Some Relationships between Factors and Components

Ogasawara, Haruhiko

doi:10.1007/bf02294372

Cited by 18 publications

(10 citation statements)

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“…They are given from acov(Ol, 02) 001 aO~l = --acov(s) ~-I , (25) as1 Is=v (X) as Is=v (X) where 01 and 02 are the vectors of different solutions (the parameter esQmmtes). Use of (25) is not limited to the solutions in PCA: 01 may be a solution in PCA while 02 a solution in factor analysis (Ogasawara, 2000b).…”

Section: Discussionmentioning

confidence: 99%

Concise Formulas for the Standard Errors of Component Loading Estimates

Ogasawara

2002

Psychometrika

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Section: Discussionmentioning

confidence: 99%

Concise Formulas for the Standard Errors of Component Loading Estimates

Ogasawara

2002

Psychometrika

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“…It is often stated that there is no model in principal component analysis and that it is a method to reduce the dimensionality of data matrices (see, e.g., Bentler, & Kano, 1990). However, the principal component "model" can be used as a saturated model for unrestricted covariance structures (see Ogasawara, 2000aOgasawara, , 2000b. By this formulation, the statistical treatment of the results for principal component analysis becomes feasible as is the case for factor analysis.…”

Section: Exploratory Second-order Component Analysismentioning

confidence: 99%

“…However, they can be obtained quite straightforwardly by extending the restrictions in the first-order component analysis (see Ogasawara, 2000aOgasawara, , 2000b. The detailed description for the restrictions including ∂r/∂1′ (Ogasawara, 2002) will be available from the author upon request.…”

Section: The Restrictions For Parametersmentioning

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Exploratory second‐order analyses for components and factors

Ogasawara

2002

Jpn Psychol Res

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Exploratory methods using second-order components and second-order common factors were proposed. The second-order components were obtained from the resolution of the correlation matrix of obliquely rotated first-order principal components. The standard errors of the estimates of the second-order component loadings were derived from an augmented information matrix with restrictions for the loadings and associated parameters. The second-order factor analysis proposed was similar to the classical method in that the factor correlations among the first-order factors were further resolved by the exploratory method of factor analysis. However, in this paper the second-order factor loadings were estimated by the generalized least squares using the asymptotic variance-covariance matrix for the first-order factor correlations. The asymptotic standard errors for the estimates of the second-order factor loadings were also derived. A numerical example was presented with simulated results.Key words: asymptotic standard errors, exploratory factor analysis, factor rotation, secondorder component analysis, second-order factor analysis.Second-order factor analysis goes back to the age of L. L. Thurstone. In the early period the correlation matrix of obliquely rotated firstorder factors was further resolved by the usual method of factor analysis (Thurstone, 1944(Thurstone, / 1947 see Gorsuch, 1983, Ch. 11). At present, second-or higher-order factor models are often treated as structural equation models (see, e.g., Bollen, 1989, Ch. 7). That is, the first-and second-order factor loadings in confirmatory factor analysis models are estimated without factor rotation. These confirmatory secondorder models are distinct from that of the exploratory stepwise method. When hypotheses are to be confirmed, confirmatory factor models which are identified without factor rotation may be useful. However, they are not necessarily of use when factor analysis is performed with exploratory purposes.Browne and Du Toit (1992) gave a method of the second-order factor analysis with a partially exploratory model. In their model, the correlation matrix of the first-order factors is described by the loadings of the secondorder factors. Under this model with structured correlation matrix of first-order factors, they imposed restrictions on the first-order factor loadings such that the loadings optimize a criterion of oblique rotation. The first-order factor loadings are simultaneously estimated with second-order loadings. This method gives an identifiability as a covariance structure model, while it has the problem that different rotation criteria in the first-order factors give different values of goodness-of-fit. They stated: 'The fit of the exploratory second order factor analysis model is affected by the rotation criterion employed ' (p. 294), where 'the exploratory second order factor analysis' is different from the classical stepwise method in that oblique factor rotation is not utilized, although criteria for oblique rotation are used. The...

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“…Furthermore, even though similarities between component and factor analyses have been well-documented (Ogasawara, 2000;Velicer & Jackson, 1990;Velicer, Peacock, & Jackson, 1982), the theoretical status of components and factors is not the same (Borsboom, Mellenbergh, & van Heerden, 2003;Gorsuch, 1990). Therefore, to examine covariance structure differences in terms of differences in underlying latent variables (i.e., unobservable variables that have a causal relationship to the observed variables), such as the previously mentioned personality traits and affect dimensions, an EFAbased method is to be preferred.…”

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Mixture Simultaneous Factor Analysis for Capturing Differences in Latent Variables Between Higher Level Units of Multilevel Data

Roover

Vermunt

Timmerman

et al. 2017

Structural Equation Modeling: A Multidisciplinary Journal

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Given multivariate data, many research questions pertain to the covariance structure: whether and how the variables (e.g., personality measures) covary. Exploratory factor analysis (EFA) is often used to look for latent variables that might explain the covariances among variables; for example, the Big Five personality structure. In the case of multilevel data, one might wonder whether or not the same covariance (factor) structure holds for each so-called data block (containing data of 1 higher level unit). For instance, is the Big Five personality structure found in each country or do cross-cultural differences exist? The well-known multigroup EFA framework falls short in answering such questions, especially for numerous groups or blocks. We introduce mixture simultaneous factor analysis (MSFA), performing a mixture model clustering of data blocks, based on their factor structure. A simulation study shows excellent results with respect to parameter recovery and an empirical example is included to illustrate the value of MSFA.Keywords: factor analysis, latent variables, mixture model clustering, multilevel data Given multivariate data, researchers often wonder whether the variables covary to some extent and in what way. For instance, in personality psychology, there has been a debate about the structure of personality measures (i.e., the Big Five vs. Big Three debate; De Raad et al., 2010). Similarly, emotion psychologists have discussed intensely whether and how emotions as well as norms for experiencing emotions can be meaningfully organized in a lowdimensional space (e.g., Ekman, 1999;Fontaine, Scherer, Roesch, & Ellsworth, 2007;Russell & Barrett, 1999;Stearns, 1994). Factor analysis (Lawley & Maxwell, 1962) is an important tool in these debates as it explains the covariance structure of the variables by means of a few latent variables, called factors. When the researchers have a priori assumptions on the number and nature of the underlying latent variables, confirmatory factor analysis (CFA) is often used, whereas exploratory factor analysis (EFA) is applied when one has no such assumptions.Research questions about the covariance structure get further ramifications when the data have a multilevel structure; for instance, when personality measures are available for inhabitants from different countries. We refer to data organized according to the higher level units (e.g., the countries) as data blocks. For multilevel data, one can wonder whether or not the same structure holds for each data block. For example, is the Big Five personality structure found in each country or not (De Raad et al., 2010) Setiadi, & Markam, 2002;MacKinnon & Keating, 1989;Rodriguez & Church, 2003).When looking for differences and similarities in covariance structures, using EFA is very advantageous because it leaves more room for finding differences than CFA does. For instance, in the emotion norm example (Eid & Diener, 2001), one might very well expect two latent variables to show up in each country corresponding to approved and disap...

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Some Relationships between Factors and Components

Abstract: component loadings, factor loadings, standardized variables, factor rotation, asymptotic correlations, standard errors,

Cited by 18 publications

References 19 publications

Concise Formulas for the Standard Errors of Component Loading Estimates

Concise Formulas for the Standard Errors of Component Loading Estimates

Exploratory second‐order analyses for components and factors

Mixture Simultaneous Factor Analysis for Capturing Differences in Latent Variables Between Higher Level Units of Multilevel Data

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