R help - Jun 2006 - NLS and fitting of x-values?

I collected eggs laid by Springtails everyday over 28 days after swich to
isotopically enriched diet. The eggs were pooled at day 7, 14, and 28 (+ day 0 =
initial value) and analyzed for isotopes. After the diet switch the isotopic
values of the adults and eggs change towards those of the new diet.
Here are the d13C values (y) of the eggs:

x      y 
0   -22.2
0   -22.2
0   -22.2
0   -22.0
7   486.9
7   498.6
7   489.6
14  820.9
14  817.4
28  895.6
28  900.7
28  890.6
28  885.8

The y values represent the mean of the sampling period.

The dataset is very small but previous experiments have shown that a exponential
asymptotic model can be used for this kind of situations.

How do I fit a model to these pooled values? The y values can be regarded as the
mean of the given sampling period.

My first guess is that the x values should be in the middle of the collection
period. I call these x-values xi:
xi=c(rep(0,4),rep(3.5,3),rep(10.5,2),rep(21,4))

If I fit them to a nonlinear regression model via least squares (NLS) I get the
parameters:
       Value Std. Error  t value 
a 900.386000  3.7839900 237.9460
b 916.630000 29.3987000  31.1792
c   0.230811  0.0102677  22.4792

How do I procede from here? I should probably use a maximum likelihood estimate
to estimate the fitted xi?
Any help would be greatly appreciated.

-------------------------------------------
Thomas Larsen, PhD student
Department of Terrestrial Ecology
National Environmental Research Institute
Vejls?vej 25, P.O. Box 314, DK-8600 Silkeborg
Phone +45 8920 1572; Fax +45 8920 1414;

"Larsen, Thomas" <thl at dmu.dk> writes:
> I collected eggs laid by Springtails everyday over 28 days after swich to
isotopically enriched diet. The eggs were pooled at day 7, 14, and 28 (+ day 0 =
initial value) and analyzed for isotopes. After the diet switch the isotopic
values of the adults and eggs change towards those of the new diet.
> Here are the d13C values (y) of the eggs:
> 
> x      y 
> 0   -22.2
> 0   -22.2
> 0   -22.2
> 0   -22.0
> 7   486.9
> 7   498.6
> 7   489.6
> 14  820.9
> 14  817.4
> 28  895.6
> 28  900.7
> 28  890.6
> 28  885.8
> 
> The y values represent the mean of the sampling period.
> 
> The dataset is very small but previous experiments have shown that a
exponential asymptotic model can be used for this kind of situations.
> 
> How do I fit a model to these pooled values? The y values can be regarded
as the mean of the given sampling period.
> 
> My first guess is that the x values should be in the middle of the
collection period. I call these x-values xi:
> xi=c(rep(0,4),rep(3.5,3),rep(10.5,2),rep(21,4))
> 
> If I fit them to a nonlinear regression model via least squares (NLS) I get
the parameters:
>        Value Std. Error  t value 
> a 900.386000  3.7839900 237.9460
> b 916.630000 29.3987000  31.1792
> c   0.230811  0.0102677  22.4792
> 
> How do I procede from here? I should probably use a maximum likelihood
estimate to estimate the fitted xi?
> Any help would be greatly appreciated.
My take is that you should just have your expected y values (which in
this case are also the eggs-values, but never mind...) modeled as what
they are, namely an integral under the curve between x[i-1] and x[i],
or for practical purposes take the sum over days (say) 15 to 28 and
divide by 14. 

-- 
   O__  ---- Peter Dalgaard             ?ster Farimagsgade 5, Entr.B
  c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
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~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)                  FAX: (+45) 35327907