Efficiency factors for natural contrasts in partially confounded factorial designs

With reference to a partially confounded design for a 3n factorial experiment, an explicit expression is derived for the efficiency factors of natural contrasts of the forms linear, linear × linear, linear × quadratic, etc. This expression involves the harmonic mean of the efficiency factors for pencils. It is also shown that such a neat result does not exist for general s-level factorials where s>3. Nevertheless, the case of such general s is also explored.