[R] lme: null deviance, deviance due to the random effects, residual deviance

[Prof Brian Ripley]

On Mon, 1 May 2006, Spencer Graves wrote:

> 	  As far as I know, the term "deviance" has no standard definition.  A
> good, fairly common definition (I think) is that the deviance is "up to
> [an additive] constant, minus twice the maximised log-likelihood. Where
> sensible, the constant is chosen so that a saturated model has deviance
> zero."  ("http://finzi.psych.upenn.edu/R/library/gnm/html/gnm.html".)
> Because of this "constant", the "the proportion of deviance 'explained'
> by the model" in not a well defined concept.  I found this definition
> using RSiteSearch("deviance define").  However, even this definition is
> not used consistently;  it's not even used for 'deviance.lm', which I
> discovered using methods("deviance") and methods("logLik") followed by
> 'getAnywhere("deviance.lm"), etc.

It _is_ used by deviance.lm.  In that case the saturated model has 
infinite log-likelihood, so it would not be sensible to choose an infinite 
constant and if you did the result for the saturated model would be NaN 
(= Inf - Inf).

McCullagh & Nelder (1989) would be the authorative reference, but the 1982 
first edition manages to use 'deviance' in three separate senses on one 
page.  In particular, what you are describing is the unscaled residual 
deviance.

> 	  This is not a "trivial and stupid question".  Instead, it's connected
> to subtle issues in statistical methods, and this reply may contribute
> more obfuscation than enlightenment.  If you describe some more specific
> application where you might want to use something like this and what you
> are trying to achieve, you might get a more useful reply.

He may be looking for extensions/analogues of R^2, of which there exist 
several.

> 	  hope this helps,
> 	  spencer graves
>
> Patrick Giraudoux wrote:
>
>> A maybe trivial and stupid question:
>>
>> In the case of a lm or glm fit, it is quite informative (to me) to have
>> a look to the null deviance and the residual deviance of a model. This
>> is generally provided in the print method or the summary, eg:
>>
>> Null Deviance:      658.8
>> Residual Deviance: 507.3
>>
>> and (a bit simpled minded) I like to think that the proportion of
>> deviance 'explained' by the model is (658.8-507.3)/658.8 = 23%
>>
>> In the case of lme models, is it possible and reasonable to try and get the:
>> - null deviance
>> - the total deviance due to the the random effect(s)
>> - the residual deviance?
>>
>> With the idea that Null deviance = Fixed effects + Random Effects +
>> Residuals
>>
>> If yes how to do it ? A lme object provides the following:
>>
>> > names(glm6)
>>  [1] "modelStruct"  "dims"         "contrasts"    "coefficients"
>>  [5] "varFix"       "sigma"        "apVar"        "logLik"
>>  [9] "numIter"      "groups"       "call"         "method"
>> [13] "fitted"       "residuals"    "fixDF"        "family"
>>
>> so no $null.deviance and $deviance elements as in glm objects...
>>
>> I tried to find out an answer on R-help & Pineihro & Bates (2000).
>> Partial success only:
>>
>> - null deviance: Response: possibly yes: see
>> http://tolstoy.newcastle.edu.au/R/help/05/12/17796.html (Spencer
>> Graves). The (null?) deviance is -2*logLik(mylme), but a personnal trial
>> with some glm objects did not led to the same numbers that the one given
>> by the print.glm method...
>>
>> - the deviance due to the the random effect(s). I was supposing that the
>> coefficients given by ranef(mylme) may be an entry... but beyond this, I
>> guess those coefficients must be weighed in some way... which is a far
>> beyond my capacities in this matter...
>>
>> - residual deviance. I was supposing that it may be
>> sum(residuals(mylme)^2). With some doubts as far as I feel that I am
>> thinking sum of squares estimation in the context of likelihood and
>> deviance estimations... So  most likely irrelevant. Moreover, in the
>> case I was exploring, this quantity is much larger than the null
>> deviance computed as above...
>>
>> Any hint appreciated,
>>
>> Patrick Giraudoux

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595