# Within subject, Multiple treatment: testing if each treatment further increases dependent variable?

Suppose I have data that looks something like the following for a single subject:

Treatments: amount of a shot they receive, such as 5cc, 10cc, 15cc, 20cc Outcome of interest: energy level

Question of interest: do increases in the amount of the shot received cause increased energy levels.

And then assume that the above is done with multiple subjects (i.e. $$n$$ subjects face all possible treatments).

Can someone point me in the direction of tests that work for such a situation? Or perhaps point me to a book/chapter that discusses this topic?

Some thoughts: What if we looked at two treatments at a time.

For example, consider the 5cc and 10cc treatment.

• Average the "energy level" of each subject for the 5cc treatment, and call this $$e_{5cc}$$.
• Average the "energy level" of each subject for the 10cc treatment, and call this $$e_{10cc}$$
• test if $$e_{10cc} \geq e_{5cc}$$ with two-sample t-test or something?

or, alternatively, just compute the sign (+ or -) of the difference in energy levels, for each subject, between the 10cc and 5cc treament, and do a Sign test?

However, if we look at two treatments at a time, how would we deal with the possibility that sometimes we see an increase and sometimes we see a decrease?

• For example, what if we see a significant increase from 5 to 10cc, a decrease from 10 to 15cc, and an increase from 15-20 cc.
• Somehow we would need to test if the increases outweigh the decreases... (i.e., we would need a way to find out which pairwise comparisons matter most)
• Would it make sense to have the cc of the treatment be a continuous variable? Then you could test the beta coefficient related to the cc variable (or some sort of transformation of cc such as cc$^2$ Feb 29, 2020 at 0:45
• @DavidVeitch Is what you are suggesting the following: Let $cc$ be continuous. Then just make a panel dataset of the data and use standard estimating techniques for panel data? Also, to directly answer your question: yes it makes sense for cc to be a continuous variable, but in any observable dataset the data will have it in discrete increments. Feb 29, 2020 at 0:52