Bayes factor between Student t and Gaussian mixed models within an animal breeding context.
ABSTRACT The implementation of Student t mixed models in animal breeding has been suggested as a useful statistical tool to effectively mute the impact of preferential treatment or other sources of outliers in field data. Nevertheless, these additional sources of variation are undeclared and we do not know whether a Student t mixed model is required or if a standard, and less parameterized, Gaussian mixed model would be sufficient to serve the intended purpose. Within this context, our aim was to develop the Bayes factor between two nested models that only differed in a bounded variable in order to easily compare a Student t and a Gaussian mixed model. It is important to highlight that the Student t density converges to a Gaussian process when degrees of freedom tend to infinity. The two models can then be viewed as nested models that differ in terms of degrees of freedom. The Bayes factor can be easily calculated from the output of a Markov chain Monte Carlo sampling of the complex model (Student t mixed model). The performance of this Bayes factor was tested under simulation and on a real dataset, using the deviation information criterion (DIC) as the standard reference criterion. The two statistical tools showed similar trends along the parameter space, although the Bayes factor appeared to be the more conservative. There was considerable evidence favoring the Student t mixed model for data sets simulated under Student t processes with limited degrees of freedom, and moderate advantages associated with using the Gaussian mixed model when working with datasets simulated with 50 or more degrees of freedom. For the analysis of real data (weight of Pietrain pigs at six months), both the Bayes factor and DIC slightly favored the Student t mixed model, with there being a reduced incidence of outlier individuals in this population.
- Joaquim Casellas, Rodrigo J Gularte, Charles R Farber, Luis Varona, Margarete Mehrabian, Eric E Schadt, Aldon J Lusis, Alan D Attie, Brian S Yandell, Juan F Medrano[Show abstract] [Hide abstract]
ABSTRACT: Transmission ratio distortion (TRD) is the departure from the expected genotypic frequencies under Mendelian inheritance. This departure can be due to multiple physiological mechanisms during gametogenesis, fertilization, fetal and embryonic development, and early neonatal life. Although a few TRD loci have been reported in mouse, inheritance patterns have never been evaluated for TRD. In this article, we developed a Bayesian binomial model accounting for additive and dominant deviation TRD mechanisms. Moreover, this model was used to perform genome-wide scans for TRD quantitative trait loci (QTL) on six F2 mouse crosses involving between 296 and 541 mice and between 72 and 1854 genetic markers. Statistical significance of each model was checked at each genetic marker with Bayes factors. Genome scans revealed overdominance TRD QTL located in mouse chromosomes 1, 2, 12, 13, and 14 and additive TRD QTL in mouse chromosomes 2, 3, and 15, although these results did not replicate across mouse crosses. This research contributes new statistical tools for the analysis of specific genetic patterns involved in TRD in F2 populations, our results suggesting a relevant incidence of TRD phenomena in mouse with important implications for both statistical analyses and biological research.Genetics 02/2012; 191(1):247-59. · 4.39 Impact Factor
Page 1
Original article
Bayes factor between Student t
and Gaussian mixed models
within an animal breeding context
Joaquim CASELLAS1*, Noelia IBA´N ˜EZ-ESCRICHE1,
LuisAlbertoGARCI´A-CORTE´S2,LuisVARONA1
1Gene `tica i Millora Animal, IRTA-Lleida, 25198 Lleida, Spain
2Departamento de Mejora Gene ´tica Animal, SGIT-INIA, Carretera de la Corun ˜a, km. 7,
28040 Madrid, Spain
(Received 2 April 2007; accepted 19 December 2007)
Abstract – The implementation of Student t mixed models in animal breeding has been
suggested as a useful statistical tool to effectively mute the impact of preferential treatment
or other sources of outliers in field data. Nevertheless, these additional sources of variation
are undeclared and we do not know whether a Student t mixed model is required or if a
standard, and less parameterized, Gaussian mixed model would be sufficient to serve the
intended purpose. Within this context, our aim was to develop the Bayes factor between
two nested models that only differed in a bounded variable in order to easily compare a
Student t and a Gaussian mixed model. It is important to highlight that the Student t
density converges to a Gaussian process when degrees of freedom tend to infinity. The two
models can then be viewed as nested models that differ in terms of degrees of freedom. The
Bayes factor can be easily calculated from the output of a Markov chain Monte Carlo
sampling of the complex model (Student t mixed model). The performance of this Bayes
factor was tested under simulation and on a real dataset, using the deviation information
criterion (DIC) as the standard reference criterion. The two statistical tools showed similar
trends along the parameter space, although the Bayes factor appeared to be the more
conservative. There was considerable evidence favoring the Student t mixed model for data
sets simulated under Student t processes with limited degrees of freedom, and moderate
advantages associated with using the Gaussian mixed model when working with datasets
simulated with 50 or more degrees of freedom. For the analysis of real data (weight of
Pietrain pigs at six months), both the Bayes factor and DIC slightly favored the Student t
mixed model, with there being a reduced incidence of outlier individuals in this population.
Bayes factor / Gaussian distribution / mixed model / Student t distribution / preferential
treatment
*Corresponding author: Joaquim.Casellas@irta.es
Genet. Sel. Evol. 40 (2008) 395–413
? INRA, EDP Sciences, 2008
DOI: 10.1051/gse:2008007
Available online at:
www.gse-journal.org
Article published by EDP Sciences
Page 2
1. INTRODUCTION
Genetic evaluations in animal breeding are generally performed using the
mixed effects models pioneered by Henderson [9]. Usually, these models assume
Gaussian distributions for most random effects, including the residuals, and in
absence of contradictory evidence, it is practical to assume normality on the basis
of both mathematical convenience and biological plausibility. Nevertheless,
departures from normality are common in animal breeding, e.g. when more valu-
able animals receive preferential treatment [14,15]. This preferential treatment
could be defined as any management practice that increases or decreases produc-
tion and is applied to one or several animals, but not to their contemporaries [14].
Amongst others, these practices may include separate housing, better (or worse)
or more (or less) feed, or better (or worse) sanitary attentions. Obviously, which
animals or productive records receive preferential treatment is not known with
any degree of certainty in real populations and this information loss could imply
substantial bias in genetic evaluations [14,15]. Other potential causes of outliers
or abnormal phenotypic records could be measurement errors, sickness, short-
term-changes in herd environment and mismanagement of data [11].
We generally lack a priori sufficient information relating to the presence or
absence of preferential treatment in our livestock data sets. It has been recently
demonstrated that the specification of heavy-tailed residual distributions (such as
the Student t distribution) instead of the usual Gaussian process in best linear
unbiased prediction (BLUP) models may effectively mute the impact of residual
outliers, particularly in situations where the preferential treatment of some breed
stock may be anticipated [16,21]. As a result, accurate statistical tests are
required to compare the mathematical simplicity of the Gaussian mixed model
with the improved goodness of fit (under preferential treatment or other
unknown sources of outliers) of the Student t mixed model.
General statistical tools such as the deviance information criterion (DIC) [20]
or other approaches to Bayes factors [6] have been used to make comparisons
between Gaussian and Student t mixed models. However, they imply high com-
putational demands because both the Gaussian and the Student t mixed model
must be analysed to calculate the corresponding comparison parameter. Within
this context, the Bayes factor developed by Garcı ´a-Corte ´s et al. [5] and Varona
et al. [23] in the animal breeding context implies a substantial simplification
because it compares two models that only differ in terms of a single bounded
variable, and therefore only the analysis of the complex model is required.
The Student t distribution converges with the Gaussian distribution when the
number of degrees of freedom tends to infinity. This property can be exploited
396
J. Casellas et al.
Page 3
to appropriately adapt Varona et al. [23] Bayes factor, generating a useful statis-
tical tool for the analysis of field data, especially when used for genetic evalu-
ation purposes. In this paper, we focused our efforts on describing the
development of this Bayes factor to make comparisons between Gaussian and
Student t processes, and we tested its performance on both simulated and real
data sets, using DIC as the standard reference criterion.
2. MATERIALS AND METHODS
2.1. Statistical background for Student t mixed models
Take as a starting point a standard linear model [9] such as
y ¼ Xb þ Wp þ Za þ e;
ð1Þ
where y is the vector with n phenotypic data, X, W, Z are the incidence matri-
ces of systematic (b), permanent environmental (p) and additive genetic
effects (a), respectively, and e is the vector of residuals. The probability
density of phenotypic data can be modeled under a multivariate Student t
distribution with m degrees of freedom (with m being equal to or greater
than 2):
p y b;p;a;r2
e;m
??
??¼
Y
n
i¼1
Cmþ1
? ?C1
ð
2
? ?m
yi? xib ? wip ? zia
??
2
C
m
2
1
2
r2
e
???1
2
? 1 þ
Þ
mr2
0yi? xib ? wip ? zia
e
ðÞ
"#?1
2mþ1
ðÞ
;
ð2Þ
where xi, wiand ziare the ith row of X, W and Z, respectively, yiis the ith
scalar element of y, r2
function with the argument as defined within parentheses. For small values
of m, the Student t distribution shows a Gaussian-like pattern with increased
probability in tails, whereas this distribution converges to a Gaussian distribu-
tion when m tends to infinity [16]. For mathematical convenience, we can
define d = 2/m (0 ? d < 1) and then, the conditional density (2) reduces to
a normal density when d = 0 (as is, m tends to infinity).
Following Strande ´n and Gianola [21], the previous model can be extended
to an alternative parameterization if the data vector is partitioned according to
eis the residual variance and C(.) is the standard gamma
Student t versus Gaussian mixed models
397
Page 4
m ‘clusters’ typified by a common factor (e.g. animal, maternal environment,
herd-year-season at birth), with the previous linear model defined as:
2
y1
...
ym
664
3
775¼
X1
...
Xm
2
664
3
775b þ
W1
...
Wm
2
664
3
775p þ
Z1
...
Zm
2
664
3
775a þ
e1
...
em
2
664
3
775;
ð3Þ
Xj,WjandZjbeingtheappropriateincidencematricesofrecordsinthejthclus-
ter (yj), and ejbeing the corresponding vector of residuals. This reparameteriza-
tion allows for an alternative description of the conditional density of y [21]:
p y b;p;a;r2
e;d
??
??¼
Y
m
j¼1
p yjb;p;a;r2
e;s2
j
???
??
p s2
jd j
??
;
ð4Þ
where p yjb;p;a;r2
e;sj
??
??is a multivariate normal distribution weighted by s2
p yjb;p;a;r2
j
? N Xjb þ Wjp þ Zja;Inj
j,
e;s2
???
??
r2
s2
e
j
!
;
ð5Þ
Injbeing an identity matrix with dimensions nj· nj, and the conditional dis-
tribution of the mixing parameter (s2
j) is a Gamma density
p s2
jd j
??
¼
1
2d
? ? s2
? ? 1
C
2d
1
2d
j
? ?
1
2d?1
ðÞexp ?s2
j
2d
??
ð6Þ
with it having an expectation of 1 when d = 0 [4,21].
2.2. Bayes factor between Student t and Gaussian linear models
The Bayes factor developed by Verdinelli and Wasserman [25], and applied
to the animal breeding context by Garcı ´a-Corte ´s et al. [5] and Varona et al. [23],
contrasts nested linear models that only differ in terms of a bounded variable.
We adapted this methodology to compare a Student t mixed linear model with
its simplification to the Gaussian mixed linear model when m tends to infinity or,
for mathematical convenience, d = 2/m = 0. Within this context, the posterior dis-
tribution of all the parameters of a Student t mixed model can be stated in two
ways, with a pure Student t Bayesian likelihood (Model T1):
?
? pTr2
pT1b;p;a;r2
p;r2
a;r2
e;d y j
?
/ pT1y b;p;a;r2
?
e;d
??
??
??pTd ð ÞpTb
a
ð ÞpTp r2
?pTr2
p
???
?
e
?
p
?
pTa A;r2
??pTr2
a
???;
ð7Þ
398
J. Casellas et al.
Page 5
or with a Gaussian · Gamma Bayesian likelihood (Model T2):
pT2b;p;a;r2
p;r2
a;r2
e;d;s2
j2 1;m
ðÞy j
??
/
Y
? pTd ð ÞpTb
? pTa A;r2
m
j¼1
pT2yjb;p;a;r2
e;s2
j
???
ð ÞpTp r2
?pTr2
??
pTr2
?
pT2s2
jd j
?
??
p
???
?
?
?
p
?
ea
??
?
a
?pTr2
?;
ð8Þ
where A is the numerator relationship matrix between individuals. Following
in part Varona et al. [23], the prior distribution assumed for the bounded
variable (d) was assumed
(
pTd ð Þ ¼
1 if d 2 0;1
otherwise:
½?;
0
ð9Þ
The permanent environmental and the additive genetic effects were assumed
to be drawn from multivariate normal distributions,
???
pTa A;r2
a
pT p r2
p
??
? N 0;Ipr2
p
??
;
ð10Þ
??
??? N 0;Ar2
a
??;
ð11Þ
with Ipbeing an identity matrix with dimensions equal to the number of
elements of p. The prior distributions for the remaining parameters of the
model were defined as:
?
0otherwise for each levellof b;
pTb
ð Þ ¼
k1
if bl2 ?
1
2k1;
1
2k1
?
;
8
>
>
:
<
ð12Þ
pTr2
p
??
¼
k2
if r2
p2 0;1
k2
??
;
0 otherwise;
8
>
>
:
<
ð13Þ
Student t versus Gaussian mixed models
399
Page 6
pTr2
a
??¼
k3
if r2
a2 0;1
k3
??
;
0otherwise
8
>
>
:
8
>
<
ð14Þ
pTr2
e
??¼
k4
if r2
e2 0;1
k4
??
;
0 otherwise;
>
:
<
ð15Þ
where k1, k2, k3and k4are four values that were small enough to ensure a flat
distribution over the parameter space [23].
The joint posterior distribution of all the parameters in the alternative Gaus-
sian mixed model (Model G) was proportional to
?
? pGr2
pGb;p;a;r2
p;r2
a;r2
ey j
?
/ pGy b;p;a;r2
?
e
??
??pGb
ð ÞpGp r2
?pGr2
p
???
?
?
?
p
?
pGa A;r2
a
??
?
a
?pGr2
e
??;
ð16Þ
where the Bayesian likelihood was defined as multivariate normal,
??
and the prior distributions pG(b), pGp r2
pGr2
e
The Bayes factor between Model T1(or Model T2) and Model G (BFT/G) can
be easily calculated from the Markov chain Monte Carlo sampler output of the
complex model (Student t mixed model). Under Model T1, the conditional
posterior distribution of all the parameters in the model did not reduce to well-
known distributions and generic sampling processes such as Metropolis-Hastings
[8] are required. Simplicity was gained under the alternative Model T2during the
sampling process. In this case, sampling from all the parameters in Model T2can
beperformedusingaGibbssampler [7],withtheexceptionofd, whichrequiresa
Metropolis-Hastings step [8]. Following Garcı ´a-Corte ´s et al. [5] and Varona et al.
[23], the posterior density pT(d = 0|y) suffices to obtain BFT/G,
pGy b;p;a;r2
e
??? N Xb þ Wp þ Za;Ir2
p
, pGr2
e
??;
ð17Þ
?and
???
??
p
??
, pGa A;r2
a
??
??, pGr2
a
?
??were identical to the prior distributions of Model T1(or Model T2).
BFT=G¼
pTd ¼ 0
pTd ¼ 0 y j
ðÞ
ðÞ¼
1
pTd ¼ 0 y j
ðÞ;
ð18Þ
400
J. Casellas et al.
Page 7
because pT(d = 0) = 1 (see equation (9)). Alternatively,
BFG=T¼pTd ¼ 0 y j
ðÞ
pTd ¼ 0
ðÞ
¼ pTd ¼ 0 y j
ðÞ:
ð19Þ
The BFT/Gcan be obtained by averaging the full conditional densities of each
cycle at d = 0 using the Rao-Blackwell argument [26]. At this point, compu-
tational simplicity is gained with Model T1(or a normal density for d = 0),
whereas Model T2tends to computationally unquantifiable extreme probabil-
ities when d is close to zero. A BFT/Ggreater than 1 indicates that the Student t
mixed model is more suitable, whereas a BFT/Gsmaller than 1 indicates that
the Gaussian mixed model is more suitable.
From the standard definition of the Bayes factor [13],
POT=G¼ BFT=G? PrOT=G¼ BFT=G?pT
pG
;
ð20Þ
where POT/Gis the posterior odds between models, PrOT/Gis the prior odds
between models, and pTand pGare the a priori probabilities for Student t
mixed model and Gaussian mixed model, respectively. In the standard devel-
opment of the Bayes factor described above, we assumed that prior odds were
1 and pTand pGwere both 0.5. Nevertheless, we could modify prior odds
depending on our a priori knowledge, e.g. Student t mixed model is a more
parameterized model and it could be easily penalized with a smaller-than-1
prior odds. Posterior odds can be viewed as the weighted value of the Bayes
factor, conditional to our a priori degree of belief.
2.3. Simulation studies
The Bayes factor methodology developed above was validated through sim-
ulation. Seven different scenarios were analyzed following a Student t residual
process, with degrees of freedom equal to 5 (d = 0.4), 10 (d = 0.2), 20 (d = 0.1),
50 (d = 0.04), 100 (d = 0.02), 200 (d = 0.01) and 300 (d = 0.007), respectively.
Twenty-five replicates were simulated for each case and each replicate included
five non-overlapping generations with 200 individuals (10 sires and 190 dams)
and random mating. Following Model T2, each individual had a phenotypic
record and was assigned its own independent cluster. Data were generated from
a normal density Nðl;Ir?
from equation (6). Note that l included a unique systematic effect (10 levels ran-
domly assigned with equal probability and sampled from a uniform distribution
between 0 and 1) and a normally distributed additive genetic effect generated
eÞ weighted by a cluster-characteristic value drawn
Student t versus Gaussian mixed models
401
Page 8
under standard rules [1]. Residual and additive genetic variances were equal to 1
and 0.5, respectively.
This simulation process generated seven different scenarios with 25 data sets
which were analyzed twice, through the previously described Bayes factor and
through a standard Gaussian model (Model G). For each analysis, a single chain
was launched that contained 100 000 rounds, after discarding the first 10 000
rounds as burn-in [19]. Comparisons between the two models were performed
through three approaches: (a) Bayes factor between nested models, (b) DIC
[20], and (c) correlation coefficient between simulated and predicted breeding
values (qa,a ˜). Note that DIC is based on the posterior distribution of the deviance
statistic [20], which is ?2 times the sampling distribution of the data as specified
in formula (2) or as the conjugated distribution of (5) and (6), p y b;p;a;r2
andQ
D b;p;a;r2
is the posterior expectation of the deviance statistic,
pD¼ D b;p;a;r2
D b;p;a;r2
h h 2 b;p;a;r2
e;d
??
??
m
j¼1p yjb;p;a;r2
gained with (2), DIC being calculated as D b;p;a;r2
e;d
e;d
e;d
e;d
e;s2
j
???
??
p s2
jd j
??
, respectively. Computational simplicity is
?
?? D?b;? p;? a;? r2
?
2.4. Analysis of weight at six months in Pietrain pigs
e;d
?? pD where
?
?
?
?is the mean of the deviance statistic and?h is the mean value of
?
?
e;?d
??is the effective number of parameters,
??.
After editing, 2330 records of live weight at six months in Pietrain pigs were
analyzed, with an average weight (± SE) of 102.9 (± 0.265) kg. These pigs
were randomly chosen from 641 litters from successive generations grouped
in 135 batches during the fattening period, and their records were collected
between years 2003 and 2006 in a purebred Pietrain farm registered in the
reference Spanish Databank (BDporc?, http://www.bdporc.irta.es). At the
beginning of the fattening period (two months of age), batches were created with
pigs from different litters in order to homogenize piglet weight, and these groups
were maintained up to slaughter (six months of age). Pigs were reared under
standard farm management during the suckling and fattening periods. Pedigree
expanded up to five generations and comprised 2601 individuals, with 109 boars
and 337 dams with known progeny.
The operational model included the additive genetic effect of each individual,
the permanent environmental effect characterized by the batch during the fatten-
ing period, and three systematic sources of variation: sex (male or female),
year · season with 11 levels, and age at weighing (180.0 ± 0.3 days) treated
as a covariate. Data were analyzed by applying the Bayes factor described above
and assuming a different cluster for each pig with phenotypic data. To easily
402
J. Casellas et al.
Page 9
compare this method with a standard Gaussian model, data were also analyzed
under Model G. The empirical correlation between estimated breeding values
(posterior mean) was calculated in the two models and, as for the simulated data
sets, DIC was calculated for Model T and Model G. Each Gibbs sampler ran
with a single chain of 450 000 rounds after discarding the first 50 000 iterations
as burn-in [19].
3. RESULTS
3.1. Simulated datasets
Summarized results of the 25 replicates for each simulated Student t process
(5, 10, 20, 50, 100, 200 and 300 degrees of freedom) are shown in Table I.
Estimates for additive genetic variance showed coherent behavior with average
estimates slightly greater than 0.5. Average residual variance estimated using the
Student t mixed model clearly agreed with the simulated value. Nevertheless,
residual variance was clearly over-estimated for simulations with few degrees
of freedom in which a Gaussian mixed model was applied, showing higher stan-
dard errors in data sets with few degrees of freedom. Simulations with 5 degrees
of freedom showed the highest average residual variance under the Gaussian
mixed model (1.664 ± 0.038), whereas the average residual variance was
reduced to 1.222 ± 0.025 for replicates with 10 degrees of freedom, and con-
verged to one for datasets with 300 degrees of freedom (showing a standard
error smaller than 0.020). Under the Student t mixed model, average estimates
of degrees of freedom fitted with true values without any noticeable bias,
although precision decreased with larger degrees of freedom (Tab. I). Substantial
discrepancies were observed between the two models in terms of predicted
breeding values in extreme heavy-tailed simulations. Although the correlation
coefficients between predicted breeding values in the Student t and Gaussian
mixed models increased quickly in line with the degrees of freedom, the empir-
ical correlation in replicates with 5 degrees of freedom was very small (0.377 ±
0.030) and average correlations greater than 0.9 were observed in simulations
with 100 or more degrees of freedom (Tab. I).
Empirical correlations between simulated and predicted breeding values
increased with degrees of freedom in both the Student t and Gaussian mixed
models, although the Student t mixed model reached higher correlations when
simulated degrees of freedom were small. As seen in Table II, simulations under
extremely heavy-tailed processes (5 degrees of freedom) showed average corre-
lations of 0.420 and 0.377 for Student t and Gaussian mixed models, respec-
tively, suggesting substantial bias for genetic evaluations performed with
Student t versus Gaussian mixed models
403
Page 10
Table I. Variance component (· 100), degrees of freedom and breeding value correlation estimates (mean ± SE).
Simulation (m) Student t mixed model
~ r2
e
106.8 ± 2.5
98.8 ± 2.1
98.0 ± 1.4
99.8 ± 1.7
101.5 ± 1.7
101.9 ± 1.8
101.5 ± 1.8
Gaussian mixed model
~ r2
a
55.2 ± 3.5
56.7 ± 2.3
52.3 ± 1.8
50.7 ± 2.5
50.8 ± 2.4
51.9 ± 1.7
52.3 ± 2.0
~ r2
a
~ m
~ r2
e
qT;G
550.7 ± 2.1
55.5 ± 2.0
52.1 ± 2.1
51.3 ± 2.1
51.7 ± 2.5
50.5 ± 1.9
51.8 ± 2.0
5.0 ± 0.3
10.4 ± 0.4
22.3 ± 0.9
53.9 ± 1.0
102.5 ± 1.2
200.2 ± 1.3
304.1 ± 1.5
166.4 ± 3.8
122.2 ± 2.5
108.8 ± 1.5
104.7 ± 2.0
103.9 ± 1.7
102.2 ± 1.8
101.6 ± 1.9
0.377 ± 0.030
0.438 ± 0.028
0.632 ± 0.025
0.862 ± 0.019
0.961 ± 0.010
0.997 ± 0.001
0.999 ± 0.001
10
20
50
100
200
300
qT,G: Empirical correlation between predicted breeding values in Student t and Gaussian mixed models.
404
J. Casellas et al.
Page 11
standard Gaussian models when normality did not hold. Differences between the
two models quickly decreased with increasing degrees of freedom and were
almost negligible for m = 20 and higher values.
As was expected, both the Bayes factor and DIC clearly favored Student t
mixed models in simulated scenarios with few degrees of freedom and showed
similar behavior throughout the analyzed framework (Fig. 1A). Datasets simu-
lated under a residual Student t process with 5 degrees of freedom reached an
average Bayes factor favoring the Student t mixed model of 4.6 · 1092, and
the average difference between DIC was also huge (? 490). Although both com-
parison criteria decreased when degrees of freedom increased, our results sug-
gest that the Student t mixed model was preferable instead of the Gaussian
model up to 20 degrees of freedom, and that even for simulations with
50 degrees of freedom, the superiority of the Student t model remained almost
total (Tab. III). Substantial discrepancies between the Bayes factor and DIC
appeared in the last two scenarios (200 and 300 degrees of freedom). While
the average Bayes factor slightly favored the Gaussian model, the DIC contin-
ued to produce smaller estimates for the Student t model, although with only a
minimal difference in the last scenario (Tab. II). This suggests that the Bayes fac-
tor was more conservative, favoring the less parameterized model. This hypoth-
esis was confirmed in Table III where the Bayes factor supported the Gaussian
mixed model in 0, 0, 0, 1, 11, 18 and 19 data sets (for simulations with 5, 10, 20,
Table II. Comparison criteria (average estimates) between Student t and Gaussian
mixed models for each simulation scenario.
Simulation (m) Student t
mixed model
Gaussian
mixed model
~ qa;~ a
0.420
0.443
0.463
0.470
0.469
0.470
0.470
DICT
3116
2895
2854
2842
2841
2840
2840
~ qa;~ a
0.377
0.431
0.460
0.468
0.469
0.469
0.470
DICG
3606
3112
2939
2887
2867
2850
2841
DICDiff.
?490
?217
?85
?45
?27
?10
?1
BFT/G
4:6 ? 1092
1:2 ? 1071
2:2 ? 1034
1:8 ? 1014
5:7 ? 104
0.569
0.420
5
10
20
50
100
200
300
~ qa;~ a: Empirical correlation between simulated and predicted breeding values.
DICT: Deviance information criterion for the Student t mixed model.
DICG: Deviance information criterion for the Gaussian mixed model.
DICDiff.= DICT– DICG.
BFT/G: Bayes factor of the Student t mixed model against the Gaussian mixed model.
Student t versus Gaussian mixed models
405
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Table III. Distribution of the Bayes factors (log10(BFT/G)) between Student t and Gaussian mixed models (and the number of Gaussian
models favored by the DIC) for each simulation scenario.
Simulation (m)
5
10
20
50
100
200
300
(?10,?1]
0 (0)
0 (0)
0 (0)
0 (0)
3 (1)
3 (2)
11 (5)
?? b:
(?1, 0]
0 (0)
0 (0)
0 (0)
1 (0)
8 (2)
15 (3)
8 (5)
(0, 1](1, 10](10, 50](50, 100](100, 150] (150, 200] Overall
0 (0)
0 (0)
0 (0)
1 (0)
6 (0)
7 (1)
6 (1)
0 (0)
1 (0)
9 (0)
20 (0)
8 (0)
0 (0)
0 (0)
2 (0)
19 (0)
16 (0)
3 (0)
0 (0)
0 (0)
0 (0)
14 (0)
4 (0)
0 (0)
0 (0)
0 (0)
0 (0)
0 (0)
6 (0)
1 (0)
0 (0)
0 (0)
0 (0)
0 (0)
0 (0)
3 (0)
0 (0)
0 (0)
0 (0)
0 (0)
0 (0)
0 (0)
25 (0)
25 (0)
25 (0)
25 (0)
25 (3)
25 (6)
25 (11)
a;b?ð¼ a < log10BFT=G
?
406
J. Casellas et al.
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50, 100, 200 and 300 degrees of freedom), whereas DIC favored the Gaussian
mixed model in 0, 0, 0, 0, 3, 5 and 10 data sets, respectively (Fig. 1B).
3.2. Analysis of Pietrain pig weight at six months
Analysis under a Student t linear mixed model placed the highest posterior
density region at 95% (HPD95) for the d parameter between 0.004 and 0.009,
with the modal value at 0.005 (Tab. IV). Degrees of freedom (m) determine
Figure 1. Plot of differences in DIC between Student t and Gaussian mixed models
against log10(BFT/G) for all replicates (A) and for combinations close to 0 (B).
Student t versus Gaussian mixed models
407
Page 14
the slope of the Student t distribution and they were consequently drawn within
a wide HPD95 (111.20–235.70) and the mode reached 191.12. The posterior
distribution of m was roughly symmetrical (Fig. 2). The posterior mean of the
weights (s2
i) for the Student t mixed model ranged from 0.924 to 1.026, although
TableIV.Summary statistics forthe analysis of pig weight at six months under Student t
and Gaussian mixed models.
HPD95 = highest posterior density region at 95%.
~ c2¼ ~ r2
~h2¼ ~ r2
p= ~ r2
aþ ~ r2
~ r2
pþ ~ r2
pþ ~ r2
e
?
e
?
.
a=
aþ ~ r2
?
:
?
MeanMode HPD95
Student t model
~ r2
a
~ r2
p
~ r2
e
~ c2
~h2
~d
~ m
Gaussian model
~ r2
a
~ r2
p
~ r2
e
~ c2
~h2
36.87
35.16
68.72
0.248
0.260
0.006
177.95
35.92
35.86
68.63
0.239
0.254
0.005
191.12
24.81–50.06
22.33–48.83
59.97–77.32
0.177–0.321
0.178–0.341
0.004–0.009
111.20–235.70
36.21
35.88
69.54
0.247
0.255
35.84
35.81
69.01
0.237
0.256
24.67–49.21
22.32–48.42
60.91–77.82
0.179–0.318
0.173–0.338
Figure 2. Posterior distribution of degrees of freedom for weight at six months in
Pietrain pigs.
408
J. Casellas et al.
Page 15
the majority were located between 0.975 and 1.025 (95.19%; Tab. V). In this
sense, values smaller than 0.925 were in a minority (0.22%), although they
could have had a substantial influence as outliers. It is important to highlight
the fact that differences in variance component estimation between the Gaussian
and Student t mixed models were minimal (Tab. IV), with similar values for her-
itability (0.256 and 0.254, respectively) and for the coefficient of common envi-
ronment (0.237 and 0.239, respectively).
The Bayes factor favored the Student t model rather than the Gaussian model
(BFT/G= 2.532), although the small size of this value is not worth more than a
bare mention according to Jeffreys’ [12] scale of evidence. He/she suggested that
differences could be minimal, although DIC reported substantial discrepancies
between the Student t mixed model (16 445) and the Gaussian mixed model
(16 450); this difference was smaller than the average difference between DIC
in simulations with 200 degrees of freedom. On the contrary, the empirical
correlation coefficient between predicted breeding values in each model was
0.999 (this was 0.993 if only breeding animals were considered) and showed
that, although the posterior probabilities of both models were slightly (Bayes
factor) or substantially (DIC) different, genetic evaluations performed with a
Student t or a Gaussian mixed model provided an almost identical genetic rank-
ing for this data set.
4. DISCUSSION
The Bayes factor as originally proposed by Verdinelli and Wasserman [25]
has been applied to various models used in animal breeding and the genetics
research field. Although the method was initially developed to test for the
genetic background of linear traits [5] and the location of quantitative trait loci
(QTL) [23], this Bayes factor has been recently modified to discriminate
Table V. Distribution of the posterior mean value of weights (s2
weight at six months under Student t mixed models.
i) for the analyses of pig
n
%
0:900 < s2
0:925 < s2
0:950 < s2
0:975 < s2
1:000 < s2
1:025 < s2
i? 0:925
i? 0:950
i? 0:975
i? 1:000
i? 1:025
i? 1:050
Overall
50.22
0.64
3.78
34.29
60.90
0.17
100.00
15
88
799
1419
4
2330
Student t versus Gaussian mixed models
409