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, BLUPf90MATVEC, 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 example


The 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)
 
R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.
 
  Natural language support but running in an English locale
 
R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.
 
Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.
 
[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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