You can use normal likelihood ratio tests. Here’s a simple example. First, let’s create observations from 10 individuals based on your parameters:
Asym = .6
xmid = 23
scal = 5
n = 10
time = seq(1,60,5)
d = data.frame(time=rep(time,10),
Asym, xmid, scal, group=0)
d$subj = factor(rep(1:n, each=length(time)))
Now let half of them have different asymptotes and midpoint parameters:
ind = (nrow(d)/2):nrow(d)
d$Asym[ind] = d$Asym[ind] + .1
d$xmid[ind] = d$xmid[ind] + 10
d$group[ind] = 1
d$group=factor(d$group)
We can simulate response values for all the individuals, based on the model:
set.seed(1)
d = transform(d, y = Asym/(1+exp((xmid-time)/scal)) +
rnorm(nrow(d), sd=.04))
library(lattice)
xyplot(y~time | group, group=subj,
data=d, type=c("g","l"), col="black")
We can see clear differences between the two groups, differences that the models should be able to pick up. Now let’s first try to fit a simple model, ignoring groups:
> fm1 = nls(y ~ SSlogis(time, Asym, xmid, scal), data=d)
> coef(fm1)
Asym xmid scal
0.6633042 28.5219166 5.8286082
Perhaps as expected, the estimates for Asym and xmid are somewhere between the real parameter values for the two groups. (That this would be the case isn’t obvious, though, since the scale parameter is also changed, to adjust for the model misspecification.) Now let’s fit a full model, with different parameters for the two groups:
> fm2 = nls(y ~ SSlogis(time, Asym[group], xmid[group], scal[group]),
data=d,
start=list(Asym=rep(.6,2), xmid=rep(23,2), scal=rep(5,2)))
> coef(fm2)
Asym1 Asym2 xmid1 xmid2 scal1 scal2
0.602768 0.714199 22.769315 33.331976 4.629332 4.749555
Since the two models are nested, we can do a likelihood ratio test:
> anova(fm1, fm2)
Analysis of Variance Table
Model 1: y ~ SSlogis(time, Asym, xmid, scal)
Model 2: y ~ SSlogis(time, Asym[group], xmid[group], scal[group])
Res.Df Res.Sum Sq Df Sum Sq F value Pr(>F)
1 117 0.70968
2 114 0.13934 3 0.57034 155.54 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
The extremely small p-value clearly shows that the simple model was too simple; the two groups do differ in their parameters.
However, the two scale parameters estimates are almost identical, with a difference of just .1. Perhaps we need only need one scale parameter? (Of course we know the answer is yes, since we have simulated data.)
(The difference between the two asymptote parameters is also just .1, but that’s a large difference when we take the standard errors into account – see summary(fm2).)
So we fit a new model, with a common scale parameter for the two groups, but different Asym and xmid parameters, as before:
> fm3 = nls(y ~ SSlogis(time, Asym[group], xmid[group], scal),
data=d,
start=list(Asym=rep(.6,2), xmid=rep(23,2), scal=5))
> coef(fm3)
Asym1 Asym2 xmid1 xmid2 scal
0.6035251 0.7129002 22.7821155 33.3080264 4.6928316
And since the reduced model is nested in the full model, we can again do a likelihood ratio test:
> anova(fm3, fm2)
Analysis of Variance Table
Model 1: y ~ SSlogis(time, Asym[group], xmid[group], scal)
Model 2: y ~ SSlogis(time, Asym[group], xmid[group], scal[group])
Res.Df Res.Sum Sq Df Sum Sq F value Pr(>F)
1 115 0.13945
2 114 0.13934 1 0.00010637 0.087 0.7685
The large p-value indicates that the reduced model fits as well as the full model, as expected.
We can of course do similar tests to check if different parameter values are needed for just Asym, just xmid or both. That said, I would not recommend doing stepwise regression like this to eliminate parameters. Instead, just test the full model (fm2) against the simple model (fm1), and be happy with the results. To quantify any differences, plots will be helpful.
