Analysis of complex or multifactorial traits
Analysis of complex or multifactorial traits
All genes affect all traits - the question is by how much?
Analysis of complex or multifactorial traits
All genes affect all traits - the question is by how much?
Infinitesimal model
Analysis of complex or multifactorial traits
All genes affect all traits - the question is by how much?
Infinitesimal model
Quasi-infinitesimal model, Oligogenic model
Population genetics
Population genetics
Quantitative genetics
Population genetics
Quantitative genetics
Phenotypes first in quantitative genetics
In the era of genomics, phenotype is king
Galton (1886). Regression towards mediocrity in hereditary stature "
Quantitative genetic model:
\begin{align*} \mathbf{y} &= \mathbf{g} + \boldsymbol{\epsilon} \\ \end{align*} where \mathbf{y} is the vector of observed phenotypes, \mathbf{g} is the vector of genetic values, and \boldsymbol{\epsilon} is the vector of residuals.
Example:
Animal ID | y | g | e |
---|---|---|---|
1 | 10 | ? | ? |
2 | 7 | ? | ? |
3 | 12 | ? | ? |
Quantitative genetic model:
\begin{align*} \mathbf{y} &= \mathbf{g} + \boldsymbol{\epsilon} \\ \end{align*} where \mathbf{y} is the vector of observed phenotypes, \mathbf{g} is the vector of genetic values, and \boldsymbol{\epsilon} is the vector of residuals.
Example:
Animal ID | y | g | e |
---|---|---|---|
1 | 10 | 5 | 5 |
2 | 7 | 6 | 1 |
3 | 12 | 2 | 10 |
Quantitative genetic model:
\begin{align*} \mathbf{y} &= \mathbf{g} + \boldsymbol{\epsilon} \\ \end{align*} where \mathbf{y} is the vector of observed phenotypes, \mathbf{g} is the vector of genetic values, and \boldsymbol{\epsilon} is the vector of residuals.
Example:
Animal ID | y | g | e |
---|---|---|---|
1 | 10 | 5 | 5 |
2 | 7 | 6 | 1 |
3 | 12 | 2 | 10 |
Phenotypes can be observed and measured but genotypic and additive genetic effects cannot
linkage disequilibrium between QTL and SNP
relationship among individuals
Repeat of numbers 0, 1, and 2
Connecting phenotypic data with genomic information
where n is the number of individuals (e.g., accessions) and m is the number of SNPs.
automated process
less labor intensive
less prone to measurement error
Image data
Connecting image data with genomic information
where n is the number of individuals (e.g., accessions) and m is the number of SNPs.
Big data are being generated in almost every field
too large to permit visual inspection
big data \ne clean data
missing observations
empty cells
confounding
outliers
n: number of individuals (records)
m number of SNPs (genetic markers)
Prediction of additive genetic effects
Prediction of total genetic effects parametrically
Prediction of total genetic effects non-parametrically
Complex traits are controlled by large number of genes with small effects, and influenced by both genetics and environments
Inference (location)
Inference (variability)
Combination of above two (e.g., estimate proportion of additive genetic variance explained by QTLs)
\mathbf{y = Xb + Zu + e}
\mathbf{X}: incidence matrix of systematic effects
\mathbf{Z}: incidence matrix of random effects
\mathbf{K}: genomic relationship matrix
\mathbf{R}: residual relationship matrix
\sigma^2_{u}: genomic variance; \sigma^2_{e}: residual variance
BLUE: \hat{\mathbf{b}} = (\mathbf{X'V^{-1}X})^{-}\mathbf{X'V^{-1}y}; BLUP: \hat{\mathbf{u}} = \mathbf{KZ'}\mathbf{V^{-1}}(\mathbf{y - X\hat{b}})
where \mathbf{V} = \mathbf{ZK\sigma^2_{u}Z' + R\sigma^2_{e}}
The corresponding mixed model equations (MME) are
\mathbf{G}^* = \sigma^2_u \mathbf{K}
\mathbf{R}^* = \sigma^2_e \mathbf{R}
If we multipy \mathbf{R}^* = \sigma^2_e\mathbf{I} to the both sides
where \lambda = \sigma^2_e / \sigma^2_u
N | Phe | Env | Gen |
---|---|---|---|
1 | 47 | E1 | G1 |
2 | 51 | E1 | G2 |
3 | 46 | E1 | G3 |
4 | 58 | E1 | G4 |
5 | 52 | E2 | G1 |
6 | 46 | E2 | G2 |
7 | 52 | E2 | G3 |
8 | 54 | E2 | G4 |
9 | 53 | E3 | G1 |
10 | 48 | E3 | G2 |
11 | 58 | E3 | G3 |
12 | 52 | E3 | G4 |
N | EnvE1 | EnvE2 | EnvE3 |
---|---|---|---|
1 | 1 | 0 | 0 |
2 | 1 | 0 | 0 |
3 | 1 | 0 | 0 |
4 | 1 | 0 | 0 |
5 | 0 | 1 | 0 |
6 | 0 | 1 | 0 |
7 | 0 | 1 | 0 |
8 | 0 | 1 | 0 |
9 | 0 | 0 | 1 |
10 | 0 | 0 | 1 |
11 | 0 | 0 | 1 |
12 | 0 | 0 | 1 |
N | GenG1 | GenG2 | GenG3 | GenG4 |
---|---|---|---|---|
1 | 1 | 0 | 0 | 0 |
2 | 0 | 1 | 0 | 0 |
3 | 0 | 0 | 1 | 0 |
4 | 0 | 0 | 0 | 1 |
5 | 1 | 0 | 0 | 0 |
6 | 0 | 1 | 0 | 0 |
7 | 0 | 0 | 1 | 0 |
8 | 0 | 0 | 0 | 1 |
9 | 1 | 0 | 0 | 0 |
10 | 0 | 1 | 0 | 0 |
11 | 0 | 0 | 1 | 0 |
12 | 0 | 0 | 0 | 1 |
\mathbf{X'} is the 3 x 12 matrix
\mathbf{X} is the 12 x 3 matrix
\mathbf{X'X} is the 3 x 3 matrix
EnvE1 | EnvE2 | EnvE3 | |
---|---|---|---|
EnvE1 | 4 | 0 | 0 |
EnvE2 | 0 | 4 | 0 |
EnvE3 | 0 | 0 | 4 |
\mathbf{X'} is the 3 x 12 matrix
\mathbf{X} is the 12 x 3 matrix
\mathbf{X'X} is the 3 x 3 matrix
EnvE1 | EnvE2 | EnvE3 | |
---|---|---|---|
EnvE1 | 4 | 0 | 0 |
EnvE2 | 0 | 4 | 0 |
EnvE3 | 0 | 0 | 4 |
\mathbf{X'} is the 3 x 12 matrix
\mathbf{Z} is the 12 x 4 matrix
\mathbf{X'Z} is the 3 x 4 matrix
GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|
EnvE1 | 1 | 1 | 1 | 1 |
EnvE2 | 1 | 1 | 1 | 1 |
EnvE3 | 1 | 1 | 1 | 1 |
\mathbf{X'} is the 3 x 12 matrix
\mathbf{Z} is the 12 x 4 matrix
\mathbf{X'Z} is the 3 x 4 matrix
GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|
EnvE1 | 1 | 1 | 1 | 1 |
EnvE2 | 1 | 1 | 1 | 1 |
EnvE3 | 1 | 1 | 1 | 1 |
\mathbf{Z'} is the 4 x 12 matrix
\mathbf{X} is the 12 x 3 matrix
\mathbf{X'Z} is the 4 x 3 matrix
EnvE1 | EnvE2 | EnvE3 | |
---|---|---|---|
GenG1 | 1 | 1 | 1 |
GenG2 | 1 | 1 | 1 |
GenG3 | 1 | 1 | 1 |
GenG4 | 1 | 1 | 1 |
\mathbf{Z'} is the 4 x 12 matrix
\mathbf{Z} is the 12 x 4 matrix
\mathbf{Z'Z} is the 4 x 4 matrix
GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|
GenG1 | 3 | 0 | 0 | 0 |
GenG2 | 0 | 3 | 0 | 0 |
GenG3 | 0 | 0 | 3 | 0 |
GenG4 | 0 | 0 | 0 | 3 |
\mathbf{Z'} is the 4 x 12 matrix
\mathbf{Z} is the 12 x 4 matrix
\mathbf{Z'Z} is the 4 x 4 matrix
\mathbf{Z'} is the 4 x 12 matrix
\mathbf{Z} is the 12 x 4 matrix
\mathbf{Z'Z} is the 4 x 4 matrix
assume \mathbf{K} = \mathbf{I} (no relationship)
\lambda = \sigma^2_e / \sigma^2_u = 1.64/9.56 = 0.17
\mathbf{Z'} is the 4 x 12 matrix
\mathbf{Z} is the 12 x 4 matrix
\mathbf{Z'Z} is the 4 x 4 matrix
assume \mathbf{K} = \mathbf{I} (no relationship)
\lambda = \sigma^2_e / \sigma^2_u = 1.64/9.56 = 0.17
GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|
GenG1 | 3.17 | 0 | 0 | 0 |
GenG2 | 0 | 3.17 | 0 | 0 |
GenG3 | 0 | 0 | 3.17 | 0 |
GenG4 | 0 | 0 | 0 | 3.17 |
EnvE1 | EnvE2 | EnvE3 | GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|---|---|---|
EnvE1 | 4 | 0 | 0 | 1 | 1 | 1 | 1 |
EnvE2 | 0 | 4 | 0 | 1 | 1 | 1 | 1 |
EnvE3 | 0 | 0 | 4 | 1 | 1 | 1 | 1 |
GenG1 | 1 | 1 | 1 | 3.17 | 0 | 0 | 0 |
GenG2 | 1 | 1 | 1 | 0 | 3.17 | 0 | 0 |
GenG3 | 1 | 1 | 1 | 0 | 0 | 3.17 | 0 |
GenG4 | 1 | 1 | 1 | 0 | 0 | 0 | 3.17 |
\mathbf{X'} is the 3 x 12 matrix
\mathbf{y} is the 12 x 1 matrix
\mathbf{X'y} is the 3 x 1 matrix
EnvE1 | 202 |
EnvE2 | 204 |
EnvE3 | 211 |
\mathbf{X'} is the 3 x 12 matrix
\mathbf{y} is the 12 x 1 matrix
\mathbf{X'y} is the 3 x 1 matrix
EnvE1 | 202 |
EnvE2 | 204 |
EnvE3 | 211 |
\mathbf{Z'} is the 4 x 12 matrix
\mathbf{y} is the 12 x 1 matrix
\mathbf{Z'y} is the 4 x 1 matrix
GenG1 | 152 |
GenG2 | 145 |
GenG3 | 156 |
GenG4 | 164 |
\mathbf{Z'} is the 4 x 12 matrix
\mathbf{y} is the 12 x 1 matrix
\mathbf{Z'y} is the 4 x 1 matrix
GenG1 | 152 |
GenG2 | 145 |
GenG3 | 156 |
GenG4 | 164 |
EnvE1 | 202 |
EnvE2 | 204 |
EnvE3 | 211 |
GenG1 | 152 |
GenG2 | 145 |
GenG3 | 156 |
GenG4 | 164 |
EnvE1 | 50.50 |
EnvE2 | 51.00 |
EnvE3 | 52.75 |
GenG1 | -0.71 |
GenG2 | -2.92 |
GenG3 | 0.55 |
GenG4 | 3.08 |
BLUE = sum / n_{x} = the sum of phenotypes in each environment / the number of phenotypes observed in each environment
BLUP = sum / n_{z} + \lambda = the sum of phenotypes for each genotype / the number of phenotypes observed for each genotype + \lambda
BLUE = sum / n_{x} = the sum of phenotypes in each environment / the number of phenotypes observed in each environment
BLUP = sum / n_{z} + \lambda = the sum of phenotypes for each genotype / the number of phenotypes observed for each genotype + \lambda
Note that \lambda = \frac{1-h^2}{h^2}
More observations \rightarrow less shrinkage
Higher heritability \rightarrow less shrinkage
1: The first type of \mathbf{G} matrix
\mathbf{G} = \frac{\mathbf{W_c}\mathbf{W_c'}}{\sum 2 p_j (1-p_j)}
\mathbf{W_c}: centered marker matrix
p_j: allele frequency at jth marker
2: The second type of \mathbf{G} matrix
\mathbf{G} = \frac{\mathbf{W_{cs}}\mathbf{W_{cs}'}}{m}
\mathbf{W_{cs}}: centered and scaled marker matrix
m: number of markers
Suppose \mathbf{K} is given by
GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|
GenG1 | 1.00 | 0.64 | 0.23 | 0.48 |
GenG2 | 0.64 | 1.00 | 0.33 | 0.67 |
GenG3 | 0.23 | 0.33 | 1.00 | 0.31 |
GenG4 | 0.48 | 0.67 | 0.31 | 1.00 |
Suppose \mathbf{K} is given by
GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|
GenG1 | 1.00 | 0.64 | 0.23 | 0.48 |
GenG2 | 0.64 | 1.00 | 0.33 | 0.67 |
GenG3 | 0.23 | 0.33 | 1.00 | 0.31 |
GenG4 | 0.48 | 0.67 | 0.31 | 1.00 |
Then \lambda \mathbf{K}^{-1} is
GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|
GenG1 | 0.15 | -0.09 | 0.00 | -0.01 |
GenG2 | -0.09 | 0.22 | -0.02 | -0.10 |
GenG3 | 0.00 | -0.02 | 0.10 | -0.02 |
GenG4 | -0.01 | -0.10 | -0.02 | 0.17 |
EnvE1 | EnvE2 | EnvE3 | GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|---|---|---|
EnvE1 | 4 | 0 | 0 | 1 | 1 | 1 | 1 |
EnvE2 | 0 | 4 | 0 | 1 | 1 | 1 | 1 |
EnvE3 | 0 | 0 | 4 | 1 | 1 | 1 | 1 |
GenG1 | 1 | 1 | 1 | 3.15 | -0.09 | 0.00 | -0.01 |
GenG2 | 1 | 1 | 1 | -0.09 | 3.22 | -0.02 | -0.10 |
GenG3 | 1 | 1 | 1 | 0.00 | -0.02 | 3.10 | -0.02 |
GenG4 | 1 | 1 | 1 | -0.01 | -0.10 | -0.02 | 3.17 |
N | Phe | Env | Gen |
---|---|---|---|
1 | 47 | E1 | G1 |
2 | 51 | E1 | G2 |
3 | NA | E1 | G3 |
4 | 58 | E1 | G4 |
5 | 52 | E2 | G1 |
6 | 46 | E2 | G2 |
7 | 52 | E2 | G3 |
8 | NA | E2 | G4 |
9 | 53 | E3 | G1 |
10 | 48 | E3 | G2 |
11 | 58 | E3 | G3 |
12 | 52 | E3 | G4 |
N | EnvE1 | EnvE2 | EnvE3 |
---|---|---|---|
1 | 1 | 0 | 0 |
2 | 1 | 0 | 0 |
4 | 1 | 0 | 0 |
5 | 0 | 1 | 0 |
6 | 0 | 1 | 0 |
7 | 0 | 1 | 0 |
9 | 0 | 0 | 1 |
10 | 0 | 0 | 1 |
11 | 0 | 0 | 1 |
12 | 0 | 0 | 1 |
Remove missing rows
\mathbf{X} is the 10 x 3 matrix
N | GenG1 | GenG2 | GenG3 | GenG4 |
---|---|---|---|---|
1 | 1 | 0 | 0 | 0 |
2 | 0 | 1 | 0 | 0 |
4 | 0 | 0 | 0 | 1 |
5 | 1 | 0 | 0 | 0 |
6 | 0 | 1 | 0 | 0 |
7 | 0 | 0 | 1 | 0 |
9 | 1 | 0 | 0 | 0 |
10 | 0 | 1 | 0 | 0 |
11 | 0 | 0 | 1 | 0 |
12 | 0 | 0 | 0 | 1 |
Remove missing rows
\mathbf{Z} is the 10 x 4 matrix
\mathbf{X'} is the 3 x 10 matrix
\mathbf{X} is the 10 x 3 matrix
\mathbf{X'X} is the 3 x 3 matrix
EnvE1 | EnvE2 | EnvE3 | |
---|---|---|---|
EnvE1 | 3 | 0 | 0 |
EnvE2 | 0 | 3 | 0 |
EnvE3 | 0 | 0 | 4 |
\mathbf{X'} is the 3 x 10 matrix
\mathbf{X} is the 10 x 3 matrix
\mathbf{X'X} is the 3 x 3 matrix
EnvE1 | EnvE2 | EnvE3 | |
---|---|---|---|
EnvE1 | 3 | 0 | 0 |
EnvE2 | 0 | 3 | 0 |
EnvE3 | 0 | 0 | 4 |
\mathbf{X'} is the 3 x 10 matrix
\mathbf{Z} is the 10 x 4 matrix
\mathbf{X'Z} is the 3 x 4 matrix
GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|
EnvE1 | 1 | 1 | 0 | 1 |
EnvE2 | 1 | 1 | 1 | 0 |
EnvE3 | 1 | 1 | 1 | 1 |
\mathbf{X'} is the 3 x 10 matrix
\mathbf{Z} is the 10 x 4 matrix
\mathbf{X'Z} is the 3 x 4 matrix
GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|
EnvE1 | 1 | 1 | 0 | 1 |
EnvE2 | 1 | 1 | 1 | 0 |
EnvE3 | 1 | 1 | 1 | 1 |
\mathbf{Z'} is the 4 x 10 matrix
\mathbf{X} is the 10 x 3 matrix
\mathbf{X'Z} is the 4 x 3 matrix
EnvE1 | EnvE2 | EnvE3 | |
---|---|---|---|
GenG1 | 1 | 1 | 1 |
GenG2 | 1 | 1 | 1 |
GenG3 | 0 | 1 | 1 |
GenG4 | 1 | 0 | 1 |
\mathbf{Z'} is the 4 x 10 matrix
\mathbf{Z} is the 4 x 10 matrix
\mathbf{Z'Z} is the 4 x 4 matrix
GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|
GenG1 | 3 | 0 | 0 | 0 |
GenG2 | 0 | 3 | 0 | 0 |
GenG3 | 0 | 0 | 2 | 0 |
GenG4 | 0 | 0 | 0 | 2 |
EnvE1 | EnvE2 | EnvE3 | GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|---|---|---|
EnvE1 | 3 | 0 | 0 | 1 | 1 | 0 | 1 |
EnvE2 | 0 | 3 | 0 | 1 | 1 | 1 | 0 |
EnvE3 | 0 | 0 | 4 | 1 | 1 | 1 | 1 |
GenG1 | 1 | 1 | 1 | 3.10 | -0.06 | 0.00 | -0.01 |
GenG2 | 1 | 1 | 1 | -0.06 | 3.15 | -0.01 | -0.07 |
GenG3 | 0 | 1 | 1 | 0.00 | -0.01 | 2.07 | -0.01 |
GenG4 | 1 | 0 | 1 | -0.01 | -0.07 | -0.01 | 2.11 |
EnvE1 | 156 |
EnvE2 | 150 |
EnvE3 | 211 |
GenG1 | 152 |
GenG2 | 145 |
GenG3 | 110 |
GenG4 | 110 |
N | Phe | Env | Gen |
---|---|---|---|
1 | NA | E1 | G1 |
2 | 51 | E1 | G2 |
3 | 46 | E1 | G3 |
4 | 58 | E1 | G4 |
5 | NA | E2 | G1 |
6 | 46 | E2 | G2 |
7 | 52 | E2 | G3 |
8 | 54 | E2 | G4 |
9 | NA | E3 | G1 |
10 | 48 | E3 | G2 |
11 | 58 | E3 | G3 |
12 | 52 | E3 | G4 |
N | EnvE1 | EnvE2 | EnvE3 |
---|---|---|---|
2 | 1 | 0 | 0 |
3 | 1 | 0 | 0 |
4 | 1 | 0 | 0 |
6 | 0 | 1 | 0 |
7 | 0 | 1 | 0 |
8 | 0 | 1 | 0 |
10 | 0 | 0 | 1 |
11 | 0 | 0 | 1 |
12 | 0 | 0 | 1 |
Remove missing rows
\mathbf{X} is the 9 x 3 matrix
N | GenG1 | GenG2 | GenG3 | GenG4 |
---|---|---|---|---|
2 | 0 | 1 | 0 | 0 |
3 | 0 | 0 | 1 | 0 |
4 | 0 | 0 | 0 | 1 |
6 | 0 | 1 | 0 | 0 |
7 | 0 | 0 | 1 | 0 |
8 | 0 | 0 | 0 | 1 |
10 | 0 | 1 | 0 | 0 |
11 | 0 | 0 | 1 | 0 |
12 | 0 | 0 | 0 | 1 |
Remove missing rows
\mathbf{Z} is the 9 x 4 matrix
\mathbf{X'} is the 3 x 9 matrix
\mathbf{X} is the 9 x 3 matrix
\mathbf{X'X} is the 3 x 3 matrix
EnvE1 | EnvE2 | EnvE3 | |
---|---|---|---|
EnvE1 | 3 | 0 | 0 |
EnvE2 | 0 | 3 | 0 |
EnvE3 | 0 | 0 | 3 |
\mathbf{X'} is the 3 x 9 matrix
\mathbf{X} is the 9 x 3 matrix
\mathbf{X'X} is the 3 x 3 matrix
EnvE1 | EnvE2 | EnvE3 | |
---|---|---|---|
EnvE1 | 3 | 0 | 0 |
EnvE2 | 0 | 3 | 0 |
EnvE3 | 0 | 0 | 3 |
\mathbf{X'} is the 3 x 9 matrix
\mathbf{Z} is the 9 x 4 matrix
\mathbf{X'Z} is the 3 x 4 matrix
GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|
EnvE1 | 0 | 1 | 1 | 1 |
EnvE2 | 0 | 1 | 1 | 1 |
EnvE3 | 0 | 1 | 1 | 1 |
\mathbf{X'} is the 3 x 9 matrix
\mathbf{Z} is the 9 x 4 matrix
\mathbf{X'Z} is the 3 x 4 matrix
GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|
EnvE1 | 0 | 1 | 1 | 1 |
EnvE2 | 0 | 1 | 1 | 1 |
EnvE3 | 0 | 1 | 1 | 1 |
\mathbf{Z'} is the 4 x 12 matrix
\mathbf{X} is the 12 x 3 matrix
\mathbf{X'Z} is the 4 x 3 matrix
EnvE1 | EnvE2 | EnvE3 | |
---|---|---|---|
GenG1 | 0 | 0 | 0 |
GenG2 | 1 | 1 | 1 |
GenG3 | 1 | 1 | 1 |
GenG4 | 1 | 1 | 1 |
\mathbf{Z'} is the 4 x 9 matrix
\mathbf{Z} is the 4 x 9 matrix
\mathbf{Z'Z} is the 4 x 4 matrix
GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|
GenG1 | 0 | 0 | 0 | 0 |
GenG2 | 0 | 3 | 0 | 0 |
GenG3 | 0 | 0 | 3 | 0 |
GenG4 | 0 | 0 | 0 | 3 |
EnvE1 | EnvE2 | EnvE3 | GenG1 | GenG2 | GenG3 | GenG4 | |
---|---|---|---|---|---|---|---|
EnvE1 | 3 | 0 | 0 | 0 | 1 | 0 | 1 |
EnvE2 | 0 | 3 | 0 | 0 | 1 | 1 | 0 |
EnvE3 | 0 | 0 | 3 | 0 | 1 | 1 | 1 |
GenG1 | 0 | 0 | 0 | 0.14 | -0.08 | 0.00 | -0.01 |
GenG2 | 1 | 1 | 1 | -0.08 | 3.19 | -0.02 | -0.09 |
GenG3 | 0 | 1 | 1 | 0.00 | -0.02 | 3.09 | -0.01 |
GenG4 | 1 | 0 | 1 | -0.01 | -0.09 | -0.01 | 3.15 |
EnvE1 | 155 |
EnvE2 | 152 |
EnvE3 | 158 |
GenG1 | 0 |
GenG2 | 145 |
GenG3 | 156 |
GenG4 | 164 |
EnvE1 | 52.06 |
EnvE2 | 51.06 |
EnvE3 | 53.06 |
GenG1 | -1.82 |
GenG2 | -3.48 |
GenG3 | -0.07 |
GenG4 | 2.38 |
s: number of unique fixed effects
q: number of unique genotypes
does not depend on n
Previous model
\mathbf{y = Xb + Zu + e}
Linear mixed model single-marker regression \mathbf{y = Xb + Wa + Zu + e}
\mathbf{W}: marker matrix
\mathbf{a}: vector of marker effect
Single marker-based mixed model association (MMA) \begin{align*} \mathbf{y} &= \mu + \mathbf{w_ja_j} + \mathbf{Zg} + \boldsymbol{\epsilon} \\ \mathbf{g} &\sim N(0, \mathbf{G}\sigma^2_{g}) \end{align*}
\mathbf{G} captures population structure and polygenic effects
Single marker-based mixed model association (MMA) \begin{align*} \mathbf{y} &= \mu + \mathbf{w_ja_j} + \mathbf{Zg} + \boldsymbol{\epsilon} \\ \mathbf{g} &\sim N(0, \mathbf{G}\sigma^2_{g}) \end{align*}
\mathbf{G} captures population structure and polygenic effects
Double counting?
Single marker-based mixed model association (MMA) \begin{align*} \mathbf{y} &= \mu + \mathbf{w_ja_j} + \mathbf{Zg} + \boldsymbol{\epsilon} \\ \mathbf{g} &\sim N(0, \mathbf{G}\sigma^2_{g}) \end{align*}
\mathbf{G} captures population structure and polygenic effects
Double counting?
Alternatively, \begin{align*} \mathbf{y} &= \mu + \mathbf{w_ja_j} + \mathbf{Zg} + \boldsymbol{\epsilon} \\ \mathbf{g} &\sim N(0, \mathbf{G}_{-k}\sigma^2_{g_{-k}}) \end{align*} where -k denotes the kth chromosome removed
See
Rincent et al. 2014. (10.1534/genetics.113.159731)
Chen and Lipka. 2016. (10.1534/g3.116.029090)
1: Mixed model equations (MME) \begin{align*} \mathbf{y} &= \mu + \mathbf{w_ja_j} + \mathbf{Zg} + \boldsymbol{\epsilon} \\ \end{align*}
The mixed model equations of Henderson (1949) are given by
2: Weighted least squares \begin{align*} \hat{\mathbf{a}} &= (\mathbf{W'U T U'W})^{-1}\mathbf{W'U} \mathbf{T} \mathbf{U'y} \end{align*} where
\begin{align*} \mathbf{T} = [\mathbf{D} + \lambda \mathbf{I} ]^{-1} \end{align*}
Animal
Plant
Keyboard shortcuts
↑, ←, Pg Up, k | Go to previous slide |
↓, →, Pg Dn, Space, j | Go to next slide |
Home | Go to first slide |
End | Go to last slide |
Number + Return | Go to specific slide |
b / m / f | Toggle blackout / mirrored / fullscreen mode |
c | Clone slideshow |
p | Toggle presenter mode |
t | Restart the presentation timer |
?, h | Toggle this help |
Esc | Back to slideshow |