R help - Feb 2008 - mixed model nested ANOVA (part two)

First of all thank you for the responses.  I appreciate the
suggestions i have received thus far.

Just to reiterate

I am trying to analyze a data set that has been collected from a
hierarchical sampling design.  The model should be a mixed model
nested ANOVA.  The purpose of my study is to analyze the variability
at each spatial scale in my design (random factors, variance
components), and say something about the variability between regions
(fixed factor, contrast of means).  The data is as follows;

region (fixed)
Location (random)
Site(random)

site nested in location nested in region.

Also i have read in Quinn and Keough 2002, design and analysis of
experiments for biologists, that a variance component analysis should
only be conducted after a rejection of the null hypothesis of no
variance at that level.

I have tried to implement
mod1<-lmer(density ~ 1 + (1|site) + (1|location) + (1|region))

However, as i understand it, this treats all my factors as random.
Plus I do not know how to extract SS or MS from this model.

anova(mod1) gives me
Analysis of Variance Table
     Df Sum Sq Mean Sq

and summary(mod1) gives me
Linear mixed-effects model fit by REML
Formula: density ~ 1 + (1 | site) + (1 | location) + (1 | region)
   AIC   BIC logLik MLdeviance REMLdeviance
 15658 15678  -7825      15662        15650
Random effects:
 Groups   Name        Variance Std.Dev.
 site     (Intercept)  22191   148.97
 location (Intercept)  33544   183.15
 region   (Intercept)  41412   203.50
 Residual             696189   834.38
number of obs: 960, groups: site, 4; location, 4; region, 3

Fixed effects:
            Estimate Std. Error t value
(Intercept)    261.3      168.7   1.549

from what i understand the variance in the penultimate column are my
variance components.  But how do i conduct my significance test?

I have also tried
mod1<-lmer(density ~ region + (1|site) + (1|location))

Which i think is the correct mixed model for my design.  However once
again i do not know how to evaluate significance for the random
factors.

Thank-you again for any additional advice i receive

Stephen Cole

Hi Stephen,
>> Also i have read in Quinn and Keough 2002, design and analysis of
>> experiments for
>> biologists, that a variance component analysis should only be conducted
>> after a rejection
>> of the null hypothesis of no variance at that level.
Once again the caveat: there are experts on this list who really know about
this stuff, and I am not one of them.  Your general strategy would be to set
up two models with the same fixed effects, one of which doesn't have random
effects.  You then test the two models using anova(mod.withRandom,
modWithoutRandom).

I haven't tried this using lmer/2(), but with lme() you do this by fitting
your fixed+random effects model using lme() and your fixed-only effects
model using lm().  If you are using weights to model heteroskedasticity,
then it's better to use gls(), as this will accept the same weights argument
as the call to lme().

Then you simply do anova(lme.model, lm/gls.model).  This tells you about the
significance of your random effects, i.e. whether you need a random-effects
component.

To recap:
mod.rand <- lme(fixed=y ~ x, random=~x|Site, data=...)
mod,fix <- lm(fixed=y ~ x, data=...)

anova(mod.rand, mod.fix)

HTH, Mark.


Stephen Cole-2 wrote:
> 
> First of all thank you for the responses.  I appreciate the
> suggestions i have received thus far.
> 
> Just to reiterate
> 
> I am trying to analyze a data set that has been collected from a
> hierarchical sampling design.  The model should be a mixed model
> nested ANOVA.  The purpose of my study is to analyze the variability
> at each spatial scale in my design (random factors, variance
> components), and say something about the variability between regions
> (fixed factor, contrast of means).  The data is as follows;
> 
> region (fixed)
> Location (random)
> Site(random)
> 
> site nested in location nested in region.
> 
> Also i have read in Quinn and Keough 2002, design and analysis of
> experiments for biologists, that a variance component analysis should
> only be conducted after a rejection of the null hypothesis of no
> variance at that level.
> 
> I have tried to implement
> mod1<-lmer(density ~ 1 + (1|site) + (1|location) + (1|region))
> 
> However, as i understand it, this treats all my factors as random.
> Plus I do not know how to extract SS or MS from this model.
> 
> anova(mod1) gives me
> Analysis of Variance Table
>      Df Sum Sq Mean Sq
> 
> and summary(mod1) gives me
> Linear mixed-effects model fit by REML
> Formula: density ~ 1 + (1 | site) + (1 | location) + (1 | region)
>    AIC   BIC logLik MLdeviance REMLdeviance
>  15658 15678  -7825      15662        15650
> Random effects:
>  Groups   Name        Variance Std.Dev.
>  site     (Intercept)  22191   148.97
>  location (Intercept)  33544   183.15
>  region   (Intercept)  41412   203.50
>  Residual             696189   834.38
> number of obs: 960, groups: site, 4; location, 4; region, 3
> 
> Fixed effects:
>             Estimate Std. Error t value
> (Intercept)    261.3      168.7   1.549
> 
> from what i understand the variance in the penultimate column are my
> variance components.  But how do i conduct my significance test?
> 
> I have also tried
> mod1<-lmer(density ~ region + (1|site) + (1|location))
> 
> Which i think is the correct mixed model for my design.  However once
> again i do not know how to evaluate significance for the random
> factors.
> 
> Thank-you again for any additional advice i receive
> 
> Stephen Cole
> 
> ______________________________________________
> 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.
> 
> 
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>> Also i have read in Quinn and Keough 2002, design and analysis of
>> experiments for
>> biologists, that a variance component analysis should only be conducted
>> after a rejection
>> of the null hypothesis of no variance at that level.
Hmmm...

This does rather assume that 'no significant result' means
'near-zero variance contribution'.

These are not identical statements if the anova has low power; an
'insignificant' term can conceal practically important sizes of effect.
So if you have a smallish number of groups (say, ten or less) you might want to
find out what that estimated between-group variance could have been before you
throw it away. That's especially important if you're expecting to say
something about standard errors or confidence intervals of fixed effects.

I may well be biased, here, though. In the kinds of nested design I get involved
in (often inter-laboratory or homogeneity studies in chemistry), there is nearly
always a between-group  effect; the only question is its size. Under those
circumstances, the null hypothesis is not a particularly compelling starting
point.  I'd rather have a variance component estimate and know how vague it
was than assume it wasn't there at all.

But if you have good power and a good reason for believing there's no case
to answer, sure; assume zero unless proven otherwise.

S


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