next up previous contents
Next: Optimality criteria Up: Statistical Properties Previous: Canonical variances   Contents

Pairwise variances

In statistical practice, some experiments focus on comparing the effect of each treatment to each other treatment; these are the elementary contrasts $\tau_i - \tau_{i'}$. The variances $v_{dii'}$ of the elementary contrasts for a connected design $d$, aside from the constant $\sigma^2$, are

v_{dii'} = c^+_{dii} + c^+_{di'i'} - 2 c^+_{dii'}

for $1 \leq i < i' \leq v$, where $c^+_{dii'}$ is the general element of $C^{+}_d$ . Several optimality criteria are based on the $v(v-1)/2$ numbers $v_{dii'}$, called pairwise variances. Moreover, partial balance properties are reflected in the $v_{dii'}$. For these reasons, pairwise_variances is also an element in the external representation. For disconnected designs some elementary contrasts are not estimable; in the external representation, the corresponding values $v_{dii'}$ are recorded as ``not_applicable.''

Peter Dobcsanyi 2003-12-15