The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child...
Transcript of The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child...
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The statistical models of genomic prediction
John M Hickey, Chris Gaynor, Gregor Gorjanc
www.alphagenes.roslin.ed.ac.uk
@hickeyjohn
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Genomic selection
Goddard & Hayes Nat. Rev. Genet. 2009
“GS is the quantitative geneticists revenge on molecular genetics” - A. Archibald
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Relationship within and between training and prediction individuals
Relationships between TP and selection candidates leveraged for prediction
Selection candidates Training Pop.
Sel
ectio
n ca
ndid
ates
Tr
aini
ng P
op.
226240244248254542316323823619523522223223723923418480765511594103105444058167881461291947117513625113811551631413148373524223416576133180916833174166158114561161016118591704359118121124102211871509739651116711947751232021922092042062501541492039216019983182208179205201207454168425121910912815916122510812513711726107848102162242201981811341431878590521771275327195714162462021171301386610014515111048916413546491701511321213182412451891402471061475022721415224319036176781127238212253213301692916218819319623332156237760249956421197218242131622281481101781711572302524531446996737417298791732002818614219118322321582251153229998622121793122126210139120
120139210126122932172218699229153251822152231831911421862820017379981727473966914435425223015717117811014822862131242218197126495249607723156322331961931881622916930213253212387211278176361902431522142275014710624714018924524118131213215117049461351648910411151451006613813017212024616145719275312717752908518714313418119822022421610884107261171371251082251611591281092195142684145207201205179208182831991609220314915425020620420919220212375471196711165399715087211102124121118594370911856110111656114158166174331689180133671653422243537483114116315581113251361757119412914688167584044105103941155576801842342392372322222351952362386323154254248244240226
Haplotypes Genomic relationship matrix
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Useful things with matrices
• Counting how many animals passing a scales
• Summing the animals weight
x =111
!
"
###
$
%
&&&
x'x = 1 1 1!"
#$
111
!
"
%%%
#
$
&&&= 1×1( )+ 1×1( )+ 1×1( ) = 3
y =101520
!
"
###
$
%
&&&
x'y = 1 1 1!"
#$
101520
!
"
%%%
#
$
&&&= 1×10( )+ 1×15( )+ 1×20( ) = 45
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Useful things with matrices
• Averaging the animals weight
x'y = 1 1 1!"
#$
101520
!
"
%%%
#
$
&&&= 1×10( )+ 1×15( )+ 1×20( ) = 45
x'x = 1 1 1!"
#$
111
!
"
%%%
#
$
&&&= 1×1( )+ 1×1( )+ 1×1( ) = 3
x'yx'x
= x'x[ ]-1 x'y = b
453=13× 45=15
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Useful things with matrices
• Summing total weight in males and females
• Weight of average male and average female
X =110
001
!
"
####
$
%
&&&&
X'y =1 1 0
0 0 1
!
"
###
$
%
&&&
101520
!
"
###
$
%
&&&= 25
20
!
"#
$
%&y =
101520
!
"
###
$
%
&&&
X'yX'X
= X'X[ ]-1 X'y = b2520
!
"#
$
%&
2 00 1
!
"#
$
%&
=
12
0
0 11
!
"
####
$
%
&&&&
2520
!
"#
$
%&=
12.520
!
"#
$
%&
b11 =12×25
"
#$
%
&'+ 0×20( ) =12.5
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Shrinkage – Random Wand
• Ridge regression • BayesA • BayesB • BayesC • BayesLasso • BayesR • FnBayesB
• All differ in the shrinkage parameter – Some measure of our belief
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Lets put in a little bit of genetics
• Diploid genomes
– Markers are AA, Aa, aA, or aa
– Label a=0 and A=1
– Thus the dosage is: • AA=2 • Aa=1 • aA=1 • aa=0
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Mixed model equations
• Sample mean 0.75 • True intercept is 0.19 • True effect is 0.50
X'X X'ZZ'X Z'Z
!
"#
$
%&
-1X'yZ'y
!
"##
$
%&&= b
u
!
"##
$
%&&
X'X X'ZZ'X Z'Z+ Iλ
!
"#
$
%&
-1X'yZ'y
!
"##
$
%&&= b
u
!
"##
$
%&&
y =
0.100.701.300.65
1.250.120.681.20
!
"
###########
$
%
&&&&&&&&&&&
Z =
0121
2012
!
"
###########
$
%
&&&&&&&&&&&
X =
1111
1111
!
"
###########
$
%
&&&&&&&&&&&
LHS = 8 99 15
!
"#
$
%&
LHS = 8 99 15.85
!
"#
$
%&
b = 0.110.57
!
"#
$
%&
RHS = 69.53
!
"#
$
%&
RHS = 69.53
!
"#
$
%&
b = 0.200.49
!
"#
$
%& λ = 0.85
TBV =
0.00.51.00.5
1.00.00.51.0
!
"
###########
$
%
&&&&&&&&&&&
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A range of shrinkage values
• If Lambda =1000 the SNP solution =0.00 • And the solution for the intercept = 0.75
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0 0.2 0.4 0.6 0.8 1 1.2
BetaHat
Lambda
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Shrinkage versus more data
• Two data sets • One with 8 animals, the other with 80 animals • Compare effect of Lambda in both
X'X X'ZZ'X Z'Z+ Iλ
!
"#
$
%&
-1X'yZ'y
!
"##
$
%&&= b
u
!
"##
$
%&&
LHS = 8 99 15
!
"#
$
%& RHS = 6
9.53
!
"#
$
%& b = 0.11
0.57
!
"#
$
%& LHS = 80 85
85 263
!
"#
$
%& RHS = 57.4
147.94
!
"#
$
%& b = 0.18
0.50
!
"#
$
%&
No Lambda
Lambda = 5.0 (extremely high value)
LHS = 8 99 20
!
"#
$
%& b = 0.43
0.28
!
"#
$
%& LHS = 80 85
85 268
!
"#
$
%& b = 0.18
0.50
!
"#
$
%&RHS = 57.4
147.94
!
"#
$
%&RHS = 6
9.53
!
"#
$
%&
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Mendelian sampling
Sire
Child 1 Child 2 Child 3 Child 4
Child 5 Child 6
In theory you can have sibs that are genetically unrelated
This is why I am different from my brother
0
20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Additive genetic relation
HSFSm=4A =
1.00 0.50 0.50 0.50 0.500.50 1.00 0.25 0.25 0.250.50 0.25 1.00 0.25 0.250.50 0.25 0.25 1.00 0.250.50 0.25 0.25 0.25 1.00
!
"
######
$
%
&&&&&&
G =
1.00 0.50 0.50 0.50 0.500.50 1.00 0.20 0.30 0.200.50 0.20 1.00 0.20 0.300.50 0.30 0.20 1.00 0.200.50 0.20 0.30 0.20 1.00
!
"
######
$
%
&&&&&&
And “hidden” relationships
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Population version
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10 animal example
• 10 animal example – 2 unrelated sire families (FamA and FamB) – Dam’s are unrelated
• In each family – 2 half sibs used in prediction set – Sire and 2 half sibs used in training set – 5 individuals from other family used in training set
• Purpose – Show prediction due to parent average versus MS – Pedigree versus genomics – Close versus distant relatives – Show shrinkage
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Pedigree
ID Sire Dam
1 0 02 1 03 1 04 1 05 1 06 0 07 6 08 6 09 6 0
10 6 0
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Genetic relationships
• Captured in a matrix A – traditionally built using pedigree – Relationship between each pair of individuals
• Range from 0 to 2
– Inbred individuals have a relationship with themselves of 2
– Pair of completely unrelated individuals have a coefficient of relationship of 0
– Full sib have a relationship of 0.5 • If parents are not related
– Half sibs have a relationship of 0.25 • If parents are not related
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“some animals are more equal than others”…….. even if the additive genetic relationship is the same
0
20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Additive genetic relation
HSFSm=4
Genomic Relationships
e.g. actual relationship between HS can vary between 0.2 and 0.3
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Lets do 5 animals first ID Sire Dam 1 0 02 1 03 1 04 1 05 1 0
A =
1.00 0.50 0.50 0.50 0.500.50 1.00 0.25 0.25 0.250.50 0.25 1.00 0.25 0.250.50 0.25 0.25 1.00 0.250.50 0.25 0.25 0.25 1.00
!
"
######
$
%
&&&&&&
G =
1.00 0.50 0.50 0.50 0.500.50 1.00 0.20 0.30 0.200.50 0.20 1.00 0.20 0.300.50 0.30 0.20 1.00 0.200.50 0.20 0.30 0.20 1.00
!
"
######
$
%
&&&&&&
Pedigree tells: Which family you belong to
Genomics tells: Which family you belong to Which sib you are more closely related to And shows “hidden” relationships (We will see the last bit with 10 animals) Linkage
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10 animals for comparison
A =
1.00 0.50 0.50 0.50 0.50 0.00 0.00 0.00 0.00 0.000.50 1.00 0.25 0.25 0.25 0.00 0.00 0.00 0.00 0.000.50 0.25 1.00 0.25 0.25 0.00 0.00 0.00 0.00 0.000.50 0.25 0.25 1.00 0.25 0.00 0.00 0.00 0.00 0.000.50 0.25 0.25 0.25 1.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 1.00 0.50 0.50 0.50 0.500.00 0.00 0.00 0.00 0.00 0.50 1.00 0.25 0.25 0.250.00 0.00 0.00 0.00 0.00 0.50 0.25 1.00 0.25 0.250.00 0.00 0.00 0.00 0.00 0.50 0.25 0.25 1.00 0.250.00 0.00 0.00 0.00 0.00 0.50 0.25 0.25 0.25 1.00
!
"
##############
$
%
&&&&&&&&&&&&&&
ID Sire Dam 1 0 02 1 03 1 04 1 05 1 06 0 07 6 08 6 09 6 0
10 6 0
G =
1.00 0.50 0.50 0.50 0.50 0.02 0.02 0.02 0.02 0.020.50 1.00 0.20 0.30 0.20 0.02 0.01 0.03 0.01 0.030.50 0.20 1.00 0.20 0.30 0.02 0.03 0.01 0.03 0.010.50 0.30 0.20 1.00 0.20 0.02 0.01 0.03 0.01 0.030.50 0.20 0.30 0.20 1.00 0.02 0.03 0.01 0.03 0.010.02 0.02 0.02 0.02 0.02 1.00 0.50 0.50 0.50 0.500.02 0.01 0.03 0.01 0.03 0.50 1.00 0.20 0.30 0.200.02 0.03 0.01 0.03 0.01 0.50 0.20 1.00 0.20 0.300.02 0.01 0.03 0.01 0.03 0.50 0.30 0.20 1.00 0.200.02 0.03 0.01 0.03 0.01 0.50 0.20 0.30 0.20 1.00
!
"
##############
$
%
&&&&&&&&&&&&&&
Family relationships Family relationships Segregation within family Missing pedigree = “Unrelated” Linkage Linkage disequilibrium
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5 animal example
X'X X'ZZ'X Z'Z+G-1λ
!
"##
$
%&&
-1X'yZ'y
!
"##
$
%&&= b
u
!
"##
$
%&&
X'X X'ZZ'X Z'Z+A-1λ
!
"##
$
%&&
-1X'yZ'y
!
"##
$
%&&= b
u
!
"##
$
%&&
A =
1.00 0.50 0.50 0.50 0.500.50 1.00 0.25 0.25 0.250.50 0.25 1.00 0.25 0.250.50 0.25 0.25 1.00 0.250.50 0.25 0.25 0.25 1.00
!
"
######
$
%
&&&&&&
G =
1.00 0.50 0.50 0.50 0.500.50 1.00 0.20 0.30 0.200.50 0.20 1.00 0.20 0.300.50 0.30 0.20 1.00 0.200.50 0.20 0.30 0.20 1.00
!
"
######
$
%
&&&&&&
LHS−1 =
1.55 -1.27 -1.18 -1.18 -0.64 -0.64-1.27 1.64 1.09 1.09 0.82 0.82-1.18 1.09 1.53 0.93 0.55 0.55-1.18 1.09 0.93 1.53 0.55 0.55-0.64 0.82 0.55 0.55 1.91 0.41-0.64 0.82 0.55 0.55 0.41 1.91
"
#
$$$$$$$
%
&
'''''''
LHS−1 =
1.52 -1.26 -1.15 -1.15 -0.63 -0.63-1.26 1.63 1.07 1.07 0.81 0.81-1.15 1.07 1.49 0.88 0.58 0.50-1.15 1.07 0.88 1.49 0.50 0.58-0.63 0.81 0.58 0.50 1.90 0.32-0.63 0.81 0.50 0.58 0.32 1.90
"
#
$$$$$$$
%
&
'''''''
Solutions =
0.000.00-1.201.200.000.00
!
"
#######
$
%
&&&&&&&
Solutions =
0.000.00-1.231.23-0.150.15
!
"
#######
$
%
&&&&&&&
TrueValues =
0.000.00−2.002.00−2.002.00
"
#
$$$$$$$$
%
&
''''''''
y =
0.00-2.002.00MissingMissing
!
"
######
$
%
&&&&&&
RHS =
0.000.00-2.002.00##
!
"
#######
$
%
&&&&&&&
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5 animal example
• When we do BLUP we get an estimated breeding value (EBV)
• An EBV is simply a weighted average of all the phenotypic data available – With simultaneous correction for all other effects
• The weightings are determined by the inverse of the LHS – This is primarily driven by the relationship matrix
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5 animal example
X'X X'ZZ'X Z'Z+G-1λ
!
"##
$
%&&
-1X'yZ'y
!
"##
$
%&&= b
u
!
"##
$
%&&
X'X X'ZZ'X Z'Z+A-1λ
!
"##
$
%&&
-1X'yZ'y
!
"##
$
%&&= b
u
!
"##
$
%&&
A =
1.00 0.50 0.50 0.50 0.500.50 1.00 0.25 0.25 0.250.50 0.25 1.00 0.25 0.250.50 0.25 0.25 1.00 0.250.50 0.25 0.25 0.25 1.00
!
"
######
$
%
&&&&&&
G =
1.00 0.50 0.50 0.50 0.500.50 1.00 0.20 0.30 0.200.50 0.20 1.00 0.20 0.300.50 0.30 0.20 1.00 0.200.50 0.20 0.30 0.20 1.00
!
"
######
$
%
&&&&&&
LHS−1 =
1.55 -1.27 -1.18 -1.18 -0.64 -0.64-1.27 1.64 1.09 1.09 0.82 0.82-1.18 1.09 1.53 0.93 0.55 0.55-1.18 1.09 0.93 1.53 0.55 0.55-0.64 0.82 0.55 0.55 1.91 0.41-0.64 0.82 0.55 0.55 0.41 1.91
"
#
$$$$$$$
%
&
'''''''
LHS−1 =
1.52 -1.26 -1.15 -1.15 -0.63 -0.63-1.26 1.63 1.07 1.07 0.81 0.81-1.15 1.07 1.49 0.88 0.58 0.50-1.15 1.07 0.88 1.49 0.50 0.58-0.63 0.81 0.58 0.50 1.90 0.32-0.63 0.81 0.50 0.58 0.32 1.90
"
#
$$$$$$$
%
&
'''''''
RHS =
0.000.00-2.002.00##
!
"
#######
$
%
&&&&&&&
Solutions =
0.000.00-1.201.200.000.00
!
"
#######
$
%
&&&&&&&
Solutions =
0.000.00-1.231.23-0.150.15
!
"
#######
$
%
&&&&&&&
TrueValues =
0.000.00−2.002.00−2.002.00
"
#
$$$$$$$$
%
&
''''''''
y =
0.00-2.002.00MissingMissing
!
"
######
$
%
&&&&&&
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5 animal example
LHS−1 =
1.55 -1.27 -1.18 -1.18 -0.64 -0.64-1.27 1.64 1.09 1.09 0.82 0.82-1.18 1.09 1.53 0.93 0.55 0.55-1.18 1.09 0.93 1.53 0.55 0.55-0.64 0.82 0.55 0.55 1.91 0.41-0.64 0.82 0.55 0.55 0.41 1.91
"
#
$$$$$$$
%
&
'''''''
LHS−1 =
1.52 -1.26 -1.15 -1.15 -0.63 -0.63-1.26 1.63 1.07 1.07 0.81 0.81-1.15 1.07 1.49 0.88 0.58 0.50-1.15 1.07 0.88 1.49 0.50 0.58-0.63 0.81 0.58 0.50 1.90 0.32-0.63 0.81 0.50 0.58 0.32 1.90
"
#
$$$$$$$
%
&
'''''''
RHS =
0.000.00-2.002.00##
!
"
#######
$
%
&&&&&&&
Solutions =
0.000.00-1.201.200.000.00
!
"
#######
$
%
&&&&&&&
Solutions =
0.000.00-1.231.23-0.150.15
!
"
#######
$
%
&&&&&&&
TrueValues =
0.000.00−2.002.00−2.002.00
"
#
$$$$$$$$
%
&
''''''''
y =
0.00-2.002.00MissingMissing
!
"
######
$
%
&&&&&&
uSon4 = LHSSon4,Mean−1 ×RHSMean( )+ LHSSon4,Sire
−1 ×RHSSire( )+ LHSSon4,Son2−1 ×RHSSon2( )+ LHSSon4,Son3
−1 ×RHSSon3( )
uSon4 = −0.63×0.00( )+ 0.81×0.00( )+ 0.58×−2.00( )+ 0.50×2.00( ) = −0.15→Genomic
uSon4 = −0.64×0.00( )+ 0.82×0.00( )+ 0.55×−2.00( )+ 0.55×2.00( ) = 0.00→ Pedigree
A =
1.00 0.50 0.50 0.50 0.500.50 1.00 0.25 0.25 0.250.50 0.25 1.00 0.25 0.250.50 0.25 0.25 1.00 0.250.50 0.25 0.25 0.25 1.00
!
"
######
$
%
&&&&&&
G =
1.00 0.50 0.50 0.50 0.500.50 1.00 0.20 0.30 0.200.50 0.20 1.00 0.20 0.300.50 0.30 0.20 1.00 0.200.50 0.20 0.30 0.20 1.00
!
"
######
$
%
&&&&&&
![Page 25: The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 In theory you can have sibs that are genetically unrelated This](https://reader034.fdocuments.in/reader034/viewer/2022042109/5e89d9ffbd840b00245090b3/html5/thumbnails/25.jpg)
10 animals
A =
1.00 0.50 0.50 0.50 0.50 0.00 0.00 0.00 0.00 0.000.50 1.00 0.25 0.25 0.25 0.00 0.00 0.00 0.00 0.000.50 0.25 1.00 0.25 0.25 0.00 0.00 0.00 0.00 0.000.50 0.25 0.25 1.00 0.25 0.00 0.00 0.00 0.00 0.000.50 0.25 0.25 0.25 1.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 1.00 0.50 0.50 0.50 0.500.00 0.00 0.00 0.00 0.00 0.50 1.00 0.25 0.25 0.250.00 0.00 0.00 0.00 0.00 0.50 0.25 1.00 0.25 0.250.00 0.00 0.00 0.00 0.00 0.50 0.25 0.25 1.00 0.250.00 0.00 0.00 0.00 0.00 0.50 0.25 0.25 0.25 1.00
!
"
##############
$
%
&&&&&&&&&&&&&&
ID Sire Dam 1 0 02 1 03 1 04 1 05 1 06 0 07 6 08 6 09 6 0
10 6 0
G =
1.00 0.50 0.50 0.50 0.50 0.02 0.02 0.02 0.02 0.020.50 1.00 0.20 0.30 0.20 0.02 0.01 0.03 0.01 0.030.50 0.20 1.00 0.20 0.30 0.02 0.03 0.01 0.03 0.010.50 0.30 0.20 1.00 0.20 0.02 0.01 0.03 0.01 0.030.50 0.20 0.30 0.20 1.00 0.02 0.03 0.01 0.03 0.010.02 0.02 0.02 0.02 0.02 1.00 0.50 0.50 0.50 0.500.02 0.01 0.03 0.01 0.03 0.50 1.00 0.20 0.30 0.200.02 0.03 0.01 0.03 0.01 0.50 0.20 1.00 0.20 0.300.02 0.01 0.03 0.01 0.03 0.50 0.30 0.20 1.00 0.200.02 0.03 0.01 0.03 0.01 0.50 0.20 0.30 0.20 1.00
!
"
##############
$
%
&&&&&&&&&&&&&&
Family relationships Family relationships Segregation within family Missing pedigree = “Unrelated” Linkage Linkage disequilibrium
Z'Z+A-1λ!" #$-1Z'y[ ] = u[ ] Z'Z+G-1λ!" #$
-1Z'y[ ] = u[ ]
![Page 26: The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 In theory you can have sibs that are genetically unrelated This](https://reader034.fdocuments.in/reader034/viewer/2022042109/5e89d9ffbd840b00245090b3/html5/thumbnails/26.jpg)
10 animal example
LHS−1 =
0.59 0.12 0.12 0.00 0.00 0.00 0.29 0.29 0.00 0.000.12 0.62 0.02 0.00 0.00 0.00 0.06 0.06 0.00 0.000.12 0.02 0.62 0.00 0.00 0.00 0.06 0.06 0.00 0.000.00 0.00 0.00 0.59 0.12 0.12 0.00 0.00 0.29 0.290.00 0.00 0.00 0.12 0.62 0.02 0.00 0.00 0.06 0.060.00 0.00 0.00 0.12 0.02 0.62 0.00 0.00 0.06 0.060.29 0.06 0.06 0.00 0.00 0.00 1.65 0.15 0.00 0.000.29 0.06 0.06 0.00 0.00 0.00 0.15 1.65 0.00 0.000.00 0.00 0.00 0.29 0.06 0.06 0.00 0.00 1.65 0.150.00 0.00 0.00 0.29 0.06 0.06 0.00 0.00 0.15 1.65
"
#
$$$$$$$$$$$$$$
%
&
''''''''''''''
RHS =
0-22201822####
!
"
##############
$
%
&&&&&&&&&&&&&&
A G1 2 3 6 7 8 4 5 9 10 1 2 3 6 7 8 4 5 9 10
1 1.00 0.50 0.50 0.00 0.00 0.00 0.50 0.50 0.00 0.00 1 1.00 0.50 0.50 0.02 0.02 0.02 0.50 0.50 0.02 0.022 0.50 1.00 0.25 0.00 0.00 0.00 0.25 0.25 0.00 0.00 2 0.50 1.00 0.20 0.02 0.01 0.03 0.30 0.20 0.01 0.033 0.50 0.25 1.00 0.00 0.00 0.00 0.25 0.25 0.00 0.00 3 0.50 0.20 1.00 0.02 0.03 0.01 0.20 0.30 0.03 0.016 0.00 0.00 0.00 1.00 0.50 0.50 0.00 0.00 0.50 0.50 6 0.02 0.02 0.02 1.00 0.50 0.50 0.02 0.02 0.50 0.507 0.00 0.00 0.00 0.50 1.00 0.25 0.00 0.00 0.25 0.25 7 0.02 0.01 0.03 0.50 1.00 0.20 0.01 0.03 0.30 0.208 0.00 0.00 0.00 0.50 0.25 1.00 0.00 0.00 0.25 0.25 8 0.02 0.03 0.01 0.50 0.20 1.00 0.03 0.01 0.20 0.304 0.50 0.25 0.25 0.00 0.00 0.00 1.00 0.25 0.00 0.00 4 0.50 0.30 0.20 0.02 0.01 0.03 1.00 0.20 0.02 0.025 0.50 0.25 0.25 0.00 0.00 0.00 0.25 1.00 0.00 0.00 5 0.50 0.20 0.30 0.02 0.03 0.01 0.20 1.00 0.01 0.039 0.00 0.00 0.00 0.50 0.25 0.25 0.00 0.00 1.00 0.25 9 0.02 0.01 0.03 0.50 0.30 0.20 0.02 0.01 1.00 0.2010 0.00 0.00 0.00 0.50 0.25 0.25 0.00 0.00 0.25 1.00 10 0.02 0.03 0.01 0.50 0.20 0.30 0.02 0.03 0.20 1.00
Solutions =
0.00-1.201.2016.4714.0916.490.000.008.248.24
!
"
##############
$
%
&&&&&&&&&&&&&&
Solutions =
0.09-1.091.3516.5813.9016.340.250.378.168.41
!
"
##############
$
%
&&&&&&&&&&&&&&
TrueBreedingValues =
0-22201822-221822
!
"
##############
$
%
&&&&&&&&&&&&&&
0.5853 0.1219 0.1219 0.0012 0.0017 0.0017 0.2926 0.2926 0.0040 0.00400.1219 0.6247 0.0094 0.0017 -0.0006 0.0053 0.0992 0.0225 -0.0014 0.01280.1219 0.0094 0.6247 0.0017 0.0053 -0.0006 0.0225 0.0992 0.0128 -0.00140.0012 0.0017 0.0017 0.5853 0.1219 0.1219 0.0040 0.0040 0.2926 0.29260.0017 -0.0006 0.0053 0.1219 0.6247 0.0094 -0.0014 0.0128 0.0992 0.02250.0017 0.0053 -0.0006 0.1219 0.0094 0.6247 0.0128 -0.0014 0.0225 0.09920.2926 0.0992 0.0225 0.0040 -0.0014 0.0128 1.6380 0.0539 0.0167 0.01080.2926 0.0225 0.0992 0.0040 0.0128 -0.0014 0.0539 1.6380 -0.0092 0.03670.0040 -0.0014 0.0128 0.2926 0.0992 0.0225 0.0167 -0.0092 1.6380 0.05390.0040 0.0128 -0.0014 0.2926 0.0225 0.0992 0.0108 0.0367 0.0539 1.6380
LHS−1 =
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10 animal example
uSonA4 = LHSSonA4,SireA−1 ×RHSSireA( )+ LHSSonA4,SonA2
−1 ×RHSSonA2( )+ LHSSonA4,SonA3−1 ×RHSSonA3( )
uSonA4 = LHSSonA4,SireA−1 ×RHSSireA( )+ LHSSonA4,SonA2
−1 ×RHSSonA2( )+ LHSSonA4,SonA3−1 ×RHSSonA3( )+ LHSSonA4,SireB
−1 ×RHSSireB( )+ LHSSonA4,SonB6−1 ×RHSSonB6( )+ LHSSonA4,SonB7
−1 ×RHSSonB7( )
uSon4 = 0.29×0.00( )+ 0.06×−2.00( )+ 0.06×2.00( ) = 0.00→ Pedigree
uSon4 = 0.2926×0.00( )+ 0.0992×−2.00( )+ 0.0225×2.00( )+ 0.0040×20.00( )+ −0.0014×18.00( )+ 0.00128×22.00( ) = 0.18→Genomic
Phenotype is missing for other non-zero coefficient
![Page 28: The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 In theory you can have sibs that are genetically unrelated This](https://reader034.fdocuments.in/reader034/viewer/2022042109/5e89d9ffbd840b00245090b3/html5/thumbnails/28.jpg)
Matrix Inversion
• http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-3-multiplication-and-inverse-matrices/
• Minute 37 of this video from Gilbert Strang
• Gauss-Jordan Elimination
![Page 29: The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 In theory you can have sibs that are genetically unrelated This](https://reader034.fdocuments.in/reader034/viewer/2022042109/5e89d9ffbd840b00245090b3/html5/thumbnails/29.jpg)
Inversion by Gauss-Jordan
A = 1 32 7
!
"#
$
%&
1 32 7
1 00 1
!
"
##
$
%
&&
A−1 = 7 −3−2 1
"
#$
%
&' I = 1 0
0 1
!
"#
$
%&
A-1A = I = AA-1
1 30 1
1 0−2 1
"
#
$$
%
&
''
1 00 1
7 −3−2 1
"
#
$$
%
&
''
Move 1 Subtract 2 of Row 1 from Row 2
Move 2 Subtract 3 of Row 2 from Row 1
![Page 30: The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 In theory you can have sibs that are genetically unrelated This](https://reader034.fdocuments.in/reader034/viewer/2022042109/5e89d9ffbd840b00245090b3/html5/thumbnails/30.jpg)
Useful things with matrices
• Averaging the animals weight
x'y = 1 1 1!"
#$
101520
!
"
%%%
#
$
&&&= 1×10( )+ 1×15( )+ 1×20( ) = 45
x'x = 1 1 1!"
#$
111
!
"
%%%
#
$
&&&= 1×1( )+ 1×1( )+ 1×1( ) = 3
x'yx'x
= x'x[ ]-1 x'y = b
453=13× 45=15
![Page 31: The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 In theory you can have sibs that are genetically unrelated This](https://reader034.fdocuments.in/reader034/viewer/2022042109/5e89d9ffbd840b00245090b3/html5/thumbnails/31.jpg)
Gauss Seidel Residual Update
• Easy efficient way to solve and understand genomic prediction equations
• Form – X’X (diagonal) – Form X’y – Initialize values for beta’s – Assume current values of beta-i’s are correct – Form new y vector (called e) based on the residuals – Estimate new solution for betai (Xi’e divided by Xi’Xi) – Repeat until convergence
• Simple extension to Bayesian model
![Page 32: The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 In theory you can have sibs that are genetically unrelated This](https://reader034.fdocuments.in/reader034/viewer/2022042109/5e89d9ffbd840b00245090b3/html5/thumbnails/32.jpg)
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Legarra and Misztal (JDS 2008)
![Page 34: The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 In theory you can have sibs that are genetically unrelated This](https://reader034.fdocuments.in/reader034/viewer/2022042109/5e89d9ffbd840b00245090b3/html5/thumbnails/34.jpg)
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Excel Sheet
• Lots of little examples with Excel
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GBLUP versus other methods
• Genomic BLUP is the simplest method to do genomic evaluations
• Algebraically identical to ridge regression
• Ridge regression treats each marker as a random effect
• Ridge regression has the same shrinkage parameter for each marker
• Other methods allow heterogeneous shrinkage parameters
![Page 37: The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 In theory you can have sibs that are genetically unrelated This](https://reader034.fdocuments.in/reader034/viewer/2022042109/5e89d9ffbd840b00245090b3/html5/thumbnails/37.jpg)
Brief description of SNP models
• Genomic selection prediction models treat markers as random effects
• MARS treats markers as fixed effects
• Random effects have two benefits – Trick to allow all markers be fitted simultaneously – Shrinkage
• Fixed effect models overestimate marker effects • Random effects models correct for this overestimation by
shrinking marker effects back towards the mean of all marker effects
• Shrinkage is proportional to the uncertainty in the marker effect (and a statistical prior)
– More uncertainty = more shrinkage towards the mean – More information to estimate effect = less shrinkage
![Page 38: The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 In theory you can have sibs that are genetically unrelated This](https://reader034.fdocuments.in/reader034/viewer/2022042109/5e89d9ffbd840b00245090b3/html5/thumbnails/38.jpg)
• Ridge regression – All SNP’s in the model – All have equal shrinkage parameter – Shrinkage parameter is estimated or set apriori
• BayesA – All SNP’s in the model – Each SNP has unique shrinkage parameter – Each shrinkage parameter estimated – Shape and scale parameters are fixed – Problem is it cannot shrink to zero
Brief description of SNP models
![Page 39: The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 In theory you can have sibs that are genetically unrelated This](https://reader034.fdocuments.in/reader034/viewer/2022042109/5e89d9ffbd840b00245090b3/html5/thumbnails/39.jpg)
• BayesLasso – Similar to BayesA – Each SNP has unique shrinkage parameter – Each shrinkage parameter estimated – Shape and scale parameters are estimated – Use inverse Gaussian distribution instead of inverse chi square
which allows greater shrinkage towards zero
Brief description of SNP models
![Page 40: The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 In theory you can have sibs that are genetically unrelated This](https://reader034.fdocuments.in/reader034/viewer/2022042109/5e89d9ffbd840b00245090b3/html5/thumbnails/40.jpg)
• BayesB – Similar to BayesA except that proportion of 1- π SNPs
are in the model – Thus can shrink SNPs to zero – Mixture model
• BayesCpi – Has similarities to SnpBlup and BayesB – Estimates π – All SNP have equal shrinkage parameter – Estimates shrinkage parameter – Mixture model
Brief description of SNP models
![Page 41: The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 In theory you can have sibs that are genetically unrelated This](https://reader034.fdocuments.in/reader034/viewer/2022042109/5e89d9ffbd840b00245090b3/html5/thumbnails/41.jpg)
Summary of SNP models
• Ridge regression SNP have EQUAL shrinkage parameter
• BayesA/BayesLasso achieve shrinkage • Unequal shrinkage parameter for all SNPs
• BayesB achieves shrinkage • Only including the proportion of SNPs in each round • These SNPs have UNEQUAL shrinkage parameter
• BayesCpi achieves shrinkage – Only including the proportion of SNPs in each round – These SNPs have EQUAL shrinkage parameter
• Models have an equivalence to genomic relationship matrix G = MHM’
![Page 42: The statistical models of genomic prediction...Mendelian sampling Sire Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 In theory you can have sibs that are genetically unrelated This](https://reader034.fdocuments.in/reader034/viewer/2022042109/5e89d9ffbd840b00245090b3/html5/thumbnails/42.jpg)
Non-linear models
• Reproducing kernel Hilbert space
• Neural networks
• Basically these bend the relationships in the genomic relationship matrix
• Close relatives get more weight
• Distant relatives less weight
• Capture epistatic interactions that may be shared by close relatives but not by distant relatives
• Perhaps useful for advanced yield trials
• I favour simpler models – Prevents a pointless debate about which model – Genetic improvement is an additive thing