Increasing power of test by transforming format of data - is this cheating? I'm working on a health care dataset that contains one row per member, measuring total number of the member's hospital inpatient (IP) visits during 2012 and 2013. The initial research question of the study asks if the percentage of high IP utilizers (3+ visits per year) in 2013 is statistically significant different from 2012. However, our power is limited by small sample of members in either year who had 3+ IP visits.
Rather than comparing the % of members who are high utilizers in 2012 and 2013, some colleagues have proposed we ask a slightly different research question which increases the power of the study by comparing the percentage of total hospital visits accounted for by the high utilizers. Now they're no longer interested in the % of members who are high utilizers, but rather the % of all IP visits which are associated with members who had 3+ visits per year.
Therefore, Member A with 9 IP visits, Member B with 2 IP visits, and Member C with 1 IP visits in 2012 are now measured as 12 observations rather than 3 observations. We've quadrupled our sample size. The % IP visits belonging to a high utilizer is 100% x (9/12) = 75%.
My question: Is this "cheating" - increasing the power of the study by inflating the number of records by the number of visits per member?
 A: The scope of the question centres around high IP utilizers, so the focus here is on people, not number of visits. 
Therefore, to answer the question posed, you can't switch this and talk about number of visits just because you want the power of your test to increase. If you do this, you would not be answering the question posed. 
It may very well be that the total number of visits in 2013 is significantly greater than in 2012 simply because more people were treated in 2013 than in 2012 - not that any given additional person visited the hospital more frequently.
However, since the focus of your problem is on people who are frequent visitors, I think the appropriate thing to do is take the percentage of frequent visitors in 2012 and compare it with the percentage of frequent visitors in 2013. 
Unless you have a really small sample (say less than 20), I wouldn't worry too much about the power of the test. If you aren't really happy with this power, then I suggest you go the non-parametric route which is robust. Sticking to the t-test (or Z-test) should otherwise be ok.
A: As rocinate points out, this modification does change the meaning of the question. My impression is that this is a substantial change in meaning and you should think carefully about what the relevant hypothesis (i.e. proposed mechanism) is and whether it is still addressed by this new analysis.
More generally, I think you also need to be careful about what tests you use -- you have increased your nominal sample size, but your individual observations are no longer independent (since they occur in groups). Your test may be more "powerful" in the sense that you have included more information and may be able to detect more variation (as in the difference between parametric and non-parametric tests), but it should not be more powerful in terms of sampling theory (at least, it should not be much more powerful). If you treat them as independent samples, you will underestimate the sampling error. Unfortunately, I do not know how to account for this... it may be addressed by a separate question.
Finally, the only behavior that I would consider truly cheating is if you are deciding to change the statistical test after looking at the results of those tests. Ideally, you will have decided upon your statistical tests prior to handling any data -- if you do decide to change your tests after working with the data, you have to be very careful that you aren't fishing for a "positive result". One option is to correct for multiple testing, but even that is tricky. The ideal solution would be to collect a new, independent dataset and test your hypothesis there (treating your first dataset as a preliminary study that was used to generate a hypothesis).
