The orthoblique rotation for factor analysis

The gradient projection algorithm (GPA) is the most used tool to rotate factor models. However, GPA is currently limited to fully orthogonal or fully oblique solutions. Keeping structural zeros in selected positions of the factor correlation matrix after rotation is currently unavailable. This limitation is relevant to analyses where theoretical considerations or study design require certain factor correlations to be zero. We therefore introduce the orthoblique rotation, which allows orthogonal and oblique relations between factors to coexist. Building on Jennrich's algorithms, we describe the geometric foundations of GPA and extend them to the orthoblique rotation. We illustrate its usefulness for exploratory bifactor models with multiple correlated general factors that remain orthogonal to group factors, and for multitrait–multimethod models in which trait and method factors may correlate within their respective sets while remaining orthogonal between sets. Finally, we make available the new orthoblique rotation in the R package latent through the lrotate() function, which extends the capabilities of the popular GPArotation package of Bernaards and Jennrich (2005). Taken together, this work helps bridge the gap between exploratory and confirmatory factor analyses and continues Jennrich's efforts to expand the flexibility and scope of factor rotation.