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A new efficient method for determining the number of components in PARAFAC models
Journal of Chemometrics, 2003
A new diagnostic called the core consistency diagnostic (CORCONDIA) is suggested for determining the proper number of components for multiway models. It applies especially to the parallel factor analysis (PARAFAC) model, but also to other models that can be considered as restricted Tucker3 models. It is based on scrutinizing the`appropriateness' of the structural model based on the data and the estimated parameters of gradually augmented models. A PARAFAC model (employing dimension-wise combinations of components for all modes) is called appropriate if adding other combinations of the same components does not improve the fit considerably. It is proposed to choose the largest model that is still sufficiently appropriate. Using examples from a range of different types of data, it is shown that the core consistency diagnostic is an effective tool for determining the appropriate number of components in e.g. PARAFAC models. However, it is also shown, using simulated data, that the theoretical understanding of CORCONDIA is not yet complete.
An External Validity Approach for Assessing Essential Unidimensionality in Correlated-Factor Models
Educational and Psychological Measurement
Many psychometric measures yield data that are compatible with (a) an essentially unidimensional factor analysis solution and (b) a correlated-factor solution. Deciding which of these structures is the most appropriate and useful is of considerable importance, and various procedures have been proposed to help in this decision. The only fully developed procedures available to date, however, are internal, and they use only the information contained in the item scores. In contrast, this article proposes an external auxiliary procedure in which primary factor scores and general factor scores are related to relevant external variables. Our proposal consists of two groups of procedures. The procedures in the first group (differential validity procedures) assess the extent to which the primary factor scores relate differentially to the external variables. Procedures in the second group (incremental validity procedures) assess the extent to which the primary factor scores yield predictive validity increments with respect to the single general factor scores. Both groups of procedures are based on a second-order structural model with latent variables from which new methodological results are obtained. The functioning of the proposal is assessed by means of a simulation study, and its usefulness is illustrated with a real-data example in the personality domain.
Psychometrika, 1994
This paper presents some results on identification in multitrait-multimethod (MTMM) confirmatory factor analysis (CFA) models. Some MTMM models are not identified when the (factorial-patterned) loadings matrix is of deficient column rank. For at least one other MTMM model, identification does exist despite such deficiency. It is also shown that for some MTMM CFA models, Howe's (1955) conditions sufficient for rotational uniqueness can fail, yet the model may well be identified and rotationally unique. Implications of these results for CFA models in general are discussed.
Component Analysis for Structural Equation Models with Concomitant Indicators
Studies in Classification, Data Analysis, and Knowledge Organization, 2013
A new approach to structural equation modelling based on so-called Extended Redundancy Analysis has been recently proposed in literature, enhanced with the added characteristic of generalizing Redundancy Analysis and Reduced-Rank Regression models for more than two blocks. However, in presence of direct effects linking exogenous and endogenous variables, the latent composite scores are estimated by ignoring the presence of the specified direct effects. In this paper, we extend Extended Redundancy Analysis, permitting us to specify and fit a variety of relationships among latent composites and endogenous variables. In particular, covariates are allowed to affect endogenous indicators indirectly through the latent composites and/or directly.
On the Detection of the Correct Number of Factors in Two-Facet Models by Means of Parallel Analysis
Educational and Psychological Measurement, 2021
Methods for optimal factor rotation of two-facet loading matrices have recently been proposed. However, the problem of the correct number of factors to retain for rotation of two-facet loading matrices has rarely been addressed in the context of exploratory factor analysis. Most previous studies were based on the observation that two-facet loading matrices may be rank deficient when the salient loadings of each factor have the same sign. It was shown here that full-rank two-facet loading matrices are, in principle, possible, when some factors have positive and negative salient loadings. Accordingly, the current simulation study on the number of factors to extract for two-facet models was based on rank-deficient and full-rank two-facet population models. The number of factors to extract was estimated from traditional parallel analysis based on the mean of the unreduced eigenvalues as well as from nine other rather traditional versions of parallel analysis (based on the 95th percentile of eigenvalues, based on reduced eigenvalues, based on eigenvalue differences). Parallel analysis based on the mean eigenvalues of the correlation matrix with the squared multiple correlations of each variable with the remaining variables inserted in the main diagonal had the highest detection rates for most of the two-facet factor models. Recommendations for the identification of the correct number of factors are based on the simulation results, on the results of an empirical example data set, and on the conditions for approximately rank-deficient and full-rank two-facet models.
Imagination, Cognition and Personality
This project applied a bifactor model, which specifies a general factor that accounts for the common variance among all scale items, and group factors that reflect additional common variance among clusters of items. This general factor is designated as “M” because of a presumption in the research literature that its origins are to be found in method. The model was applied in eight samples using nine datasets and across three different personality measures, including the Big Five and the HEXACO. Inclusion of M significantly increased model fit and increased the variance explained of items. Evidence showed that M did not reflect aspects of method such as random error or an acquiescent response bias. M correlated positively with variables suggesting psychological adjustment and negatively with variables pointing toward maladjustment. M showed unique relationships with constructs suggesting psychological adjustment over and beyond the Big Five. Data supported an interpretation of M as a...
Indeterminacy problems and the interpretation of factor analysis results
Statistica Neerlandica, 1978
This paper reviews indeterminacy problems for the factor analysis model and their consequences for the interpretation of the results. Two types of indeterminacy are discerned: indeterminacy of the parameters in the model (the number of factors, the specific variances and the factorloadings) and the indeterminacy of the factors, given the parameters in the model. It is argued that parameter indeterminacy is partly to be overcome, provided that a strong underlying theory for the subject matter under research is present. Factor indeterminacy remains a major stumbling-block for the interpretation of results. The GUTTMAN criterion is advocated as a measure of factor indeterminacy.
An Exploratory Study of the Three Phases Analysis of Factor
2021
In this study, we examine factor analysis as a multivariate statistical tool, starting from the origin of factor analysis with regards to Spearman's approach of 1904 to the three phases of factor analysis. This is done with a view of determining the similarities and individual contributions of each of the three phases of factor analysis. This was achieved by examining the algorithms used in parameter estimations of the three phases of factor analysis. By inputting data into the algorithms and examining their outcomes and proffering recommendations based on the respective findings.
On the Interpretation of Factor Analysis
The importance of the researcher’s interpretation of factor analysis is illustrated by means of an example. The results from this example appear to be meaningful and easily interpreted. The example omits any measure of reliability or validity. If a measure of reliability had been included, it would have indicated the worthlessness of the results. A survey of 46 recent papers from 6 journals supported the claim that the example is typical, two-thirds of the papers provide no measure of reliability. In fact, some papers did not even provide sufficient information to allow for replication. To improve the current situation some measure of factor reliability should accompany applied studies that utilize factor analysis. Three operational approaches are suggested for obtaining measures of factor reliability: use of split samples, Monte Carlo simulation, and a priori models.