a note on the computation of the second order derivatives of the elementary symmetric functions in the rasch model jansen 1984 gave explicit formulas for the computation of the second order derivatives of the elementary symmetric functions but they are only applicable to those pairs of items which have unequal parameters it is shown here that for equal parameters similar explicit formulas do exist too facilitating the application of the newton raphson procedure to estimate the parameters in the rasch model and related models according to the conditional maximum likelihood principle

symbolic computation of normal forms for semi simple cases this paper presents a method and computer programs for computing the normal forms of ordinary differential equations whose jacobian matrix evaluated at an equilibrium involves semi simple eigenvalues the method can be used to deal with systems which are not necessarily described on a center manifold an iterative procedure is developed for finding the closed form expressions of the normal forms and associated nonlinear transformations computer programs using a symbolic computer language maple are developed to facilitate the application of the method the programs can be conveniently executed on a main frame a workstation or a pc machine without any interaction a number of examples are presented to demonstrate the applicability of the method and the computation efficiency of the maple programs

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

an item response model with internal restrictions on item difficulty an irt model based on the rasch model is proposed for composite tasks that is tasks that are decomposed into subtasks of different kinds there is one subtask for each component that is discerned in the composite tasks a component is a generic kind of subtask of which the subtasks resulting from the decomposition are specific instantiations with respect to the particular composite tasks under study the proposed model constrains the difficulties of the composite tasks to be linear combinations of the difficulties of the corresponding subtask items which are estimated together with the weights used in the linear combinations one weight for each kind of subtask although the model does not belong to the exponential family its parameters can be estimated using conditional maximum likelihood estimation the approach is demonstrated with an application to spelling tasks

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

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

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

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

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

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

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

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

mixture analysis of longitudinal binary data the dependence of longitudinal binary outcomes on covariates and the covariation observed between them is often modelled by multivariate logistic and probit models respectively assuming specified association structure or random effects alternatively latent class models may be used that capture the covariation by assuming heterogeneity of the observational units regarding their reaction tendencies while postulating independence within classes in the presence of a few categorical covariates the multi group method of latent class analysis allows one to relate the class sizes and the class specific response probabilities to these covariates wheeze data from the harvard six cities study on respiratory health are a typical example for such a situation at four occasions the wheeze status of 537 children was examined 187 among them exposed to maternal smoking and 350 not exposed thus there is a single binary covariate maternal smoking versus no maternal smoking making easily applicable the multi group method of latent class analysis based on a series of unrestricted and restricted models having up to three classes for the exposed and not exposed subgroup each no statistically significant effect of maternal smoking on children s wheeze status could be substantiated moreover it was not possible to show statistically significant difference at all between the two distributions of wheeze patterns collected from exposed and not exposed children

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

some neglected problems in irt the paper addresses three neglected questions from irt in section 1 the properties of the measurement of ability or trait parameters and item difficulty parameters in the rasch model are discussed it is shown that the solution to this problem is rather complex and depends both on general assumptions about properties of the item response functions and on assumptions about the available item universe section 2 deals with the measurement of individual change or modifiability based on a rasch test a conditional likelihood approach is presented that yields a an ml estimator of modifiability for given item parameters b allows one to test hypotheses about change by means of a clopper pearson confidence interval for the modifiability parameter or c to estimate modifiability jointly with the item parameters uniqueness results for all three methods are also presented in section 3 the mantel haenszel method for detecting dif is discussed under a novel perspective what is the most general framework within which the mantel haenszel method correctly detects dif of a studied item the answer is that this is a 2pl model where however all discrimination parameters are known and the studied item has the same discrimination in both populations since these requirements would hardly be satisfied in practical applications the case of constant discrimination parameters that is the rasch model is the only realistic framework a simple pearsonx2 test for dif of one studied item is proposed as an alternative to the mantel haenszel test moreover this test is generalized to the case of two items simultaneously studied for dif

remarks on equivalent linear logistic test models by bechger verstralen and verhelst 2002 this paper discusses a new form of specifying and normalizing a linear logistic test model lltm as suggested by bechger verstralen and verhelst psychometrika 2002 it is shown that there are infinitely many ways to specify the same normalization moreover the relationship between some of their results and equivalent previous results in the literature is clarified and it is shown that the goals of estimating and testing a single element of the weight matrix for which they propose new methods can be reached by means of simple well known tools already implemented in published lltm software

equivalent mirid models it is shown that in the context of the model with internal restrictions on the item difficulties mirid different componential theories about an item set may lead to equivalent models furthermore we provide conditions for the identifiability of the mirid model parameters and it will be shown how the mirid model relates to the linear logistic test model lltm while it is known that the lltm is a special case of the mirid we show that it is possible to construct an lltm that encompasses the mirid the mirid model places a bilinear restriction on the item parameters of the rasch model it is explained how this fact is used to simplify the results of bechger verhelst and verstralen 2001 and bechger verstralen and verhelst 2002 and extend their scope to a wider class of models

on prescribing geodesic curvature on d2 abstract some new results are obtained for the problem of prescribing geodesic curvature k on d when k possesses some kinds of symmetries

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

symbolic numerical methods for the computation of normal forms of pdes the center manifold and the normal forms are effective tools for the study of local bifurcations occurring in evolution equations the computation of the center manifold and the normal form amounts after more or less complex algebraic transformations to solve in a recursive way a hierarchy of linear equations we present a method and computer programs for the computation of normal forms of some nonlinear parabolic pdes these computations are performed using the symbolic algebra system maple matlab and exploiting the compatibility of these two systems here the linear equations to be solved are infinite dimensional and we use the finite element method for this purpose the use of the finite element method allows to consider problems with complex shape domains in our programs maple takes care of the algebraic manipulations delivering the set of linear equations to be solved and writes some parts of the matlab code for their resolution we give three applications a pitchfork bifurcation in a semilinear parabolic equation a hopf bifurcation and a bifurcation to rotating and standing waves in a reaction diffusion system

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

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

identifiability and equivalence of gllirm models the generalized logit linear item response model gllirm is a linearly constrained nominal categories model ncm that computes the scale and intercept parameters for categories as a weighted sum of basic parameters this paper addresses the problems of the identifiability of the basic parameters and the equivalence between different gllirm models it is shown that the identifiability of the basic parameters depends on the size and rank of the coefficient matrix of the linear functions moreover two models are observationally equivalent if the product of the respective coefficient matrices has full column rank finally the paper also explores the relations between the parameters of nested models

a small note on symmetric geodesic curvature on d2 some new results are obtained for the problem of prescribing geodesic curvature k on d when k possesses some kinds of symmetries

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

remarks on prescribing gauss curvature we study the nonlinear partial differential equation for the problem of prescribing gauss curvature k on s 2 we give an example of a rotationally symmetric k for which the kazdan warner obstruction is satisfied but the equation has no rotationally symmetric solution on the other hand we give a simple sufficient condition for solvability of the equation when k is rotationally symmetric finally we give a sufficient condition for solvability when k is not necessarily rotationally symmetric

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented