## Problems

For the following problems, assume that e has a standard normal distribution.

2.1. Simulate the distribution of y for a backcross design. Do the same for an intercross design. Consider various levels of a and 5.

2.2. Consider the two major genes additive model:

where Xi denote the number of Ai alleles at locus i (i = 1, 2). Assume the genes corresponding to xi and x2 lie on two different chromosomes, so by Mendel's laws x1 and x2 are independent.

(a) Investigate the distribution of the phenotype for various values of the model parameters (including the probabilities pi of the allele Ai).

(b) Assume that the indicated genes are the only genes contributing to the trait, so e can be regarded as the environmental effect E. Find expressions for aj and ay2. (It will be helpful to rewrite the model as in (2.3).)

(c) Assume in addition that the parental strains are inbred. How would you estimate H2 if you know the phenotypes of samples from both parental strains and from the intercross?

(d) Extend the model to include k independent genes, k > 2. Assume that ai = a, for all i and that the parental strains are inbred. Can you figure out a way to estimate k and a from phenotypic data involving both parental strains and the intercross progeny?

2.3. Show that (2.3) follows from (2.2). Verify that the two terms involving xM and xF on the right-hand side of (2.3) are uncorrelated. Hence verify (2.4) and (2.5). Hint: To facilitate algebraic manipulations, it may be helpful to observe that \xM — xF \ = xM + xF — 2 XMXF •

2.4. The model (2.3) is customarily written in the somewhat different form y = m + a(x — 2p) + S[I{x=1y — 2p(1 — p) — (1 — 2p)(x — 2p)] + e , where a = a +(1 — 2p)S and (5 = —2 S. Show that this is the same as (2.3). The form given in (2.3) seems slightly easier to manipulate computationally and illustrates that what geneticists call "dominance" is exactly what statisticians call "interaction," in this case the interaction of the allele inherited from the mother with that inherited from the father.

2.5. Generalize the model of Prob. 2.2 to include interaction between loci, as follows: Starting from y = ¡j, + aixi + a2x2 + yx\x2 + e, where as above the two loci are assumed to lie on different chromosomes, re-write the model in the form:

y = m + ai(xi — 2pi) + a2(x2 — 2p2) + y(xi — 2pi)(x2 — 2p2) + e .

What are ai and a2? Find an expression for a^. The term 72pi(1— pi)p2(1— p2) is called the interaction variance, or more precisely the additive-additive interaction variance to distinguish this form of interaction from other possibilities.