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.

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.