Small pedigree based mixed model example
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Pedigree based mixed models (often called animal models, due to modelling animal performance) are the cornerstone of animal breeding and quantitative genetics. There are many programs that can be used for analyzing your data with these models, e.g., ASREML, BLUPf90, MATVEC, MiXBLUP & MiX99, SurvivalKit, PEST/VCE, WOMBAT, …). There are also R packages you can use: pedigreemm and MCMCglmm. If you want to run your own program you can take the example code bellow and start from it. The code shows the essence of building the system of equations that needs to be solved on a simple example. Note that this is mean only for demonstration purposes and small scale analyses. In addition, variance components are assumed known here. In order to understand the model for this simple example a bit better the graphical model view is shown first.
Graphical model view of simple pedigree based mixed model exampleThe code:
options(width=200)
### --- Required packages ---
## install.packages(pkg=c("pedigreemm", "MatrixModels"))
library(package="pedigreemm") ## pedigree functions
library(package="MatrixModels") ## sparse matrices
### --- Data ---
## NOTE:
## - some individuals have one or both parents (un)known
## - some individuals have phenotype (un)known
## - some indididuals have repeated phenotype observations
example <- data.frame(
individual=c( 1, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10),
father=c(NA, NA, NA, 2, 2, 4, 2, 5, 5, NA, 8),
mother=c(NA, NA, NA, 1, NA, 3, 3, 6, 6, NA, 9),
phenotype=c(NA, 103, 106, 98, 101, 106, 93, NA, NA, NA, 109),
group=c(NA, 1, 1, 1, 2, 2, 2, NA, NA, NA, 1))
## Variance components
sigma2e <- 1
sigma2a <- 3
(h2 <- sigma2a / (sigma2a + sigma2e))
### --- Setup data ---
## Make sure each individual has only one record in pedigree
ped <- example[!duplicated(example$individual), 1:3]
## Factors (this eases buliding the design matrix considerably)
example$individual <- factor(example$individual)
example$group <- factor(example$group)
## Phenotype data
dat <- example[!is.na(example$phenotype), ]
### --- Setup MME ---
## Phenotype vector
(y <- dat$phenotype)
## Design matrix for the "fixed" effects (group)
(X <- model.Matrix( ~ group, data=dat, sparse=TRUE))
## Design matrix for the "random" effects (individual)
(Z <- model.Matrix(~ individual - 1, data=dat, sparse=TRUE))
## Inverse additive relationship matrix
ped2 <- pedigree(sire=ped$father, dam=ped$mother, label=ped$individual)
TInv <- as(ped2, "sparseMatrix")
DInv <- Diagonal(x=1/Dmat(ped2))
AInv <- crossprod(sqrt(DInv) %*% TInv)
## Variance ratio
alpha <- sigma2e / sigma2a
## Mixed Model Equations (MME)
## ... Left-Hand Side (LHS) without pedigree prior
(LHS0 <- rBind(cBind(crossprod(X), crossprod(X, Z)),
cBind(crossprod(Z, X), crossprod(Z, Z))))
## ... Left-Hand Side (LHS) with pedigree prior
round(
LHS <- rBind(cBind(crossprod(X), crossprod(X, Z)),
cBind(crossprod(Z, X), crossprod(Z, Z) + AInv * alpha)), digits=1)
## ... Right-Hand Side (RHS)
(RHS <- rBind(crossprod(X, y),
crossprod(Z, y)))
### --- Solutions ---
## Solve
LHSInv <- solve(LHS)
sol <- LHSInv %*% RHS
## Standard errors
se <- diag(LHSInv) * sigma2e
## Reliabilities
r2 <- 1 - diag(LHSInv) * alpha
## Accuracies
r <- sqrt(r2)
## Printout
cBind(sol, se, r, r2)And the transcript:
R version 2.14.2 (2012-02-29)
Copyright (C) 2012 The R Foundation for Statistical Computing
ISBN 3-900051-07-0
Platform: x86_64-pc-linux-gnu (64-bit)
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Type 'license()' or 'licence()' for distribution details.
Natural language support but running in an English locale
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Type 'contributors()' for more information and
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Type 'demo()' for some demos, 'help()' for on-line help, or
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[Previously saved workspace restored]
>
> options(width=200)
>
> ### --- Required packages ---
>
> ## install.packages(pkg=c("pedigreemm", "MatrixModels"))
>
> library(package="pedigreemm") ## pedigree functions
Loading required package: lme4
Loading required package: Matrix
Loading required package: lattice
Attaching package: ‘Matrix’
The following object(s) are masked from ‘package:base’:
det
Attaching package: ‘lme4’
The following object(s) are masked from ‘package:stats’:
AIC, BIC
> library(package="MatrixModels") ## sparse matrices
>
> ### --- Data ---
>
> ## NOTE:
> ## - some individuals have one or both parents (un)known
> ## - some individuals have phenotype (un)known
> ## - some indididuals have repeated phenotype observations
> example <- data.frame(
+ individual=c( 1, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10),
+ father=c(NA, NA, NA, 2, 2, 4, 2, 5, 5, NA, 8),
+ mother=c(NA, NA, NA, 1, NA, 3, 3, 6, 6, NA, 9),
+ phenotype=c(NA, 103, 106, 98, 101, 106, 93, NA, NA, NA, 109),
+ group=c(NA, 1, 1, 1, 2, 2, 2, NA, NA, NA, 1))
>
> ## Variance components
> sigma2e <- 1
> sigma2a <- 3
> (h2 <- sigma2a / (sigma2a + sigma2e))
[1] 0.75
>
> ### --- Setup data ---
>
> ## Make sure each individual has only one record in pedigree
> ped <- example[!duplicated(example$individual), 1:3]
>
> ## Factors (this eases buliding the design matrix considerably)
> example$individual <- factor(example$individual)
> example$group <- factor(example$group)
>
> ## Phenotype data
> dat <- example[!is.na(example$phenotype), ]
>
> ### --- Setup MME ---
>
> ## Phenotype vector
> (y <- dat$phenotype)
[1] 103 106 98 101 106 93 109
>
> ## Design matrix for the "fixed" effects (group)
> (X <- model.Matrix( ~ group, data=dat, sparse=TRUE))
"dsparseModelMatrix": 7 x 2 sparse Matrix of class "dgCMatrix"
(Intercept) group2
2 1 .
3 1 .
4 1 .
5 1 1
6 1 1
7 1 1
11 1 .
@ assign: 0 1
@ contrasts:
$group
[1] "contr.treatment"
>
> ## Design matrix for the "random" effects (individual)
> (Z <- model.Matrix(~ individual - 1, data=dat, sparse=TRUE))
"dsparseModelMatrix": 7 x 10 sparse Matrix of class "dgCMatrix"
[[ suppressing 10 column names ‘individual1’, ‘individual2’, ‘individual3’ ... ]]
2 . 1 . . . . . . . .
3 . 1 . . . . . . . .
4 . . 1 . . . . . . .
5 . . . 1 . . . . . .
6 . . . . 1 . . . . .
7 . . . . . 1 . . . .
11 . . . . . . . . . 1
@ assign: 1 1 1 1 1 1 1 1 1 1
@ contrasts:
$individual
[1] "contr.treatment"
>
> ## Inverse additive relationship matrix
> ped2 <- pedigree(sire=ped$father, dam=ped$mother, label=ped$individual)
> TInv <- as(ped2, "sparseMatrix")
> DInv <- Diagonal(x=1/Dmat(ped2))
> AInv <- crossprod(sqrt(DInv) %*% TInv)
>
> ## Variance ratio
> alpha <- sigma2e / sigma2a
>
> ## Mixed Model Equations (MME)
>
> ## ... Left-Hand Side (LHS) without pedigree prior
> (LHS0 <- rBind(cBind(crossprod(X), crossprod(X, Z)),
+ cBind(crossprod(Z, X), crossprod(Z, Z))))
12 x 12 sparse Matrix of class "dgCMatrix"
[[ suppressing 12 column names ‘(Intercept)’, ‘group2’, ‘individual1’ ... ]]
(Intercept) 7 3 . 2 1 1 1 1 . . . 1
group2 3 3 . . . 1 1 1 . . . .
individual1 . . . . . . . . . . . .
individual2 2 . . 2 . . . . . . . .
individual3 1 . . . 1 . . . . . . .
individual4 1 1 . . . 1 . . . . . .
individual5 1 1 . . . . 1 . . . . .
individual6 1 1 . . . . . 1 . . . .
individual7 . . . . . . . . . . . .
individual8 . . . . . . . . . . . .
individual9 . . . . . . . . . . . .
individual10 1 . . . . . . . . . . 1
>
> ## ... Left-Hand Side (LHS) with pedigree prior
> round(
+ LHS <- rBind(cBind(crossprod(X), crossprod(X, Z)),
+ cBind(crossprod(Z, X), crossprod(Z, Z) + AInv * alpha)), digits=1)
12 x 12 sparse Matrix of class "dgCMatrix"
[[ suppressing 12 column names ‘(Intercept)’, ‘group2’, ‘individual1’ ... ]]
(Intercept) 7 3 . 2.0 1.0 1.0 1.0 1.0 . . . 1.0
group2 3 3 . . . 1.0 1.0 1.0 . . . .
individual1 . . 0.5 0.2 -0.3 . . . . . . .
individual2 2 . 0.2 2.8 -0.2 -0.2 . -0.3 . . . .
individual3 1 . -0.3 -0.2 2.0 0.2 -0.3 -0.3 . . . .
individual4 1 1 . -0.2 0.2 1.6 -0.3 . . . . .
individual5 1 1 . . -0.3 -0.3 2.1 0.4 -0.4 -0.4 . .
individual6 1 1 . -0.3 -0.3 . 0.4 2.1 -0.4 -0.4 . .
individual7 . . . . . . -0.4 -0.4 0.8 . . .
individual8 . . . . . . -0.4 -0.4 . 1.0 0.2 -0.4
individual9 . . . . . . . . . 0.2 0.5 -0.4
individual10 1 . . . . . . . . -0.4 -0.4 1.8
>
> ## ... Right-Hand Side (RHS)
> (RHS <- rBind(crossprod(X, y),
+ crossprod(Z, y)))
12 x 1 Matrix of class "dgeMatrix"
[,1]
(Intercept) 716
group2 300
individual1 0
individual2 209
individual3 98
individual4 101
individual5 106
individual6 93
individual7 0
individual8 0
individual9 0
individual10 109
>
> ### --- Solutions ---
>
> ## Solve
> LHSInv <- solve(LHS)
> sol <- LHSInv %*% RHS
>
> ## Standard errors
> se <- diag(LHSInv) * sigma2e
>
> ## Reliabilities
> r2 <- 1 - diag(LHSInv) * alpha
>
> ## Accuracies
> r <- sqrt(r2)
Warning message:
In sqrt(r2) : NaNs produced
>
> ## Printout
> cBind(sol, se, r, r2)
12 x 4 Matrix of class "dgeMatrix"
se r r2
[1,] 104.76444809 1.901479 0.6051228 0.3661736
[2,] -4.65061921 1.376348 0.7356748 0.5412175
[3,] -2.62685846 2.432448 0.4349528 0.1891839
[4,] -0.77797260 1.983301 0.5821509 0.3388997
[5,] -4.32927399 1.979138 0.5833415 0.3402874
[6,] 1.54997883 2.316720 0.4772421 0.2277600
[7,] 3.18706240 2.604540 0.3630701 0.1318199
[8,] -5.07852788 2.508673 0.4046922 0.1637758
[9,] -0.94573274 3.459904 NaN -0.1533013
[10,] -0.08765651 3.534636 NaN -0.1782121
[11,] 2.11218764 2.511203 0.4036487 0.1629323
[12,] 2.82742681 2.009539 0.5745901 0.3301538
>
> proc.time()
user system elapsed
3.350 0.070 3.425
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