assouad type dimensions and homogeneity of fractals we investigate several aspects of the assouad dimension and the lower dimension which together form a natural dimension pair in particular we compute these dimensions for certain classes of self affine sets and quasiself similar sets and study their relationships with other notions of dimension such as the hausdorff dimension for example we also investigate some basic properties of these dimensions including their behaviour regarding unions and products and their set theoretic complexity

fitting and testing conditional multinormal partial credit models a multinormal partial credit model for factor analysis of polytomously scored items with ordered response categories is derived using an extension of the dutch identity holland in psychometrika 55 5 18 1990 in the model latent variables are assumed to have a multivariate normal distribution conditional on unweighted sums of item scores which are sufficient statistics attention is paid to maximum likelihood estimation of item parameters multivariate moments of latent variables and person parameters it is shown that the maximum likelihood estimates can be found without the use of numerical integration techniques more general models are discussed which can be used for testing the model and it is shown how models with different numbers of latent variables can be tested against each other in addition multi group extensions are proposed which can be used for testing both measurement invariance and latent population differences models and procedures discussed are demonstrated in an empirical data example

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

sample size determination within the scope of conditional maximum likelihood estimation with special focus on testing the rasch model this paper refers to the exponential family of probability distributions and the conditional maximum likelihood cml theory it is concerned with the determination of the sample size for three groups of tests of linear hypotheses known as the fundamental trinity of wald score and likelihood ratio tests the main practical purpose refers to the special case of tests of the class of rasch models the theoretical background is discussed and the formal framework for sample size calculations is provided given a predetermined deviation from the model to be tested and the probabilities of the errors of the first and second kinds

rasch models for item bundles this paper discusses the application of a class of rasch models to situations where test items are grouped into subsets and the common attributes of items within these subsets brings into question the usual assumption of conditional independence the models are all expressed as particular cases of the random coefficients multinomial logit model developed by adams and wilson this formulation allows a very flexible approach to the specification of alternative models and makes model testing particularly straightforward the use of the models is illustrated using item bundles constructed in the framework of the solo taxonomy of biggs and collis

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

sufficiency and conditional estimation of person parameters in the polytomous rasch model rasch models are characterised by sufficient statistics for all parameters in the rasch unidimensional model for two ordered categories the parameterisation of the person and item is symmetrical and it is readily established that the total scores of a person and item are sufficient statistics for their respective parameters in contrast in the unidimensional polytomous rasch model for more than two ordered categories the parameterisation is not symmetrical specifically each item has a vector of item parameters one for each category and each person only one person parameter in addition different items can have different numbers of categories and therefore different numbers of parameters the sufficient statistic for the parameters of an item is itself a vector in estimating the person parameters in presently available software these sufficient statistics are not used to condition out the item parameters this paper derives a conditional pairwise pseudo likelihood and constructs estimates of the parameters of any number of persons which are independent of all item parameters and of the maximum scores of all items it also shows that these estimates are consistent although rasch s original work began with equating tests using test scores and not with items of a test the polytomous rasch model has not been applied in this way operationally this is because the current approaches in which item parameters are estimated first cannot handle test data where there may be many scores with zero frequencies a small simulation study shows that when using the estimation equations derived in this paper such a property of the data is no impediment to the application of the model at the level of tests this opens up the possibility of using the polytomous rasch model directly in equating test scores

a mixed effects regression model for longitudinal multivariate ordinal data summary a mixed effects item response theory model that allows for three level multivariate ordinal outcomes and accommodates multiple random subject effects is proposed for analysis of multivariate ordinal outcomes in longitudinal studies this model allows for the estimation of different item factor loadings item discrimination parameters for the multiple outcomes the covariates in the model do not have to follow the proportional odds assumption and can be at any level assuming either a probit or logistic response function maximum marginal likelihood estimation is proposed utilizing multidimensional gauss hermite quadrature for integration of the random effects an iterative fisher scoring solution which provides standard errors for all model parameters is used an analysis of a longitudinal substance use data set where four items of substance use behavior cigarette use alcohol use marijuana use and getting drunk or high are repeatedly measured over time is used to illustrate application of the proposed model

a latent class model for rating data a latent class model for rating data is presented which is the analogue of andrich s binomial rasch model for lazarsfeld s latent class analysis lca the response probabilities for the rating categories follow a binomial distribution and depend on class specific item parameters the em algorithm for parameter estimation as well as goodness of fit tests for the model are described an example using questionnaire items on interest in physics illustrates the use of the model as an alternative to the latent trait approach of analyzing test data

decomposition of a rasch partial credit item into independent binary and indecomposable trinary items for each rasch masters partial credit item there exists a set of independent rasch binary and indecomposable trinary items for which the sum of the scores and the partial credit score have identical probability density functions if each indecomposable trinary item is further expressed as the sum of two binary items then the binary items are positively dependent and cannot be both of the rasch type

the quasi assouad dimension of stochastically self similar sets the class of stochastically self similar sets contains many famous examples of random sets for example mandelbrot percolation and general fractal percolation under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used we show that the quasi assouad dimension of self similar random recursive sets is almost surely equal to the almost sure hausdorff dimension of the set we further comment on random homogeneous and v variable sets and the removal of overlap conditions

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

on the existence and uniqueness of jml estimates for the partial credit model a necessary and sufficient condition is given in this paper for the existence and uniqueness of the maximum likelihood the so called joint maximum likelihood estimate of the parameters of the partial credit model this condition is stated in terms of a structural property of the pattern of the data matrix that can be easily verified on the basis of a simple iterative procedure the result is proved by using an argument of haberman 1977

bayes factors for evaluating latent monotonicity in polytomous item response theory models the assumption of latent monotonicity is made by all common parametric and nonparametric polytomous item response theory models and is crucial for establishing an ordinal level of measurement of the item score three forms of latent monotonicity can be distinguished monotonicity of the cumulative probabilities of the continuation ratios and of the adjacent category ratios observable consequences of these different forms of latent monotonicity are derived and bayes factor methods for testing these consequences are proposed these methods allow for the quantification of the evidence both in favor and against the tested property both item level and category level bayes factors are considered and their performance is evaluated using a simulation study the methods are applied to an empirical example consisting of a 10 item likert scale to investigate whether a polytomous item scoring rule results in item scores that are of ordinal level measurement

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

assouad dimensions of complementary sets given a positive decreasing sequence a whose sum is l we consider all the closed subsets of 0 l such that the lengths of their complementary open intervals are in one to one correspondence with the sequence a the aim of this paper is to investigate the possible values that assouad type dimensions can attain for this class of sets in many cases the set of attainable values is a closed interval whose endpoints we determine

sequential analysis of latent variables using mixed effect latent variable models impact of non informative and informative missing data sequential methods allowing for early stopping of clinical trials are widely used in various therapeutic areas these methods allow for the analysis of different types of endpoints quantitative qualitative time to event and often provide in average substantial reductions in sample size as compared with single stage designs while maintaining pre specified type i and ii errors sequential methods are also used when analysing particular endpoints that cannot be directly measured such as depression quality of life or cognitive functioning which are often measured through questionnaires these types of endpoints are usually referred to as latent variables and should be analysed with latent variable models in addition in most clinical trials studying such latent variables incomplete data are not uncommon and the missing data process might also be non ignorable we investigated the impact of informative or non informative missing data on the statistical properties of the double triangular test dtt combined with the mixed effects rasch model mrm for dichotomous responses or the traditional method based on observed patient s scores s to the questionnaire the achieved type i errors for the dtt were usually close to the target value of 0 05 for both methods but increased slightly for the mrm when informative missing data were present the dtt was very close to the nominal power of 0 95 when the mrm was used but substantially underpowered with the s method reduction of about 23 per cent irrespective of whether informative missing data were present or not moreover the dtt using the mrm allowed for reaching a conclusion under h0 or h1 with fewer patients than the s method the average sample number for the latter increasing importantly when the proportion of missing data increased incorporating mrm in sequential analysis of latent variables might provide a more powerful method than the traditional s method even in the presence of non informative or informative missing data copyright 2007 john wiley sons ltd

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

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

almost sure assouad like dimensions of complementary sets given a non negative decreasing sequence a with sum 1 we consider all the closed subsets of 0 1 such that the lengths of their complementary open intervals are given by the terms of a these are the so called complementary sets or rearrangements of the cantor set constructed from a in this paper we determine the almost sure value of the varphi dimension of these sets given a natural model of randomness the varphi dimensions are intermediate assouad like dimensions which include the assouad and quasi assouad dimensions as special cases the answers depend on the size of varphi with one size behaving almost surely like the assouad dimensions of the associated cantor set and the other like the quasi assouad dimensions these results are new even for the assouad dimensions

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

extensions of the partial credit model the partial credit model developed by masters 1982 is a unidimensional latent trait model for responses scored in two or more ordered categories in the present paper some extensions of the model are presented first a marginal maximum likelihood estimation procedure is developed which allows for incomplete data and linear restrictions on both the item and the population parameters secondly two statistical tests for evaluating model fit are presented the former test has power against violation of the assumption about the ability distribution the latter test offers the possibility of identifying specific items that do not fit the model

constraint programming for computing non stationary r s inventory policies abstract this paper proposes a constraint programming model for computing the finite horizon single item inventory problem with stochastic demands in discrete time periods with service level constraints under the non stationary version of the periodic review order up to level policy i e non stationary r s or simply r n s n it is observed that the modeling process is more natural and the required number of variables is smaller compared to the mip formulation of the same problem the computational tests show that the cp approach is more tractable than the conventional mip formulation two different domain reduction methods are proposed to improve the computational performance of solution algorithms the numerical experiments confirmed the effectiveness of these methods

item information and discrimination functions for trinary pcm items for trinary partial credit items the shape of the item information and the item discrimination function is examined in relation to the item parameters in particular it is shown that these functions are unimodal if 2 1 4 ln 2 and bimodal otherwise the locations and values of the maxima are derived furthermore it is demonstrated that the value of the maximum is decreasing in 2 1 consequently the maximum of a unimodal item information function is always larger than the maximum of a bimodal one and similarly for the item discrimination function

em algorithm estimation of a structural equation model for the longitudinal study of the quality of life health related quality of life hrqol data are measured via patient questionnaires completed by the patients themselves at different time points we focused on oncology data gathered through the use of european organization for research and treatment of cancer questionnaires which decompose hrqol into several functional dimensions several symptomatic dimensions and the global health status ghs we aimed to perform a global analysis of hrqol and reduce the number of analyses required by using a two step approach first a structural equation model sem was used for each time point in these models the ghs is explained by two latent variables each latent variable is a factor that summarizes respectively the functional dimensions and the symptomatic dimensions to the global measurement this is achieved through the maximization of the likelihood of each sem using the em algorithm which has the advantage of giving an estimation of the subject specific factors and the influence of additional explanatory variables then to consider the longitudinal aspect the ghs variable and the two factors were concatenated for each patient visit at which the questionnaire was completed the ghs and the two factors estimated in the first step can then be explained by additional explanatory variables using a linear mixed model

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

finding optimal s s policies is about as simple as evaluating a single policy in this paper a new algorithm for computing optimal s s policies is derived based upon a number of new properties of the infinite horizon cost function c s s as well as a new upper bound for optimal order up to levels s and a new lower bound for optimal reorder levels s the algorithm is simple and easy to understand its computational complexity is only 2 4 times that required to evaluate a specific single s s policy the algorithm applies to both periodic review and continuous review inventory systems

loss of information in estimating item parameters in incomplete designs in this paper the efficiency of conditional maximum likelihood cml and marginal maximum likelihood mml estimation of the item parameters of the rasch model in incomplete designs is investigated the use of the concept of f information eggen 2000 is generalized to incomplete testing designs the scaled determinant of the f information matrix is used as a scalar measure of information contained in a set of item parameters in this paper the relation between the normalization of the rasch model and this determinant is clarified it is shown that comparing estimation methods with the defined information efficiency is independent of the chosen normalization the generalization of the method to other models than the rasch model is discussed in examples information comparisons are conducted it is found that for both cml and mml some information is lost in all incomplete designs compared to complete designs a general result is that with increasing test booklet length the efficiency of an incomplete design compared to a complete design is increasing as is the efficiency of cml compared to mml the main difference between cml and mml is seen in the effect of the length of the test booklet it will be demonstrated that with very small booklets there is a substantial loss in information about 35 with cml estimation while this loss is only about 10 in mml estimation however with increasing test length the differences between cml and mml quickly disappear

the asymptotic posterior normality of the latent trait for polytomous irt models chang and stout 1993 presented a derivation of the asymptotic posterior normality of the latent trait given examinee responses under nonrestrictive nonparametric assumptions for dichotomous irt models this paper presents an extention of their results to polytomous irt models in a fairly straightforward manner in addition a global information function is defined and the relationship between the global information function and the currently used information functions is discussed an information index that combines both the global and local information is proposed for adaptive testing applications

nonparametric estimation of item and respondent locations from unfolding type items unlike their monotone counterparts nonparametric unfolding response models which assume the item response function is unimodal have seen little attention in the psychometric literature this paper studies the nonparametric behavior of unfolding models by building on the work of post 1992 the paper provides rigorous justification for a class of nonparametric estimators of respondents latent attitudes by proving that the estimators consistently rank order the respondents the paper also suggests an algorithm for the rank ordering of items along the attitudes scale finally the methods are evaluated using simulated data

an efficient algorithm for computing optimal s s policies this paper presents an algorithm to compute an optimal s s policy under standard assumptions stationary data well behaved one period costs discrete demand full backlogging and the average cost criterion the method is iterative starting with an arbitrary given s s policy and converging to an optimal policy in a finite number of iterations any of the available approximations can thus be used as an initial solution each iteration requires only modest computations also a lower bound on the true optimal cost can be computed and used in a termination test empirical testing suggests very fast convergence

a new specification of generalized linear models for categorical responses many regression models for categorical responses have been introduced motivated by different paradigms but it is difficult to compare them because of their different specifications in this paper we propose a unified specification of regression models for categorical responses based on a decomposition of the link function into an inverse continuous cumulative distribution function and a ratio of probabilities this allows us to define a new family of reference models for nominal responses comparable to the families of adjacent cumulative and sequential models for ordinal responses a new equivalence between cumulative and sequential models is shown invariances under permutations of the categories are studied for each family of models we introduce a reversibility property that distinguishes adjacent and cumulative models from sequential models the new family of reference models is tested on three benchmark classification datasets

analysis of multiple partially ordered responses to belief items with don t know option understanding beliefs values and preferences of patients is a tenet of contemporary health sciences this application was motivated by the analysis of multiple partially ordered set poset responses from an inventory on layman beliefs about diabetes the partially ordered set arises because of two features in the data first the response options contain a don t know dk option and second there were two consecutive occasions of measurement as predicted by the common sense model of illness beliefs about diabetes were not necessarily stable across the two measurement occasions instead of analyzing the two occasions separately we studied the joint responses across the occasions as a poset response few analytic methods exist for data structures other than ordered or nominal categories poset responses are routinely collapsed and then analyzed as either rank ordered or nominal data leading to the loss of nuanced information that might be present within poset categories in this paper we developed a general class of item response models for analyzing the poset data collected from the common sense model of diabetes inventory the inferential object of interest is the latent trait that indicates congruence of belief with the biomedical model to apply an item response model to the poset diabetes inventory we proved that a simple coding algorithm circumvents the requirement of writing new codes such that standard irt software could be directly used for the purpose of item estimation and individual scoring simulation experiments were used to examine parameter recovery for the proposed poset model

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

measuring change in latent subgroups using dichotomous data unconditional conditional and semiparametric maximum likelihood estimation abstract changes in dichotomous data caused by treatments can be analyzed by means of the so called linear logistic model with relaxed assumptions llra in contrast to models of the rasch type the llra allows incidental multidimensional parameters describing the response behavior of the subjects so that each observable criterion used for assessing the changes is governed by its own subject parameter because generalizability of the treatment effects over subjects and criteria is scientifically desirable statistical tests for this desideratum play an important role in practical situations whereas generalizability over the criteria can easily be tested generalizability over subjects can be tested only by stratifying the sample according to some further observable subject variables however one cannot be sure to detect nongeneralizability over the subjects by this method with certainty therefore the mixture llra is proposed that allows directly unobservable types of subjects reacting differently on

the assouad dimensions of projections of planar sets the first named author is supported by a leverhulme trust research fellowship and the second named author is supported by the academy of finland through the grant restricted families of projections and connections to kakeya type problems grant number 274512

constraint satisfaction problems algorithms and applications a constraint satisfaction problem csp requires a value selected from a given finite domain to be assigned to each variable in the problem so that all constraints relating the variables are satisfied many combinatorial problems in operational research such as scheduling and timetabling can be formulated as csps researchers in artificial intelligence ai usually adopt a constraint satisfaction approach as their preferred method when tackling such problems however constraint satisfaction approaches are not widely known amongst operational researchers the aim of this paper is to introduce constraint satisfaction to the operational researcher we start by defining csps and describing the basic techniques for solving them we then show how various combinatorial optimization problems are solved using a constraint satisfaction approach based on computational experience in the literature constraint satisfaction approaches are compared with well known operational research or techniques such as integer programming branch and bound and simulated annealing

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

a repetitive branch and bound procedure for minimum within cluster sums of squares partitioning minimization of the within cluster sums of squares wcss is one of the most important optimization criteria in cluster analysis although cluster analysis modules in commercial software packages typically use heuristic methods for this criterion optimal approaches can be computationally feasible for problems of modest size this paper presents a new branch and bound algorithm for minimizing wcss algorithmic enhancements include an effective reordering of objects and a repetitive solution approach that precludes the need for splitting the data set while maintaining strong bounds throughout the solution process the new algorithm provided optimal solutions for problems with up to 240 objects and eight well separated clusters poorly separated problems with no inherent cluster structure were optimally solved for up to 60 objects and six clusters the repetitive branch and bound algorithm was also successfully applied to three empirical data sets from the classification literature

protective estimation of mixed effects logistic regression when data are not missing at random we consider estimation of mixed effects logistic regression models for longitudinal data when missing outcomes are not missing at random a typology of missingness mechanisms is presented that includes missingness dependent on observed or missing current outcomes observed or missing lagged outcomes and subject specific effects when data are not missing at random consistent estimation by maximum marginal likelihood generally requires correct parametric modelling of the missingness mechanism which hinges on unverifiable assumptions we show that standard maximum conditional likelihood estimators are protective in the sense that they are consistent for monotone or intermittent missing data under a wide range of missingness mechanisms our approach requires neither specification of parametric models for the missingness mechanism nor refreshment samples and is straightforward to implement in standard software 2013 biometrika trust

a proportional odds model with subject specific effects for repeated ordered categorical responses summary suppose subjects make repeated responses on the same ordered categorical scale we propose a generalization of the rasch model that expresses the cumulative logit of the response distribution using subject parameters and a proportional odds structure for item effects parameters in the model describe subject specific rather than populationaveraged effects consistent estimation of the effects requires eliminating the subject parameters we accomplish this by simultaneous fitting of rasch models conditional on sufficient statistics for those parameters for the possible binary collapsings of the response the fitting process uses an improved newton raphson algorithm for fitting generalized loglinear models by maximum likelihood estimation subject to constraints for the case of two items we give simple expressions for an effect estimate and its standard error and suggest a test of marginal homogeneity for ordinal matched pairs

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

an extension of the partial credit model with an application to the measurement of change the partial credit model is considered under the assumption of a certain linear decomposition of the item category parameters ih into basic parameters j this model is referred to as the linear partial credit model a conditional maximum likelihood algorithm for estimation of the j is presented based on a recurrences for the combinatorial functions involved and b using a quasi newton approach the so called broyden fletcher goldfarb shanno bfgs method a guarantees numerically stable results b avoids the direct computation of the hesse matrix yet produces a sequence of certain positive definite matricesb k k 1 2 converging to the asymptotic variance covariance matrix of the hat alpha j the practicality of these numerical methods is demonstrated both by means of simulations and of an empirical application to the measurement of treatment effects in patients with psychosomatic disorders

on measurement properties of continuation ratio models three classes of polytomous irt models are distinguished these classes are the adjacent category models the cumulative probability models and the continuation ratio models so far the latter class has received relatively little attention the class of continuation ratio models includes logistic models such as the sequential model tutz 1990 and nonlogistic models such as the acceleration model samejima 1995 and the nonparametric sequential model hemker 1996 four measurement properties are discussed these are monotone likelihood ratio of the total score stochastic ordering of the latent trait by the total score stochastic ordering of the total score by the latent trait and invariant item ordering these properties have been investigated previously for the adjacent category models and the cumulative probability models and for the continuation ratio models this is done here it is shown that stochastic ordering of the total score by the latent trait is implied by all continuation ratio models while monotone likelihood ratio of the total score and stochastic ordering on the latent trait by the total score are not implied by any of the continuation ratio models only the sequential rating scale model implies the property of invariant item ordering also we present a venn diagram showing the relationships between all known polytomous irt models from all three classes

a person fit test for irt models for polytomous items a person fit test based on the lagrange multiplier test is presented for three item response theory models for polytomous items the generalized partial credit model the sequential model and the graded response model the test can also be used in the framework of multidimensional ability parameters it is shown that the lagrange multiplier statistic can take both the effects of estimation of the item parameters and the estimation of the person parameters into account the lagrange multiplier statistic has an asymptotic 2 distribution the type i error rate and power are investigated using simulation studies results show that test statistics that ignore the effects of estimation of the persons ability parameters have decreased type i error rates and power incorporating a correction to account for the effects of the estimation of the persons ability parameters results in acceptable type i error rates and power characteristics incorporating a correction for the estimation of the item parameters has very little additional effect it is investigated to what extent the three models give comparable results both in the simulation studies and in an example using data from the neo personality inventory revised

identifiability of nonlinear logistic test models the linear logistic test model lltm specifies the item parameters as a weighted sum of basic parameters the lltm is a special case of a more general nonlinear logistic test model nltm where the weights are partially unknown this paper is about the identifiability of the nltm sufficient and necessary conditions for global identifiability are presented for a nltm where the weights are linear functions while conditions for local identifiability are shown to require a model with less restrictions it is also discussed how these conditions are checked using an algorithm due to bekker merckens and wansbeek 1994 several illustrations are given

new dimension spectra finer information on scaling and homogeneity abstract we introduce a new dimension spectrum motivated by the assouad dimension a familiar notion of dimension which for a given metric space returns the minimal exponent 0 such that for any pair of scales 0 r r any ball of radius r may be covered by a constant times r r balls of radius r to each 0 1 we associate the appropriate analogue of the assouad dimension with the restriction that the two scales r and r used in the definition satisfy log r log r the resulting dimension spectrum as a function of thus gives finer geometric information regarding the scaling structure of the space and in some precise sense interpolates between the upper box dimension and the assouad dimension this latter point is particularly useful because the spectrum is generally better behaved than the assouad dimension we also consider the corresponding lower spectrum motivated by the lower dimension which acts as a dual to the assouad spectrum we conduct a detailed study of these dimension spectra including analytic geometric and measureability properties we also compute the spectra explicitly for some common examples of fractals including decreasing sequences with decreasing gaps and spirals with sub exponential and monotonic winding we also give several applications of our results including dimension distortion estimates under bi holder maps for assouad dimension and the provision of new bi lipschitz invariants

sample size determination for rasch model tests this paper is concerned with supplementing statistical tests for the rasch model so that additionally to the probability of the error of the first kind type i probability the probability of the error of the second kind type ii probability can be controlled at a predetermined level by basing the test on the appropriate number of observations an approach to determining a practically meaningful extent of model deviation is proposed and the approximate distribution of the wald test is derived under the extent of model deviation of interest

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

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

measuring growth in a longitudinal large scale assessment with a general latent variable model the aim of the research presented here is the use of extensions of longitudinal item response theory irt models in the analysis and comparison of group specific growth in large scale assessments of educational outcomes a general discrete latent variable model was used to specify and compare two types of multidimensional item response theory mirt models for longitudinal data a a model that handles repeated measurements as multiple correlated variables over time and b a model that assumes one common variable over time and additional variables that quantify the change using extensions of these mirt models we approach the issue of modeling and comparing group specific growth in observed and unobserved subpopulations the analyses presented in this paper aim at answering the question whether academic growth is homogeneous across types of schools defined by academic demands and curricular differences in order to facilitate answering this research question a a model with a single two dimensional ability distribution was compared to b a model assuming multiple populations with potentially different two dimensional ability distributions based on type of school and to c a model that assumes that the observations are sampled from a discrete mixture of unobserved populations allowing for differences across schools with respect to mixing proportions for this purpose we specified a hierarchical mixture distribution variant of the two mirt models the latter model c is a growth mixture mirt model that allows for variation of the mixing proportions across clusters in a hierarchically organized sample we applied the proposed models to the pisa i plus data for assessing learning and change across multiple subpopulations the results of this study support the hypothesis of differential growth

the unique correspondence of the item response function and item category response functions in polytomously scored item response models the item response function irf for a polytomously scored item is defined as a weighted sum of the item category response functions icrf the probability of getting a particular score for a randomly sampled examinee of ability this paper establishes the correspondence between an irf and a unique set of icrfs for two of the most commonly used polytomous irt models the partial credit models and the graded response model specifically a proof of the following assertion is provided for these models if two items have the same irf then they must have the same number of categories moreover they must consist of the same icrfs as a corollary for the rasch dichotomous model if two tests have the same test characteristic function tcf then they must have the same number of items moreover for each item in one of the tests an item in the other test with an identical irf must exist theoretical as well as practical implications of these results are discussed

polytomous latent scales for the investigation of the ordering of items we propose three latent scales within the framework of nonparametric item response theory for polytomously scored items latent scales are models that imply an invariant item ordering meaning that the order of the items is the same for each measurement value on the latent scale this ordering property may be important in for example intelligence testing and person fit analysis we derive observable properties of the three latent scales that can each be used to investigate in real data whether the particular model adequately describes the data we also propose a methodology for analyzing test data in an effort to find support for a latent scale and we use two real data examples to illustrate the practical use of this methodology

a person fit index for polytomous rasch models latent class models and their mixture generalizations a normally distributed person fit index is proposed for detecting aberrant response patterns in latent class models and mixture distribution irt models for dichotomous and polytomous data this article extends previous work on the null distribution of person fit indices for the dichotomous rasch model to a number of models for categorical data a comparison of two different approaches to handle the skewness of the person fit index distribution is included

generalized latent trait models in this paper we discuss a general model framework within which manifest variables with different distributions in the exponential family can be analyzed with a latent trait model a unified maximum likelihood method for estimating the parameters of the generalized latent trait model will be presented we discuss in addition the scoring of individuals on the latent dimensions the general framework presented allows not only the analysis of manifest variables all of one type but also the simultaneous analysis of a collection of variables with different distributions the approach used analyzes the data as they are by making assumptions about the distribution of the manifest variables directly

conditions for rasch dichotomizability of the unidimensional polytomous rasch model jansen and roskam 1986 discussed the compatibility of the unidimensional polytomous rasch model with dichotomization of the response continuum they derived a rather strict condition in which dichotomization of multicategory data that fit the unidimensional polytomous rasch model results in dichotomous data which fit the dichotomous research model with effectively the same subject parameter in this paper a more general dichotomization condition is derived for the polytomous rasch model which appears less restrictive but upholds that the intrinsic logic of the unidimensional polytomous rasch model defies dichotomization in general the robustness of dichotomous analysis investigated in a simulation study it shows a close relation with the two parameters birnbaum model theoretical and methodological implications are discussed

assessing item fit for unidimensional item response theory models using residuals from estimated item response functions residual analysis e g hambleton swaminathan item response theory principles and applications kluwer academic boston 1985 hambleton swaminathan rogers fundamentals of item response theory sage newbury park 1991 is a popular method to assess fit of item response theory irt models we suggest a form of residual analysis that may be applied to assess item fit for unidimensional irt models the residual analysis consists of a comparison of the maximum likelihood estimate of the item characteristic curve with an alternative ratio estimate of the item characteristic curve the large sample distribution of the residual is proved to be standardized normal when the irt model fits the data we compare the performance of our suggested residual to the standardized residual of hambleton et al fundamentals of item response theory sage newbury park 1991 in a detailed simulation study we then calculate our suggested residuals using data from an operational test the residuals appear to be useful in assessing the item fit for unidimensional irt models

power and sample size determination for group comparison of patient reported outcomes using polytomous rasch models the analysis of patient reported outcomes or other psychological traits can be realized using the rasch measurement model when the objective of a study is to compare groups of individuals it is important before the study to define a sample size such that the group comparison test will attain a given power the raschpower procedure rp allows doing so with dichotomous items the rp is extended to polytomous items several computational issues were identified and adaptations have been proposed the performance of this new version of rp is assessed using simulations this adaptation of rp allows obtaining a good estimate of the expected power of a test to compare groups of patients in a large number of practical situations a stata module as well as its implementation online is proposed to perform the rp two versions of the rp for polytomous items are proposed deterministic and stochastic versions these two versions produce similar results in all of the tested cases we recommend the use of the deterministic version when the measure is obtained using small questionnaires or items with a few number of response categories and the stochastic version elsewhere so as to optimize computing time

a note on weighted likelihood and jeffreys modal estimation of proficiency levels in polytomous item response models warm in psychometrika 54 427 450 1989 established the equivalence between the so called jeffreys modal and the weighted likelihood estimators of proficiency level with some dichotomous item response models the purpose of this note is to extend this result to polytomous item response models first a general condition is derived to ensure the perfect equivalence between these two estimators second it is shown that this condition is fulfilled by two broad classes of polytomous models including among others the partial credit rating scale graded response and nominal response models

a simple heuristic for computing nonstationary s s policies nonstationary inventory problems with set up costs proportional ordering costs and stochastic demands occur in a large number of industrial distribution and service contexts it is well known that nonstationary s s policies are optimal for such problems in this paper we propose a simple myopic heuristic for computing the policies the heuristic involves approximating the future problem at each period by a stationary one and obtaining the solution to the corresponding stationary problem we numerically compare our heuristic with an earlier myopic heuristic and the optimal dynamic programming solution procedure over all problems tested the new heuristic averaged 1 7 error compared with 2 0 error for the old procedure and is on average 399 times as fast as the d p and 2062 as fast as the old heuristic moreover our heuristic owing to its myopic nature requires the demand data only a few periods into the future while the dynamic programming solution needs the demand data for the entire time horizon which are typically not available in most practical situations

an efficient computational method for a stochastic dynamic lot sizing problem under service level constraints abstract we provide an efficient computational approach to solve the mixed integer programming mip model developed by tarim and kingsman 8 for solving a stochastic lot sizing problem with service level constraints under the static dynamic uncertainty strategy the effectiveness of the proposed method hinges on three novelties i the proposed relaxation is computationally efficient and provides an optimal solution most of the time ii if the relaxation produces an infeasible solution then this solution yields a tight lower bound for the optimal cost and iii it can be modified easily to obtain a feasible solution which yields an upper bound in case of infeasibility the relaxation approach is implemented at each node of the search tree in a branch and bound procedure to efficiently search for an optimal solution extensive numerical tests show that our method dominates the mip solution approach and can handle real life size problems in trivial time

multicategorical spline model for item response theory additional information contained in incorrect responses calls for a multicategorical rather than a binary analysis of multiple choice data a nonparametric divided by total model for joint maximum likelihood estimation of probability of choice functions for particular responses and of latent ability is proposed the model approximates probability functions by rational splines some illustrative examples of real test data analysis and the results of a monte carlo study are presented

some results on mean square error for factor score prediction for the confirmatory factor model a series of inequalities is given with respect to the mean square error mse of three main factor score predictors the eigenvalues of these mse matrices are a monotonic function of the eigenvalues of the matrix p 1 2 p p 1 p 1 2 this matrix increases with the number of observable variables p a necessary and sufficient condition for mean square convergence of predictors is divergence of the smallest eigenvalue of p or equivalently divergence of signal to noise schneeweiss mathes 1995 the same condition is necessary and sufficient for convergence to zero of the positive definite mse differences of factor predictors convergence to zero of the distance between factor predictors and convergence to the unit value of the relative efficiencies of predictors various illustrations and examples of the convergence are given as well as explicit recommendations on the problem of choosing between the three main factor score predictors

almost bi lipschitz embeddings and almost homogeneous sets this paper is concerned with embeddings of homogeneous spaces into euclidean spaces we show that any homogeneous metric space can be em bedded into a hilbert space using an almost bi lipschitz mapping bi lipschitz to within logarithmic corrections the image of this set is no longer homo geneous but almost homogeneous we therefore study the problem of em bedding an almost homogeneous subset x of a hilbert space h into a finite dimensional euclidean space in fact we show that if x is a compact subset of a banach space and x x is almost homogeneous then for n sufficiently large a prevalent set of linear maps from x into r n are almost bi lipschitz between x and its image we are then able to use the kuratowski embedding of x d into l x to prove a similar result for compact metric spaces

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 comparison of latent trait and latent class analyses of likert type data this paper brings together and compares two developments in the analysis of likert attitude scales the first is the generalization of latent class models to ordered response categories the second is the introduction of latent trait models with multiplicative parameter structures for the analysis of rating scales key similarities and differences between these two methods are described and illustrated by applying a latent trait model and a latent class model to the analysis of a set of life satisfaction data the way in which the latent trait model defines a unit of measurement takes into account the order of the response categories and scales the latent classes is discussed while the latent class model provides better fit to these data this is achieved at the cost of a logically inconsistent assignment of individuals to latent classes

modelling and computing rn sn policies for inventory systems with non stationary stochastic demand abstract this paper addresses the single item non stationary stochastic demand inventory control problem under the non stationary r s policy in non stationary r s policies two sets of control parameters the review intervals which are not necessarily equal and the order up to levels for replenishment periods are fixed at the beginning of the planning horizon to minimize the expected total cost it is assumed that the total cost is comprised of fixed ordering costs and proportional direct item inventory holding and shortage costs with the common assumption that the actual demand per period is a normally distributed random variable about some forecast value a certainty equivalent mixed integer linear programming model is developed for computing policy parameters the model is obtained by means of a piecewise linear approximation to the non linear terms in the cost function numerical examples are provided

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

a gibbs sampler for the extended marginal rasch model in their seminal work on characterizing the manifest probabilities of latent trait models cressie and holland give a theoretically important characterization of the marginal rasch model because their representation of the marginal rasch model does not involve any latent trait nor any specific distribution of a latent trait it opens up the possibility for constructing a markov chain monte carlo method for bayesian inference for the marginal rasch model that does not rely on data augmentation such an approach would be highly efficient as its computational cost does not depend on the number of respondents which makes it suitable for large scale educational measurement in this paper such an approach will be developed and its operating characteristics illustrated with simulated data

latent trait models and dichotomization of graded responses this paper discusses thecompatibility of the polychotomous rasch model with dichotomization of the response continuum it is argued that in the case of graded responses the response categories presented to the subject are essentially an arbitrary polychotomization of the response continuum ranging for example from total rejection or disagreement to total acceptance or agreement of an item or statement because of this arbitrariness the measurement outcome should be independent of the specific polychotomization applied for example presenting a specific multicategory response format should not affect the measurement outcome when such is the case the original polychotomous model is called compatible with dichotomization

nervousness in inventory management comparison of basic control rules abstract using a rolling horizon planning framework in inventory control leads to nervousness in the planning system caused by instability of order release decisions in successive planning cycles for a single stage inventory system with arbitrary stochastic demand it is shown analytically how planning stability is affected by policy parameters if s nq s s and t s control rules are applied it turns out that the reorder point s does not influence stability whereas the lot size determining parameters q s s and t can have a considerable impact however this influence turns out to be quite different for different measures of stability regarding order setup or order quantity deviations respectively using these planning stability results the priority of the different control rules under the aspect of nervousness is discussed

stochastic ordering of the latent trait by the sum score under various polytomous irt models the sum score is often used to order respondents on the latent trait measured by the test therefore it is desirable that under the chosen model the sum score stochastically orders the latent trait it is known that unlike dichotomous item response theory irt models most polytomous irt models do not imply stochastic ordering it is unknown however 1 whether stochastic ordering is often or rarely violated and 2 whether violations yield a serious problem for practical data analysis these are the central issues of this paper first some unanswered questions that pertain to polytomous irt models implying stochastic ordering were investigated second simulation studies were conducted to evaluate stochastic ordering in practical situations it was found that for most polytomous irt models that do not imply stochastic ordering the sum score can be used safely to order respondents on the latent trait

a rasch model for continuous ratings a unidimensional latent trait model for continuous ratings is developed this model is an extension of andrich s rating formulation which assumes that the response process at latent thresholds is governed by the dichotomous rasch model item characteristic functions and information functions are used to illustrate that the model takes ceiling and floor effects into account both the dichotomous rasch model and a linear model with normally distributed error can be derived as limiting cases the separability of the structural and incidental parameters is demonstrated and a procedure for estimating the parameters is outlined