Ability estimation for conventional tests

Five different ability estimators—maximum likelihood [MLE ()], weighted likelihood [WLE ()], Bayesian modal [BME ()], expected a posteriori [EAP ()] and the standardized number-right score [Z ()]—were used as scores for conventional, multiple-choice tests. The bias, standard error and reliability of the five ability estimators were evaluated using Monte Carlo estimates of the unknown conditional means and variances of the estimators. The results indicated that ability estimates based on BME (), EAP () or WLE () were reasonably unbiased for the range of abilities corresponding to the difficulty of a test, and that their standard errors were relatively small. Also, they were as reliable as the old standby—the number-right score.     

A modification of kendall's tau for measuring association in contingency tables

A coefficient of association is described for a contingency table containing data classified into two sets of ordered categories. Within each of the two sets the number of categories or the number of cases in each category need not be the same.=+1 for perfect positive association and has an expectation of 0 for chance association. In many cases also has –1 as a lower limit. The limitations of Kendall's _a and _b and Stuart's _c are discussed, as is the identity of these coefficients to' under certain conditions. Computational procedure for is given.     

Estimating latent distributions

Consider vectors of item responses obtained from a sample of subjects from a population in which ability is distributed with densityg(), where the are unknown parameters. Assuming the responses depend on through a fully specified item response model, this paper presents maximum likelihood equations for the estimation of the population parameters directly from the observed responses; i.e., without estimating an ability parameter for each subject. Also provided are asymptotic standard errors and tests of fit, computing approximations, and details of four special cases: a non-parametric approximation, a normal solution, a resolution of normal components, and a beta-binomial solution.The author would like to thank R. Darrell Bock for his comments, suggestions, and encouragement during the course of this work.     

Unbiased estimators of ability parameters,of their variance,and of their parallel-forms reliability

Given known item parameters, unbiased estimators are derived i) for an examinee's ability parameter and for his proportion-correct true score, ii) for the variances of and across examinees in the group tested, and iii) for the parallel-forms reliability of the maximum likelihood estimator .This work was supported in part by contract N00014-80-C-0402, project designation NR 150-453 between the Office of Naval Research and Educational Testing Service. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Parameter estimation in latent trait models

Latent trait models for binary responses to a set of test items are considered from the point of view of estimating latent trait parameters=( ₁, , _n ) and item parameters=( ₁, , _k ), where _j may be vector valued. With considered a random sample from a prior distribution with parameter, the estimation of (, ) is studied under the theory of the EM algorithm. An example and computational details are presented for the Rasch model.This work was supported by Contract No. N00014-81-K-0265, Modification No. P00002, from Personnel and Training Research Programs, Psychological Sciences Division, Office of Naval Research. The authors wish to thank an anonymous reviewer for several valuable suggestions.     

A central limit theorem for families of stochastic processes indexed by a small average step size parameter,and some applications to learning models

Let > 0 be a measure of the average step size of a stochastic process {p _n ⁽⁾ } _n=1 ⁽⁾. Conditions are given under whichp _n ⁽⁾ is approximately normally distributed whenn is large and is small. This result is applied to a number of learning models where is a learning rate parameter andp _n ⁽⁾ is the probability that the subject makes a certain response on thenth experimental trial. Both linear and stimulus sampling models are considered.     

A measure for the distance between an interval hypothesis and the truth

The problem of distance from the truth, and more generally distance between hypotheses, is considered here with respect to the case of quantitative hypotheses concerning the value of a given scientific quantity.Our main goal consists in the explication of the concept of distance D(I, ) between an interval hypothesis I and a point hypothesis . In particular, we attempt to give an axiomatic foundation of this notion on the basis of a small number of adequacy conditions.Moreover, the distance function introduced here is employed for the reformulation of the approach to scientific inference — developed by Hintikka, Levi and other scholars — labelled cognitive decision theory. In this connection, we supply a concrete illustration of the rules for inductive acceptance of interval hypotheses that can be obtained on the basis of D(I, ).Lastly, our approach is compared with other proposals made in literature about verisimilitude and distance from the truth.I am grateful for the possibility of spending a period of some months at the Department of Philosophy of University of Gröningen. I profited there a lot from the supervision by Dr. Theo Kuipers. I am also grateful to Prof. Ilkka Niiniluoto who gave rise to my interest in this kind of research during a period spent at Department of Philosophy of University of Helsinki, and who made a number of useful suggestions in the final stage of this paper. Moreover, I would like to thank Dr. Carlo Buttasi for suggesting a better formulation of the proof of (T.24).     

A note on the equivalence of the graded response model and the sequential model

This paper concerns items that consist of several item steps to be responded to sequentially. The item scoreX is defined as the number of correct responses until the first failure. Samejima's graded response model states that each steph=1,...,m is characterized by a parameterb _h , and, for a subject with ability, Pr(Xh; )=F(–b _h ). Tutz's general sequential model associates with each step a parameterdh, and it states that Pr(Xh;)= _r ^=1h G(–d _r ). Tutz's (1991, 1997) conjectures that the models are equivalent if and only ifF(x)=G(x) is an extreme value distribution. This paper presents a proof for this conjecture.     

A further note on freeman's measure of association

This note extends and elaborates Hubert's attempt to provide an interpretation of Freeman's measure of association,. The measure is used in a a contingency table when observations are ordered on one variable and unordered on the other. No attempt is made explore the distribution of.     

Natural deduction rules for English

A system of natural deduction rules is proposed for an idealized form of English. The rules presuppose a sharp distinction between proper names and such expressions as the c, a (an) c, some c, any c, and every c, where c represents a common noun. These latter expressions are called quantifiers, and other expressions of the form that c or that c itself, are called quantified terms. Introduction and elimination rules are presented for any, every, some, a (an), and the, and also for any which, every which, and so on, as well as rules for some other concepts. One outcome of these rules is that Every man loves some woman is implied by, but does not imply, Some woman is loved by every man, since the latter is taken to mean the same as Some woman is loved by all men. Also, Jack knows which woman came is implied by Some woman is known by Jack to have come, but not by Jack knows that some woman came.     

On superintuitionistic logics as fragments of proof logic extensions

Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI set up correspondingly by the calculiFor any calculus we denote by the set of all formulae of the calculus and by the lattice of all logics that are the extensions of the logic of the calculus, i.e. sets of formulae containing the axioms of and closed with respect to its rules of inference. In the logiclG the sign is decoded as follows: A = (A & A). The result of placing in the formulaA before each of its subformula is denoted byTrA. The maps are defined (in the definitions of x and the decoding of is meant), by virtue of which the diagram is constructedIn this diagram the maps, x and are isomorphisms, thereforex ^–1 = ; and the maps and are the semilattice epimorphisms that are not commutative with lattice operation +. Besides, the given diagram is commutative, and the next equalities take place: ^–1 = ^–1 and = ^–1 x. The latter implies in particular that any superintuitionistic logic is a superintuitionistic fragment of some proof logic extension.     

Логистическая система, исходными связками которой являются эквивалентность и дизъюнкция

- Allatum est die 5 Martii 1970     

A new proof for monotone likelihood ratio for the sum of independent bernoulli random variables

By use of an inequality of Marcus and Lopes for elementary symmetric functions, a new proof is presented for the following result by Ghurye and Wallace: Given that the independent random variablesX _j are Bernoulli with success probabilityp _j () strictly between 0 and 1 and nondecreasing in, the sum X _j has monotone likelihood ratio.     

A reduction rule for Peirce formula

A reduction rule is introduced as a transformation of proof figures in implicational classical logic. Proof figures are represented as typed terms in a -calculus with a new constant P ⁽⁽⁾⁾. It is shown that all terms with the same type are equivalent with respect to -reduction augmented by this P-reduction rule. Hence all the proofs of the same implicational formula are equivalent. It is also shown that strong normalization fails for P-reduction. Weak normalization is shown for P-reduction with another reduction rule which simplifies of (( ) ) into an atomic type.This work was partially supported by a Grant-in-Aid for General Scientific Research No. 05680276 of the Ministry of Education, Science and Culture, Japan and by Japan Society for the Promotion of Science. Hiroakira Ono     

Mathematics of Totalities: an alternative to mathematics of sets

I dare say, a set is contranatural if some pair of its elements has a nonempty intersection. So, we consider only collections of disjoint nonempty elements and call them totalities. We propose the propositional logicTT, where a proposition letters some totality. The proposition is true if it letters the greatest totality. There are five connectives inTT: , , , , # and the last is called plexus. The truth of # means that any element of the totality has a nonempty intersection with any element of the totality . An imbeddingG of the classical predicate logicCPL inTT is defined. A formulaf ofCPL is a classical tautology if and only ifG(f) is always true inTT. So, mathematics may be expounded inTT, without quantifiers.     

Mathematical quantum theory I: Random ultrafilters as hidden variables

The basic purpose of this essay, the first of an intended pair, is to interpret standard von Neumann quantum theory in a framework of iterated measure algebraic truth for mathematical (and thus mathematical-physical) assertions — a framework, that is, in which the truth-values for such assertions are elements of iterated boolean measure-algebras (cf. Sections 2.2.9, 5.2.1–5.2.6 and 5.3 below).The essay itself employs constructions of Takeuti's boolean-valued analysis (whose origins lay in work of Scott, Solovay, Krauss and others) to provide a metamathematical interpretation of ideas sometimes considered disparate, heuristic, or simply ill-defined: the collapse of the wave function, for example; Everett's many worlds'-construal of quantum measurement; and a natural product space of contextual (nonlocal) hidden variables.More precisely, these constructions permit us to write down a category-theoretically natural correlation between ideal outcomes of quantum measurements u of a universal wave function, and possible worlds of an Everett-Wheeler-like many-worlds-theory.The universal wave function, first, is simply a pure state of the Hilbert space (L ₂([0, 1]) ^M in a model M an appropriate mathematical-physical theory T, where T includes enough set-theory to derive all the analysis needed for von Neumann-algebraic formulations of quantum theory.The worlds of this framework can then be given a genuine model-theoretic construal: they are random models M(u) determined by M-random elements u of the unit interval [0, 1], where M is again a fixed model of T.Each choice of a fixed basis for a Hilbert space H in a model of M of T then assigns ideal spectral values for observables A on H (random ultrafilters on the range of A regarded as a projection-valued measure) to such M-random reals u. If is the universal Lebesgue measure-algebra on [0, 1], these assignments are interrelated by the spectral functional calculus with value 1 in the boolean extension (V( )) ^M , and therefore in each M(u).Finally, each such M-random u also generates a corresponding extension M(u) of M, in which ideal outcomes of measurements of all observables A in states are determined by the assignments just mentioned from the random spectral values u for the universal position-observable on L ₂([0, 1]) in M.At the suggestion of the essay's referee, I plan to draw on its ideas in the projected sequel to examine more recent modal and decoherence-interpretations of quantum theory, as well as Schrödinger's traditional construal of time-evolution. A preliminary account of the latter — an obvious prerequisite for any serious many-worlds-theory, given that Everett's original intention was to integrate time-evolution and wave-function collapse — is sketched briefly in Section 5.3. The basic idea is to apply results from the theory of iterated measure-algebras to reinterpret time-ordered processes of measurements (determined, for example, by a given Hamiltonian observable H in M) as individual measurements in somewhat more complexly defined extensions M(u) of M.In plainer English: if one takes a little care to distinguish boolean- from measure-algebraic tensor-products of the universal measure-algebra L, one can reinterpret formal time-evolution so that it becomes internal to the universal random models M(u).     

Cut-free tableau calculi for some propositional normal modal logics

We give sound and complete tableau and sequent calculi for the prepositional normal modal logics S4.04, K4B and G ⁰(these logics are the smallest normal modal logics containing K and the schemata A A, A A and A ( A); A A and AA; A A and ((A A) A) A resp.) with the following properties: the calculi for S4.04 and G ⁰are cut-free and have the interpolation property, the calculus for K4B contains a restricted version of the cut-rule, the so-called analytical cut-rule.In addition we show that G ⁰is not compact (and therefore not canonical), and we proof with the tableau-method that G ⁰is characterised by the class of all finite, (transitive) trees of degenerate or simple clusters of worlds; therefore G ⁰is decidable and also characterised by the class of all frames for G ⁰.Research supported by Fonds zur Förderung der wissenschaftlichen Forschung, project number P8495-PHY.Presented by W. Rautenberg     

Third Men: The Logic of the Sophisms at Arist. SE 22, 178b36–170a10

This article aims at elucidating the logic of Arist. SE 22, 178b36–179a10 and, in particular, of the sophism labelled "Third Man" discussed in it. I suggest that neither the sophistic Walking Man argument, proposed by ancient commentators, nor the Aristotelian Third Man of the , suggested by modern interpreters, can be identified with the fallacious argument Aristotle presents and solves in the passage. I propose an alternative reconstruction of the Third Man sophism and argue that an explanation of the lines regarding the identity of Coriscus and Coriscus the musician (178b39–179a3) is indispensable for its correct understanding, since they hint at another sophism in some important aspects analogous. Finally, I show that two contradictions concerning spotted by scholars in the passage are only apparent and can be dissolved once the assumption that the anti-Platonic Third Man argument is at stake here is discarded, and once the passage is read in the light of its agonistic context.     

An asymptotic likelihood ratio test for a markov chain with bernoullian or contingent input

A Markov chain with transition probabilitiesp _ij () that are functions of a parameter vector is defined. One oft input values is delivered to the chain on each trial. Under the hypothesis H₀ the parameter vector is independent of the input; under the hypothesis H₁ the vector is in general different for each different input. A likelihood ratio test for a single observation on a chain of great length is given for testing H₀ against H₁, given that the distribution of the inputs depends at most on the previous input and the present state of the chain. The test is therefore one of stationarity of the transition probabilities against a specific alternative form of nonstationarity. Application of the approach to a statistical test of lumpability of the states of a chain is indicated. Tests for other related hypotheses are suggested. Application of the test to within-subject effects for individual subjects is considered. Finally, some applications of the results in psychophysical contexts are suggested.Part of the final version of this paper was written at Stanford University during work sessions of a 1967 Summer Conference on Mathematical Models for Perception and Learning. Support was provided by the National Science Foundation through the Committee on Mathematics in the Social Sciences of the Institute for Advanced Study in the Behavioral Sciences.     

Utility theory with inexact preferences and degrees of preference

a–b* c–d is taken to mean that your degree of preference for a over b is less than your degree of preference for c over d. Various properties of the strength-of-preference comparison relation * are examined along with properties of simple preferences defined from *. The investigation recognizes an individual's limited ability to make precise judgments. Several utility theorems relating a–b * c–d to u(a)–u(b) are included.