How to explain HLM (response to an email)

Hello XXX, thanks for your email! I recently taught a premier for HLM, so I hope I can update my website with new material.

This time, I tried to say that HLM is simple, but equations make it difficult to learn (which is also the case for learning music instruments, like piano).

In the attached graph, I tried to compare my intuitive model (mental representation model) and a formal equation. When it is in my mind, coefficients are either random (red font) or fixed (blue). The red coefficients are adjusted for precision/reliability, while the blue ones are not (like OLS coefficients). I tried to show how the equation expresses the same idea, but I feel it quickly gets annoying.

HLM slide sample

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