[R] multicollinearity in nlme models

[Daniel O'Shea]

I am working on a nlme model that has multiple fixed effects (linear and nonlinear) with a nonlinear (asymptotic) random effect.

asymporig<-function(x,th1,th2)th1*(1-exp(-exp(th2)*x))
asymporigb<-function(x,th1b,th2b)th1b*(1-exp(-exp(th2b)*x))

mod.vol.nlme<-nlme(fa20~(ah*habdiv+ads*ds+ads2*ds2+at*trout)+asymporig(da.p,th1,th2)+
    asymporigb(vol,th1b,th2b),
    fixed=ah+ads+ads2+at+th1+th2+th1b+th2b~1,
    random=pdBlocked(list(th1~1,th2~1)),
    start=c(ah=.5524,ads=.8,ads2=-.1,at=-1,th1=2.542,th2=-7.117,th1b=2,th2b=-7),
    data=pca1.grouped,verbose=T)

I am looking at potential multicollinearity among the fixed effects, in particular I am concerned about multicollinearity between da.p (drainage area) and vol (volume).  How do I interpret the correlation reported in the summary command for th1 and th1b, which are the asymptotes for fa20~da.p and fa20~vol.  It is -.50, but how is the correlation calculated?  

I have run the above model with out vol and the coefficients for the remaining variables are very similar (within the approx. 95% conf. interv.) to the coefficients in the above model and vol and da.p are significant, both suggesting multicollinearity is not severe??  I am interested in which variables influence fa20 (richness) not necessarily the model with the smallest residual sd.

I do have Pinheiro and Bates, but do not find much reference to this type of information.  Thanks for any suggestions or help.

Dan


####summary
#################
Nonlinear mixed-effects model fit by maximum likelihood
  Model: fa20 ~ (ah * habdiv + ads * ds + ads2 * ds2 + at * trout) + asymporig(da.p,      th1, th2) + asymporigb(vol, th1b, th2b) 
 Data: pca1.grouped 
       AIC      BIC    logLik
  3151.665 3248.518 -1555.832

Random effects:
 Composite Structure: Blocked

 Block 1: th1
 Formula: th1 ~ 1 | bas
              th1
StdDev: 0.8125094

 Block 2: th2
 Formula: th2 ~ 1 | bas
              th2 Residual
StdDev: 0.9468531 1.028757

Variance function:
 Structure: Different standard deviations per stratum
 Formula: ~1 | bas 
 Parameter estimates:
       LS        CD        MS        DM        RN        LM        UM        RD 
1.0000000 0.7884995 1.2107482 1.4159803 1.0463657 1.3982966 1.2195945 1.1978807 
       MN        SC 
1.3858409 1.2006228 
Fixed effects: ah + ads + ads2 + at + th1 + th2 + th1b + th2b ~ 1 
         Value Std.Error  DF    t-value p-value
ah    0.597032 0.1330044 920   4.488812       0
ads   1.283297 0.0874561 920  14.673614       0
ads2 -0.125186 0.0130289 920  -9.608281       0
at   -0.731506 0.1394553 920  -5.245451       0
th1   2.363269 0.3385592 920   6.980373       0
th2  -3.910520 0.3575392 920 -10.937318       0
th1b  1.402536 0.2188125 920   6.409764       0
th2b -6.765038 0.2931669 920 -23.075723       0
 Correlation: 
     ah     ads    ads2   at     th1    th2    th1b  
ads  -0.595                                          
ads2  0.571 -0.974                                   
at   -0.092 -0.104  0.104                            
th1   0.010 -0.153  0.147 -0.020                     
th2  -0.012 -0.139  0.105 -0.015 -0.071              
th1b  0.043 -0.110  0.070  0.084 -0.500  0.163       
th2b -0.038 -0.032 -0.030 -0.016 -0.017 -0.225 -0.056

Standardized Within-Group Residuals:
        Min          Q1         Med          Q3         Max 
-3.73841391 -0.63008005  0.03189713  0.68903314  3.90583424 

Number of Observations: 937
Number of Groups: 10