Comparison Between Linear and Non-parametric Regression Models for Genome-Enabled Prediction in Wheat

Abstract

In genome-enabled prediction, parametric, semi-parametric, and non-parametric regression models have been used. This study assessed the predictive ability of linear and non-linear models using dense molecular markers. The linear models were linear on marker effects and included the Bayesian LASSO, Bayesian ridge regression, Bayes A, and Bayes B. The non-linear models (this refers to non-linearity on markers) were reproducing kernel Hilbert space (RKHS) regression, Bayesian regularized neural networks (BRNN), and radial basis function neural networks (RBFNN). These statistical models were compared using 306 elite wheat lines from CIMMYT genotyped with 1717 diversity array technology (DArT) markers and two traits, days to heading (DTH) and grain yield (GY), measured in each of 12 environments. It was found that the three non-linear models had better overall prediction accuracy than the linear regression specification. Results showed a consistent superiority of RKHS and RBFNN over the Bayesian LASSO, Bayesian ridge regression, Bayes A, and Bayes B models.

Full text

INVESTIGATION
Comparison Between Linear and Non-parametric
Regression Models for Genome-Enabled Prediction
in Wheat
Paulino Pérez-Rodríguez,*
,
1
Daniel Gianola,
†
Juan Manuel González-Camacho,* José Crossa,
‡
Yann Manès,
‡
and Susanne Dreisigacker
‡
*Colegio de Postgraduados, Montecillo, Texcoco 56230, México, †Departments of Animal Sciences, Dairy Science,
and Biostatistics and Medical Informatics, University of Wisconsin-Madison, Madison, Wisconsin 53706, and ‡Biometrics
and Statistics Unit and Global Wheat Program, International Maize and Wheat Improvement Center (CIMMYT), 06600
Mexico, D.F., México
ABSTRACT In genome-enabled prediction, parametric, semi-parametric, and non-parametric regression
models have been used. This study assessed the predictive ability of linear and non-linear models using
dense molecular markers. The linear models were linear on marker effects and included the Bayesian
LASSO, Bayesian ridge regression, Bayes A, and Bayes B. The non-linear models (this refers to non-linearity
on markers) were reproducing kernel Hilbert space (RKHS) regression, Bayesian regularized neural networks
(BRNN), and radial basis function neural networks (RBFNN). These statistical models were compared using
306 elite wheat lines from CIMMYT genotyped with 1717 diversity array technology (DArT) markers and two
traits, days to heading (DTH) and grain yield (GY), measured in each of 12 environments. It was found that
the three non-linear models had better overall prediction accuracy than the linear regression specification.
Results showed a consistent superiority of RKHS and RBFNN over the Bayesian LASSO, Bayesian ridge
regression, Bayes A, and Bayes B models.
KEYWORDS
GenPred
Shared data
resources
Genome-enabled prediction of complex traits based on marker data
are becoming important in plant and animal breeding, personalized
medicine, and evolutionary biology (Meuwissen et al. 2001; Bernardo
and Yu 2007; de los Campos et al. 2009, 2010; Crossa et al. 2010, 2011;
Ober et al. 2012). In the standard, infinitesimal, pedigree-based model
of quantitative genetics, the family structure of a population is re-
flected in some expected resemblance between relatives. The latter is
measured as an expected covariance matrix among individuals and is
used to predict genetic values (e.g. Crossa et al. 2006; Burgueño et al.
2007, 2011). Whereas pedigree-based models do not account for Men-
delian segregation and the expected covariance matrix is constructed
using assumptions that do not hold (e.g. absence of selection and
mutation and random mating), the marker-based models allow trac-
ing Mendelian segregation at several positions of the genome and
observing realized (as opposed to expected) covariances. This enhan-
ces the potential for improving the accuracy of estimates of genetic
values, thus increasing the genetic progress attainable when these
predictions are used for selection purposes in lieu of pedigree-based
predictions. Recently, de los Campos et al. (2009, 2010) and Crossa
et al. (2010, 2011) used Bayesian estimates from genomic parametric
and semi-parametric regressions, and they found that models that
incorporate pedigree and markers simultaneously had better predic-
tion accuracy for several traits in wheat and maize than models based
only on pedigree or only on markers.
The standard linear genetic model represents the phenotypic
response of the i
th
individual (yi) as the sum of a genetic value, gi,and
of a model residual, ei, such that the linear model for nindividuals
ði¼1; :::; nÞis represented as yi¼giþei. However, building predic-
tive models for complex traits using a large number of molecular markers
(p) with a set of lines comprising individuals (n)withpnis
challenging because individual marker effects are not likelihood-
identified. In this case, marker effects can be estimated via penal-
ized parametric or semi-parametric methods or their Bayesian
counterparts, rather than via ordinary least squares. This reduces
Copyright © 2012 Pérez-Rodríguez et al.
doi: 10.1534/g3.112.003665
Manuscript received July 9, 2012; accepted for publication October 5, 2012
This is an open-access article distributed under the terms of the Creative
Commons Attribution Unported License (http://creativecommons.org/licenses/
by/3.0/), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Supporting information is available online at http://www.g3journal.org/lookup/
suppl/doi:10.1534/g3.112.003665/-/DC1
1
Corresponding author: Colegio de Postgraduados, Montecillo, Texcoco 56230,
México. E-mail: perpdgo@gmail.com
Volume 2 | December 2012 | 1595

the mean-squared error of estimates; it also increases prediction
accuracy of out-of-sample cases and prevents over-fitting (de los
Campos et al. 2010). In addition to the well-known Bayes A and B
linear regression models originally proposed by Meuwissen et al.
(2001) for incorporating marker effects into gi, there are several pe-
nalized parametric regression methods for estimating marker effects,
such as ridge regression, the least absolute shrinkage and selection
operator (LASSO), and the elastic net (Hastie et al. 2009). The Bayes-
ian counterparts of these models have proved to be useful because
appropriate priors can be assigned to the regularization parameter(s),
and uncertainty in the estimations and predictions can be measured
directly by applying the Bayesian paradigm.
Regression methods assume a linear relationship between pheno-
type and genotype, and they typically account for additive allelic
effects only; however, evidence of epistatic effects on plant traits is vast
and well documented (e.g. Holland 2001, 2008). In wheat, for instance,
detailed analyses have revealed a complex circuitry of epistatic inter-
actions in the regulation of heading time involving different vernali-
zation genes, day-length sensitivity genes, and earliness per se genes, as
well as the environment (Laurie et al. 1995; Cockram et al. 2007).
Epistatic effects have also been found to be an important component
of the genetic basis of plant height and bread-making quality traits
(Zhang et al. 2008; Conti et al. 2011). It is becoming common to study
gene ·gene interactions by using a paradigm of networks that
includes aggregating gene ·gene interaction that exists even in the
absence of main effects (McKinney and Pajewski 2012). Interactions
between alleles at two or more loci could theoretically be represented
in a linear model via use of appropriate contrasts. However, this does
not scale when the number of markers (p) is large, as the number of
2-locus, 3-locus, etc., interactions is mind boggling.
An alternative approach to the standard parametric modeling of
complex interactions is provided by non-linear, semi-parametric
methods, such as kernel-based models (e.g. Gianola et al. 2006;
Gianola and van Kaam 2008) or artificial neural networks (NN) (Okut
et al. 2011; Gianola et al. 2011), under the assumption that such
procedures can capture signals from high-order interactions. The po-
tential of these methods, however, depends on the kernel chosen and
on the neural network architecture. In a recent study, Heslot et al.
(2012) compared the predictive accuracy of several genome-enabled
prediction models, including reproducing kernel Hilbert space
(RKHS) and NN, using barley and wheat data; the authors found
that the non-linear models gave a modest but consistent predictive
superiority (as measured by correlations between predictions and
realizations) over the linear models. In particular, the RKHS model
had a better predictive ability than that obtained using the para-
metric regressions.
The use of RKHS for predicting complex traits was first proposed
by Gianola et al. (2006) and Gianola and van Kaam (2008). de los
Campos et al. (2010) further developed the theoretical basis of RHKS
with “kernel averaging”(simultaneous use of various kernels in the
model) and showed its good prediction accuracy. Other empirical
studies in plants have corroborated the increase in prediction accuracy
of kernel methods (e.g. Crossa et al. 2010, 2011; de los Campos et al.
2010; Heslot et al. 2012). Recently, Long et al. (2010), using chicken
data, and González-Camacho et al. (2012), using maize data, showed
that NN methods provided prediction accuracy comparable to that
obtained using the RKHS method. In NN, the bases functions (adap-
tive “covariates”) are inferred from the data, which gives the NN great
potential and flexibility for capturing complex interactions between
input variables (Hastie et al. 2009). In particular, Bayesian regularized
neural networks (BRNN) and radial basis function neural networks
(RBFNN) have features that make them attractive for use in genomic
selection (GS).
In this study, we examined the predictive ability of various linear
and non-linear models, including the Bayes A and B linear regression
models of Meuwissen et al. (2001); the Bayesian LASSO, as in Park
and Casella (2008) and de los Campos et al. (2009); RKHS, using the
“kernel averaging”strategy proposed by de los Campos et al. (2010);
the RBFNN, proposed and used by González-Camacho et al. (2012);
and the BRNN, as described by Neal (1996) and used in the context of
GS by Gianola et al. (2011). The predictive ability of these models was
compared using a cross-validation scheme applied to a wheat data set
from CIMMYT’s Global Wheat Program.
MATERIALS AND METHODS
Experimental data
The data set included 306 elite wheat lines, 263 lines that are
candidates for the 29
th
Semi-Arid Wheat Screening Nursery (SAWSN),
and 43 lines from the 18
th
Semi-Arid Wheat Yield Trial (SAWYT) from
CIMMYT’s Global Wheat Program. These lines were genotyped
with 1717 diversity array technology (DArT) markers generated
by Triticarte Pty. Ltd. (Canberra, Australia; http://www.triticarte.
com.au). Two traits were analyzed: grain yield (GY) and days to
heading (DTH) (see Supporting Information, File S1).
The traits were measured in a total of 12 different environments
(1–12) (Table 1): GY in environments 1–7 and DTH in environments
1–5 and 8–12 (10 in all). Different agronomic practices were used.
Yield trials were planted in 2009 and 2010 using prepared beds and
flat plots under controlled drought or irrigated conditions. Yield data
from experiments in 2010 were replicated, whereas data from trials
in 2009 were adjusted means from an alpha lattice incomplete
block design with adjustment for spatial variability in the direction
of rows and columns using the autoregressive model fitted in both
directions.
Data used to train the models for GY and DTH in 2009 were the
best linear unbiased estimator (BLUE) after spatial analysis, whereas
the BLUE data for 2010 were obtained after performing analyses in
each of the 12 environments and combined. The experimental designs
in each location consisted of alpha lattice incomplete block designs of
different sizes, with two replicates each.
Broad-sense heritability at individual environments was calculated
as h2¼s2
g=ðs2
gþs2
e
nrepsÞ,wheres2
gand s2
eare the genotype and error var-
iance components, respectively, and nreps is the number of replicates.
For the combined analyses across environments, broad-sense herita-
bility was calculated as h2¼s2
g=ðs2
gþs2
ge
nenv þs2
e
nenv ·nrepsÞÞ, where the term s2
ge is
the genotype ·environment interaction variance component, and
nenv is the number of environments included in the analysis.
Statistical models
One method for incorporating markers is to define gias a para-
metric linear regression on marker covariates xij with form
gi¼P
p
j¼1
xijbj, such that yi¼P
p
j¼1
xijbjþei(j= 1,2,...,pmarkers); here, bjis
the partial regression of yion the j
th
marker covariate (Meuwissen
et al. 2001). Extending the model to allow for an intercept
yi¼mþX
p
j¼1
xijbjþei(1)
We adopted Gaussian assumptions for model residuals; specifi-
cally, the joint distribution of model residuals in Equation 1 was
1596 | P. Pérez-Rodríguez et al.

assumed normal with mean zero and variance s2
e. The likelihood
function is
pyjm;g;s2
e¼Y
n
i¼1
N0
@yimþX
p
j¼1
xij bj;s2
e1
A(2)
where NyijmþP
p
j¼1
xijbj;s2
eis a normal density for random variable
yicentered at mþP
p
j¼1
xijbjand with variance s2
e.Dependingon
how priors on the marker effects are assigned, different Bayesian
linear regression models result.
Linear models: Bayesian ridge regression, Bayesian
LASSO, Bayes A, and Bayes B
A standard penalized regression method is ridge regression (Hoerl
and Kennard 1970); its Bayesian counterpart, Bayesian ridge regres-
sion (BRR), uses a prior density of marker effects, pðbjjvÞ,thatis,
Gaussian, centered at zero and with variance common to all the
markers, that is, pðbjjs2
bÞ¼Nðbjj0;s2
bÞ,wheres2
bis a prior-variance
of marker effects. Marker effects are assumed independent and iden-
tically distributed apriori. We assigned scaled inverse chi distributions
x22ðdf:;s:Þto the variance parameters s2
eand s2
b. The prior degrees
of freedom parameters were set to df:¼4ands:¼1. It can be shown
that the posterior mean of marker effects is the best linear unbiased
predictor (BLUP) of marker effects, so Bayesian ridge regression is
often referred to as RR-BLUP (de los Campos et al. 2012).
The Bayesian LASSO, Bayes A, and Bayes B relax the assumption
of common prior variance to all marker effects. The relationship
among these three models is as follows: Bayes B can be considered as
the most general of the three, in the sense that Bayes A and Bayesian
ridge regression can be viewed as special cases of Bayes B. This is
because Bayes A is obtained from Bayes B by setting p¼0(the
proportion of markers with null effects), and Bayesian ridge regression
is obtained from Bayes B by setting p¼0 and assuming that all the
markers have the same variance.
Bayes B uses a mixture distribution with a mass at zero, such that
the (conditional) prior distribution of marker effects is given by
bjs2
j;p¼0 with probability p
Nð0;s2
jÞwith probability 1-p(3)
The prior assigned to s2
j; j¼1; ::::; pis the same for all markers,
i.e. a scaled inverted chi squared distribution x22ðdfb;sbÞ,wheredfb
are the degrees of freedom and sbis a scaling parameter. Bayes B
becomes Bayes A by setting p=0.
In the case of Bayes B, we took p¼0:95;dfb¼4, and
sb¼~s2
aðdfb22Þ=dfbwith ~s2
a¼~s2
S=hð12pÞP
p
j¼1
2qjð12qjÞi,where
qjis the allele frequency for marker jand ~s2
Sis the additive genetic
variance explained by markers [see Habier et al. (2011) and Resende
et al. (2012) for more details]. In the case of s2
e,weassignedaflat prior
as in Wang et al. (1994).
The Bayesian LASSO assigns a double exponential (DE) distribu-
tion to all marker effects (conditionally on a regularization parameter
l), centered at zero and with marker-specific variance, that is,
pðbjjl;seÞ¼DEbjj0;l
s2
e. The DE distribution does not conjugate
with the Gaussian likelihood, but it can be represented as a mixture of
scaled normal densities, which allows easy implementation of the model
(Park and Casella 2008; de los Campos et al. 2009). The priors used
were exactly the same as those used in González-Camacho et al. (2012).
The models used in this study, the Bayesian ridge regression,
Bayesian LASSO (BL), Bayes A, and Bayes B, are explained in detail
in several articles; for example, Bayes A and Bayes B are described
in Meuwissen et al. (2001), Habier et al. (2011), and Resende et al.
(2012), and an account of BL is given in de los Campos et al. (2009,
2012), Crossa et al. (2010, 2011), Perez et al. (2010), and González-
Camacho et al. (2012).
Non-linear models: RBFNN, BRNN, and RKHS
In this section, we describe the basic structure of the non-linear single
hidden layer feed-forward neural network (SLNN) with two of its
variants, the radial basis function neural network and the Bayesian
regularized neural network. We also give a brief explanation of RKHS
with the averaging kernel method at the end of this section.
Single hidden layer feed-forward neural network: In a single-layer
feed-forward (SLNN), the non-linear activation functions in the
hidden layer enable a NN to have universal approximation ability,
giving it great potential and flexibility in terms of capturing complex
patterns. The structure of the SLNN is depicted in Figure 1, which
illustrates the structure of the method for a phenotypic continuous
response. This NN can be thought of as a two-step regression (e.g.
Hastie et al. 2009). In the first step, in the non-linear hidden layer, S
data-derived basis functions (k=1,2,...,Sneurons), fz½k
ig,are
inferred, and in the second step, in the linear output layer, the re-
sponse is regressed on the basis functions (inferred in the hidden
layer). The inner product between the input vector and the weight
nTable 1 Twelve environments representing combinations of diverse agronomic management (drought or full irrigation, sowing in
standard, bed, or flat systems), sites in Mexico, and years for two traits, grain yield (GY) and days to heading (DTH), with their broad-
sense heritability (h
2
) measured in 2010
Environment Code Agronomic Management Site in Mexico Year Trait Measured h
2
(GY) h
2
(DTH)
1 Drought-bed Cd. Obregon 2009 GY, DTH ——
2 Drought-bed Cd. Obregon 2010 GY, DTH 0.833 0.991
3 Drought-flat Cd. Obregon 2010 GY, DTH 0.465 0.984
4 Full irrigation-bed Cd. Obregon 2009 GY, DTH ——
5 Full irrigation-bed Cd. Obregon 2010 GY, DTH 0.832 0.086
6 Heat-bed Cd. Obregon 2010 GY 0.876 —
7 Full irrigation-flat melga Cd. Obregon 2010 GY 0.876 —
8 Standard Toluca 2009 DTH ——
9 Standard El Batan 2009 DTH ——
10 Small observation plot Cd. Obregon 2009 DTH ——
11 Small observation plot Cd. Obregon 2010 DTH —0.950
12 Standard Agua Fria 2010 DTH —0.990
Volume 2 December 2012 | Linear and Non-parametric Regression Models for GS | 1597

vector (b½k) of each neuron of the hidden layer, plus a bias (intercept
bk), is performed, that is, u½k
i¼bkþP
p
j¼1
xij b½k
j;(j=1,...,pmarkers);
this is then transformed using a non-linear activation function
gkðu½k
iÞ. One obtains z½k
i¼gkbkþP
p
j¼1
xij b½k
j,wherebkis an in-
tercept and (b
1[1]
,...,b
p[1]
;...,b
1[S]
,...,b
p[S]
)9is a vector of re-
gression coefficients or “weights”of each neuron kin the hidden
layer. The gkð:Þis the activation function, which maps the inputs
into the real line in the closed interval [21,1]; for example,
gkðxÞ¼expð2xÞ21
expð2xÞþ1is known as the tangent hyperbolic function.
Finally, in the linear output layer, phenotypes are regressed on the
data-derived features, fz½k
ig, according to
yi¼mþX
S
k¼1
wkz½k
iþei¼mþX
S
k¼1
wkgk0
@bkþX
p
j¼1
xij b½k
j1
Aþei:
(4)
Radial basis function neural network: The RBFNN was first
proposed by Broomhead and Lowe (1988) and Poggio and Girosi
(1990). Figure 2 shows the architecture of a single hidden layer
RBFNN with Snon-linear neurons. Each non-linear neuron in the
hidden layer has a Gaussian radial basis function (RBF) defined as
z½k
i¼exp½2hkkxi2ckk2,where kxi2ckkis the Euclidean norm
between the input vector xiand the center vector ckand hkis the
bandwidth of the Gaussian RBF. Subsequently, in the linear output layer,
phenotypes are regressed on the data-derived features, fz½k
ig,accord-
ing to yi¼mþP
S
k¼1
wkz½k
iþei,whereeiis a model residual.
Estimating the parameters of the RBFNN: The vector of
weights v¼fw1; :::; wSgof the linear output layer is obtained using
the ordinary least-squares fit that minimizes the mean squared differ-
ences between the ^
yi (from RBFNN) and the observed responses yi in
the training set, provided that the Gaussian RBFs for centers ckand hk
of the hidden layer are defined. The centers are selected using an
orthogonalization least-squares learning algorithm, as described by
Chen et al. (1991) and implemented in Matlab 2010b. The centers
are added iteratively such that each new selected center is orthogonal
to the others. The selected centers maximize the decrease in the mean-
squared error of the RBFNN, and the algorithm stops when the
number of centers (neurons) added to the RBFNN attains a desired
precision (goal error) or when the number of centers is equal to the
number of input vectors, that is, when S=n. The bandwidth hkof the
Gaussian RBF is defined in terms of a design parameter of the net
spread,thatis,hk¼0:8326
spread2
for each Gaussian RBF of the hidden
layer. To select the best RBFNN, a grid for training the net was gener-
ated, containing different values of spread and different precision values
(goal error). The initial value of the spread was the median of the
Euclidean distances between each pair of input vectors (xi), and an initial
value of 0.02 for the goal error was considered. The parameter spread
allows adjusting the form of the Gaussian RBF such that it is sufficiently
large to respond to overlapping regions of the input space but not so big
that it might induce the Gaussian RBF to have a similar response.
Bayesian regularized neural networks: The difference between
SLNN and BRNN is in the function to be minimized (see the
penalized function below); therefore, the basic structure of a BRNN
can be represented in Figure 1 as well. The SLNN described above is
flexible enough to approximate any non-linear function; this great
flexibility allows NN to capture complex interactions among predictor
Figure 1 Structure of a single-layer feed-
forward neural network (SLNN) adapted from
González-Camacho et al. (2012). In the hid-
den layer, input variables xi=ðxi1; :::; xipÞ
(j=1,...,p markers) are combined for each
neuron (k=1,...,Sneurons) using a linear func-
tion, u½k
i¼bkþP
p
j¼1
xij b½k
j, and subsequently trans-
formed using a non-linear activation function,
yielding a set of inferred scores, z½k
i¼gkðu½k
iÞ.
These scores are used in the output layer as
basis functions to regress the response using
the linear activation function on the data-
derived predictors yi¼mþP
S
k¼1
wkz½k
iþei.
1598 | P. Pérez-Rodríguez et al.

variables (Hastie et al. 2009). However, this flexibility also leads to two
important issues: (1) as the number of neurons increases, the number
of parameters to be estimated also increases; and (2) as the number of
parameters rises, the risk of over-fitting also increases. It is common
practice to use penalized methods via Bayesian methods to prevent or
palliate over-fitting.
MacKay (1992, 1994) developed a framework for obtaining esti-
mates of all the parameters in a feed-forward single neural network
by using an empirical Bayes approach. Let u¼(w
1
,...,w
S
;b
1
,...,b
S
;
b
1[1]
,...,b
p[1]
;...,b
1[S]
,...,b
p[S]
,m)9be the vector containing all the
weights, biases, and connection strengths. The author showed that the
estimation problem can be solved in two steps, followed by iteration:
(1) Obtain the conditional posterior modes of the elements in u
assuming that the variance components s2
eand s2
uare known
and that the prior distribution for the all the elements in uis given
by pðujs2
uÞ¼MNð0;s2
uIÞ. It is important to note that this ap-
proach assigns the same prior to all elements of u, even though this
may not always be the best thing to do. The density of the condi-
tional (given the variance parameters) posterior distribution of the
elements of u,accordingtoBayes’theorem, is given by
pðuy;s2
e;s2
uÞ¼ pðyu;s2
eÞpðus2
uÞ
pðys2
e;s2
uÞ(5)
The conditional modes can be obtained by maximizing Equation 5
over u. However, the problem is equivalent to minimizing the following
penalized sum of squares [see Gianola et al. (2011) for more details]
FðuÞ¼bX
n
i¼1
e2
iþaX
m
j¼1
u2
j
where b¼1=ð2s2
eÞ,a¼1=ð2s2
uÞ,eiis the difference between ob-
served and predicted phenotypes for the fitted model, and uj
(j¼1; :::; m) is the j
th
element of vector u.
(2) Update s2
eand s2
u. The updating formulas are obtained by max-
imizing an approximation to the marginal likelihood of the data
pðyjs2
e;s2
uÞ(the “evidence”) given by the denominator of Equa-
tion 5.
(3) Iterate between (1) and (2) until convergence.
The original algorithm developed by MacKay was further im-
proved by Foresee and Hagan (1997) and adopted by Gianola et al.
(2011) in the context of genome and pedigree-enabled prediction. The
algorithm is equivalent to estimation via maximum penalized likeli-
hood estimation when “weight decay”is used, but it has the advan-
tage of providing a way of setting the extent of “weight decay”
through the variance component s2
u. Neal (1996) pointed out that
the procedure of MacKay (1992, 1994) can be further generalized.
For example, there is no need to approximate probabilities via
Gaussian assumptions; furthermore, it is possible to estimate the
entire posterior distributions of all the elements in u, not only their
(conditional) posterior modes. Next, we briefly review Neal’s ap-
proach to solving the problem; a comprehensive revision can be
found in Lampinen and Vehtari (2001).
Prior distributions:
a) Variance component of the residuals: Neal (1996) used a
conjugate inverse Gamma distribution as a prior for the variance
associated with the residual, ei, given in Equation 4, that is, s2
e
Inv-Gammaðse;dfeÞ,whereseand dfeare the scale and degrees of
freedom parameters, respectively. These parameters can be set to
the default values given by Neal (1996), se=0.05, dfe=0.5. These values
were also used by Lampinen and Vehtari (2001).
b) Connection strengths, weights, and biases: Neal (1996) sug-
gested dividing the network parameters in uinto groups and then
using hierarchical models for each group of parameters; for example,
connection strengths (b
1[1]
,...,b
p[1]
;...;b
1[S]
,...,b
p[S]
), biases
(b
1
,...,b
S
) of the hidden layer, and output weights (w
1
,...,w
S
), and
general mean or bias (m) of the linear output layer. Suppose that
u
1
,...,u
k
are parameters of a given group; then assume
Figure 2 Structure of a radial basis function
neural network adapted from González-
Camacho et al. (2012). In the hidden layer,
information from input variables ðxi1; :::; xipÞ
(j=1,...,p markers) is first summarized by
means of the Euclidean distance between
each of the input vectors fxigwith respect
to S(data-inferred) (k=1,...,Sneurons) centers
fckg, that is, u½k
i¼hkjjxi2ckjj2. These dis-
tances are then transformed using the
Gaussian function z½k
i¼expð2u½k
iÞ. These
scores are used in the output layer as basis
functions for the linear regression
yi¼mþP
S
k¼1
wkz½k
iþei.
Volume 2 December 2012 | Linear and Non-parametric Regression Models for GS | 1599

pðu1; :::; uks2
uÞ¼ð2pÞ2k=2sk
uexp(21
2s2
uX
S
k¼1
u2
k)
And, at the last stage of the model, assign the prior s2
u
Inv-Gammaðsu;dfuÞ. The scale parameter of the distribution asso-
ciated with the group of parameters containing the connection
strengths (b
1[1]
,...,b
p[1]
;...;b
1[S]
,...,b
p[S]
) changes according
to the number of inputs, in this case, su¼ð0:05=p1=dfuÞ2with
dfu¼0:5andpis the number of markers in the data set.
By using Markov chain Monte Carlo (MCMC) techniques through
an algorithm called hybrid Monte Carlo, Neal (1996) developed a soft-
ware termed flexible Bayesian modeling (FBM) capable of obtaining
samples from the posterior distributions of all unknowns in a neural
network (as in Figure 1).
Reproducing kernel Hilbert spaces regression: RKHS models have
been suggested as an alternative to multiple linear regression for
capturing complex interaction patterns that may be difficult to
account for in a linear model framework (Gianola et al. 2006). In
RKHS model, the regression function takes the form
fðxiÞ¼mþX
n
i9¼1
ai9Kðxi;xi9Þ(6)
where xi¼ðxi1; :::; xip Þ
9and xi9¼ðxi91; :::; xi9pÞ
9are input vectors of
marker genotypes in individuals iand i9;ai9are regression coeffi-
cients; and Kðxi;xi9Þ¼expð2hkxi2xi9k2Þis the reproducing kernel
defined (here) with a Gaussian RBF, where his a bandwidth param-
eter and kxi2xi9kis the Euclidean norm between each pair of input
vectors. The strategy termed “kernel averaging”for selecting optimal
values of hwithin a set of candidate values was implemented using
the Bayesian approach described in de los Campos et al. (2010).
Similarities and connections between the RKHS and the RBFNN
are given in González-Camacho et al. (2012).
Assessment of the models’predictive ability
The predictive ability of the models given above was compared using
Pearson’s correlation and predictive mean-squared error (PMSE)
using predicted and realized values. A total of 50 random partitions
were generated for each of the data sets, and each partition randomly
assigned 90% of the lines to the training set and the remaining 10% to
the validation set. The partition scheme used was similar to that in
Gianola et al. (2011) and González-Camacho et al. (2012).
All scripts were run in a Linux work station; for Bayesian ridge
regression and Bayesian LASSO, we used the R package BLR (de los
Campos and Perez 2010), whereas for RKHS, we used the R imple-
mentation described in de los Campos et al. (2010), which was kindly
provided by the authors. In the case of Bayes A and Bayes B, we used
a program described by Hickey and Tier (2009), which is freely
available at http://sites.google.com/site/hickeyjohn/alphabayes.For
the BRNN, we used the FMB software available at http://www.cs.
toronto.edu/~radford/fbm.software.html. Because the computational
time required to evaluate the predictive ability of the BRNN network
was great, we used the Condor high throughput computing system at
the University of Wisconsin-Madison (http://research.cs.wisc.edu/
condor). The RBFNN model was run using Matlab 2010b for Linux.
The differences in computing times between the models were great.
The computing times for evaluating the prediction ability of the 50
partitions for each trait were as follows, 10 min for RBFNN, 1.5 hr for
RKHS, 3 hr for BRR, 3.5 hr for BL, 4.5 hr for Bayes B, 5.5 hr for Bayes
A, and 30 days for BRNN. In the case of RKHS, BRR, BL, Bayes A,
and Bayes B, inferences were based on 35,000 MCMC samples, and on
10,000 samples for BRNN. The estimated computing times were
obtained using, as reference, a single Intel Xeon CPU 5330 2.4 GHz
and 8 Gb of RAM memory. Significant reduction in computing time
was achieved by parallelizing the tasks.
RESULTS
Data from replicated experiments in 2010 were used to calculate the
broad-sense heritability for each trait in each environment (Table 1).
Broad-sense heritability across locations for 2010 data were 0.67 for
GY and 0.92 for DTH. These high estimates can be explained, at least
in part, by the strict environmental control of trials conducted at
CIMMYT’s experiment station at Ciudad Obregon. The heritability
of the two traits for 2009 was not estimated because the only available
phenotypic data were adjusted means for each environment.
Predictive assessment of the models
The predictive ability of the different models for GY and DTH varied
among the 12 environments. The model deemed best using correla-
tions (Table 2) tended to be the one with the smallest average PMSE
(Table 3). The three non-parametric models had higher predictive
correlations and smaller PMSE than the linear models for both GY
and DTH. Within the linear models, the results are mixed, and all
models gave similar predictions. Within the non-parametric models,
RBFNN and RKHS always gave higher correlations between predicted
values and realized phenotypes, and a smaller average PMSE than the
BRNN. The mean of the correlations and the associated standard
errors can be used to test for statistically significant improvements
in the predictability of the non-linear models vs. the linear models.
The t-test (with a¼0:05) showed that RKHS gave significant
improvements in prediction in 13/19 cases (Table 3) compared with
the BL, whereas RBFNN was significantly better than the BL in 10/19
cases. Similar results were obtained when comparing RKHS and
RBFNN with Bayes A and Bayes B.
Correlations between observed and predicted values for DTH were
lowest overall in environments 4 and 8, in Cd. Obregon, 2009, and in
Toluca, 2009. Average PMSE was in agreement with the findings
based on correlations. Although accuracies in environment 4 were
much lower than in other environments, the higher accuracy of the
non-parametric models (RKHS, RBFNN, and BRNN) over that of the
linear models (BL, BRR, Bayes A, and Bayes B) was consistent with
what was observed in the other environments. Figures 3 and 4 give
scatter plots of the correlations obtained with the three non-parametric
models vs. the BL for DTH and GY, respectively; each circle repre-
sents the estimated correlations for each of the two models included
in the plot. In Figure 3, A–C, DTH had a total of 500 points (10
environments and 50 random training-testing partitions). In Figure 4,
A–C, GY had a total of 350 points (7 environments and 50 random
partitions in each environment). A point above the 45-degree line
represents an analysis where the method whose predictive correlation
is given on the vertical axis (RKHS, RBFNN, BRNN) outperformed
the one whose correlation is given on the horizontal axis (BL). Both
figures show that although there is a great deal of variability due to
partition, for both DTH and GY, the overall superiority of RKHS and
RBFNN over the linear model BL is clear. For both traits, BL had
slightly better prediction accuracy than the BRNN in terms of the
number of individual correlation points. It is interesting to note that
some cross-validation partitions picked subsets of training data that
had negative, zero, or very low correlations with the observed values in
1600 | P. Pérez-Rodríguez et al.

the validation set. These results indicate that lines in the training set
are not necessarily related to those in the validation set.
DISCUSSION AND CONCLUSIONS
Understanding the impact of epistasis on quantitative traits remains
a major challenge. In wheat, several studies have reported significant
epistasis for grain yield and heading or flowering time (Goldringer
et al. 1997). Detailed analyses have shown that vernalization, day-
length sensitivity, and earliness per se genes are mainly responsible
for regulating heading time. The vernalization requirement relates to
the sensitivity of the plant to cold temperatures, which causes it to
accelerate spike primordial formation. Transgenic and mutant analyses,
for example, have suggested a pathway involving epistatic interactions
that combines environment-induced suppression and upregulation of
several genes, leading to final floral transition (Shimada et al. 2009).
There is evidence that the aggregation of multiple gene ·gene
interactions (epistasis) with small effects into small epistatic networks
nTable 2 Average correlation (SE in parentheses) between observed and predicted values for grain yield (GY) and days to heading (DTH)
in 12 environments for seven models
Trait Environment BL BRR Bayes A Bayes B RKHS RBFNN BRNN
1 0.59 (0.11) 0.59 (0.11) 0.59 (0.11) 0.56 (0.11) 0.66 (0.09) 0.66 (0.10) 0.64 (0.11)
2 0.58 (0.14) 0.57 (0.14) 0.61 (0.12) 0.57 (0.13) 0.63 (0.13) 0.61 (0.13) 0.62 (0.13)
3 0.60 (0.13) 0.60 (0.12) 0.62 (0.11) 0.60 (0.12) 0.68 (0.10) 0.69 (0.10) 0.67 (0.11)
4 0.02 (0.18) 0.07 (0.17) 0.06 (0.17) 0.06 (0.17) 0.12 (0.18) 0.16 (0.18) 0.02 (0.19)
DTH 5 0.65 (0.09) 0.64 (0.10) 0.66 (0.09) 0.66 (0.09) 0.69 (0.08) 0.68 (0.08) 0.68 (0.08)
8 0.36 (0.15) 0.37 (0.15) 0.36 (0.15) 0.35 (0.14) 0.46 (0.13) 0.46 (0.14) 0.39 (0.15)
9 0.59 (0.12) 0.59 (0.11) 0.53 (0.12) 0.52 (0.11) 0.62 (0.11) 0.63 (0.11) 0.61 (0.12)
10 0.54 (0.14) 0.52 (0.14) 0.56 (0.13) 0.54 (0.14) 0.61 (0.13) 0.62 (0.12) 0.57 (0.13)
11 0.52 (0.15) 0.52 (0.16) 0.53 (0.13) 0.51 (0.13) 0.58 (0.14) 0.59 (0.13) 0.55 (0.14)
12 0.45 (0.19) 0.42 (0.18) 0.45 (0.18) 0.45 (0.18) 0.47 (0.18) 0.39 (0.19) 0.35 (0.19)
Average 0.59 (0.12) 0.58 (0.12) 0.60 (0.12) 0.57 (0.12) 0.65 (0.10) 0.48 (0.14) 0.48 (0.14)
1 0.48 (0.13) 0.43 (0.14) 0.48 (0.13) 0.46 (0.13) 0.51 (0.12) 0.51 (0.12) 0.50 (0.13)
2 0.48 (0.14) 0.41 (0.17) 0.48 (0.14) 0.48 (0.14) 0.50 (0.14) 0.43 (0.16) 0.43 (0.16)
3 0.20 (0.21) 0.29 (0.22) 0.20 (0.22) 0.18 (0.22) 0.37 (0.20) 0.42 (0.21) 0.32 (0.24)
GY 4 0.45 (0.15) 0.46 (0.13) 0.43 (0.15) 0.42 (0.15) 0.53 (0.12) 0.55 (0.11) 0.49 (0.14)
5 0.59 (0.14) 0.56 (0.16) 0.75 (0.11) 0.74 (0.12) 0.64 (0.13) 0.66 (0.13) 0.63 (0.13)
6 0.70 (0.10) 0.67 (0.11) 0.73 (0.08) 0.71 (0.08) 0.73 (0.08) 0.71 (0.08) 0.69 (0.10)
7 0.46 (0.14) 0.50 (0.14) 0.42 (0.14) 0.40 (0.15) 0.53 (0.13) 0.54 (0.14) 0.50 (0.14)
Average 0.62 (0.10) 0.57 (0.14) 0.69 (0.10) 0.70 (0.09) 0.67 (0.09) 0.56 (0.12) 0.65 (0.10)
Fitted models were Bayesian LASSO (BL), RR-BLUP (BRR), Bayes A, Bayes B, reproducing kernel Hilbert spaces regression (RKHS), radial basis function neural networks
(RBFNN) and Bayesian regularized neural networks (BRNN) across 50 random partitions of the data with 90% in the training set and 10% in the validation set. The
models with highest correlations are underlined.
nTable 3 Predictive mean- squared error (PMSE) between observed and predicted values for grain yield (GY) and days to heading (DTH)
in 12 environments for seven models
Trait Environment BL BRR Bayes A Bayes B RKHS RBFNN BRNN
1 13.02 13.18 12.72 13.23 11.02 10.85 11.52
2 11.89 12.37 10.65 11.28 10.19 10.72 10.44
3 8.18 8.44 7.31 7.59 6.29 6.25 6.63
4 21.59 22.27 21.79 21.67 21.14 22.64 21.49
DTH 5 8.86 9.23 8.48 8.37 7.95 8.02 8.21
8 14.72 15.22 14.54 14.58 13.12 13.19 14.81
9 21.38 21.44 23.71 23.93 20.50 19.84 20.62
10 7.72 8.51 7.27 7.57 6.66 6.51 7.36
11 6.83 7.12 6.59 6.74 6.03 5.96 6.51
12 13.60 14.42 13.56 13.46 13.25 14.86 15.75
Average 6.09 6.47 5.99 6.28 5.31 9.12 9.25
1 0.07 0.09 0.07 0.07 0.07 0.07 0.07
2 0.06 0.08 0.06 0.06 0.06 0.07 0.07
3 0.06 0.07 0.06 0.06 0.05 0.05 0.05
GY 4 0.22 0.24 0.23 0.23 0.20 0.19 0.21
5 0.39 0.44 0.26 0.27 0.35 0.33 0.36
6 0.13 0.15 0.12 0.13 0.12 0.13 0.13
7 0.40 0.41 0.43 0.44 0.38 0.37 0.39
Average 0.06 0.07 0.05 0.05 0.05 0.07 0.06
Fitted models were Bayesian LASSO (BL), RR-BLUP (BRR), Bayes A, Bayes B, reproducing kernel Hilbert space regression (RKHS), radial basis function neural networks
(RBFNN) and Bayesian regularized neural networks (BRNN) across 50 random partitions of the data with 90% in the training set and 10% in the validation set. The
models with lowest PMSE are underlined.
Volume 2 December 2012 | Linear and Non-parametric Regression Models for GS | 1601

is important for explaining the heritability of complex traits in ge-
nome-wide association studies (McKinney and Pajewski 2012). Epi-
static networks and gene ·gene interactions can also be exploited for
GS via suitable statistical-genetic models that incorporate network
complexities. Evidence from this study, as well as from other research
involving other plant and animal species, suggests that models that are
non-linear in input variables (e.g. SNPs) predict outcomes in testing
sets better than standard linear regression models for genome-enabled
prediction. However, it should be pointed out that better predictive
ability can have several causes, one of them the ability of some non-
linear models to capture epistatic effects. Furthermore, the random
cross-validation scheme used in this study was not designed to
Figure 3 Plots of the predictive correlation for each of 50 cross-validation partitions and 10 environments for days to heading (DTH) in different
combinations of models. (A) When the best non-parametric model is RKHS, this is represented by an open circle; when the best linear model is BL,
this is represented by a filled circle. (B) When the best non-parametric model is RBFNN, this is represented by an open circle; when the best linear
model is BL, this is represented by a filled circle. (C) When the best non-parametric model is BRNN, this is represented by an open circle; when the
best linear model is BL, this is represented by a filled circle. The histograms depict the distribution of the correlations in the testing set obtained
from the 50 partitions for different models. The horizontal (vertical) dashed line represents the average of the correlations for the testing set in the
50 partitions for the model shown on the Y (X) axis. The solid line represents Y = X; i.e. both models have the same prediction ability.
1602 | P. Pérez-Rodríguez et al.

specifically assess epistasis but rather to compare the models’predic-
tive ability.
It is interesting to compare results from different predictive
machineries when applied to either maize or wheat. Differences in
the prediction accuracy of non-parametric and linear models (at least
for the data sets included in this and other studies) seem to be more
pronounced in wheat than in maize. Although differences depend,
among other factors, on the trait-environment combination and the
number of markers, it is clear from González-Camacho et al. (2012)
that for flowering traits (highly additive) and traits such as grain yield
(additive and epistatic) in maize, the BL model performed very sim-
ilarly to the RKHS and RBFNN. On the other hand, in the present
study, which involves wheat, the RKHS, RBFNN, and BRNN models
clearly had a markedly better predictive accuracy than BL, BRR, Bayes
A, or Bayes B. This may be due to the fact that, in wheat, additive ·
additive epistasis plays an important role in grain yield, as found by
Figure 4 Plot of the correlation for each of 50 cross-validation partitions and seven environments for grain yield (GY) in different combinations of
models. (A) When the best model is RKHS, this is represented by an open circle; when the best model is BL, this is represented by a filled circle. (B)
When best model is RBFNN, this is represented by an open circle; when the best model is BL, this is represented by a filled circle. (C) When the
best model is BRNN, this is represented by an open circle; when the best model is BL, this is represented by a filled circle. The histograms depict
the distribution of the correlations in the testing set obtained from the 50 partitions for different models. The horizontal (vertical) dashed line
represents the average of the correlations for the testing set in the 50 partitions for the model shown on the Y (X) axis. The solid line represents Y =
X; i.e. both models have the same prediction ability.
Volume 2 December 2012 | Linear and Non-parametric Regression Models for GS | 1603

Crossa et al. (2006) and Burgueño et al. (2007, 2011) when assessing
additive, additive ·additive, additive ·environment, and additive ·
additive ·environment interactions using a pedigree-based model
with the relationship matrix A.
As pointed out first by Gianola et al. (2006) and subsequently by
Long et al. (2010), non-parametric models do not impose strong
assumptions on the phenotype-genotype relationship, and they have
the potential of capturing interactions among loci. Our results with
real wheat data sets agreed with previous findings in animal and plant
breeding and with simulated experiments, in that a non-parametric
treatment of markers may account for epistatic effects that are not
captured by linear additive regression models. Using extensive maize
data sets, González-Camacho et al. (2012) found that RBFNN and
RKHS had some similarities and seemed to be useful for predicting
quantitative traits with different complex underlying gene action un-
der varying types of interaction in different environmental conditions.
These authors suggested that it is possible to make further improve-
ments in the accuracy of the RKHS and RBFNN models by introduc-
ing differential weights in SNPs, as shown by Long et al. (2010) for
RBFs.
The training population used here was not developed specifically
for this study; it was made up of a set of elite lines from the CIMMYT
rain-fed spring wheat breeding program. Our results show that it
is possible to achieve good predictions of line performance by com-
bining phenotypic and genotypic data generated on elite lines. As
genotyping costs decrease, breeding programs could make use of
genome-enabled prediction models to predict the values of new
breeding lines generated from crosses between elite lines in the
training set before they reach the yield testing stage. Lines with the
highest estimated breeding values could be intercrossed before being
phenotyped. Such a “rapid cycling”scheme would accelerate the fix-
ation rate of favorable alleles in elite materials and should increase the
genetic gain per unit of time, as described by Heffner et al. (2009).
It is important to point out that proof-of-concept experiments are
required before genome-enabled selection can be implemented
successfully in plant breeding programs. It is necessary to test
genomic predictions on breeding materials derived from crosses
between lines of the training population. If predictions are reliable
enough, an experiment using the same set of parental materials
could be carried out to compare the field performance of lines coming
from a genomic-assisted recurrent selection program scheme vs. lines
coming from a conventional breeding scheme. The accuracies
reported in this study represent prediction of wheat lines using a train-
ing set comprising lines with some degree of relatedness to lines in the
validation set. When the validation and the training sets are not
genetically related (unrelated families) or represent populations with
different genetic structures and different linkage disequilibrium pat-
terns, then negligible accuracies are to be expected. It seems that
successful application of genomic selection in plant breeding requires
some genetic relatedness between individuals in the training and val-
idation sets, and that linkage disequilibrium information per se does
not suffice (e.g. Makowsky et al. 2011).
ACKNOWLEDGMENTS
Financial support by the Wisconsin Agriculture Experiment Station
and the AVIAGEN, Ltd. (Newbridge, Scotland) to Paulino Pérez and
Daniel Gianola is acknowledged. We thank the Centro Internacional
de Mejoramiento de Maíz y Trigo (CIMMYT) researchers who carried
out the wheat trials and provided the phenotypic data analyzed in this
article.
LITERATURE CITED
Bernardo, R., and J. M. Yu, 2007 Prospects for genome-wide selection for
quantitative traits in maize. Crop Sci. 47(3): 1082–1090.
Broomhead, D. S., and D. Lowe, 1988 Multivariable functional interpola-
tion and adaptive networks. Complex Systems 2: 321–355.
Burgueño, J., J. Crossa, P. L. Cornelius, R. Trethowan, G. McLaren et al.,
2007 Modeling additive ·environment and additive ·additive ·en-
vironment using genetic covariances of relatives of wheat genotypes. Crop
Sci. 47(1): 311–320.
Burgueño, J., J. Crossa, J. M. Cotes, F. San Vicente, and B. Das, 2011 Pre-
diction assessment of linear mixed models for multienvironment trials.
Crop Sci. 51(3): 944–954.
Chen, S., C. F. N. Cowan, and P. M. Grant, 1991 Orthogonal least squares
learning algorithm for radial basis function networks. Neural Networks,
IEEE Transactions on 2(2): 302–309.
Cockram, J., H. Jones, F. J. Leigh, D. O’Sullivan, W. Powell et al.,
2007 Control of flowering time in temperate cereals: genes, domesti-
cation, and sustainable productivity. J. Exp. Bot. 58(6): 1231–1244.
Conti, V., P. F. Roncallo, V. Beaufort, G. L. Cervigni, R. Miranda et al.,
2011 Mapping of main and epistatic effect QTLs associated to grain
protein and gluten strength using a RIL population of durum wheat.
J. Appl. Genet. 52(3): 287–298.
Crossa, J., J. Burgueño, P. L. Cornelius, G. McLaren, R. Trethowan et al.,
2006 Modeling genotype ·environment interaction using additive ge-
netic covariances of relatives for predicting breeding values of wheat
genotypes. Crop Sci. 46(4): 1722–1733.
Crossa, J., G. de los Campos, P. Perez, D. Gianola, J. Burgueño et al.,
2010 Prediction of genetic values of quantitative traits in plant breeding
using pedigree and molecular markers. Genetics 186(2): 713–724.
Crossa, J., P. Perez, G. de los Campos, G. Mahuku, S. Dreisigacker et al.,
2011 Genomic selection and prediction in plant breeding. J. Crop Im-
prov. 25(3): 239–261.
de los Campos, G., and P. Perez, 2010. BLR: Bayesian Linear Regression
R package, version 1.2.
de los Campos, G., H. Naya, D. Gianola, J. Crossa, A. Legarra et al.,
2009 Predicting quantitative traits with regression models for dense
molecular markers and pedigree. Genetics 182(1): 375–385.
de los Campos, G., D. Gianola, G. J. M. Rosa, K. A. Weigel, and J. Crossa,
2010 Semi-parametric genomic-enabled prediction of genetic values
using reproducing kernel Hilbert spaces methods. Genet. Res. 92(4):
295–308.
de los Campos, G., J. M. Hickey, R. Pong-Wong, H. D. Daetwyler, and M. P. L.
Calus, 2012 Whole genome regression and prediction methods applied to
plant and animal breeding. Genetics DOI: 10.1534/genetics.112.14331.
Foresee, D., and M. T. Hagan, 1997. Gauss-Newton approximation to
Bayesian learning. International Conference on Neural Networks, June
9–12, Houston, TX.
Gianola, D., and J. B. C. H. M. van Kaam, 2008 Reproducing kernel Hilbert
spaces regression methods for genomic assisted prediction of quantitative
traits. Genetics 178(4): 2289–2303.
Gianola, D., R. L. Fernando, and A. Stella, 2006 Genomic-assisted predic-
tion of genetic value with semiparametric procedures. Genetics 173(3):
1761–1776.
Gianola, D., H. Okut, K. A. Weigel, and G. J. M. Rosa, 2011 Predicting
complex quantitative traits with Bayesian neural networks: a case study
with Jersey cows and wheat. BMC Genet. 12: 87.
Goldringer, I., P. Brabant, and A. Gallais, 1997 Estimation of additive
and epistatic genetic variances for agronomic traits in a population of
doubled-haploid lines of wheat. Heredity 79: 60–71.
González-Camacho, J. M., G. de los Campos, P. Perez, D. Gianola, J. Cairns
et al., 2012 Genome-enabled prediction of genetic values using radial
basis function. Theor. Appl. Genet. 125: 759–771.
Habier, D., R. L. Fernando, K. Kizilkaya, and D. J. Garrik, 2011 Extension
of the Bayesian alphabet for genomic selection. BMC Bioinformatics 12:
186.
Hastie, T., R. Tibshirani, and J. Friedman, 2009 The Elements of Statistical
Learning: Data Mining,Inference and Prediction, Ed. 2. Springer, New York.
1604 | P. Pérez-Rodríguez et al.

Heffner, E. L., M. E. Sorrells, and J. L. Jannink, 2009 Genomic selection for
crop improvement. Crop Sci. 49(1): 1–12.
Heslot, N., H. P. Yang, M. E. Sorrells, and J. L. Jannink, 2012 Genomic
selection in plant breeding: a comparison of models. Crop Sci. 52(1):
146–160.
Hickey, J. M., and B. Tier, 2009 AlphaBayes (Beta): Software for Polygenic
and Whole Genome Analysis. User Manual. University of New England,
Armidale, Australia.
Hoerl, A. E., and R. W. Kennard, 1970 Ridge regression: biased estimation
for nonorthogonal problems. Technometrics 12(1): 55–67.
Holland, J. B., 2001 Epistasis and plant breeding. Plant Breeding Reviews
21: 27–92.
Holland, J. B., 2008 Theoretical and biological foundations of plant
breeding, pp. 127–140 in Plant Breeding: The Arnel R. Hallauer Inter-
national Symposium, edited by K. R. Lamkey and M. Lee. Blackwell
Publishing, Ames, IA.
Lampinen, J., and A. Vehtari, 2001 Bayesian approach for neural networks -
review and case studies. Neural Netw. 14(3): 257–274.
Laurie, D. A., N. Pratchett, J. W. Snape, and J. H. Bezant, 1995 RFLP
mapping of five major genes and eight quantitative trait loci controlling
flowering time in a winter ·spring barley (Hordeum vulgare L.) cross.
Genome 38(3): 575–585.
Long, N. Y., D. Gianola, G. J. M. Rosa, K. A. Weigel, A. Kranis et al.,
2010 Radial basis function regression methods for predicting quanti-
tative traits using SNP markers. Genet. Res. 92(3): 209–225.
MacKay, D. J. C., 1992 A practical Bayesian framework for backpropaga-
tion networks. Neural Comput. 4(3): 448–472.
MacKay, D. J. C., 1994 Bayesian non-linear modelling for the prediction
competition. ASHRAE Transactions 100(Pt. 2): 1053–1062.
Makowsky, R., N. M. Pajewski, Y. C. Klimentidis, A. I. Vazquez, C. W. Duarte
et al., 2011 Beyond missing heritability: prediction of complex traits.
PLoS Genet. 7(4): e1002051.
McKinney, B. A., and N. M. Pajewski, 2012. Six degrees of epistasis: sta-
tistical network models for GWAS. Front. Genet. 2: 109.
Meuwissen, T. H. E., B. J. Hayes, and M. E. Goddard, 2001 Prediction of
total genetic value using genome-wide dense marker maps. Genetics 157
(4): 1819–1829.
Neal, R. M., 1996. Bayesian Learning for Neural Networks (Lecture Notes in
Statistics), Vol. 118. Springer-Verlag, NY.
Ober, U., J. F. Ayroles, E. A. Stone, S. Richards, D. Zhu et al., 2012 Using
whole-genome sequence data to predict quantitative trait phenotypes in
Drosophila melanogaster. PLoS Genet. 8(5): e1002685.
Okut, H., D. Gianola, G. J. Rosa, and K. A. Weigel, 2011 Prediction of body
mass index in mice using dense molecular markers and a regularized
neural network. Genet. Res. Camb. 93: 189–201.
Park, T., and G. Casella, 2008 The Bayesian LASSO. J. Am. Stat. Assoc. 103:
681–686.
Perez, P., G. de los Campos, J. Crossa, and D. Gianola, 2010 Genomic-
enabled prediction based on molecular markers and pedigree using the
Bayesian linear regression package in R. Plant Genome 3(2): 106–116.
Poggio, T., and F. Girosi, 1990 Networks for approximation and learning.
Proc. IEEE 78(9): 1481–1497.
Resende, M. F. R., P. Muñoz, M. D. V. Resende, D. J. Garrick, R. L. Fernando
et al., 2012 Accuracy of genomic selection methods in a standard data
set of loblolly pine (Pinus taeda L.). Genetics 4: 1503–1510.
Shimada, S., T. Ogawa, and S. Kitagawa, 2009 A genetic network of
flowering-time genes in wheat leaves, in which an APETALA1/FRUITFULL-
like gene, VRN-1, is upstream of FLOWERING LOCUS T.PlantJ.58:
668–681.
Wang, C. S., J. J. Rutledge, and D. Gianola, 1994 Bayesian analysis of mixed
linear models via Gibbs sampling with an application to litter size in
Iberian pigs. Genet. Sel. Evol. 26: 91–115.
Zhang, K., J. Tian, L. Zhao, and S. Wang, 2008 Mapping QTLs with epi-
static effects and QTL ·environment interactions for plant height using
a doubled haploid population in cultivated wheat. J. Genet. Genomics 35
(2): 119–127.
Communicating editor: J. B. Holland
Volume 2 December 2012 | Linear and Non-parametric Regression Models for GS | 1605