tensors of nonnegative rank two abstract a nonnegative tensor has nonnegative rank at most 2 if and only if it is supermodular and has flattening rank at most 2 we prove this result then explore the semialgebraic geometry of the general markov model on phylogenetic trees with binary states and comment on possible extensions to tensors of higher rank

total positivity in markov structures we discuss properties of distributions that are multivariate totally positive of order two mtp2 related to conditional independence in particular we show that any independence model generated by an mtp2 distribution is a compositional semi graphoid which is upward stable and singletontransitive in addition we prove that any mtp2 distribution satisfying an appropriate support condition is faithful to its concentration graph finally we analyze factorization properties of mtp2 distributions and discuss ways of constructing mtp2 distributions in particular we give conditions on the log linear parameters of a discrete distribution which ensure mtp2 and characterize conditional gaussian distributions which satisfy mtp2

a likelihood ratio test for mtp_2 within binary variables multivariate totally positive mtp 2 binary distributions have been studied in many fields such as statistical mechanics computer storage and latent variable models we show that mtp 2 is equivalent to the requirement that the parameters of a saturated log linear model belong to a convex cone and we provide a fisher scoring algorithm for maximum likelihood estimation we also show that the asymptotic distribution of the log likelihood ratio is a mixture of chi squares a distribution known as chi bar squared in the literature on order restricted inference for this we derive tight bounds which turn out to have very simple forms a potential application of this method is for item response theory irt models which are used in educational assessment to analyse the responses of a group of subjects to a collection of questions items an important issue within irt is whether the joint distribution of the manifest variables is compatible with a single latent variable representation satisfying local independence and monotonicity which in turn imply that the joint distribution of item responses is mtp 2

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

the foundations of factor analysis summary a new approach to factor analysis and related latent variable methods is proposed which is based on data reduction using the idea of bayesian sufficiency considerations of symmetry invariance and independence are used to determine an appropriate family of models the results are expressed in terms of linear functions of the manifest variables after the manner of principal components analysis the approach justifies some of the practices based on the normal theory factor model and lays a foundation for the treatment of nonnormal including categorical variables 1 background factor analysis is a widely used statistical technique but its theoretical foundations are somewhat obscure and subject to dispute it is one of a family of multivariate methods which also includes latent structure and latent trait analysis the common feature of the models underlying these methods is that the observed random variables are assumed to depend on latent that is unobserved random variables there is sometimes debate about whether these latent variables are real in any sense but they can be viewed simply as constructs designed to simplify and summarize the complex web of interrelated variables with which nature confronts us by expressing these relationships in terms of a small number of latent variables or factors the models effect a reduction in dimensionality which aids comprehension an early theoretical account of the subject is by anderson rubin 1956 and more recent and comprehensive treatments are provided by lawley maxwell 1971 and harman 1968 the present paper is an attempt to provide a coherent framework within which existing methodology can be evaluated and a base from which new methods can be developed our approach is to start from a very general statement of the problem in terms of the distributions of the random variables involved and then to invoke ideas of symmetry invariance and conditional independence to determine the class of models that it is reasonable to consider this not only exhibits the unity of the various models already in existence but resolves many of the ambiguities and obscurities with which the subject has been bedevilled an early approach on these lines is given by anderson 1959 and was recognized by birnbaum in his contribution to lord novick 1968 the same line was followed by bartholomew 1980 in spite of this the practical implications do not seem to have been made fully explicit or generally recognized

additive conjoint isotonic probabilistic models adisop the isop model or model of twodimensional or bi isotonicity scheiblechner 1995 postulates that the probabilities of ordered response categories increase isotonically in the order of subject ability and item easiness adding a conventional cancellation axiom for the factors of subjects and items gives the adisop model where the c d f s of response categories are functions of an additive item and subject parameter and an ordinal category parameter extending cancellation to the interactions of subjects and categories as well as of items and categories independence axiom of the category factor from the subject and item factor gives the cadisop model completely additive model in which the parallel c d f s are functions of the sum of subject item and category parameters the cadisop model is very close to the unidimensional version of the polytomous rasch model with the logistic item category characteristic s replaced by nonparametric axioms and statistics the axioms representation theorems and algorithms for model fitting of the additive models are presented

a note on multiple solutions of some semilinear elliptic problems abstract we discuss the existence of multiple solutions of nonlinear elliptic equations by a combination of variational topological methods and the generalized conley index theory we obtain several positive solutions and sign changing solutions our main point is to show the usefulness of the morse inequalities for morse decompositions in the generalized conley index theory

some recent development in a concept of causality abstract the paper considers three separate but related topics i what is the relationship between causation and co integration if a pair of i 1 series are co integration there must be causation in at least one direction an implication is that some tests of causation based on different series may have missed one source of causation ii is there a need for a definition of instantaneous causation in a decision science it is argued that no such definition is required iii can causality tests be used for policy evaluation it is suggested that these tests are useful but that they should be evaluated with care

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

latent class analysis of two way contingency tables by bayesian methods summary latent class analysis in two way contingency tables usually suffers from unidentifiability problems these can be overcome by using bayesian techniques in which prior distributions are assumed on the latent parameters application of these techniques however involves some unusual computational difficulties bayesian techniques suitable for latent class analyses together with a remedy to the computational difficulties are discussed in this paper

nonparametric polytomous irt models for invariant item ordering with results for parametric models it is often considered desirable to have the same ordering of the items by difficulty across different levels of the trait or ability such an ordering is an invariant item ordering iio an iio facilitates the interpretation of test results for dichotomously scored items earlier research surveyed the theory and methods of an invariant ordering in a nonparametric irt context here the focus is on polytomously scored items and both nonparametric and parametric irt models are considered

solutions of semilinear elliptic equations with asymptotic linear nonlinearity and and bare positive real numbers the solutions structure of 1 is quite dierent when is equal to zero or not when is equal to zero or a nonzero real number but small enough there are some resultsof solutions structure of 1 in 1 9 here we extend these results to the case when is an arbitrary positive real number forconvenience wedenotef u

exact multiplicity of positive solutions for a class of semilinear problem ii we consider the positive solutions to the semilinear problem u f u 0 in b n u 0 on b n where bn is the unit ball in rn n 1 and is a positive parameter it is well known that if is a smooth function then any positive solution to the equation is radially symmetric and all solutions can be parameterized by their maximum values we develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions u and the precise shape of the global bifurcation diagrams are rigorously proved our technique combines the bifurcation analysis stability analysis and topological methods we show that the shape of the bifurcation curve depends on the shape of the graph of function u u as well as the growth rale of

multiple solutions for a class of semilinear elliptic problems with robin boundary condition abstract in this paper we show the existence of at least four nontrivial solutions for a class of semilinear elliptic problems with robin boundary condition and jumping nonlinearities solvability of oscillating equations with robin boundary condition is also investigated we prove the conclusions by using sub super solution method fucik spectrum theory mountain pass theorem in order intervals and morse theory

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

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

limits on log odds ratios for unidimensional item response theory models bounds are established for log odds ratios log cross product ratios involving pairs of items for item response models first expressions for bounds on log odds ratios are provided for one dimensional item response models in general then explicit bounds are obtained for the rasch model and the two parameter logistic 2pl model results are also illustrated through an example from a study of model checking procedures the bounds obtained can provide an elementary basis for assessment of goodness of fit of these models

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

likelihood inference on the underlying structure of irt models the assumptions underlying item response theory irt models may be expressed as a set of equality and inequality constraints on the parameters of a latent class model it is well known that the same assumptions imply that the parameters of the manifest distribution have to satisfy a more complicated set of inequality constraints which however are necessary but not sufficient in this paper we describe how the theory for likelihood based inference under equality and inequality constraints may be used to test the underlying assumptions of irt models it turns out that the analysis based directly on the latent structure is simpler and more flexible in particular we indicate how several interesting extensions of the rasch model may be obtained by partial relaxation of the basic constraints an application to a data set provided by educational testing service is used to illustrate the approach

mcmc estimation and some model fit analysis of multidimensional irt models a bayesian procedure to estimate the three parameter normal ogive model and a generalization of the procedure to a model with multidimensional ability parameters are presented the procedure is a generalization of a procedure by albert 1992 for estimating the two parameter normal ogive model the procedure supports analyzing data from multiple populations and incomplete designs it is shown that restrictions can be imposed on the factor matrix for testing specific hypotheses about the ability structure the technique is illustrated using simulated and real data

an inequality for correlations in unidimensional monotone latent variable models for binary variables it is shown that a unidimensional monotone latent variable model for binary items implies a restriction on the relative sizes of item correlations the negative logarithm of the correlations satisfies the triangle inequality this inequality is not implied by the condition that the correlations are nonnegative the criterion that coefficient h exceeds 0 30 or manifest monotonicity the inequality implies both a lower bound and an upper bound for each correlation between two items based on the correlations of those two items with every possible third item it is discussed how this can be used in mokken s a theory and procedure of scale analysis mouton the hague 1971 scale analysis

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

nonexistence of solutions for prescribed mean curvature equations on a ball abstract we prove two nonexistence results of radial solutions to the prescribed mean curvature type problem on a ball div d u 1 d u 2 f u x b r r n u 0 x b r where is a positive parameter f is a continuous function with f 0 0 under suitable assumptions on f we show that the problem with superlinear f has no nontrivial positive solutions for small while the problem with sublinear f has no nontrivial positive solutions for large the former covers many well known nonexistence results by finn serrin narukawa and suzuki ishimura pan and xing to the best of the authors knowledge the latter is the first nonexistence result involving sublinear mean curvature type equations in higher dimensions in particular the sublinear cases contain some important logistic type nonlinearities these nonexistence results differ greatly from those of semilinear problems

structure of the solution set for a class of semilinear elliptic equations with asymptotic linear nonlinearity we consider a semilinear elliptic equation with asymptotic linear nonlinearity applying bifurcation theory and spectral analysis we obtain the exact multiplicity of the positive solutions and a very precise structure of the solution set which improves the previous knowledge of the problem

independence via uncorrelatedness under certain dependence structures a characterization of independence via uncorrelatedness is shown to hold for the families satisfying positive and negative dependence conditions for the associated random variables the bounds on covariance functions obtained by lebowitz comm math phys 28 1972 313 321 readily yield such a characterization an elementary proof for the same characterization is also given for a condition weaker than association labeled as strong positive negative orthant dependence this condition is compared with the linear positive dependence under which newman and wright ann probab 9 1981 671 675 obtained the characterization

exact multiplicity of solutions to superlinear and sublinear problems f2 lim u f u u f 0 lim u f u u f 0 for the de5niteness we assume f f and when f f we use f to represent it we will consider f being either superlinear or sublinear f is said to be superlinear if f u u is decreasing in 0 and is increasing in 0 and f is said to be sublinear if f u u is increasing in 0 and is decreasing in 0 the semilinear equation 1 1 with f satisfying f1 f2 has been studied extensively since early 1970s several di9erent approaches like variational methods morse

solutions of superlinear elliptic equations and their morse indices we consider here solutions of semilinear second order elliptic equations with superlinear nonlinearities and we present some relationships between their morse indices and some qualitative properties in particular we show that for uo subcritical nonlinearities bounds on solutions are equivalent to bounds on their morse indices

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

differential effects and generic biases in observational studies there are two treatments each of which may be applied or withheld yielding a 2 2 factorial arrangement with three degrees of freedom between groups the differential effect of the two treatments is the effect of applying one treatment in lieu of the other in randomised experiments the differential effect is of no more or less interest than other treatment contrasts differential effects play a special role in certain observational studies in which treatments are not assigned to subjects at random where differing outcomes may reflect biased assignments rather than effects caused by the treatments differential effects are immune to certain types of unobserved bias called generic biases which are associated with both treatments in a similar way this is explored using several examples and models differential effects are not immune to differential biases whose possible consequences are examined by sensitivity analysis

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

statistical analysis for multiplicatively modulated nonlinear autoregressive model and its applications to electrophysiological signal analysis in humans modulating the dynamics of a nonlinear autoregressive model with a radial basis function rbf of exogenous variables is known to reduce the prediction error here rbf is a function that decays to zero exponentially if the deviation between the exogenous variables and a center location becomes large this paper introduces a class of rbf based multiplicatively modulated nonlinear autoregressive mmnar models first we establish the local asymptotic normality lan for vector conditional heteroscedastic autoregressive nonlinear charn models which include the mmnar and many other well known time series models as special cases asymptotic optimality for estimation and testing is described in terms of lan properties the mmnar model indicates goodness of fit for surface electromyograms emg using electrocorticograms ecog as the exogenous variables concretely it is found that the negative potential of the motor cortex forces change in the frequency of emg which is reasonable from a physiological point of view the proposed mmnar model fitting is both useful and efficient as a signal processing technique for extracting information on the action potential which is associated with the postsynaptic potential

modelling nonlinear random vibrations using an amplitude dependent autoregressive time series model summary the behaviour of nonlinear deterministic vibrations has been studied by many authors and may typically include such features as jump phenomena and limit cycles nonlinear random vibrations in continuous time have also been studied and these may commonly give rise to the phenomenon of amplitude dependent frequency a discrete time series model is introduced which may be demonstrated to have properties similar to those of nonlinear random vibrations this model is of autoregressive form with amplitudedependent coefficients and may be estimated using an extension of a method for estimating linear time series models the model is fitted to the canadian lynx data and demonstrates that it may be possible to regard the periodic behaviour of this series as being generated by some underlying self exciting mechanism

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

isotonic ordinal probabilistic models isop the concept of an ordinal instrumental probabilistic comparison is introduced it relies on an ordinal scale given a priori and on the concept of stochastic dominance it is used to define a weakly independently ordered system or isotonic ordinal probabilistic isop model which allows the construction of separate sample free ordinal scales on a set of subjects and a set of items the isop model is a common nonparametric theoretical structure for unidimensional models for quantitative ordinal and dichotomous variables

persistence and bifurcation of degenerate solutions abstract we consider a nonlinear equation f e u 0 where f is a differentiable mapping from r r x to y and x y are banach spaces when e varies from a fixed e e 0 bifurcation occurs to the solution curve s u s we study the degenerate solutions of the equation and we obtain several bifurcation theorems on the degenerate solutions which can be applied in many nonlinear problems to obtain precise global bifurcation diagrams

when do latent class models overstate accuracy for diagnostic and other classifiers in the absence of a gold standard summary latent class models are increasingly used to assess the accuracy of medical diagnostic tests and other classifications when no gold standard is available and the true state is unknown when the latent class is treated as the true class the latent class models provide measures of components of accuracy including specificity and sensitivity and their complements type i and type ii error rates the error rates according to the latent class model differ from the true error rates however and empirical comparisons with a gold standard suggest the true error rates often are larger we investigate conditions under which the true type i and type ii error rates are larger than those provided by the latent class models results from uebersax 1988 psychological bulletin 104 405 416 are extended to accommodate random effects and covariates affecting the responses the results are important for interpreting the results of latent class analyses an error decomposition is presented that incorporates an error component from invalidity of the latent class model

solution curves and exact multiplicity results for 2mth order boundary value problems abstract for any integer m 2 we consider the 2mth order boundary value problem 1 m u 2m x g u x u x x 1 1 u i 1 u i 1 0 i 0 m 1 where r and the function g r r is c1 and satisfies g 0 0 g 0 0 together with some further conditions as we obtain curves of nontrivial solutions of this problem bifurcating from u 0 at the eigenvalues of the linearised problem and obtain the exact number of solutions of the problem for lying in various intervals in r

on bifurcation critical groups and exact multiplicity of solutions to semilinear elliptic boundary value problems in this paper we use a bifurcation approach morse theory to study asymptotically linear and superlinear elliptic boundary value problems and obtain new results on the exact multiplicity of solutions and sign changing solutions to asymptotically linear and superlinear problems and the properties of the solutions

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

unidimensional factor models imply weaker partial correlations than zero order correlations in this paper we present a new implication of the unidimensional factor model we prove that the partial correlation between two observed variables that load on one factor given any subset of other observed variables that load on this factor lies between zero and the zero order correlation between these two observed variables we implement this result in an empirical bootstrap test that rejects the unidimensional factor model when partial correlations are identified that are either stronger than the zero order correlation or have a different sign than the zero order correlation we demonstrate the use of the test in an empirical data example with data consisting of fourteen items that measure extraversion

on nontrivial solutions of an asymptotically linear dirichlet problem let be a bounded domain in rn n 2 with smooth boundary let r r be a continuous function such that 0 0 and lim exist we discuss the existence of nontrivial solutions of the dirichlet problem u u in u 0 on our hypotheses seem to allow significantly greater freedom for the numbers lim than is usually found in the literature in that the interval with end points is here permitted to contain an eigenvalue of subject to zero dirichlet data furthermore we do not require any differentiability of and we weaken other technical conditions on that are generally assumed in other papers dealing with this problem

m matrices as covariance matrices of multinormal distributions abstract the class of multivariate normal densities n 0 whose inverse covariance matrix 1 is an m matrix is equivalent to this normal density being multivariate totally positive of order 2 mtp 2 equivalent characterizations are given in terms of certain partial correlation coefficients being positive it is further shown that related partial and multiple regression coefficients and canonical correlation are positive when is an m matrix the corresponding normal random vector components are negatively associated this concept and some extensions are discussed

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

remarks and a correction of ligtvoet s treatment of the isotonic partial credit model this note contains some remarks on ligtvoet s psychometrika 77 479 494 2012 treatment of the isotonic partial credit model the proposition relating to the observable property mwi is shown to be false

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

criterion related construct validity established results on latent variable models are applied to the study of the validity of a psychological test when the test predicts a criterion by measuring a unidimensional latent construct not only must the total score predict the criterion but the joint distribution of criterion scores and item responses must exhibit a certain pattern the presence of this population pattern may be tested with sample data using the stratified wilcoxon rank sum test often criterion information is available only for selected examinees for instance those who are admitted or hired three cases are discussed i selection at random ii selection based on the current test and iii selection based on other measures of the latent construct discriminant validity is also discussed

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

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

comparing item characteristic curves test items are often evaluated and compared by contrasting the shapes of their item characteristics curves icc s or surfaces the current paper develops and applies three general i e nonparametric comparisons of the shapes of two item characteristic surfaces i proportional latent odds ii uniform relative difficulty and iii item sensitivity two items may be compared in these ways while making no assumption about the shapes of item characteristic surfaces for other items and no assumption about the dimensionality of the latent variable also studied is a method for comparing the relative shapes of two item characteristic curves in two examinee populations

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

adjusting for information inflation due to local dependency in moderately large item clusters when multiple items are clustered around a reading passage the local independence assumption in item response theory is often violated the amount of information contained in an item cluster is usually overestimated if violation of local independence is ignored and items are treated as locally independent when in fact they are not in this article we provide a general method that adjusts for the inflation of information associated with a test containing item clusters a computational scheme was presented for the evaluation of the factor of adjustment for clusters in the restrictive case of two items per cluster and the general case of more than two items per cluster the methodology was motivated by a study of the naep reading assessment we present a simulated study along with an analysis of a naep data set

exact multiplicity and global structure of solutions for a class of semilinear elliptic equations abstract in this paper we establish an exact multiplicity result of solutions for a class of semilinear elliptic equation we also obtain a precise global bifurcation diagram of the solution set as a result an open problem presented by c h hsu and y w shih c h hsu y w shih solutions of semilinear elliptic equations with asymptotic linear nonlinearity nonlinear anal 50 2002 275 283 is completely solved our argument is mainly based on bifurcation theory and continuation method

an isotonic partial credit model for ordering subjects on the basis of their sum scores in practice the sum of the item scores is often used as a basis for comparing subjects for items that have more than two ordered score categories only the partial credit model pcm and special cases of this model imply that the subjects are stochastically ordered on the common latent variable however the pcm is very restrictive with respect to the constraints that it imposes on the data in this paper sufficient conditions for the stochastic ordering of subjects by their sum score are obtained these conditions define the isotonic nonparametric pcm model the isotonic pcm is more flexible than the pcm which makes it useful for a wider variety of tests also observable properties of the isotonic pcm are derived in the form of inequality constraints it is shown how to obtain estimates of the score distribution under these constraints by using the gibbs sampling algorithm a small simulation study shows that the bayesian p values based on the log likelihood ratio statistic can be used to assess the fit of the isotonic pcm to the data where model data fit can be taken as a justification of the use of the sum score to order subjects

the control of the false discovery rate in multiple testing under dependency benjamini and hochberg suggest that the false discovery rate may be the appropriate error rate to control in many applied multiple testing problems a simple procedure was given there as an fdr controlling procedure for independent test statistics and was shown to be much more powerful than comparable procedures which control the traditional familywise error rate we prove that this same procedure also controls the false discovery rate when the test statistics have positive regression dependency on each of the test statistics corresponding to the true null hypotheses this condition for positive dependency is general enough to cover many problems of practical interest including the comparisons of many treatments with a single control multivariate normal test statistics with positive correlation matrix and multivariate t furthermore the test statistics may be discrete and the tested hypotheses composite without posing special difficulties for all other forms of dependency a simple conservative modification of the procedure controls the false discovery rate thus the range of problems for which a procedure with proven fdr control can be offered is greatly increased 1 1 simultaneous hypotheses testing the control of the increased type i error when testing simultaneously a family of hypotheses is a central issue in the area of multiple comparisons rarely are we interested only in whether all hypotheses are jointly true or not which is the test of the intersection null hypothesis in most applications we infer about the individual hypotheses realizing that some of the tested hypotheses are usually true we hope not all and some are not we wish to decide which ones are not true indicating statistical discoveries an important such problem is that of multiple endpoints in a clinical trial a new treatment is compared with an existing one in terms of a large number of potential benefits endpoints

generalized residuals for general models for contingency tables with application to item response theory generalized residuals are a tool employed in the analysis of contingency tables to examine possible sources of model error they have typically been applied to log linear models and to latent class models a general approach to generalized residuals is developed for a very general class of models for contingency tables to illustrate their use generalized residuals are applied to models based on item response theory irt models such models are commonly applied to analysis of standardized achievement or aptitude tests to obtain a realistic perspective on application of generalized residuals actual testing data are employed

copula functions for residual dependency most item response theory models are not robust to violations of conditional independence however several modeling approaches e g conditioning on other responses additional random effects exist that try to incorporate local item dependencies but they have some drawbacks such as the nonreproducibility of marginal probabilities and resulting interpretation problems in this paper a new class of models making use of copulas to deal with local item dependencies is introduced these models belong to the bigger class of marginal models in which margins and association structure are modeled separately it is shown how this approach overcomes some of the problems associated with other local item dependency models

distractor selection ratios a latent variable representation for multiple choice item and option characteristic curves is presented under standard assumptions of conditional independence of item responses and monotonicity of item characteristic curves a criterion for distractors is proposed based on distractor selection ratios a connection is made between the proposed criterion and the theory of individual choice behavior providing new insight the main results allow for the testing of the criterion from observable data without first specifying a parametric form for the characteristic curves a series of examples apply the method

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

positive association among three binary variables and cross product ratios we show that when the three way association level among the three binary variables x u 1 and u 2 is fixed dp pr x 1 u 1 1 pr x 11 u 1 0 increases as the cross product ratio of u 1 and u 2 increases under the assumption that x is positively associated with u 1 and u 2 we then discuss some implications of this property

conditional covariance structure of generalized compensatory multidimensional items some nonparametric dimensionality assessment procedures such as dimtest and detect use nonparametric estimates of item pair conditional covariances given an appropriately chosen subtest score as their basic building blocks such conditional covariances given some subtest score can be regarded as an approximation to the conditional covariances given an appropriately chosen unidimensional latent composite where the composite is oriented in the multidimensional test space direction in which the subtest score measures best in this paper the structure and properties of such item pair conditional covariances given a unidimensional latent composite are thoroughly investigated assuming a semiparametric irt modeling framework called a generalized compensatory model it is shown that such conditional covariances are highly informative about the multidimensionality structure of a test the theory developed here is very useful in establishing properties of dimensionality assessment procedures current and yet to be developed that are based upon estimating such conditional covariances in particular the new theory is used to justify the dimtest procedure because of the importance of conditional covariance estimation a new bias reducing approach is presented a byproduct of likely independent importance beyond the study of conditional covariances is a rigorous score information based definition of an item s and a score s direction of best measurement in the multidimensional test space

a multidimensional item response model constrained latent class analysis using the gibbs sampler and posterior predictive checks in this paper it will be shown that a certain class of constrained latent class models may be interpreted as a special case of nonparametric multidimensional item response models the parameters of this latent class model will be estimated using an application of the gibbs sampler it will be illustrated that the gibbs sampler is an excellent tool if inequality constraints have to be taken into consideration when making inferences model fit will be investigated using posterior predictive checks checks for manifest monotonicity the agreement between the observed and expected conditional association structure marginal local homogeneity and the number of latent classes will be presented

conditional covariance theory and detect for polytomous items this paper extends the theory of conditional covariances to polytomous items it has been proven that under some mild conditions commonly assumed in the analysis of response data the conditional covariance of two items dichotomously or polytomously scored given an appropriately chosen composite is positive if and only if the two items measure similar constructs besides the composite the theory provides a theoretical foundation for dimensionality assessment procedures based on conditional covariances or correlations such as detect and dimtest so that the performance of these procedures is theoretically justified when applied to response data with polytomous items various estimators of conditional covariances are constructed and special attention is paid to the case of complex sampling data such as those from the national assessment of educational progress naep as such the new version of detect can be applied to response data sets not only with polytomous items but also with missing values either by design or at random detect is then applied to analyze the dimensional structure of the 2002 naep reading samples of grades 4 and 8 the detect results show that the substantive test structure based on the purposes for reading is consistent with the statistical dimensional structure for either grade

functional coefficient regression models for nonlinear time series abstract the local linear regression technique is applied to estimation of functional coefficient regression models for time series data the models include threshold autoregressive models and functional coefficient autoregressive models as special cases but with the added advantages such as depicting finer structure of the underlying dynamics and better postsample forecasting performance also proposed are a new bootstrap test for the goodness of fit of models and a bandwidth selector based on newly defined cross validatory estimation for the expected forecasting errors the proposed methodology is data analytic and of sufficient flexibility to analyze complex and multivariate nonlinear structures without suffering from the curse of dimensionality the asymptotic properties of the proposed estimators are investigated under the mixing condition both simulated and real data examples are used for illustration

on the positive solutions of semilinear equations δ ℎ 0 on the compact manifolds in this paper we study the existence nonexistence and uniqueness of positive solutions of semilinear equations u u hu p 0 on compact riemannian manifolds as well as on bounded smooth domains in r n with homogeneous dirichlet or neumann boundary conditions

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

stochastic order in dichotomous item response models for fixed adaptive and multidimensional tests dichotomous irt models can be viewed as families of stochastically ordered distributions of responses to test items this paper explores several properties of such distributions in particular it is examined under what conditions stochastic order in families of conditional distributions is transferred to their inverse distributions from two families of related distributions to a third family or from multivariate conditional distributions to a marginal distribution the main results are formulated as a series of theorems and corollaries which apply to dichotomous irt models one part of the results holds for unidimensional models with fixed item parameters the other part holds for models with random item parameters as used for example in adaptive testing or for tests with multidimensional abilities

separating two sources of dependence in repeated influenza outbreaks summary two or more responses measured on a randomly selected individual can be dependent for two quite different reasons in the first place they can be dependent within that is conditional upon each individual secondly dependence is created simply from the heterogeneity between individuals this paper addresses a rare instance of being able to measure separately the dependence from the two sources the data comprise frequencies of the infection profiles for a sample of individuals exposed to four influenza outbreaks two of the outbreaks were caused by the same virus and gave rise to negative withinindividual dependence because infection at the first occasion leads to a reduced probability of infection at the second this negative dependence cannot be estimated from the 2 x 2 table of frequencies for these two outbreaks because it is masked by positive dependence due to between individual heterogeneity however the negative dependence can be estimated using the joint frequencies for three or four outbreaks the two caused by the same virus and one or both of the other two