Crossed random effects - Beyond NLME;
The orange data revisited



ADMB Files
Code: orange2.tpl
Data: orange2.dat
Initial values: orange2.pin
All required files (DOS): orange2.zip
All required files (linux): orange2.tar.gz
Results: orange2.par

Running ADMB-executables
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Model description

The orange data were considered by Millar (2004, Aust NZ J. Stat, 46, p. 543-554). A "day effect" (v) was added to the original model formulation, yielding
yij = f1,ij /(1 + exp[-(t-f2)/f3]) ] + eij,
f1,ij = f1 + ui + vj
where u is a tree-effect and v is a day-effect. This is an example of a model where the random effects u and v are crossed. Such models cannot easily be fit in nlme (Pinheiro & Bates, 2000), while the inclusion of the random effect v requires only 2-3 lines of extra code in the ADMB-RE program.

Comparison with Millar (2004)

Millar (2004) used simulated likelihood to evaluate the marginal likelihood. The following table shows a comparison of point estimates and standard deviations (SD):
Millar SD ADMB-RE SD
f1 195.9 14.5 196.2 19.4
f2 747.6 59.1 748.4 62.3
f3 352.7 32.0 352.9 33.3
Var(e 28.1 8.2 28.1 8.2
Var(u) 1059.8 684.5 1061.0 687.6
Var(v) 109.1 88.8 109.9 90.7

The differences between the two approaches are minor.