A generalized rasch model for manifest predictors

A logistic regression model is suggested for estimating the relation between a set of manifest predictors and a latent trait assumed to be measured by a set ofk dichotomous items. Usually the estimated subject parameters of latent trait models are biased, especially for short tests. Therefore, the relation between a latent trait and a set of predictors should not be estimated with a regression model in which the estimated subject parameters are used as a dependent variable. Direct estimation of the relation between the latent trait and one or more independent variables is suggested instead. Estimation methods and test statistics for the Rasch model are discussed and the model is illustrated with simulated and empirical data.     

Improper solutions in the analysis of covariance structures: Their interpretability and a comparison of alternate respecifications

A Monte Carlo approach was employed to investigate the interpretability of improper solutions caused by sampling error in maximum likelihood confirmatory factor analysis. Four models were studied with two sample sizes. Of the overall goodness-of-fit indices provided by the LISREL VI program significant differences between improper and proper solutions were found only for the root mean square residual. As expected, indicators of the factor on which the negative uniqueness estimate occurred had biased loadings, and the correlations of its factor with other factors were also biased. In contrast, the loadings of indicators on other factors and those factor intercorrelations did not have any bias of practical significance. For initial solutions with one negative uniqueness estimate, three respecifications were studied: Fix the uniqueness at .00, fix it at .20, or constrain the domain of the solution to be proper. For alternate, respecified solutions that were converged and proper, the constrained solutions and uniqueness fixed at .00 solutions were equivalent. The mean goodness-of-fit and pattern coefficient values for the original improper solutions were not meaningfully different from those obtained under the constrained and uniqueness fixed at .00 respecifications.This investigation was supported in part by a grant from the Baylor University Research Committee (#018-F83-URC). The authors gratefully acknowledge the comments and suggestions of Claes Fornell and Roger E. Kirk, and the assistance of Timothy J. Vance with the analysis.     

Maximum likelihood estimation of multivariate polyserial and polychoric correlation coefficients

The method of finding the maximum likelihood estimates of the parameters in a multivariate normal model with some of the component variables observable only in polytomous form is developed. The main stratagem used is a reparameterization which converts the corresponding log likelihood function to an easily handled one. The maximum likelihood estimates are found by a Fletcher-Powell algorithm, and their standard error estimates are obtained from the information matrix. When the dimension of the random vector observable only in polytomous form is large, obtaining the maximum likelihood estimates is computationally rather labor expensive. Therefore, a more efficient method, the partition maximum likelihood method, is proposed. These estimation methods are demonstrated by real and simulated data, and are compared by means of a simulation study.     

Maximum likelihood estimation of polyserial correlations

This paper considers a multivariate normal model with one of the component variables observable only in polytomous form. The maximum likelihood approach is used for estimation of the parameters in the model. The Newton-Raphson algorithm is implemented to obtain the solution of the problem. Examples based on real and simulated data are reported.The research of the first author was supported in part by a research grant (DA01070) from the US Public Health Service. We are indebted to the referees and the editor for some very valuable comments and suggestions.     

A new estimator of the uniqueness in factor analysis

A closed form estimator of the uniqueness (unique variance) in factor analysis is proposed. It has analytically desirable properties—consistency, asymptotic normality and scale invariance. The estimation procedure is given through the application to the two sets of Emmett's data and Holzinger and Swineford's data. The new estimator is shown to lead to values rather close to the maximum likelihood estimator.     

On the estimation of parameters in latent structure analysis

In this note, we describe the iterative procedure introduced earlier by Goodman to calculate the maximum likelihood estimates of the parameters in latent structure analysis, and we provide here a simple and direct proof of the fact that the parameter estimates obtained with the iterative procedure cannot lie outside the allowed interval. Formann recently stated that Goodman's algorithm can yield parameter estimates that lie outside the allowed interval, and we prove in the present note that Formann's contention is incorrect.This research was supported in part by Research Contract No. NSF SOC 76-80389 from the Division of the Social Sciences of the National Science Foundation. The author is indebted to C. C. Clogg for helpful comments and for the numerical results reported here (see, e.g., Table 1).     

Estimating the standard errors of rotated factor loadings by jackknifing

The jackknife by groups and modifications of the jackknife by groups are used to estimate standard errors of rotated factor loadings for selected populations in common factor model maximum likelihood factor analysis. Simulations are performed in whicht-statistics based upon these jackknife estimates of the standard errors are computed. The validity of thet-statistics and their associated confidence intervals is assessed. Methods are given through which the computational efficiency of the jackknife may be greatly enhanced in the factor analysis model.Computing assistance was obtained from the Health Sciences Computing Facility, UCLA, sponsored by NIH Special Research Resources Grant RR-3.The author wishes to thank his doctoral committee co-chairmen, Drs James W. Frane and Robert I. Jennrich, UCLA, for their contributions to this research.     

A feasible method for standard errors of estimate in maximum likelihood factor analysis

A jackknife-like procedure is developed for producing standard errors of estimate in maximum likelihood factor analysis. Unlike earlier methods based on information theory, the procedure developed is computationally feasible on larger problems. Unlike earlier methods based on the jackknife, the present procedure is not plagued by the factor alignment problem, the Heywood case problem, or the necessity to jackknife by groups. Standard errors may be produced for rotated and unrotated loading estimates using either orthogonal or oblique rotation as well as for estimates of unique factor variances and common factor correlations. The total cost for larger problems is a small multiple of the square of the number of variables times the number of observations used in the analysis. Examples are given to demonstrate the feasibility of the method.The research done by R. I. Jennrich was supported in part by NSF Grant MCS 77-02121. The research done by D. B. Clarkson was supported in part by NSERC Grant A3109.     

Likelihood methods for testing group problem solving models with censored data

Problem solving models relating individual and group solution times under time limit censoring are presented. Maximum likelihood estimates of parameters of the resulting censored distributions are derived and goodness of fit tests for the individual-group models are constructed. The methods are illustrated on data previously analyzed by Restle and Davis.This research was supported by grants from the Graduate School, University of Minnesota. Also, work of the second author was aided by a Single Quarter Leave granted by the Regents of the University of Minnesota.     

Confidence regions for multidimensional scaling analysis

Techniques are developed for surrounding each of the points in a multidimensional scaling solution with a region which will contain the population point with some level of confidence. Bayesian credibility regions are also discussed. A general theorem is proven which describes the asymptotic distribution of maximum likelihood estimates subject to identifiability constraints. This theorem is applied to a number of models to display asymptotic variance-covariance matrices for coordinate estimates under different rotational constraints. A technique is described for displaying Bayesian conditional credibility regions for any sample size.The research reported here was supported by grant number APA 320 to the author by the National Research Council of Canada.