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Suppose I would want to say something about "explained variance" at level 1 and level 2 in my models, after adding a bunch of variables.

For instance, I first run a model with binary outcome Y with only cons (and variance at individual and school level).
Then I run a model with the same outcome Y, cons (variance at individual and school level), and 2 predictors (x1, x2).
I want to say something about the effect size, so I want to say how much variance in Y I have explained by x1 and x2 (i.e., how much variance at individual level and school level).

How could I do this (and test for significance as well) when I run models with MCMC with R2MLwin?


Follow up question:
I at first thought I might be able to use VPC for this (but that doesn't make sense, because it always adds up to 1 for the 2 levels).
But now I am wondering how I can calculate the VPC at all for MCMC logistic models (e.g., for model binomialMCMC from the manual).

I would be happy with any advise!

Professor Kelvyn Jones asked me to pass on the following:

This is a surprisingly hard question to answer as the level 1 binomial or multinomial variance cannot go down as additional predictors are entered into the model. The attached (rather long document I am afraid) considers the issue and discusses some R-squared approaches for the discrete outcome multilevel model.

In passing it does not matter how the estimates were obtained – quasi-likelihood or MCMC.

Kelvyn Jones

Thank you, I will look into this.

I also read something about "scaling" (Fielding, 2004, quality & quantity 38: 425-433, Scaling for residual variance components of ordered category responses in generalized linear mixed multilevel models).
Do you have an idea about whether this is something that can be achieved with R2MLwiN?

Dear Gerine,
I will agree with Kelvyn that this is indeed a challenging problem due to the scaling that occurs. I don't know Tony Fielding's paper that well so don't think this has been implemented in MLwiN unless he wrote macros himself. With regard VPC estimation there are 2 references (Goldstein, Browne and Rasbash in Understanding statistics 2002 and Browne et al. 2005 in JRSS Series A both of which you can get from google scholar via https://scholar.google.com/citations?us ... AAAJ&hl=en
Best wishes,
Bill.