theorem for it. The cut-elimination result ensures the subterm property for those formulas which contain only terms corresponding to sets in ZF^–. This implies that LZF is a conservative extension of ZF^– and therefore the former is consistent relative to the latter. Hiroakira Ono     

Some necessary conditions for common-factor analysis

LetR be any correlation matrix of ordern, with unity as each main diagonal element. Common-factor analysis, in the Spearman-Thurstone sense, seeks a diagonal matrixU ² such thatG = R – U ² is Gramian and of minimum rankr. Lets ₁ be the number of latent roots ofR which are greater than or equal to unity. Then it is proved here thatr s ₁. Two further lower bounds tor are also established that are better thans ₁. Simple computing procedures are shown for all three lower bounds that avoid any calculations of latent roots. It is proved further that there are many cases where the rank of all diagonal-free submatrices inR is small, but the minimum rankr for a GramianG is nevertheless very large compared withn. Heuristic criteria are given for testing the hypothesis that a finiter exists for the infinite universe of content from which the sample ofn observed variables is selected; in many cases, the Spearman-Thurstone type of multiple common-factor structure cannot hold.This research was made possible in part by an uncommitted grant-in-aid from the Behavioral Sciences Division of the Ford Foundation.     

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).     

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.     

Indeterminism, Explanation, and Luck

I first adumbrate pertinent aspectsof Robert Kane's libertarian theory of free choice oraction and an objection of luck that has been levelledagainst the theory. I then consider Kane's recentresponses to this objection. To meet these responses,I argue that the view that undetermined choices (ofthe sort implied by Kane's theory) are a matter ofluck is associated with a view about actionexplanation, to wit: when Jones does A and hisdoing of A is undetermined, and when hiscounterpart, Jones*, in the nearest possibleworld in which the past and the laws are held constantuntil the moment of choice does B instead, thereis no explanation (deterministic or indeterministic)of the difference in outcome – Jones's A-ing butJones*'s B-ing – in terms of prior reasonsor motives of either agent. Absence of such anexplanation is one crucial factor that underliesthe charge that Jones's A-ing and Jones*'sB-ing are matters of luck. I argue that thissort of luck is incompatible with responsibility.     

The Undecidability of Iterated Modal Relativization

In dynamic epistemic logic and other fields, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the transitive closure operation, due to its connection to common knowledge. We show that for three fragments of the logic of iterated relativization and transitive closure, the satisfiability problems are fi1 ¹₁–complete. Two of these fragments do not include transitive closure. We also show that the question of whether a sentence in these fragments has a finite (tree) model is fi0 ⁰₁–complete. These results go via reduction to problems concerning domino systems.     

Is There A Logic Of Confirmation Transfer?

This article begins by exploring a lost topic in the philosophy of science:the properties of the relations evidence confirming h confirmsh' and, more generally, evidence confirming each ofh₁, h₂, ..., h_m confirms at least one of h₁, h₂,ldots;, h_n'.The Bayesian understanding of confirmation as positive evidential relevanceis employed throughout. The resulting formal system is, to say the least, oddlybehaved. Some aspects of this odd behaviour the system has in common withsome of the non-classical logics developed in the twentieth century. Oneaspect – its ``parasitism' on classical logic – it does not, and it is this featurethat makes the system an interesting focus for discussion of questions in thephilosophy of logic. We gain some purchase on an answer to a recently prominentquestion, namely, what is a logical system? More exactly, we ask whether satisfaction of formal constraints is sufficient for a relation to be considered a (logical) consequence relation. The question whether confirmation transfer yields a logical system is answered in the negative, despite confirmation transfer having the standard properties of a consequence relation, on the grounds that validity of sequents in the system is not determined by the meanings of the connectives that occur in formulas. Developing the system in a different direction, we find it bears on the project of ``proof-theoretic semantics': conferring meaning on connectives by means of introduction (and possibly elimination) rules is not an autonomous activity, rather it presupposes a prior, non-formal,notion of consequence. Some historical ramifications are alsoaddressed briefly.     

On Some Categories of Involutive Centered Residuated Lattices

Motivated by an old construction due to J. Kalman that relates distributive lattices and centered Kleene algebras we define the functor K ^• relating integral residuated lattices with 0 (IRL₀) with certain involutive residuated lattices. Our work is also based on the results obtained by Cignoli about an adjunction between Heyting and Nelson algebras, which is an enrichment of the basic adjunction between lattices and Kleene algebras. The lifting of the functor to the category of residuated lattices leads us to study other adjunctions and equivalences. For example, we treat the functor C whose domain is cuRL, the category of involutive residuated lattices M whose unit is fixed by the involution and has a Boolean complement c (the underlying set of C M is the set of elements greater or equal than c). If we restrict to the full subcategory NRL of cuRL of those objects that have a nilpotent c, then C is an equivalence. In fact, C M is isomorphic to C _e M, and C _e is adjoint to , where assigns to an object A of IRL₀ the product A × A ⁰ which is an object of NRL.     

To what extent can communalities reduce rank?

The question is raised as to whether the null hypothesis concerning the number of common factors underlying a given set of correlations should be that this number is small. Psychological and algebraic evidence indicate that a more appropriate null hypothesis is that the number is relatively large, and that smallness should be but an alternative hypothesis. The question is also raised as to why approximation procedures should be aimed primarily at the observed correlation matrixR and not at, say,R ^–1. What may be best forR may be worst forR ^–1, and conversely, yetR ^–1 is directly involved in problems of multiple and partial regressions. It is shown that a widely accepted inequality for the possible rank to whichR can be reduced, when modified by communalities, is indeed false.This research was facilitated by a noncommitted grant-in-aid to the writer from the Ford Foundation.     

Discovery algorithms for hierarchical relations

Two algorithms based on a latent class model are presented for discovering hierarchical relations that exist among a set ofK dichotomous items. The two algorithms, stepwise forward selection and backward elimination, incorporate statistical criteria for selecting (or deleting) 0–1 response pattern vectors to form the subset of the total possible 2 ^k vectors that uniquely describe the hierarchy. The performances of the algorithms are compared, using computer-constructed data, with those of three competing deterministic approaches based on ordering theory and the calculation of Phi/Phi-max coefficients. The discovery algorithms are also demonstrated on real data sets investigated in the literature.     

On the size of refutation Kripke models for some linear modal and tense logics

LetL be any modal or tense logic with the finite model property. For eachm, definer _L (m) to be the smallest numberr such that for any formulaA withm modal operators,A is provable inL if and only ifA is valid in everyL-model with at mostr worlds. Thus, the functionr _L determines the size of refutation Kripke models forL. In this paper, we will give an estimation ofr _L (m) for some linear modal and tense logicsL.     

Rules in relevant logic — II: Formula representation

This paper surveys the various forms of Deduction Theorem for a broad range of relevant logics. The logics range from the basic system B of Routley-Meyer through to the system R of relevant implication, and the forms of Deduction Theorem are characterized by the various formula representations of rules that are either unrestricted or restricted in certain ways. The formula representations cover the iterated form,A ₁ .A ₂ . ... .A _n B, the conjunctive form,A ₁&A ₂ & ...A _n B, the combined conjunctive and iterated form, enthymematic version of these three forms, and the classical implicational form,A ₁&A ₂& ...A _n B. The concept of general enthymeme is introduced and the Deduction Theorem is shown to apply for rules essentially derived using Modus Ponens and Adjunction only, with logics containing either (A B)&(B C) .A C orA B .B C .A C.I acknowledge help from anonymous referees for guidance in preparing Part II, and especially for the suggestion that Theorem 9 could be expanded to fully contraction-less logics.     

Interpretations of the first-order theory of diagonalizable algebras in peano arithmetic

For every sequence |p _n } _n of formulas of Peano ArithmeticPA with, every formulaA of the first-order theory diagonalizable algebras, we associate a formula ⁰ A, called the value ofA inPA with respect to the interpretation. We show that, ifA is true in every diagonalizable algebra, then, for every, ⁰ A is a theorem ofPA.     

Measuring preschool cognitive growth while it's still happening: The Learning Express

Educators need accurate assessments of preschool cognitive growth to guide curriculum design, evaluation, and timely modification of their instructional programs. But available tests do not provide content breadth or growth sensitivity over brief intervals. This article details evidence for a multiform, multiscale test criterion-referenced to national standards for alphabet knowledge, vocabulary, listening comprehension and mathematics, developed in field trials with 3433 3–5¹/₂-year-old Head Start children. The test enables repeated assessments (20–30 min per time point) over a school year. Each subscale is calibrated to yield scaled scores based on item response theory and Bayesian estimation of ability. Multilevel modeling shows that nearly all score variation is associated with child performance rather than examiner performance and individual growth-curve modeling demonstrates the high sensitivity of scores to child growth, controlled for age, sex, prior schooling, and language and special needs status.