[R-sig-ME] 3-level binomial model
Andrew Robinson
A.Robinson at ms.unimelb.edu.au
Thu Apr 17 23:39:27 CEST 2008
Hi Harold,
On Thu, Apr 17, 2008 at 07:59:17AM -0400, Doran, Harold wrote:
> 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.
Ok -- I personally find log-odds of success hard to interpret
meaningfully, but that probably reflects my inexperience with glm.
> 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.
But it can be assumed to be true about the linear predictor,
conditional on your model. The variance components are assumed to
have constant variance in the linear predictor. Dealing with epsilon
is a bit strange, I assume that the MlWin manual provided that
insight. And, the linear predictor for the logit link function is the
log-odds of success.
So, if you can interpret the log-odds of success meaningfully then I
think that the variance components, and/or some monotonic function of
them, can be interpreted. I don't, but maybe others do.
> 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.
Within limits. Journal guardians can be negotiated with.
Andrew
> > -----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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> > > End of R-sig-mixed-models Digest, Vol 16, Issue 36
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> > >
> > >
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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/
> >
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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/
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