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I am asking to find out if an idea that popped into my head is a real thing or just silly.

EFA often presumes the data are static. I know there are forms of EFA that take time into account, but I have no clue how they work. Do they measure how the structure of the factors change over time and map that, even to the extent of data jumping to other factors? Do they create a structural equation model then track how the data deviates from it over time?

I have zero clue.

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I wouldn't say that EFA presumes the data are static, but rather, that most people apply EFA in research using cross-sectional designs, where the possibility of change in latent measurement models cannot be indulged/explored. I haven't seen too much of this kind of longitudinal work done with EFA, but longitudinal CFA and SEM are well-established approaches. Little's book (2013) provides a very nice introduction--I would just focus on the latent CFA and latent panel model for starters. And just as between multiple groups (e.g., Vandenberg and Lance, 2000), one can use longitudinal CFA to examine whether measurement model parameters--loadings ($\lambda$s), intercepts ($\tau$s), etc.,--are equivalent between waves of assessment (i.e., longitudinal invariance). As in other forms of comparative analyses (Meredith, 1993) , longitudinal invariance is necessary if you're interested in making strong comparisons between structural parameters (e.g., latent means, $\alpha$s) across time.

tl;dr: Little's book is not super recent (pretty sure I heard a new edition is forthcoming) but it still is a great place to start as an introduction to CFA/SEM in general, and especially so for the longitudinal context.

References

Little, T. D. (2013). Longitudinal structural equation modeling. New York, NY: Guilford Press.

Meredith, W. (1993). Measurement invariance, factor analysis and factorial invariance. Psychometrika, 58(4), 525-543.

Vandenberg, R. J., & Lance, C. E. (2000). A review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational research. Organizational Research Methods, 3, 4-70.

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  • $\begingroup$ What if the design is inherently exploratory? A fundamental assumption of CFA and of SEM is flat-out false. There is nothing to confirm. There is no known structure to apply a-priori. What then? $\endgroup$
    – Bryan
    Commented Apr 11, 2019 at 14:32
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    $\begingroup$ The same principles would apply; whatever the exploratory model you elect to fit (an indiscriminate pattern of loadings onto however many factors you extract) could still be evaluated in terms of invariance of measurement parameters across waves of assessment. Asparouhov & Muthén (2009) discuss this possibility in their formative paper on exploratory structural equation modelling. $\endgroup$
    – jsakaluk
    Commented Apr 11, 2019 at 20:34

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