Rotational Uniqueness Conditions Under Oblique Factor Correlation Metric
Carel F. W. Peeters
TL;DR
This paper addresses whether Jöreskog's C1–C3 conditions guarantee global rotational uniqueness in oblique factor analysis under the oblique correlation metric. It shows that C1–C3 alone yield only local rotational uniqueness and introduces an amended condition set including a per-column polarity truncation (C4) to achieve global rotational uniqueness when $\mathrm{diag}(\mathbf{\Phi})=\mathbf{I}_{m}$. The main result proves that, with C1–C4, the rotation must satisfy $\mathbf{R}=\mathbf{I}_{m}$, whereas C1–C3 do not ensure global unicity and are not equivalent to the C2–C* pairing. The paper further discusses practical implications for estimation, identifiability, and modeling flexibility, highlighting advantages for Bayesian inference and unit-invariance considerations. Overall, the work clarifies the conditions necessary for global rotational uniqueness in oblique factor models and advocates using C1–C4 for improved identifiability and practical estimation.
Abstract
In an addendum to his seminal 1969 article Jöreskog stated two sets of conditions for rotational identification of the oblique factor solution under utilization of fixed zero elements in the factor loadings matrix. These condition sets, formulated under factor correlation and factor covariance metrics, respectively, were claimed to be equivalent and to lead to global rotational uniqueness of the factor solution. It is shown here that the conditions for the oblique factor correlation structure need to be amended for global rotational uniqueness, and hence, that the condition sets are not equivalent in terms of unicity of the solution.