R help - May 2012 - mgcv: inclusion of random intercept in model - based on p-value of smooth or anova?

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

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
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>

-- 
Simon Wood, Mathematical Science, University of Bath BA2 7AY UK
+44 (0)1225 386603               http://people.bath.ac.uk/sw283

Having looked at this further, I've made some changes in mgcv_1.7-17 to 
the p-value computations for terms that can be penalized to zero during 
fitting (e.g. s(x,bs="re"), s(x,m=1) etc).

The Wald statistic based p-values from summary.gam and anova.gam (i.e. 
what you get from e.g. anova(a) where a is a fitted gam object) are 
quite well founded for smooth terms that are non-zero under full 
penalization (e.g. a cubic spline is a straight line under full 
penalization). For such smooths, an extension of Nychka's (1988) result 
on CI's for splines gives a well founded distributional result on which 
to base a Wald statistic. However, the Nychka result requires the 
smoothing bias to be substantially less than the smoothing estimator 
variance, and this will often not be the case if smoothing can actually 
penalize a term to zero (to understand why, see argument in appendix of 
Marra & Wood, 2012, Scandinavian Journal of Statistics, 39,53-74).

Simulation testing shows that this theoretical concern has serious 
practical consequences. So for terms that can be penalized to zero, 
alternative approximations have to be used, and these are now 
implemented in mgcv_1.7-17 (see ?summary.gam).

The approximate test performed by anova(a,b) (a and b are fitted "gam"
objects) is less well founded. It is a reasonable approximation when 
each smooth term in the models could in principle be well approximated 
by an unpenalized term of rank approximately equal to the edf of the 
smooth term, but otherwise the p-values produced are likely to be much 
too small. In particular simulation testing suggests that the test is 
not to be trusted with s(...,bs="re") terms, and can be poor if the 
models being compared involve any terms that can be penalized to zero 
during fitting. (Although the mechanisms are a little different, this is 
similar to the problem we would have if the models were viewed as 
regular mixed models and we tried to use a GLRT to test variance 
components for equality to zero).

These issues are now documented in ?anova.gam and ?summary.gam...

Simon

On 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
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>

-- 
Simon Wood, Mathematical Science, University of Bath BA2 7AY UK
+44 (0)1225 386603               http://people.bath.ac.uk/sw283