Direct Schmid–Leiman Transformations and Rank-Deficient Loadings Matrices

The Schmid–Leiman (S–L; Psychometrika 22: 53–61, 1957) transformation is a popular method for conducting exploratory bifactor analysis that has been used in hundreds of studies of individual differences variables. To perform a two-level S–L transformation, it is generally believed that two separate factor analyses are required: a first-level analysis in which k obliquely rotated factors are extracted from an observed-variable correlation matrix, and a second-level analysis in which a general factor is extracted from the correlations of the first-level factors. In this article, I demonstrate that the S–L loadings matrix is necessarily rank deficient. I then show how this feature of the S–L transformation can be used to obtain a direct S–L solution from an unrotated first-level factor structure. Next, I reanalyze two examples from Mansolf and Reise (Multivar Behav Res 51: 698–717, 2016) to illustrate the utility of ‘best-fitting’ S–L rotations when gauging the ability of hierarchical factor models to recover known bifactor structures. Finally, I show how to compute direct bifactor solutions for non-hierarchical bifactor structures. An online supplement includes R code to reproduce all of the analyses that are reported in the article.     

Visually induced gains in pitch discrimination: Linking audio-visual processing with auditory abilities

Perception is fundamentally a multisensory experience. The principle of inverse effectiveness (PoIE) states how the multisensory gain is maximal when responses to the unisensory constituents of the stimuli are weak. It is one of the basic principles underlying multisensory processing of spatiotemporally corresponding crossmodal stimuli that are well established at behavioral as well as neural levels. It is not yet clear, however, how modality-specific stimulus features influence discrimination of subtle changes in a crossmodally corresponding feature belonging to another modality. Here, we tested the hypothesis that reliance on visual cues to pitch discrimination follow the PoIE at the interindividual level (i.e., varies with varying levels of auditory-only pitch discrimination abilities). Using an oddball pitch discrimination task, we measured the effect of varying visually perceived vertical position in participants exhibiting a wide range of pitch discrimination abilities (i.e., musicians and nonmusicians). Visual cues significantly enhanced pitch discrimination as measured by the sensitivity index d’, and more so in the crossmodally congruent than incongruent condition. The magnitude of gain caused by compatible visual cues was associated with individual pitch discrimination thresholds, as predicted by the PoIE. This was not the case for the magnitude of the congruence effect, which was unrelated to individual pitch discrimination thresholds, indicating that the pitch-height association is robust to variations in auditory skills. Our findings shed light on individual differences in multisensory processing by suggesting that relevant multisensory information that crucially aids some perceivers’ performance may be of less importance to others, depending on their unisensory abilities.