kac moody groups and cosheaves on davis building abstract we investigate smooth representations of complete kac moody groups we approach representation theory via geometry in particular the group action on the davis realisation of its bruhat tits building our results include an estimate on projective dimension localisation theorem unimodularity and homological duality

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

psychometrics behind computerized adaptive testing the paper provides a survey of 18 years progress that my colleagues students both former and current and i made in a prominent research area in psychometrics computerized adaptive testing cat we start with a historical review of the establishment of a large sample foundation for cat it is worth noting that the asymptotic results were derived under the framework of martingale theory a very theoretical perspective of probability theory which may seem unrelated to educational and psychological testing in addition we address a number of issues that emerged from large scale implementation and show that how theoretical works can be helpful to solve the problems finally we propose that cat technology can be very useful to support individualized instruction on a mass scale we show that even paper and pencil based tests can be made adaptive to support classroom teaching

a nonparametric approach for assessing latent trait unidimensionality assuming a nonparametric family of item response theory models a theory based procedure for testing the hypothesis of unidimensionality of the latent space is proposed the asymptotic distribution of the test statistic is derived assuming unidimensionality thereby establishing an asymptotically valid statistical test of the unidimensionality of the latent trait based upon a new notion of dimensionality the test is shown to have asymptotic power 1 a 6300 trial monte carlo study using published item parameter estimates of widely used standardized tests indicates conservative adherence to the nominal level of significance and statistical power averaging 81 out of 100 rejections for examinee sample sizes and psychological test lengths often incurred in practice

analysis of residuals for the multinomial item response model using the item response model as developed on the multinomial distribution asymptotic variances are obtained for residuals associated with response patterns and first and second order marginal frequencies of manifest variables when the model does not fit well an examination of these residuals may reveal the source of the poor fit finally a limited information test of fit for the model is developed by using residuals defined for the first and second order marginals model evaluation based on residuals for these marginals is particularly useful when the response pattern frequencies are sparse

classical test theory as a first order item response theory application to true score prediction from a possibly nonparallel test we give an account of classical test theory ctt in terms of the more fundamental ideas of item response theory irt this approach views classical test theory as a very general version of irt and the commonly used irt models as detailed elaborations of ctt for special purposes we then use this approach to ctt to derive some general results regarding the prediction of the true score of a test from an observed score on that test as well from an observed score on a different test this leads us to a new view of linking tests that were not developed to be linked to each other in addition we propose true score prediction analogues of the dorans and holland measures of the population sensitivity of test linking functions we illustrate the accuracy of the first order theory using simulated data from the rasch model and illustrate the effect of population differences using a set of real data

marginal maximum likelihood estimation for the one parameter logistic model two algorithms are described for marginal maximum likelihood estimation for the one parameter logistic model the more efficient of the two algorithms is extended to estimation for the linear logistic model numerical examples of both procedures are presented

stochastic ordering using the latent trait and the sum score in polytomous irt models in a restricted class of item response theory irt models for polytomous items the unweighted total score has monotone likelihood ratio mlr in the latent trait mlr implies two stochastic ordering so properties denoted som and sol which are both weaker than mlr but very useful for measurement with irt models therefore these so properties are investigated for a broader class of irt models for which the mlr property does not hold

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

on the bock aitkin procedure from an em algorithm perspective the relationship between the em algorithm and the bock aitkin procedure is described with a continuous distribution of ability latent trait from an em algorithm perspective previous work has been restricted to the discrete case from a probit analysis perspective

psychometrics from practice to theory and back the paper surveys 15 years of progress in three psychometric research areas latent dimensionality structure test fairness and skills diagnosis of educational tests it is proposed that one effective model for selecting and carrying out research is to chose one s research questions from practical challenges facing educational testing then bring to bear sophisticated probability modeling and statistical analyses to solve these questions and finally to make effectiveness of the research answers in meeting the educational testing challenges be the ultimate criterion for judging the value of the research the problem solving power and the joy of working with a dedicated focused and collegial group of colleagues is emphasized finally it is suggested that the summative assessment testing paradigm that has driven test measurement research for over half a century is giving way to a new paradigm that in addition embraces skills level formative assessment opening up a plethora of challenging exciting and societally important research problems for psychometricians

semiparametric estimation in the rasch model and related exponential response models including a simple latent class model for item analysis abstract the rasch model for item analysis is an important member of the class of exponential response models in which the number of nuisance parameters increases with the number of subjects leading to the failure of the usual likelihood methodology both conditional likelihood methods and mixture model techniques have been used to circumvent these problems in this article we show that these seemingly unrelated analyses are in fact closely linked to each other despite dramatic structural differences between the classes of models implied by each approach we show that the finite mixture model for j dichotomous items having t latent classes gives the same estimates of item parameters as conditional likelihood on a set whose probability approaches one if t j 1 2 unconditional maximum likelihood estimators for the finite mixture model can be viewed as keifer wolfowitz estimators for the random effects version of the rasch model latent class versions of the model are especially attractive when t is

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

generalizations of paradoxical results in multidimensional item response theory maximum likelihood and bayesian ability estimation in multidimensional item response models can lead to paradoxical results as proven by hooker finkelman and schwartzman psychometrika 74 3 419 442 2009 changing a correct response on one item into an incorrect response may produce a higher ability estimate in one dimension furthermore the conditions under which this paradox arises are very general and may in fact be fulfilled by many of the multidimensional scales currently in use this paper tries to emphasize and extend the generality of the results of hooker et al by 1 considering the paradox in a generalized class of irt models 2 giving a weaker sufficient condition for the occurrence of the paradox with relations to an important concept of statistical association and by 3 providing some additional specific results for linearly compensatory models with special emphasis on the factor analysis model

the dutch identity a new tool for the study of item response models the dutch identity is a useful way to reexpress the basic equations of item response models that relate the manifest probabilities to the item response functions irfs and the latent trait distribution the identity may be exploited in several ways for example a to suggest how item response models behave for large numbers of items they are approximate submodels of second order loglinear models for 2 j tables b to suggest new ways to assess the dimensionality of the latent trait principle components analysis of matrices composed of second order interactions from loglinear models c to give insight into the structure of latent class models and d to illuminate the problem of identifying the irfs and the latent trait distribution from sample data

properties of and algorithms for fitting three way component models with offset terms prior to a three way component analysis of a three way data set it is customary to preprocess the data by centering and or rescaling them harshman and lundy 1984 considered that three way data actually consist of a three way model part which in fact pertains to ratio scale measurements as well as additive offset terms that turn the ratio scale measurements into interval scale measurements they mentioned that such offset terms might be estimated by incorporating additional components in the model but discarded this idea in favor of an approach to remove such terms from the model by means of centering then estimates for the three way component model parameters are obtained by analyzing the centered data in the present paper the possibility of actually estimating the offset terms is taken up again first it is mentioned in which cases such offset terms can be estimated uniquely next procedures are offered for estimating model parameters and offset parameters simultaneously as well as successively i e providing offset term estimates after the three way model parameters have been estimated in the traditional way on the basis of the centered data these procedures are provided for both the candecomp parafac model and the tucker3 model extended with offset terms the successive and the simultaneous approaches for estimating model and offset parameters have been compared on the basis of simulated data it was found that both procedures perform well when the fitted model captures at least all offset terms actually underlying the data the simultaneous procedures performed slightly better than the successive procedures if fewer offset terms are fitted than there are underlying the model the results are considerably poorer but in these cases the successive procedures performed better than the simultaneous ones all in all it can be concluded that the traditional approach for estimating model parameters can hardly be improved upon and that offset terms can sufficiently well be estimated by the proposed successive approach which is a simple extension of the traditional approach

the missing data assumptions of the neat design and their implications for test equating the non equivalent groups with anchor test neat design involves missingdata that are missing by design three nonlinear observed score equating methods used with a neat design are the frequency estimation equipercentile equating feee the chain equipercentile equating cee and the item response theory observed score equating irt ose these three methods each make different assumptions about the missing data in the neat design the feee method assumes that the conditional distribution of the test score given the anchor test score is the same in the two examinee groups the cee method assumes that the equipercentile functions equating the test score to the anchor test score are the same in the two examinee groups the irt ose method assumes that the irt model employed fits the data adequately and the items in the tests and the anchor test do not exhibit differential item functioning across the two examinee groups this paper first describes the missing data assumptions of the three equating methods then it describes how the missing data in the neat design can be filled in a manner that is coherent with the assumptions made by each of these equating methods implications on equating are also discussed

on holland s dutch identity conjecture the manifest probabilities of observed examinee response patterns resulting from marginalization with respect to the latent ability distribution produce the marginal likelihood function in item response theory under the conditions that the posterior distribution of examinee ability given some test response pattern is normal and the item logit functions are linear holland 1990a gives a quadratic form for the log manifest probabilities by using the dutch identity further holland conjectures that this special quadratic form is a limiting one for all smooth unidimensional item response models as test length tends to infinity the purpose of this paper is to give three counterexamples to demonstrate that holland s dutch identity conjecture does not hold in general the counterexamples suggest that only under strong assumptions can it be true that the limits of log manifest probabilities are quadratic three propositions giving sets of such strong conditions are given

conditional association and unidimensionality in monotone latent variable models latent variable models represent the joint distribution of observable variables in terms of a simple structure involving unobserved or latent variables usually assuming the conditional independence of the observable variables given the latent variables these models play an important role in educational measurement and psychometrics in sociology and in population genetics and are implicit in some work on systems reliability we study a broad class of latent variable models namely the monotone unidimensional models in which the latent variable is a scalar the observable variables are conditionally independent given the latent variable and the conditional distribution of the observables given the latent variable is stochastically increasing in the latent variable all models in this class imply a new strong form of positive dependence among the observable variables namely conditional positive association this positive dependence condition may be used to test whether any model in this class can provide an adequate fit to observed data various applications generalizations and a numerical example are discussed

an analysis of item response theory and rasch models based on the most probable distribution method the most probable distribution method is applied to derive the logistic model as the distribution accounting for the maximum number of possible outcomes in a dichotomous test while introducing latent traits and item characteristics as constraints to the system the item response theory logistic models with a particular focus on the one parameter logistic model or rasch model and their properties and assumptions are discussed for both infinite and finite populations

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

a characterization of monotone unidimensional latent variable models recently the problem of characterizing monotone unidimensional latent variable models for binary repeated measures was studied by ellis and van den wollenberg and by junker we generalize their work with a de finetti like characterization of the distribution of repeated measures x x 1 x 2 that can be represented with mixtures of likelihoods of independent but not identically distributed random variables where the data satisfy a stochastic ordering property with respect to the mixing variable the random variables x j may be arbitrary real valued random variables we show that the distribution of x can be given a monotone unidimensional latent variable representation that is useful in the sense of junker if and only if this distribution satisfies conditional association ca and a vanishing conditional dependence vcd condition which asserts that finite subsets of the variables in x become independent as we condition on a larger and larger segment of the remaining variables in x it is also interesting that the mixture representation is in a certain ordinal sense unique when ca and vcd hold the characterization theorem extends and simplifies the main result of junker and generalizes methods of ellis and van den wollenberg to a much broader class of models exchangeable sequences of binary random variables also satisfy both ca and vcd as do exchangeable sequences arising as location mixtures in the same way that de finetti s theorem provides a path toward justifying standard i i d mixture components in hierarchical models on the basis of our intuitions about the exchangeability of observations this theorem justifies one dimensional latent variable components in hierarchical models in terms of our intuitions about positive association and redundancy between observations because these conditions are on the joint distribution of the observable data x they may also be used to construct asymptotically power 1 tests for unidimensional latent variable models

manifest characterization and testing for certain latent properties work due to junker and more recently due to junker and ellis characterized desired latent properties of an educational testing procedure in terms of a collection of other manifest properties this is important because one can only propose tests for manifest quantities not latent ones here we complete the conversion of a pair of latent properties to equivalent conditions in terms of four manifest quantities and identify a general method for producing tests for manifest properties 1 introduction item response theory irt is the statistical theory of standardized tests which are commonly used in educational testing applications the goal is to combine the data generated by many examinees answering a collection of test items so as to estimate the value of a parameter or trait say for each of the examinees the parameter is intended to quantify a latent trait such as mathematics ability the traits are latent in the sense that they cannot be measured directly more specifically desired properties of suchtesting procedures are called latent when their statement depends explicitly on the the latent trait being estimated most efforts at modeling involve the introduction of latent parameters one specifies parameters to reflect aspects of the physical problem such as the individual achievement of each examinee the difficulty of each test item specialized knowledge that certain examinees might possess etc this is in contrast to the manifest properties of a testing procedure which can be phrased in a way that does not explicitly depend on the latent trait the limitation of a latent model is that it contains parameters for the latent trait the experimenter wants to estimate each examinee s parameter value and so cannot perform hypothesis tests on a conjectured model in the usual way the solution to this problem is to convert latent models or latent statements about them to physically meaningful manifest properties which are provably mathematically equivalent to the latent properties actually wanted manifest statements are amenable to conventional hypothesis testing and the problem then becomes the identification of optimal hypothesis tests for the manifest quantities in this way one can test hypotheses that are independent of the latent structure at least in principle the goal of characterizing latent properties in terms of manifest properties to be tested begins with stout 1987 stout considers the requirement that

on supremum of bounded quantum observable in this paper we present a new necessary and sufficient condition for which the supremum a b exists with respect to the logic order moreover we give out a new and much simpler representation of a b with respect to our results have nice physical meanings

local homogeneity in latent trait models a characterization of the homogeneous monotone irt model the stochastic subject formulation of latent trait models contends that within a given subject the event of obtaining a certain response pattern may be probabilistic ordinary latent trait models do not imply that these within subject probabilities are identical to the conditional probabilities specified by the model the latter condition is called local homogeneity it is shown that local homgeneity is equivalent to subpopulation invariance of the model in case of the monotone irt model local homogeneity implies absence of item bias absence of item specific traits and the possibility to join overlapping subtests the following characterization theorem is proved the homogeneous monotone irt model holds for a finite or countable item pool if and only if the pool is experimentally independent and pairwise nonnegative association holds in every positive subpopulation

tail measurability in monotone latent variable models we consider latent variable models for an infinite sequence or universe of manifest observable variables that may be discrete continuous or some combination of these the main theorem is a general characterization by empirical conditions of when it is possible to construct latent variable models that satisfy unidimensionality monotonicity conditional independence andtail measurability tail measurability means that the latent variable can be estimated consistently from the sequence of manifest variables even though an arbitrary finite subsequence has been removed the characterizing necessary and sufficient conditions that the manifest variables must satisfy for these models are conditional association and vanishing conditional dependence as one conditions upon successively more other manifest variables our main theorem considerably generalizes and sharpens earlier results of ellis and van den wollenberg 1993 holland and rosenbaum 1986 and junker 1993 it is also related to the work of stout 1990 the main theorem is preceded by many results for latent variable modelsin general not necessarily unidimensional and monotone they pertain to the uniqueness of latent variables and are connected with the conditional independence theorem of suppes and zanotti 1981 we discuss new definitions of the concepts of true score and subpopulation which generalize these notions from the stochastic subject random sampling and domain sampling formulations of latent variable models e g holland 1990 lord novick 1968 these definitions do not require the a priori specification of a latent variable model

item response theory with estimation of the latent population distribution using spline based densities the purpose of this paper is to introduce a new method for fitting item response theory models with the latent population distribution estimated from the data using splines a spline based density estimation system provides a flexible alternative to existing procedures that use a normal distribution or a different functional form for the population distribution a simulation study shows that the new procedure is feasible in practice and that when the latent distribution is not well approximated as normal two parameter logistic 2pl item parameter estimates and expected a posteriori scores eaps can be improved over what they would be with the normal model an example with real data compares the new method and the extant empirical histogram approach

a new item response theory modeling approach with applications to unidimensionality assessment and ability estimation using an infinite item test framework it is argued that the usual assumption of local independence be replaced by a weaker assumption essential independence a fortiori the usual assumption of unidimensionality is replaced by a weaker and arguably more appropriatestatistically testable assumption of essential unidimennsionality essential unidimennsionnality implies the existence of a unique unidimensional latent ability essential unidimensionality is equivalent to the consistent estimation of this latnet ability in an ordinal scaling sense using anybalanced empirical scaling a variation of this estimation approach allows consistent estimation of ability on the given latent ability scale

comparing latent structures of the grade of membership rasch and latent class models this paper focuses on model interpretation issues and employs a geometric approach to compare the potential value of using the grade of membership gom model in representing population heterogeneity we consider population heterogeneity manifolds generated by letting subject specific parameters vary over their natural range while keeping other population parameters fixed in the marginal space based on marginal probabilities and in the full parameter space based on cell probabilities the case of a 2 2 contingency table is discussed in detail and a generalization to 2j tables with j 3 is sketched our approach highlights the main distinction between the gom model and the probabilistic mixture of classes by demonstrating geometrically the difference between the concepts of partial and probabilistic memberships by using the geometric approach we show that in special cases the gom model can be thought of as being similar to an item response theory irt model in representing population heterogeneity finally we show that the gom item parameters can provide quantities analogous to more general logistic irt item parameters as a latent structure model the gom model might be considered a useful alternative for a data analysis when both classes of extreme responses and additional heterogeneity that cannot be captured by those latent classes are expected in the population

on the sampling theory roundations of item response theory models item response theory it models are now in common use for the analysis of dichotomous item responses this paper examines the sampling theory foundations for statistical inference in these models the discussion includes some history on the stochastic subject versus the random sampling interpretations of the probability in irt models the relationship between three versions of maximum likelihood estimation for irt models estimating versus estimating predictors irt models and loglinear models the identifiability of irt models and the role of robustness and bayesian statistics from the sampling theory perspective

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

some remarks on scheiblechner s treatment of isop models scheiblechner 1995 proposes a probabilistic axiomatization of measurement called isop isotonic ordinal probabilistic models that replaces rasch s 1980 specific objectivity assumptions with two interesting ordinal assumptions special cases of scheiblechner s model include standard unidimensional factor analysis models in which the loadings are held constant and the rasch model for binary item responses closely related are the doubly monotone item response models of mokken 1971 see also mokken lewis 1982 sijtsma 1988 molenaar 1991 sijtsma junker 1996 and sijtsma hemker in press more generally strictly unidimensional latent variable models have been considered in some detail by holland and rosenbaum 1986 ellis and van den wollenberg 1993 and junker 1990 1993 the purpose of this note is to provide connections with current research in foundations and nonparametric latent variable and item response modeling that are missing from scheiblechner s 1995 paper and to point out important related work by hemker sijtsma molenaar junker 1996 1997 ellis and junker in press and junker and ellis 1997 we also discuss counterexamples to three major theorems in the paper by carrying out these three tasks we hope to provide researchers interested in the foundations of measurement and item response modeling the opportunity to give the isop approach the careful attention it deserves

consistent estimation in the rasch model based on nonparametric margins consider the class of two parameter marginal logistic rasch models for a test ofm true false items where the latent ability is assumed to be bounded using results of karlin and studen we show that this class of nonparametric marginal logistic nml models is equivalent to the class of marginal logistic models where the latent ability assumes at most m 2 2 values this equivalence has two implications first estimation for the nml model is accomplished by estimating the parameters of a discrete marginal logistic model second consistency for the maximum likelihood estimates of the nml model can be shown whenm is odd using the results of kiefer and wolfowitz an example is presented which demonstrates the estimation strategy and contrasts the nml model with a normal marginal logistic model