Rotation to Simple Loadings Using Component Loss Functions: The Oblique Case

Asparouhov and Muthén (2014) presented a new method for multiple-group confirmatory factor analysis (CFA), referred to as the alignment method. The alignment method can be used to estimate group-specific factor means and variances without requiring exact measurement invariance. A strength of the method is the ability to conveniently estimate models for many groups, such as with comparisons of countries. This paper focuses on IRT applications of the alignment method. An empirical investigation is made of binary knowledge items administered in two separate surveys of a set of countries. A Monte Carlo study is presented that shows how the quality of the alignment can be assessed.

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“…CLF has been used in EFA analysis, see for example Jennrich (2006) and it is used similarly here. One good choice for the CLF is…”

Section: The Alignment Methodsmentioning

confidence: 99%

IRT studies of many groups: the alignment method

2014

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“…As suggested by their name, they count the variables, close to certain hyperplane, and then try to maximize their number. Recently the rationale behind these methods inspired the introduction of a class of rotation criteria called component loss functions (CLF) (Jennrich, 2004(Jennrich, , 2006. The most intriguing of them is given by the 1 matrix norm of the rotated loadings, defined for any p × r matrix A as A 1 = p i r j |a i j |.…”

Section: Rotated Component Loadingsmentioning

confidence: 99%

“…The orthogonal CLF simple structure rotation (2) does not always produce satisfying results. For this reason, the oblique CLF is usually applied instead (Jennrich, 2006). It is defined as…”

Section: Rotated Component Loadingsmentioning

confidence: 99%

“…The sparse loadings obtained by SPARSIMAX will be compared with the corresponding rotated (classic) loadings using standard VARIMAX (MATLAB, 2011), and orthogonal and oblique CLF (Jennrich, 2004(Jennrich, , 2006. Thurstone's 26 Box Problem: Thurstone constructed an artificial data set with 30 observations and 26 variables in the following way.…”

Section: Numerical Examplesmentioning

confidence: 99%

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Sparse Versus Simple Structure Loadings

Trendafilov

Adachi

2014

Psychometrika

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The component loadings are interpreted by considering their magnitudes, which indicates how strongly each of the original variables relates to the corresponding principal component. The usual ad hoc practice in the interpretation process is to ignore the variables with small absolute loadings or set to zero loadings smaller than some threshold value. This, in fact, makes the component loadings sparse in an artificial and a subjective way. We propose a new alternative approach, which produces sparse loadings in an optimal way. The introduced approach is illustrated on two well-known data sets and compared to the existing rotation methods.

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“…For orthogonal rotations, the raw varimax (Kaiser, 1958) and the linear component loss (Jennrich, 2006) criteria were used. For oblique rotations, the quartimin (Carroll, 1953) Jennrich (2006) proposed a modification for component loss functions (e.g., linear) that are not differentiable at λ ij = 0. The modified loss function is differentiable and continuous at all points.…”

Section: Numerical Comparisonsmentioning

confidence: 99%