[R-sig-ME] 3-level binomial model

[Andrew Robinson]

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: +61-3-8344-4599
http://www.ms.unimelb.edu.au/~andrewpr
http://blogs.mbs.edu/fishing-in-the-bay/