Group-level variance estimates of zero often arise when fitting multilevel or hierarchical linear models, especially when the number of groups is small. For situations where zero variances are implausible a priori, we propose a maximum penalized likelihood approach to avoid such boundary estimates. This approach is equivalent to estimating variance parameters by their posterior mode, given a weakly informative prior distribution. By choosing the penalty from the log-gamma family with shape parameter greater than 1, we ensure that the estimated variance will be positive. We suggest a default log-gamma(2, λ) penalty with λ → 0, which ensures that the maximum penalized likelihood estimate is approximately one standard error from zero when the maximum likelihood estimate is zero, thus remaining consistent with the data while being nondegenerate. We also show that the maximum penalized likelihood estimator with this default penalty is a good approximation to the posterior median obtained under a noninformative prior.Our default method provides better estimates of model parameters and standard errors than the maximum likelihood or the restricted maximum likelihood estimators. The log-gamma family can also be used to convey substantive prior information. In either case-pure penalization or prior information-our recommended procedure gives nondegenerate estimates and in the limit coincides with maximum likelihood as the number of groups increases.

Diagnostic classification models have recently gained prominence in educational assessment, psychiatric evaluation, and many other disciplines. Central to the model specification is the so-called Q-matrix that provides a qualitative specification of the item-attribute relationship. In this paper, we develop theories on the identifiability for the Q-matrix under the DINA and the DINO models. We further propose an estimation procedure for the Q-matrix through the regularized maximum likelihood. The applicability of this procedure is not limited to the DINA or the DINO model and it can be applied to essentially all Q-matrix based diagnostic classification models. Simulation studies are conducted to illustrate its performance. Furthermore, two case studies are presented. The first case is a data set on fraction subtraction (educational application) and the second case is a subsample of the National Epidemiological Survey on Alcohol and Related Conditions concerning the social anxiety disorder (psychiatric application).

Iterative imputation, in which variables are imputed one at a time each given a model predicting from all the others, is a popular technique that can be convenient and flexible, as it replaces a potentially difficult multivariate modeling problem with relatively simple univariate regressions. In this paper, we begin to characterize the stationary distributions of iterative imputations and their statistical properties. More precisely, when the conditional models are compatible (defined in the text), we give a set of sufficient conditions under which the imputation distribution converges in total variation to the posterior distribution of a Bayesian model. When the conditional models are incompatible but are valid, we show that the combined imputation estimator is consistent. arXiv:1012.2902v2 [math.ST] 3 Apr 2012 imputation algorithms are not well understood. Even if, as we would prefer, the fitting of each imputation model and the imputations themselves are performed using conditional Bayesian inference, the stationary distribution of the algorithm (if it exists) does not in general correspond to Bayesian inference on any specified multivariate distribution. Key questions are: (1) Under what conditions does the algorithm converge to a stationary distribution? (2) What statistical properties does the procedure admit given that a unique stationary distribution exists?Regarding the first question, researchers have long known that the Markov chain may be nonrecurrent ("blowing up" to infinity or drifting like a nonstationary random walk), even if each of

We develop a latent variable selection method for multidimensional item response theory models. The proposed method identifies latent traits probed by items of a multidimensional test. Its basic strategy is to impose an [Formula: see text] penalty term to the log-likelihood. The computation is carried out by the expectation-maximization algorithm combined with the coordinate descent algorithm. Simulation studies show that the resulting estimator provides an effective way in correctly identifying the latent structures. The method is applied to a real dataset involving the Eysenck Personality Questionnaire.

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