A new procedure for detection of crossing DIF

The purpose of this paper is to present a hypothesis testing and estimation procedure, Crossing SIBTEST, for detecting crossing DIF. Crossing DIF exists when the difference in the probabilities of a correct answer for the two examinee groups changes signs as ability level is varied. In item response theory terms, crossing DIF is indicated by two crossing item characteristic curves. Our new procedure, denoted as Crossing SIBTEST, first estimates the matching subtest score at which crossing occurs using least squares regression analysis. A Crossing SIBTEST statistic then is used to test the hypothesis of crossing DIF. The performance of Crossing SIBTEST is evaluated in this study.This research was partially supported by a grant from the Law School Admission Council and by National Science Foundation Mathematics Grant NSF-DMS-94-04327. The research reported here is collaborative in every respect and the order of authorship is alphabetical. The authors thank Jeff Douglas and Louis Roussos for their useful comments and discussions.     

Improved bounds for optimal intervals in weakly constrained regression problems

The solution of weakly constrained regression problems typically requires the iterative search, in a given interval, of a point where a certain function has a zero derivative. This note deals with improved bounds for the interval to be searched.     

A multidimensional measure of neighboring

The study reports the development of a short and easily administered questionnaire aiming at measuring dimensions of social life within neighborhoods. Principal-components analysis consistently extracted four factors replicated in three independent samples (N=96 to 1,060). The factors emerged as theoretically meaningful dimensions tapping the concepts of supportive acts of neighboring, neighbor annoyance, neighborhood attachment, and weak social ties. Factor invariance and factor replicability were high. Internal consistency, test-retest reliability, and construct validity yielded acceptable results. The principal virtue of the measure is that it may be applied to discern qualitative differences between neighborhoods by simultaneous assessments of several dimensions of neighboring. This research was financially supported by The Norwegian National Research Council. We thank Jostein Rise for valuable comments, and David R. Jacobs, Jr., and Melanie Young for advice in translating the questionnaire.     

More factors than subjects,tests and treatments: An indeterminacy theorem for canonical decomposition and individual differences scaling

Some methods that analyze three-way arrays of data (including INDSCAL and CANDECOMP/PARAFAC) provide solutions that are not subject to arbitrary rotation. This property is studied in this paper by means of the triple product [A, B, C] of three matrices. The question is how well the triple product determines the three factors. The answer: up to permutation of columns and multiplication of columns by scalars—under certain conditions. In this paper we greatly expand the conditions under which the result is known to hold. A surprising fact is that the nonrotatability characteristic can hold even when the number of factors extracted is greater thanevery dimension of the three-way array, namely, the number of subjects, the number of tests, and the number of treatments.This paper is being published in place of Dr. Kruskal's presidential address to the Psychometric Society, April, 1975. Further results like those in this paper, as well as a surprising connection with an area of mathematics called arithmetic complexity theory, will be found in a more recent paper [Kruskal, in press].     

Principal component analysis of three-mode data by means of alternating least squares algorithms

A new method to estimate the parameters of Tucker's three-mode principal component model is discussed, and the convergence properties of the alternating least squares algorithm to solve the estimation problem are considered. A special case of the general Tucker model, in which the principal component analysis is only performed over two of the three modes is briefly outlined as well. The Miller & Nicely data on the confusion of English consonants are used to illustrate the programs TUCKALS3 and TUCKALS2 which incorporate the algorithms for the two models described.     

A model and algorithm for multidimensional scaling with external constraints on the distances

A method for externally constraining certain distances in multidimensional scaling configurations is introduced and illustrated. The approach defines an objective function which is a linear composite of the loss function of the point configurationX relative to the proximity dataP and the loss ofX relative to a pseudo-data matrixR. The matrixR is set up such that the side constraints to be imposed onX's distances are expressed by the relations amongR's numerical elements. One then uses a double-phase procedure with relative penalties on the loss components to generate a constrained solutionX. Various possibilities for constructing actual MDS algorithms are conceivable: the major classes are defined by the specification of metric or nonmetric loss for data and/or constraints, and by the various possibilities for partitioning the matricesP andR. Further generalizations are introduced by substitutingR by a set ofR matrices,R _i ,i=1, ...r, which opens the way for formulating overlapping constraints as, e.g., in patterns that are both row- and column-conditional at the same time.     

Constrained multidimensional scaling inN spaces

A gradient method is used to obtain least squares estimates of parameters of them-dimensional euclidean model simultaneously inN spaces, given the observation of all pairwise distances ofn stimuli for each space. The procedure can estimate an additive constant as well as stimulus projections and the metric of the reference axes of the configuration in each space. Each parameter in the model can be fixed to equal some a priori value, constrained to be equal to any other parameter, or free to take on any value in the parameter space. Two applications of the procedure are described.     

A monte carlo investigation of recovery of structure by alscal

A Monte Carlo study was carried out to investigate the ability of ALSCAL to recover true structure inherent in simulated proximity measures. The nature of the simulated data varied according to (a) number of stimuli, (b) number of individuals, (c) number of dimensions, and (d) level of random error. Four aspects of recovery were studied: (a) SSTRESS, (b) recovery of true distances, (c) recovery of stimulus dimensions, and (d) recovery of individual weights. Results indicated that all four measures were rather strongly affected by random error. Also, SSTRESS improved with fewer stimuli in more dimensions, but the other three indices behaved in the opposite fashion. Most importantly, it was found that the number of individuals, over the range studied, did not have a substantial effect on any of the four measures of recovery. Practical implications and suggestions for further research are discussed.The authors wish to thank Drs. Forrest W. Young, Paul D. Isaac and Thomas E. Nygren, who provided many helpful comments during this project.     

Recovery of structure in incomplete data by alscal

A Monte Carlo study was carried out in order to investigate the ability of ALSCAL to recover true structure inherent in simulated proximity measures when portions of the data are missing. All sets of simulated proximity measures were based on 30 stimuli and three dimensions, and selection of missing elements was done randomly. Properties of the simulated data varied according to (a) the number of individuals, (b) the level of random error, (c) the proportion of missing data, and (d) whether the same entries or different entries were deleted for each individual. Results showed that very accurate recovery of true distances, stimulus coordinates, and weight vectors could be achieved with as much as 60% missing data as long as sample size was sufficiently large and the level of random error was low.     

Validation of the Preschool and Early Childhood Functional Assessment Scale (PECFAS)

Efforts to determine the prevalence of serious emotional disturbance in preschool-aged children have been hampered by the lack of a validated measure. The Preschool and Early Childhood Functional Assessment Scale (PECFAS) is a multi-dimensional measure that assesses the psychosocial functioning of children aged 3–7 years. The concurrent validity and reliability of the PECFAS were assessed in a sample of 30 preschool-aged children in a large Head Start program in Ventura, California. PECFAS ratings based on in-depth interviews were significantly related to parental ratings that the children had mental health problems, psychiatric diagnoses, teacher ratings of the child's need for mental health evaluations, teacher ratings of behavior problems on a standardized screening inventory (DIAL-R), and actual referrals for mental health evaluations. Interrater reliability for the total PECFAS score was high (r = .90) as was internal consistency of the five subscales (alpha = .86). Using the PECFAS scores as a standard, the weighted prevalence of serious emotional disturbance in this West Coast Head Start program was 17%, at the lower end of the current estimated rate of SED for older children in low income samples (18–26%).