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Biometrika 1999 86(2):321-332; doi:10.1093/biomet/86.2.321
© 1999 by Biometrika Trust
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An importance sampling algorithm for exact conditional tests in log-linear models

JG BoothA1 and RW ButlerA2

A1 Department of Statistics, University of Florida, Gainesville, FL 32611, USA E-mail: jbooth@stat.ufl.edu A2 Department of Statistics, Colorado State University, Fort Collins, Colorado 80525, USA E-mail: walrus@stat.colostate.edu

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 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.

Key Words: Delta method; Markov chain; Monte Carlo error; Normal approximation; Ratio estimate.


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