Contributions to quantitative genetic models by Yule and by Weinberg prior to Fisher 1918
This year we mark and celebrate the centenary of R. A. Fisher's 1918 (Phil. Trans. R. Soc. Edin. 52: 393–433) paper on “The correlation between relatives on the supposition of Mendelian inheritance,” which has had a fundamental impact on the understanding and utilization of what we now call quantitative genetics. Here, I discuss briefly some of the backgrounds to the science and the scientists that would have contributed more to its development had their work been known. W. Provine (1971, The origins of theoretical population genetics. University of Chicago Press, Chicago) gives a broader history.
The basic data recording what we now call quantitative traits, and on which most modelling was based, were obtained by Francis Galton and his collaborators. A biography by M. G. Bulmer 2003 (Francis Galton. Pioneer of Heredity and Biometry. Johns Hopkins University Press, Baltimore) discusses Galton's work and includes further history on its impact and interpretation than is possible here. The analyses focussed on human stature (height), which was a wise and, presumably partly fortuitous, trait to take as a model because (we now know) its heritability is high, with little confounding variation due to common environment of sibs or non-additive gene action. In view of the inferences made subsequently from the correlations among sibs and between generations, the history of the field would probably have been different had he used a less heritable trait.
Galton estimated, inter alia, the regression of height of child on height of parent. Like Darwin before him, he was troubled by the fact that the variation among the offspring generation was maintained, whereas any averaging of parental contributions would have been expected to halve, because of its dependence on the parental mean. To address this problem, Galton speculated about particulate inheritance (Bulmer 2003, ibid.), but was unaware of any biological foundation for his models. When Mendel's findings became known and provided this key fact, Galton did not fully appreciate their relevance. Further, Mendel's finding of dominance, showing that effects on phenotype were not all linearly related to number of gene copies itself, had a major impact subsequently on both arguments and developments.
Karl Pearson, leader of the UK biometric school, set out to use Galton's findings on human height to assess the relevance of Mendel's laws. He deduced from theory the correlations among relatives incorporating the discrete Mendelian rules with dominance and then tested the fit of the models. Pearson (1904, Phil. Trans. R. Soc. London A203:53–65, 84–86) concluded, with my clarification shown as [ ]: “First. – That the parental correlation appears to be markedly greater than 1/3, nearer to 0.45 to 0.5. [1/3 is the correlation Pearson computed for the correlation with complete dominance] Secondly – That this correlation appears to vary slightly from character to character [contrary to the precise figures expected from Mendel's rules without including environmental effects.] Thirdly. – That it does not appear to be absolutely the same for all species.” Based on such analyses, Pearson rejected Mendelian genetics as the foundation for the inheritance of quantitative traits: his forcefulness, dogmatism and devotion to continuous models were overriding. We have, of course, simple explanations for these findings under the Mendelian hypothesis, because he assumed complete dominance, no environmental variance, and therefore equal heritability of each trait. Subsequent to this dogmatic start, ideas evolved, with the most insightful developments being by Yule and by Weinberg, which preceded those of Fisher.
So far as the coefficients of correlation alone are concerned, it is accordingly impossible to distinguish between the effect of the heterozygote giving rise to forms that are not strictly intermediate and the effect of the environment in causing somatic variations which are not heritable.
The case taken is a limited one, but the results are sufficient to show that the theory of the pure gamete, as applied to compound characters, is much more flexible than would appear from Professor Pearson's work, and can hardly be summarily dismissed as inapplicable to cases in which the coefficients of correlation approximate to 0.5.
Yule's model provides the basis of those we have used subsequently for gene action in the absence of epistasis, but his contribution has largely been ignored.
Wilhelm Weinberg was a German physician (a gynaecologist), but with considerable mathematical skills, and was a highly prolific scientist with an interest in inheritance. Inter alia, Weinberg showed how to calculate the proportion of identical twins from the same sex twinning rate and worked on ascertainment bias. He is best known, of course, for discovering independently of the English mathematician G. H. Hardy what we know as the Hardy–Weinberg law. Their respective papers were both published in 1908 (Hardy, G.H. 1908 Science 28:49–50; Weinberg, W. 1908 Jahres Wurtt. Ver. Vaterl. Naturkind. 64:369–382). Hardy's analysis became known quickly, Weinberg's remained unrecognized, likely because of the obscure provincial journal in which Weinberg's paper was published. The result was known as Hardy's law until Stern (1943, Science 97: 137–138) drew attention to Weinberg's paper, and the law was then renamed. Weinberg's papers were written in German, and Crow (1999, Genetics 152: 821–825) suggested that was why they were generally ignored, but their difficulty probably also contributed.
Weinberg also proposed quantitative genetic models and deduced their implications. He published his work in two long papers in 1909 and summarized and further developed them in the following year (Weinberg 1910 Arch. Rassen. Ces. Biol. 7:35–49). A translation into English by Karin Meyer of the 1910 paper has been published in a book of collected papers, which is now out of print (Hill 1984, Quantitative Genetics. Part 1. Explanation and analysis of continuous variation. Benchmark Papers in Genetics. Van Nostrand Reinhold Co.), alongside the other papers mentioned in the present article. Here, citations to Weinberg are taken from this translation.
First, we find a surprise in Weinberg's paper: “Under assumptions of no environmental influence and therefore equal values for a trait for all individuals of the same heritable (endogenous) type have to be assumed. In this case panmixis leads to a constant composition of successive generations. This was shown first by Pearson for a sequence of generations originating form a single cross, while Hardy and myself, more or less at the same time and independently of each other, proved it for any composition of the generation in which panmixis begins.” (Weinberg 1910, ibid.). So Weinberg knew about Hardy's law, but presumably never informed Hardy, Pearson or anyone else who would appreciate it. Otherwise, it could have been called the “Hardy–Weinberg Law” over 30 years before Stern's suggestion. Weinberg seems not to have minded whether people read his work!
In the area of quantitative genetics, Weinberg's (1910, ibid.) paper shows he knew Pearson's work and conclusions. He makes no mention of Yule but his models fully accommodate those of Yule. Weinberg developed formulae for variation due to single loci and for multiple loci, “polyhybridism.” Although he refers also to “complicated polyhybridism,” that is epistasis, he does not derive any specific formulae for its possible influence on covariances among relatives. Weinberg drew attention to the environmental component of variation, and in particular the sharing of environment by relatives and consequent environmental correlations, both within and across generations, that are confounded with genetic contributions. Weinberg's paper is difficult, even in English.
To show his conclusions, I quote from the English translation of Weinberg's paper: “The dominance rule is not, therefore, the stumbling block in the dispute between Pearson and representatives of experimental research in inheritance, but it is due to an insufficient accounting for the influence of the environment by the English Biometrical School and an overvaluation of biometrics generally. The latter can only supplement with its statistical results the experimental and individual analysis but cannot replace it. The great advantages of an experimental and individual analysis of the phenomena of inheritance over the statistical treatment of averages obtain, on the contrary, fresh support through my investigations.”
Fisher was presumably unaware of this work by Weinberg (perhaps even of his existence). In any case, Fisher, as we all know, introduced so many concepts in his 1918 paper relevant to and beyond the work on the correlation of relatives, such as the partition of variance, the foundations of experimental design and of evolutionary theory, his place at the top table is not at risk. Had Weinberg's work been known, it is likely that he would have been there too.
