Among the class of nonparametric techniques is a group of methods called permutation, or randomization, methods.

**EXACT METHODS**

Among the class of nonparametric techniques is a
group of methods called permutation, or randomization, methods. The methods
have the advantage that conditioned on some aspect of the data at hand, they
have a significance level that is exactly the specified level. The
conditioning we refer to is conditioning on the marginal totals in a 2 × 2
table. In a two-sample problem, we condition on observing the combined
observations without regard to which population they came from.

For the parametric techniques that we have studied
in this course, achieving the correct significance level is simply a matter of
finding the correct critical value(s) in a table of the sampling distribution
under the null hypothesis. For more complicated testing situations in which
nonparametric methods are used or approximate distributions are applied, the
test may not be exact. For example, many bootstrap testing procedures provide
useful nonparametric tests but they are not exact over the entire range of
distributions that we consider under the null hypothesis. For such hypothesis
tests that have a large set of possible distributions for the population being
sampled, this exactness property is not obtainable.

We saw that Fisher’s exact test, an alternative to
the chi-square test for a 2 × 2 contingency table, is one example of an exact
permutation test. Cytel Corp. is one of the few companies that produce software
specializing in exact methods. Cyrus Mehta, Cytel’s president, began to develop
the corporation’s main products, StatXact and LogXact, in 1987. Cytel provides
the most extensive and best algorithms for performing exact probability
calculations. The software programs em-ploy fast algorithms based on network
optimization algorithms that were originally developed for operations research
problems. Cytel’s current products are described on their website at
www.cytel.com. The latest version of StatXact includes sample size and power calculations.

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