The lower-bound estimate of sphericity is equal to

• A. 1/(k+1)
• B. 1
• C. k-1
• D. 1/(k-1)

Answer: (D)

Sphericity is about the equality of variances of the difference scores between every pair of related conditions in a repeated-measures design. The degree to which that equality holds is summarized by epsilon, which ranges from a minimum value up to 1. The smallest possible value—when sphericity is most violated—occurs at 1 divided by (the number of levels minus one). In other words, with k treatment levels, the lower-bound estimate of sphericity is 1/(k−1). This is why the lower bound is 1/(k−1): it reflects the extreme case of violation given the number of conditions. For example, with four levels the lower bound is 1/3, and with two levels it would be 1 (which makes sense because two levels never violate sphericity). The other options do not represent this minimal bound.