The Power of a Few Large Blocks: A credible assumption with incredible efficiency

The most powerful assumption in data analysis is that of independence. Unfortunately, it is almost impossible to check, so it often must stay as an assumption. One way out of this trap is to assume less independence than usual: Instead of assuming all n observations are i.i.d. (independently and identically distributed), it is often more credible to assume that several blocks of the data are independent of each other. This gain in credibility, however, might come with a loss of statistical power compared to an analysis that correctly assumed (if it did) the full i.i.d. assumption. The main goal of this paper is to address this trade-off. We show that in the true i.i.d. case, little power is lost if the weaker block independence is assumed, as long as there are at least a handful of independent blocks; on the other hand, if block-dependent data is treated independently, the risk of making type I errors can be severe. Hence, we advocate using the weaker block independence assumption rather than the stronger full independence assumption, if the former leads to a gain in credibility.