Nested Linear Mixed Effects Model for Region-of-Interest to Region-of-Interest (ROI-to-ROI) Analysis I am trying to examine the effects of language (English/Mother Tongue) on functional connectivity between different regions of interest (ROI). Hence, I employed a nested Linear Mixed Effects Model approach to examine the fixed effect of language nested in each ROI. Wondering if this is a crossed-factored model or whether nested mixed effects would be even appropriate. Any clarification would help!
For reference, my model is specified as:
lmer(FCM ~ ROI_Pair / language + cov_1 + cov_2 + (1|subject), data = df)

The dataframe looks like this:

 A: Your Model
It looks like your model syntax is doing the following:

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*Use FCM as a DV with a normally distributed error

*Model an interaction between ROI and language

*Add two main effects from covariates

*Estimate the random intercepts only for subjects (no random slopes are included here)

Nested Random Effects
This is neither nested nor crossed random effects. Nested effects typically have a hierarchy involved (students within schools or schools within districts). A typical hierarchical cluster has a random effect specification such as (1|g1/g2). Yours is just (1|g), which isn't inherently wrong, but simply doesn't model clusters within other clusters.
Crossed Random Effects
Crossed random effects expand the data frame by each random effect. If you have 20 subjects and 10 items, while modeling both as random intercepts, you would get 200 observations because each item is crossed with each subject. Here you only have one random effect, random intercepts, so it doesn't qualify.
Final Remarks
Whether or not this model is appropriate for you is based off your assumptions of the data. Keep in mind that this is essentially a repeated measures regression (you are estimating the repeated observations of each subject) and assumes that each subject has similar mean responses to the outcome. I'm not sure what each variable means here, but if you suspect any of them vary among subjects by a lot, you may consider including it as a random slope. Some examples are modeling something like (1+cov1|subject) for correlated slopes/intercepts or (1+cov1||subject) for uncorrelated slopes/intercepts. I'm also assuming FCM has a normal error (the residual distribution has a Gaussian shape). If not, consider also using the glmer function, which models different families of distributions.
Useful Citations
I've listed some useful articles if you are confused about random effects structures. The first is written by lme4's creator Douglas Bates and there is a page specifically showcasing model syntax. Another by Baayen et al., 2008 explains how crossed random effects work in a straightforward way. There are a few other "user-friendly" guides to using mixed models below as well.

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*Baayen, R. H., Davidson, D. J., & Bates, D. M. (2008). Mixed-effects modeling with crossed random effects for subjects and items. Journal of Memory and Language, 59, 390–412. https://doi.org/10.1016/j.jml.2007.12.005

*Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1). https://doi.org/10.18637/jss.v067.i01

*Brown, V. A. (2021). An introduction to linear mixed-effects modeling in R. Advances in Methods and Practices in Psychological Science, 4(1), 1–19. https://doi.org/10.1177/2515245920960351

*Harrison, X. A., Donaldson, L., Correa-Cano, M. E., Evans, J., Fisher, D. N., Goodwin, C. E. D., Robinson, B. S., Hodgson, D. J., & Inger, R. (2018). A brief introduction to mixed effects modelling and multi-model inference in ecology. PeerJ, 6, e4794. https://doi.org/10.7717/peerj.4794

*Meteyard, L., & Davies, R. A. I. (2020). Best practice guidance for linear mixed-effects models in psychological science. Journal of Memory and Language, 112, 104092. https://doi.org/10.1016/j.jml.2020.104092
