The presented results imply that a combination of the ESEM approach and regularized esti- mation may provide a data analytic framework for ERP data that is less prone to all three causes of variance misallocation. Recently, regularized structural equation modeling (regSEM) has been proposed as a method that offers such a combination (Jacobucci et al., 2016). In the following, a temporal regSEM for ERP data will be outlined in more detail and discussed in the light of alternative suggestions to reduce the rotation bias. Finally, further research questions and potential extensions of ESEM/regSEM are discussed.
More specifically, an ESEM can be conceptualized as a regSEM by using the same structural model as in the ESEM and specifying a measurement model in which all factors are allowed to load on all observed variables. Crucially, as explained in more detail in chapter 4, all factor loadings enter the penalty term during estimation to achieve a sparse measurement model. How can regSEM be applied to ERP data? Building on the findings regarding ESEM in chapter 3, a structural model should be specified in which the indicator variables for electrode sites and conditions as well as their interactions predict the latent factors. The factors should be allowed to correlate because even after controlling for the factor topography, the biases resulting from a violated orthogonality assumption are profound (Chapter 3). Furthermore, all factors are allowed to load on each sampling point, and all factor loadings are considered
to-be-regularized. In addition, it may be considered to include the factor correlations into the penalty term as well to counteract the inflation of factor correlation estimates due to temporal overlap (Chapters 2 & 3). The findings in chapter 4 suggest that an elastic net penalty could provide a reasonable default choice for the penalty function. A regSEM specified in this way both properly considers the structure of ERP data sets and it should be less prone to variance misallocation than simple structure rotation. Despite the promising results reported here, a direct empirical test of this notion is a necessary task for future research.
It may be argued that the proposed regSEM addresses the problem of rotation bias only indi- rectly because it is maximally agnostic about the expected time courses of the latent factors. Alternatively, it was suggested to develop ERP-specific rotation criteria that either make use of a priori knowledge of condition effects (Beauducel & Leue, 2015), or that include assump- tions regarding the time courses of the factors that are motivated directly from ERP research (Beauducel, 2018). Given that a priori knowledge of condition effects is rarely available, the latter approach is arguably more generally applicable than the former. Specifically, assuming that ERPs are transient voltage deflections, Beauducel (2018) proposed ERP-specific rotation techniques that take a two-step approach. First, an initial Varimax-rotated solution is found. Second, a rotation target is derived either by fixing successive loadings with small slopes to zero (Event-related orthogonal partial Procrustes rotation, EPP rotation) or by fitting an opti- mal Gaussian shape to the Varimax solution (Gaussian event-related Procrustes rotation, GEP rotation). The rotation target is then supplied to a target rotation algorithm yielding the final rotated solution. Both EPP and GEP rotation aim at reducing typical distortions of the factor loading estimates that have been observed in the literature on variance misallocation (Wood & McCarthy, 1984). Beauducel (2018) found that both EPP and GEP rotation were able to recover factor loading patterns that included slow-wave potentials (i.e., factors with non-zero loadings spread over the whole epoch) which are known to be a challenging ground for simple structure rotation techniques (Verleger & Möcks, 1987).
1In regSEM, the initial model may be over-parameterized. As long as a sufficient number of parameters is
penalized, identification can still be achieved because some of parameters are shrunken to zero (Jacobucci, 2017; Jacobucci et al., 2016).
ERP-specific rotation criteria can easily be combined with the ESEM approach presented in chapter 3, offering an alternative approach to reduce the rotation bias within the ESEM frame- work. A direct comparison of ESEM with simple structure rotation, ESEM with ERP-specific rotation, and regSEM is required to explore which of the methods performs best under a vari- ety of conditions. In order to judge in how far the methods are generally useful, it is crucial for such a comparison to include conditions that are challenging enough. For instance, on the one hand, slow-wave potentials may be challenging to recover for regSEM (just as for simple structure rotation) because the ideal factor solution may not be sparse any more. On the other hand, ERP-specific rotation may be challenged by factors with bi-phasic time courses because it assumes that the targeted time course shape is approximately correct (e.g., mono-phasic, transient factors for GEP rotation). These examples illustrate the importance of a representa- tive choice of the simulation conditions because otherwise no generalizable conclusions can be drawn.
Apart from the question which of the methods proposed here has the best relative performance, an important question is how these methods perform when compared with other frequently used decomposition techniques such as ICA, multimode PCAs (Möcks, 1988), or temporospatial PCA (i.e., applying a spatial PCA to the results of a temporal PCA; Dien, 2010). Some efforts have been made in that direction (Bugli & Lambert, 2007; Delorme, Palmer, Onton, Oostenveld, & Makeig, 2012; Dien, Khoe, & Mangun, 2007; Makeig et al., 1999; Verleger, Paulick, Möcks, Smith, & Keller, 2013), but these studies were often limited to real data examples making it hard to judge the generalizability (but see Groppe et al., 2008, for a notable exception). In this context, it is important to investigate the mathematical relationships between the underlying models in order to establish generalizable conclusions about their relative strengths and weak- nesses. For instance, it would be interesting to learn under which conditions different methods can be expected to give (approximately) equivalent results.
Finally, it would be interesting to explore in how far some of the assumptions of the temporal EFA/ESEM approach can be relaxed. Arguably the most restrictive assumption is that the fac- tor loading patterns are fixed across participants and conditions.2 Especially, the consequences
of latency-jitter (i.e., time-shifted factor loadings patterns between participants and condi- tions) have been investigated by previous research (Donchin, 1978; Möcks, 1986), showing that latency-jitter results in additional factors in the solution that have a certain shape. However, it is hard to distinguish whether genuine factors from additional factors due to latency-jitter in empirical applications. Therefore, an extension of ESEM that allows for variation in the factor loading patterns (see Marsh et al., 2017, for a solution in the context of confirmatory factor analysis) or even allows to attribute differences in factor loading patterns to other variables (De Roover, Timmerman, & Ceulemans, 2017) would be very useful for ERP researchers (e.g., Barry et al., 2016).