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Rotating Factors to Simplify Their Structural Paths

Abstract

Applications of structural equation modeling (SEM) may encounter issues like inadmissible parameter estimates, nonconvergence, or unsatisfactory model fit. We propose a new factor rotation method that reparameterizes the factor correlation matrix in exploratory factor analysis (EFA) such that factors can be either exogenous or endogenous. The proposed method is an oblique rotation method for EFA, but it allows directional structural paths among factors. We thus referred it to as FSP (factor structural paths) rotation. In particular, we can use FSP rotation to “translate” an SEM model to incorporate theoretical expectations on both factor loadings and structural parameters. We illustrate FSP rotation with an empirical example and explore its statistical properties with simulated data. The results include that (1) EFA with FSP rotation tends to fit data better and encounters fewer Heywood cases than SEM does when there are cross-loadings and many small nonzero loadings, (2) FSP rotated parameter estimates are satisfactory for small models, and (3) FSP rotated parameter estimates are more satisfactory for large models when the structural parameter matrices are sparse.

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Notes

  1. We count diagonal elements of a correlation matrix as unique elements because they are standardized variances. Rotational indeterminacy requires \(m^2\) constraints in EFA. In orthogonal rotation, there are \(m(m+1)/2\) constraints in the factor correlation matrix and \(m(m-1)/2\) in the factor loading matrix (Archer & Jennrich 1973, Eq. (17)). The \(m(m+1)/2\) constraints in the factor correlation matrix are m diagonal elements of ones and \(m(m-1)/2\) above-diagonal (or below-diagonal) elements of zeros. In oblique rotation, there are m constraints in the factor correlation matrix and \(m(m-1)\) constraints imposed on the factor loading matrix and the factor correlation matrix jointly (Jennrich, 1973, Eq. (28)). The m constraints in the factor correlation matrix are m diagonal elements of ones. Therefore, we need to set the factor variances to be one for both simple interpretation and model identification.

  2. The two discrepancy functions are \(f_{OLS}(\varvec{r}, \varvec{\theta }) = \text {trace} [\varvec{R}- \varvec{P}(\varvec{\theta })]^2\) and \( f_{ML}(\varvec{r}, \varvec{\theta }) = \log |\varvec{P}(\varvec{\theta })|- \log |\varvec{R}| + \text {trace}[\varvec{R}\varvec{P}(\varvec{\theta })^{-1}] - p\), respectively. The matrix function “trace” sums together the diagonal elements of a square matrix.

  3. The derivation presented in Appendix A of Zhang et al. (2019) is also applicable to FSP rotation, and their Equation (20) directly implies the current constraint functions.

  4. The target values for factor loadings and structural parameters are presented in an online support file (https://drive.google.com/file/d/1u-XHXlFHyjDumxJC8UN-JNzjAISuC9t4/view?usp=sharing,Table B1). The file also includes the R code for the illustration.

  5. We also computed mean RMSEAs with all samples. Including or excluding samples with Heywood cases produced essentially the same results.

  6. The FSP.target rotated population values and the FSP.geomin rotated population values of Model II are reported as supplementary materials (Table B4).

  7. The FSP.geomin rotated population values and the FSP.target rotated population values of Model III are reported as supplementary materials (Table B5).

  8. Tables B6 and B7 of Supplemental materials report such an example.

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Correspondence to Guangjian Zhang.

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Appendices

Appendix

Derivatives of the FSP.target Rotation Criterion

The derivatives of the FSP.target rotation criterion Q with regard to the rotation matrix \(\varvec{T}\) involve four terms. The first term involves factor loadings, and it was described in Jennrich (2002). The fourth term involves correlations among exogenous factors, and it was described in Zhang et al. (2019). We now derive the second term tr\(\left( \frac{\mathrm{d}Q}{\mathrm{d} \varvec{B}'} \frac{\mathrm{d} \varvec{B}}{\mathrm{d} t_{kl}}\right) \) and the third term tr\(\left( \frac{\mathrm{d}Q}{\mathrm{d} \varvec{\Gamma }'} \frac{\mathrm{d} \varvec{\Gamma }}{\mathrm{d} t_{kl}} \right) \).

We compute both terms using the chain rule, which involves the product of the derivatives of the semtarget rotation criterion function Q with regard to \(\varvec{B}\) or \(\varvec{\Gamma }\) and the derivatives of \(\varvec{B}\) or \(\varvec{\Gamma }\) with regard to \(\varvec{T}\). The derivatives of the semtarget rotation criterion function Q with regard to \(\varvec{B}\) and \(\varvec{\Gamma }\) are

$$\begin{aligned} \frac{\mathrm{d}Q}{\mathrm{d} \varvec{B}'} = 2 \left( \varvec{W}_{\varvec{B}} \bigodot ( \varvec{B} - \varvec{H}_{\varvec{B}} )\right) ' \text {, and } \frac{\mathrm{d}Q}{\mathrm{d} \varvec{\Gamma }'} = 2 \left( \varvec{W}_{\varvec{\Gamma }} \bigodot ( \varvec{\Gamma } - \varvec{H}_{\varvec{\Gamma }} )\right) ' \text {.} \end{aligned}$$
(22)

We use the chain rule again to derive the derivatives of \(\varvec{B}\) and \(\varvec{\Gamma }\) with regard to \(t_{kl}\),

$$\begin{aligned} \frac{\mathrm{d} \varvec{B}}{\mathrm{d} t_{kl}} = \frac{\mathrm{d} \varvec{B}}{\mathrm{d} \varvec{\Phi }_z } \frac{\mathrm{d} \varvec{\Phi }_z}{\mathrm{d} t_{kl}} \text {, and } \frac{\mathrm{d} \varvec{\Gamma }}{\mathrm{d} t_{kl}} = \frac{\mathrm{d} \varvec{\Gamma }}{\mathrm{d} \varvec{\Phi }_z } \frac{\mathrm{d} \varvec{\Phi }_z}{\mathrm{d} t_{kl}} \text {.} \end{aligned}$$
(23)

Let \(\varvec{w}_i\) contains the all regression weights of predicting \(\eta _i\) from all its predictors. These regression weights correspond to the elements at the ith row of \(\varvec{B}\) and \(\varvec{\Gamma }\). The derivatives \(\frac{\mathrm{d} w_{ij}}{\mathrm{d} \phi _{kl}}\) are

$$\begin{aligned} \frac{\mathrm{d} w_{ij}}{\mathrm{d} \phi _{kl}} = \left\{ \begin{array}{cl} \left[ \varvec{\Phi }^{-1}_{i+,i+ } \right] _{j-i,l-i} &{} \text {, if } k = i \text { and } l> i \text {,} \\ - w_{ik} \left[ \varvec{\Phi }^{-1}_{i+,i+ } \right] _{j-i,l-i} &{} \text {, if } k> i \text { and } l > i \text {,} \\ 0 &{} \text {, other else.} \\ \end{array} \right. \end{aligned}$$
(24)

Here, \(w_{ik}\) is the kth element of the vector \(\varvec{w}_i\), and \(\varvec{\Phi }_{i+, i+}\) is defined in Eq. (4).

The partial derivatives of EFA factor correlations \(\varvec{\Phi }_z\) with regard to \(\varvec{T}\) are

$$\begin{aligned} \frac{\mathrm{d} \phi _{ij}}{\mathrm{d} t_{kl}} = \delta _ {il} t_{kl} + \delta _ {jl} t_{ki} \text {.} \end{aligned}$$
(25)

Here, \(\phi _{ij}\) is a typical element of \(\varvec{\Phi }_z\). If \(i=l\), the value \(\delta _ {il}\) is 1, and it is zero otherwise.

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Zhang, G., Hattori, M. & Trichtinger, L.A. Rotating Factors to Simplify Their Structural Paths. Psychometrika (2022). https://doi.org/10.1007/s11336-022-09877-3

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Keywords

  • structural equation modeling
  • SEM
  • factor analysis
  • EFA
  • factor rotation
  • oblique rotation