Regression for cross-over trial with small sample size I am in a bit of a pickle with some data that I have obtained in a collaboration. This is a crossover trial in which 8 elite cyclists were assigned to two experimental conditions at different timepoints. I am unsure as to how to carry out the analyses. The study can be summarized in the following points: 


*

*All participants went through a bicycle test to exhaustion

*One of two supplements were consumed directly after the bicycle test

*Blood amino acid concentrations were measured at 6 timepoints after the bicycle test

*The cyclists returned the day after and underwent a performance test (time trial)

*The same procedure was repeated after crossover to the other group. 

*The main outcome was thus to compare the two supplements' effect on performance


However, what I want to do is to explore whether the changes observed in different amino acids (point 2) predicts performance (point 4). I have performed simple linear regression in both groups with calculated AUCs for the amino acids as predictors but to me this seems problematic because of the small sample size and the rules of thumb concerning sample size in regression. 
So, my question is whether any of you are aware of suitable alternatives that lets me explore this in more detail?
 A: You have a paired experiment, that is, both treatments done on each subject. If you were to simply compare treatments you could use a paired test. (You can get some information about sequence by using a two factor ANOVA.) This strongly suggests using the pairing in regression analyses.
Ignore the rules of thumb about sample size. You have a well-structured experiment and the rules don't really apply. If you had more observations you would just have a more powerful experiment. Each pair of observations is difficult to obtain and expensive. Elite athletes are hard to find unless you are in a training facility. Your experiment must use 1-2 hours of their time and professionals' time to collect data. If the experiment you have done shows useful results then you can probably justify additional data collection. 
I do not know if AUC is a good measure of uptake or absorption or metabolism. You have to decide on scientific grounds. Choosing an alternative would probably require a large sample size.
I suggest the following two analyses, both using the difference in the dependent variable (performance test), or y1 - y2, where the difference is for supplement 1 minus supplement 2.
Also compute the corresponding difference for AUC1 - AUC2, or whatever independent variable you use.
Do one regression of the difference on y (dependent variable) with the difference in AUC as an independent variable. The intercept is interpreted as the mean difference due to treatment. Since you are using AUC difference also, this is the difference in treatments adjusted for AUC as a covariate. This regression uses two df out of your 8 observations leaving 6 to estimate error.
Do another regression of the difference in y on both AUC1 and AUC2. This will estimate coefficients for the two supplements and you can test if these two coefficients are equal or not. If they are equal, use the first regression. If they are different then you might have to figure out why. Keep in mind that there might be modest differences, possibly even statistically significant, that aren't scientifically interesting. This regression uses 3 df to estimate parameters leaving 5 to estimate error. 
If you suspect that there is an effect of sequence or time then put in an indicator (dummy) variable coded 0 for the sequence 1 then 2 and coded 1 for the sequence 2 then 1. This uses 1 degree of freedom leaving either 5 or 4 to estimate error. I would prefer not to include it unless it is highly significant.
A: What you want to do looks more like Variance analysis than regression.
You can have a look at that to understand it better : https://www.accountingtools.com/articles/what-is-variance-analysis.html
and this one : http://blog.minitab.com/blog/adventures-in-statistics-2/understanding-analysis-of-variance-anova-and-the-f-test
In a first step you will be able to look how your two supplements have affected the cyclists.
The prediction will be a bit different but it possible  to do it with an ANOVA.
