© 1999, 2000 Gregory Carey Chapter 19: Advanced Topics - 1Quantitative Genetics: II - Advanced Topics:In this section, mathematical models are developed for the computation of differenttypes of genetic variance. Several substantive points about genetic variance components andtheir effect on the analysis of behavioral data are also made. The reader uninterested in themathematics can read the two text boxes to gain the substantive conclusions.Quantitative geneticists partition total genetic variance into three types—additive,dominance, and epistatic variance. Additive genetic variance measures the extent to whichphenotypic individual differences are predictable from the additive effects of allelicsubstitutions. Dominance genetic variance is variance associated with dominant geneaction—the fact that the genetic value for a heterozygote is not exactly the average of thegenetic value of the two homozygotes. Epistatic genetic variance is the variance associatedwith the statistical interaction among loci—gene by gene interaction as it is often called.Additive and dominance variance may be illustrated by examining a single locus,designated here as M locus, with two alleles M1 and M2. The additive effect of allele M2 isthe average change in genotypic values seen by substituting an M2 allele for an M1 allele.To find this effect, simply construct a new variable, called X1 here, that equals the number ofM2 alleles for the individual’s genotype. For genotype M1M1, X1 = 0; for M1M2, X1 = 1; andfor M2M2, X1 = 2. To account for dominance, construct another new variable, X2, with valuesof 0, 1, and 0 for, respectively, genotypes M1M1, M1M2, and M2M2. Table 18A.1 provideshypothetical data set up with the new variables1. It is assumed that the phenotype is scaledso that the population mean is 100 and the population standard deviation is 15.[Insert Table 18A.1 here] 1 This coding system can be used to account for any number of alleles at a locus. For example, to modeladditive effects of allele M3, construct another variable giving the number of M3 alleles in a genotype.There would then be three dominance variables, one for the heterozygote M1M2, a second for the© 1999, 2000 Gregory Carey Chapter 19: Advanced Topics - 2If we had actual data on individuals we would calculate the additive and dominanceeffects and their variance components by performing two regression. In the first, we wouldregress the phenotypic score, noted as Y herein, on X1. The squared multiple correlationfrom this regression equals the additive heritability (hA2), the proportion of phenotypicvariance associated with additive gene action at this locus. For the data in Table 18A.1, R2 =.137 = hA2, so 13.7% of phenotypic variance is predicted from additive gene effects at thislocus. The regression line for this equation will be the strait line of best fit that through thethree genotypic values for the three genotypes. It is illustrated in Figure 18A.1.The second regression model would be of the formYbbXbX=+ +01122.The intercept, b0, will equal the genotypic value for M1M1. The regressioncoefficient b1 equals the average effect of substituting allele M2 for M1 in any genotype.Finally, the coefficient b2 equals the genotypic value of the heterozygote less the average ofthe genotypic values of the two homozygotes. This measures dominant gene action. Forthe present example, b0 = 94, b1 = 7, and b2 = -5. The value of coefficient b1 informs us thaton average one M2 allele increases phenotypic values by 7 units. Because the value of b2 isnot 0, we can conclude that there is some degree of dominance. Because b2 = -5, we canconclude that there is partial dominance for allele M1 so that the genotypic value of theheterozygote is moved 5 units away from the midpoint of the two homozygotes and towardgenotype M1M1.The multiple correlation from this model equals the additive heritability plus thedominance heritability (hhA2D2+). For the present example, R2 = .162. Dominanceheritability can be found by subtracting the R2 from the first regression model from thisvalue: hD2=− =.. .162 137 025. heterozygote M1M3, and the third for the heterozygote M2M3. For each of the three dominance variables, the© 1999, 2000 Gregory Carey Chapter 19: Advanced Topics - 3The regression coefficients can also be used to calculate additive and dominanceheritability. Let p1 denote the frequency of allele M1 and p2, the frequency of M2; and let Vdenote the variance of the phenotype (which would be 225 for this example). Then,hpp b p p bVA2=+−212 1 1 2 22[( )],andhppbVD2=()21222.You should verify that these equations give the same heritability estimates and theregression procedure.To examine epistasis, consider the N locus with alleles N1 and N2. Just as wecreated two new variables for the M locus, we could also create two new variables to modelthe additive effect and the dominance effect at this locus. Call these variables X3 and X4.For genotypes N1N1, N1N2,and N2N2, the respective values for X3 will be 0, 1, and 2; therespective values for X4 would be 0, 1, and 0. The coding for the additive and the dominanceeffects at both the M and the N loci are given in Table 18A.2.[Insert Table 18A.2 here]In regression, an interaction between two predictor variables is modeled by creatinga new variable that is the product of the two predictor variables and entering this variable.Genetic epistasis is modeled in the same way. Multiplying the additive variable for the Mlocus (X1) by the additive variable for the N locus (X3) gives a new variable (X5 in Table18A.2) that geneticists call additive by additive epistasis. The variance associated with this istermed additive by additive epistatic variance.There are two different ways to model the interaction between an additive effect atone locus and a dominance effect at a second locus. First, we could multiply the additivevariable for the M locus by the dominance variable for the N locus. This new variable is appropriate heterozygote would have a value of 1 and all other genotypes would be assigned a value of 0.© 1999, 2000 Gregory Carey Chapter 19: Advanced Topics - 4given as X6 in Table 18A.2. The second way is to multiply the dominance variable for M bythe additive variable for N, giving variable X7 in Table 18A.2. Together variables X6 and X7model additive by dominance epistasis and the variance associated with these two variablesis called additive by dominance epistatic