Martin,
I've just submitted mgcv_1.7-19 to CRAN, which includes a major upgrade
of the p-value computation for random effect terms (and any other smooth
term which can be penalized to zero as part of estimation). The new
p-values are still conditional on the smoothing parameter/variance
component estimates, but given those estimates are based on the null
distribution of the restricted likelihood ratio statistic. Basically, if
you are going to condition on the smoothing parameters, then this is the
right thing to do, but if smoothing parameter estimates are very
uncertain, then the p-values may be underestimates (in simulations the
underestimation seems to be rather modest).
In the Gaussian mixed additive model case, the alternative is to use the
RLRsim package, while for GLMMs the only way to further improve things
is by writing code to simulate for the null distribution of the test
statistic...
Thanks for pushing be into action on this!
best,
Simon
On 25/06/12 16:31, Simon Wood wrote:
Martin,
I had a nagging feeling that there must be more to this than my previous
reply suggested, and have been digging a bit further. Basically I would
expect these p-values to not be great if you had nested random effects
(such as main effects + their interaction), but your case looked rather
straightforward...
After some digging it turns out that the p-value for Placename is wrong
in this case, as you suspected. R routine `qr' has a `tol' argument that
by default is set to 1e-7. In the computation of the test statistic for
"re" terms I had called qr without changing this default to the value 0
that is actually appropriate (I should have noticed this, I've been
caught out by it before). With the correct 'tol', Placement is
non-significant. This will be fixed in the next release, but that won't
happen until I've tested the random effects p-value approximations a bit
more thoroughly.
best,
Simon
On 11/06/12 14:56, Martijn Wieling wrote:
Dear Simon,
I ran an additional analysis using bam (mgcv 1.7-17) with three random
intercepts and no non-linearities, and compared these to the results
of lmer (lme4). Using bam results in a significant random intercept
(even though it has a very low edf-value), while the lmer results show
no variance associated to the random intercept of Placename. Should I
drop the random intercept of Placename and if so, how is this apparent
from the results of bam?
Summaries of both models are shown below.
With kind regards,
Martijn
#### l1 = lmer(RefPMIdistMeanLog.c ~ geogamfit + (1|Word) + (1|Key) +
(1|Placename), data=wrddst); print(l1,cor=F)
Linear mixed model fit by REML
Formula: RefPMIdistMeanLog.c ~ geogamfit + (1 | Word) + (1 | Key) + (1
| Placename)
Data: wrddst
AIC BIC logLik deviance REMLdev
-44985 -44927 22498 -45009 -44997
Random effects:
Groups Name Variance Std.Dev.
Word (Intercept) 0.0800944 0.283009
Key (Intercept) 0.0013641 0.036933
Placename (Intercept) 0.0000000 0.000000
Residual 0.0381774 0.195390
Number of obs: 112608, groups: Word, 357; Key, 320; Placename, 40
Fixed effects:
Estimate Std. Error t value
(Intercept) -0.00342 0.01513 -0.23
geogamfit 0.99249 0.02612 37.99
#### m1 = bam(RefPMIdistMeanLog.c ~ geogamfit + s(Word,bs="re") +
s(Key,bs="re") + s(Placename,bs="re"), data=wrddst,method="REML");
summary(m1,freq=F)
Family: gaussian
Link function: identity
Formula:
RefPMIdistMeanLog.c ~ geogamfit + s(Word, bs = "re") + s(Key,
bs = "re") + s(Placename, bs = "re")
Parametric coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.00342 0.01513 -0.226 0.821
geogamfit 0.99249 0.02612 37.991<2e-16 ***
---
Signif. codes: 0 â***â 0.001 â**â 0.01 â*â 0.05 â.â 0.1 â â 1
Approximate significance of smooth terms:
edf Ref.df F p-value
s(Word) 3.554e+02 347 634.716<2e-16 ***
s(Key) 2.946e+02 316 23.054<2e-16 ***
s(Placename) 1.489e-04 38 7.282<2e-16 ***
---
Signif. codes: 0 â***â 0.001 â**â 0.01 â*â 0.05 â.â 0.1 â â 1
R-sq.(adj) = 0.693 Deviance explained = 69.4%
REML score = -22498 Scale est. = 0.038177 n = 112608
On Wed, May 23, 2012 at 11:30 AM, Simon Wood-4 [via R]
<ml-node+s789695n4631060...@n4.nabble.com> wrote:
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
[hidden email]
+31(0)614108622
*******************************************
University of Groningen
http://www.rug.nl/staff/m.b.wieling
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