Hi,

When running a 2-level logistic null-model (intercept only) the icc can be derived e.g. using the threshold approach, which sets the level-1 variance at 3.29. There's is another method, however, proposed by Goldstein, which is based on simulation. This method is e.g. available as macro "vpc" in mlwin but is easily implemented in R. My question has to do with this alternative method to derive an icc value.

In the mlwin user's guide (and several papers) this alternative method is explained as producing an icc measure "on the probability scale". My question is twofold.

• Does "on the probability scale" mean that this icc measure does NOT provide an estimate of the intraclass correlation of the Y-values themselves? I mean, the correlation between two Y-values (0 or 1) of two randomly chosen, say, pupils in a randomly chosen class, is the quantity that I would expect an icc measure to estimate. The remark "on the probability scale" makes me doubt whether the alternative icc measure does indeed estimate this correlation as meant above.
  

• In this alternative icc measure, the formula  var(2) / (var(1) + var(2)) is used to calculate the icc value, with var(2) and (var(1) representing the level-2 and level-1 variances. The level-1 variance of the simulation method, it seems to me, is related to variance of Y-values (0/1) while the level-2 variance is a variance of probabilties, not of Y-values. At this point, it gets hard for me to keep up.
  

• Also, I don't see how the formule var(2) / ((var1) + var(2)) which is easily proven to indeed represent the correlation between to Y-values in case of continous Y and a linear model, can simply be applied to the simulated quantities resulting from a logistic model.