[R-sig-ME] A question about LME4

Steve Walker steve.walker at utoronto.ca
Tue Aug 11 19:05:45 CEST 2015

On 2015-08-11 11:37 AM, Douglas Bates wrote:
> As Ben mentioned in his off-list reply to you, the paper
> http://arxiv.org/abs/1406.5823 describes in some detail the computational
> methods that are used in lme4.

This arxiv paper (in press at JSS) is pretty long, although hopefully 
comprehensive.  To help pinpoint things a bit check out equations 34 and 
41.  These give the ML and REML criteria (on the deviance scale) that 
are used within lmer.

> The fact that they don't correspond to
> those in the 2006 paper by Gurka in The American Statistician is not an
> oversight.  They are quite superior to any methods based on the equations
> in that paper, as they should be.  We have spent the last 20 years or so
> developing them.

To expand on these remarks a bit, both the fixed effects coefficient 
vector and residual variance parameter are profiled out, which makes 
computations more efficient.  Also, correlations among random effects 
are partly accounted for in these equations using the log determinant of 
a sparse Cholesky factor, which is very efficiently updated over 
iterations of the nonlinear optimizer.


> On Tue, Aug 11, 2015 at 10:31 AM <Subhash.Chandra at ecodev.vic.gov.au> wrote:
>> Appreciate your help and advice on (hopefully a simple) question. This
>> relates to the mathematical expressions that LME4 uses to estimate log
>> likelihood under maximum likelihood (ML) and under residual maximum
>> likelihood (ReML) for fitting linear mixed models (LMM). What I am
>> 'exactly' looking for is mathematical expressions for log likelihood in
>> the form of expressions (2), (4) and (5) in the attached paper by Gurka
>> (2006).
>> I have gone through the LME4 documentation
>> https://cran.r-project.org/web/packages/lme4/vignettes/lmer.pdf
>> https://cran.r-project.org/web/packages/lme4/vignettes/Theory.pdf
>> I must admit I am unable to figure out the answer to my question.
>> Regards,
>> Subhash
>> Dr Subhash Chandra | Chief Biometrician
>> Agriculture Research & Development Division
>> Department of Economic Development, Jobs, Transport & Resources
>> 255 Ferguson Road, Tatura 3616, Victoria, Australia
>> T:  03 5833 5397 | M: 0427 277 560 | E: Subhash.Chandra at ecodev.vic.gov.au
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