U-SHAPE: This is called the bath tube function or Teguchi loss function type. It generally is used in failure rate modeling. If your data distribution fallows this characteristic, the modeling test that comes close to it is the Weibull.

POWER: The relationship of sample size are explained in the attached articles---Beta Probability. See attached figure. Power = 1 - beta. The power formula is also given at the link below.

Power and samples size are related. If you select to adopt Weibull as the tool of analysis, the Gou et al. article gives sample size calculation that would meet your power requirement.

The shape of U or non-U is not relevant to beta probability because even if you deal with the loss function U shape, the cumulative probability distribution still follows the bell shape, i.e the bottom of the U (minima) matches the bell's maxima when the two curves are super-imposed (see figure: loss function). The beta probability (testing for power) is the verification of how much does your observed tail intrudes into the (1 - a) region.

REFERENCES: See attached and links.

http://www.columbia.edu/~cjd11/charles_dimaggio/DIRE/styled-4/code-12/

http://www.nature.com/hdy/journal/v106/n4/full/hdy201081a.html

This is an interesting topic and thanks for the explanation on modelling a U shaped distribution. I have published a theoretical network physiology perspective on the correlation between high and low ABPI and CVD. If you are interested the link is: http://journal.frontiersin.org/article/10.3389/fphys.2015.00101/abstract

It's open access.

Paul, you may not be able to pre-specify a power calculation for ABI vs. CV events from a purely mathematical direction.  The U-shaped curves for SBP or DBP vs CV events were determined from repeat evaluation of very large data sets.  Possibly you could interrogate large, publicly available data sets such as the NHANEs or ARIC databases (not sure if either contain ABI data) or other databse to explore your hypothesis, to evaluate whether or not there is a "signal" of a u-shaped distribution, then prepare your estimate with a reference to an established data set.  Best of luck

Thanks all for your responses. I'll explore the data using the above suggestion and see if I can make any headway. 

Cheers.