
Dear forum members,
 
What is the formula to calculate denominator degrees of freedom (den df) for nonlinear mixed-effect models with covariates? A reviewer of my manuscript raised a question of the inflated den df in a model, which is similar to a CO2 uptake example from  Pinheiro and Bates (2000, page 376). In this CO2 dataset, there are two treatments and two types, but DF is 64? In fact my model is a cosine wave with covariates, but DFs for covariates are much higher than the levels of the covariates as in the CO2 example.
 
Summary of the CO2 example:
 
> summary(fm4CO2.nlme)
Nonlinear mixed-effects model fit by maximum likelihood
  Model: uptake ~ SSasympOff(conc, Asym, lrc, c0) 
 Data: CO2 
       AIC      BIC    logLik
  388.4185 420.0191 -181.2092
 
Random effects:
 Formula: list(Asym ~ 1, lrc ~ 1)
 Level: Plant
 Structure: General positive-definite, Log-Cholesky parametrization
                 StdDev   Corr  
Asym.(Intercept) 2.349640 As.(I)
lrc.(Intercept)  0.079597 -0.92 
Residual         1.791962       
 
Fixed effects: list(Asym + lrc ~ Type * Treatment, c0 ~ 1) 
                                          Value Std.Error DF   t-value p-value
Asym.(Intercept)                       41.81756  1.562426 64  26.76451  0.0000
Asym.TypeMississippi                  -10.53045  2.208318 64  -4.76854  0.0000
Asym.Treatmentchilled                  -2.96943  2.213172 64  -1.34171  0.1844
Asym.TypeMississippi:Treatmentchilled -10.90037  3.112220 64  -3.50244  0.0008
lrc.(Intercept)                        -4.55724  0.096291 64 -47.32785  0.0000
lrc.TypeMississippi                    -0.10412  0.121683 64  -0.85570  0.3954
lrc.Treatmentchilled                   -0.17124  0.111959 64  -1.52953  0.1311
lrc.TypeMississippi:Treatmentchilled    0.74188  0.221742 64   3.34570  0.0014
c0                                     50.51075  4.364727 64  11.57249  0.0000
 Correlation: 
                                      As.(I) Asy.TM Asym.T A.TM:T lr.(I) lrc.TM
Asym.TypeMississippi                  -0.703                                   
Asym.Treatmentchilled                 -0.701  0.496                            
Asym.TypeMississippi:Treatmentchilled  0.497 -0.709 -0.711                     
lrc.(Intercept)                       -0.627  0.415  0.407 -0.278              
lrc.TypeMississippi                    0.458 -0.680 -0.322  0.482 -0.535       
lrc.Treatmentchilled                   0.500 -0.351 -0.717  0.509 -0.594  0.445
lrc.TypeMississippi:Treatmentchilled  -0.262  0.375  0.362 -0.547  0.365 -0.553
c0                                    -0.086  0.014  0.001  0.019  0.590 -0.033
                                      lrc.Tr l.TM:T
Asym.TypeMississippi                               
Asym.Treatmentchilled                              
Asym.TypeMississippi:Treatmentchilled              
lrc.(Intercept)                                    
lrc.TypeMississippi                                
lrc.Treatmentchilled                               
lrc.TypeMississippi:Treatmentchilled  -0.511       
c0                                    -0.057  0.140
 
Standardized Within-Group Residuals:
        Min          Q1         Med          Q3         Max 
-2.86206487 -0.49445730 -0.04217037  0.56599012  3.04061332 
 
Number of Observations: 84
Number of Groups: 12 
 
Link to the book:
http://books.google.com/books?id=N3WeyHFbHLQC&pg=PA139&lpg=PA139&dq=mixed-effect+model+building+first+step&source=bl&ots=pR7PWIuKu8&sig=TLhq-k5O4ZNwkBWcyQI8VZk9Umk&hl=en&ei=1HguSrKaPJi0Nb3DnfUJ&sa=X&oi=book_result&ct=result&resnum=1#PPA376,M1
 
 

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