An importance sampling algorithm for exact conditional tests in log-linear models

A simple but quite general simulation method for conducting exact conditional lack-of-fit tests in log-linear models is proposed. Our Monte Carlo approximation utilises an importance sampling method motivated by the crude normal approximation to the Poisson distribution. Examples considered include tests of quasi-symmetry and related models for square tables and tests concerning higher-order interactions in multi-way tables. The method is competitive with direct simulation from the exact conditional distribution when this is feasible and outperforms alternative Monte Carlo procedures when direct simulation is infeasible provided the number of degrees of freedom of the test is not too large. Extension of the method to tests against non-saturated alternatives is straightforward and is briefly discussed and illustrated.