is multicollinearity of fixed effects resolved by random effects

I think grouping them into families is a very good idea.
This makes use of prior insight, reduces the number of
parameters to an extent that it becomes more reasonable
to think about fixed effects, and you can look for individual
species that stray from the path laid out for them by their
families.  For the fixed effects analyses that I suggested,
you might do these by families.

Conceptually, there are taxonomic factors that surely
function, no doubt in interaction with location variables,
as fixed effects.  I consider that one ought to start by thinking
of them as fixed effects, unless it can be demonstrated that
data are indistinguishable from random variation.

Maybe however those effects operate more as the level of
genera or families than at the level of species.  Responding
to this point may be a useful aim for the study.

John.

John Maindonald             email: john.maindonald at anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.


On 19 May 2008, at 4:04 AM, Jordan Mayor wrote:

Thank you JM, RK, and DB for your thoughtful responses to my  
questions.  I am working on posting the file on the web and will  
respond as soon as it is publicly available - (I could email it  
direct to those concerned as well).  Until then here is the output  
requested by DB and RK as well as more information regarding sites,  
species, and the data:

ds=d[d$TYPE=="s",] #separates into just sap fungi as used in the  

example below

dm=d[d$TYPE=="m",] #separates into just myc fungi (treated as  

distinct populations in my analyses)

table(table(ds$SPECIES))

    0      1    2   3   5   6
393 161  45   7   1   1
# As seen all species are not present at all the sites - this is  
partly why I thought to nest SPECIES within SITE.  The sites range  
widely from boreal tundra, to temperate California, to tropical  
forests of Borneo and Guyana.  Most fungi are not so cosmopolitan as  
to even potentially be present everywhere - let alone be collected  
during brief collecting forays.  One site (pine forest in CA) didn't  
even list  fungal names.  All fungi were listed as mycorrhizal (m),  
saprotrophic (s), or of unknown ecological roles (unk).  Most of the  
genera (or families) are present at all sites but that biologically  
coarseness concerns me.  If, as pointed out by RK the ideal  
situation would be every species present at every site then perhaps  
creating a new column containing families or genera will move the  
dataset toward that direction.

xtabs(~SITE,ds)

SITE
                             Aheden                             
Aheden 2
                                   
4                                   0
   Ashiu (temp deciduous broadleaf)        Betsele
                                  
21                                   0
                     Breuil, France                       Chiba
                                  
14                                   6
                                  d                              
Flakaliden
                                  
23                                   0
                Glacier Bay, Alaska                Guyana
                                  4                                   
20
                                  h                     heath  
tundra, subarctic Sweden
                                  
38                                   4
                          Kagoshima                        
Kulbacksliden
                                   
1                                   0
                              Kyoto                        Lamar  
Haines
                                 26                                   
13
          Lambir (lowland tropical)               Miyajima
                                  
14                                   1
                           Norikura                        Norrliden
                                   
3                                   0
                            Okinawa       Ontake (subalpine  
coniferous)
                                   
0                                   8
                              Oodai                  pine forests in  
CA
                                  2                                   
25
                          Shirahama                     nowbowl
                                  2                                   
13
                  Spruce plantation                   Stadsskogen
                                 17                                   
13
                        Svartberget                     Tanigawa
                                   
0                                   6
tussock tundra near Toolik Lake, AK      Vilan
                                   
5                                   0
                        Woods Creek
                                 25                     # the zeros  
here are because some sites have no X13C data - only X15N

I did try running models with centered predictors, as suggested by  
RK - see below.  I found that the AIC scores were much larger  
however when the predictors were centered using my method.  I  
centered by taking the mean of mycorrhizal and saprotrophic fungal  
groups at each site - then I subtracted each (m) or (s) fungus from  
those same group means within each site.  Because the (m) and (s)  
fungi have unique sources of carbon, and nitrogen and these are  
reflected in their isotope values (X13C, X15N), I deemed this  
centering level to be appropriatein order to preserve the magnitude  
of difference between the groups.

mix.model1  # Raw isotope values used

Linear mixed model fit by maximum likelihood
Formula: X13C ~ MAT + MAP + LAT + (1 | SPECIES) + (1 | SITE)
   Data: ds
  AIC   BIC    logLik     deviance  REMLdev
 1016 1041 -500.9     1002         1031
Random effects:
 Groups   Name        Variance Std.Dev.
 SPECIES  (Intercept) 0.62149  0.78835
 SITE     (Intercept) 0.34885  0.59063
 Residual             1.28911  1.13539
Number of obs: 283, groups: SPECIES, 215; SITE, 24

Fixed effects:
              Estimate Std. Error t value
(Intercept) -2.207e+01  1.121e+00 -19.695
MAT         -6.418e-02  3.833e-02  -1.675
MAP          3.137e-05  2.100e-04   0.149
LAT         -8.690e-03  1.778e-02  -0.489

Correlation of Fixed Effects:
          (Intr)       MAT     MAP
MAT  -0.707
MAP  -0.401  -0.247
LAT   -0.959  0.692   0.272

mix.model1a # Group centered within each site

Linear mixed model fit by maximum likelihood
Formula: STND_13c ~ MAT + MAP + LAT + (1 | SPECIES) + (1 | SITE)
   Data: ds
  AIC   BIC    logLik     deviance  REMLdev
 2623 2649  -1305     2609         2620
Random effects:
 Groups   Name        Variance Std.Dev.
 SPECIES  (Intercept) 301.9720 17.3773
 SITE     (Intercept)   5.1294  2.2648
 Residual             327.3857 18.0938
Number of obs: 283, groups: SPECIES, 215; SITE, 24

Fixed effects:
             Estimate Std. Error t value
(Intercept) 3.463e+01  1.166e+01  2.9713
MAT         3.899e-01  4.330e-01  0.9006
MAP         2.543e-05  1.811e-03  0.0140
LAT         3.235e-01  1.867e-01  1.7324

Correlation of Fixed Effects:
         (Intr)       MAT     MAP
MAT -0.779
MAP -0.230  -0.300
LAT  -0.955   0.732   0.116

anova(mix.model1,mix.model1a)

Data: ds
Models:
mix.model1: X13C ~ MAT + MAP + LAT + (1 | SPECIES) + (1 | SITE)
mix.model1a: STND_13c ~ MAT + MAP + LAT + (1 | SPECIES) + (1 | SITE)
                            Df      AIC         BIC          logLik   
Chisq Chi Df  Pr(>Chisq)
mix.model1.p    7      1015.85  1041.37  -500.93
mix.model1a.p  7      2623.49  2649.00 -1304.74     0      0       <  
2.2e-16 ***

# As seen above, the model using my centered values performed  
poorly.  My interpretation is that the centering removed the very  
variability associated with climate I am trying to predict!

#In addition, I compared the other models mentioned by RK using the  
raw isotope values:

anova(mix.model1,mix.model2,mix.model3)

Data: ds
Models:
mix.model1: X13C ~ MAT + MAP + LAT + (1 | SPECIES) + (1 | SITE)
mix.model2: X13C ~ (MAT + MAP + LAT)^2 + (1 | SPECIES) + (1 | SITE)
mix.model3: X13C ~ MAT * MAP * LAT + (1 | SPECIES) + (1 | SITE)
                         Df     AIC         BIC            
logLik      Chisq    Chi Df   Pr(>Chisq)
mix.model1.p  7     1015.85  1041.37    -500.93
mix.model2.p 10    978.21    1014.66    -479.10   43.645       
3       1.796e-09 ***
mix.model3.p 11    980.18    1020.28    -479.09   0.023         
1        0.8794

# I have refrained from trying the models mentioned by DB in order  
to protect his rights to not commit the "capital mistake" ;)  
however, the three predictors (MAT, MAP, LAT) do indeed seem to  
overplot - I will try to select only one perhaps in my final models.

# Again - thank you all for your help on this.

-- 
Jordan Mayor, Ph.D. Candidate
Ecosystem Dynamics Research Lab
Department of Botany, University of Florida
Gainesville, FL 32611