A couple of weeks ago there was a question regarding apparent
convergence in nls when using the SSfol selfStart model for fitting a
first-order pharmacokinetic model. I can't manage to find the original
message either in my archive or in the list archives but the data were
time conc dose
0.50 5.40 1
0.75 11.10 1
1.00 8.40 1
1.25 13.80 1
1.50 15.50 1
1.75 18.00 1
2.00 17.00 1
2.50 13.90 1
3.00 11.20 1
3.50 9.90 1
4.00 4.70 1
5.00 5.00 1
6.00 1.90 1
7.00 1.90 1
9.00 1.10 1
12.00 0.95 1
14.00 0.46 1
and the attempted fit looked like
> nls(conc ~ SSfol(dose, time, lKe, lKa, lCl), testdat, trace = 1)
nls(conc ~ SSfol(dose, time, lKe, lKa, lCl), testdat, trace = 1)
99.15824 : -1.2061792 0.1296156 -4.3020997
86.07567 : -0.7053265 -0.3873204 -4.1278009
85.19743 : -0.5548499 -0.5333776 -4.1173627
Error in nls(conc ~ SSfol(dose, time, lKe, lKa, lCl), testdat, trace = 1) :
step factor 0.000488281 reduced below `minFactor' of 0.000976562
If one allows much smaller step factors you can get more iterations but
nls still doesn't converge.
It took me a long time to find out why. The lack of convergence is
related to the form of the SSfol model. When the first two parameters
(lKe and lKa) are equal the model degenerates to a different analytic
form. If you were to express the corresponding system of ordinary
differential equations in the matrix form (as described in Appendix 5 of
Bates and Watts (1988)), it would be a Schur triangular block and the
special techniques for nondiagonalizable matrices, described in section
A5.2 of that appendix, must be used.
The clue here is that the first two parameter values are getting very
close to each other and this shows up as a singularity. The fitted
model is determined by two parameters, not three.
