Assumptions of multilevel analysis In response to another question StasK writes:

In multilevel analysis, you have to make strong assumptions: (i) that
  your random effects are normal (or, if you have random slopes as long
  as random intercepts, that the joint distribution is multivariate
  normal), (ii) that your model contains all relevant variables, so that
  you are safe assuming that errors and regressors are uncorrelated at
  all levels, (iii) you have enough observations at each level to really
  utilize the asymptotic theory results concerning the likelihood ratio
  test statistics and inverse of the information matrix as the estimator
  of the variances of the parameter estimates. These assumptions are
  swept under the carpet, most of the time, and rarely if ever checked.



*

*Is this an exhaustive list of the assumptions?

*Why are these the assumptions?

*What are the best ways to test those assumptions?

*What should be done if the assumptions fail? 

 A: I have a partial answer based on a course I just finished. Here is the list of assumptions I was provided:


*

*Level 1 residuals ($_{}$) are independent and normally distributed. This can be tested with a histogram or normality test.

*Level 1 predictors ($_{}$) are independent of Level 1 residuals ($_{}$). Do a scatter plot between the two or test the correlation.

*Level 2 residuals ($_{}$) are multivariate normal. This means that each one is normally distributed an all linear combinations are normally distributed. I'm not familiar with how to test the latter.

*Level 2 predictors ($_$) are independent of Level 2 errors ($_{}$). You can do a scatter plot between the predictor(s) and Level 2 errors.

*The Level 1 residuals ($_{}$) are independent of the Level 2 residuals ($_{}$). This one is difficult to test. I don't have any notes on it.

*The  predictors at Level 1 are independent of the residuals at Level 2, and the predictors at Level 2 are independent of the residuals at Level 1. Same for this one.

*Level 1 residuals are homogenous across the clusters. You can do a Level 1 residual box plot and visually look for similar spreads by cluster, or do a test for homogeneity of level-1 variance. HLM has this option.
In response to your #4, if the normality assumptions aren't met, one option (at least within HLM) is to use the "with robust standard errors" output.
