## Testing and Measuring

Usually a test statistic is evaluated under the hypothesis of compliance with HWE. Numerous alternatives for testing HWE have been proposed in the literature and frequentist approaches are most common. The chi-square test compares the observed with the expected numbers under HWE genotype frequencies and remains the most popular option. However, it is now commonly recognized that large sample goodness of fit tests, such as the chi-squared test and the likelihood ratio test, are inefficient and too liberal and may produce contradictory results when the sample size is small or some genotype frequencies very low.

The exact test is an alternative to the chi-square test and can be performed easily for biallelic loci when the asymptotic assumptions for the former test do not hold. Its performance (power and outbreeding detection) is comparable to the chi-square test, but has the advantage of being able to deal with low genotype frequencies, when the chi-square asymptotic distribution is inadequate. The test assesses the conditional probability to observe the given number of homozygotes for fixed sample size. Given that only a finite number of combinations (and thus a finite number of probabilities) occur, the 'achievable significance level' is selected to be as close as possible to the nominal 5%. For the multiallelic loci the exact test proposed by Guo and Thompson can be applied. They proposed two relatively simple inference algorithms based on Markov Chain Monte Carlo methods. More sophisticated approaches for exact inference have been recently proposed, as the use of the Bayes factor which is however restricted for the case of a biallelic locus.

We need to keep in mind that a nonsignificant test result is equivalent to 'non-rejection' of the HWE assumption but it does not prove that the locus exhibits HWE. Besides, the usually applied tests have low power.

Bayesian approaches have also become popular. They can deal with small samples and low or zero frequencies for some genotypes better than the frequentist approaches do. However, the main feature that makes them attractive is their ability to evaluate the 'degree' of departure from HWE. Shoemaker et al. provide a very useful outline of the Bayesian tests as well as a comparative study with the frequentist approaches. The core idea is to parametrize the HWE law; popular choices are the disequilibrium parameter and the fixation coefficient.

The disequilibrium parameter is the difference between observed and expected numbers of homozygotes. The fixation coefficient F can be estimated as one minus the ratio of the number of observed heterozygotes to the number expected under HWE, and can take values between

1 - n and 1. For populations in HWE, the expectation of F is zero. Negative values imply outbreeding and positive values inbreeding.

There is no strict consensus on what constitutes a 'large' fixation coefficient; a value outside the interval [—0.03, 0.03] are conventionally considered an important deviation. For a biallelic locus the observed genotype frequencies can be parametrized as aa = — n)F, a A = 2^(1 — ^)(1 — F), AA = (1 — ^)2+^(1 — ■k)F. Posterior distributions samples with MCMC are used to estimate the probability that the parameter F lies within or outside a given interval.

The package gap for the freely available software R© provides a range of options for testing HWE. Other commercial software such as Stata also have packages for HWE testing.

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