I want to perform a mediation analysis in the context of ESEM (specifically, ESEM-within-CFA). My understanding of my dataset was that I must include the cross-loadings between all variables in the model; when I fit an ESEM, the correlation among variables is much lower.
However, In the Handbook of Structural Equation Modeling, Morin states:
"When cross-loadings need to be included, they should only be included across constructs located at the same position in the predictive model under investigation… . Incorporating cross-loading between variables located at different stages of a theoretical "causal" chain would create a paradoxical nonrecursive situation in which the same indicator would define two constructs specified as predictive one another."
I understand that the same component of variance in an indicator cannot be attributed to two constructs in a causal chain, but I have difficulty understanding why different components of variance in one indicator cannot be attributed to two such constructs. For instance, scores on the item "I laugh easily" (a NEO item from the Extraversion factor) have reflections from individuals' Extraversion trait. But Depression may also reflect in how one answers to this item, without any relation to one’s Extraversion (random life events may lead to a good mood). Then in a predictive model, I assume, if I ignore this cross-loading, it will overestimate the predictive power of Extraversion on Depression.
Am I missing something here? Should I include the cross-loadings between these variables in such a situation?
Thank you,
Ali
When conducting a mediation analysis in the context of ESEM-within-CFA, the inclusion of cross-loadings can depend on the research question and the specific model being tested. In general, ESEM models can allow for the inclusion of cross-loadings, which may improve the fit of the model and increase the accuracy of the parameter estimates.
However, it is important to note that the decision to include cross-loadings in the ESEM model should not be based solely on the magnitude of the correlations among variables. Rather, the inclusion of cross-loadings should be guided by theoretical or empirical evidence that supports the idea that the latent factors are not strictly orthogonal.
In the context of mediation analysis, it is also important to consider the direction of the causal effects between the variables. Specifically, the presence of cross-loadings in the model may affect the interpretation of the mediation effects, and it may be necessary to use more advanced mediation analysis techniques that can account for non-orthogonal latent factors, such as the cross-lagged panel model or the latent growth curve model.
In summary, the inclusion of cross-loadings in ESEM models can depend on the specific research question and the underlying theoretical assumptions. When conducting a mediation analysis in the context of ESEM-within-CFA, it is important to carefully consider the direction of the causal effects and to use appropriate mediation analysis techniques that can account for non-orthogonal latent factors.
Hi,
if you have an indicator with a double-loading, you have to incorporate the double loading in the structural model. I cannot follow the reasoning of Morin here. The indicator does not create a "paradoxical recursive situqation", as there is no effect of the latent dependent variable back to the independent variable. And even if the model would become recursive (as in models with a feedback loop), simply ignoring that would surely not solve the problem.
Let's try a little simulation. I have this model here in mind when I read your post:
http://dagitty.net/maHqHEQ
In this model, there are two latent variables where x_latent causally affects y_latent. Both latents have 2 unique loadings and share an indicator (z).
In the attached R-simulation, I created data for this model (N=1,000) and tested two models. You should be able to copy-paste the code. Be sure that you only install the lavaan packge (which is currently in the pdf but de-activated by the #-symbole).
Model 1 is the correct one and fits cleanly. Moreover, you get your effect of the latent variable. Model 2 omits the loading of the z indicator. The consequence is that the model does not fit any more and more importantly, you effect of x_latent is strongly biased. The reason for that is, that by not-adressing the relationship between y_latent and z (by removing the loading), the model tries to compensate for that by upward biasing the only left connection between y_latent and z--and this is the structural effect.
Quite simple, actually. Simple path tracing.
HTH
Holger
Holger Steinmetz
Thank you very much; Morin's reasoning was in contrast to my understanding of latent variable modeling, but I needed expert approval.
Ma'Mon Abu Hammad
Thank you for your notes.
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