Thank you so much, Douglas and Ben.
This was very helpful.


Asaf


On 22 February 2014 16:01, Douglas Bates <bates@stat.wisc.edu> wrote:

> Yes, the order is determined by decreasing numbers of levels in the
> grouping factors.  For nested grouping factors there is an advantage in
> ordering the terms in this way in that Z'Z has a big diagonal block in the
> upper-left corner.  I thought it would be advantageous for non-nested
> grouping factors when creating a fill-reducing permutation but it turns out
> this is not as simple as I had hoped.  It is generally not bad but it isn't
> always the best.
>
> The goal is to find an ordering of the columns of Z that reduces fill-in
> when calculating the sparse Cholesky factorization.  Getting the best
> ordering can be shown to be NP-hard (http://en.wikipedia.org/wiki/NP-hard)
> so it is not feasible if you have many columns in Z.  There are heuristics
> that can be used.  The Matrix and lme4 packages use the Approximate Minimal
> Degree (AMD) method or the column-oriented approximate minimal degree
> method (COLAMD) developed by Tim Davis.  Other software uses graph
> partitioning algorithms like Metis.
>
> In many cases for mixed-effects models, these algorithms are unnecessary.
> One characteristic of models with nested grouping factors (including the
> trivial case of a model with a single grouping factor) is that there is no
> fill-in regardless of the permutation chosen.  For the rare case of
> completely crossed grouping factors (say Subject and Item in item response
> situations) the ordering with the greatest number of levels first is
> optimal because the Cholesky factor is dense in the columns corresponding
> to the second and subsequent factors.  It's in the middle ground of
> partially crossed grouping factors that the various fill-reducing methods
> are useful.
>
>
>
> On Sat, Feb 22, 2014 at 11:45 AM, Ben Bolker <bbolker@gmail.com> wrote:
>
>> I *believe* that the order is determined by the number of levels of the
>> grouping variables (decreasing? increasing? don't remember), in order to
>> optimize the internal structure of the random effects design matrices,
>> i.e.
>> to minimize the amount of fill-in.  Don't remember the details at the
>> moment, it will be mentioned somewhere in Doug Bates's various
>> presentations on lme4/mixed models. View shared post<
>> https://www.google.co.uk/#>
>>
>>
>>
>> On Wed, Feb 19, 2014 at 7:30 PM, Asaf Weinstein
>> <asafw.at.wharton@gmail.com>wrote:
>>
>> > Hi,
>> >
>> > I am running an (unbalanced) two-way random effects model.
>> > I was not able to figure out myself how lmer decides which of the
>> variance
>> > components (of the two random effects) is reported first in the output
>> (in
>> > other words, how the order in getME(fm, 'theta') is determined?).
>> >
>> > Thanks so much,
>> > Asaf
>> >
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>> >
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>> >
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