Contributions to factor analysis of dichotomous variables

A new method is proposed for the factor analysis of dichotomous variables. Similar to the method of Christoffersson this uses information from the first and second order proportions to fit a multiple factor model. Through a transformation into a new set of sample characteristics, the estimation is considerably simplified. A generalized least-squares estimator is proposed, which asymptotically is as efficient as the corresponding estimator of Christoffersson, but which demands less computing time.This research was supported by the Bank of Sweden Tercentenary Foundation under project Structural Equation Models in the Social Sciences, project director Karl G. Jöreskog.     

Simultaneous factor analysis in several populations

This paper is concerned with the study of similarities and differences in factor structures between different groups. A common situation occurs when a battery of tests has been administered to samples of examinees from several populations.A very general model is presented, in which any parameter in the factor analysis models (factor loadings, factor variances, factor covariances, and unique variances) for the different groups may be assigned an arbitrary value or constrained to be equal to some other parameter. Given such a specification, the model is estimated by the maximum likelihood method yielding a large samplex ² of goodness of fit. By computing several solutions under different specifications one can test various hypotheses.The method is capable of dealing with any degree of invariance, from the one extreme, where nothing is invariant, to the other extreme, where everything is invariant. Neither the number of tests nor the number of common factors need to be the same for all groups, but to be at all interesting, it is assumed that there is a common core of tests in each battery that is the same or at least content-wise comparable.This research was supported by grant NSF-GB-12959 from National Science Foundation. My thanks are due to Michael Browne for his comments on an earlier draft of this paper and to Marielle van Thillo who checked the mathematical derivations and wrote and debugged the computer program SIFASP.Now at Statistics Department, University of Uppsala, Sweden.     

Simultaneous factor analysis of several Gramian matrices

Given several Gramian matrices, a least-square fit to all the matrices by one factor matrix, with a predetermined number of factors, is shown to be the principal axes solution of the average of the matrices.     

Measurement invariance,factor analysis and factorial invariance

Several concepts are introduced and defined: measurement invariance, structural bias, weak measurement invariance, strong factorial invariance, and strict factorial invariance. It is shown that factorial invariance has implications for (weak) measurement invariance. Definitions of fairness in employment/admissions testing and salary equity are provided and it is argued that strict factorial invariance is required for fairness/equity to exist. Implications for item and test bias are developed and it is argued that item or test bias probably depends on the existence of latent variables that are irrelevant to the primary goal of test constructers.Presidential address delivered at the Annual Meeting of the Psychometric Society in Berkeley, California, June 18–20, 1993.     

Simultaneous optimization of factor assessibility and representativeness

P. M. Bentler has shown that Rao's canonical factor analysis is in effect a psychometric analysis, leading to factors that are maximally assessible from the data. He contrasts this with Kaiser and Caffrey's alpha factor analysis that leads to factors that maximally represent the true factors in the content domain. Noting the problems associated with factors that may be highly assessible, but not very representative, or vice versa, Bentler suggests the need for a technique that would, insofar as possible, be optimal with respect to both criteria. Such a technique is presented here, and is shown to resolve into a traditional scaling method, which in turn acquires a richer psychometric interpretation.The author would like to gratefully acknowledge support for this research from the Workshop in Political Theory and Policy Analysis (Indiana University) and Esco Research Associates. The paper benefitted substantially from the criticisms and suggestions of an anonymous referee.     

The disintegrative factor in therapeutic groups as illuminated by kibbutz life

This article deals with the way individuals cope with the threat they experience in the framework of psychodynamic group psychotherapy. This threat, which is understood to be a disintegrative threat, is intensified in the therapeutic group. We see the therapeutic group as possessing attributes of a potential space, in which the participants are enabled to process the defense and coping mechanisms intended to preserve the sense of integration. This will be illustrated by the experiences of individuals who grew up in a kibbutz (an Israeli society in which infants were raised with their peer group) and participated in psychodynamic groups.     

The effect of additional variables on factor indeterminacy in models with a single common factor

Determinate solutions for the indeterminate common factor ofp variables satisfying the single common factor model are not unique. Therefore an infinite sequence of additional variables that conform jointly with the originalp variables to the original single common factor model does not determine a unique solution for the indeterminate factor of thep variables (although the solution is unique for the factor of the infinite sequence). Other infinite sequences may be found to determine different solutions for the factor of the originalp variables. The paper discusses a number of theorems about the effects of additional variables on factor indeterminacy in a model with a single common factor and draws conclusions from them for factor theory in general.     

An analysis of several variables influencing the efficacy of flash card instruction

We conducted four experiments using an alternating treatments design to assess the influence of several variables on children's acquisition of number facts during one-on-one flash card instruction. In Experiment 1, we manipulated the order of stimulus presentation; in most cases a higher percentage of correct responses occurred, and mastery criterion was reached in somewhat fewer sessions, when a missed problem was re-presented after the next item than when problems were re-presented in a sequential order. In Experiment 2 a comparison of error-contingent reprimands with no reprimands yielded similar results in favor of the former procedure. Experiment 3 indicated that a higher percentage of correct responses and more rapid acquisition occurred when the tutor and student were seated knee-to-knee than when a desk was positioned between them. Experiment 4 compared the rate of learning in the presence and absence of a treatment package consisting of all three variables identified as leading to more rapid learning in the previous three experiments. Children reached criterion in less than half the time when the treatment package was in effect.     

Selection of variables in exploratory factor analysis: An empirical comparison of a stepwise and traditional approach

The purpose of this study was to investigate and compare the performance of a stepwise variable selection algorithm to traditional exploratory factor analysis. The Monte Carlo study included six factors in the design; the number of common factors; the number of variables explained by the common factors; the magnitude of factor loadings; the number of variables not explained by the common factors; the type of anomaly evidenced by the poorly explained variables; and sample size. The performance of the methods was evaluated in terms of selection and pattern accuracy, and bias and root mean squared error of the structure coefficients. Results indicate that the stepwise algorithm was generally ineffective at excluding anomalous variables from the factor model. The poor selection accuracy of the stepwise approach suggests that it should be avoided.     

A Comparison of Computer-Assisted Psychotherapy and Cognitive-Behavioral Therapy in Groups

This study is the first independent, randomized controlled study in a clinical setting of a computer-assisted therapy program (Therapeutic Learning Program; TLP) developed to permit individualized therapy in a group setting. This software differs from other computer-assisted approaches in that targets for change are chosen by the patient rather than determined by the software. TLP was compared to a standard cognitive-behavioral treatment in 109 patients with 6-month follow-up. Although both treatments were associated with a significant reduction in symptoms of anxiety and depression at posttreatment and follow-up, no differences were found between treatments with respect to patient-rated satisfaction and effectiveness, clinician ratings of improvement, and scores on measures of anxiety and depression. At treatment midpoint patients tended to give more neutral ratings of satisfaction and treatment effectiveness of the TLP treatment, perhaps suspending judgment about this more novel treatment until its completion. It is concluded that at least one form of computer-assisted psychotherapy is as effective and acceptable to patients in a clinical setting as standard short-term cognitive-behavioral therapy conducted by experienced clinicians.     

A note on sufficient conditions that a common factor will be determinate in an infinite domain of variables

It is proved for the common factor model withr common factors that under certain condition s which maintain the distinctiveness of each common factor a given common factor will be determinate if there exists an unlimited number of variables in the model each having an absolute correlation with the factor greater than some arbitrarily small positive quantity.The author is indebted to R. P. McDonald for suggesting the proof of Guttman's determinantal equation for the squared multiple correlation in predicting a factor from the observed variables used in the parenthetical note.     

Two-step weighted least squares factor analysis of dichotomized variables

A two-step weighted least squares estimator for multiple factor analysis of dichotomized variables is discussed. The estimator is based on the first and second order joint probabilities. Asymptotic standard errors and a model test are obtained by applying the Jackknife procedure.     

Measures of invariance and comparability in factor analysis for fixed variables

New procedures are presented for measuring invariance and matching factors for fixed variables and for fixed or different subjects. Two of these, the coefficient of invariance for factor loadings and the coefficient of factor similarity, utilize factor scores computed from the different sets of factor loadings and one of the original standard score matrices. Another, the coefficient of subject invariance, is obtained by using one of the sets of factor loadings in conjunction with the different standard score matrices. These coefficients are correlations between factor scores of the appropriate matrices. When the best match of factors is desired, rather than degree of resemblance, the method of assignment is proposed.     

Comparison of several shortened versions of the EMBU: exploratory and confirmatory factor analyses

The aim of the present study was to analyze the psychometric properties of different versions of the EMBU (Egna Minnen av Barndoms Uppfostran – My memories of upbringing) in a non-clinical sample of adolescents. The factor structure of the questionnaire was studied by means of more adequate exploratory and confirmatory factor procedures in order to ascertain which version achieves a better fit to data. The 64-item version of the EMBU was administered to a non-clinical sample of 832 adolescents. Exploratory and Confirmatory factor analyses were carried out for successive versions of 64, 37, 24 and 23 items. The theoretical four-factor structure of the 64-item version is not replicated in a factor analysis through principal axis extraction with direct oblimin rotation. The "Favoring Subject" factor was not found. 37 items present loadings equal to or larger than 0.30 for the "Rejection", "Emotional Warmth" and "Overprotection" factors. A re-analysis of these items shows a clear three-factor structure. When Arrindell et al .'s (1999 ) 23-item version was analysed, the three expected factors were also found. Confirmatory factor analyses allow us to reduce the 36-item version to a shorter one of 24 items. This last version achieves the best fit to data, including the 23-item version. The new 24-item version improves the construct validity of the EMBU and presents similar reliability coefficients.     

A Monte Carlo study comparing PIV,ULS and DWLS in the estimation of dichotomous confirmatory factor analysis

We conducted a Monte Carlo study to investigate the performance of the polychoric instrumental variable estimator (PIV) in comparison to unweighted least squares (ULS) and diagonally weighted least squares (DWLS) in the estimation of a confirmatory factor analysis model with dichotomous indicators. The simulation involved 144 conditions (1,000 replications per condition) that were defined by a combination of (a) two types of latent factor models, (b) four sample sizes (100, 250, 500, 1,000), (c) three factor loadings (low, moderate, strong), (d) three levels of non‐normality (normal, moderately, and extremely non‐normal), and (e) whether the factor model was correctly specified or misspecified. The results showed that when the model was correctly specified, PIV produced estimates that were as accurate as ULS and DWLS. Furthermore, the simulation showed that PIV was more robust to structural misspecifications than ULS and DWLS.     

Age comparative factor analysis of ability variables in adulthood and old age

The generality of ability factor structure in adulthood and old age was investigated using simultaneous maximum likelihood procedures. Data were analyzed for 198 young individuals (age range of 15–32 years), 156 younger old individuals (age range of 53–68 years), and 156 older old individuals (69–91 years). Variables were nine tests marking three ability factors: Verbal Comprehension, Sensitivity to Problems and Semantic Redefinition. Results indicated no changes in the number of factors and no psychologically important shifts in salient factor loadings. Increasingly larger factor covariances, however, were obtained in the two older groups. The results supported the generalizability of ability structures to late life at the level of factor loadings. The implications of these results for the dedifferentiation hypothesis were not clear cut. The consistency of the number of factors across age groups precluded strong support for this hypothesis, The increased factor covariances could be construed as modest support for this hypothesis, but other interpretations were possible.     

Selecting polychoric instrumental variables in confirmatory factor analysis: An alternative specification test and effects of instrumental variables

The polychoric instrumental variable (PIV) approach is a recently proposed method to fit a confirmatory factor analysis model with ordinal data. In this paper, we first examine the small-sample properties of the specification tests for testing the validity of instrumental variables (IVs). Second, we investigate the effects of using different numbers of IVs. Our results show that specification tests derived for continuous data are extremely oversized at all sample sizes when applied to ordinal variables. Possible modifications for ordinal data are proposed in the present study. Simulation results show that the modified specification tests with all available IVs are able to detect model misspecification. In terms of estimation accuracy, the PIV approach where the IVs outnumber the endogenous variables by one produces a lower bias but a higher variation than the PIV approach with more IVs for correctly specified factor loadings at small samples.     

Counselling former members of charismatic groups: considering pre-involvement variables,reasons for joining the group and corresponding values

A majority of counsellors and other health professionals who work with former members of charismatic groups appear to give little consideration to pre-involvement variables such as reasons for initially joining the group and corresponding values. This study explores reasons for joining a charismatic group from the perspective of former members through the use of qualitative methods. Seven participants from four different groups were recruited via purposive sampling and modified snowball sampling. In-depth interviews were used to explore the participants’ accounts of why they joined a charismatic group. The findings of this study suggest that participants of this study joined charismatic groups because of the ability of such groups to provide ‘certainty’ and meet needs of friendship, meaning, and belonging. The need for health professionals to consider pre-involvement variables when working with former members is discussed.