Trans-generational effect of maternal lactation during pregnancy: A Holstein cow model.
Authors: Oscar González-Recio, Eva Ugarte, Alex Bach
File S1. Supplemental statistical procedure information.
The following underlying statistical model was considered:
y = 1¢ m + Xb + Zu + e
here, the ith component of the n-vector y depicts the phenotypic value
of individual i. The  vector corresponds to the environmental forces affecting
the trait, with X being the corresponding incidence matrix. Then, u = {ui }
represents a vector of additive genetic effects, with ui being the additive
genetic merit of animal i in the pedigree. A priori, the additive genetic effects
, with G = As a2 , where A is a
were assumed to be distributed as
(txt) additive relationship matrix between animals, with t=131,308, and s a2
corresponds to the additive genetic variance. The matrix Z is an (nxg) incidence
matrix, which rows consist of unit vectors with one component being 1 and all
the others zero, indicating the respective position of the individual with record
in the g-vector of genetic merit of all individuals in the relationship matrix.
The residuals e were assumed to be distributed as N(0, s e2 ), where s e2
is the residual variance
A hierarchical Bayesian model was implemented to estimate the
residuals of the model as described next:
(
n
)
(
Likelihood: p y | m, b ,u,G,s e2 = Õ N yi | m + xi b + z i u, s e2
i=1
1
)
Trans-generational effect of maternal lactation during pregnancy: A Holstein cow model.
Authors: Oscar González-Recio, Eva Ugarte, Alex Bach
Prior:
p ( m, b ,u,G, s e2 ) = p ( m | s e2 ) p ( b | s e2 ) p ( u | G ) p ( G ) p (s e2 )
(
) (
)
(s | d ,S ) c (s | d ,S ) c (G | d ,S ) c (s
= N m | 0, s m2 N b | 0,Is b2 N ( u | 0,G )
´ c -2
2
m
m
m
-2
2
b
b
b
-2
u
(
N ( yi | m + xi b + zi u,s e2 ) ,
Above,
-2
u
)
N m | 0,s m2 ,
2
e
| de ,Se )
(
)
N b | 0,Is b2 , and
N ( u | 0,G) are normal densities centered at m + xi b + z i u , or 0, with variances
c -2 (s e2 | de ,Se ) , c -2 (s m2 | dm ,Sm ) ,
s e2 , s m2 , s b2 and G, respectively.
c -2 (s b2 | db ,Sb ) and c -2 ( G | du ,Su ) are scaled-inverted chi-square densities,
with d· degrees of freedom and scale S·, respectively. The role of u is to adjust
the additive action of the genes inherited in the individual by means of a covariance matrix between individuals using the rules of the numerator
relationship matrix [1]. In this model, all fully conditionals have closed form,
thus a Gibbs sampler algorithm can be used to draw samples from the joint
posterior distributions [2], with standard derivations are then used for Bayesian
linear models [3]. The joint posterior distribution is:
n
p ( m, b ,u,G, s e2 | y ) = Õ N ( yi | m + x i b + z i u, s e2 )
(
i=1
) (
)
(s | d ,S ) c (s | d ,S ) c (G | d ,S ) c (s
´N m | 0,s m N b | 0,Is b2 N ( u | 0,G )
´ c -2
2
2
m
m
m
-2
2
b
b
-2
b
-2
u
2
u
2
e
| de ,Se )
Trans-generational effect of maternal lactation during pregnancy: A Holstein cow model.
Authors: Oscar González-Recio, Eva Ugarte, Alex Bach
The fully conditional distribution of any unknown is obtained by
removing from the right-hand side of the equation above the components that
do not involve such an unknown [3]. The remaining components are kernels of
known distribution as conjugate priors were chosen.
The fully conditional distribution for each unknown is given next.
1. Intercept
é n
ù
p ( m | else) µ ê Õ N ( yi | m + xi b + z i u,s e2 ) ú N m | 0, s m2
ë i=1
û
(
é n
ù
µ ê Õ N ( yi* | m, s e2 ) ú N m | 0,s m2
ë i=1
û
(
)
)
This is recognized as the kernel of a normal distribution with mean
( ns
-2
e
+ s m-2
n
) åy s
-1
*
i
-2
e
(
and variance ns e-2 + s m-2
i=1
)
-1
, where yi* = yi - xi b - ui .
2. Regression coefficients for environmental effects ( b )
é n
ù
p b j | else µ ê Õ N ( yi | m + xi b + z i u,s e2 ) ú ´ N b j | 0, s b2
ë i=1
û
(
)
(
é n
ù
µ ê Õ N yi** | x i b j ,s e2 ú N b j | 0, s b2
ë i=1
û
(
) (
)
)
This is recognized as the kernel of a normal distribution with mean and
-1
n
n
æ
ö
variance equal to the solution of ç s e-2 å xij2 + s b-2 ÷ b̂ j = s e-2 å xij yi** , where
è
ø
i=1
i=1
3
Trans-generational effect of maternal lactation during pregnancy: A Holstein cow model.
Authors: Oscar González-Recio, Eva Ugarte, Alex Bach
yi** = yi - m - å xip b p - ui . In practice one can set s b2 large enough so that an
p¹ j
efficient effectively flat prior is assigned to these coefficients.
3. Infinitesimal additive genetic effect (u)
p ( G | else) µ N ( u | 0,G) ´ c -2 ( G | du ,Su )
This is recognized as a multivariate normal distribution with mean
vector
( Is
-2
e
(co-variance
matrix)
equal
to
the
solution
+ G-1 ) û = s e-2 y*** , where yi*** = yi - m - xi b .
-1
4. Infinitesimal additive genetic variance ( s a2 )
Given that G = As a2
p ( G | else) µ N ( u | 0,G) ´ c -2 ( G | du ,Su )
or equivalently,
p (s a2 | else) µ N ( u | 0, s a2 ) ´ c -2 (s a2 | da ,Sa )
µ c -2 (s a2 | d. f . = da + t,S = Sa + u¢A -1u )
where t is the order of the square matrix A.
5. Residual variance ( s e2 )
é n
ù
p (s e2 | else) µ ê Õ N ( yi | m + x i b + z i u, s e2 ) ú ´ c -2 (s e2 | de ,Se )
ë i=1
û
µ c -2 (s e2 | d. f . = de + n,S = Se + e¢e )
where e = y - 1¢ m - xi b - z i u .
4
of
the
system,
Trans-generational effect of maternal lactation during pregnancy: A Holstein cow model.
Authors: Oscar González-Recio, Eva Ugarte, Alex Bach
References for S1
1. Henderson CR (1976) A Simple Method for Computing the Inverse of a
Numerator Relationship Matrix Used in Prediction of Breeding
Values. Biometrics 32 (1): 69–83. DOI:10.2307/2529339
2. Gelfand A, Smith AFM (1990) Sampling based approaches to
calculating marginal densities. J. Anim. Stat. Assoc. 85: 398-409.
3. Sorensen DA, Gianola D (2002) Likelihood, Bayesian and MCMC
Methods in Quantitative Genetics. Springer-Verlag, New York, pp. 588595.
5