non archimedean abelian polish groups and their actions abstract in this paper we consider non archimedean abelian polish groups whose orbit equivalence relations are all borel such groups are called tame we show that a non archimedean abelian polish group g is tame if and only if there does not exist a continuous surjective homomorphism from a closed subgroup of g onto z or z p for any prime p in addition to determining the structure of tame groups we also consider the actions of such groups and study the complexity of their orbit equivalence relations in the borel reducibility hierarchy it is shown that if such an orbit equivalence relation is essentially countable then it must be essentially hyperfinite we also find an upper bound in the borel reducibility hierarchy for the orbit equivalence relations of all tame non archimedean abelian polish groups

baire measurable paradoxical decompositions via matchings abstract we show that every locally finite bipartite borel graph satisfying a strengthening of hall s condition has a borel perfect matching on some comeager invariant borel set we apply this to show that if a group acting by borel automorphisms on a polish space has a paradoxical decomposition then it admits a paradoxical decomposition using pieces having the baire property this strengthens a theorem of dougherty and foreman who showed that there is a paradoxical decomposition of the unit ball in r 3 using baire measurable pieces we also obtain a baire category solution to the dynamical von neumann day problem if a is a nonamenable action of a group on a polish space x by borel automorphisms then there is a free baire measurable action of f 2 on x which is lipschitz with respect to a

finite versus infinite an insufficient shift abstract the shift graph g s is defined on the space of infinite subsets of natural numbers by letting two sets be adjacent if one can be obtained from the other by removing its least element we show that this graph is not a minimum among the graphs of the form g f defined on some polish space x where two distinct points are adjacent if one can be obtained from the other by a given borel function f x x this answers the primary outstanding question from 8

characterizing the manifest probabilities of latent trait models the problem of characterizing the manifest probabilities of a latent trait model is considered the item characteristic curve is transformed to the item passing odds curve and a corresponding transformation is made on the distribution of ability this results in a useful expression for the manifest probabilities of any latent trait model the result is then applied to give a characterization of the rasch model as a log linear model for a 2 j contingency table partial results are also obtained for other models the question of the identifiability of guessing parameters is also discussed

measurable versions of vizing s theorem abstract we establish two versions of vizing s theorem for borel multi graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively and the approximate version states that for any borel probability measure on the edge set and any 0 we can properly colour all but fraction of edges with colours in a borel way the measurable version which is our main result states that if additionally the measure is invariant then there is a measurable proper edge colouring of the whole edge set with at most colours

loglinear multidimensional irt models for polytomously scored items a loglinear irt model is proposed that relates polytomously scored item responses to a multidimensional latent space the analyst may specify a response function for each response indicating which latent abilities are necessary to arrive at that response each item may have a different number of response categories so that free response items are more easily analyzed conditional maximum likelihood estimates are derived and the models may be tested generally or against alternative loglinear irt models

a latent class model for rating data a latent class model for rating data is presented which is the analogue of andrich s binomial rasch model for lazarsfeld s latent class analysis lca the response probabilities for the rating categories follow a binomial distribution and depend on class specific item parameters the em algorithm for parameter estimation as well as goodness of fit tests for the model are described an example using questionnaire items on interest in physics illustrates the use of the model as an alternative to the latent trait approach of analyzing test data

expanding the rasch model to a general model having more than one dimension the well known rasch model is generalized to a multicomponent model so that observations of component events are not needed to apply the model it is shown that the generalized model has retained the property of the specific objectivity of the rasch model for a restricted variant of the model maximum likelihood estimates of its parameters and a statistical test of the model are given the results of an application to a mathematics test involving six components are described

the generalized logit linear item response model for binary designed items this paper introduces the generalized logit linear item response model gllirm which represents the item solving process as a series of dichotomous operations or steps the gllirm assumes that the probability function of the item response is a logistic function of a linear composite of basic parameters which describe the operations and the coefficients depend on three design matrices x y and z the gllirm provides a tool for testing hypotheses on the item solving process and generalizes existing models an empirical application is included in which the model is applied to evaluate sources of difficulty and pairwise item interactions in a logical analysis test

asymmetric item characteristic curves and item complexity insights from simulation and real data analyses while item complexity is often considered as an item feature in test development it is much less frequently attended to in the psychometric modeling of test items prior work suggests that item complexity may manifest through asymmetry in item characteristics curves iccs samejima in psychometrika 65 319 335 2000 in the current paper we study the potential for asymmetric irt models to inform empirically about underlying item complexity and thus the potential value of asymmetric models as tools for item validation both simulation and real data studies are presented some psychometric consequences of ignoring asymmetry as well as potential strategies for more effective estimation of asymmetry are considered in discussion

a glimm effros dichotomy for borel equivalence relations a basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by glimm g12 in the locally compact group action case and extended by effros e 1 e2 in the polish group action case when additionally the induced equivalence relation is f it is the purpose of this paper to extend the glimm effros dichotomy to the very general context of an arbitrary borel equivalence relation not even necessarily induced by a group action despite the totally classical descriptive set theoretic nature of our result our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability or recursion theory on the integers

optimal item difficulty for the three parameter normal ogive response model in tailored testing it is important to determine the optimal difficulty of the next item to present to the examinee this paper shows that the difference that maximizes information for the three parameter normal ogive response model is approximately 1 7 times the optimal difference b for the three parameter logistic model under the normal model calculation of the optimal difficulty for minimizing the bayes risk is equivalent to maximizing an associated information function

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

item bias detection using loglinear irt a method is proposed for the detection of item bias with respect to observed or unobserved subgroups the method uses quasi loglinear models for the incomplete subgroup test score item 1 itemk contingency table if subgroup membership is unknown the models are haberman s incomplete latent class models the conditional rasch model is formulated as a quasi loglinear model the parameters in this loglinear model that correspond to the main effects of the item responses are the conditional estimates of the parameters in the rasch model item bias can then be tested by comparing the quasi loglinear rasch model with models that contain parameters for the interaction of item responses and the subgroups

logistic latent trait models with linear constraints two linearly constrained logistic models which are based on the well known dichotomous rasch model the linear logistic test model lltm and the linear logistic model with relaxed assumptions llra are discussed necessary and sufficient conditions for the existence of unique conditional maximum likelihood estimates of the structural model parameters are derived methods for testing composite hypotheses within the framework of these models and a number of typical applications to real data are mentioned

a rasch model for partial credit scoring a unidimensional latent trait model for responses scored in two or more ordered categories is developed this partial credit model is a member of the family of latent trait models which share the property of parameter separability and so permit specifically objective comparisons of persons and items the model can be viewed as an extension of andrich s rating scale model to situations in which ordered response alternatives are free to vary in number and structure from item to item the difference between the parameters in this model and the category boundaries in samejima s graded response model is demonstrated an unconditional maximum likelihood procedure for estimating the model parameters is developed

essential countability of treeable equivalence relations we establish a dichotomy theorem characterizing the circumstances under which a treeable borel equivalence relation e is essentially countable under additional topological assumptions on the treeing we in fact show that e is essentially countable if and only if there is no continuous embedding of e1 into e our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations and allow us to show that up to continuous kakutani embeddability there is a minimum borel function which is not essentially countable to one

a class of multidimensional irt models for testing unidimensionality and clustering items we illustrate a class of multidimensional item response theory models in which the items are allowed to have different discriminating power and the latent traits are represented through a vector having a discrete distribution we also show how the hypothesis of unidimensionality may be tested against a specific bidimensional alternative by using a likelihood ratio statistic between two nested models in this class for this aim we also derive an asymptotically equivalent wald test statistic which is faster to compute moreover we propose a hierarchical clustering algorithm which can be used when the dimensionality of the latent structure is completely unknown for dividing items into groups referred to different latent traits the approach is illustrated through a simulation study and an application to a dataset collected within the national assessment of educational progress 1996

essential independence and likelihood based ability estimation for polytomous items a definition ofessential independence is proposed for sequences of polytomous items for items satisfying the reasonable assumption that the expected amount of credit awarded increases with examinee ability we develop a theory ofessential unidimensionality which closely parallels that of stout essentially unidimensional item sequences can be shown to have a unique up to change of scale dominant underlying trait which can be consistently estimated by a monotone transformation of the sum of the item scores in more general polytomous response latent trait models with or without ordered responses anm estimator based upon maximum likelihood may be shown to be consistent for under essentially unidimensional violations of local independence and a variety of monotonicity identifiability conditions a rigorous proof of this fact is given and the standard error of the estimator is explored these results suggest that ability estimation methods that rely on the summation form of the log likelihood under local independence should generally be robust under essential independence but standard errors may vary greatly from what is usually expected depending on the degree of departure from local independence an index of departure from local independence is also proposed

extension of feldt s approach to testing homogeneity of coefficients of reliability limitations and extensions of feldt s approach to testing the equality of cronbach s alpha coefficients in independent and matched samples are discussed in particular this approach is used to test equality of intraclass correlation coefficients

a unified approach to duality in convex programming abstract bifunctional duality lagrange duality and fenchel duality in convex programming are presented under a common point of view stability criteria weaker and more natural than the usual slater type constraint qualification hypotheses are obtained for any of these types of duality

baire class xi colorings the first three levels the mathbb g 0 dichotomy due to kechris solecki and todor v cevic characterizes the analytic relations having a borel measurable countable coloring we give a version of the mathbb g 0 dichotomy for boraxi measurable countable colorings when xi leq 3 a boraxi measurable countable coloring gives a covering of the diagonal consisting of countably many boraxi squares this leads to the study of countable unions of boraxi rectangles we also give a hurewicz like dichotomy for such countable unions when xi leq 2

an irt based model for dichotomous longitudinal data the llra linear logistic model with relaxed assumptions fischer 1974 1977a 1977b 1983a was developed within the framework of generalized rasch models for assessing change in dichotomous item score matrices between two points in time it allows to quantify change on latent trait dimensions and to explain change in terms of treatment effects treatment interactions and a trend effect a remarkable feature of the model is that unidimensionality of the item set is not required the present paper extends this model to designs with any number of time points and even with different sets of items presented on different occasions provided that one unidimensional subscale is available per latent trait thus unidimensionality assumptions within subscales are combined with multidimensionality of the item set conditional maximum likelihood methods for parameter estimation and hypothesis testing are developed and a necessary and sufficient condition for unique identification of the model given the data is derived finally a sample application is presented

measurable circle squaring laczkovich proved that if bounded subsets a and b of r k have the same non zero lebesgue measure and the box dimension of the boundary of each set is less than k then there is a partition of a into finitely many parts that can be translated to form a partition of b here we show that it can be additionally required that each part is both baire and lebesgue measurable as special cases this gives measurable and translation only versions of tarski s circle squaring and hilbert s third problem

on the existence and uniqueness of maximum likelihood estimates in the rasch model necessary and sufficient conditions for the existence and uniqueness of a solution of the so called unconditional uml and the conditional cml maximum likelihood estimation equations in the dichotomous rasch model are given the basic critical condition is essentially the same for uml and cml estimation for complete data matricesa it is formulated both as a structural property ofa and in terms of the sufficient marginal sums in case of incomplete data the condition is equivalent to complete connectedness of a certain directed graph it is shown how to apply the results in practical uses of the rasch model

a response model for multiple choice items in the mid 1960s samejima initiated the development of item response models that involve separate response functions for all of the alternatives in the multiple choice and likert type formats her work in this area began at the educational testing service and continued during a visit to the l l thurstone psychometric laboratory at the university of north carolina both samejima s 1969 this volume original model for graded item responses and bock s 1972 this volume model for nominal responses were originally intended to produce response functions for all of the alternatives of multiple choice items for various reasons neither model has proved entirely satisfactory for that purpose although both have been applied in other contexts using a combination of ideas suggested by bock 1972 and samejima 1968 1979 a multiple choice model was developed that produces response functions that fit unidimensional multiple choice tests better thissen and steinberg 1984 that model is the subject of this chapter

weighted minimum trace factor analysis in the last decade several authors discussed the so called minimum trace factor analysis mtfa which provides the greatest lower bound g l b to reliability however the mtfa fails to be scale free in this paper we propose to solve the scale problem by maximization of the g l b as the function of weights closely related to the primal problem of the g l b maximization is the dual problem we investigate the primal and dual problems utilizing convex analysis techniques the asymptotic distribution of the maximal g l b is obtained provided the population covariance matrix satisfies sone uniqueness and regularity assumptions finally we outline computational algorithms and consider numerical examples

statistical bias in maximum likelihood estimators of item parameters asymptotic formulas are derived for the bias in the maximum likelihood estimators of the item parameters in the logistic item response model when examinee abilities are known numerical results are given for a typical verbal test for college admission

a sample coverage approach to multiple system estimation with application to census undercount abstract the concept of sample coverage used in animal abundance estimation is modified to evaluate the undercount of a census lack of independence between the census and its postenumeration survey leads to correlation bias for the standard estimator of population size an additional recapture sample besides the census and postenumeration survey can be used to estimate the correlation bias due to two types of dependences this work expresses the correlation bias for the three sample model as a function of expected sample coverage and measures of dependence between lists a nonparametric population size estimator that incorporates the correlation bias is proposed a simulation study investigates the performance of the proposed procedure data from a 1988 dress rehearsal study for the 1990 census conducted by the u s bureau of the census are used to illustrate the proposed three system estimation procedure and to compare the resulting estimates with those given by darroch fienberg glonek and junker

test theory and psychometrika the past twenty five years on the occasion ofpsychometrika s fiftieth anniversary the past twenty five years developments in mental test theory are reviewed with special emphasis on the topics receiving attention in the pages of this journal analogous reviews forpsychometrika s first quarter century were given by gulliksen and guilford in 1961 much of the recent progress in test theory and in other branches of psychometrics as well has been made by treating the problems in this field as being essentially ones of statistical inference it is concluded that a research in test theory is in a healthy state and b psychometrika is an important source of information about that research

applying the principles of specific objectivity and of generalizability to the measurement of change a formal framework for measuring change in sets of dichotomous data is developed and implications of the principle of specific objectivity of results within this framework are investigated building upon the concept of specific objectivity as introduced by g rasch three equivalent formal definitions of that postulate are given and it is shown that they lead to latent additivity of the parametric structure if in addition the observations are assumed to be locally independent realizations of bernoulli variables a family of models follows necessarily which are isomorphic to a logistic model with additive parameters determining an interval scale for latent trait measurement and a ratio scale for quantifying change adding the further assumption of generalizability over subsets of items from a given universe yields a logistic model which allows a multidimensional description of individual differences and a quantitative assessment of treatment effects as a special case a unidimensional parameterization is introduced also and a unidimensional latent trait model for change is derived as a side result the relationship between specific objectivity and additive conjoint measurement is clarified

finite graphs and amenability hyperfiniteness or amenability of measurable equivalence relations and group actions has been studied for almost fifty years recently unexpected applications of hyperfiniteness were found in computer science in the context of testability of graph properties in this paper we propose a unified approach to hyperfiniteness we establish some new results and give new proofs of theorems of schramm lovasz newman sohler and ornstein weiss

an extension of the rating scale model with an application to the measurement of change the polytomous unidimensional rasch model with equidistant scoring also known as the rating scale model is extended in such a way that the item parameters are linearly decomposed into certain basic parameters the extended model is denoted as the linear rating scale model lrsm a conditional maximum likelihood estimation procedure and a likelihood ratio test of hypotheses within the framework of the lrsm are presented since the lrsm is a generalization of both the dichotomous rasch model and the rating scale model the present algorithm is suited for conditional maximum likelihood estimation in these submodels as well the practicality of the conditional method is demonstrated by means of a dichotomous rasch example with 100 items of a rating scale example with 30 items and 5 categories and in the light of an empirical application to the measurement of treatment effects in a clinical study

marginal maximum likelihood estimation of item parameters application of an em algorithm maximum likelihood estimation of item parameters in the marginal distribution integrating over the distribution of ability becomes practical when computing procedures based on an em algorithm are used by characterizing the ability distribution empirically arbitrary assumptions about its form are avoided the em procedure is shown to apply to general item response models lacking simple sufficient statistics for ability this includes models with more than one latent dimension

loglinear rasch model tests existing statistical tests for the fit of the rasch model have been criticized because they are only sensitive to specific violations of its assumptions contingency table methods using loglinear models have been used to test various psychometric models in this paper the assumptions of the rasch model are discussed and the rasch model is reformulated as a quasi independence model the model is a quasi loglinear model for the incomplete subgroup score item 1 item 2 itemk contingency table using ordinary contingency table methods the rasch model can be tested generally or against less restrictive quasi loglinear models to investigate specific violations of its assumptions

estimating latent correlations between repeated testings a model for longitudinal latent structure analysis is proposed we assume that test scores for a given mental or attitudinal test are observed for the same individuals at two different points in time the purpose of the analysis is to fit a model that combines the values of the latent variable at the two time points in a two dimensional latent density the correlation coefficient between the two values of the latent variable can then be estimated the theory and methods are illustrated by a danish dataset concerning psychic vulnerability

tau equivalence and equipercentile equating test scores that are not perfectly reliable cannot be strictly equated unless they are strictly parallel lord 1980 this fact implies that tau equivalence can be lost if an equipercentile equating is applied to observed scores that are not strictly parallel seventy two simulated testing conditions are produced to simulate equating tests with different difficulties and discriminations number correct and trait metrics are examined when an equipercentile equating is applied to these data locally biased i e non tau equivalent results are produced for tests of unequal difficulty differences between the criteria of tau equivalence and equipercentile equivalence are discussed

continuous affine support mappings for convex operators abstract we investigate the problem of existence of continuous affine resp linear support mappings for convex resp sublinear partially defined operators we prove that a weakly borel measurable convex operator defined on a closed convex subset of a locally convex frechet space with values in an ordered locally convex vector space with the least upper bound property and an order unit inducing the topology in fact has continuous affine support mappings we give an example indicating that the order unit being present is essential a step towards our result is provided by a variant of the sandwich theorem obtained for the case of weakly borel measurable convex and concave operators finally we obtain necessary and sufficient conditions for the existence of a continuous affine support mapping h for a convex operator such that h x 0 x 0 holds for a fixed point x 0 in terms of the upper directional derivate x 0 for at x 0

computing maximum likelihood estimates of loglinear models from marginal sums with special attention to loglinear item response theory in this paper algorithms are described for obtaining the maximum likelihood estimates of the parameters in loglinear models modified versions of the iterative proportional fitting and newton raphson algorithms are described that work on the minimal sufficient statistics rather than on the usual counts in the full contingency table this is desirable if the contingency table becomes too large to store special attention is given to loglinear irt models that are used for the analysis of educational and psychological test data to calculate the necessary expected sufficient statistics and other marginal sums of the table a method is described that avoids summing large numbers of elementary cell frequencies by writing them out in terms of multiplicative model parameters and applying the distributive law of multiplication over summation these algorithms are used in the computer program logimo the modified algorithms are illustrated with simulated data

folner tilings for actions of amenable groups we show that every probability measure preserving action of a countable amenable group g can be tiled modulo a null set using finitely many finite subsets of g shapes with prescribed approximate invariance so that the collection of tiling centers for each shape is borel this is a dynamical version of the downarowicz huczek zhang tiling theorem for countable amenable groups and strengthens the ornstein weiss rokhlin lemma as an application we prove that for every countably infinite amenable group g the crossed product of a generic free minimal action of g on the cantor set is mathcal z stable

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

nonparametric goodness of fit tests for the rasch model a monte carlo algorithm realizing a family of nonparametric tests for the rasch model is introduced which are conditional on the item and subject marginals the algorithm is based on random changes of elements of data matrices without changing the marginals most powerful tests against all alternative hypotheses are given for which a monotone characteristic may be computed from the data matrix alternatives may also be composed computation times are long but exactp values are approximated with the quality of approximation only depending on calculation time but not on the number of persons the power and the flexibility of the procedure is demonstrated by means of an empirical example where among others indicators for increased item similarities the existence of subscales violations of sufficiency of the raw score as well as learning processes were found many of the features described are implemented in the program t rasch 1 0 by ponocny and ponocny seliger 1999

a dichotomy characterizing analytic digraphs of uncountable borel chromatic number in any dimension we study the extension of the kechris solecki todorcevic dichotomy on analytic graphs to dimensions higher than 2 we prove that the extension is possible in any dimension finite or infinite the original proof works in the case of the finite dimension we first prove that the natural extension does not work in the case of the infinite dimension for the notion of continuous homomorphism used in the original theorem then we solve the problem in the case of the infinite dimension finally we prove that the natural extension works in the case of the infinite dimension but for the notion of baire measurable homomorphism

countable abelian group actions and hyperfinite equivalence relations an equivalence relation e on a standard borel space is hyperfinite if e is the increasing union of countably many borel equivalence relations e n where all e n equivalence classs are finite in this article we establish the following theorem if a countable abelian group acts on a standard borel space in a borel manner then the orbit equivalence relation is hyperfinite the proof uses constructions and analysis of borel marker sets and regions in the space 2 mathbb z omega this technique is also applied to a problem of finding borel chromatic numbers for invariant borel subspaces of 2 mathbb z n

testing the conditional independence and monotonicity assumptions of item response theory when item characteristic curves are nondecreasing functions of a latent variable the conditional or local independence of item responses given the latent variable implies nonnegative conditional covariances between all monotone increasing functions of a set of item responses given any function of the remaining item responses this general result provides a basis for testing the conditional independence assumption without first specifying a parametric form for the nondecreasing item characteristic curves the proposed tests are simple have known asymptotic null distributions and possess certain optimal properties in an example the conditional independence hypothesis is rejected for all possible forms of monotone item characteristic curves

an extension of the partial credit model with an application to the measurement of change the partial credit model is considered under the assumption of a certain linear decomposition of the item category parameters ih into basic parameters j this model is referred to as the linear partial credit model a conditional maximum likelihood algorithm for estimation of the j is presented based on a recurrences for the combinatorial functions involved and b using a quasi newton approach the so called broyden fletcher goldfarb shanno bfgs method a guarantees numerically stable results b avoids the direct computation of the hesse matrix yet produces a sequence of certain positive definite matricesb k k 1 2 converging to the asymptotic variance covariance matrix of the hat alpha j the practicality of these numerical methods is demonstrated both by means of simulations and of an empirical application to the measurement of treatment effects in patients with psychosomatic disorders

krieger s finite generator theorem for actions of countable groups i for an ergodic p m p action g curvearrowright x mu of a countable group g we define the rokhlin entropy h mathrm rok g x mu to be the infimum of the shannon entropies of countable generating partitions it is known that for free ergodic actions of amenable groups this notion coincides with classical kolmogorov sinai entropy it is thus natural to view rokhlin entropy as a close analogue to classical entropy under this analogy we prove that krieger s finite generator theorem holds for all countably infinite groups specifically if h mathrm rok g x mu log k then there exists a generating partition consisting of k sets

measurable versions of the lovasz local lemma and measurable graph colorings abstract in this paper we investigate the extent to which the lovasz local lemma an important tool in probabilistic combinatorics can be adapted for the measurable setting in most applications the lovasz local lemma is used to produce a function f x y with certain properties where x is some underlying combinatorial structure and y is a typically finite set can this function f be chosen to be borel or measurable for some probability borel measure on x assuming that x is a standard borel space in the positive direction we prove that if the set of constraints put on f is in a certain sense locally finite then there is always a borel choice for f that is e close to satisfying these constraints for any e 0 moreover if the combinatorial structure on x is induced by the 0 1 shift action of a countable group then even without any local finiteness assumptions there is a borel choice for f which satisfies the constraints on an invariant conull set i e with e 0 a direct corollary of our results is an upper bound on the measurable chromatic number of the graph g n generated by the shift action of the free group f n that is asymptotically tight up to a factor of at most 2 which answers a question of lyons and nazarov on the other hand our result for structures induced by measure preserving group actions is at least for amenable groups sharp in the following sense a probability measure preserving action of a countably infinite amenable group satisfies the measurable version of the lovasz local lemma if and only if it admits a factor map to the 0 1 shift action to prove this we combine the tools of the ornstein weiss theory of entropy for actions of amenable groups with concepts from computability theory specifically kolmogorov complexity

an extension of the rasch model for ratings providing both location and dispersion parameters an elaboration of a psychometric model for rated data which belongs to the class of rasch models is shown to provide a model with two parameters one characterising location and one characterising dispersion the later parameter derived from the idea of a unit of scale is also shown to reflect the shape of rating distributions ranging from unimodal through uniform and then to u shaped distributions a brief case is made that when a rating distribution is treated as a random error distribution then the distribution should be unimodal

computational aspects of the greatest lower bound to the reliability and constrained minimum trace factor analysis in the last decade several algorithms for computing the greatest lower bound to reliability or the constrained minimum trace communality solution in factor analysis have been developed in this paper convergence properties of these methods are examined instead of using lagrange multipliers a new theorem is applied that gives a sufficient condition for a symmetric matrix to be gramian whereas computational pitfalls for two methods suggested by woodhouse and jackson can be constructed it is shown that a slightly modified version of one method suggested by bentler and woodward can safely be applied to any set of data a uniqueness proof for the solution desired is offered

increasing item complexity a possible cause of scale shrinkage for unidimensional item response theory when the three parameter logistic model is applied to tests covering a broad range of difficulty there frequently is an increase in mean item discrimination and a decrease in variance of item difficulties and traits as the tests become more difficult to examine the hypothesis that this unexpected scale shrinkage effect occurs because the items increase in complexity as they increase in difficulty an approximate relationship is derived between the unidimensional model used in data analysis and a multidimensional model hypothesized to be generating the item responses scale shrinkage is successfully predicted for several sets of simulated data

modification indices for the 2 pl and the nominal response model in this paper it is shown that various violations of the 2 pl model and the nominal response model can be evaluated using the lagrange multiplier test or the equivalent efficient score test the tests presented here focus on violation of local stochastic independence and insufficient capture of the form of the item characteristic curves primarily the tests are item oriented diagnostic tools but taken together they also serve the purpose of evaluation of global model fit a useful feature of lagrange multiplier statistics is that they are evaluated using maximum likelihood estimates of the null model only that is the parameters of alternative models need not be estimated as numerical examples an application to real data and some power studies are presented

a dynamic generalization of the rasch model in the present paper a model for describing dynamic processes is constructed by combining the common rasch model with the concept of structurally incomplete designs this is accomplished by mapping each item on a collection of virtual items one of which is assumed to be presented to the respondent dependent on the preceding responses and or the feedback obtained it is shown that in the case of subject control no unique conditional maximum likelihood cml estimates exist whereas marginal maximum likelihood mml proves a suitable estimation procedure a hierarchical family of dynamic models is presented and it is shown how to test special cases against more general ones furthermore it is shown that the model presented is a generalization of a class of mathematical learning models known as luce s beta model

a note on the asymptotic variance covariance matrix of item parameter estimates in the rasch model lord and wingersky have developed a method for computing the asymptotic variance covariance matrix of maximum likelihood estimates for item and person parameters under some restrictions on the estimates which are needed in order to fix the latent scale the method is tedious but can be simplified for the rasch model when one is only interested in the item parameters this is demonstrated here under a suitable restriction on the item parameter estimates

loglinear rasch models for the analysis of stability and change loglinear unidimensional and multidimensional rasch models are considered for the analysis of repeated observations of polytomous indicators with ordered response categories reparameterizations and parameter restrictions are provided which facilitate specification of a variety of hypotheses about latent processes of change models of purely quantitative change in latent traits are proposed as well as models including structural change a conditional likelihood ratio test is presented for the comparison of unidimensional and multiple scales rasch models in the context of longitudinal research this renders possible the statistical test of homogeneity of change against subject specific change in latent traits applications to two empirical data sets illustrate the use of the models

locally dependent latent trait model and the dutch identity revisited in this paper we propose a class of locally dependent latent trait models for responses to psychological and educational tests typically item response models treat an individual s multiple response to stimuli as conditional independent given the individual s latent trait in this paper instead the focus is on models based on a family of conditional distributions or kernel that describes joint multiple item responses as a function of student latent trait not assuming conditional independence specifically we examine a hybrid kernel which comprises a component for one way item response functions and a component for conditional associations between items given latent traits the class of models allows the extension of item response theory to cover some new and innovative applications in psychological and educational research an em algorithm for marginal maximum likelihood of the hybrid kernel model is proposed furthermore we delineate the relationship of the class of locally dependent models and the log linear model by revisiting the dutch identity holland 1990

minimum rank and minimum trace of covariance matrices this paper considers some mathematical aspects of minimum trace factor analysis mtfa the uniqueness of an optimal point of mtfa is proved and necessary and sufficient conditions for a point x to be optimal are established finally some results about the connection between mtfa and the classical minimum rank factor analysis will be presented

a multidimensional latent trait model for measuring learning and change a latent trait model is presented for the repeated measurement of ability based on a multidimensional conceptualization of the change process a simplex structure is postulated to link item performance under a given measurement condition or occasion to initial ability and to one or more modifiabilities that represent individual differences in change since item discriminations are constrained to be equal within a measurement condition the model belongs to the family of multidimensional rasch models maximum likelihood estimators of the item parameters and abilities are derived and an example provided that shows good recovery of both item and ability parameters properties of the model are explored particularly for several classical issues in measuring change

a determinacy approach to borel combinatorics we introduce a new method involving infinite games and borel determinacy which we use to answer several well known questions in borel combinatorics

some improved diagnostics for failure of the rasch model although several goodness of fit tests have been developed for the rasch model for dichotomous items most of them are of a global asymptotic and confirmatory type this paper based on ideas from a recent thesis by van den wollenberg offers some suggestions for local small sample and exploratory techniques difficulty plots for person groups scoring right and wrong on a specific item a slope test per item based on a binomial distribution per score group and a unidimensionality check based on an extended hypergeometric distribution per score group

rank reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis one of the intriguing questions of factor analysis is the extent to which one can reduce the rank of a symmetric matrix by only changing its diagonal entries we show in this paper that the set of matrices which can be reduced to rankr has positive lebesgue measure if and only ifr is greater or equal to the ledermann bound in other words the ledermann bound is shown to bealmost surely the greatest lower bound to a reduced rank of the sample covariance matrix afterwards an asymptotic sampling theory of so called minimum trace factor analysis mtfa is proposed the theory is based on continuous and differential properties of functions involved in the mtfa convex analysis techniques are utilized to obtain conditions for differentiability of these functions

conjunctive item response theory kernels conjunctive item response models are introduced such that a sufficient statistics for latent traits are not necessarily additive in item scores b items are not necessarily locally independent and c existing compensatory additive item response models including the binomial rasch logistic and general locally independent model are special cases simple estimates and hypothesis tests for conjunctive models are introduced and evaluated as well conjunctive models are also identified with cognitive models that assume the existence of several individually necessary component processes for a global ability it is concluded that conjunctive models and methods may show promise for constructing improved tests and uncovering conjunctive cognitive structure it is also concluded that conjunctive item response theory may help to clarify the relationships between local dependence multidimensionality and item response function form

two new test statistics for the rasch model it is shown that the presently available statistical tests for the rasch model are insensitive to violation of the unidimensionality axiom two new test statistics are presented the first one q 1 is sensitive to the same effects as the presently available statistics but has some desirable properties of a nonstatistical nature the second statistic q 2 is sensitive to violation of local stochastic independence and unidimensionality and thus fills an existing gap

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 hat theta

a comment on some standard errors in item response theory in maximum likelihood estimation the standard error of the location parameter of the three parameter logistic model can be large due to inaccurate estimation of the lower asymptote thissen and wainer who demonstrated this effect suggested that the introduction of a prior distribution for the lower asymptote might alleviate the problems here it is demonstrated in some detail that the standard error of the location parameter can be made acceptably small in this way

validating a selection test a predictive probability approach the problem of inferring the validity of a selection test x as a predictor of some criterion y when completexy data are not available is investigated the basic approach is to construct the predictive probability distribution of the unobservedy scores and then derive interval estimates of the least squares regression weights the difference in averagey scores for selected and unselected cases and the residual variance in predictingy fromx further an approximation to the predictive distribution of the squared correlation betweenx andy in a future group is derived

some standard errors in item response theory the mathematics required to calculate the asymptotic standard errors of the parameters of three commonly used logistic item response models is described and used to generate values for some common situations it is shown that the maximum likelihood estimation of a lower asymptote can wreak havoc with the accuracy of estimation of a location parameter indicating that if one needs to have accurate estimates of location parameters say for purposes of test linking equating or computerized adaptive testing the sample sizes required for acceptable accuracy may be unattainable in most applications it is suggested that other estimation methods be used if the three parameter model is applied in these situations

the structure of hyperfinite borel equivalence relations we study the structure of the equivalence relations induced by the orbits of a single borel automorphism on a standard borel space we show that any two such equivalence relations which are not smooth i e do not admit borel selectors are borel embeddable into each other this utilizes among other things work of effros and weiss using this and also results of dye varadarajan and recent work of nadkarni we show that the cardinality of the set of ergodic invariant measures is a complete invariant for borel isomorphism of aperiodic nonsmooth such equivalence relations in particular since the only possible such cardinalities are the finite ones countable infinity and the cardinality of the continuum there are exactly countably infinitely many isomorphism types canonical examples of each type are also discussed

maximum entropy reconstruction using derivative information part 1 fisher information and convex duality maximum entropy spectral density estimation is a technique for reconstructing an unknown density function from some known measurements by maximizing a given measure of entropy of the estimate here we present a variety of new entropy measures which attempt to control derivative values of the densities our models apply among others to the inference problem based on the averaged fisher information measure the duality theory we develop resembles models used in convex optimal control problems we present a variety of examples including relaxed moment matching with fisher information and best interpolation on a strip

sandwich theorem and supportability for partially defined convex operators abstract given a convex operator e f and a concave operator e f both defined an a locally convex frechet space and having values in an ordered locally convex vector space f with the least upper bound property and satisfying we ask for conditions ensuring the existence of a continuous affine mapping h e f satisfying h besides this continuous sandwich problem we discuss the problem of existence of continuous linear support mappings for partially defined sublinear operators by reducing it to a sandwich problem

binary programming and test design an algorithmic approach to test design using information functions is presented the approach uses a special branch of linear programming i e binary programming in addition results of some benchmark problems are presented within the same framework it is also possible to formulate the problem of individualized testing

a comparison of latent trait and latent class analyses of likert type data this paper brings together and compares two developments in the analysis of likert attitude scales the first is the generalization of latent class models to ordered response categories the second is the introduction of latent trait models with multiplicative parameter structures for the analysis of rating scales key similarities and differences between these two methods are described and illustrated by applying a latent trait model and a latent class model to the analysis of a set of life satisfaction data the way in which the latent trait model defines a unit of measurement takes into account the order of the response categories and scales the latent classes is discussed while the latent class model provides better fit to these data this is achieved at the cost of a logically inconsistent assignment of individuals to latent classes

banach tarski paradox using pieces with the property of baire in 1924 banach and tarski using ideas of hausdorff proved that there is a partition of the unit sphere s2 into sets a1 ak b1 bl and a collection of isometries sigma1 sigmak rho1 rhol so that sigma1a1 sigmakak and rho1b1 rholbl both are partitions of s2 the sets in these partitions are constructed by using the axiom of choice and cannot all be lebesgue measurable in this note we solve a problem of marczewski from 1930 by showing that there is a partition of s2 into sets a1 ak b1 bl with a different strong regularity property the property of baire we also prove a version of the banach tarski paradox that involves only open sets and does not use the axiom of choice

latent change classes in dichotomous data changes in dichotomous data caused by treatments can be analyzed by means of the so called linear logistic model with relaxed assumptions llra the llra does not require observable criteria representing a single underlying latent trait but it postulates the generalizability of the treatment effects over criteria and subjects to test this latter crucial assumption the mixture llra was proposed that allows directly unobservable types of subjects to have different treatment effects as the earlier methods for estimating the parameters of the mixture llra have specific drawbacks a further method based on the conditional maximum likelihood principle will be presented here in contrast to the earlier conditional methods it uses all of the dichotomous change data while having fewer parameters further its goodness of fit tests become more sensitive to a falsely specified number of change types even though the treatment effects are biased for typically occurring small to moderate sample sizes however parametric bootstrapping of the distributions of the fit statistics is recommended for performing hypotheses tests finally three applications of the new method to empirical data are described first about the effect of the so called trager psychophysical integration second about the effect of autogenic therapy on patients with psychosomatic symptoms and third about the effect of religious education on the attitude towards sects

when are item response models consistent with observed data the problem of deciding whether a set of mental test data is consistent with any one of a large class of item response models is considered the classical assumption of locla independence is weakened to a new condition local nonnegative dependence lnd necessary and sufficient conditions are derived for a lnd item response model to fit a set of data this leads to a condition that a set of data must satisfy if it is to be representable by any item response model that assumes both local independence and monotone item characteristic curves an example is given to show that lnd is strictly weaker than local independence thus rejection of lnd models implies rejection of all item response models that assume local independence for a given set of data

borel chromatic numbers we study in this paper graph coloring problems in the context of descriptive set theory we consider graphs g x r where the vertex set x is a standard borel space i e a complete separable metrizable space equipped with its algebra of borel sets and the edge relation r x 2 is definable i e borel analytic co analytic etc a borel n coloring of such a graph where 1 n n 0 is a borel map c x y with card y n such that xry c x c y if such a borel coloring exists we define the borel chromatic number of g in symbols x b g to be the smallest such n otherwise we say that g has uncountable borel chromatic number in symbols x b g n 0