Latent Variable Selection In Multidimensional Item Response Theory Models

In the fields of educational assessment and psychometric measurement,researchers usually use tests to evaluate individuals’ latent traits or latent abilities.The responses of subjects to items are often governed by multiple latent traits.Multidimensional item response theory（MIRT）serves as a theoretical framework for describing the interaction between multiple latent traits and responses to test items.Among researches on MIRT,it is a concern to determine the relationships between items and multiple latent traits,which is essentially the problem of estimating the loading structure in item factor analysis（IFA）.In nature,the determination of non-zero loadings is to find the subset of latent traits associated with a specific item and thus can be regarded as the latent variable selection problem in statistics.To obtain the sparse loading structure with better interpretation,Sun et al.（2016）^[1] proposed the EM-based L₁-penalized likelihood method（EML1）.Compared with the traditional exploratory IFA methods,the L₁-penalized likelihood method is able to obtain the sparse loadings more objectively.However,this method treats the covariance of latent traits as known,and is inefficient in computation.Thus,in this thesis,the L₁-penalized likelihood method is improved and the L₀-penalized likelihood method is used to study the latent variable selection problem for MIRT models and a series of algorithms are provided.The main contributions of this thesis are summarized as follows.1.For the problem of latent variable selection in the multidimensional two-parameter logistic（M2PL）model,we propose an improved EML1（IEML1）method.IEML1 treats the covariance matrix of latent traits as unknown and updates it simultaneously with item parameters in EM iterations.In terms of calculation,IEML1 adopts the numerical quadrature with fixed grid points to approximate the posterior expectation in the E-step,and then obtains a new weighted L₁-penalized log-likelihood based on the constructed artificial data.Therefore,the computational complexity of the coordinate descent algorithm in the M-step of IEML1 is significantly reduced compared with EML1.Simulation studies show that IEML1 performs better than the two-stage method and the exploratory IFA method in terms of both latent variable selection and parameter estimation.2.To avoid tuning penalty parameters in the L₁-penalized likelihood method,we consider the L₀-penalized likelihood method to select latent variables for M2PL models.To be specific,the expectation model selection（EMS）algorithm is used to minimize the observed Bayesian information criterion（BIC）.Under mild regularity conditions,we prove the numerical convergence of the EMS algorithm for model selection in the presence of missing data,and verify its convergence in the latent variable selection for M2PL models.Under the local independence assumption of MIRT models,EMS does not need to enumerate all possible models in each iteration,but only calculates all possible sub-models for each item.Therefore,for the case with no more than 5 latent traits,EMS can efficiently select models.Simulation studies show that the EMS outperforms EML1 in terms of latent variable selection,parameter estimation and running time.3.To improve the computational efficiency of EMS,we propose a generalized expectation model selection（GEMS）algorithm to address the model selection problem in the presence of missing data.Under mild regularity conditions,we prove the convergence of GEMS,which implies the convergence of EMS.For the latent variable selection in M2PL models,we skillfully implement the GEMS algorithm and verify its convergence.To further reduce the complexity of MS-step in GEMS,we propose the stepwise GEMS（StepGEMS）algorithm.Simulation studies show that the GEMS and StepGEMS run faster than EMS in general,and are superior to the EML1 method and the exploratory IFA combined with rotation in terms of latent variable selection and parameter estimation.