# Testing hypothesis about the location of the maximum point on a curve

I have data from an experiment on the relationship between PPI (the dependent variable, a measure of startle reflex attentuation by weak stimuli) and SOA (the main independent variable; it's the time between the onsets of two stimuli). Both are continuous numerical variables. We measured PPI at 8 values of SOA in each of 30 subjects. Each subject's PPI measurements show a more or less "upside-down u" shape when plotted against log(SOA): there thus appears to be a value of SOA that produces maximum PPI.

The subjects were split into 3 groups, each receiving stimuli of a different intensity (78, 82 or 86 dB). My hypothesis (and graphical inspection of the data seems to bear this out) was that the "peak" of the PPI ~ log(SOA) curve (i.e. the SOA producing maximum PPI) will be shifted to the left by increasing stimulus-intensity. A linear mixed effects cubic regression model (using the nlme library in R) shows a significant interaction between intensity and poly(log.soa, 3), thus confirming that intensity does affect the shape of the PPI ~ log(SOA) curve in some way.

However, I want to know if there's some statistically valid way to explicitly test the hypothesis that intensity changes the shape of the curve by shifting the maximum to the left or right. Using the augPred function with my cubic linear mixed effects regression model I've estimated the location of each subject's maximum, and just looking at the graph of estimated peak SOA ~ intensity it does seem that this is indeed the case.

I hope I've explained everything clearly enough. Thanks for any help you can offer.

1. A very common way of performing this type of analysis is to simply determine the value of the SOA (or $\log$ SOA) for each subject that is associated with the maximum PPI value. Then perform an analysis of variance on those values.

An alternative method of finding the subject-specific value would be to fit separate smooth curves for each subject and interpolate the maximum PPI along with its accompanying SOA.

You could use the results that you obtained from the augPred() function, for that matter.

This might be somewhat unsatisfying, but it has the virtue of being feasible and interpretable. Think of it as a first-order approximation. I might start with that even if I were going to do something else, just to give me a back-up plan as well as a sanity check.

Since the groups represent differing intensities, you could also consider a regression analysis as an alternative. It might be interesting to plot the data by decibels or log-decibels, anyway.

2. You mention that "upside-down u" shapes fit reasonable well for each subject. Since you fit cubics, perhaps the fit was not great.

Anyway, if you have a parametric form for the curve, you could reparameterize the curve in terms of the quantity that you want to test --- in this case, the SOA producing the maximum PPI. Use nlme to fit the model again. Then, the interaction effect of group with your new parameter would be the interesting test.

3. Another option would be to fit a fully Bayesian model and determine posterior distributions for the contrasts of interest.