Single fitted parameter from multiple data Let's say I have to perform more than one non-linear fit over experimental replicates, each of them being an exponential decay (y <- 50*exp(-Ax)). What I'm doing right now is to use the R function nls for each decay, to obtain the estimated parameter A together with associated standard error of the fit.
What I would like to get is a single estimate (and not one for each decay) with associated error that takes into account both the inter-sample variability and the error associated with each fit.
One approach I tried is to take the average of the estimates as the best fitted parameter, and their standard deviation as the resulting error. However, with this approach I am not taking into account that each fitted parameter comes with a standard error due to the fitting procedure.
Is there a way to tackle this issue?
Thanks in advance
 A: I can think of a few different approaches.


*

*Stack all of the experiments up and treat them all as one dataset. You will get one estimate and one standard error. This will likely underestimate your standard error.

*Stack all of the experiments up, but use a mixed effects model to account for the different experiments. You would probably want to take a log to make the estimation feasible. The model you fit would look like:
$$\log(y_{ij}) - \log(50) = -(A + \eta_i) x_{ij} + \epsilon_{ij},$$
where $y_{ij}$ is the $j$th observation in experiment $i$. The random slope of $\eta_i$ will account for the correlation that may be present among observations from the same experiment.

*Borrow some ideas from multiple imputation and treat each experiment as a different imputed data set. For each experiment, you will get an estimated $\hat{A}_i$ and standard error $\sqrt{W_i} = \sqrt{var(\hat{A}_i)}$. The overall estimate of $A$ will be that average of $\hat{A}_i$ ($\bar{A}$). From the law of total variance, an estimate of the squared standard error may be:
$$\frac{1}{m} \sum_i W_i + \frac{1}{m-1}\sum_i (\hat{A}_i - \bar{A})^2,$$
wherer $m$ is the number of experiments.
