A constrained spline estimator of a hazard function

A constrained quadratic spline is proposed as an estimator of the hazard function of a random variable. A maximum penalized likelihood procedure is used to fit the estimator to a sample of psychological response times. The results of a small simulation study suggest that, with a sample size of 500, the procedure may provide a reasonably precise estimate of the shape of a hazard function.This research was conducted under the auspices of the Naval Postgraduate School during the author's sabbatical from Vanderbilt University and was partially supported by the Navy Personnal Research and Development Center. The author wishes to thank Jules Borack, Richard Sorenson, and two anonymous reviewers for a number of useful and stimulating comments on the work reported here. Thanks are also due to David Kohfeld for providing the data which were used in the empirical example.     

A gauss-newton algorithm for exploratory factor analysis

It is shown that the scoring algorithm for maximum likelihood estimation in exploratory factor analysis can be developed in a way that is many times more efficient than a direct development based on information matrices and score vectors. The algorithm offers a simple alternative to current algorithms and when used in one-step mode provides the simplest and fastest method presently available for moving from consistent to efficient estimates. Perhaps of greater importance is its potential for extension to the confirmatory model. The algorithm is developed as a Gauss-Newton algorithm to facilitate its application to generalized least squares and to maximum likelihood estimation.This research was supported by NSF Grant MCS-8301587.     

A latent trait model for dichotomous choice data

The PARELLA model is a probabilistic parallelogram model that can be used for the measurement of latent attitudes or latent preferences. The data analyzed are the dichotomous responses of persons to stimuli, with a one (zero) indicating agreement (disagreement) with the content of the stimulus. The model provides a unidimensional representation of persons and items. The response probabilities are a function of the distance between person and stimulus: the smaller the distance, the larger the probability that a person will agree with the content of the stimulus. An estimation procedure based on expectation maximization and marginal maximum likelihood is developed and the quality of the resulting parameter estimates evaluated.I gratefully acknowledge Ivo Molenaar and Wijbrandt van Schuur for their advice and encouragement during the course of the investigation, Derk-Jan Kiewiet who constructed the program for the ML estimator for the person parameter and Anne Boomsma, Wendy Post, Tom Snijders, and David Thissen for their comments on smaller aspects of the investigation.     

A loss function for alpha factor analysis

Most of the factor solutions can be got by minimizing a corresponding loss function. However, up to now, a loss function for the alpha factor analysis (AFA) has not been known. The present paper establishes such a loss function for the AFA. Some analogies to the maximum likelihood factor analysis are discussed.The author is greatly indebted to Prof. Henry F. Kaiser (University of California, Berkeley) for his kind encouragement. He is also indebted to an anonymous referee ofPsychometrika for having confronted him with the problem in 1977. Financial support by the Wiener Hochschuljubiläumsstiftung is gratefully acknowledged.     

EM algorithms for ML factor analysis

The details of EM algorithms for maximum likelihood factor analysis are presented for both the exploratory and confirmatory models. The algorithm is essentially the same for both cases and involves only simple least squares regression operations; the largest matrix inversion required is for aq ×q symmetric matrix whereq is the matrix of factors. The example that is used demonstrates that the likelihood for the factor analysis model may have multiple modes that are not simply rotations of each other; such behavior should concern users of maximum likelihood factor analysis and certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation.     

Some critical observations of the test information function as a measure of local accuracy in ability estimation

The test information function serves important roles in latent trait models and in their applications. Among others, it has been used as the measure of accuracy in ability estimation. A question arises, however, if the test information function is accurate enough for all meaningful levels of ability relative to the test, especially when the number of test items is relatively small (e.g., less than 50). In the present paper, using the constant information model and constant amounts of test information for a finite interval of ability, simulated data were produced for eight different levels of ability and for twenty different numbers of test items ranging between 10 and 200. Analyses of these data suggest that it is desirable to consider some modification of the test information function when it is used as the measure of accuracy in ability estimation.     

On the estimation of polychoric correlations and their asymptotic covariance matrix

A general theory for parametric inference in contingency tables is outlined. Estimation of polychoric correlations is seen as a special case of this theory. The asymptotic covariance matrix of the estimated polychoric correlations is derived for the case when the thresholds are estimated from the univariate marginals and the polychoric correlations are estimated from the bivariate marginals for given thresholds. Computational aspects are also discussed.The research was supported by the Swedish Council for Research in the Humanities and Social Sciences (HSFR) under the programMultivariate Statistical Analysis. The author thanks a reviewer for pointing out an error in the original version of the paper.     

A new proof for monotone likelihood ratio for the sum of independent bernoulli random variables

By use of an inequality of Marcus and Lopes for elementary symmetric functions, a new proof is presented for the following result by Ghurye and Wallace: Given that the independent random variablesX _j are Bernoulli with success probabilityp _j () strictly between 0 and 1 and nondecreasing in, the sum X _j has monotone likelihood ratio.     

Full-information item bi-factor analysis

A plausibles-factor solution for many types of psychological and educational tests is one that exhibits a general factor ands − 1 group or method related factors. The bi-factor solution results from the constraint that each item has a nonzero loading on the primary dimension and at most one of thes − 1 group factors. This paper derives a bi-factor item-response model for binary response data. In marginal maximum likelihood estimation of item parameters, the bi-factor restriction leads to a major simplification of likelihood equations and (a) permits analysis of models with large numbers of group factors; (b) permits conditional dependence within identified subsets of items; and (c) provides more parsimonious factor solutions than an unrestricted full-information item factor analysis in some cases. Supported by the Cognitive Science Program, Office of Naval Research, Under grant #N00014-89-J-1104. We would like to thank Darrell Bock for several helpful suggestions.