Transcript pedigreemm - Waisman Laboratory for Brain Imaging and Behavior

Pedigree-induced correlation
pedigreemm
http://r-forge.r-project.org/projects/pedigreemm/
www.r-project.org
Ana Ines Vazquez
University of Wisconsin
install.packages(“pedigreemm”)
library(pedigreemm)
pedigreemm:
Pedigree-based mixed-effects models
Uses:
•
•
•
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Sire, Animal model with repeated measures.
Multiple random (nested or cross classified) or fixed effects.
Random regression.
Generalized linear models (Poisson, binomial, etc).
Genetic model
pij  ui  eij
Phenotype,
Genetic effects,
y  Xβ  Zu  e
 0   G 0 
u

p   N  , 
e
 0   0 R 
py β   N Xβ, V 
V  ZGZ  R
Model residual
Mixed Model
y β ~ N Xβ, V 
1
1
1
ˆβ




XV y
BLUE  X V X
1

Eu y,β, G, V  Eu  covu,y V y  y  Xβ
uˆ BLUP  GZV1 y  Xβ
Henderson, 1963
Example:
Sire linear model
Example:
y  Xβ  Zu  e
Phenotypic measures:

y   y1,1 , y1, 2 , y2,1 , y2, 2 ,..., y17 ,1 , y17 , 2 
Fixed effect (gender and herds):
Sire effects:

u  s1 , s2 ,...,s17 

β  m, f , h1 , h2 , h3 
Data
sire
gender
herd
y
1
m
0
257
1
m
0
304
2
m
0
271
2
m
0
340
3
m
0
445
3
m
0
413
4
f
0
425
4
f
0
378
5
f
0
278
5
f
0
278
6
f
0
341
6
f
0
367
7
f
0
244
7
f
0
249
.
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.
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.
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.
17
f
3
238
17
f
3
298
Sire
Gender,
X=
herd
s1
s2
s3
s4
1
s17
0
0
0
.
0
1
0
0
0
.
0
0
1
0
0
.
0
0
1
0
0
.
0
0
0
1
0
.
0
0
0
1
0
.
0
0
0
0
1
.
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
.
0
0
1
0
0
0
0
0
0
0
.
0
0
1
0
0
0
0
0
0
0
.
0
0
1
0
0
0
0
0
0
0
.
0
0
1
0
0
0
0
0
0
0
.
0
0
1
0
0
0
0
0
0
0
.
0
0
1
0
0
0
0
0
0
0
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0
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0
1
0
1
0
0
1
0
0
1
0
1
0
0
1
,
Z=
.
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0
0
0
0
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0
0
0
0
0
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1
0
0
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0
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1
The random effects (sires) are not independent between them.
• They are related, then covariance between two of them is not cero.
• The covariance = the relationship between animals times  2
which can be measured.
V u  G  A A2
y  Xβ  Zu  e
u ~ N 0, A u2 
A
Sire-pedigree
u ~ N 0, A
2
u

A: Relationships between sires
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
1
1
0.5
0
0
0
0
0
0
0.5
0
0.5
0
0.5
0
0
0.25
0
2
0.5
1
0
0
0
0
0
0.5
0
0.5
0
0
0
0.25
0.5
0.5
0.25
3
0
0
1
0.5
0
0
0
0
0.25
0
0
0.5
0
0.5
0
0
0.5
4
0
0
0.5
1
0
0
0
0
0.5
0
0
0.25
0
0.25
0
0
0.25
5
0
0
0
0
1
0
0
0
0
0.5
0
0
0
0
0
0
0
6
0
0
0
0
0
1
0
0
0
0
0.5
0
0
0
0
0
0
7
0
0
0
0
0
0
1
0
0
0
0
0
0.5
0
0
0.25
0
8
0
0.5
0
0
0
0
0
1
0
0.25
0
0
0
0.5
0.25
0.25
0.13
9
0.5
0
0.25
0.5
0
0
0
0
1
0
0.25
0.13
0.25
0.13
0
0.13
0.13
10
0
0.5
0
0
0.5
0
0
0.25
0
1
0
0
0
0.13
0.25
0.25
0.13
11
0.5
0
0
0
0
0.5
0
0
0.25
0
1
0
0.25
0
0
0.13
0
12
0
0
0.5
0.25
0
0
0
0
0.13
0
0
1
0
0.25
0
0
0.25
13
0.5
0
0
0
0
0
0.5
0
0.25
0
0.25
0
1
0
0
0.5
0
14
0
0.25
0.5
0.25
0
0
0
0.5
0.13
0.13
0
0.25
0
1
0.13
0.13
0.31
15
0
0.5
0
0
0
0
0
0.25
0
0.25
0
0
0
0.13
1
0.25
0.5
16
0.25
0.5
0
0
0
0
0.25
0.25
0.13
0.25
0.13
0
0.5
0.13
0.25
1
0.13
17
0
0.25
0.5
0.25
0
0
0
0.13
0.13
0.13
0
0.25
0
0.31
0.5
0.13
1
Example… u estimates
id
Phenotype
(average)
u w/o
Pedigree
u w/
Pedigree
1
280
-41
-29
2
305
-22
-11
3
429
69
66
4
402
60
68
5
278
-31
-26
6
354
24
26
7
247
-54
-46
8
315
-5
-11
9
424
10
19
10
276
5
-2
11
257
-9
-7
12
394
-12
3
13
399
2
-9
14
237
-13
-1
15
270
1
5
16
276
16
7
17
268
-1
21
L:
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u*=L-1u
0.5
then u_est = Lu*
Non-linear model
(Logistic regression)
 pi 
li  ln 
  xi β  zi u
 1  pi 

l  l1 , l2 ,...,ln   Xβ  Zu
To sum up…
•
•
•
•
Sire, Animal model with repeated measures
Multiple random or fixed effects.
Random regression.
Generalized linear models (Poisson, binomial, etc).