Generalized Monte Carlo significance tests

Simple Monte Carlo significance testing has many applications, particularly in the preliminary analysis of spatial data. The method requires the value of the test statistic to be ranked among a random sample of values generated according to the null hypothesis. However, there are situations in which a sample of values can only be conveniently generated using a Markov chain, initiated by the observed data, so that independence is violated. This paper describes two methods that overcome the problem of dependence and allow exact tests to be carried out. The methods are applied to the Rasch model, to the finite lattice Ising model and to the testing of association between spatial processes. Power is discussed in a simple case.