# Identify difference from baseline in complex dataset

I have a dataset that is structured as follows:

• 6 Subjects were observed
• 2 measurements (continuous values X and Y) were taken for each subject at 7 different time points (baseline and 6 additional times)
• At baseline, all the measurements for X for all subjects is the same (actually 0), while the Y measurement varies.
• The measurements, X and Y, are know to be related.
• If we take all the data and plot X vs Y, we see a nonlinear relationship between the two measurements (see hand drawn, example plot below).

The interest is determining at what point, X, can we say that Y is significantly different from baseline? Using the baseline measurements, we can calculate a confidence interval to establish something to test against.

Seems an easy enough question to answer, but there are some complexities that limit the approaches I've take thus far. Here's what I've done and the limitations of each approach:

• Find the the intersection of the fit line to the upper 95% confidence interval of the baseline. This is simple to do, but this ignores the uncertainty in the fit and that this is a repeated measures design.
• Calculate a confidence or prediction interval for the fit line. This takes into account uncertainty in the fit, but ignores the repeated measures part.
• Use a mixed effects approach to the fit. This takes into account the repeated measures part, but now I can't calculate confidence intervals to assess uncertainty of the fit overall as I have 6 independent fitted curves, one for each subject.

What approach am I missing that allows to account for both the uncertainty of the fit and the repeated measures design? • As a first cut at the problem, how about using the intersection point for the two dotted lines on your graph? Mar 3 '18 at 14:08
• Thanks for the comment. Yes, that's an option, but it doesn't account for uncertainty in the fit of the fact that these are repeated measures, correct? As I mentioned, I can compute confidence or prediction intervals of the fit and look for a similar intersection, but again, the repeated measures question remains. Mar 3 '18 at 14:41
• I just modified my question to describe what approaches I've already tried and the limitations of each. Mar 3 '18 at 15:00