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Testing for Fit to Hardy-Weinberg Equilibrium
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There are several ways to determine whether a given population conforms to the Hardy-Weinberg equilibrium at a particular locus. However, the question usually arises when there is just a single sample from a population, representing only one generation. Can the existence of the Hardy-Weinberg equilibrium be determined with just one sample The answer is that we can determine whether the three genotypes (AA, Aa, and aa) occur with the frequencies p2, 2pq, and q2. If they do, then the population is considered to be in Hardy-Weinberg proportions; if not, then the population is not considered to be in Hardy-Weinberg proportions.
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MN Blood Types
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To determine whether observed and expected allelic frequencies are the same, we can use the chi-square statistical test. In a chi-square test, we compare an observed number with an expected number. In this case, the observed values are the actual numbers of the three genotypes in the sample, and the expected values come from the prediction that the genotypes will occur in the p2, 2pq, and q2 proportions. An analysis for the Ohio MN blood-type data is presented in table 19.2.The agreement between observed and expected numbers is very good, obvious even before the calculation of the chi-square value. Since the critical chi-square for one degree of freedom at the 0.05 level is 3.841 (see table 4.4), we nd that
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the Ohio population does not deviate from HardyWeinberg proportions at the MN locus. Earlier (chapter 4), we used the chi-square statistic to test how well real data t an expected data set based on a ratio predicted before the test. For example, we tested the data against a 3:1 ratio in table 4.2. In that case, the number of degrees of freedom was simply the number of independent categories: the total number of categories minus one. Here, however, our expected ratio is derived from the data set itself. The values p2, 2pq, and q2 came from p and q, which were estimated from the data. In this case, we lose one additional degree of freedom for every independent value we estimate from the data. If we calculate p from a sample, we lose one degree of freedom. However, we do not lose a degree of freedom for estimating q, since q is no longer an independent variable: q 1 p. So in the previous case, we lose two degrees of freedom one for estimating p and one for independent categories. The general rule of thumb in using chisquare analysis to test for data t to Hardy-Weinberg proportions is that the number of degrees of freedom must equal the number of phenotypes minus the number of alleles (in this case, 3 2 1). The chi-square analysis in table 19.2 may seem paradoxical. Because the observed allelic frequencies calculated from the original genotypic data are used to calculate the expected genotypic frequencies, it may appear to some individuals that the analysis must, by its very nature, show that the population is in HardyWeinberg proportions. To demonstrate that this is not necessarily the case, a counterexample appears in table 19.3. We use data similar to the Ohio sample, except that the original number of heterozygotes has been distributed equally among the two homozygote classes. The same allelic frequencies are maintained, yet the genotypic distribution differs. The chi-square value of 200.00 for these data demonstrates that the population represented in table 19.3 is not in Hardy-Weinberg proportions. Thus, a chi-square analysis of t to the HardyWeinberg proportions by no means represents circular reasoning.
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Table 19.2 Chi-Square Test of Goodness-of-Fit to the Hardy-Weinberg Proportions of a Sample of 200
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Persons for MN Blood Types for Which p
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MM Observed Numbers Expected Proportions Expected Numbers
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MN 76 2pq (0.3648) 72.96 0.127
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NN 10 q2 (0.0576) 11.52 0.201 Total 200 1.0 1.0 200.0 0.348
114 p2 (0.5776) 115.52 0.020
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Tamarin: Principles of Genetics, Seventh Edition
IV. Quantitative and Evolutionary Genetics
19. Population Genetics: The Hardy Weinberg Equilibrium and Mating Systems
The McGraw Hill Companies, 2001
Nineteen
Population Genetics: The Hardy-Weinberg Equilibrium and Mating Systems
Table 19.3 Chi-Square Test of Goodness-of-Fit to the Hardy-Weinberg Proportions of a Second Sample of 200