3-level binomial model

I haven't really followed this thread, but I'd disagree and say that the
variance components have a very meaningful interpretation. If the fixed
effects are the log-odds of success, then the variance component would
be the variability in the log-odds for whatever units are of interest. 

On the issue of the ICC for the GLMM, to me this is all hocus-pocus.
This is a meaningful statistic in the world of linear models because the
within-person variance (or your level 1 variance) is assumed
homoskedastic. But, this is not true with generalized linear models.

Now, you can compute it as you did by fixing the level 1 variance at the
logistic scale and you can give reviewers whatever they want, but this
doesn't make it meaningful. So, waving a magic wand to make GLMM
estimates look like linear estimates is neat, but I think the better
path is to show your reviewers why this isn't a meaningful statistic. 

On the other hand, if you job is to get past the journal guardians for
tenure, do whatever they ask.

-----Original Message-----
From: r-sig-mixed-models-bounces at r-project.org 
[mailto:r-sig-mixed-models-bounces at r-project.org] On Behalf 
Of Andrew Robinson
Sent: Wednesday, April 16, 2008 6:40 PM
To: Iasonas Lamprianou
Cc: r-sig-mixed-models at r-project.org
Subject: Re: [R-sig-ME] 3-level binomial model

Hi Iasonas,

my interpretation of what you are doing by computing those 
quantities is that you are estimating the proportion of 
variance explained in the linear predictor.  

A complication with that strategy is that the non-linearity 
in the relationship between the linear predictor and the 
probability estimate induces an interaction between the 
components of variance in terms of their effect upon the 
probability.  Also, the linear predictor is commonly 
interpreted in the context of odds ratios (via 
exponentiation), which again doesn't line up with these 
variance components because of the non-linearity in the function.

So, it's not clear to me that the variance components have a 
direct useful interpretation in this model, although I may be 
mistaken.

I seem to recall that Gelman and Hill say sensible things 
about what to do either in this case or in a similar case, 
although again I may be mistaken.  I don't have my copy here.

So it seems to me that the reviewers are right to be 
cautious, and you might take a look in G&H.

I hope that  this helps.

Andrew


On Wed, Apr 16, 2008 at 05:51:07AM -0700, Iasonas Lamprianou wrote:

Thank you all for your suggestions. My question, however, 

is how to compute the % of the variance at the level of the 
school and at the level of the pupils. In other words, does 
the concept of  intraclass correlation hold in my context? If 
yes, then how can this be computed for the pupils and the 
schools? Is the decomposistion below reasonable? 

Prof. Bates, maybe you could suggesting something using the lmer?

VPCschool = VARschool/(VARschool+VARpupil+3.29) and 
  VPCpupil = VARpupil/(VARschool+VARpupil+3.29)
 
Dr. Iasonas Lamprianou
Department of Education
The University of Manchester
Oxford Road, Manchester M13 9PL, UK
Tel. 0044 161 275 3485
iasonas.lamprianou at manchester.ac.uk



On 16/04/2008, at 12:11 PM, David Duffy wrote:

I computed the school-level and the pupil-level variance 

like that 

(as described for 2-level models in MlWin manual): I 

assumed that 

my dependent variable is based on a continuous 

unobserved variable 

(perfectly valid according to my theoretical model). Therefore, 
eijk follows a logistic distribution with variance 

pi2/3=3.29. So,

VPCschool=VARschool/(VARschool+3.29)= 

0.17577/(0.17577+3.29)=6.4% 

and VPCpupil=VPCpupil 

/(VPCpupil+3.29)=0.19977/(0.19977+3.29)=7.3%.

The reviewers of my paper are not sure if this is the 

best way to 

do it. They may reject my paper and I worry because I have spent 
3months!!!! writing it. Any ideas to support my method 

or to use a 

better one?

Would an IRT model for seven "items" be more to their taste?  I 
don't think the substantive conclusions would be much different.

Multi-level IRT is more appropriate, this allows for the nesting 
within schools. There is a package mlirt that fits these 

models in a 

Bayesian framework, but I haven't tried it. There are commercial 
programs which will fit these, Mplus is advertised to and 

Latent Gold 

with the Syntax module will, at least for a unidimensional latent 
variable.

What is more worrying is the assumption of a single latent 

variable to 

model the correlation between tests.

Ken



------------------------------

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Andrew Robinson  
Department of Mathematics and Statistics            Tel: 
+61-3-8344-6410
University of Melbourne, VIC 3010 Australia         Fax: 
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http://www.ms.unimelb.edu.au/~andrewpr
http://blogs.mbs.edu/fishing-in-the-bay/