My actual project is a bit complicated, but I'll explain by analogy (which I hope facilitates response):
I have 3 substances, say water, motor oil, and ethanol. For each substance, I have 5 samples in a beaker (total 15 beakers). I heat all the beakers on a hot-plate up to 70 degrees Celsius, and over the next hour, I measure the temperature of the fluid in each beaker at 5 minute intervals.
Newtonian cooling provides me with a good prediction about these temperature data, namely that the temperature of the fluid in each cup should follow an exponential distribution: y = a + e^(-kt) where a is room temperature.
I want to estimate the value k for each substance and test the hypothesis that k1 > k2 > k3 (1, 2, 3 corresponding to my three substances). The natural method of estimating k seems to be computing a non-linear regression on each substance's data, or possibly log-transforming all the data and then just computing a simple linear regression. However, there are problems.
- Given the obvious autocorrelation in the longitudinal data (confirmed by my (P)ACF plots of course), must I compute an AR term and filter my data prior to computing the regression?
- Assuming I compute this autoregression term, how do I compute it for five independent sets of data (the five beakers of a give substance)? I could average the five beakers together and then compute the regression, but this screws up my AR term (assuming I need one) and also throws off my estimate of the actual within-beaker variance from the model.
- What completely wrong-headed assumption(s) have I worked in here...?