Approximate distributions of the sample coefficient alpha under nonnormality as well as normality are derived by using the single- and two-term Edgeworth expansions up to the term of order 1/n. The case of the standardized coefficient alpha including the weights for the components of a test is also considered. From the numerical illustration with simulation using the normal and typical nonnormal distributions with different types/degrees of nonnormality, it is shown that the variances of the sample coefficient alpha under nonnormality can be grossly different from those under normality. The corresponding biases and skewnesses are shown to be negative under various conditions. The method of developing confidence intervals of the population coefficient alpha using the Cornish-Fisher expansion with sample cumulants is presented.
A fixed effect item response theory (IRT) model is developed for modeling group specific item parameters. Two applications are presented. The first application is that the proposed model can be used to detect whether a response mechanism is ignorable using the splitter item technique. The second application is the detection of differential item functioning. In the latter application, the fixed effect item parameters can model item parameter differences between groups. Simulation studies are presented to show the feasibility and performance of the method on both applications.
In maximum likelihood estimation of latent class models, it often occurs that one or more of the parameter estimates are on the boundary of the parameter space; that is, that estimated probabilities equal 0 (or 1) or, equivalently, that logit coefficients equal minus (or plus) infinity. This not only causes numerical problems in the computation of the variance-covariance matrix, it also makes the reported confidence intervals and significance tests for the parameters concerned meaningless. Boundary estimates can, however, easily be prevented by the use of prior distributions for the model parameters, yielding a Bayesian procedure called posterior mode or maximum a posteriori estimation. This approach is implemented in, for example, the Latent GOLD software packages for latent class analysis (Vermunt & Magidson, 2005). Little is, however, known about the quality of posterior mode estimates of the parameters of latent class models, nor about their sensitivity for the choice of the prior distribution. In this paper, we compare the quality of various types of posterior mode point and interval estimates for the parameters of latent class models with both the classical maximum likelihood estimates and the bootstrap estimates proposed by De Menezes (1999). Our simulation study shows that parameter estimates and standard errors obtained by the Bayesian approach are more reliable than the corresponding parameter estimates and standard errors obtained by maximum likelihood and parametric bootstrapping.
Ordinary least squares estimation is considered for fitting a factor analysis model to polychoric correlation matrices. A parametric bootstrap procedure is proposed for obtaining test statistics, standard error estimates, and confidence intervals associated with the OLS estimates. The adequacy of the proposed procedure is demonstrated using a simulation study.
We give a historical introduction to item response theory, which places the work of Thurstone, Lord, Guttman and Coombs in a present-day perspective. The general assumptions of modern item response theory, local independence and monotonicity of response functions, are discussed, followed by a general framework for estimating item response models. Six classes of well-known item response models and recent developments are discussed: (1) models for dichotomous item scores; (2) models for polytomous item scores; (3) nonparametric models; (4) unfolding models; (5) multidimensional models; and (6) models with restrictions on the parameters. Finally, it is noted that item response theory has evolved from unidimensional scaling of items and measurement of persons to data analysis tools for complicated research designs.