1. Module 8: Estimating Genetic Variances Nested design GCA, SCA Diallel
PBG 650 Advanced Plant Breeding
Module 8: Estimating Genetic Variances
Nested design
GCA, SCA
Diallel

2. Also called Nested design Two types of families Females
North Carolina Design 1
Hierarchical design
Two types of families
Half sibs (male groups)
Full-sibs (females/males)
Females
Males 1 2 3 4 1 5 6 7 8 2 . m . f

3. Nested design – one location
Linear Model Yijk=  + Bi + Mj + Fk(j) + eijk
Source df MS
Expected Mean Square
Blocks
r-1
MSR
Males
m-1
MSM
Females/males
m(f-1)
MSF
Error
(r-1)(mf-1)
MSE
Might also have sets and multiple environments
See Bernardo, pg 164, for ANOVA with sets and environments

4. Variance components from the nested design
(if the parents are not inbred)
Design III – primary use is to estimate average level of dominance in F2 populations, and to determine possible bias in VA and VD due to linkage
(if the parents are not inbred)

5. Expected Mean Squares in SAS
Random statement generates expected mean squares
Test option obtains appropriate F tests for the model specified
In the example below, cultivars are fixed, all other effects are random
Proc GLM;
Class Loc Rep Cultivar;
Model Yield=Loc Rep(Loc) Cultivar Loc*Cultivar;
Random Loc Rep(Loc) Loc*Cultivar/Test;
Run;
controversial (could be dropped)
Source Type III Expected Mean Square
Loc Var(Error) + 3 Var(Loc*Cultivar) + 7 Var(Rep(Loc)) + 21 Var(Loc)
Rep(Loc) Var(Error) + 7 Var(Rep(Loc))
Cultivar Var(Error) + 3 Var(Loc*Cultivar) + Q(Cultivar)
Loc*Cultivar Var(Error) + 3 Var(Loc*Cultivar)
fixed effect
Proc Mixed may give better estimates of variance components

6. Combining Ability
General combining ability (GCA)– the average of all F1 crosses from a line (or genotype), expressed as a deviation from the population mean
The expected value of a cross is the sum of the combining ability of its two parents
Specific combining ability (SCA)– the deviation of a cross from its expected value
Where X is the performance of the cross

7. Estimation of combining ability
GCA
polycross method - allow all lines to intermate naturally
top crossing - a line is crossed to a random sample of plants from a reference population
GCA and SCA
Factorial design (NC Design II) – a group of ‘male’ parents is crossed to a group of ‘female’ parents
requires mxf crosses (e.g. 5x5=25)
can be applied to two heterotic populations
Diallel – all possible crosses among a set of parents
n(n-1)/2 possible crosses without parents or reciprocals
(e.g. 10x9/2=45)

8. Variations on the Diallel
Type of cross-classified design
With or without the parents
With or without reciprocal crosses
bulk seed from both parents if maternal effects are not important
Genotypes may be random or fixed
For random model, need many parents to adequately sample the population
Large number of crosses!
Can be divided into sets
Partial diallels can be conducted
If parents are inbred, can make paired row crosses to obtain more seed
Hallauer, Carena, and Miranda (2010) pg

9. Griffing’s Methods (Diallels)
all possible crosses, including selfs
Method 2
no reciprocals
Method 3
no parents
Method 4
no parents or reciprocals
most common, because parents often inbred and less vigorous
For each Method, genotypes may be
Model I = Fixed
Model II = Random

10. Diallel crossing A B C D ……. a+a a+b a+c a+d a+n a b+b b+c b+d b+n b
Parent A B C D
……. N
Mean
a+a
a+b
a+c
a+d
a+n a
b+b
b+c
b+d
b+n b
c+c
c+d
c+n c
d+d
d+n d
n+n n
…..
…..

11. Diallel analysis

12. Random model Usually does not include parents and reciprocals
Can be divided into sets
Source df MS
Expected Mean Square
Blocks
r-1
Crosses
[n(n-1)/2] -1
MS2
GCA
n-1
MS21
SCA
n(n-3)/2
MS22
Error
(r-1){[n(n-1)/2] -1}
MS1
Griffing (1956) is classic reference

13. Genetic variances from random model
General form for variance of
a variance component
k=coefficient of MS
fg=df of the mean square

14. Fixed model GCA effects SCA effects Lattice designs are useful
Advantage: first order effects (means)
are estimated with greater precision
than variances

15. Diallel analysis with parents
Gardner-Eberhart Analysis II
Source df
Blocks
r-1
Entries
[n(n+1)/2]-1
Parents
n-1
Parents vs crosses 1
Crosses
[n(n-1)/2]-1
GCA
SCA
n(n-3)/2
Error
(r-1){[n(n+1)/2] -1}
Source df
Blocks
r-1
Entries
[n(n+1)/2]-1
Varieties
n-1
Heterosis
n(n-1)/2
Average 1
Variety
n -1
Specific
n(n-3)/2
Error
(r-1){[n(n+1)/2] -1}
Gardner-Eberhart partitioning of Sums of Squares is non-orthogonal
Fit model sequentially

16. Factorial Mating Design
Diallel
Factorial (Design II)
Parents
Diallel
Factorial 4 6 15 9 10 45 25 20
190
100
4950
2500 n
n(n-1)/2
n2/4

17. General formula for covariance of relatives
A B C D X Y
r = 2XY
 = ACBD + ADBC
Extended to include epistasis:

18. Epistatic Variance
Often assumed to be absent, but could bias estimates of A2 and D2 upwards
Estimation requires more complex mating designs
Expected to be smaller than A2 and D2, so larger experiments are needed for adequate precision
Coefficients are correlated with those for A2 and D2, which leads to multicollinearity problems
For most crops, experimental estimates of epistatic variance have been small

19. Example of mating design to estimate epistatic variance
Design I experiment from ‘Jarvis’ and ‘Indian Chief’ maize populations
Obtained random inbred lines from each population, which were used as parents in a Design II experiment
A comparison of these values can be made to estimate epistatic variances
Eberhart et al., 1966

20. Precision of variance components
Minimum of progeny to adequately sample population (Bernardo’s advice, some would say more!)
Large numbers of progeny do not guarantee precise estimates of variance
Confidence intervals can be determined for estimates of variance (sets lower and upper bounds)
It’s possible in practice to obtain negative estimates of variance components, but they are theoretically impossible
large error variance
true estimate of genetic variance is close to zero
Report as zero? (may lead to bias when results are compiled across many experiments)
See Bernardo, pg 166, for further details on confidence intervals

21. Resampling methods are useful when
Confidence interval calculations assume that the underlying distribution is normal. Work best for balanced data.
Resampling methods are useful when
underlying distributions are unknown or are not normal
we don’t know how to estimate the confidence interval
Examples
Bootstrap – resampling with replacement
Jackknife – systematically delete data points
Permutation test – data scrambling
only works when there are two or more types of families