Specifically, how does subject-level random intercept and random slope influence the goodness-of-fit (R-squared) of the model?
And, if subject A contributes 10 data-points and subject B contributes 5 to the whole dataset, wouldn't A account for more of the total residual error than B? How do multilevel models control for this?
See, for example, https://www.amazon.com/Multilevel-Statistical-Models-Harvey-Goldstein/dp/0470748656/
It is hard to tell, how much random slope and intercept influence the modelfit. It really depends on the data. But I think tha there is no genuine R^2 for MLM, but pseudo R^2's. Therefore, it might be helpful to use information criteria like AIC or BIC for model comparison, since they can also incorporate model complexity.
What do you mean with "account for", in the sense "makes the residual variance larger"? If you meant this, then I would say "no", since it really depends on the data delivered by each cluster/participant itself. Subject A might have very small deviations from the specific estimate (e.g. slope), whereas B has large differences.
Here are a couple more resources you may find helpful.
- https://www.bristol.ac.uk/cmm/
- https://www.pilesofvariance.com/
As Rainer Duesing noted there is no such thing as R^2 for a multilevel model - at least in the sense that there is no single statistic that has all the properties of R^2. Various alternatives exist that have some of the same properties, but need to be used carefully. Also note that R^2 isn't necessarily the best goodness of fit measure in a linear model and tends to be overemphasised.
Your second question is probably more interesting because it gets at a fundamental property of multilevel models - which is they incorporate shrinkage. If we have two units A and B with 5 and 10 observations respectively we traditionally think of there being two options: (mean_A+mean_B)/2 to estimate the mean or (sum of all observations)/15. What a multilevel model does is use a kind of compromise estimate between these - effectively closer to the mean of the less noisy (often larger) unit than the (mean_A+mean_B)/2. This shrunken estimator has nice statistical properties for some purposes - arguably it is more efficient and generalises better.
There are lots of shrinkage estimators (I think) but in multilevel modeling the frequentist models use Empirical Bayes estimates.
https://en.wikipedia.org/wiki/Empirical_Bayes_method#:~:text=Empirical%20Bayes%20methods%20are%20procedures,before%20any%20data%20are%20observed.
This behaves a bit like a Bayesian prior but is estimated from the data. True Bayes also incorporates shrinkage through the prior and lots of multilevel modeling these days is Bayesian.
(The above is an attempt at a very simple non-technical explanation - there is of course more to it than this, but you really need to get a good text book for this)
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