1. A few Simple Applications of Index Selection Theory
Focus is on using multiple sources
ff information to improve a single
trait

2. We measure n traits, but our goal is to improve the
response of trait one only.
In this case, the vector of weights has a simple
form and Ga simplifies to the the vector g1
of genetic covariances of all traits with trait 1
Hence, the Smith-Hazel weights and resulting
index are just (using g to denote g1)

3. The expected response
using this index is
The expected increase in
response over mass selection
The increase in accuracy in
predicting BV using the
index vs. phenotype alone

6. Application: Repeated records
Model: repeated measures on an individual include
their common genetic variance plus a common
permanent environmental effect,
Here, the repeatability r =
Smith-Hazel
weights become

7. Accuracy
Response

8. Indices with information from relatives
With n traits, need to estimate (n+1)(n+4)/2
parameters. With relatives only need trait
variance components
We can express g and P in terms of correlations,
Accuracy
Response

9. Simple case: Individual + one relative
Index
Improve of response

10. Combined Selection of both individual and family information
Individual selection: selection on z
Within-family selection: selection on (z - zf)
Between-family selection: selection zf
Note that individual selection places equal weight
on within- and between-family components
What is the optimal allocation of selection?

11. In an index-selection framework, we can phrase
this as the amount b1 of within- family vs. the
amount of between-family selection b2.
Equivalently, we can express this as the relative
weights on individual (b1) vs. family (b2 - b1) selection

12. Let t denote the phenotypic correlations among
sibs in an infinite population:
Likewise, the rA = 1/2 (FS), 1/4 (HS) denote the
genetic correlation.
With n sibs in a family, the finite-size corrections are

13. The Lush index Lush (1947) applied the Smith-Hazel index to this
problem. Here, the two traits are an individual’s
value and their family mean.
The resulting weights become

14. The resulting index can be expressed as either
the weights on within vs. between family components,
Or on individual vs. family selection,
Improvement of Response:

15. Relative weight on between-family vs. within-
family selection under Lush Index

18. Response under general combined selection
Under general weighting of
individual vs. family or
within- vs. -between family

19. MAS - Marker Assisted Selection
One very popular application of index selection is
MAS, or marker assisted selection.
The idea to use genetic markers to improve selection
is not new, going back to Neimann-Sorensen and Robertson
(1961) and Smith (1967).
We now have a very rich set of dense molecular markers
and so can actually apply some of the theory, developed
by Smith (1967) for single markers and by Lande and
Thompson (1990) for n markers
Idea: we have k molecular markers that show marker-
trait associations. Define a marker score m = Si aimi

20. ( ( ) - I = z + µ 1 ° h (1 Ω ) ∂ ¢ m R i = æ h ¢ s 1 + (1 ° ) Ω -
We wish to construct a Smith-Hazel index to maximize response in a trait given trait value z and the marker score m for an individual. That is, we wish to find the weights b for I = b1z + b2m that maximize response.
The result weights can be found to be (see notes) I s = b T z g P 1 h 2 Ω ∂ + m - ( ) I s = z + 1 h 2 (1 Ω ) ∂ m - ( *
which reduces to
Where r = fraction of s2A accounted for by the marker score m
The resulting response becomes R i = æ A h s 1 + (1 2 ) Ω

21. Performance of MAS relative to mass selection