As increasingly complex hypothesis-testing scenarios are considered in many scientific fields, analytic derivation of null distributions is often out of reach. To the rescue comes Monte Carlo testing, which may appear deceptively simple: as long as you can sample test statistics under the null hypothesis, the $p$-value is just the proportion of sampled test statistics that exceed the observed test statistic. Sampling test statistics is often simple once you have a Monte Carlo null model for your data, and defining some form of randomization procedure is also, in many cases, relatively straightforward. However, there may be several possible choices of a randomization null model for the data and no clear-cut criteria for choosing among them. Obviously, different null models may lead to very different $p$-values, and a very low $p$-value may thus occur due to the inadequacy of the chosen null model. It is preferable to use assumptions about the underlying random data generation process to guide selection of a null model. In many cases, we may order the null models by increasing preservation of the data characteristics, and we argue in this paper that this ordering in most cases gives increasing $p$-values, that is, lower significance. We denote this as the null complexity principle. The principle gives a better understanding of the different null models and may guide in the choice between the different models.

Published at http://dx.doi.org/10.1214/14-STS484 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)