[R] Comparing nonlinear, non-nested models

[Tom Shatwell]

Dear R users,

Could somebody please help me to find a way of comparing nonlinear, non-nested
models in R, where the number of parameters is not necessarily different? Here
is a sample (growth rates, y, as a function of internal substrate
concentration, x):

x <- c(0.52, 1.21, 1.45, 1.64, 1.89, 2.14, 2.47, 3.20, 4.47, 5.31, 6.48)
y <- c(0.00, 0.35, 0.41, 0.49, 0.58, 0.61, 0.71, 0.83, 0.98, 1.03, 1.06)

model1 <- function(x, xo, ym)    ym * (x-xo)/x
model2 <- function(x, xo, ym, k) ym * (x-xo)/(k+x-xo)
model3 <- function(x, xo, ym)    ym * (1-exp(-log(2)*(x-xo)/xo))
model4 <- function(x, xo, ym, k) ym * (1-exp(-log(2)*(x-xo)/k))

fit1 <- nls(y~model1(x, xo, ym),    start=list(xo=0.5, ym=1))
fit2 <- nls(y~model2(x, xo, ym, k), start=list(xo=0.5, ym=1, k=1))
fit3 <- nls(y~model3(x, xo, ym),    start=list(xo=0.5, ym=1))
fit4 <- nls(y~model4(x, xo, ym, k), start=list(xo=0.5, ym=1, k=1))

anova(fit1, fit2)
anova(fit3, fit4)

Models 1 and 2 are nested, as are models 3 and 4 (set k=xo), so they can be
compared using anova. I am looking for a way to compare the non-nested models
(ie models 1 and 3, and models 2 and 4), or better still, I would like to
compare all 4 at once. A significance test would be ideal, but I am beginning
to think that this may not make statistical sense. In that case, is there an
appropriate measure of goodness of fit? I’d be very grateful if someone could
put me on the right track.

Thanks,
Tom

R version 2.14.2 (2012-02-29)
Platform: x86_64-pc-mingw32/x64 (64-bit)

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[5] LC_TIME=German_Germany.1252