Masanori Ichikawa

Application of the bootstrap methods in factor analysis

A Monte Carlo experiment is conducted to investigate the performance of the bootstrap methods in normal theory maximum likelihood factor analysis both when the distributional assumption is satisfied and unsatisfied. The parameters and their functions of interest include unrotated loadings, analytically rotated loadings, and unique variances. The results reveal that (a) bootstrap bias estimation performs sometimes poorly for factor loadings and nonstandardized unique variances; (b) bootstrap variance estimation performs well even when the distributional assumption is violated; (c) bootstrap confidence intervals based on the Studentized statistics are recommended; (d) if structural hypothesis about the population covariance matrix is taken into account then the bootstrap distribution of the normal theory likelihood ratio test statistic is close to the corresponding sampling distribution with slightly heavier right tail.

Bayesian Information Criterion and Selection of the Number of Factors in Factor Analysis Models

In maximum likelihood exploratory factor analysis, the estimates of unique variances can often turn out to be zero or negative, which makes no sense from a statistical point of view. In order to overcome this diculty, we employ a Bayesian approach by specifying a prior distribution for the variances of unique factors. The factor analysis model is estimated by EM algorithm, for which we provide the expectation and maximization steps within a general framework of EM algorithms. Crucial issues in Bayesian factor analysis model are the choice of adjusted parameters including the number of factors and also the hyper-parameters for the prior distribution. The choice of these parameters can be viewed as a model selection and evaluation problem. We derive a model selection criterion for evaluating a Bayesian factor analysis model. Monte Carlo simulations are conducted to investigate the eectiveness of the proposed procedure. A real data example is also given to illustrate our procedure. We observe that our modeling procedure prevents the occurrence of improper solutions and also chooses the appropriate number of factors objectively.

A new approach based on a covariance structure model to source apportionment of indoor fine particles in Tokyo

Abstract A new receptor model is developed to estimate percent source contribution of indoor fine particles. Under the model, the covariance matrix of the elemental concentrations has a linear structure expressed as a weighted sum of known matrices and the percent contribution of each source is given by a function of the parameters involved. Given data, the percent source contributions are estimated using the information provided by the sample covariance matrix. Our approach makes it possible to selecta a set of sources that “best” describes the variation of the data. The standard errors of the percent source contribution estimates are evaluated by employing the bootstrap method. The proposed methods is applied to indoor air pollution data collected in the Tokyo Metropolitan area. Cigarette smoke, heavy oil combustion and diesel-powered automobiles were identified as important sources for both winter and summer data.

Model evaluation and information criteria in covariance structure analysis

The normal theory based likelihood ratio test statistic, often used to evaluate the goodness of fit of a model in covariance structure analysis, has several problems that may make it inflexible and sometimes unreliable. We introduce criteria for evaluating the models in covariance structure analysis from an information-theoretic point of view. The basic idea behind the present approach is to express a model in the form of a probability distribution and then evaluate the model by the Kullback-Leibler information. We consider four types of information criteria that are constructed by correcting the upward bias of the sample based log-likelihood as a natural estimate of the Kullback-Leibler information or, equivalently, the expected log-likelihood. Monte Carlo experiments are conducted to examine the performance of the information criteria under various sample sizes and degrees of deviations from both structural and distributional assumptions. We show that the variance of the bootstrap bias estimate caused by bootstrap simulation can be considerably reduced without any analytical derivations.

New lower and upper bounds for communality in factor analysis

We derive several relationships between communalities and the eigenvalues for ap ×p correlation matrix σ under the usual factor analysis model. For suitable choices ofj, λj(σ), where λj(σ) is thej-th largest eigenvalue of σ, provides either a lower or an upper bound to the communalities for some of the variables. We show that for at least one variable, 1 - λp (σ) improves on the use of squared mulitiple correlation coefficient as a lower bound.

Constructing second-order accurate confidence intervals for communalities in factor analysis

In an effort to find accurate alternatives to the usual confidence intervals based on normal approximations, this paper compares four methods of generating second-order accurate confidence intervals for non-standardized and standardized communalities in exploratory factor analysis under the normality assumption. The methods to generate the intervals employ, respectively, the Cornish-Fisher expansion and the approximate bootstrap confidence (ABC), and the bootstrap-t and the bias-corrected and accelerated bootstrap (BC(a)). The former two are analytical and the latter two are numerical. Explicit expressions of the asymptotic bias and skewness of the communality estimators, used in the analytical methods, are derived. A Monte Carlo experiment reveals that the performance of central intervals based on normal approximations is a consequence of imbalance of miscoverage on the left- and right-hand sides. The second-order accurate intervals do not require symmetry around the point estimates of the usual intervals and achieve better balance, even when the sample size is not large. The behaviours of the second-order accurate intervals were similar to each other, particularly for large sample sizes, and no method performed consistently better than the others.

Asymptotic Distributions of the Estimators of Communalities in Factor Analysis.

Asymptotic distributions of the estimators of communalities are derived for the maximum likelihood method in factor analysis. It is shown that the common practice of equating the asymptotic standard error of the communality estimate to the unique variance estimate is correct for standardized communality but not correct for unstandardized communality. In a Monte Carlo simulation the accuracy of the normal approximation to the distributions of the estimators are assessed when the sample size is 150 or 300.