多维项目反应模型的参数估计
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摘要
统计在教育、心理学研究中的重要应用价值已达成共识,尤其是近年来项目反应理论显示出其绝对优势,已成为教育与心理研究领域的热点问题之一.以项目反应理论为代表的考试理论的研究取得了很大进展,具体表现在三个方面,即出现了多维项目反应理论,非参数项目反应理论,以及认知诊断理论。这些理论对考试的实践产生了深远的影响。本文主要研究多维项目反应理论。
本文主要利用贝叶斯统计理论和拟似然理论的相关工具。首先,基于数据扩充技术,采用Gibbs抽样方法,给出多维三参数Logistic模型的贝叶斯后验估计。其次,在此基础上,针对缺失反应数据问题,对不同的缺失过程建模。当缺失机制不可忽略时,引入缺失模型,我们用一个二级评分反应模型来拟合缺失指标,从而减小了由于忽略缺失数据估计参数时产生的偏差。然后基于数据扩充技术的Gibbs抽样方法,同时给出多维等级反应数据模型和缺失指标模型的后验估计。再次,对纵向反应数据的建模。在教育与心理研究中,通常要通过对被试在几个时间点的反应值进行纵向分析,以得到被试相关能力随时问的变化情况。例如,如果要研究学生对数学知识掌握的进展情况,那么我们就在几个时间点对学生进行一次或多次数学测验。纵向设计中,常常假定学生的能力是随时间变化的,且被试能力之间具有相关性,因此采用联合建模(Joint modeling)方法进行拟合时,时刻点越多,能力维数越高,估计起来越困难。为了解决这一问题,我们给出两种解决方法,一种是MCMC估计法,此外基于拟似然的理论,采用成对建模(Pairwise Modeling)的方法来拟合纵向数据,无需考虑能力的维数问题,对于各个时刻点的能力之间的相关程度,可以直接给出估计。另外,该模型还可以用于各级学校的纵向教学质量评估或同类院校之间的阶段性教学横向比较。
Estimation of the Multidimensional Item Response Model
At 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),-∞ We introduce two independent random variables corresponding to each data, point yij as follows. The first is a Bernoulli random variable, denoted by rij with success probability ci. The second augmented variable is uij, which has a Uniform(0,1) distri-bution. The response yij restricts the augmented random variables rij and uij so that model (1) is obtained as a consequence. Thus we set it is easy to see that (3) follows from (4) by taking expectations.
With the introduction of r和u, the full conditional distribution of each parameter is given by where whereθj(κ) is the vectorθj withθjκdeleted. where In general, from the simulation it can be concluded that the parameter estimates and the standard errors produced by our procedure are comparable with those obtained by Bilog-MG for the present data set.
·Chapter Four mainly discussed Bayesian estimation of the multidimensional graded response model with nonignorable missing data. In educational measurement, it often happens that item nonresponses are nonignorable missing data. An example, is a test with a time limit condition, where examinees of lower ability do not reach the items at the end. Thus, the pattern of missingness in this case depends on the ability that is measured and hence the missing data are not generally ignorable. However, if the given data set contains missing observations, the mechanism causing this missingness can be characterized by its variety of randomness (Rubin,1976) as missing at random (MAR) and missing completely at random (MCAR).
Supposeθandξare the parameters of the observed data and the missing data process, respectively, and D is the missing data indicator matrix with elements diκ= 1 if a realizationχiκwas observed and diκ= 0 ifχiκwas missing for person i and itemκ. Following Rubin's definition, missing data are said to be MAR if the probability of D given the observed dataχobs, missing data,χmis, some parameterζand observed covariates y does not depend on the missing data,χmis, that is, if Furthermore, the parametersζandθare distinct if there are no functional dependen-cies, that is, restrictions on the parameter space (frequentist version) or if the prior distributions ofθandζare independent (Bayesian case). If these two components (MAR and distinctness) are satisfied then the missing data are said to be ignorable, otherwise the missing data are nonignorable. If the missing data cannot be ignored, a concurrent probability model must be defined for the observed and missing data, and inferences are made averaging over the missing data. In this Chapter we use a dichotomous IRT model to fit the missing indicator. The model is closely related to the models proposed by Moustaki & O'Muircheartaigh (2000) (see also, Moustaki & Knott,2000; Bernaards & Sijtsma,1999,2000; Holman & Glas,2005).
SupposeθandζC are the person's latent variables related to the observed data and missing data, with densities g1(θ) and g2(ζ) respectively. Let p(χiκ\diκ,θi,ακ,βκ) be the measurement model for the observed data. It is the probability of the response (observed) variable conditioned on the latent variable of the observed data, the design variable (missing data indicator) and item parameters. We consider the multidimen-sional graded response model (MGRM), where i=1,…, N, g=1,…, mκandκ=1,…, K. Here we useΦκg* (θi) to denote the boundary probability for examinee i having a score larger or equal to g on item k, and the boundary curve is given by where g=0,1,…, mκ,Φk0*(θi)= 1, andΦκ,mk+1* (θi)=0.
Let p(diκ\ζi,γκ,δκ) be the measurement model for the missing data indicator. It is the probability of the design variable conditioned on the latent variable and item parameters for missing data process. we use a Q2-dimensional IRT model proposed by Reckase (1997) and Ackerman (1996). This model, which is in logistic form, has the probability of an observation given by
We will use two models in our estimation procedure. The first model, which we call the MAR model, is given in likelihood form as It is the model that ignores the missing data process, and we ignore the model for the missing-data process p(diκ= 1\ζi,γκ,δκ)。in the estimation process. The latent variables for the observed data and the missing data process are not related in the MAR model. The second model, which we call the nonignorable model (NONMAR), is the model where missing data process is included in the estimation process. In this model, the latent variables for both the observed and missing data, processθandζare correlated by∑. This model is written in likelihood form as where g(·) is the density ofθi, andζi. It is assumed to follow a multivariate normal distribution with mean vectorμand variance-covariance∑which can be used to index the extent to which ignorability holds. Expressions (14) and (15) will be used in our Bayesian procedure to make inference on the estimation of the model parameters. Beguin and Glas (2001) give conditions for the identification of the model. From their conclusions it follows that the basis of the two-dimensional latent space can always be transformed in such a way that both the model for the observations and the model for the missing data indicators depend on the same two latent variables. Therefore, the latent parameters of the two models are not distinct. In other words, within the framework of the model they are functionally dependent.
In this Chapter, a Gibbs sampling procedure will be used to sample the posterior distributions of interest for the MGRM and multidimensional 2PLM. We augments the data by introducing independent random variables Uiκand Viκ, each having a Uniform(0,1) distribution. It is assumed that For the missing indicator diκ, we define Viκ≤φiκif diκ=1 and Viκ>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 points
Let 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,s
Fitting 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.
本文主要利用贝叶斯统计理论和拟似然理论的相关工具。首先,基于数据扩充技术,采用Gibbs抽样方法,给出多维三参数Logistic模型的贝叶斯后验估计。其次,在此基础上,针对缺失反应数据问题,对不同的缺失过程建模。当缺失机制不可忽略时,引入缺失模型,我们用一个二级评分反应模型来拟合缺失指标,从而减小了由于忽略缺失数据估计参数时产生的偏差。然后基于数据扩充技术的Gibbs抽样方法,同时给出多维等级反应数据模型和缺失指标模型的后验估计。再次,对纵向反应数据的建模。在教育与心理研究中,通常要通过对被试在几个时间点的反应值进行纵向分析,以得到被试相关能力随时问的变化情况。例如,如果要研究学生对数学知识掌握的进展情况,那么我们就在几个时间点对学生进行一次或多次数学测验。纵向设计中,常常假定学生的能力是随时间变化的,且被试能力之间具有相关性,因此采用联合建模(Joint modeling)方法进行拟合时,时刻点越多,能力维数越高,估计起来越困难。为了解决这一问题,我们给出两种解决方法,一种是MCMC估计法,此外基于拟似然的理论,采用成对建模(Pairwise Modeling)的方法来拟合纵向数据,无需考虑能力的维数问题,对于各个时刻点的能力之间的相关程度,可以直接给出估计。另外,该模型还可以用于各级学校的纵向教学质量评估或同类院校之间的阶段性教学横向比较。
Estimation of the Multidimensional Item Response Model
At 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),-∞
With the introduction of r和u, the full conditional distribution of each parameter is given by where whereθj(κ) is the vectorθj withθjκdeleted. where In general, from the simulation it can be concluded that the parameter estimates and the standard errors produced by our procedure are comparable with those obtained by Bilog-MG for the present data set.
·Chapter Four mainly discussed Bayesian estimation of the multidimensional graded response model with nonignorable missing data. In educational measurement, it often happens that item nonresponses are nonignorable missing data. An example, is a test with a time limit condition, where examinees of lower ability do not reach the items at the end. Thus, the pattern of missingness in this case depends on the ability that is measured and hence the missing data are not generally ignorable. However, if the given data set contains missing observations, the mechanism causing this missingness can be characterized by its variety of randomness (Rubin,1976) as missing at random (MAR) and missing completely at random (MCAR).
Supposeθandξare the parameters of the observed data and the missing data process, respectively, and D is the missing data indicator matrix with elements diκ= 1 if a realizationχiκwas observed and diκ= 0 ifχiκwas missing for person i and itemκ. Following Rubin's definition, missing data are said to be MAR if the probability of D given the observed dataχobs, missing data,χmis, some parameterζand observed covariates y does not depend on the missing data,χmis, that is, if Furthermore, the parametersζandθare distinct if there are no functional dependen-cies, that is, restrictions on the parameter space (frequentist version) or if the prior distributions ofθandζare independent (Bayesian case). If these two components (MAR and distinctness) are satisfied then the missing data are said to be ignorable, otherwise the missing data are nonignorable. If the missing data cannot be ignored, a concurrent probability model must be defined for the observed and missing data, and inferences are made averaging over the missing data. In this Chapter we use a dichotomous IRT model to fit the missing indicator. The model is closely related to the models proposed by Moustaki & O'Muircheartaigh (2000) (see also, Moustaki & Knott,2000; Bernaards & Sijtsma,1999,2000; Holman & Glas,2005).
SupposeθandζC are the person's latent variables related to the observed data and missing data, with densities g1(θ) and g2(ζ) respectively. Let p(χiκ\diκ,θi,ακ,βκ) be the measurement model for the observed data. It is the probability of the response (observed) variable conditioned on the latent variable of the observed data, the design variable (missing data indicator) and item parameters. We consider the multidimen-sional graded response model (MGRM), where i=1,…, N, g=1,…, mκandκ=1,…, K. Here we useΦκg* (θi) to denote the boundary probability for examinee i having a score larger or equal to g on item k, and the boundary curve is given by where g=0,1,…, mκ,Φk0*(θi)= 1, andΦκ,mk+1* (θi)=0.
Let p(diκ\ζi,γκ,δκ) be the measurement model for the missing data indicator. It is the probability of the design variable conditioned on the latent variable and item parameters for missing data process. we use a Q2-dimensional IRT model proposed by Reckase (1997) and Ackerman (1996). This model, which is in logistic form, has the probability of an observation given by
We will use two models in our estimation procedure. The first model, which we call the MAR model, is given in likelihood form as It is the model that ignores the missing data process, and we ignore the model for the missing-data process p(diκ= 1\ζi,γκ,δκ)。in the estimation process. The latent variables for the observed data and the missing data process are not related in the MAR model. The second model, which we call the nonignorable model (NONMAR), is the model where missing data process is included in the estimation process. In this model, the latent variables for both the observed and missing data, processθandζare correlated by∑. This model is written in likelihood form as where g(·) is the density ofθi, andζi. It is assumed to follow a multivariate normal distribution with mean vectorμand variance-covariance∑which can be used to index the extent to which ignorability holds. Expressions (14) and (15) will be used in our Bayesian procedure to make inference on the estimation of the model parameters. Beguin and Glas (2001) give conditions for the identification of the model. From their conclusions it follows that the basis of the two-dimensional latent space can always be transformed in such a way that both the model for the observations and the model for the missing data indicators depend on the same two latent variables. Therefore, the latent parameters of the two models are not distinct. In other words, within the framework of the model they are functionally dependent.
In this Chapter, a Gibbs sampling procedure will be used to sample the posterior distributions of interest for the MGRM and multidimensional 2PLM. We augments the data by introducing independent random variables Uiκand Viκ, each having a Uniform(0,1) distribution. It is assumed that For the missing indicator diκ, we define Viκ≤φiκif diκ=1 and Viκ>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 points
Let 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,s
Fitting 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.
引文
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