Estimation Of The Multidimensional Item Response Model

Estimation of the Multidimensional Item Response ModelAt present, there are two guide theories for tests. One is classical test theory (CTT), the other is Item Response Theory (IRT). CTT is based on the true score the-ory, and concept system such as reliability, validity etc, which is used to assess a test tool or the quality of items. Since the sixties and seventies of the 20th century, Along with the rapid development of educational measurement, modern test theory, with the core of item response theory, has been the main object of the study of education measurement. It has excited more and more attention of the researchers and educa-tors (Lord,1980; Hambleton,1989; Hambleton & Swaminathan,1985,1991; Baker & Kim,2004). Compared to CTT, IRT has obvious advantage. The item parameters it takes will not be influenced by samples, and the acquisition of the parameters will not change along with the change of the samples which were taken as testing items. Meanwhile, the estimate of ability will not change along with different test questions. Just because of these advantages, it is widely used in psychological and educational measurements. IRT has been widely used in many fields due to the development of computer technology. At present, some big exams such as TOFEL,GRE,GMAT, etc. all used Computerized Adaptive Testing based on IRT in succession (CAT, Wainer, 1990). However,IRT is not a perfect measurement theory, it has some defects need to be improved. these defects originated from the three basic assumption of the theory itself:unidimensionality, local independence, monotonicity. In view of above draw-back, some new model and method must be presented. In recent years, the central research topics are Multidimensional IRT, Nonparametric IRT,Cognitive diagnostic theory. Our country's educational and psychology evaluation method is still at an early stage. The present exam mainly tests the students'grasping knowledge,which cannot reflect the learning ability of the examinee, this is because there are little re-searchers working on educational and psychological measurement (Tao Xin,2005), let alone MIRT. Research shows use unidimensional model to fit multidimensional data will increase measurement error and make wrong inference to students'ability (Walker & Berevtas,2000). Just because of this, researchers extend the unidimensional IRT to the multidimensional IRT from different perspectives (McDonald 1967, Lord & Novick, 1968; Rechase,1997).This article mainly discussed the parameter estimation of MIRT model. Fraster (1988) presented NOHARM (Normal-Ogive Harmonic Analysis Robust Method). An alternative approach using all information in the data, and therefore labeled "Full-Information Factor Analysis", was developed by Bock, Gibbons, and Muraki (1988). This approach is a generalization of the marginal maximum likelihood (MML) and Bayes modal estimation procedures for unidimensional IRT models. TESTFACT claims that it can deal with the analysis of up to fifteen factors in its manual, but re-search shows it performs badly when the number of dimensions is higher than three (De-Mars,2005). In NOHARM a polynomial approximation method and the unweighted least squares method are used. It can only give approximate results. It also shares robustness concerns for non-normally distributed ability parameters. Both NOHARM and TESTFACT are unable to estimate the guessing parameter in the model. MCMC was first applied to IRT by Albert (1992) for estimating the two-parameter normal ogive model by Gibbs sampling. Based on an efficient data augmentation scheme (DAGS), Sahu (2002) fitted the unidimensional three parameter normal ogive (3PNO) model and provided a much faster implementation of the Gibbs sampler.This article are mainly based on the Bayesian statistical theory and pseudo-likelihood theory. Firstly, Based on an efficient data augmentation scheme, we fit-ted the multidimensional three parameter Logistic (3PL) model and provided a much faster implementation of the Gibbs sampler. Secondly, on that basis, we studied the missing data issue. IRT models on the missing data indicators will be considered for taking nonignorable missing data mechanisms into account to reduce the estimating bias by ignoring the missing data process. Once more, modeling longitudinal data. In educational and psychological research, changes over time are often investigated by performing longitudinal analysis for observations collected at several time points. For example, in educational research a goal can be to determine the development of achievement in mathematics of pupils over time.To investigate such a development, pupils may be presented one or more mathematics tests at several time points.The item responses on these tests can be related to a latent variable math achievement via. an IRT model(see,for instance,Lord,1980).In longitudinal designs,it is usually as-sumed that the positions of the students on the latent scale change over time.However, repeated measures have the complication that the responses on different time points are not independent For the Joint modeling method,For more scales and time points,this procedure may become infeasible.To circumvent the restriction, we consider MCMC method and a pairwise likelihood unnecessary to account for the dimensions of ability, and insight can be gained in the association structure of the latent traits over T times, so the trends and dependence over time will be investigated..Chapter Three mainly studied Bayesian estimation in the multidimensional three-parameter Logistic model.Suppose that each of N examinees is given n items (questions).The response yij for the jth examinee and the ith item is recorded as 1 or 0 according to whether the examinee answers the item correctly or incorrectly.Let pij denote the probability that the jth examinee is able to answer the ith item correctly. The model is the following multivariate logistic response function: where i=1,2,…,n,j.=1,2,…,N,θj=(θj1,…,θjk,…,θjm)is a vector of m ability variables with -∞<θjk<∞j=1,2,…,N,k=1, 2,…,m；ai= (ai1,…,aik,…,aim),aik>O, i=1,2,…,n,k=1,2,…,m is the vector of loadings of item i on these abilities(item discriminations)；bi=(bi1,…,bik,…,bim),-∞φiκotherwise. With the introduction of the latent variables, samplers can be drawn from the joint posterior distribution. Our simulation results show that ignoring the missing-data process results in considerable bias in the estimates of the item parameters. This bias increases as a function of the correlation between the proficiency to be measured and latent variable governing the missing-data process. Further, it was shown that this bias can be reduced using the NONMAR model (15).·Chapter Five discussed the application of MIRT model to the longitudinal item response data. In educational and psychological research, changes over time are often investigated by performing longitudinal analysis for observations collected at several time points. We use a MIRT model to analyze longitudinal data. We assume the group of individuals, selected randomly from one population, is evaluated at T pre-specified instants. For instance, one group of students evaluated at the end of 4th to 8th grades. At pre-specified times t, t=1,2,…, T, the group of individuals are administered a test composed of nt multiple-choice items, scored as right/ wrong. The total number of items n, is less than nc=∑t=1T nt because of common items among the tests. Joint modeling on T time pointsLet T be the total number of time points to be modeled jointly. At each time point t, let be a twice-differentiable item response function that describes the conditional proba-bility of a correct response to item i,i= 1,2,…, n, of individual j, j= 1,2,…,N, in test t, t=1,2,…,T, where Ujit represents the (binary) response,θjt the ability (latent trait) andξi the known vector of the item parameters. Examples of such a function are the logistic 1,2 and 3 parameters models (see Baker & Kim (2004) for details). Assuming the conditional independence of the responses to the items in test t, givenθt, we have that where It represents the set of the indexes of those items presented in test t, Uj.t= (Uj1t Uj2t,…,m,Ujntt)is the (nt×1) vector of responses of individual j in test t, andξ= (ξ1T,ξ2T,…,ξnT)T (here and hereafter, the Roman superscript T means transpose operation) Furthermore, assuming the conditional longitudinal independence of the responses to the items along the T tests, given the abilities in the T tests, we will have with (?) being the (nc×1) vector of responses of individual j in all tests andθ= (θi,θ2,…,θT)T.Finally, in this article, change over time is modeled by assuming that the latent ability parameters 6 have a multivariate normal distribution with T-dimensional vector of meansμ= (μ1,μ2,…,μT)T and a T×T covariance matrix∑= (σij)T×T-This assumption pertains to the dependence between these parameters over time points. The density will be denoted by g(θ\μ,∑). In the sequel, the covariance matrix can be the covariance over time points for a specific scale. Letλ= (ξ,μ,∑) and assuming independence of the subject j, j= 1,2,…,N, the marginal likelihood is estimates can be obtained from maximizing the above likelihood function, and infer-ences immediately follow from classical maximum likelihood theory. MML is probably the most used technique for estimation of the parameters. The theory was developed by Bock and Aitkin (1981), Thissen (1982), Rigdon and Tsutakawa (1983), and Mislevy (1984), among others. This estimation method is illustrated in detail in Andrade and Tavares (2005) to cope with longitudinal data, we will not discuss here. In contrast with linear mixed models, the marginal distribution of Uj.. cannot be derived analyti-cally. Obviously, the higher the dimension ofθ, the more difficult the approximation of the integral (Diggle et al.,2002).Firstly, We mitigate this problem by using MCMC method in Section Three, which is the product of the bivariate likelihood. To get the Gibbs sampler, We introduce two independent random variables corresponding to each data point Ujit as follows. The first is a Bernoulli random variable, denoted by rjit with success probability ci, that is, rjit-Binomial(1, ci). The second augmented variable is vjit which has a Uniform(0,1) distribution. The sampling process is similar to Chapter One.Secondly, in Section four, we mitigate this problem by using the pairwise likelihood (Lindsay,1988)instead of the joint likelihood, which is the product of the bivariate likelihoods here where gr,s is the bivariate normal density function for ability vector (θr,θs)T and with Take the logarithm for both sides, and let then we get Let A be the stacked vector combining all pair-specific parameter vectorsλr,sFitting all possible pairwise models is equivalent to maximizing a function of the form pl(λ) Although each part in. equation (5) is maximized separately, its form (a joint log-likelihood replaced by a sum of log-likelihoods) is typically encountered within pseudo-likelihood theory (Arnold & Strauss,1991; Geys et al.,1999). Therefore, results from pseudo-likelihood theory can be used for inference forλ. Note that some parameters inλwill have a. single counterpart inλ, whereas other elements in A will have multiple counterparts in A. Therefore estimates for the parameters in A are obtained by taking averages over all pairs.In Section Five EM algorithm was used for pairwise likelihood. In E-step, if the expectation cannot be expressed in closed form, it must be approximated numerically. In pairwise likelihood maximization, the expectation step is a sum of double integrals. Double integrals are more efficiently evaluated by Gauss-Hermite quadrature. At last, from the simulation, it can be concluded that the parameter estimates and the standard deviations produced by our pairwise procedure are comparable with those obtained by Joint method.