  Optimal block designs for additive models achieve their efficiency by
dividing experimental units among relatively homogenous blocks and allocating
treatments equally to blocks. Responses in many modern experiments, however,
are drawn from distributions such as the one- and two-parameter exponential
families, e.g., RNA sequence counts from a negative binomial distribution.
These violate additivity. Yet, designs generated by assuming additivity
continue to be used, because better approaches are not available, and because
the issues are not widely recognised. We solve this problem for single-factor
experiments in which treatments, taking categorical values only, are arranged
in blocks and responses drawn from a Poisson distribution. We derive
expressions for two objective functions, based on D_A- and C-optimality, with
efficient estimation of linear contrasts of the fixed effects parameters in a
Poisson generalised linear mixed model (GLMM) being the objective. These
objective functions are shown to be computational efficient, requiring no
matrix inversion. Using simulated annealing to generate Poisson GLMM-based
locally optimal designs, we show that the replication numbers of treatments in
these designs are inversely proportional to the relative magnitudes of the
treatments' expected counts. Importantly, for non-negligible treatment effect
sizes, Poisson GLMM-based optimal designs may be substantially more efficient
than their classically optimal counterparts.
