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Lecture 25:

Index Selection

Bruce Walsh lecture notes

Synbreed course

version 10 July 2013

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Selection on an Index of Traits

I =!

bjzj = bTz

!2I = !(bTz,bT z) = bT !(z, z)b = bTPb

!2AI

= !A(bTz,bTz) = bT !A(z, z)b = bTGb

A common way to select on a number of traits at once

is to base selection decisions on a simple index of

trait values,

The resulting phenotypic and additive-genetic variance

for this synthetic trait are;

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!2I = !(bTz,bTz) = bT !(z, z)b = bTPb

!2AI

= !A(bT z,bT z) = bT !A(z, z)b = bTGb

h2I =

!2AI

!2I

= bTGbbTPb

This gives the resulting heritability of the index as

If iI is the selection intensity on the index, then the

response in the index from the breeder’s eq. becomes

RI = iI hI2 !I

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Class problem

For

Compute additive and phenotypic variance for I, and h2 for the

index.

Consider the following data set (which will be used often)

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"RI = ı h2I !I = ı·b

T GbbTPb

"bT Pb = ı· bTGb

bTPb

Sj =ı

!I

!

k

bkPjk S =ı

!IPb

The response in the index due to selection is

How does selection on I translate into selection on the

underlying component traits making up the index?

Since R = GP-1S, the response vector for the underlying

component means becomes

R = GP!1S =ı

!IGb = ı· Gb"

bTPb

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Class problem

For a selection intensity of i = 2.06 (upper 5% selected)

compute the response in for the index in the last

problem.

Compute the vectors of selection differentials and

responses (S and R) for the vector of component traits.

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Changes in the additive variance of the index

Bulmer’s equation holds for the index:

We can also apply multivariate Bulmer to follow the changes

in the variance-covariance matrix of the components:

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Direction of selection vs. direction of

response

Recall from our discussion of multivariate response that if we

select in a particular direction ", the actual vector R of

response is in a different direction

"

R A selection index represents a desired

direction of response. Hence, if we want

optional response in a particular

direction (vector a), we might best

achieve this by selection in a different

direction (vector b) such that the

response is in the direction of vector a

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a

Ra Selection on I = aTz,

get a response in

direction away from a

Rb

b Can we find a direction (index

weights b) such that selection on

I = bTz, gives a response in the

desired direction of a

Yes. These are the Smith-Hazel weights.

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The Smith-Hazel Index

Suppose we wish the maximize the response on the

linear combination aTz of traits

This is done by selecting those individuals with

the largest breeding value for this index, namely

Hence, we wish to find a phenotypic index I = bTz

so that the correlation between H and I in an

individual is maximized.

Key: This vector of weights b is typically rather

different from a!

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The Smith-Hazel Index

Smith and Hazel show that the maximal gain is obtained

by selection on the index bTz where

Thus, to maximize the response in aTz in the next generation,

we choose those parents with the largest breeding value for

this trait, H = aTg. This is done by choosing individuals with the

largest values of I = bsTz.

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Response under Smith-Hazel

Select on Ib = bTz, interest is response in Ia = aTz.

Response in the

index of interest

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In-class problem

We wish to improve the index

Compute the Smith-Hazel weights (using previous

P, G)

Compare response in Ia = aTz with response in aTz given

by selection on bsTz

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Problems with Smith-Hazel

• The major issue is that both P (easy) and G(harder, less precise) must be estimated.– Especially problematic (for G) with a large

number of component traits.

– Errors in either can have major impact on theindex, significantly reducing precision

– P and G change each generation from LD

• One estimation issue is that H = P-1G canhave negative eigenvalues.– The method of bending (regressing eigenvalues

back to their means) provides one solution

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Bending:

Tune (slowly increase) bending parameter # until all eigenvalues of

H* are positive.

1.00.80.60.40.20.0

0.2

0.4

0.6

0.8

1.0V

alu

e o

f E

igen

va

lue

!

!

!

Value of bending coefficient

1

2

3

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Other indices

• Because of these estimation issueswith Smith-Hazel, several otherindices have been proposed

• Idea is to weight information withhigher confidence more

– e.g., weight by heritabilities and ignoregenetic correlations

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Heritability index : bi = a ih2i

Other indices

Estimated index:

Base-index:

Elston (or weight-free) index:

Retrospective index: since R = Gb, b = G-1R

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Restricted and Desired-gains Indices

Instead of maximizing the response on a index, we

may (additionally) wish to restrict change in certain

traits or have a desired gain in specific traits in mind

Suppose for our soybean bean data, we wish no

response in trait 1

Class Problem: Suppose weights are aT = (0,1,1), i.e,

no weight on trait 1. Compute the response in trait

one under this index.

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Morely (1955): The simplest case of want to change

trait z1 while z2 remains constant

Want b1, b2 such that

no response in trait 2

Setting b1 = 1 and

solving gives weights

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Kempthorne-Nordskog restricted index

Suppose we wish the first k (or m) traits to be unchanged.

Since response of an index is proportional to Gb,

the constraint is CGbr = 0, where C is an m X k matrix

with ones on the diagonals and zeros elsewhere.

Kempthorne-Nordskog showed that br is obtained by

where Gr = CG

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Cost of using a restricted index is less total response

in the unrestricted components of the index.

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Tallis restriction index

More generally, suppose we specify the desired

gains for k combinations of traits,

Here the constraint is CGbr = d

Solution is

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Desired-gain index

Suppose we desire a gain of R, then since R = Gb,

the desired-gains index is b = G-1R

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Class problem

Compute b for desired gains (in our soybean traits)

of (1,-2,0)

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Non-linear indicesCare is required in considering the improvement goals when using

a nonlinear merit function (index), as apparently subtle differences

in the desired outcomes can become critical

A related concern is whether we wish to maximize the additive

genetic value in merit in the parents or in their offspring. Again,

with a linear index these are equivalent, as the mean breeding value

of the parents u equals the mean value in their offspring. This

is not the case with nonlinear merit.

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Quadratic selection index

Consider the optimal weights on a quadratic index

The matrix A of quadratic

weights is given by

Optimal weights

Smith-Hazel

weights

Quadratic

effects

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Linearization

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Linear weights

change with

the current

mean

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Fitting the best linear index to a nonlinear

merit function

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Sets of response vectors and selection differentials

Given a fixed selection intensity i, we would like to

know the set of possible R (response vectors) or

S (selection vectors)

Substituting the above value of b into

recovers

Hence,

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Sets of response vectors and selection differentials

This equation describes a quadratic surface of possible, R values,

Similarly, using R = GP-1S, gives the surface

These give surfaces of all possible values of either

R or S given the selection intensity. Nonlinear

index selection uses these results

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Goddard’s method• Mike Goddard (1983) suggested a simple approach

for ANY nonlinear index.– First, the planned total selection intensity over the

course of the experiment in used in the previous resultsto obtain a surface of responses (response values in thevector of traits given the intensity)

– Next, contours of each value for the nonlinear functionare overlaid on this response surface. The largest valuethat intersects the response surfaces gives the optimalresponse

• Namely the final trait values we are trying to achieve

– Using these values (final weights), a desired gains indexis used to construct the optimal weights to achieve thisfinal target

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Contours for different

selection intensities (dashed

lines)

Contours of equal value for

The merit function (solid

lines)

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Key: Optimal weights a function of intensity of selection

As total selection intensity changes, so do the optimal weights

Hence, optimal weights change as a function of the

length of selection.

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Sequential Approaches for Multitrait selection

• Often not all traits can be selected at once.

• Index selection generally optimal, but may also beeconomic reasons for other approaches.

• Sequential approaches are used in this case.– Multistage selection

• Selecting on different traits (or indexes) over differentlife stages

– Tandem Selection

• Select different traits in different generations

– Independent Culling

• Cull the first trait, then select on second, etc

– Selection of extremes

• Selection an individual with ANY trait value in theuppermost fraction p of the population

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Multistage selection

Selecting on traits that appear in different life

stages

Optimal selection schemes

Cotterill and James optimal 2-stage. Select 1st on x,

(save p1), then then on y (save p2 = p/p1). Goal

optimize the breeding value for merit g.