How many samples are needed for a paired trial with multiple tries?

Question

I'm having problem understanding how big sample size is needed for a situation where there are pairs in the before/after groups but these have all multiple data points.

Problem

Pose a situation where you have X dart throwers. Each dart thrower have thrown $n_i$ darts for some average $m_i$ points with a "bad" dart. Now an expected better dart is introduced, so the dart throwers all start throwing with the new dart. How many throws are needed in order to be able to say that the new darts are better at say 5% significance with 80% power? (Note that this will be a function of the actual difference in average points.)

Thoughts on solution

I am aware of the power.t.test, which can be used for to compare all the points before and after. However, as the dart throwing expertise could vary substantially amongst the throwers, it should probably be a paired test. But, when reading up on the power.t.test(..., type = "paired"), it tells you how many pairs (throwers) are needed (and presumably assumes only one data point with the bad and good dart).

So, I'm considering to do a power.t.test for each of the authors but how would I assemble the results into one? Is there a paired power test which could answer how big sample is needed in the situation where you have pairs but also multiple data points for each pair?

Does the original distribution matter?

Answer 1

I will mention a few things that you will need.

First, if you use software, you will need to input the correlation between the throws with each of the two types of darts, which may be very hard to estimate without any previous data. You will also need to know the variability for each of the two types of darts.

This isn't a paired-t test problem because there are repeated measurements within each thrower for each dart. If appropriately treated as a repeated measurement problem, you will still need to know the correlation between throws with the two types of darts.