  Under a generalised estimating equation analysis approach, approximate design
theory is used to determine Bayesian D-optimal designs. For two examples,
considering simple exchangeable and exponential decay correlation structures,
we compare the efficiency of identified optimal designs to balanced
stepped-wedge designs and corresponding stepped-wedge designs determined by
optimising using a normal approximation approach. The dependence of the
Bayesian D-optimal designs on the assumed correlation structure is explored;
for the considered settings, smaller decay in the correlation between outcomes
across time periods, along with larger values of the intra-cluster correlation,
leads to designs closer to a balanced design being optimal. Unlike for normal
data, it is shown that the optimal design need not be centro-symmetric in the
binary outcome case. The efficiency of the Bayesian D-optimal design relative
to a balanced design can be large, but situations are demonstrated in which the
advantages are small. Similarly, the optimal design from a normal approximation
approach is often not much less efficient than the Bayesian D-optimal design.
Bayesian D-optimal designs can be readily identified for stepped-wedge cluster
randomised trials with binary outcome data. In certain circumstances,
principally ones with strong time period effects, they will indicate that a
design unlikely to have been identified by previous methods may be
substantially more efficient. However, they require a larger number of
assumptions than existing optimal designs, and in many situations existing
theory under a normal approximation will provide an easier means of identifying
an efficient design for binary outcome data.
