Li (Reference Li1988) coined the term ‘pseudo-random mating’ to apply to his model which demonstrated that Hardy–Weinberg proportions can be maintained with nonrandom mating for an autosomal locus with two alleles.

Stark (Reference Stark1980) gave the following mating system, which was used to classify some systems of partial inbreeding, given here in the original notation to avoid confusion with the notation in the main part of the article:

(1) ${f_{ij}} = {f_i}{f_j}(1 + \mu {d_i}{d_j}/S + \nu {e_i}{e_j}/T), i = 0, 1, 2, j = 0, 1, 2,$

where ${d_0} = - 2p$ , ${d_1} = q - p$ , ${d_2} = 2q$ , $S = 2pq(1 + \lambda )$ , ${e_0} = - p(1 - \lambda )/(q + \lambda p) \;{e_1} = 1$ , ${e_2} = - q(1 - \lambda )/(p + \lambda q),$ $T = pq(1 - \lambda )(1 + \lambda )/((q + \lambda p)$ $(p + \lambda q)),{f_0} = {q^2} + \lambda pq,$

${f_1} = 2pq(1 - \lambda )$ , ${f_2} = {p^2} + \lambda pq$ , $\mu = 2\lambda /(1 + \lambda )$ . Terms ${f_0}$ etc are the genotype frequencies in equilibrium, and so are the Hardy–Weinberg frequencies when $\lambda = 0$ . When $\lambda = 0$ , the component involving $\lambda $ in Eqn. (1) drops out but leaves the term involving $\nu $ so that the mating frequencies $\{ {f_{ij}}\} $ are not random frequencies unless $\nu = 0$ . This demonstrates the fact that Hardy–Weinberg frequencies can be maintained with nonrandom mating. Also, it clearly identifies the separation between Hardy–Weinberg frequencies and frequencies maintained by systems of mating with inbreeding.

We have shown for an autosomal locus how, with either two or three alleles, the parental distribution can be reproduced among offspring (Stark, Reference Stark2021; Stark & Seneta, Reference Stark and Seneta2012, Reference Stark and Seneta2013). A corollary of this is the fact that Hardy–Weinberg proportions can be maintained by nonrandom mating. One of the requirements of paternity experts is expressed as follows: ‘knowledge of genotype frequencies in defined populations in which the polymorphism is in Hardy-Weinberg equilibrium and random mating occurs’ (Geserick & Wirth, Reference Geserick and Wirth2012, p. 164). In his definition, Buckleton (Reference Buckleton, Buckleton, Triggs and Walsh2005b, p. 68) comes very close to one of the main points of this article when he writes ‘the Hardy-Weinberg law is a statement of independence between alleles at one locus’ but then includes random mating as one of the conditions that make the law true.

In the scenario sketched by Clayton and Buckleton (Reference Clayton, Buckleton, Buckleton, Triggs and Walsh2005, pp. 224−226), fingernail clippings have been taken from a woman who has been assaulted and claims to have scratched her attacker. Suppose that evidence (E) consists of DNA from two individuals and can be fully explained by the presence of DNA from both the woman and her suspected attacker. The authors explain the forensic approach by considering the case when the woman’s DNA at a particular locus is A ₁ A ₂ and that of the other person is A ₃ A ₄. They calculate the value of the likelihood ratio, a ratio of probabilities:

$$LR = {{{\rm{pr}}(E|{G_s}, {G_v}, {H_p})} \over {{\rm{pr}}(E|{G_v}, {H_d})}}, $$

where H _p is the hypothesis that the nail clippings contain the DNA of the complainant and the suspect, H _d is the hypothesis that the nail clippings contain the DNA of the complainant and an unrelated person, G _s is the genotype of the suspect and G _v is the genotype of the complainant. Clayton and Buckleton find the likelihood ratio to be $1/(2{p_3}{p_4})$ by invoking the Hardy–Weinberg proportion for the suspect. The importance of the LR is seen in the relation:

$${\rm{posterior}}\;{\rm{odds = likelihood}}\;{\rm{ratio}} \times {\rm{priorodds}}.$$

The prior odds are the odds on the hypothesis H _p before DNA evidence, and the posterior odds are the odds after DNA evidence. If the [subjective] probability of an event is p, the odds in favor of the event are p/(1 − p). Good (Reference Good1950, p. 62) has a note on terminology applied to odds (o): ‘If o = m/n it is often said that the odds are “m to n on” or “n to m against”’. Probability can be calculated from odds by $$p = o/(1 + o).$$ As Buckleton (Reference Buckleton, Buckleton, Triggs and Walsh2005a) points out, the weighing of evidence is subjective, so, while the calculation of the likelihood ratio may have a scientific basis, the subjective element might be disputable.

In this article, we give the conditions for maintaining equilibrium for a system with four alleles. The principle could be extended to any number of alleles.

Boyd (Reference Boyd and Burdette1962, p. 335) begins his survey of blood groups, starting with ABO, as follows:

In his tribute to Felix Bernstein, Crow (Reference Crow1993, p. 7) refers to ‘that genetically refractory species Homo sapiens’, citing two of Bernstein’s major papers relating to the ABO system (Bernstein, Reference Bernstein1924, Reference Bernstein1925). Westhoff (Reference Westhoff2019) summarizes the evolving methodology and applications of genotyping.

The ABO system with four alleles as set out by Penrose (Reference Penrose1973, pp. 25–27; p. 132) and Boyd (Reference Boyd and Burdette1962, pp. 335–337) is a relevant example. Ostrowski et al. (Reference Ostrowski, Rutkowski and Rutkowski2020) outline the important role played by Ludwik Hirszfeld (1884–1954) in the introduction of ABO into medical practice. Apart from its importance in, for example, organ transplantation, it is studied in other specialities. Kahr et al. (Reference Kahr, Franke, Brun, Wisser, Zimmermann and Haslinger2018) compared postpartum blood loss in type O and non-O women. They found a statistically significant but clinically minor increase in blood loss following delivery in women of type O. They suggest that O carriers may suffer from aggravated bleeding in the presence of additional obstetric bleeding pathologies.

Adapting the notation of Crew (Reference Crew1947, p. 65) the alleles would be written as H^A1, H^A2, H^B and H^O, but to conform with the general notation used with three alleles (in an earlier paper) are written as A, B, C and D. Genotypes are taken in order AA, BB, CC, DD, AB, AC, AD, BC, BD, CD, which are numbered from 1 to 10 and their frequencies denoted by $${G_i},i, = 1,2, \ldots ,10.$$

This article gives the conditions on the matrix of mating proportions such that the distribution of genotypes in the parents is reproduced in the offspring. The included numerical example illustrates how a parental distribution that follows the Hardy–Weinberg form can be maintained with nonrandom mating.

A Stable Population and Hardy–Weinberg Frequencies

Phenotypic identities are ignored so that the focus is on the 4 genes and 10 genotypes. There are 100 possible mating combinations, and the proportions are set out in a symmetric matrix with elements c _i,j , i, j = 1, 2, …, 10.

The parental distribution is reproduced if the mating matrix obeys the following constraints:

$${{{c_{55}} = 4{c_{12}};{c_{66}} = 4{c_{13}};{c_{77}} = 4{c_{14}};{c_{88}} = 4{c_{23}};{c_{99}} = 4{c_{24}};{c_{10, 10}} = 4{c_{34}}}};$$

$${c_{56}} = 2{c_{18}};{c_{57}} = 2{c_{19}};{c_{58}} = 2{c_{26}};{c_{59}} = 2{c_{27}};{c_{67}} = 2{c_{1,10}};{c_{68}} = 2{c_{35}};$$

$${{c_{6, 10}} = 2{c_{37}};{c_{79}} = 2{c_{45}};{c_{7, 10}} = 2{c_{46}};{c_{89}} = 2{c_{2, 10}};{c_{8, 10}} = 2{c_{39}};{c_{9, 10}} = 2{c_{48}}};$$

$${c_{5,10}} = {c_{69}} = {c_{78}}.$$

These restraints can be verified by calculating the progeny of each genotype and showing that it is the same as the proportion in the parents. For example, for genotype AB, the proportion in progeny is calculated from

$$\eqalign{ & 2{c_{12}} + {c_{15}} + {c_{18}} + {c_{19}} + {c_{25}} + {c_{26}} + {c_{27}} + \cr & + \;({c_{55}} + {c_{56}} + {c_{57}} + {c_{58}} + {c_{59}} + {c_{68}} + {c_{69}} + {c_{78}} + {c_{79}})/2. \cr} $$

Referring to the constraints and taking account of the symmetry of the mating matrix, this expression is equal to

$$\eqalign{ & {1 \over 2}{c_{55}} + {c_{51}} + {1 \over 2}{c_{56}} + {1 \over 2}{c_{57}} + {c_{52}} + {1 \over 2}{c_{58}} + {1 \over 2}{c_{59}} + \cr & + {1 \over 2}{c_{55}} + {1 \over 2}{c_{56}} + {1 \over 2}{c_{57}} + {1 \over 2}{c_{58}} + {1 \over 2}{c_{59}} + {c_{53}} + {1 \over 2}{c_{5,10}} + {1 \over 2}{c_{5,10}} + {c_{54}}. \cr} $$

Collating the terms gives the sum of the elements in the 5th row of the mating matrix and so the frequency of genotype AB in the parents.

Table 1 is an example that illustrates how Hardy–Weinberg frequencies can be maintained with nonrandom mating. The gene frequencies are 6/32, 7/32, 9/32 and 10/32, and the genotype frequencies 36/1024, 49/1024, 81/1024, 100/1024, 84/1024, 108/1024, 120/1024, 126/1024, 140/1024 and 180/1024. Each element in the table is to be divided by 8192 to convert to a fraction.

Table 1 Nonrandom mating proportions which produce the same Hardy–Weinberg frequencies in offspring as in parents (multiplied by 8192)

Genotypes are taken in order, AA, BB, CC, DD, AB, AC, AD, BC, BD, CD.

Estimating Gene Frequencies

The frequencies of genes H^A1, H^A2, H^B and H^O are $$\{ {p_1},{p_2},{p_3},{p_4}\} $$ and are defined in terms of the parental frequencies as

$${p_1} = (2{G_1} + {G_5} + {G_6} + {G_7})/2, $$

$${p_2} = (2{G_2} + {G_5} + {G_8} + {G_9})/2, $$

$${p_3} = (2{G_3} + {G_6} + {G_8} + {G_{10}})/2, $$

$${p_4} = (2{G_4} + {G_7} + {G_9} + {G_{10}})/2.$$

In Crew’s (1947, p. 65) notation, the correspondence between genotypes and phenotypes is as follows: A₁ ˜ (H^A1H^A1, H^A1H^A2 & H^A1H^O); A₂ ˜ (H^A2H^A2 & H^A2H^O); B ˜ (H^BH^B & H^BH^O); O ˜ (H^OH^O); A₁B ˜ (H^A1H^B); A₂B ˜ (H^A2H^B). The procedure for estimating gene frequencies, from a sample, used by Hartl & Clark (Reference Hartl and Clark1989, pp. 40–42) can be adapted for four alleles. Wherever necessary, the Hardy–Weinberg frequencies can be used to split the phenotype counts according to the following correspondences:

$${{\rm{A}}_1}\sim p_1^2:2{p_1}{p_2}:2{p_1}{p_4};{A_2}\;\sim p_2^2:2{p_2}{p_4};B\sim p_3^2:2{p_3}{p_4}.$$

Penrose (Reference Penrose1973, p. 132) gives the following phenotypic counts per thousand:

$${{\rm{A}}_1}\sim 349;{{\rm{A}}_2}\sim 97;{\rm{B}}\;\sim 85;{\rm{O}}\sim 436;\,{{\rm{A}}_1}{\rm{B\sim 25;}}\,{{\rm{A}}_2}{\rm{B\sim 8}}.$$

The following gene frequencies are compatible with the phenotypic counts:

$${{\rm{H}}^{{\rm{A1}}}}\sim 0.2088;\,{{\rm{H}}^{{\rm{A2}}}}\sim 0.0694;\,{{\rm{H}}^{\rm{B}}}\sim 0.0609;\,{{\rm{H}}^{\rm{O}}}\sim 0.6609$$

The method of Hartl and Clark (Reference Hartl and Clark1989), like other methods, assumes that zygotes are formed by the union of independently drawn gametes.