I am running a SEM, and I have 6 variables which will load onto two factors (positive and negative mental health). However, this model does not meet a good model fit. When I allow depression to load onto both factors, the model fit meets model fit criteria. I want to use this as a latent variable in a SEM.
I have been told previously that a variable can be allowed to factor onto two factors, but I am struggling to find anything to back this up. Does anyone have any more knowledge on this?
Thank you in advance.
In the literature, a distinction is made between confirmatory (CSEM) and exploratory (ESEM) structural equation modelling. The latter refers to cases in which all items are allowed to load on all factors of a scale. ESEM can be performed using Mplus. You can accordingly consider ESEM.
If you however want to stick to CSEM, then letting one item load on two factors in order to obtain better fit may be problematic unless you have got strong theory supporting it. The question becomes rather a theoretical than a statistical one.
Speaking of theory, I would consider positive and negative mental health two closely related phenomena. One idea could be to correlate the residuals of the items reflecting these two factors. Such a modification does often improve model fit measures as well as being justified on theoretical grounds.
Hello Danielle,
Yes, a variable may be affiliated with more than one factor, especially if the factors are correlated as suggested by Mehmet Mehmetoglu 's response. While not as "clean" a structure as Thurstone's idealized, "simple structure" solution (each variable loads near perfectly with one factor, and not at all on any other), cross-loadings do occur and can often be justified. A number of personality measures incorporate cross-loadings in the developers' model/s. Consequences of insisting on the simple structure solution can include jettisoning variables that might be theoretically justifiable, altered factor score reliability, and a more narrow set of construct indicators.
The threat of positing a structure in which many/most/all variables load on multiple factors is that you might not be specifying a distinctive set of factors at all, and that the resultant model complexity and excessive variable-factor links can bias model fit indices upward, resulting in acceptance of an otherwise not very convincing model. Perhaps the positive and negative mental health factors would "fit" in a second-order model under a common parent factor. There should, of course, always be a strong rationale for whatever model you decide to evaluate via confirmatory methods.
Good luck with your work.
Hello Danielly,
my 2c (beyond that what Mehmet and David said): Technically, a double loading means that one indicator has a higher-than-expected correlation with all indicators of the "other" latent variable. The occurence of the need to estimate such a loading can be both theoretically appropriate (there is an additional effect of the other latent on the specific indicator) but of course (as any other surprising parameter) a sign of a deeper problem (i.e., that the factor structure is misspecified) --especially when you did not specify the loading prior to the initial test but posthoc based on the model test (did you TEST the model? How's the result?).
In your case, I would expect a double loading as depression reflects not only "negative affect" but also lack of positive affect. Hence, I guess the loading on positive health is negative.
However, I have some deeper concerns whether the latents themselves are meaningful (i.e. what is "negative health"?) which makes reason number two above not unlikely. My recommendation would be to investigate what happens to the model fit (again: a test of the structure) if you incorporate other variables into the model that have affects ON your latents or are affected BY your latents (optimally with differential patterns of effects)? If reason no 2 is true this will blow up the fit even in the double loading case.
Would like to know the results out of curiosity.
Best,
Holger
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