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target="_blank" rel="nofollow" href="#ulink_e1d4b590-5c81-5ff2-b273-7d8448887d5d">Table 2.5 χ² values and associated cumulative probabilities in the right‐hand tail of the distribution for one through five degrees of freedom.

	Probability
df	0.5	0.25	0.10	0.05	0.01	0.001
1	0.4549	1.3233	2.7055	3.8415	6.6349	10.8276
2	1.3863	2.7726	4.6052	5.9915	9.2103	13.8155
3	2.3660	4.1083	6.2514	7.8147	11.3449	16.2662
4	3.3567	5.3853	7.7794	9.4877	13.2767	18.4668
5	4.3515	6.6257	9.2364	11.0705	15.0863	20.5150

       Assuming Hardy–Weinberg to test alternative models of inheritance

      Biologists are all probably familiar with the ABO blood group and are aware that mixing blood of different types can cause blood cell lysis and possibly result in death. Although we take this for granted now, there was a time when blood types and their patterns of inheritance defined an active area of clinical research. It was in 1900 that Karl Landsteiner of the University of Vienna mixed the blood of the people in his laboratory to study the patterns of blood cell agglutination (clumping). Landsteiner was awarded the Nobel Prize for Medicine in 1930 for his discovery of human ABO blood groups. Not until 1925, due to the research of Felix Bernstein, was the genetic basis of the ABO blood groups resolved (see Crow 1993a).

      Landsteiner observed the presence of four blood phenotypes A, B, AB, and O. A logical question was then, “what is the genetic basis of these four blood group phenotypes?” We will test two hypotheses (or models) to explain the inheritance of ABO blood groups that coexisted for 25 years. The approach will use the frequency of genotypes in a sample population to test the two hypotheses rather than an approach such as examining pedigrees. The hypotheses are that the four blood group phenotypes are explained by either two independent loci with two alleles each with one allele completely dominant at each locus (hypothesis 1) or a single locus with three alleles where two of the alleles show no dominance with each other but both are completely dominant over a third allele (hypothesis 2). Throughout, we will assume that Hardy–Weinberg expected genotype frequencies are met in order to determine which hypothesis best fits the available data.

Our first task is a straightforward application of Hardy–Weinberg in order to determine the expected frequencies of the blood group genotypes. The genotypes and the expected genotype frequencies are shown in Table 2.6. Look at the table but cover up the expected frequencies with a sheet of paper. The genotypes given for the two hypotheses would both explain the observed pattern of four blood groups. Hypothesis 1 requires complete dominance of the A and B alleles at their respective loci. Hypothesis 2 requires A and B to have no dominance with each other but complete dominance when paired with the O allele.

Now, let's construct several of the expected genotype frequencies (before you lift that sheet of paper). The O blood group under hypothesis 1 is the frequency of a homozygous genotype at two loci (aa bb). The frequency of one homozygote is the square of the allele frequency: fa² and fb² if we use fx to indicate the frequency of allele x. Using the product rule or Mendel's second law, the expected frequency of the two‐locus genotype is the product of frequencies of the one‐locus genotypes, fa² and fb². For the next genotype under hypothesis 1 (A_ bb), we use a little trick to simplify the amount of notation. The genotype A_ means AA or Aa: in other words, any genotype but aa. Since the frequencies of the three genotypes at one locus must sum to 1, we can write fA_ as 1 − faa or 1 − fa². Then, the frequency of the A_ bb genotype is (1 − fa²)(fb)². You should now work out and write down the other six expected genotype frequency expressions: then lift the paper and compare your work with Table 2.6.

Table 2.6 Hardy–Weinberg expected genotype frequencies for the ABO blood groups under the hypotheses of 1) two loci with two alleles each, and 2) one locus with three alleles. Both hypotheses have the potential to explain the observation of four blood group phenotypes. The notation “fx” is used to refer to the frequency of allele x. The underscore (“_”) indicates any allele, for example, A_ means both AA and Aa genotypes. The observed blood type frequencies were determined for Japanese people living in Korea (from Berstein (1925) as reported in Crow (1993b)).

Blood	Genotype	Expected genotype frequency	Observed
Type	Hypothesis 1	Hypothesis 2	Hypothesis 1	Hypothesis 2	(total = 502)
O	aa bb	OO	fa2fb2	(fO)2	148
A	A_ bb	AA, AO	(1‐fa2)(fb)2	fA2 + 2fAfO	212
B	aa B_	BB, BO	fa2(1‐fb2)	fB2 + 2fBfO	103
AB	A_ B_	AB	(1‐fa2)(1‐fb2)	2fAfB	39

      The next step is to compare the expected genotype frequencies for the two hypotheses with observed genotype frequencies. To do this, we will need to estimate allele frequencies under each hypothesis and use these to compute the expected genotype frequencies. (Although these allele frequencies are parameter estimates, the “hat” notation is not used for readability.) For the hypothesis of two loci (hypothesis 1), fb² =

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