[R-sig-ME] 3-level binomial model
Steven J. Pierce
pierces1 at msu.edu
Thu Apr 17 23:50:22 CEST 2008
Hi folks,
For GLMMs, I think the median odds ratio (MOR) as presented in the articles
listed below is more meaningful and interpretable than the ICC. Here's a
snippet of code I wrote to calculate that statistic. Perhaps this could be
refined to deal with a 3-level structure.
# Set up functions for calculating the median odds ratio (MOR) and the
interval
# odds ratio (IOR) from a multilevel logistic regression, as defined by
these
# articles/presentations:
#
# Larsen, K. (2006, November 24). New measures for understanding the
multilevel
# logistic regression model [Electronic version]. Seminar at the workshop
# “Statistische Methoden für korrelierte Daten”, Bochum, Germany.
# http://www.bgfa.ruhr-uni-bochum.de/statmetepi/2006/Vortrag_Larsen.pdf
# Larsen, K., & Merlo, J. (2005). Appropriate assessment of neighborhood
effects
# on individual health: Integrating random and fixed effects in multilevel
# logistic regression [Electronic version]. American Journal of
Epidemiology,
# 161(1), 81-88.
# http://aje.oxfordjournals.org/cgi/content/abstract/161/1/81
# Larsen, K., Petersen, J. H., Bundtz-Jørgensen, E., & Endahl, L. (2000).
# Interpreting parameters in the logistic regression model with random
effects
# [Electronic version]. Biometrics, 56, 909-914.
# http://www.jstor.org/view/0006341x/di020233/02p0159d/0
#
# These functions assume that you have run a multilevel logistic regression
# and now know the variance component associated with the clustering
variable.
# The lmer() command from the lme4 package can estimate that variance and
# will report it in the Random Effects matrix shown in the model's summary.
# Both functions as written here calculate correctly from the examples in
# Larsen & Merlo (2005).
MOR <- function(my.var, digits = 2)
{ # MOR arguments: my.var = variance associated with level 2
clustering variable
# digits = number of decimal places to which MOR value will be
rounded.
Median.OR <- round(exp(sqrt(2*my.var)*qnorm(.75)), digits)
paste("Median Odds-Ratio (MOR) = ", Median.OR) }
IOR <- function(my.var, my.coef, my.diff, digits = 2)
{ # IOR arguments: my.var = variance associated with level 2
clustering variable
# my.coef = fixed effect coefficient associated with a level 2
covariate
# my.diff = difference between 2 different possible values on
the level
# 2 covariate associated with my.coef
# digits = number of decimal places to which MOR value will be
rounded.
IOR.LL <- round(exp(my.coef*my.diff +
sqrt(2*my.var)*qnorm(.10)), digits)
IOR.UL <- round(exp(my.coef*my.diff +
sqrt(2*my.var)*qnorm(.90)), digits)
print("80% Interval Odds Ratio lower & upper limits (IOR.LL &
IOR.UL)")
cbind(my.var, my.coef, my.diff, IOR.LL, IOR.UL)
}
Steven J. Pierce
E-mail: pierces1 at msu.edu
-----Original Message-----
From: Doran, Harold [mailto:HDoran at air.org]
Sent: Thursday, April 17, 2008 7:59 AM
To: Andrew Robinson; Iasonas Lamprianou
Cc: r-sig-mixed-models at r-project.org
Subject: Re: [R-sig-ME] 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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> >
> >
> > End of R-sig-mixed-models Digest, Vol 16, Issue 36
> > **************************************************
> >
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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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