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WRIGHT's Inbreeding Coefficient

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WRIGHT's Inbreeding Coefficient Copyright 0 1995 by the Genetics Society of America Perspectives Anecdotal, Historical And Critical Commentaries on Genetics Edited 4 James F. Crow and William F. Dove SEWALL WRIGHT’S “Systems of Mating” William G. Hill Institute of Cell, Animal and Population Biology, University of Edinburgh, Edinburgh EH9 3JT, Scotland INETEEN ninety-six marksthe 75th anniversary of widely accepted for discrete characters, but controversy N the publication in this journal of the series of five remained as to the basis of variation in continuous traits papers by SEWALL WRIGHT(1921a-e) on “Systems of (PROVINE 1971). WEINBERG (1910),FISHER (1918), and Mating.” The definitions, methods, and results he pre- WRIGHT(ignorant of the work of the other two) set sented in these and two other papers published at about what became the accepted framework. WRIGHTalso suc- the same time (WRIGHT 1920,1922) have had a major cessfully addressed the problem of how genotype fre- and lasting effect on the theory and application of pop- quencies change in populations as a result of nonran- ulation genetics. My aim in this paper is to review the dom mating based on relationship or performance. work in these papers and to consider their influence Previous analysis of, for example, full-sib mating had mostly, but not exclusively, on animal improvement, involved the laborious evaluation of genotype probabili- but without being comprehensive in either topics or ties, whereas WRIGHT’Smethods were computationally references. Indeed, were I to cite only WRIGHT’Sworks simple and general. They were then and are still not that developed ideas from his 1921 papers, there would easy to understand, partly because of his use of path be a long bibliography, and it is easy to see why the coefficients with analysis in terms of continuous vari- reference lists in WRIGHT’S ownpapers looked so ego- ables rather than genotypes and partly because critical centric. PROVINE’S(1986) excellent biography provides points were dealt with very tersely. a full review ofWRIGHT’S role in genetics and evolution- WRIGHThad invented and used path coefficients be- ary biology, while WRIGHT’S(1968- 1978) four-volume fore the 1921 papers. The initial steps in their develop- treatise describes much of his work and views. ment, to describe general and specific size factors in- Background: WRIGHThad a broad training in biol- fluencing bone size of rabbits, had been taken while he ogy and came into genetics as a graduate student of was a graduate student. He had computed all partial Castle at the Bussey Institute of Harvard University. He correlations, but found that such an analysis provided worked mainly on coat color patterns in guinea pigs no understanding of causation. By computing the path for his Ph.D., which stimulated among other things his coefficients, which in statistical terms are standardized interest in multilocus inheritance and interactions. His partial regression coefficients, he aimed to describe the firstjob, from 1915, was as Senior Animal Husbandman effect of causalcomponents. In1918 WRIGHTpublished at the U.S. Department of Agriculture in Washington, his first version of the method of path coefficients (al- with a particular remit to analyze extensivedata on traits though notcalling them such), partitioning variation in such as color and body weight collected by ROMMELon length of an individual bone to independent hereditary inbred lines of guinea pigs to test the value, or other- causes including general size, length of leg bones, wise, of using inbreeding in animal improvement pro- length of hind limbs, and an independent term. grams. WRIGHT alsoundertook a substantial amount of In the first paper in which he undertook a genetic biometrical work and a widespread correspondence on analysis of a quantitative trait, WRIGHT(1920) used path animal breeding. It was, therefore, important that he coefficients to analyze data on the proportion of white could interpret data on quantitative traits from both color in spotted guinea pigs, and attributed variation inbred lines and livestock populations. These and other to three principal causes: heredity (h),environment problems he tackled in the “Systems of Mating” series common to litter mates (e), and other factors, largely while still with the U.S. Department of Agriculture. “ontogenic irregularity” or developmental noise (d). The Mendelian theory of inheritance had become The quantity h is the path coefficient from genotype to phenotype and equals the correlation between them. Address for cmrespondence: William G. Hill, Institute of Cell, Animal and Population Biology, University of Edinburgh, West Mains Rd., WRIGHTshows that h2 is the proportion of variance at- Edinburgh EH9 3JT, Scotland. E-mail: [email protected] tributable to heredity; this is, of course, the heritability, Genetics 143: 1499-1506 (August, 1996) 1500 W. G. Hill \d C. D. FIGURE 1.-WRIGHT’S famous diagramfrom his 1921 papers. The original legend is included. / Dam 3‘ FIG- 2.-A diagram illustrating the relations between two mated individuals and their progeny. E, H’,H” and If”’ arc the genetic constitutions of the four individaaab. G, C’, G“ and C”’ are four genn~elb. E and D ~p-t tangible external conditions and chance imgu- laritia as facton in development. C rcprcscntscbmce at segregation as a factor in determining the composition of the germ-cells. Path coefficients are represented by small letters. a term that he did not coin (see BELL 1977 for a discus- thereby setting the basic framework for the analysis of sion). In his partition of variation, WRIGHT had a term the effects of inbreeding and assortative mating on the for common environment of sibs but not dominance, correlation among relatives. In particular,WRIGHT com- whereas FISHER (1918) hadone for dominance but not putes how, for a locus with two equallyfrequent alleles, common environment. WEINBERG(1910) had already the frequency with which A alleles unite with A, a with included both (see translationby MEYER in HILL1984), a, and A with a depends on the correlation f and shows but had been ignored! that the percentage of heterozygosis is p = (1 - f)/2. The “Systems of Mating” series: Although the path The result is then used to show how h2 depends on diagram for coat color in guinea pigs first appears in heterozygosity and thus f: WRIGHT thereby made the the 1920 paper, it returns in part I of the better known critical transitions from a correlation f of continuous “Systems of Mating” the following year with one sig- variables to a functionp of frequencies of discrete geno- nificant change, which took WRIGHT beyond the work types and back to another quantitative measure h2. of WEINBERGand FISHER,the addition of a term (m) Subsequently (p. 121), WRIGHTdefines m more ex- describing the correlation of the parents, which en- plicitly as “the correlation betweenthe genetic constitu- abled him to analyzethe effects of nonrandom mating. tions of the parents” and provides many formulas, in- The figure fromWRIGHT (1921a), surelyone of the best cluding that for the correlation between full sibs when known diagrams in biological science, is reproduced there is nonrandom mating, i.e., m f 0. [Unfortunately, here (Figure 1).Important definitions in the 1921 pa- his results for dominance were incorrect and were cor- per are indeed obscure. He introduced and defined f rected subsequently (WRIGHT1931 and by hand on the (called the inbreeding coefficient in a later paper) as reprinted copy of the 1921 papers issuedby Iowa State follows (p. 118): College Press). FISHER (1918) had given the right for- If there is assortative mating fromany cause, there will mulas for the dominance contribution to the sib corre- be some correlation between the gametes which unite. lation.] WRIGHT alsopoints out that if an expression Represent this correlation by f: can be provided form, the correlations and path coeffi- Later (p. 119) he introduces m: cients in eachgeneration can be expressed in termsof those in the previous generation. The correlation between the egg and the sperm de- pends on that between the parental formulae which we This he does in the second paper (WRIGHT1921b), will represent by m . The correlation between the par- initially deriving resultsfor repeated brother-sister mat- ents is greater or less than that between their genetic ings: “As the reader may feel some doubt as to the constitutions, depending on whether the assortative mat- validity of the method of analysis used here, it will be ing is based on somatic resemblance or consanguinity. well to begin with a case in which the results have al- For these two cases he gives formulas relating f to m, ready been determined by direct methods. In this Perspectives 1501 case the correlation between mates is simply that be- cient of relationship of two individuals as the correla- tween a brother and sister, produced by the preceding tion between them. He is not explicit as to what mea- generation.” He gives arecurrence formula for m, sure on the individuals is being taken, but the formulas which he shows can be written in terms off in preceding imply it is the sum of values of the gametes, regarded generations (f and f ‘I) as the now familiar result f = as quantitative traits, of an individual. Hence the rela- (1 + 2f + f ”)/4. WRIGHTprovided such recurrence tionship is referred to by LUSH(1948) as the correlation formulas for a range of repeated mating systems and of genic values. The relationship rsd between sire (S) thereby essentially solved the problem of computing and dam (D)in terms of their own and their offspring’s the consequences on genotype frequencies for mating (0)inbreeding coefficient is given by systems based on consanguinity. The third paper deals with assortative mating based TYd 2f/J(1 + J)(1 fd) on phenotype, in which WRIGHT(1921~) showed that, and the numerator of the expression (to which we shall in contrast to mating of relatives, the consequences de- return) is the covariance of genic values.WRIGHT pend on the numberof lociaffecting the trait.
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