First we fit a nonlinear Regression without weighting.

So we fit the model

to our data.

In the R formula ZINC corresponds to $C_t$, INITACT to , KE to , 0.00283 is the decay rate constant for 65-Zn $k_{e2}$,
and DAY to $t$.

Were are going to estimate KE and INITACT and also supplied some start-values for the algorithm.

We can look a the summary to get the estimates and standard error:

The resulting estimates of and are and .

We can investigate the residuals, which show a clear pattern:

Secondly, we run a nonlinear regression with day-squared weighting:

We use day^2 as weights and add there a column to our data:

We run again nls, but now we supply this new column as weights:

The estimates and are quite similar to the non-weighted regression.

We could plot the two models and the data:

Finally we can also fit a linear model to the transformed Zinc-Concentrations:

First we ln-transform the concentrations:

We see that the data has now linear trend:

And fit a linear regression:

which is fitting the model
,
with a = -0.0053 and intercept = 6.07

Now plot data and model, as well as the residuals:

The mean square error can be calculated from the summary:

From which we can get an unbiased estimate of :

where

extracts the intercept from the summary.

The estimated in the summary output is , and

This result is similar to the weighted and non weighted nonlinear regression.
Again we have the same results as with SAS :) [Small deviations may be due to rounding error]

Code and data are available at my github-repo under file name ‘p85’.