The geometry of factorial indeterminancy

TheN-dimensional geometry of a Spearman-Thurstone factor solution reveals two sources for the indeterminancy of factor scores: indeterminancy of total factor space and a rotational indeterminancy within a given total factor space. The analytical papers of Ledermann [4] and Guttman [2] on indeterminancy of factor scores are related to these findings and a simple vector model is developed to reveal the properties of rotational indeterminancy. The significance of factor-score indeterminancy is discussed in light of these findings.     

A note on identification in the oblique and orthogonal factor analysis models

Conditions for removing the indeterminancy due to rotation are given for both the oblique and orthogonal factor analysis models. The conditions indicate why published counterexamples to conditions discussed by Jöreskog are not identifiable.The author would like to thank Gordon Bechtel and the reviewers for their comments and suggestions.     

Toward a generative theory of the therapeutic field

In the various models of family therapy, family systems are described as being patterned and programmed in their operations and development. The regularities and patterns found in family life are believed to be the expression of a program that determines the behavior of the family system and each of its members. Therapists holding this view will use "programming methods" in order to map family phenomena, and therapeutic work will consist of disrupting the problematic sequences in a deterministic way. However, if we take into consideration the impossibility of always linking inputs and outputs and the inability to evaluate consistently the behaviors in a system, problems associated with indeterminancy appear. Under such conditions, it may be impossible for therapists to formulate programs of family functioning. Therefore, we have operationalized a set of methodological principles that can be applied in such conditions. In this article we present the theoretical and practical implications for the field of therapy of "nonprogramming methods." A videotaped session is analyzed in order to show the application of the nonprogramming methods in clinical practice.     

Parceling Cannot Reduce Factor Indeterminacy in Factor Analysis: A Research Note

Parceling—using composites of observed variables as indicators for a common factor—strengthens loadings, but reduces the number of indicators. Factor indeterminacy is reduced when there are many observed variables per factor, and when loadings and factor correlations are strong. It is proven that parceling cannot reduce factor indeterminacy. In special cases where the ratio of loading to residual variance is the same for all items included in each parcel, factor indeterminacy is unaffected by parceling. Otherwise, parceling worsens factor indeterminacy. While factor indeterminacy does not affect the parameter estimates, standard errors, or fit indices associated with a factor model, it does create uncertainty, which endangers valid inference.

     

COMMENTS ON FACTORING DICHOTOMOUS SCORE MATRICES

The fact that the Holley-Guilford G index is a mean cross product between individuals has two implications. First, standard procedures are available for the estimation of R factor loadings. Second, the factor space contains a factor of means. This factor is not unique when item polarities are arbitrary. The factor structure in such cases is a result of both qualitative and quantitative interindividual similarity.     

Model‐related factor score predictors for confirmatory factor analysis

The present paper introduces model‐related (MR) factor score predictors, which reflect specific aspects of confirmatory factor models. The development is mainly based on Schönemann and Steiger's regression score components, but it can also be applied to the factor score coefficients. It is shown that the rotation of factor score predictors has no impact on the covariance matrix reproduced from the corresponding regression component patterns. Thus, regression score components or factor score coefficients can be rotated in order to obtain the required properties. This idea is the basis for MR factor score predictors, which are computed by means of a partial Procrustes rotation towards a target pattern representing the interesting properties of a confirmatory factor model. Two examples demonstrate the construction of MR factor score predictors reflecting specific constraints of a factor model.     

Descriptive axioms for common factor theory,image theory and component theory

Through an extension of work by Guttman, common factor theory, image theory, and component theory are derived from distinct minimum subsets of assumptions chosen out of a set of five possible assumptions. It is thence shown that the problem of indeterminacy of factor scores in the common factor model is precisely reflected in the problem of the non-orthogonality of anti-images. Indeed, image scores are determinate for the same reason that the usual estimates of factor scores are determinate, and image scores cannot be used as though they were factor scores for the same reason that factor score estimates cannot be used as though they were factor scores.     

Factor structure in groups selected on observed scores

A new method is proposed for estimating factor means and factor covariances in a group of individuals selected on their observed scores. The selection variable is, for example, the total score on an admissions test. Given a factor model for the test items based on the group of test takers, we may be interested in the factor structure for those in the top quartile. The differences in factor means and covariances between this selected group and the full group gives useful information both on successful test performance and on test validity. The new method draws on the classic Pearson-Lawley selection formulas. It avoids the fallacy of factor analysis on the selected group, which would lead to incorrect estimates. The new method is applied to a simple factor structure model for the GMAT test. Although the majority of the GMAT items test verbal skills, it is found that a quantitative factor shows the greatest change in moving from average to top quartile test takers.     

The role of nonlinear factor-to-indicator relationships in tests of measurement equivalence

Measurement invariance is a necessary condition for the evaluation of factor mean differences over groups or time. This article considers the potential problems that can arise for tests of measurement invariance when the true factor-to-indicator relationship is nonlinear (quadratic) and invariant but the linear factor model is nevertheless applied. The factor loadings and indicator intercepts of the linear model will diverge across groups as the factor mean difference increases. Power analyses show that even apparently small quadratic effects can result in rejection of measurement invariance at moderate sample sizes when the factor mean difference is medium to large. Recommendations include the identification of nonlinear relationships using diagnostic plots and consideration of newly developed methods for fitting nonlinear factor models.     

Psychometric characteristics of the Parenting Scale in a Vietnamese sample

The Parenting Scale (PS) is a well‐established instrument for measuring discipline practices in Western populations. However, whether the PS is a valid and reliable measure in Eastern populations is not known. Thus, this study examined the psychometric properties of the PS in a sample of 433 Vietnamese parents of children aged 2–7 years. Confirmatory factor analyses (CFAs) examined the commonly used Reitman et al.'s two‐factor and Rhoades and O'Leary's three‐factor solutions. Neither factor structure fit the data. An exploratory factor analysis identified a 12‐item one‐factor and 14‐item two‐factor solutions that overlapped substantially with established factor structures. The one‐factor solution reflected Lax/Overreactive parenting and the two‐factor solution consisted of Lax/Overreactive and Hostile subscales. The factor structures were confirmed via multigroup CFA. Internal consistencies were acceptable and ranged between .70 and .85. Each factor was positively associated with parent anger, dysfunctional cognitions about child behaviour, and externalising child behaviour problems. However, when entered simultaneously in a regression, Hostile discipline was not uniquely associated with child behaviour. Overall, results support the potential utility of the 12‐item one‐factor PS as a measure of dysfunctional parenting practices for Vietnamese parents.     

Assessing stress in cancer patients: a second-order factor analysis model for the Perceived Stress Scale

Using the Perceived Stress Scale (PSS), perceptions of global stress were assessed in 111 women following breast cancer surgery and at 12 and 24 months later This is the first study to factor analyze the PSS. The PSS data were factor analyzed each time using exploratory factor analysis with oblique direct quartimin rotation. Goodness-of-fit indices (root mean square error of approximation [RMSEA]), magnitude and pattern of factor loadings, and confidence interval data revealed a two-factor solution of positive versus negative stress items. The findings, replicated across time, also indicate factor stability. Hierarchical factor analyses supported a second-order factor of "perceived stress." This alternative factor model of the PSS is presented along with observations regarding the measure's use in cancer research.     

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.     

Personality and religion: Applying cattell’s model among secondary school pupils

A sample of 1,070 secondary school pupils between the ages of eleven and eighteen years completed the High School Personality Questionnaire alongside the Francis Scale of Attitude toward Christianity. The data demonstrate that religiosity is significantly correlated with five of the fourteen personality factors and with two of the four second-order factors. A positive attitude toward Christianity is associated with high scores on factor G (conformity), factor I (tendermindedness) and factor Q3 (self discipline), and low scores on factor E (submissiveness), factor F (sobriety), second-order factor of extraversion (indicating a relationship with introversion), and second-order factor of tough poise (indicating a relationship with emotionality).     

Mixture Factor Analysis for Approximating a Nonnormally Distributed Continuous Latent Factor With Continuous and Dichotomous Observed Variables

Mixture factor analysis is examined as a means of flexibly estimating nonnormally distributed continuous latent factors in the presence of both continuous and dichotomous observed variables. A simulation study compares mixture factor analysis with normal maximum likelihood (ML) latent factor modeling. Different results emerge for continuous versus dichotomous outcomes. For dichotomous outcomes, normal ML path estimates have bias that worsens as latent factor skew/kurtosis increases and does not diminish as sample size increases, whereas the mixture factor analysis model produces nearly unbiased estimators as sample sizes increase (500 and greater) and offers near nominal coverage probability. For continuous outcome variables, both methods produce factor loading estimates with minimal bias regardless of latent factor skew, but the mixture factor analysis is more efficient. The method is demonstrated using data motivated by a study on youth with cystic fibrosis examining predictors of treatment adherence. In summary, mixture factor analysis provides improvements over normal ML estimation in the presence of skewed/kurtotic latent factors, but due to variability in the estimator relating the latent factor to dichotomous outcomes and computational issues, the improvements were only fully realized, in this study, at larger sample sizes (500 and greater).     

Exploratory factor analysis for small samples

Traditionally, two distinct approaches have been employed for exploratory factor analysis: maximum likelihood factor analysis and principal component analysis. A third alternative, called regularized exploratory factor analysis, was introduced recently in the psychometric literature. Small sample size is an important issue that has received considerable discussion in the factor analysis literature. However, little is known about the differential performance of these three approaches to exploratory factor analysis in a small sample size scenario. A simulation study and an empirical example demonstrate that regularized exploratory factor analysis may be recommended over the two traditional approaches, particularly when sample sizes are small (below 50) and the sample covariance matrix is near singular.     

On the factor score controversy

A summary and interpretation of the recent literature on the indeterminacy of factor scores is given in simple terms. A good index of factor score determinacy is the squared multiple correlation of the factor with the observed variables.     

Effects of inbreeding on mental-ability factors

The differential effects of inbreeding on 11 subtests of the WISC, in a Japanese (Hiroshima) population, are related to the factor structure of the tests. The degree of inbreeding depression on mental abilities is most strongly correlated with the subtests' loadings on the General factor, g, which is common to all of the subtests. Loadings on the Verbal factor are also positively correlated with inbreeding depression. The (non-verbal) Performance factor, however, is slightly enhanced by inbreeding. One possible inference from these findings is that at least the General factor shows genetic dominance, which is theoretically consistent with natural selection for g in the course of human evolution. It is also noted that the factor structure of the WISC in the Japanese population is highly similar to that of the U.S. standardization sample.