[R] mgcv: inclusion of random intercept in model - based on p-value of smooth or anova?

[Simon Wood]

Dear Martijn,

Thanks for the off line code and data: very helpful.

The answer to this is something of a 'can of worms'. Starting with the 
p-value inconsistency. The problem here really is that neither test is 
well justified in the case of s(...,"re") terms (and not having realised 
the extent of the problem it's not flagged properly).

In the case of the p-value from `summary', the p-value is computed as if 
the random effect were any other smooth. However the theory on which the 
p-values for smooths rests does not hold for "re" terms (basically the 
usual notion of smoothing bias is meaningless in the "re" case, and "re" 
terms can not usually be well approximated by truncated eigen 
approximations). The upshot is that you can get bias toward accepting 
the null. I'll revert to doing something more sensible for "re" terms 
for the next release, but it still won't be great, I guess.

The p-value from the comparison of models via 'anova' is equally suspect 
for "re" terms. Basically, this test is justified as a rough 
approximation in the case of usual smooth models, by the fact that we 
can approximate the model smooths by unpenalized reduced rank eigen 
approximations having degrees of freedom set to the effective degrees of 
freedom of the smooths. Again, however, such reduced rank approximations 
are generally not available for "re" terms, and I don't know if there is 
then a decent justification for the test in this case.

'AIC' might then be seen as the answer for model selection, but Greven 
and Kneib (2010, Biometrika), show that this is biased towards selecting 
the larger random effects model in this case (they provide a correction, 
but I'm not sure how easy it is to apply here).

You are left with a couple of sensible possibilities that are easy to 
use, if it's not clear from the estimates that the term is zero. Both 
involve using gam(...,method="REML") or gam(...,method="ML").

1. use gam.vcomp to get a confidence interval for the "re" variance 
component. If this is bounded well away from zero, then the result is 
clear.

2. Run a glrt test based on twice the difference in ML/REML score 
reported for the 2 models (c.f. chisq on 1 df for your case). This 
suffers from the usual problem of using a glrt test to test a variance 
component for equality to zero. (AIC based on this marginal likelihood 
doesn't fix the problem either --- see Greven and Kneib, again).

The second issue, that adding a fixed effect can reduce the EDF, while 
improving the fit, is less of a problem, I think. If I'm happy to select 
the degree of smoothness of a model by GCV, REML or whatever, then I 
should also be happy to accept that the model with the fewer degrees of 
freedom, but more variables, is better than the one with more degrees of 
freedom and fewer variables. (The converse that I would ever reject the 
better fitting, less complex model is obviously perverse).

You can get similar effects in ordinary linear modelling: adding an 
important predictor gives such an improvement in fit that you can drop 
polynomial dependencies on other predictors, so a model with more 
degrees of freedom but fewer variables does worse than one with fewer 
degrees of freedom and more variables... the issue is just a bit more 
prominent when fitting GAMs because part of model selection is 
integrated with fitting in this case.

best,
Simon




 > 08/05/12 15:01, Martijn Wieling wrote:
> Dear useRs,
>
> I am using mgcv version 1.7-16. When I create a model with a few
> non-linear terms and a random intercept for (in my case) country using
> s(Country,bs="re"), the representative line in my model (i.e.
> approximate significance of smooth terms) for the random intercept
> reads:
>                          edf       Ref.df     F          p-value
> s(Country)       36.127 58.551   0.644    0.982
>
> Can I interpret this as there being no support for a random intercept
> for country? However, when I compare the simpler model to the model
> including the random intercept, the latter appears to be a significant
> improvement.
>
>> anova(gam1,gam2,test="F")
> Model 1: ....
> Model 2: .... + s(BirthNation, bs="re")
>    Resid. Df Resid. Dev     Df Deviance      F    Pr(>F)
> 1    789.44     416.54
> 2    753.15     373.54 36.292   43.003 2.3891 1.225e-05 ***
>
> I hope somebody could help me in how I should proceed in these
> situations. Do I include the random intercept or not?
>
> I also have a related question. When I used to create a mixed-effects
> regression model using lmer and included e.g., an interaction in the
> fixed-effects structure, I would test if the inclusion of this
> interaction was warranted using anova(lmer1,lmer2). It then would show
> me that I invested 1 additional df and the resulting (possibly
> significant) improvement in fit of my model.
>
> This approach does not seem to work when using gam. In this case an
> apparent investment of 1 degree of freedom for the interaction, might
> result in an actual decrease of the degrees of freedom invested by the
> total model (caused by a decrease of the edf's of splines in the model
> with the interaction). In this case, how would I proceed in
> determining if the model including the interaction term is better?
>
> With kind regards,
> Martijn Wieling
>
> --
> *******************************************
> Martijn Wieling
> http://www.martijnwieling.nl
> wieling at gmail.com
> +31(0)614108622
> *******************************************
> University of Groningen
> http://www.rug.nl/staff/m.b.wieling
>
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-- 
Simon Wood, Mathematical Science, University of Bath BA2 7AY UK
+44 (0)1225 386603               http://people.bath.ac.uk/sw283