1. Lecture 7: Correlated Characters

2. Genetic vs. Phenotypic correlations
Within an individual, trait values can be positively or negatively correlated,
height and weight -- positively correlated
Weight and lifespan -- negatively correlated
Such phenotypic correlations can be directly measured,
rP denotes the phenotypic correlation
Phenotypic correlations arise because genetic and/or environmental values within an individual are correlated.

3. The phenotypic values between traits x and y
within an individual are correlated
Likewise, the environmental values for the
two traits within the individual could
also be correlated
Correlations between the breeding values of
x and y within the individual can generate a
phenotypic correlation

4. Genetic & Environmental Correlations
rA = correlation in breeding values (the genetic correlation) can arise from
pleiotropic effects of loci on both traits
linkage disequilibrium, which decays over time
rE = correlation in environmental values
includes non-additive genetic effects
arises from exposure of the two traits to the same individual environment

5. The relative contributions of genetic and environmental correlations to the phenotypic correlation
If heritability values are high for both traits, then
the correlation in breeding values dominates the
phenotypic corrrelation
If heritability values in EITHER trait are low, then
the correlation in environmental values dominates the
phenotypic corrrelation
In practice, phenotypic and genetic correlations often
have the same sign and are of similar magnitude, but
this is not always the case

6. Estimating Genetic Correlations
Similarly, we can estimate VA(x,y), the covariance in the
breeding values for traits x and y, by the regression of
trait x in the parent and trait y in the offspring
Trait x in parent
Trait y in
offspring
Slope =
(1/2) VA(x,y)/VP(x)
VA(x,y) = 2 *slope * VP(x)
Recall that we estimated VA from the regression of
trait x in the parent on trait x in the offspring,
Trait x in parent
Trait x in
offspring
Slope =
(1/2) VA(x)/VP(x)
VA(x) = 2 *slope * VP(x)

7. Thus, one estimator of VA(x,y) is
2 *by|x * VP(x) + 2 *bx|y * VP(y) 2
VA(x,y) =
by|x VP(x) + bx|y VP(y)
VA(x,y) =
Put another way,
Cov(xO,yP) = Cov(yO,xP) = (1/2)Cov(Ax,Ay)
Cov(xO,xP) = (1/2) VA (x) = (1/2)Cov(Ax, Ax)
Cov(yO,yP) = (1/2) VA (y) = (1/2)Cov(Ay, Ay)
Likewise, for half-sibs,
Cov(xHS,yHS) = (1/4) Cov(Ax,Ay)
Cov(xHS,xHS) = (1/4) Cov(Ax,Ax) = (1/4) VA (x)
Cov(yHS,yHS) = (1/4) Cov(Ay,Ay) = (1/4) VA (y)

8. G X E and Genetic Correlations
One way to deal with G x E is to treat the same trait
measured in two (or more) different environments as
correlated characters.
If no G x E is present, the genetic correlation should be 1
Example: 94 half-sib families of seed beetles were placed
in petri dishes which either contained (Environment 1)
or lacked seeds (Environment 2)
Total number of eggs laid and longevity in days were
measured and their breeding values estimated

9. Positive genetic correlations Negative genetic correlationsd s P r e s e n t . S e e d s A b s e n t . 3 3 2 2 1 1
Fecundity - 1 - 2 - 1 - 3 - 2 - 2 - 1 1 2 - 4 - 2 2 4 L o n g e v i t y i n D a y s F e c u n d i t y . L o n g e v i t y . 3 6 2 4 1 2
Seeds Absent - 1 - 2 - 2 - 4 - 3 - 2 - 1 1 2 3 - 2 - 1 1 2 S e e d s P r e s e n t

10. Correlated Response to Selection
Direct selection of a character can cause a within-
generation change in the mean of a phenotypically
correlated character.
Direct selection on
x also changes the
mean of y

11. Phenotypic correlations induce within-generation
changes
Trait y
Trait x
Phenotypic values Sy Sx
For there to be a between-generation change, the
breeding values must be correlated. Such a change
is called a correlated response to selection

12. Trait y
Trait x
Breeding values
Trait y
Trait x
Breeding values
Trait y
Trait x
Breeding values
Trait y
Trait x
Breeding values Rx
Ry = 0

13. Predicting the correlated response
The change in character y in response to selection
on x is the regression of the breeding value of y
on the breeding value of x,
Ay = bAy|Ax Ax
where
bAy|Ax =
Cov(Ax,Ay)
Var(Ax)
= rA
s(Ax)
s(Ay)
If Rx denotes the direct response to selection on x,
CRy denotes the correlated response in y, with
CRy = bAy|Ax Rx

14. We can rewrite CRy = bAy|Ax Rx as follows
Recall that ix = Sx/sP (x) is the selection intensity
First, note that Rx = h2xSx = ixhx sA (x)
Since bAy|Ax = rA sA(x) / sA(y),
We have CRy = bAy|Ax Rx = rA sA (y) hxix
Substituting sA (y)= hy sP (y) gives our final result:
shows that hx hy rA in the corrected response plays the
same role as hx2 does in the direct response. As a result,
hx hy rA is often called the co-heritability
CRy = ix hx hy rA sP (y)
Noting that we can also express the direct response as
Rx = ixhx2 sp (x)

15. Estimating the Genetic Correlation from Selection Response
Suppose we have two experiments:
Direct selection on x, record Rx, CRy
Direct selection on y, record Ry, CRx
Simple algebra shows that
rA2 =
CRx CRy
Rx Ry
This is the realized genetic correlation, akin to the
realized heritability, h2 = R/S

16. Example: A double selection experiment in bristle number in fruit flies
In one line, direct selection on abdominal (AB) bristles. The direct RAB and
correlated CRST responses measured
In the other line, direct selection on sternopleural (ST)
Bristle, with the direct RST and correlated CRAB responses measured
Mean Bristle Number
Selection Line
abdominal
sternopleural
High AB
33.4
26.4
Low AB
2.4
12.8
High ST
22.2
45.0
Low ST
11.1
9.5
Direct responses in red, correlated in blue
RAB = = 31.0, RST = = 35.5
CRAB = = 13.6, C RST = = 11.1

17. Direct vs. Indirect Response
We can change the mean of x via a direct response Rx
or an indirect response CRx due to selection on y
Hence, indirect selection gives a large response when
• Character y has a greater heritability than x, and the genetic
correlation between x and y is high. This could occur if x is difficult to
measure with precison but x is not.
• The selection intensity is much greater for y than x. This would be true
if y were measurable in both sexes but x measurable in only one sex.

18. Matrices ( )
The identity matrix I ) (

19. ( ) ) ( ( ) The identity matrix serves the role of one in
matrix multiplication: AI =A, IA = A

20. ( ) ( ) The Inverse Matrix, A-1 For
For a square matrix A, define the Inverse of A, A-1, as
the matrix satisfying ( )
For ( )
If this quantity (the determinant)
is zero, the inverse does not exist.

21. The inverse serves the role of division in matrix multiplication
Suppose we are trying to solve the system Ax = c for x.
A-1 Ax = A-1 c. Note that A-1 Ax = Ix = x, giving x = A-1 c

22. Multivariate trait selection
Vector of
responses
Vector of selection differentials
P = phenotypic covariance matrix. Pij = Cov(Pi,Pj)
G = Genetic covariance matrix. Gij = Cov(Ai,Aj) ( ) ( )

23. The multidimensional breeders' equation
R = G P-1 S
R= h2S = (VA/VP) S
Natural parallels
with univariate
breeders equation
P-1 S = b is called the selection gradient
and measures the amount of direct selection
on a character
The gradient version of the breeders’ Equation is
R = G b

24. Sources of within-generation change in the mean
Since b = P-1 S, S = P b,
Change in mean from phenotypically
correlated characters under direct selection
Change in mean from direct selection on trait j
Within-generation change in trait j
Indirect response from genetically
correlated characters under direct selection
Response from direct selection on trait j
Between-generation change in trait j