# Comparing sets of parameters extracted from different exponential fits to each other for significance

I take two simultaneous measurements from two different regions of a biological system, where both signals can be modelled by simple exponential rise and decay: $$I(t)=(1-e^{-t/\tau_{rise}})(e^{-t/\tau_{decay}})$$ First, by kernel matching to get initial coefficients, and later by nonlinear regression (nlinfit in MATLAB), I am able to get the fits for the two signals.The system is relatively noisy. Also, since every acquisition is different and each should be used independently for analysis, I cannot have means of repetitions. So, strictly, I only have 50 time points for each curve.

My aim is to be able to compare these two fits. In short, I would like to be able to say whether the two are statistically different. One would perhaps, right this moment, suggest ANOVA or alike (I'm not the statistics expert at all!) to tell whether these two populations are different from each other. I request more:

From the fits, I am able to extract their peak value, and when this peak occurs. I would like to have a measure on how reliable these extracted values are, considering the noise. For a noisier acquisition, my extracted values should have a lower confidence. My first question is: What is the best method to assess the confidence to my fits, and to the values that I can extract from these fits? I believe one cannot simply look at $R^2$ for nonlinear regression.

Second and the real question is related to the fit comparison, or rather peak value comparison: So, I have the peak information. What matters most for me is if the peak of fit A is greater and appears earlier than of fit B. In cases when I have strong signal and low noise, even if the peak values are not so different, by intuition, I think I should be able to compare the peaks for difference. In cases when noise is high (for any of the two signals or both) and signal is low, again by intuition, the values need to be separated more to be able to have sound a statement, otherwise I shouldn't be able to have a confident remark. How do I do this? I have a hunch that this is related to goodness of fit.

Over a quick dinner conversation in a meeting, someone hinted me towards bootstrapping. However, I'm not sure whether I was concise enough to deliver my needs, nor have any idea on how to apply bootstrapping.

Your help is greatly appreciated. I hope I was clear.

• see here for an approach Commented Aug 20, 2013 at 0:37