[R-sig-ME] Strange logLik's

[Douglas Bates]

On Thu, Jan 8, 2009 at 11:39 AM, De Smedt Sebastiaan
<Sebastiaan.DeSmedt at ua.ac.be> wrote:
> Dear list members,

> I sampled a total of about 2000 leaves, nested in trees (10 per tree).
> In the field, those trees are nested in 9 different provenances. On the
> leaves, I measured some morphological parameters, for example 'length'.

> Every provenance is characterised by some environmental factors, as for
> example 'rainfall' and 'number of dry months'. These factors are the
> same within avery provenance. Further, each tree is characterised by a
> (arbitrary) degree of human pressure ('MG').

> The objective of the study is to see in which way the environmental and
> human factors are influencing leaf variables.

> I constructed a linear mixed-effects model:

>> lme1<-lme(lenght~MG+dry.months+rainfall, random=~1|provenance/tree)

> I've chosen this random structure because of the field reality (leaves
> are nested in trees, are nested in provenances).

> Further, I constructed two other models with different random
> structures:

>> lme2<-update(lme1,random=~1|provenance)

>> lme3<-update(lme1,random=~1|nr.boom)

> When I compare these models with an anova, I have this output:

>     Model df       AIC       BIC    logLik   Test  L.Ratio p-value
> lme1     1  6 -3123.050 -3089.028 1567.5252
> lme2     2  5 -1653.025 -1624.673  831.5125 1 vs 2 1472.025  <.0001
> lme3     3  5 -2947.977 -2919.625 1478.9884

> I have some questions about this output:

> 1)      Why are the logLik-values positive?

Are you asking this because you believe that log-likelihoods must be
negative?  If so, you are mistaken.  When the response is modeled by a
continuous random variable the likelihood is the value of the
probability density evaluated at the observed data.  A probability
density can exceed unity - all that is required is that it be
non-negative and that it integrate to unity.  Thus the log-likelihood
can be positive, as in this case.

> 2)      I always heard: 'The smaller the logLik, the better the model'.
> Is this also the case with positive logLik's? The residual errors of
> model lme1 and lme3 are much smaller than the residual error of lme2, so
> the 'tree'-effect is really important!

Again, I think you are laboring under the misconception that a
log-likelihood must be negative and, furthermore, when you say
"smaller is better" you mean smaller magnitude, not smaller
algebraically.

The method of estimation is "maximum likelihood" so larger values of
the likelihood or log-likelihood are preferred.  The likelihood ratio
test statistic for model 2 versus model 1 is positive, as it should
be, because model 2 is a specialization of model 1.
> 3)      The variables 'dry.months' and 'rainfall' are the same within
> every provenance. What means this for the model?
>
>
>
> Can anyone help me?
>
>
>
> Thanks!!!
>
>
>
>
>
> Sebastiaan De Smedt
>
> Dept. Bioscience Engineering
>
> University of Antwerp
>
> Groenenborgerlaan 171-V616
>
> B-2020 Antwerpen
>
> T: +32 (0)3/265.35.17
>
> F: +32 (0)3/265.32.25
>
>
>
>
>
> P  Help to save paper - do you really need to print this e-mail???  P
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