Exact Tests for Hardy–Weinberg Proportions

Abstract

Exact conditional tests are often required to evaluate statistically whether a sample of diploids comes from a population with Hardy-Weinberg proportions or to confirm the accuracy of genotype assignments. This requirement is especially common when the sample includes multiple alleles and sparse data, thus rendering asymptotic methods, such as the common x 2 -test, unreliable. Such an exact test can be performed using the likelihood ratio as its test statistic rather than the more commonly used probability test. Conceptual advantages in using the likelihood ratio are discussed. A substantially improved algorithm is described to permit the performance of a full-enumeration exact test on sample sizes that are too large for previous methods. An improved Monte Carlo algorithm is also proposed for samples that preclude full enumeration. These algorithms are about two orders of magnitude faster than those currently in use. Finally, methods are derived to compute the number of possible samples with a given set of allele counts, a useful quantity for evaluating the feasibility of the full enumeration procedure. Software implementing these methods, ExactoHW, is provided. W HEN studying the genetics of a population, one of the first questions to be asked is whether the genotype frequencies fit Hardy-Weinberg (HW) expectations. They will fit HW if the population is behaving like a single randomly mating unit without intense viability selection acting on the sampled loci. In addition, testing for HW proportions is often used for quality control in genotyping, as the test is sensitive to misclassifications or undetected null alleles. Traditionally, geneticists have relied on test statistics with asymptotic x 2 -distributions to test for goodness-of-fit with respect to HW proportions. However, as pointed out by several authors (Elston and Forthofer 1977;Emigh 1980;Louis and Dempster 1987;Hernandez and Weir 1989;Guo and Thompson 1992;Chakraborty and Zhong 1994;Rousset and Raymond 1995;Aoki 2003;Maiste and Weir 2004;Wigginton et al. 2005;Kang 2008;Rohlfs and Weir 2008), these asymptotic tests quickly become unreliable when samples are small or when rare alleles are involved. The latter situation is increasingly common as techniques for detecting large numbers of alleles become widely used. Moreover, loci with large numbers of alleles are intentionally selected for use in DNA identification techniques (e.g., Weir 1992). The result is often sparse-matrix data for which the asymptotic methods cannot be trusted.A solution to this problem is to use an exact test (Levene 1949;Haldane 1954) analogous to Fisher's exact test for independence in a 2 3 2 contingency table and its generalization to rectangular tables (Freeman and Halton 1951). In this approach, one considers only potential outcomes that have the same allele frequencies as observed, thus greatly reducing the number of outcomes that must be analyzed. One then identifies all such outcomes that deviate from the HW null hypothesis by at least as much the observed sample...

CYP2A6Longitudinal Effects in Young Smokers

“…Diplotype counts did not deviate substantially from the null hypothesis of Hardy-Weinberg equilibrium using a multi-allelic exact conditional test (likelihood ratio P value = .01) implemented in the ExactoHW software package. 15 …”

Taxonomic Revision of Desmognathus wrighti (Caudata: Plethodontidae)

“…We then classified each individual's genotype as homozygous for N. lewisi-like alleles, heterozygous, or homozygous for N. maculosus-like alleles. We used the exact test (Engels, 2009) and randomization to test for deviations from single-locus Hardy-Weinberg expectations. To assess linkage disequilibrium (co-occurrence of N. aff.…”

A Note on Exact Conditional and Unconditional Tests for Hardy-Weinberg Equilibrium

“…[2] . Testing for HWE has been studied extensively for decades [4][5][6][7] , and many improved algorithms have been proposed to overcome the computational burden [7] . Traditionally, the testing procedure relies on asymptotic approaches using either the Pearson χ 2 test or the likelihood ratio (LR) test [8,9] .…”

“…The power of a study would be overestimated due to the overestimated type I error rate. These asymptotic approaches may become unreliable for small to medium sample sizes or for alleles with low probability [7] .…”

“…The exact conditional approach (referred to as the C approach) [7,10] is the most frequently used approach among exact approaches. This approach is based upon the conditional probability of genotype counts with fixed allelic counts from the observed data, and the associated p value is calculated by enumerating all possible genotype counts.…”