There are many approaches to formulate multiple correspondence analysis of multi-item categorical data. A lower-rank approximation approach gives the freedom for the oblique rotation of axes. In the current paper, we apply the oblique rotation of axes to a variable-by-dimension matrix to arrive at simple structure. This matrix is defined in either of two different manners, that is, by treating categories as variables or, alternatively, by treating items as variables. For each of these two options, the standardized inner products between dimensions and variables are used as the elements of the component structure matrix. We adopt a promax method for the oblique rotation. In this method, scores are rotated in such a way that the above matrix is matched with the target matrix obtained from the result of the varimax rotation.
Consider a case where cause-effect relationships between variables can be described as a causal diagram and the corresponding Gaussian linear structural equation model. In order to identify total effects in studies with an unobserved response variable, this paper proposes graphical criteria for selecting both covariates and variables caused by the response variable. The results enable us not only to judge from the graph structure whether a total effect can be expressed through the observed covariances, but also to provide its closed-form expression in case where its answer is affirmative. The graphical criteria of this paper are helpful to infer total effects when it is difficult to observe a response variable.
A new Bayesian method was proposed to obtain the optimal solution when the prior knowledge was given as a target factor pattern matrix. This method (Bayesian Procrustes Solution) chooses the best target matrix in terms of the posterior probability or Bayes factor when there are two or more possible target matrices. Then the posterior distributions of the relevant parameters are approximated by a sufficient large sample of parameters simulated by the Gibbs sampler. The estimates of the parameters are given as the means of these distributions. The simulation study demonstrated that this method has more robustness with respect to misspecification of the target matrix than the traditional method. Finally, the application of this method to the real data showed the validity of this method.
Multilevel modeling is often used in the social sciences for analyzing data that has a hierarchical structure, e.g., students nested within schools. In an earlier study, we investigated the performance of various prediction rules for predicting a future observable within a hierarchical data set (Afshartous & de Leeuw, 2004). We apply the multilevel prediction approach to the NELS:88 educational data in order to assess the predictive performance on a real data set; four candidate models are considered and predictions are evaluated via both cross-validation and bootstrapping methods. The goal is to develop model selection criteria that assess the predictive ability of candidate multilevel models. We also introduce two plots that 1) aid in visualizing the amount to which the multilevel model predictions are “shrunk” or translated from the OLS predictions, and 2) help identify if certain groups exist for which the predictions are particularly good or bad.
Various fit indices exist in structural equation models. Most of these indices are related to the noncentrality parameter (NCP) of the chi-square distribution that the involved test statistic is implicitly assumed to follow. Existing literature suggests that few statistics can be well approximated by chi-square distributions. The meaning of the NCP is not clear when the behavior of the statistic cannot be described by a chi-square distribution. In this paper we define a new measure of model misfit (MMM) as the difference between the expected values of a statistic under the alternative and null hypotheses. This definition does not need to assume that the population covariance matrix is in the vicinity of the proposed model, nor does it need for the test statistic to follow any distribution of a known form. The MMM does not necessarily equal the discrepancy between the model and the population covariance matrix as has been assumed in existing literature. Bootstrap approaches to estimating the MMM and a related quantity are developed. An algorithm for obtaining bootstrap confidence intervals of the MMM is constructed. Examples with practical data sets contrast several measures of model misfit. The quantile-quantile plot is used to illustrate the unrealistic nature of chi-square distribution assumptions under either the null or an alternative hypothesis in practice.