I have a linear mixed model, from which I extracted the slopes for each individual participant (using coef(model)$participant). Then, out of curiosity, I ran a linear regression (equivalent in structure to the original linear mixed model) on each participant's data and saved the estimates for each participant.
Why do these two approaches of looking at individual participants' estimates result in different values and distributions? What are linear mixed models "doing" in the background that makes participant-level slopes different from what we would get in a linear regression on that participant's data? It might be important to note that the participant was the only unit of grouping the data in the linear mixed model (e.g., the random structure was 1 + Independent variable|Participant).
Any pointers would be much appreciated, thank you in advance! (P.S. This question is not about whether doing individual linear regressions is a better or a worse approach to linear mixed models as was discussed here, I would just like to better understand how different estimates are calculated.)
As an example, see the distribution of individual participants' slopes from the (a) linear mixed model (coef(model)$participant) and (b) participant-level linear regressions:
a)
b) 
