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

some existence principles and some general results for singular nonlinear two point boundary value problems abstract existence results for singular second order nonlinear two point boundary value problems of the form 1 p t p t y q t f t y 0 0 where limy 0 f t y are presented here we may allow singularities at y 0 t 0 and or t 1 also existence principles are presented for the general two point boundary value problem 1 p py qf t y py 0 0 where f 0 1 r 2 r is continuous the proof uses the nonlinear alternative of leray and schauder

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

paradoxical results in multidimensional item response theory in multidimensional item response theory mirt it is possible for the estimate of a subject s ability in some dimension to decrease after they have answered a question correctly this paper investigates how and when this type of paradoxical result can occur we demonstrate that many response models and statistical estimates can produce paradoxical results and that in the popular class of linearly compensatory models maximum likelihood estimates are guaranteed to do so in light of these findings the appropriateness of multidimensional item response methods for assigning scores in high stakes testing is called into question

gaussian processes kinematic formulae and poincare s limit we consider vector valued unit variance gaussian processes defined over stratified manifolds and the geometry of their excursion sets in particular we develop an explicit formula for the expectation of all the lipschitz killing curvatures of these sets whereas our motivation is primarily probabilistic with statistical applications in the background this formula has also an interpretation as a version of the classic kinematic fundamental formula of integral geometry all of these aspects are developed in the paper particularly novel is the method of proof which is based on a an approximation to the canonical gaussian process on the n sphere the n limit which gives the final result is handled via recent extensions of the classic poincare limit theorem

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

testing for a signal with unknown location and scale in a stationary gaussian random field we suppose that our observations can be decomposed into a fixed signal plus random noise where the noise is modelled as a particular stationary gaussian random field in n dimensional euclidean space the signal has the form of a known function centered at an unknown location and multiplied by an unknown amplitude and we are primarily interested in a test to detect such a signal there are many examples where the signal scale or width is assumed known and the test is based on maximising a gaussian random field over all locations in a subset of n dimensional euclidean space the novel feature of this work is that the width of the signal is also unknown and the test is based on maximising a gaussian random field in n 1 dimensions n dimensions for the location plus one dimension for the width two convergent approaches are used to approximate the null distribution one based on the method of knowles and siegmund which uses a version of weyl s tube formula for manifolds with boundaries and the other based on some recent work by worsley which uses the hadwiger characteristic of excursion sets as introduced by adler finally we compare the power of our method with one based on a fixed but perhaps incorrect signal width

some new results for singular problems with sign changing nonlinearities we present some general existence theorems for second order boundary value problems with sign changing nonlinearities both the upper and lower solution approach and the growth restriction approach are discussed

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

multiplicity positive solutions to periodic problems for first order impulsive differential equations this paper is devoted to studying the existence of single and multiple positive solutions to periodic problems for first order impulsive differential equations in addition we apply our existence theorems to nonlinear periodic problems with singularity at the origin our proofs are based on krasnoselskii fixed point theorem in cones

a new explanation and proof of the paradoxical scoring results in multidimensional item response models in multidimensional item response models paradoxical scoring effects can arise wherein correct answers are penalized and incorrect answers are rewarded for the most prominent class of irt models the class of linearly compensatory models a general derivation of paradoxical scoring effects based on the geometry of item discrimination vectors is given which furthermore corrects an error in an established theorem on paradoxical results this approach highlights the very counterintuitive way in which item discrimination parameters and also factor loadings have to be interpreted in terms of their influence on the latent ability estimate it is proven that despite the error in the original proof the key result concerning the existence of paradoxical effects remains true although the actual relation to the item parameters is shown to be a more complicated function than previous results suggested the new proof enables further insights into the actual mathematical causation of the paradox and generalizes the findings within the class of linearly compensatory models

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

paradoxical results and item bundles hooker finkelman and schwartzman psychometrika 2009 in press defined a paradoxical result as the attainment of a higher test score by changing answers from correct to incorrect and demonstrated that such results are unavoidable for maximum likelihood estimates in multidimensional item response theory the potential for these results to occur leads to the undesirable possibility of a subject s best answer being detrimental to them this paper considers the existence of paradoxical results in tests composed of item bundles when compensatory models are used we demonstrate that paradoxical results can occur when bundle effects are modeled as nuisance parameters for each subject however when these nuisance parameters are modeled as random effects or used in a bayesian analysis it is possible to design tests comprised of many short bundles that avoid paradoxical results and we provide an algorithm for doing so we also examine alternative models for handling dependence between item bundles and show that using fixed dependency effects is always guaranteed to avoid paradoxical results

on separable tests correlated priors and paradoxical results in multidimensional item response theory this paper presents a study of the impact of prior structure on paradoxical results in multidimensional item response theory paradoxical results refer to the possibility that an incorrect response could be beneficial to an examinee we demonstrate that when three or more ability dimensions are being used paradoxical results can be induced by using priors in which all abilities are positively correlated where they would not occur if the abilities were modeled as being independent in the case of separable tests we demonstrate the mathematical causes of paradoxical results develop a computationally feasible means to check whether they can occur in any given test and demonstrate a class of prior covariance matrices that can be guaranteed to avoid them

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

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

singular problems an upper and lower solution approach this paper presents an upper and lower solution theory for singular boundary value problems where the nonlinearity may be singular in the dependent variable in addition our nonlinearity is allowed to change sign

nonlinear boundary value problems for shallow membrane caps ii abstract suppose a shallow membrane cap with an undeformed shape described in cylindrical coordinates by z c 1 r where 0 r 1 and 1 is subjected to a uniform vertical pressure p if the resulting deformed shape is radially symmetric then under certain assumptions the radial stress sr satisfies the ordinary differential equation r 2 s r 3rs r 2 r 2y 2 2 vr 2 s r r 2 8s r 2 for 0 0 if the boundary stress s is specified or s r 1 1 v s r 1 if the boundary displacement is specified here v 0 v 1 a radially symmetric solution sr r positive for 0 s 1 4 v or 1 v 1 4 v the solution is unique in the case 4 3 if is fairly large it may happen that sr r 0 as r 0 in all other cases sr r has a positive limit as r 0 rather detailed information on the behavior of solutions sr r is provided conditions are obtained which guarantee monotonicity of sr in any case sr has at most one critical point and is monotone in some neighborhood of r 0 a computational algorithm making use of the qualitative behavior of sr is discussed and some numerical results are included

singular boundary value problems for the one dimension p laplacian abstract the singular boundary value problem u g t u u 0 for t 0 1 u 0 u 1 0 where s s p 2s p 1 is studied in this paper the singularity may appear at u 0 t 0 and t 1 and the function g may change sign the existence of solutions is obtained via an upper and lower solution method

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

nonuniform nonresonant singular dirichlet boundary value problems for the one dimensional p laplacian with sign changing nonlinearity abstract existence results are presented for the nonuniform nonresonant singular boundary value problem u f t u u 0 for t 0 1 u 0 u 1 0 where s s p 2 s p 1 the singularity may appear at u 0 t 0 or t 1 and the function f may change sign the existence of solutions is obtained via an upper and lower solutions method

nonuniform nonresonance at the first eigenvalue for singular boundary value problems with sign changing nonlinearities new nonresonant results are presented for the boundary value problem y f t y y 0 0 t 1 with dirichlet boundary data our nonlinearity may be singular in its dependent variable and is allowed to change sign

estimating the number of peaks in a random field using the hadwiger characteristic of excursion sets with applications to medical images excursion sets has been studied by adler who defined the ig integral geometry characteristic of excursion sets as a multidimensional analogue of the number of upcrossings of the threshold by a unidimensional process the ig characteristic equals the euler characteristic of an excursion set provided that the set does not touch the boundary of the volume and adler found the expected ig characteristic for a stationary random field inside a fixed volume worsley evans marrett and neelin used the ig characteristic as an estimator of the number of regions of activation of positron emission tomography pet images of blood flow in the brain unfortunately the ig characteristic is only defined on intervals it is not invariant under rotations and it can fail to count connected regions that touch the boundary this is important since activation often occurs in the cortical regions near the boundary of the brain in this paper we study the hadwiger characteristic which is defined on arbitrary sets is invariant under rotations and does count connected regions whether they touch the boundary or not our main result is a simple expression for the expected hadwiger characteristic for an isotropic stationary random field in two and three dimensions and on a smooth surface embedded in three dimensions results are applied to pet studies of pain perception and word recognition

nonnegative solutions of singular boundary value problems with sign changing nonlinearities the paper presents sufficient conditions for the existence of positive solutions of the equation x t q t f t x t x t 0 with the dirichlet conditions x 0 0 x 1 0 and of the equation p t x t p t q t f t x t p t x t 0 with the boundary conditions limt o p t x t 0 x 1 0 our nonlinearity f is allowed to change sign and f may be singular at x 0 the proofs are based on a combination of the regularity and sequential techniques and the method of lower and upper functions

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

euler characteristics for gaussian fields on manifolds we are interested in the geometric properties of real valued gaussian random fields defined on manifolds our manifolds m are of class c3 and the random fields f are smooth our interest in these fields focuses on their excursion sets f 1 u and their geometric properties specifically we derive the expected euler characteristic ee f 1 u of an excursion set of a smooth gaussian random field part of the motivation for this comes from the fact that ee f 1 u relates global properties of m to a geometry related to the covariance structure of f of further interest is the relation between the expected euler characteristic of an excursion set above a level u and pp supp mf p u our proofs rely on results from random fields on rrn as well as differential and riemannian geometry