How to - regression of a noisy titration curve? I'd appreciate advice on the correct statistical method to analyse a dataset - 
Dataset is basically a titration curve consisting of [0.5, 1, 2, 3, 4, 5, 6] pg of starting material and 8 replicates in each 'pg bin'.  In 'stage 1' of the process each bin is labeled separately, in 'stage 2' all bins are pooled into one tube and amplified together and in 'stage 3' this is thrown into a DNA sequencer and bins are separated based on the labels.
So: the original input is a defined quantity of material and the final output is a number and what I would like to know is how accurately the output reflects the input.  This feels like some kind of regression but I'm not sure what to use.  The mean of 8 replicates?  Their median?  Something to do with the variability between them?  Something based on a binomial sampling (with different bins having a different expected value?)?
Below is an example dataset - the x axis is the titration curve bins ('0 pg' is background process noise), the y axis is the 'output number' and the barplots are over the 8 replicates.  There are 2 batches which are basically a repetition of the same experiment.
Thanks (and apologies if this has been asked before)!

 A: I did not yet understand what the task at hand is. But I'll try to start towards an answer. 

the original input is a defined quantity of material and the final output is a number and what I would like to know is how accurately the output reflects the input.

This is part of the validation of the analytical method. 

This feels like some kind of regression

This is one step before: you're talking about the modeling here. 
Yes: as the dependent variate is metric, this is a regression 

but I'm not sure what to use.

Without further information (such as I expect it to be linear from theory - see comments) we won't be able to answer this: the type of regression will usually be concluded from the expert knowledge about application, measurements and the actual data (including theory).

The mean of 8 replicates? Their median? Something to do with the variability between them? Something based on a binomial sampling (with different bins having a different expected value?)?

You need to do exactly the same that the method asks you to do for real analyses. So if you want to use the mean of the replicates here, also analyses are required to do 8 replicates and then take the mean. This is rather unusual: one typically goes for the individual measuements, and I'd recommend to go through the whole data analysis (including validation) process for that easiest setup before thinking about more complicated protocols.

Sidenote: typically the calibration solutions are prepared from one stock solution or one stock solution for each of the replicates. Strictly speaking this has the consequence that the data points are not truly independent. While a mixed model would be a better description of the situation, my guess is that in the end you may be better off sticking to a least squares model unless you have specific questions that are only answered by the mixed model (e.g. variance between stock solutions).
IMHO you can do this if you validate with properly independent validation samples, i.e.


*

*validation samples prepared from unknown stock solution (replicate stock solution)

*randomize the order of measurements at least for the valdation samples


IMHO it is even acceptable if your judgment of the whole analytical method concludes that the likely sources of error are not this dependence of the samples. But you should report this. 

Many chemical-analytical methods follow a sigmoid shape, even if they are expected to be "linear": at very low concentrations you may get a flat signal, e.g. due to cross sensitivities. At very high concentrations your detection saturates, thus the signal levels off. One part of method development is to find out the concentration range where linearity can be assumed. 
Is that your task at hand?


 If I do regular linear regression can I then solve a 'reverse problem'? E.g given a datapoint assign its most likely bin and give a confidence interval for that assignment? 

Most probably, you don't want to assign a most likely bin but instead want to predict a metric concentration. You can set up the regression model either as signal = f (concentration) or as concentration = f (signal). These ways are called  classical (aka ordinary) least squares regression and inverse regression, and they have different properties and assumptions.
Confidence and also prediction intervals can be calculated for both classical and inverse models.

Suggested reading: Here are some technical terms which I think may be of interest:


*

*limit of detection/quantitation (LOD/LOQ) is used to characterize the low end of possible concentrations for qualitative/quantitative analysis.
There are norms how to measure these. 

*classical (aka ordinary) regression vs. inverse regression. In particular, they differ in their assumption of error: classical least squares assumes instrument noise on signal >> concentration error in calibration samples; inverse least squares assumen concentration error >> noise on signal 

*confidence and prediction intervals



Some rules (of thumb):


*

*One way of calculating the LOD is to use the concentration/amount corresponding to the mean + 3 standard deviations of the blank signal (attention: works only if the blanks are independent, measured properly, and cover all non-analyte influences that occur in the real samples). A quick and dirty guess from your plots puts this somewhere between 2 and 3 pg.
A rule of thumb for LOD and LOQ is that the LOQ is typically ca. 3 LOD, which would put it somewhere around 5 - 10 pg, and thus basically outside the concentration/amount range your experiment covers. With the variance you show, you probably need to add measurements at higher concentrations/amounts.

*Concentration series for proving linearity should at least contain 10 equally spaced concentration steps (thus covering one order of magnitude).
