SOLUTIONS for Optional Problems in Quantitative Genetics

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since 26 Nov 1999

Suppose Var(x) = 10, Var(y) = 25, and Cov(x,y) = 10
1. What is the slope for the regression of y on x (i.e., what is b for y = a + bx)?
  - SOLUTION: b = Cov(x,y)/ Var(x) = 10/10 = 1
2. What is the correlation between x and y?
  - SOLUTION: p(x,y) = Cov(x,y)/[Var(x) Var(y)]^1/2 = 10 / (250)^1/2 = 0.634
3. What fraction of the variation in y is accounted for by knowing x?
  - SOLUTION: Fraction of variation = the square of the correlation or 0.632² = 0.4
4. What is the variance in y if we know the x value?
  - SOLUTION: Var(y | x ) = Var(y) [ 1-p(x,y)²] = 25*0.6 = 15
Consider two relatives where for any locus, Prob(1 allele ibd) = 1/3 and Prob(both alleles ibd) = 1/6. What is the genetic correlation between such relatives?
- SOLUTION: [ (1/3)*(1/2) + (1/6)*1] Var(A) + (1/6)Var (D) = (1/3)Var(A) + (1/6)Var(D)
Suppose the mean number of nose hairs in the population is 30. Alas, your dad has 40 and your mom has 60. If the heritability is 0.75, what is your expected number of nose hairs?
- SOLUTION: your mean = 30 + (0.75/2)(60-30) + (0.75/2)(40-30) = 30+ 0.375*(30+10) = 45
Suppose in a crop plant that we allow only the tallest plants to produce pollen, while female plants are chosen at random. Suppose (i) the average height of all plants is 60, (ii) the mean height of pollen donors is 100, and (iii) the average height of their offspring is 70. What is the heritability?
- SOLUTION: The key is to remember that for selection on one parent, R = (h²/2)*S. Here R = 70-60 = 10 and S = 100-60 = 40, giving 10 = (h²/2)*40 = 20*h^2 or h² = 1/2