Robust nonorthogonal analyses revisited: An update based on trimmed means

Three approaches to the analysis of main and interaction effect hypotheses in nonorthogonal designs were compared in a 2×2 design for data that was neither normal in form nor equal in variance. The approaches involved either least squares or robust estimators of central tendency and variability and/or a test statistic that either pools or does not pool sources of variance. Specifically, we compared the ANOVA F test which used trimmed means and Winsorized variances, the Welch-James test with the usual least squares estimators for central tendency and variability and the Welch-James test using trimmed means and Winsorized variances. As hypothesized, we found that the latter approach provided excellent Type I error control, whereas the former two did not.Financial support for this research was provided by grants to the first author from the National Sciences and Engineering Research Council of Canada (#OGP0015855) and the Social Sciences and Humanities Research Council (#410-95-0006). The authors would like to express their appreciation to the Associate Editor as well as the reviewers who provided valuable comments on an earlier version of this paper.     

An investigation of hierarchical Bayes procedures in item response theory

Hierarchical Bayes procedures for the two-parameter logistic item response model were compared for estimating item and ability parameters. Simulated data sets were analyzed via two joint and two marginal Bayesian estimation procedures. The marginal Bayesian estimation procedures yielded consistently smaller root mean square differences than the joint Bayesian estimation procedures for item and ability estimates. As the sample size and test length increased, the four Bayes procedures yielded essentially the same result.The authors wish to thank the Editor and anonymous reviewers for their insightful comments and suggestions.     

Mueller-Lyer illusion and size estimation performance in schizophrenics compared to normal controls

A size estimation (SE) paradigm and the Mueller-Lyer (ML) illusion were used to examine perceptual disturbances in schizophrenics. 35 reliably diagnosed (DSM III-R) schizophrenics were compared to 20 subjects with no history of psychiatric illness. Perceptual distortions found in previous studies of schizophrenics were only to a certain extent confirmed in the present results. More overestimators were found among the schizophrenics than among the normals on the SE task. The schizophrenics, first of all the chronic patients, also proved to be more prone to the Mueller-Lyer illusion. A reason why the very clear differences between schizophrenics and normals found in previous examinations were not confirmed in the present study, might be that a reliable diagnostic instrument was for the first time used in this kind of study.     

Binary action based estimation of propensities

An observer is to make inference statements about a quantityp, called apropensity and bounded between 0 and 1, based on the observation thatp does or does not exceed a constantc. The propensityp may have an interpretation as a proportion, as a long-run relative frequency, or as a personal probability held by some subject. Applications in medicine, engineering, political science, and, most especially, human decision making are indicated. Bayes solutions for the observer are obtained based on prior distributions in the mixture of beta distribution family; these are then specialized to power-function prior distributions. Inference about logp and log odds is considered. Multiple-action problems are considered in which the focus of inference shifts to theprocess generating the propensitiesp, both in the case of a process parameter known to the subject and unknown. Empirical Bayes techniques are developed for observer inference aboutc when is known to the subject. A Bayes rule, a minimax rule and a beta-minimax rule are constructed for the subject when he is uncertain about.This research was partially supported by the Defense Advanced Research Projects Agency of the Department of Defense and was monitored by ONR under Contract No. N00014-77-C-0095. Any opinions, findings, conclusions or recommendations expressed herein are those of the author and do not necessarily reflect the views of the Defense Advanced Research Projects Agency, the Office of Naval Research, or Carnegie-Mellon University.     

Relating tests given to different samples

Suppose a collection of standard tests is given to all subjects in a random sample, but a different new test is given to each group of subjects in nonoverlapping subsamples. A simple method is developed for displaying the information that the data set contains about the correlational structure of the new tests. This is possible to some extent, even though each subject takes only one new test. The method uses plausible values of the partial correlations among the new tests given the standard tests in order to generate plausible simple correlations among the new tests and plausible multiple correlations between composites of the new tests and the standard tests. The real data example included suggests that the method can be useful in practical problems.     

Estimation in the power law

Most psychophysicists neglect to consider how error should be characterized in applications of the power law. Failures of the power law to agree with certain theoretical predictions may be due to bias of parameter estimates. A power law with lognormal product structure is proposed and approximately unbiased parameter estimates are given for several common estimation situations.     

Grading distractor-identification tests

On distractor-identification tests students mark as many distractors as possible on each test item. A grading scale is developed for this type testing. The scale is optimal in that it is the unique scale giving an unbiased estimate of the student's true score, i.e., the score that would result if no guessing occurred. If the test is administered as a usual multiple choice test and graded using the usual correction for guessing scale, the expected item score is the same as for the distractor-identification testing using the optimal grading scale. However, the variance of the item score is shown to be less for distractor-identification testing than for usual multiple choice testing under certain conditions.     

The bias function of the maximum likelihood estimate of ability for the dichotomous response level

Samejima has recently given an approximation for the bias function for the maximum likelihood estimate of the latent trait in the general case where item responses are discrete, generalizing Lord's bias function in the three-parameter logistic model for the dichotomous response level. In the present paper, observations are made about the behavior of this bias function for the dichotomous response level in general, and also with respect to several widely used mathematical models. Some empirical examples are given.     

An approximation for the bias function of the maximum likelihood estimate of a latent variable for the general case where the item responses are discrete

Lord developed an approximation for the bias function for the maximum likelihood estimate in the context of the three-parameter logistic model. Using Taylor's expansion of the likelihood equation, he obtained an equation that includes the conditional expectation, given true ability, of the discrepancy between the maximum likelihood estimate and true ability. All terms of orders higher thann ^?1 are ignored wheren indicates the number of items. Lord assumed that all item and individual parameters are bounded, all item parameters are known or well-estimated, and the number of items is reasonably large. In the present paper, an approximation for the bias function of the maximum likelihood estimate of the latent trait, or ability, will be developed using the same assumptions for the more general case where item responses are discrete. This will include the dichotomous response level, for which the three-parameter logistic model has been discussed, the graded response level and the nominal response level. Some observations will be made for both dichotomous and graded response levels.     

A numerical approach to the approximate and the exact minimum rank of a covariance matrix

A concept of approximate minimum rank for a covariance matrix is defined, which contains the (exact) minimum rank as a special case. A computational procedure to evaluate the approximate minimum rank is offered. The procedure yields those proper communalities for which the unexplained common variance, ignored in low-rank factor analysis, is minimized. The procedure also permits a numerical determination of the exact minimum rank of a covariance matrix, within limits of computational accuracy. A set of 180 covariance matrices with known or bounded minimum rank was analyzed. The procedure was successful throughout in recovering the desired rank.The authors are obliged to Paul Bekker for stimulating and helpful comments.