multi symplectic integration of the camassa holm equation the camassa holm equation is rich in geometric structures it is completely integrable bi hamiltonian and it represents geodesics for a certain metric in the group of diffeomorphism here two new multi symplectic formulations for the camassa holm equation are presented and the associated local conservation laws are shown to correspond to certain well known hamiltonian functionals the multi symplectic discretisation of each formulation is exemplified by means of the euler box scheme numerical experiments show that the schemes have good conservative properties and one of them is designed to handle the conservative continuation of peakon antipeakon collisions

quartic non polynomial spline approach to the solution of a system of third order boundary value problems quartic non polynomial splines are used to develop a new numerical method for computing approximations to the solution of a system of third order boundary value problems associated with obstacle unilateral and contact problems it is shown that the new method gives approximations which are better than those produced by other collocation finite difference and spline methods convergence analysis of the method is discussed through standard procedures a numerical example is given to illustrate the applicability and efficiency of the novel method

minimizing the number of observations a generalization of the spearman brown formula a new method for determining the minimum number of observations per subject needed to achieve a specific generalizability coefficient is presented this method which consists of a branch and bound algorithm allows for the employment of constraints specified by the investigator

a mixed integer programming model for multiple stage adaptive testing abstract the last decade has seen paper and pencil p p tests being replaced by computerized adaptive tests cats within many testing programs a cat may yield several advantages relative to a conventional p p test a cat can determine the questions or test items to administer allowing each test form to be tailored to a test taker s skill level subsequent items can be chosen to match the capability of the test taker by adapting to a test taker s ability a cat can acquire more information about a test taker while administering fewer items a multiple stage adaptive test mst provides a means to implement a cat that allows review before the administration the mst format is a hybrid between the conventional p p and cat formats this paper presents mixed integer programming models for mst assembly problems computational results with commercial optimization software will be given and advantages of the models evaluated

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

quintic nonpolynomial spline method for the solution of a second order boundary value problem with engineering applications abstract nonpolynomial quintic spline functions are used to develop a numerical algorithm for computing an approximation to the solution of a system of second order boundary value problems associated with heat transfer we show that the approximate solutions obtained by our algorithm are better than those produced by other spline and domain decomposition methods a comparison of our algorithm with nonpolynomial quadratic spline method is discussed with the help of two numerical examples

a maximin model for test design with practical constraints a maximin model for irt based test design is proposed in the model only the relative shape of the target test information function is specified it serves as a constraint subject to which a linear programming algorithm maximizes the information in the test in the practice of test construction several demands as linear constraints in the model a worked example of a text construction problem with practical constraints is presented the paper concludes with a discussion of some alternative models of test construction

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

a comparison of three simple test theory models in very simple test theory models such as the rasch model a single parameter is used to represent the ability of any examinee or the difficulty of any item simple models such as these provide very important points of departure for more detailed modeling when a substantial amount of data are available and are themselves of real practical value for small or even medium samples they can also serve a normative role in test design as an alternative to the rasch model or the rasch model with a correction for guessing a simple model is introduced which characterizes strength of response in terms of the ratio of ability and difficulty parameters rather than their difference this model provides a natural account of guessing and has other useful things to contribute as well it also offers an alternative to the rasch model with the usual correction for guessing the three models are compared in terms of statistical properties and fits to actual data the goal of the paper is to widen the range of minimal models available to test analysts

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

geometric integration of the paraxial equation abstract evolution of solitary waves in photovoltaic photorefractive crystal satisfy the paraxial equation the paraxial equation is transformed into the symplectic structure of the infinite dimensional hamiltonian system the symplectic structure of the paraxial equation is discretizated by the symplectic method the corresponding symplectic scheme preserves conservation of discrete energy which reflects conservation of energy of the paraxial equation the symplectic scheme is applied to simulate the solitary wave behaviors of the paraxial equation evolution of the solitary waves with the different applied electric field and the different photovoltaic fields are investigated

an entropy approach to the scaling of ordinal categorical data this paper studies the problem of scaling ordinal categorical data observed over two or more sets of categories measuring a single characteristic scaling is obtained by solving a constrained entropy model which finds the most probable values of the scales given the data a kullback leibler statistic is generated which operationalizes a measure for the strength of consistency among the sets of categories a variety of data of two and three sets of categories are analyzed using the entropy approach

minimax d optimal designs for item response theory models various different item response theory irt models can be used in educational and psychological measurement to analyze test data one of the major drawbacks of these models is that efficient parameter estimation can only be achieved with very large data sets therefore it is often worthwhile to search for designs of the test data that in some way will optimize the parameter estimates the results from the statistical theory on optimal design can be applied for efficient estimation of the parameters a major problem in finding an optimal design for irt models is that the designs are only optimal for a given set of parameters that is they are locally optimal locally optimal designs can be constructed with a sequential design procedure in this paper minimax designs are proposed for irt models to overcome the problem of local optimality minimax designs are compared to sequentially constructed designs for the two parameter logistic model and the results show that minimax design can be nearly as efficient as sequentially constructed designs

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

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

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

a model for evaluating sensitivity and specificity for correlated diagnostic tests in efficacy studies with an imperfect reference test abstract the purpose of a diagnostic efficacy study is to evaluate and compare the sensitivities and specificities of several diagnostic tests usually the diagnostic tests are correlated conditional on disease status and the reference test is subject to error in the chlamydia trachomatis study five screening tests for detecting chlamydia in endocervical specimens were compared the five tests are correlated and the reference test the cell culture test has less than 100 sensitivity the conventional method ignores both the correlations between the tests and the misclassification of the reference test and thus cannot provide a valid analysis we propose a model to evaluate and compare the sensitivities and specificities of correlated diagnostic tests when there is either an imperfect reference test or even no reference test the model also can estimate the effects of covariates it is a generalized linear mixed model with two unobserved variables one continuous and one dichotomous we use a hybrid

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

sextic spline method for the solution of a system of obstacle problems we use sextic spline function to develop numerical method for the solution of system of second order boundary value problems associated with obstacle unilateral and contact problems we show that the approximate solutions obtained by the present method are better than those produced by other collocation finite difference and spline methods a numerical example is given to illustrate practical usefulness of our method

observed score equating as a test assembly problem a set of linear conditions on item response functions is derived that guarantees identical observed score distributions on two test forms the conditions can be added as constraints to a linear programming model for test assembly that assembles a new test form to have an observed score distribution optimally equated to the distribution on an old form for a well designed item pool and items fitting the irt model use of the model results into observed score pre equating and prevents the necessity ofpost hoc equating by a conventional observed score equating method an empirical example illustrates the use of the model for an item pool from the law school admission test

non polynomial spline approach to the solution of a system of third order boundary value problems we use a quartic spline equivalent nonpolynomial spline functions to develop a numerical method for computing approximations to the solution of a system of third order boundary value problems associated with obstacle unilateral and contact problems we show that the present method gives approximations which are better than those produced by other collocation finite difference and spline methods a numerical example is given to illustrate the applicability and efficiency of the new method

bayesian estimation of a multilevel irt model using gibbs sampling in this article a two level regression model is imposed on the ability parameters in an item response theory irt model the advantage of using latent rather than observed scores as dependent variables of a multilevel model is that it offers the possibility of separating the influence of item difficulty and ability level and modeling response variation and measurement error another advantage is that contrary to observed scores latent scores are test independent which offers the possibility of using results from different tests in one analysis where the parameters of the irt model and the multilevel model can be concurrently estimated the two parameter normal ogive model is used for the irt measurement model it will be shown that the parameters of the two parameter normal ogive model and the multilevel model can be estimated in a bayesian framework using gibbs sampling examples using simulated and real data are given

weighted likelihood estimation of ability in item response theory applications of item response theory which depend upon its parameter invariance property require that parameter estimates be unbiased a new method weighted likelihood estimation wle is derived and proved to be less biased than maximum likelihood estimation mle with the same asymptotic variance and normal distribution wle removes the first order bias term from mle two monte carlo studies compare wle with mle and bayesian modal estimation bme of ability in conventional tests and tailored tests assuming the item parameters are known constants the monte carlo studies favor wle over mle and bme on several criteria over a wide range of the ability scale

structure preservation of exponentially fitted runge kutta methods the preservation of some structure properties of the flow of differential systems by numerical exponentially fitted runge kutta efrk methods is considered a complete characterisation of efrk methods that preserve linear or quadratic invariants is given and following the approach of bochev and scovel on quadratic invariants and symplectic structure bit 34 1994 337 345 the sufficient conditions on symplecticity of efrk methods derived by van de vyver a fourth order symplectic exponentially fitted integrator comput phys comm 174 2006 255 262 are obtained further a family of symplectic efrk two stage methods with order four has been derived it includes the symplectic efrk method proposed by van de vyver as well as a collocation method at variable nodes that can be considered as the natural collocation extension of the classical rk gauss method finally the results of some numerical experiments are presented to compare the relative merits of several fitted and nonfitted fourth order methods in the integration of oscillatory systems

tage iterative algorithm and nonpolynomial spline basis for the solution of nonlinear singular second order ordinary differential equations abstract in the present paper we discuss three point difference method based on nonpolynomial spline basis for the second order ordinary differential equation difference schemes are derived for linear and nonlinear case and are used to solve via two parameter alternating group explicit iterative algorithm the schemes have a fourth and second order of uniform convergence for the choice of the parameters involved in the method computational results are presented comparing the two methods in terms of accuracy and execution times the results indicate the advantage of using parallel implementation of the new method

nonpolynomial spline approach to the solution of a system of second order boundary value problems we use a cubic spline equivalent nonpolynomial spline functions to develop a numerical method for computing approximations to the solution of a system of second order boundary value problems associated with obstacle unilateral and contact problems we show that the present method gives approximations which are better than those produced by other collocation finite difference and spline methods convergence analysis of the method is discussed a numerical example is given to illustrate practical usefulness of our method

non polynomial spline methods for the solution of a system of obstacle problems we use a non polynomial quintic spline functions to develop numerical methods for the solution of system of second order boundary value problems associated with obstacle unilateral and contact problems we show that the approximate solutions obtained by the present methods are better than those produced by other collocation finite difference and spline methods a numerical example is given to illustrate practical usefulness of our methods

new conservation schemes for the nonlinear schrodinger equation new explicit square conservation schemes of any order for the nonlinear schrodinger equation are presented the basic idea is to discrete the space variable of the nonlinear schrodinger equation approximately so that the resulting semi discrete equation can be cast into an ordinary differential equation dy dt a t y y a t y is a skew symmetry matrix then the lie group methods which can preserve the modulus square conservation property of the ordinary differential equation are applied to the ordinary differential equation numerical results show the effective of the lie group method preserving the modulus square conservation of the discrete nonlinear schrodinger equation

some aspects of variational inequalities abstract in this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence uniqueness computational properties various generalizations sensitivity analysis and their applications we also propose some open problems with sufficient information and references so that someone may attempt solution s in his her area of special interest we also include some new results which we have recently obtained

some studies on nonpolynomial interpolation and error analysis abstract in this paper we propose nonpolynomial and hermite nonpolynomial interpolation with multiple parameters and present method to determine optimal value of parameters which generate minimum error in approximation the generalized error analysis results for nonpolynomial and hermite nonpolynomial interpolations are derived we establish theoretical relationship among nonpolynomial polynomial interpolation and the fourier series and propose solution to runge s phenomenon through nonpolynomial interpolation the hermite nonpolynomial cubic spline nonpolynomial cubic spline interpolation methods and their error analysis are presented numerical simulations are carried out for the analysis of error in cubic spline interpolations proposed method is applied to the analysis of various time series to show comparison in errors between polynomial and nonpolynomial spline interpolations and to empirical mode decomposition emd to illustrate practical usefulness of the results

on the approximation of the quadratic exponential distribution in a latent variable context summary following cox wermuth 1994 2002 we show that the distribution of a set of binary observable variables induced by a certain discrete latent variable model may be approximated by a quadratic exponential distribution this discrete latent variable model is equivalent to the latent class version of the two parameter logistic model of birnbaum 1968 which may be seen as a generalized version of the rasch model rasch 1960 1961 on the basis of this result we develop an approximate maximum likelihood estimator of the item parameters of the two parameter logistic model which is very simply implemented the proposed approach is illustrated through an example based on a dataset on educational assessment

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

symplectic integration of hamiltonian wave equations the numerical integration of a wide class of hamiltonian partial differential equations by standard symplectic schemes is discussed with a consistent hamiltonian approach we discretize the hamiltonian and the poisson structure separately then form the the resulting ode s the stability accuracy and dispersion of different explicit splitting methods are analyzed and we give the circumstances under which the best results can be obtained in particular when the hamiltonian can be split into linear and nonlinear terms many different treatments and examples are compared

achievement test construction using 0 1 linear programming in educational testing the work of professional test agencies has shown a trend towards item banking achievement test construction is viewed as selecting items from a test item bank such that certain specifications are met as the number of possible tests is large and practice usually imposes various constraints on the selection process a mathematical programming approach is obvious in this paper it is shown how to formulate achievement test construction as a 0 1 linear programming problem a heuristic for solving the problem is proposed and two examples are given it is concluded that a 0 1 linear programming approach fits the problem of test construction in an appropriate way and offers test agencies the possibility of computerizing their services

high dimensional multivariate probit analysis a computationally practical form of probit analysis for multiple response variables based on an assumed common factor model for the latent tolerances is proposed numerical integration over the factor space provides maximum likelihood estimation of the probit regression parameters and of the probabilities of response combinations under the model the procedure is applied to five variables from the pneumoconiosis field trial two variables of which were previously analyzed by ashford and sowden 1970 biometrics 26 535 546

maximizing the coefficient of generalizability under the constraint of limited resources a procedure for maximizing the coefficient of generalizability under the constraint of limited resources is presented the procedure uses optimization techniques that offer an investigator or test constructor the possibility of employing practical constraints the procedure is illustrated for the two facet random model crossed design

on structural equation modeling with data that are not missing completely at random a general latent variable model is given which includes the specification of a missing data mechanism this framework allows for an elucidating discussion of existing general multivariate theory bearing on maximum likelihood estimation with missing data here missing completely at random is not a prerequisite for unbiased estimation in large samples as when using the traditional listwise or pairwise present data approaches the theory is connected with old and new results in the area of selection and factorial invariance it is pointed out that in many applications maximum likelihood estimation with missing data may be carried out by existing structural equation modeling software such as lisrel and liscomp several sets of artifical data are generated within the general model framework the proposed estimator is compared to the two traditional ones and found superior

construction of symplectic schemes for wave equations via hyperbolic functions sinh x cosh x and tanh x abstract hamiltonian systems are canonical systems on phase space endowed with symplectic structures the dynamical evolutions i e the phase flow of the hamiltonian systems are symplectic transformations which are area preserving the importance of the hamiltonian systems and their special property require the numerical algorithms for them should preserve as much as possible the relevant symplectic properties of the original systems feng kang 1 3 proposed in 1984 a new approach to computing hamiltonian systems from the view point of symplectic geometry he systematically described the general method for constructing symplectic schemes with any order accuracy via generating functions a generalization of the above theory and methods for canonical hamiltonian equations in infinite dimension can be found in 4 using self adjoint schemes we can construct schemes of arbitrary even order 5 these schemes can be applied to wave equation 6 7 and the stability of them can be seen in 7 8 in this paper we will use the hyperbolic functions sinh x cosh x and tanh x to construct symplectic schemes of arbitrary order for wave equations and stabilities of these constructed schemes are also discussed