[R-sig-ME] glmm with correlated residuals

Ben Bolker bbolker at gmail.com
Fri Dec 21 21:05:58 CET 2012


  The model I had in mind was hierarchical:

  Y_i ~ Poisson(lambda_i)
  lambda_i = exp(eta_i)
  eta ~ MVN(mu=X beta, Sigma=sigma^2*C(rho))

where C(rho) is [for example] the correlation matrix corresponding to an
AR1 model with correlation rho. So the variance-covariance structure of
Y is *not* AR1 (but presumably there's some reasonable one-to-one
relationship between the variance-covariance structures of eta and of Y
-- I'm guessing that the correlation is just diluted by the additional
Poisson variance; this might be reasonable to approach with a GEE for
the marginal variance-covariance structure).  It also captures (I think)
the *biological* goal, which is to characterize count data where there
is evidently serial autocorrelation [which may be of primary interest,
or a statistical nuisance to be accounted for].

  I think that's perfectly well-defined and sensible, and
straightforward to estimate with Bayesian or frequentist hierarchical
modeling tools, although I'm happy to be corrected ...

  I completely agree that it's not clear what glmmPQL and friends are
actually doing when one specifies a model like

glmmPQL(y~x,random=...,correlation=corAR1())  ;

I wish someone (not me!) would sit down and figure it out. Dormann et al
2007 suggest on the basis of simulation studies that one can use glmmPQL
in this way, but I don't recall that they offer much of a theoretical
justification of what model is *actually* being fitted ...

Dormann, Carsten F. et al. 2007. “Methods to Account for Spatial
Autocorrelation in the Analysis of Species Distributional Data: a
Review.” Ecography 30 (5): 609–628.
doi:10.1111/j.2007.0906-7590.05171.x.
http://dx.doi.org/10.1111/j.2007.0906-7590.05171.x.

 Ben B.

On 12-12-21 02:31 PM, Douglas Bates wrote:
> <rant>
> I realize I am coming in late to the discussion and I may end up
> sounding like a grumpy old man spoiling everyone's fun, but I don't
> think there is a model with the properties being discussed.  For a model
> in which the conditional distribution of the response given the random
> effects is Gaussian (i.e. a linear mixed-effects model or certain types
> of nonlinear mixed-effects models as described in Pinheiro and Bates,
> 2000) it is possible to modify the variance-covariance structure
> separately from the conditional mean.  Doing so will also carry over to
> the marginal distribution of the response.  But this property is not
> general - it is quite specifically a property of the Gaussian (or
> "normal") distribution.
> 
> So, to me, there is no point in discussing whether the parameters in
> this model should be estimated by quasi-likelihood (i.e. something that
> sort-of, kind-of looks like a likelihood) or by defining a series of
> conditional distributions for an MCMC iterative scheme or ...  This may
> sound priggish or the argument of an abstracted theoretician with no
> connection to actual data analysis but, if you want to define maximum
> likelihood estimators or whatever criterion you want to use, you must
> first have a likelihood and to get a likelihood you need a probability
> model.  And, to the best of my knowledge, there is not probability model
> with these properties. 
> 
> When you don't have a probability model you have moved from statistical
> analysis to "data mining" algorithms, which is okay but you should be
> explicit about that and not couch is as if it were a generalized linear
> mixed model.  All you have is an algorithm - you put numbers in and you
> get numbers out and if you like the numbers you get out then more power
> to you
> 
> To me this is one of the big pitfalls of generalized linear mixed models
> - there is an assumption that they generalize all the properties of
> linear mixed models but they don't.  There are properties of the
> Gaussian distribution that do not generalize and for years people have
> concentrated on the details of how to fit or approximately fit models
> that don't exist.  Concentrating on the low-level details of the
> mechanism without considering whether such a model could even exist is
> bound to lead you into trouble.
> </rant>
> 
> 
> 
> 
> On Thu, Dec 20, 2012 at 6:59 PM, Adam Smith <raptorbio at hotmail.com
> <mailto:raptorbio at hotmail.com>> wrote:
> 
> 
>     I believe the gamm function in mgcv can also fit this model.  IIRC,
>     the function also permits use of the negative binomial distribution
>     (but you have to provide theta)...
>     It also uses PQL, so the same concerns apply.
>     Regards,
>     Adam
> 
>     > Date: Thu, 20 Dec 2012 22:02:23 +0000
>     > From: j.hadfield at ed.ac.uk <mailto:j.hadfield at ed.ac.uk>
>     > To: bbolker at gmail.com <mailto:bbolker at gmail.com>
>     > CC: r-sig-mixed-models at r-project.org
>     <mailto:r-sig-mixed-models at r-project.org>
>     > Subject: Re: [R-sig-ME] glmm with correlated residuals
>     >
>     > Hi Emilio,
>     >
>     > asreml will fit such a model but the PQL method it uses for
>     > approximating the likelihood can give dodgy results in some cases. For
>     > Poisson data, particularly if it has high mean, it might be OK though.
>     > It is fast enough that you can check the degree of bias using
>     > simulation.
>     >
>     > You can fit spatial simultaneous autoregressive lag models in MCMCglmm
>     > but only for Gaussian data. It would be very difficult for me to
>     > implement these for  non-Gaussian data.
>     >
>     > AR(1) models with nugget effect could be fitted, but only if the
>     > correlation parameter is known. The inverse of the correlation matrix
>     > could then be passed to ginverse and the scale estimated, but this is
>     > not that much use I guess.
>     >
>     >
>     > Cheers,
>     >
>     > Jarrod
>     >
>     >
>     >
>     >
>     >
>     > Quoting Ben Bolker <bbolker at gmail.com <mailto:bbolker at gmail.com>>
>     on Thu, 20 Dec 2012 20:53:50
>     > +0000 (UTC):
>     >
>     > > Joshua Wiley <jwiley.psych at ...> writes:
>     > >
>     > >>
>     > >> Hi Emilio,
>     > >>
>     > >> I would suggest doing it in MCMCglmm.  It can handle residual
>     > >> structures too.  This is a nice starting guide:
>     > >>
>     http://cran.r-project.org/web/packages/MCMCglmm/vignettes/CourseNotes.pdf
>     > >>
>     > >> Cheers,
>     > >>
>     > >> Josh
>     > >
>     > >   Josh, are you sure?  I don't find "autoreg" anywhere in the
>     course notes.
>     > > I don't personally know of an easy way to do this other than
>     using INLA
>     > > (which I haven't tried much); glmmPQL (easy but perhaps dicey
>     depending
>     > > on the circumstances, and sometimes hard to figure out whether it's
>     > > fitting a well-defined model); or hand-coding in
>     WinBUGS/JAGS/Stan or
>     > > AD Model Builder ...  You could also give up on Poisson-ness
>     (you said
>     > > there was heterogeneity of variance -- not quite sure what that
>     means
>     > > in this context?) and fit a GLS model with appropriate variance
>     > > structure (using gls() with weights= and correlation= arguments
>     set).
>     > >
>     > >   I'd love to hear other answers.
>     > >
>     > >   Ben
>     > >
>     > >
>     > >>
>     > >> On Thu, Dec 20, 2012 at 8:37 AM, Emilio A. Laca <ealaca at ...> wrote:
>     > >> > Fellow R users,
>     > >
>     > >> > what package would you recommend for fitting a poisson glmm with
>     > >     one random effect and residuals that are correlated (most likely
>     > >     AR(1)) and have heterogeneity of variance?  I have successfully
>     > >     fitted the model without addressing the structured residuals in
>     > >     MCMCglmm.  I would appreciate it very much if you could point me
>     > >     in the direction of an example.
>     > >
>     > >> > Emilio A. Laca, Professor
>     > >
>     > >  [snip]
>     > >
>     > > _______________________________________________
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>     <mailto:R-sig-mixed-models at r-project.org> mailing list
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>     > >
>     > >
>     >
>     > --
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