  We consider conditional exact tests of factor effects in designed experiments
for discrete response variables. Similarly to the analysis of contingency
tables, a Markov chain Monte Carlo method can be used for performing exact
tests, when large-sample approximations are poor and the enumeration of the
conditional sample space is infeasible. For designed experiments with a single
observation for each run, we formulate log-linear or logistic models and
consider a connected Markov chain over an appropriate sample space. In
particular, we investigate fractional factorial designs with $2^{p-q}$ runs,
noting correspondences to the models for $2^{p-q}$ contingency tables.
