Prediction Theory 1 Introduction 2 General Linear Mixed Model

18 EXERCISES
1. Below are data on progeny of 6 rams used in 5 sheep flocks (for some trait). The rams
were unrelated to each other and to any of the ewes to which they were mated. The first
number is the number of progeny in the herd, and the second (within parentheses) is the
sum of the observations.
Let the model equation be
Ram
Flocks
ID 1 2 3 4 5
1 6(638) 8(611) 6(546) 5(472) 0(0)
2 5(497) 5(405) 5(510) 0(0) 4(378)
3 15(1641) 6(598) 5(614) 6(639) 5(443)
4 6(871) 11(1355) 0(0) 3(412) 3(367)
5 2(235) 4(414) 8(874) 4(454) 6(830)
6 0(0) 0(0) 4(460) 12(1312) 5(558)
y ijk = µ + F i + R j + e ijk
where F i is a flock effect, R j is a ram effect, and e ijk is a residual effect. There are a total
of 149 observations and the total sum of squares was equal to 1,793,791. Assume that
σ 2 e = 7σ 2 f = 1.5σ2 r when doing the problems below.
(a) Set up the mixed model equations and solve. Calculate the SEPs and reliabilities of
the ram solutions.
(b) Repeat the above analysis, but assume that flocks are a fixed factor (i.e. do not add
any variance ratio to the diagonals of the flock equations). How do the evaluations,
SEP, and reliabilities change from the previous model
(c) Assume that rams are a fixed factor, and flocks are random.
similarly to the previous two models
Do the rams rank
2. When a model has one random factor and its covariance matrix is an identity matrix times
a scalar constant, then prove the the solutions for that factor from the MME will sum to
zero. Try to make the proof as general as possible.
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