Multilevel model vs. separate models for each level What are the advantages and disadvantages of running separate models vs. multilevel modeling?
More particularly, suppose a study examined patients nested within doctors' practices nested within countries. What are the advantages/disadvantages of running separate models for each country vs. a three level nested model? 
 A: The question is dated but I think it's very important. The best answer I can get is from Joop J Hox (2010) book "Multilevel Analysis Techniques and Applications, Second Edition". 
Suppose two-level hierarchical data with $p$ explanatory variables at the lowest level and $q$ explanatory variables at the highest level. Then, at page 55, he writes:

An ordinary single-level regression model for the same data would
  estimate only the intercept, one error variance, and p + q regression
  slopes. The superiority of the multilevel regression model is clear,
  if we consider that the data are clustered in groups. If we have 100
  groups, estimating an ordinary multiple regression model in each group
  separately requires estimating 100 × (1 regression intercept + 1
  residual variance + p regression slopes) plus possible interactions
  with the q group-level variables. Multilevel regression replaces
  estimating 100 intercepts by estimating an average intercept plus its
  residual variance across groups, assuming a normal distribution for
  these residuals. Thus, multilevel regression analysis replaces
  estimating 100 separate intercepts by estimating two parameters (the
  mean and variance of the intercepts), plus a normality assumption. The
  same simplification is used for the regression slopes. Instead of
  estimating 100 slopes for the explanatory variable pupil gender, we
  estimate the average slope along with its variance across groups, and
  assume that the distribution of the slopes is normal. Nevertheless,
  even with a modest number of explanatory variables, multilevel
  regression analysis implies a complicated model. Generally, we do not
  want to estimate the complete model, first because this is likely to
  get us into computational problems, but also because it is very
  difficult to interpret such a complex model. We prefer more limited
  models that include only those parameters that have proven their worth
  in previous research, or are of special interest for our theoretical
  problem.

That's for the description. Now the pages 29-30 will answer your question more accurately.

The predicted intercepts and slopes for the 100 classes are not
  identical to the values we would obtain if we carried out 100 separate
  ordinary regression analyses in each of the 100 classes, using
  standard ordinary least squares (OLS) techniques. If we were to
  compare the results from 100 separate OLS regression analyses to the
  values obtained from a multilevel regression analysis, we would find
  that the results from the separate analyses are more variable. This is
  because the multilevel estimates of the regression coefficients of the
  100 classes are weighted. They are so-called Empirical Bayes (EB) or
  shrinkage estimates: a weighted average of the specific OLS estimate
  in each class and the overall regression coefficient, estimated for
  all similar classes.
As a result, the regression coefficients are shrunk back towards the
  mean coefficient for the whole data set. The shrinkage weight depends
  on the reliability of the estimated coefficient. Coefficients that are
  estimated with small accuracy shrink more than very accurately
  estimated coefficients. Accuracy of estimation depends on two factors:
  the group sample size, and the distance between the group-based
  estimate and the overall estimate. Estimates for small groups are less
  reliable, and shrink more than estimates for large groups. Other
  things being equal, estimates that are very far from the overall
  estimate are assumed less reliable, and they shrink more than
  estimates that are close to the overall average. The statistical
  method used is called empirical Bayes estimation. Because of this
  shrinkage effect, empirical Bayes estimators are biased. However, they
  are usually more precise, a property that is often more useful than
  being unbiased (see Kendall, 1959).

I hope it's satisfying.
A: Advantage: The ability to explicitly test for differences in parameters by cluster (i.e. differences in significance do not mean significant differences). 
A: Specifying a random effect involves assuming that the means of those levels are samples from a normal distribution.  Better to specify them as fixed effects, AKA dummy variables if this assumption doesn't fit your data.  In this way you are controlling for groupwise heterogeneity in the mean (at that level), but you are NOT allowing for heterogeneity in responses to your lower-level variables.
If you expect heterogeneity in response to your lower-level explanatory variables, separate models make sense, unless you want to run some sort of random coefficient model (which again involves the assumption that coefficients are normally distributed).
(I believe there are methods for non-normal random effects, but nothing as widely used or accessible as lme)
