How to get standards errors of the parameters of a non-linear model (R and Excel) I am working on the movement of fish species from the centre of a protected area to a non-protected area. Based on the article by R. Abesamis, itself inspired by the work of B. Kaunda-Arar (page 91), I applied to my dataset a logistic decay D = 1/(1+exp(S*(I-d)) where D is the proportion of biomass in the middle of the reserve, d is the distance from the centre of the protected area, S is the slope and I is the inflection point. The parameters to be estimated in this model are S and I. However, I had to use Excel to estimate these parameters because the nls function in R was giving me constant error messages (the well-known singular gradient error). I successfully estimated S and I using the Excel solver, but I need to know the standard errors of these parameters. For this, the article states that a linearization of my logistic model is necessary, but despite my research on this subject I do not understand how to do this.
My questions are therefore the following:

*

*Is it possible to get the standard error of my parameters directly from Excel? If so, how?

*If this is not possible in Excel, how can I get the standard errors of my S and I parameters in R? (note: using the nls function, even when applying the values obtained by my Excel model as starting values for S and I, the gradient error message is still displayed. Same thing when using nlsLM.)


If your model is
$$
D=\frac{1}{1+e^{S(I-d)}},
$$
then you get
$$
1-D=1-\frac{1}{1+e^{S(I-d)}} = \frac{1+
e^{S(I+d)}-1}{1+e^{S(I+d)}} =\frac{e^{S(I-d)}}{1+e^{S(I-d)}}
$$
Which means that
$$
\frac{1-D}{D}=e^{S(I-d)}
$$
and
$$
\ln e^{S(I-d)} = S(I-d),
$$
i.e., a linear function. You can apply methods for estimating parameters of a linear model now.
