the em algorithm an old folk song sung to a fast new tune celebrating the 20th anniversary of the presentation of the paper by dempster laird and rubin which popularized the em algorithm we investigate after a brief historical account strategies that aim to make the em algorithm converge faster while maintaining its simplicity and stability e g automatic monotone convergence in likelihood first we introduce the idea of a working parameter to facilitate the search for efficient data augmentation schemes and thus fast em implementations second summarizing various recent extensions of the em algorithm we formulate a general alternating expectation conditional maximization algorithm aecm that couples flexible data augmentation schemes with model reduction schemes to achieve efficient computations we illustrate these methods using multivariate t models with known or unknown degrees of freedom and poisson models for image reconstruction we show through both empirical and theoretical evidence the potential for a dramatic reduction in computational time with little increase in human effort we also discuss the intrinsic connection between em type algorithms and the gibbs sampler and the possibility of using the techniques presented here to speed up the latter the main conclusion of the paper is that with the help of statistical considerations it is possible to construct algorithms that are simple stable and fast

the heteroscedastic graded response model with a skewed latent trait testing statistical and substantive hypotheses related to skewed item category functions the graded response model grm samejima estimation of ability using a response pattern of graded scores psychometric monograph no 17 richmond va the psychometric society 1969 can be derived by assuming a linear regression of a continuous variable z on the trait to underlie the ordinal item scores takane de leeuw in psychometrika 52 393 408 1987 traditionally a normal distribution is specified for z implying homoscedastic error variances and a normally distributed in this paper we present the heteroscedastic grm with skewed latent trait which extends the traditional grm by incorporation of heteroscedastic error variances and a skew normal latent trait an appealing property of the extended grm is that it includes the traditional grm as a special case this enables specific tests on the normality assumption of z we show how violations of normality in z can lead to asymmetrical category response functions the ability to test this normality assumption is beneficial from both a statistical and substantive perspective in a simulation study we show the viability of the model and investigate the specificity of the effects we apply the model to a dataset on affect and a dataset on alexithymia

structural modeling and psychometrika an historical perspective on growth and achievements the field of linear structural equation modeling with continuous variables is reviewed trends in psychometric theory and data analysis across the five decades of publication ofpsychometrika are discussed especially the clarification of concepts of population and sample explication of the parametric structure of models delineation of concepts of exploratory and confirmatory data analysis expansion of statistical theory in psychometrics estimation via optimization of an explicit objective function and implementation of general function minimization methods developments in the ideas of factor analysis latent variables as well as structural and causal modeling are noted some major conceptual achievements involving general covariance structure representations multiple population models and moment structures are reviewed the major statistical achievements of normal theory generalized least squares estimation elliptical and distribution free estimation and higher moment estimation are discussed computer programs that implement some of the theoretical developments are described

characterizing the manifest probabilities of latent trait models the problem of characterizing the manifest probabilities of a latent trait model is considered the item characteristic curve is transformed to the item passing odds curve and a corresponding transformation is made on the distribution of ability this results in a useful expression for the manifest probabilities of any latent trait model the result is then applied to give a characterization of the rasch model as a log linear model for a 2 j contingency table partial results are also obtained for other models the question of the identifiability of guessing parameters is also discussed

a generalized rasch model for manifest predictors a logistic regression model is suggested for estimating the relation between a set of manifest predictors and a latent trait assumed to be measured by a set ofk dichotomous items usually the estimated subject parameters of latent trait models are biased especially for short tests therefore the relation between a latent trait and a set of predictors should not be estimated with a regression model in which the estimated subject parameters are used as a dependent variable direct estimation of the relation between the latent trait and one or more independent variables is suggested instead estimation methods and test statistics for the rasch model are discussed and the model is illustrated with simulated and empirical data

a random effects model for binary data summary this paper presents a method based on maximizing the marginal likelihood for analyzing binary data with random effects with the assumption of a parametric family that allows for a wide variety of shapes for the distribution of the random effects the marginal likelihood can be computed without numerical integrations the method uses local independence models as well as those that incorporate additional dependence among the responses two examples a panel study with binary responses and an analysis of item response data will be used to illustrate the method

online calibration via variable length computerized adaptive testing item calibration is an essential issue in modern item response theory based psychological or educational testing due to the popularity of computerized adaptive testing methods to efficiently calibrate new items have become more important than that in the time when paper and pencil test administration is the norm there are many calibration processes being proposed and discussed from both theoretical and practical perspectives among them the online calibration may be one of the most cost effective processes in this paper under a variable length computerized adaptive testing scenario we integrate the methods of adaptive design sequential estimation and measurement error models to solve online item calibration problems the proposed sequential estimate of item parameters is shown to be strongly consistent and asymptotically normally distributed with a prechosen accuracy numerical results show that the proposed method is very promising in terms of both estimation accuracy and efficiency the results of using calibrated items to estimate the latent trait levels are also reported

the effect of uncertainty of item parameter estimation on ability estimates the conventional method of measuring ability which is based on items with assumed true parameter values obtained from a pretest is compared to a bayesian method that deals with the uncertainties of such items computational expressions are presented for approximating the posterior mean and variance of ability under the three parameter logistic 3pl model a 1987 american college testing program act math test is used to demonstrate that the standard practice of using maximum likelihood or empirical bayes techniques may seriously underestimate the uncertainty in estimated ability when the pretest sample is only moderately large

identification of the 1pl model with guessing parameter parametric and semi parametric results in this paper we study the identification of a particular case of the 3pl model namely when the discrimination parameters are all constant and equal to 1 we term this model 1pl g model the identification analysis is performed under three different specifications the first specification considers the abilities as unknown parameters it is proved that the item parameters and the abilities are identified if a difficulty parameter and a guessing parameter are fixed at zero the second specification assumes that the abilities are mutually independent and identically distributed according to a distribution known up to the scale parameter it is shown that the item parameters and the scale parameter are identified if a guessing parameter is fixed at zero the third specification corresponds to a semi parametric 1pl g model where the distribution g generating the abilities is a parameter of interest it is not only shown that after fixing a difficulty parameter and a guessing parameter at zero the item parameters are identified but also that under those restrictions the distribution g is not identified it is finally shown that after introducing two identification restrictions either on the distribution g or on the item parameters the distribution g and the item parameters are identified provided an infinite quantity of items is available

statistical inference for multiple choice tests finite sample inference procedures are considered for analyzing the observed scores on a multiple choice test with several items where for example the items are dissimilar or the item responses are correlated a discrete p parameter exponential family model leads to a generalized linear model framework and in a special case a convenient regression of true score upon observed score techniques based upon the likelihood function akaike s information criteria aic an approximate bayesian marginalization procedure based on conditional maximization bcm and simulations for exact posterior densities importance sampling are used to facilitate finite sample investigations of the average true score individual true scores and various probabilities of interest a simulation study suggests that when the examinees come from two different populations the exponential family can adequately generalize duncan s beta binomial model extensions to regression models the classical test theory model and empirical bayes estimation problems are mentioned the duncan keats and matsumura data sets are used to illustrate potential advantages and flexibility of the exponential family model and the bcm technique

variance component models with binary response interviewer variability summary interviewer variability in a binary response is an example of a problem requiring variance component estimation in a non normal family the maximum likelihood estimation procedure is derived and used to examine some binary items on a large questionnaire this raises some interesting questions about the use of unbalanced ano va methods with these data two interviewers asking the same question of the same respondent can obtain different responses such interviewer effects create problems in the design and analysis of survey questionnaires collins 1980 gives a review of these problems together with a variety of partial remedies these include pilot studies and careful design to eliminate difficulties with wording of questions and instructions on the questionnaires in spite of these precautions interviewer effects remain questionnaire items are usually categorical and occasionally binary but usually for analysis the questionnaire item is reduced to a binary response it is the variability of the population of interviewers that is of interest and not just the individual interviewers thus this variability is best modelled as a random effect rather than a fixed effect the number of interviews completed by each interviewer varies so that the design is unbalanced thus we have an unbalanced variance component model with a categorical or binary response the intraclass correlation the ratio of the interviewer variance to the sum of all variances in the model is of great interest as a measure of interviewer variability in estimating the variance components the categorical or binary nature of the response is usually ignored and the analysis carried out using analysis of variance this is not maximum likelihood but analysis of variance does have some desirable properties when the error variables are non normal the rule of thumb generally applied is that the analysis of variance is reasonably accurate as long as the proportions in each of the categories of the response are between 0 1 and 0 9 scheffe 1959 p 343 discusses the effects of non normality in a one way balanced model very little theoretical work has been

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

structural modeling of measurement error in generalized linear models with rasch measures as covariates this paper proposes a structural analysis for generalized linear models when some explanatory variables are measured with error and the measurement error variance is a function of the true variables the focus is on latent variables investigated on the basis of questionnaires and estimated using item response theory models latent variable estimates are then treated as observed measures of the true variables this leads to a two stage estimation procedure which constitutes an alternative to a joint model for the outcome variable and the responses given to the questionnaire simulation studies explore the effect of ignoring the true error structure and the performance of the proposed method two illustrative examples concern achievement data of university students particular attention is given to the rasch model

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

bayesian factor analysis for multilevel binary observations multilevel covariance structure models have become increasingly popular in the psychometric literature in the past few years to account for population heterogeneity and complex study designs we develop practical simulation based procedures for bayesian inference of multilevel binary factor analysis models we illustrate how markov chain monte carlo procedures such as gibbs sampling and metropolis hastings methods can be used to perform bayesian inference model checking and model comparison without the need for multidimensional numerical integration we illustrate the proposed estimation methods using three simulation studies and an application involving student s achievement results in different areas of mathematics

a note on the reliability coefficients for item response model based ability estimates assuming item parameters on a test are known constants the reliability coefficient for item response theory irt ability estimates is defined for a population of examinees in two different ways as a the product moment correlation between ability estimates on two parallel forms of a test and b the squared correlation between the true abilities and estimates due to the bias of irt ability estimates the parallel forms reliability coefficient is not generally equal to the squared correlation reliability coefficient it is shown algebraically that the parallel forms reliability coefficient is expected to be greater than the squared correlation reliability coefficient but the difference would be negligible in a practical sense

analysis of residuals for the multinomial item response model using the item response model as developed on the multinomial distribution asymptotic variances are obtained for residuals associated with response patterns and first and second order marginal frequencies of manifest variables when the model does not fit well an examination of these residuals may reveal the source of the poor fit finally a limited information test of fit for the model is developed by using residuals defined for the first and second order marginals model evaluation based on residuals for these marginals is particularly useful when the response pattern frequencies are sparse

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

high dimensional maximum marginal likelihood item factor analysis by adaptive quadrature although the bock aitkin likelihood based estimation method for factor analysis of dichotomous item response data has important advantages over classical analysis of item tetrachoric correlations a serious limitation of the method is its reliance on fixed point gauss hermite g h quadrature in the solution of the likelihood equations and likelihood ratio tests when the number of latent dimensions is large computational considerations require that the number of quadrature points per dimension be few but with large numbers of items the dispersion of the likelihood given the response pattern becomes so small that the likelihood cannot be accurately evaluated with the sparse fixed points in the latent space in this paper we demonstrate that substantial improvement in accuracy can be obtained by adapting the quadrature points to the location and dispersion of the likelihood surfaces corresponding to each distinct pattern in the data in particular we show that adaptive g h quadrature combined with mean and covariance adjustments at each iteration of an em algorithm produces an accurate fast converging solution with as few as two points per dimension evaluations of this method with simulated data are shown to yield accurate recovery of the generating factor loadings for models of upto eight dimensions unlike an earlier application of adaptive gibbs sampling to this problem by meng and schilling the simulations also confirm the validity of the present method in calculating likelihood ratio chi square statistics for determining the number of factors required in the model finally we apply the method to a sample of real data from a test of teacher qualifications

marginal maximum likelihood estimation for the one parameter logistic model two algorithms are described for marginal maximum likelihood estimation for the one parameter logistic model the more efficient of the two algorithms is extended to estimation for the linear logistic model numerical examples of both procedures are presented

stochastic approximation em for large scale exploratory irt factor analysis a stochastic approximation em algorithm saem is described for exploratory factor analysis of dichotomous or ordinal variables the factor structure is obtained from sufficient statistics that are updated during iterations with the robbins monro procedure two large scale simulations are reported that compare accuracy and cpu time of the proposed saem algorithm to the metropolis hasting robbins monro procedure and to a generalized least squares analysis of the polychoric correlation matrix a smaller scale application to real data is also reported including a method for obtaining standard errors of rotated factor loadings a simulation study based on the real data analysis is conducted to study bias and error estimates the saem factor algorithm requires minimal lines of code no derivatives and no large matrix inversion it is programmed entirely in r

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

on the bock aitkin procedure from an em algorithm perspective the relationship between the em algorithm and the bock aitkin procedure is described with a continuous distribution of ability latent trait from an em algorithm perspective previous work has been restricted to the discrete case from a probit analysis perspective

em and beyond the basic theme of the em algorithm to repeatedly use complete data methods to solve incomplete data problems is also a theme of several more recent statistical techniques these techniques multiple imputation data augmentation stochastic relaxation and sampling importance resampling combine simulation techniques with complete data methods to attack problems that are difficult or impossible for em

information matrices and standard errors for mles of item parameters in irt the paper clarifies the relationship among several information matrices for the maximum likelihood estimates mles of item parameters it shows that the process of calculating the observed information matrix also generates a related matrix that is the middle piece of a sandwich type covariance matrix monte carlo results indicate that standard errors ses based on the observed information matrix are robust to many but not all conditions of model distribution misspecifications ses based on the sandwich type covariance matrix perform most consistently across conditions results also suggest that ses based on other matrices are either not consistent or perform not as robust as those based on the sandwich type covariance matrix or the observed information matrix

a latent trait finite mixture model for the analysis of rating agreement this article presents a latent distribution model for the analysis of agreement on dichotomous or ordered category ratings the model includes parameters that characterize bias category definitions and measurement error for each rater or test parameter estimates can be used to evaluate rater performance and to improve classification or measurement with use of multiple ratings a simple maximum likelihood estimation procedure is described two examples illustrate the approach although considered in the context of analyzing rater agreement the model provides a general approach for mixture analysis using two or more ordered caregory measures

a general maximum likelihood analysis of variance components in generalized linear models this paper describes an em algorithm for nonparametric maximum likelihood ml estimation in generalized linear models with variance component structure the algorithm provides an alternative analysis to approximate mql and pql analyses mcgilchrist and aisbett 1991 biometrical journal 33 131 141 breslow and clayton 1993 journal of the american statistical association 88 9 25 mcgilchrist 1994 journal of the royal statistical society series b 56 61 69 goldstein 1995 multilevel statistical models and to gee analyses liang and zeger 1986 biometrika 73 13 22 the algorithm first given by hinde and wood 1987 in longitudinal data analysis 110 126 is a generalization of that for random effect models for overdispersion in generalized linear models described in aitkin 1996 statistics and computing 6 251 262 the algorithm is initially derived as a form of gaussian quadrature assuming a normal mixing distribution but with only slight variation it can be used for a completely unknown mixing distribution giving a straightforward method for the fully nonparametric ml estimation of this distribution this is of value because the ml estimates of the glm parameters can be sensitive to the specification of a parametric form for the mixing distribution the nonparametric analysis can be extended straightforwardly to general random parameter models with full npml estimation of the joint distribution of the random parameters this can produce substantial computational saving compared with full numerical integration over a specified parametric distribution for the random parameters a simple method is described for obtaining correct standard errors for parameter estimates when using the em algorithm several examples are discussed involving simple variance component and longitudinal models and small area estimation

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

a variational maximization maximization algorithm for generalized linear mixed models with crossed random effects we present a variational maximization maximization algorithm for approximate maximum likelihood estimation of generalized linear mixed models with crossed random effects e g item response models with random items random raters or random occasion specific effects the method is based on a factorized variational approximation of the latent variable distribution given observed variables which creates a lower bound of the log marginal likelihood the lower bound is maximized with respect to the factorized distributions as well as model parameters with the proposed algorithm a high dimensional intractable integration is translated into a two dimensional integration problem we incorporate an adaptive gauss hermite quadrature method in conjunction with the variational method in order to increase computational efficiency numerical studies show that under the small sample size conditions that are considered the proposed algorithm outperforms the laplace approximation

generalized fiducial inference for binary logistic item response models generalized fiducial inference gfi has been proposed as an alternative to likelihood based and bayesian inference in mainstream statistics confidence intervals cis can be constructed from a fiducial distribution on the parameter space in a fashion similar to those used with a bayesian posterior distribution however no prior distribution needs to be specified which renders gfi more suitable when no a priori information about model parameters is available in the current paper we apply gfi to a family of binary logistic item response theory models which includes the two parameter logistic 2pl bifactor and exploratory item factor models as special cases asymptotic properties of the resulting fiducial distribution are discussed random draws from the fiducial distribution can be obtained by the proposed markov chain monte carlo sampling algorithm we investigate the finite sample performance of our fiducial percentile ci and two commonly used wald type cis associated with maximum likelihood ml estimation via monte carlo simulation the use of gfi in high dimensional exploratory item factor analysis was illustrated by the analysis of a set of the eysenck personality questionnaire data

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

generalizing logistic regression by nonparametric mixing abstract logistic regression is a common technique for analyzing the effect of a covariate vector x on the number of successes y in m trials when y has a binomial distribution but at times either the logistic curve does not describe the probability of success p x adequately or m is larger than 1 and y is more variable than the binomial distribution allows overdispersion relative to the binomial distribution is possible if the m trials in a set or litter are positively correlated an important covariate is omitted or x is measured with error a simple way to accommodate departures from the logit link and overdispersion is to introduce a random intercept and thus permit a random propensity toward success when varies between individual binary trials according to a discrete or multimodal distribution p x has smooth steps and y has a binomial m p x distribution when the random is constant for a set of m binary trials and varies between sets of m trials according to a discrete or multimodal

the dutch identity a new tool for the study of item response models the dutch identity is a useful way to reexpress the basic equations of item response models that relate the manifest probabilities to the item response functions irfs and the latent trait distribution the identity may be exploited in several ways for example a to suggest how item response models behave for large numbers of items they are approximate submodels of second order loglinear models for 2 j tables b to suggest new ways to assess the dimensionality of the latent trait principle components analysis of matrices composed of second order interactions from loglinear models c to give insight into the structure of latent class models and d to illuminate the problem of identifying the irfs and the latent trait distribution from sample data

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

item response theory observed score kernel equating item response theory irt observed score kernel equating is introduced for the non equivalent groups with anchor test equating design using either chain equating or post stratification equating t

asymptotic standard errors of irt observed score equating methods a method of the irt observed score equating using chain equating through a third test without equating coefficients is presented with the assumption of the three parameter logistic model the asymptotic standard errors of the equated scores by this method are obtained using the results given by m liou and p e cheng the asymptotic standard errors of the irt observed score equating method using a synthetic examinee group with equating coefficients which is a currently used method are also provided numerical examples show that the standard errors by these observed score equating methods are similar to those by the corresponding true score equating methods except in the range of low scores

a three dimensional parabolic equation model of sound propagation using higher order operator splitting and pade approximants an alternating direction implicit adi three dimensional fluid parabolic equation solution method with enhanced accuracy is presented the method uses a square root helmholtz operator splitting algorithm that retains cross multiplied operator terms that have been previously neglected with these higher order cross terms the valid angular range of the parabolic equation solution is improved the method is tested for accuracy against an image solution in an idealized wedge problem computational efficiency improvements resulting from the adi discretization are also discussed

joint consistency of nonparametric item characteristic curve and ability estimation the simultaneous and nonparametric estimation of latent abilities and item characteristic curves is considered the asymptotic properties of ordinal ability estimation and kernel smoothed nonparametric item characteristic curve estimation are investigated under very general assumptions on the underlying item response theory model as both the test length and the sample size increase a large deviation probability inequality is stated for ordinal ability estimation the mean squared error of kernel smoothed item characteristic curve estimates is studied and a strong consistency result is obtained showing that the worst case error in the item characteristic curve estimates over all items and ability levels converges to zero with probability equal to one

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

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

multiple equating of separate irt calibrations when test forms are calibrated separately item response theory parameters are not comparable because they are expressed on different measurement scales the equating process includes the conversion of item parameter estimates on a common scale and the determination of comparable test scores various statistical methods have been proposed to perform equating between two test forms this paper provides a generalization to multiple test forms of the mean geometric mean the mean mean the haebara and the stocking lord methods the proposed methods estimate simultaneously the equating coefficients that permit the scale transformation of the parameters of all forms to the scale of the base form asymptotic standard errors of the equating coefficients are derived a simulation study is presented to illustrate the performance of the methods

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

probability matrix decomposition models in this paper we consider a class of models for two way matrices with binary entries of 0 and 1 first we considerboolean matrix decomposition conceptualize it as alatent response model lrm and by making use of this conceptualization generalize it to a larger class of matrix decomposition models second probability matrix decomposition pmd models are introduced as a probabilistic version of this larger class of deterministic matrix decomposition models third an algorithm for the computation of the maximum likelihood ml and the maximum a posteriori map estimates of the parameters of pmd models is presented this algorithm is an em algorithm and is a special case of a more general algorithm that can be used for the whole class of lrms and fourth as an example a pmd model is applied to data on decision making in psychiatric diagnosis

modification indices for the 2 pl and the nominal response model in this paper it is shown that various violations of the 2 pl model and the nominal response model can be evaluated using the lagrange multiplier test or the equivalent efficient score test the tests presented here focus on violation of local stochastic independence and insufficient capture of the form of the item characteristic curves primarily the tests are item oriented diagnostic tools but taken together they also serve the purpose of evaluation of global model fit a useful feature of lagrange multiplier statistics is that they are evaluated using maximum likelihood estimates of the null model only that is the parameters of alternative models need not be estimated as numerical examples an application to real data and some power studies are presented

fitting full information item factor models and an empirical investigation of bridge sampling abstract based on item response theory bock and aitken introduced a method of item factor analysis termed full information item factor fiif analysis by bartholomew because it uses all distinct item response vectors as data but a limitation of their fitting algorithm is its reliance on fixed point gauss hermite quadrature which can produce appreciable numerical errors especially in high dimension problems the first purpose of this article is to offer more reliable methods by using recent advances in statistical computation specifically we illustrate two ways of implementing monte carlo expectation maximization em algorithm to fit a fiif model using the gibbs sampler to carry out the computation for the e steps we also show how to use bridge sampling to simulate the likelihood ratios for monitoring the convergence of a monte carlo em a strategy that is useful in general simulations demonstrate substantial improvement over bock and aitken s algorithm in recovering known factor loadings in hig

how should we assess the fit of rasch type models approximating the power of goodness of fit statistics in categorical data analysis we investigate the performance of three statistics r 1 r 2 glas in psychometrika 53 525 546 1988 and m 2 maydeu olivares joe in j am stat assoc 100 1009 1020 2005 psychometrika 71 713 732 2006 to assess the overall fit of a one parameter logistic model 1pl estimated by marginal maximum likelihood ml r 1 and r 2 were specifically designed to target specific assumptions of rasch models whereas m 2 is a general purpose test statistic we report asymptotic power rates under some interesting violations of model assumptions different item discrimination presence of guessing and multidimensionality as well as empirical rejection rates for correctly specified models and some misspecified models all three statistics were found to be more powerful than pearson s x 2 against two and three parameter logistic alternatives 2pl and 3pl and against multidimensional 1pl models the results suggest that there is no clear advantage in using goodness of fit statistics specifically designed for rasch type models to test these models when marginal ml estimation is used

bootstrap calibrated interval estimates for latent variable scores in item response theory in most item response theory applications model parameters need to be first calibrated from sample data latent variable lv scores calculated using estimated parameters are thus subject to sampling error inherited from the calibration stage in this article we propose a resampling based method namely bootstrap calibration bc to reduce the impact of the carryover sampling error on the interval estimates of lv scores bc modifies the quantile of the plug in posterior i e the posterior distribution of the lv evaluated at the estimated model parameters to better match the corresponding quantile of the true posterior i e the posterior distribution evaluated at the true model parameters over repeated sampling of calibration data furthermore to achieve better coverage of the fixed true lv score we explore the use of bc in conjunction with jeffreys prior we investigate the finite sample performance of bc via monte carlo simulations and apply it to two empirical data examples

estimation of latent group effects abstract conventional methods of multivariate normal analysis do not apply when the variables of interest are not observed directly but must be inferred from fallible or incomplete data a method of estimating such effects by marginal maximum likelihood implemented by means of an em algorithm is proposed asymptotic standard errors and likelihood ratio tests of fit are provided the procedures are illustrated with data from the administration of the armed services vocational aptitude battery to a probability sample of american youth

likelihood analysis of multivariate probit models using a parameter expanded mcem algorithm multivariate binary data arise in a variety of settings in this article we propose a practical and efficient computational framework for maximum likelihood estimation of multivariate probit regression models this approach uses the monte carlo expectation maximization mcem algorithm with parameter expansion to complete the m step to avoid the direct evaluation of the intractable multivariate normal orthant probabilities the parameter expansion not only enables a closed form solution in the m step but also improves efficiency using the simulation studies we compare the performance of our approach with the mcem algorithms developed by chib and greenberg 1998 and song and lee 2005 as well as the iterative approach proposed by li and schafer 2008 our approach is further illustrated using a real world example

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

item response theory with estimation of the latent population distribution using spline based densities the purpose of this paper is to introduce a new method for fitting item response theory models with the latent population distribution estimated from the data using splines a spline based density estimation system provides a flexible alternative to existing procedures that use a normal distribution or a different functional form for the population distribution a simulation study shows that the new procedure is feasible in practice and that when the latent distribution is not well approximated as normal two parameter logistic 2pl item parameter estimates and expected a posteriori scores eaps can be improved over what they would be with the normal model an example with real data compares the new method and the extant empirical histogram approach

bayesian estimation of item response curves item response curves for a set of binary responses are studied from a bayesian viewpoint of estimating the item parameters for the two parameter logistic model with normally distributed ability restricted bivariate beta priors are used to illustrate the computation of the posterior mode via the em algorithm the procedure is illustrated by data from a mathematics test

a mixture cure rate model for responses and response times in time limit tests many large scale standardized tests are intended to measure skills related to ability rather than the rate at which examinees can work time limits imposed on these tests make it difficult to distinguish between the effect of low proficiency and the effect of lack of time this paper proposes a mixture cure rate model approach to address this issue maximum likelihood estimation is proposed for parameter and variance estimation for three cases when examinee parameters are to be estimated given precalibrated item parameters when item parameters are to be calibrated given known examinee parameters and when item parameters are to be estimated without assuming known examinee parameters large sample properties are established for the cases under suitable regularity conditions simulation studies suggest that the proposed approach is appropriate for inferences concerning model parameters in addition not distinguishing between the effect of low proficiency and the effect of lack of time is shown to have considerable consequences for parameter estimation a real data example is presented to demonstrate the new model choice of survival models for the latent power times is also discussed

psychometric engineering as art the psychometric society is devoted to the development of psychology as a quantitative rational science engineering is often set in contradistinction with science art is sometimes considered different from science why then juxtapose the words in the title psychometric engineering andart because an important aspect of quantitative psychology is problem solving and engineering solves problems and an essential aspect of a good solution is beauty hence art in overview and with examples this presentation describes activities that are quantitative psychology as engineering and art that is as design extended illustrations involve systems for scoring tests in realistic contexts allusions are made to other examples that extend the conception of quantitative psychology as engineering and art across a wider range of psychometric activities

estimating multiple classification latent class models this paper presents a new class of models for persons by items data the essential new feature of this class is the representation of the persons every person is represented by its membership tomultiple latent classes each of which belongs to onelatent classification the models can be considered as a formalization of the hypothesis that the responses come about in a process that involves the application of a number ofmental operations two algorithms for maximum likelihood ml and maximum a posteriori map estimation are described they both make use of the tractability of the complete data likelihood to maximize the observed data likelihood properties of the map estimators i e uniqueness and goodness of recovery and the existence of asymptotic standard errors were examined in a simulation study then one of these models is applied to the responses to a set of fraction addition problems finally the models are compared to some related models in the literature

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

on the relationship between item response theory and factor analysis of discretized variables equivalence of marginal likelihood of the two parameter normal ogive model in item response theory irt and factor analysis of dichotomized variables fa was formally proved the basic result on the dichotomous variables was extended to multicategory cases both ordered and unordered categorical data pair comparison data arising from multiple judgment sampling were discussed as a special case of the unordered categorical data a taxonomy of data for the irt and fa models was also attempted

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

the role of collateral information about examinees in item parameter estimation standard procedures for estimating item parameters in item response theory irt ignore collateral information that may be available about examinees such as their standing on demographic and educational variables this paper describes circumstances under which collateral information about examineesmay be used to make inferences about item parameters more precise and circumstances under which itmust be used to obtain correct inferences

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

a mixed effects multinomial logistic regression model a mixed effects multinomial logistic regression model is described for analysis of clustered or longitudinal nominal or ordinal response data the model is parameterized to allow flexibility in the choice of contrasts used to represent comparisons across the response categories estimation is achieved using a maximum marginal likelihood mml solution that uses quadrature to numerically integrate over the distribution of random effects an analysis of a psychiatric data set in which homeless adults with serious mental illness are repeatedly classified in terms of their living arrangement is used to illustrate features of the model

high dimensional exploratory item factor analysis by a metropolis hastings robbins monro algorithm a metropolis hastings robbins monro mh rm algorithm for high dimensional maximum marginal likelihood exploratory item factor analysis is proposed the sequence of estimates from the mh rm algorithm converges with probability one to the maximum likelihood solution details on the computer implementation of this algorithm are provided the accuracy of the proposed algorithm is demonstrated with simulations as an illustration the proposed algorithm is applied to explore the factor structure underlying a new quality of life scale for children it is shown that when the dimensionality is high mh rm has advantages over existing methods such as numerical quadrature based em algorithm extensions of the algorithm to other modeling frameworks are discussed

mixed effects analyses of rank ordered data this paper presents a synthesis of bock s 1972 nominal categories model and luce s 1959 choice model for mixed effects analyses of rank ordered data it is shown that the proposed ranking model is both parsimonious and flexible in accounting for preference heterogeneity as well as fixed and random effects of covariates relationships to other approaches including takane s 1987 ideal point discriminant model and croon s 1989 latent class version of luce s ranking model are also discussed the application focuses on a ranking study of behavioral traits that parents find desirable in children

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

modeling and testing differential item functioning in unidimensional binary item response models with a single continuous covariate a functional data analysis approach differential item functioning dif referring to between group variation in item characteristics above and beyond the group level disparity in the latent variable of interest has long been regarded as an important item level diagnostic the presence of dif impairs the fit of the single group item response model being used and calls for either model modification or item deletion in practice depending on the mode of analysis methods for testing dif with continuous covariates rather than categorical grouping variables have been developed however they are restrictive in parametric forms and thus are not sufficiently flexible to describe complex interaction among latent variables and covariates in the current study we formulate the probability of endorsing each test item as a general bivariate function of a unidimensional latent trait and a single covariate which is then approximated by a two dimensional smoothing spline the accuracy and precision of the proposed procedure is evaluated via monte carlo simulations if anchor items are available we proposed an extended model that simultaneously estimates item characteristic functions icfs for anchor items icfs conditional on the covariate for non anchor items and the latent variable density conditional on the covariate all using regression splines a permutation dif test is developed and its performance is compared to the conventional parametric approach in a simulation study we also illustrate the proposed semiparametric dif testing procedure with an empirical example

latent variable models for ordered categorical data abstract the multivariate analysis of ordered categorical data is an important growth point in statistics especially as applied to the social sciences this paper is concerned with the situation in which the associations between a set of categorical variables are assumed to arise from their dependence on a small number of latent variables the problem of modelling this relationship is first posed in very general terms and then specialized to particular cases for which statistical methods are available or in course of development such models can be viewed as tools for data reduction or for scaling and both aspects are reviewed the possible relevance of the method in economics and their relationship to other methods for categorical variables is briefly discussed

latent variable structural equation modeling with categorical data abstract structural equation modeling with latent variables is overviewed for situations involving a mixture of dichotomous ordered polytomous and continuous indicators of latent variables special emphasis is placed on categorical variables models in psychometrics econometrics and biometrics are interrelated via a general model due to muthen limited information least squares estimators and full information estimation are discussed an example is estimated with a model for a four wave longitudinal data set where dichotomous responses are related to each other and a set of independent variables via latent variables with a variance component structure

the local minima problem in hierarchical classes analysis an evaluation of a simulated annealing algorithm and various multistart procedures hierarchical classes models are quasi order retaining boolean decomposition models for n way n mode binary data to fit these models to data rationally started alternating least squares or equivalently alternating least absolute deviations algorithms have been proposed extensive simulation studies showed that these algorithms succeed quite well in recovering the underlying truth but frequently end in a local minimum in this paper we evaluate whether or not this local minimum problem can be mitigated by means of two common strategies for avoiding local minima in combinatorial data analysis simulated annealing sa and use of a multistart procedure in particular we propose a generic sa algorithm for hierarchical classes analysis and three different types of random starts the effectiveness of the sa algorithm and the random starts is evaluated by reanalyzing data sets of previous simulation studies the reported results support the use of the proposed sa algorithm in combination with a random multistart procedure regardless of the properties of the data set under study

consistent estimation in the rasch model based on nonparametric margins consider the class of two parameter marginal logistic rasch models for a test ofm true false items where the latent ability is assumed to be bounded using results of karlin and studen we show that this class of nonparametric marginal logistic nml models is equivalent to the class of marginal logistic models where the latent ability assumes at most m 2 2 values this equivalence has two implications first estimation for the nml model is accomplished by estimating the parameters of a discrete marginal logistic model second consistency for the maximum likelihood estimates of the nml model can be shown whenm is odd using the results of kiefer and wolfowitz an example is presented which demonstrates the estimation strategy and contrasts the nml model with a normal marginal logistic model