Validation of an approximate REML algorithm for parameter estimation in a multitrait, multiple across-country evaluation model: a simulation study.

Abstract

A multitrait, multiple across-country evaluation (MT-MACE) model permitting a variable number of correlated traits per country allows international genetic evaluation models to more closely match national models. Before the MT-MACE evaluation can be applied, genetic (co)variance components within and across country must be estimated. An approximate REML algorithm for parameter estimation was developed and was validated via simulation. This method is based on the expectation maximization REML (EM-REML) algorithm. Because obtaining the inverse of co-efficient matrix is not usually feasible for large amounts of data, an algorithm using the multiple-trait effective daughter contribution (EDC) is proposed to provide approximate diagonal elements of the inverse matrix. The accuracy of the approximate EM-REML was tested with simulated data and compared with an average information REML (AI-REML) from available software. Two simulation studies were performed. First, data of 2 countries were simulated using a single-trait model. Estimates of across-country genetic correlations with the developed algorithm were unbiased and very precise. The precision, however, depended on the percentage of bulls with data in both countries. The results obtained with the approximate EM-REML software were very close to those obtained with the AI-REML software regarding estimated genetic correlations and bulls' estimated breeding values. The second simulation assumed a multiple trait model and the same number of traits, pedigree structure, EDC, and pattern of missing records as for actual observations for milk yield obtained from French and German national Holstein evaluations. As with the single-trait scenarios, the approximate EM-REML gave nearly unbiased and very precise estimates of within- and across-country genetic correlations. The results obtained in both simulation studies confirmed the suitability of the MT-MACE model and approximate EM-REML software in a wide range of situations. Even when the genetic trend was incorrectly estimated by the national evaluations, a joint analysis including a time effect in the MT-MACE model adequately corrected for this bias.

Full text

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J. Dairy Sci. 90:4846–4855
doi:10.3168/jds.2007-0072
© American Dairy Science Association, 2007.
Validation of an Approximate REML Algorithm for Parameter
Estimation in a Multitrait, Multiple Across-Country
Evaluation Model: A Simulation Study
J. Tarre ´s,*†1Z. Liu,* V. Ducrocq,† F. Reinhardt,* and R. Reents*
*Vereinigte Informationssysteme Tierhaltung w.v., Heideweg 1, 27283 Verden, Germany
†UR337, Station de Ge ´ne ´tique Quantitative et Applique ´e, Institut National de la Recherche Agronomique, 78352 Jouy-en-Josas Cedex, France
ABSTRACT
A multitrait, multiple across-country evaluation
(MT-MACE) model permitting a variable number of
correlated traits per country allows international ge-
netic evaluation models to more closely match national
models. Before the MT-MACE evaluation can be ap-
plied, genetic (co)variance components within and
across country must be estimated. An approximate
REML algorithm for parameter estimation was devel-
oped and was validated via simulation. This method is
based on the expectation maximization REML (EM-
REML) algorithm. Because obtaining the inverse of co-
efficientmatrixisnotusuallyfeasibleforlargeamounts
of data, an algorithm using the multiple-trait effective
daughter contribution (EDC) is proposed to provide ap-
proximate diagonal elements of the inverse matrix. The
accuracyoftheapproximateEM-REMLwastestedwith
simulateddata andcomparedwithan averageinforma-
tion REML (AI-REML) from available software. Two
simulation studies were performed. First, data of 2
countries were simulated using a single-trait model.
Estimates of across-country genetic correlations with
the developed algorithm were unbiased and very pre-
cise. The precision, however, depended on the percent-
age of bulls with data in both countries. The results
obtained with the approximate EM-REML software
were very close to those obtained with the AI-REML
software regarding estimated genetic correlations and
bulls’estimatedbreedingvalues.Thesecondsimulation
assumedamultipletraitmodelandthesamenumberof
traits, pedigree structure, EDC, and pattern of missing
records as for actual observations for milk yield ob-
tainedfromFrenchandGermannationalHolsteineval-
uations. As with the single-trait scenarios, the approxi-
mate EM-REML gave nearly unbiased and very precise
estimates of within-and across-country genetic correla-
Received February 2, 2007.
Accepted June 11, 2007.
1Corresponding author: joaquim.tarres@dga.jouy.inra.fr
4846
tions. The results obtained in both simulation studies
confirmed the suitability of the MT-MACE model and
approximate EM-REML software in a wide range of
situations.Evenwhenthegenetictrendwasincorrectly
estimated by the national evaluations, a joint analysis
including a time effect in the MT-MACE model ade-
quately corrected for this bias.
Key words: multiple across-country evaluation, re-
stricted maximum likelihood, daughter yield deviation,
effective daughter contribution
INTRODUCTION
The multiple across-country evaluation (MACE;
Schaeffer, 1994) methodology is used for international
dairy bull comparisons. Estimated breeding values in
national genetic evaluations are deregressed within
each country to obtain the values of the dependent vari-
able for bulls that have daughters with records. Cur-
rently, a single EBV per bull is permitted for each coun-
try ininternational geneticevaluations byInterbull. As
more and more countries have upgraded their national
genetic evaluation system to a multiple-trait model or
a multiple-lactation random regression test-day model
(RRTDM), differences among models for national and
international evaluations have become increasingly ev-
ident. To optimize genetic evaluation models for both
national and international evaluations, Sullivan and
Wilton (2001) proposed a multiple-trait MACE (MT-
MACE) with a variable number of traits per country.
This model extended the current single-trait MACE
(ST-MACE) to multiple lactations, or traits for coun-
tries using a multiple trait model in national genetic
evaluation. Schaeffer (2001) developed a multiple trait
de-regression method for MACE evaluation. More re-
cently, the simulation study by Sullivan et al. (2005)
confirmed the theoretical expectation that MT-MACE
methods should be preferred over methods that allow
only one trait per country when assuming true genetic
parameters. Mark and Sullivan (2006) applied MT-
MACE to field data for udder health to quantify the
benefits in terms of reliability and ability to predict

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VALIDATION OF AN APPROXIMATE REML ALGORITHM
4847
international genetic merit of MT-MACE compared
with ST-MACE, and to discuss the practical implemen-
tation of MT-MACE. They concluded that the MT-
MACE method is recommended for international ge-
netic udder health evaluations.
In parallel to the development involving de-regres-
sion, Ducrocq et al. (2001) suggested using corrected
records for a 2-step multitrait genetic evaluation. In-
deed, this approach can also be applied to international
genetic evaluations. Yield deviations (YD) of cows and
daughter yield deviations (DYD) of bulls are, in fact,
corrected records. For single-trait models, VanRaden
and Wiggans (1991) developed formulas for the calcula-
tion of YD and DYD and its daughter equivalent, which
were later called effective daughter contribution
(EDC). Liu et al. (2004a) extended the YD and DYD
derivation to multiple trait models, including RRTDM.
Also, Liu et al. (2004a) developed a method for approxi-
matingreliabilitiesofEBV undergeneralmultipletrait
models, which can also be used to approximate EDC
associated with bulls’ DYD.
Based on DYD, Liu et al. (2004b) presented an MT-
MACE model for international bull comparison. This
model allowed a variable number of correlated traits
per country. For countries using multiple trait models
in national genetic evaluations, a vector of DYD and
its corresponding EDC matrix are needed for each bull
with daughter performance information. An approxi-
mate REML algorithm was developed to estimate
across country genetic correlations based on multitrait
EDC (MTEDC, Liu et al., 2004a). For solving the large
equation system of the MT-MACE model, a precondi-
tioned conjugate gradients algorithm (PCG, Strande ´n
and Lidauer, 1999) was applied together with the itera-
tion on data technique. This MACE model can also be
used to analyze YD of cows or de-regressed estimates
of breeding values.
Theaimofthispaperwastovalidatetheapproximate
REML method for estimating across-country genetic
correlations via simulation. In addition to the analysis
with a ST-MACE model, a MT-MACE analysis was per-
formed to check the suitability of this approximate
REML algorithm in a wide range of situations.
MATERIALS AND METHODS
The MT-MACE Model
For a country j using a multitrait model in national
genetic evaluation, the following statistical model was
applied to the data of a bull i from the country j:
yij= fj+ aij+ eij
[1]
Journal of Dairy Science Vol. 90 No. 10, 2007
where yijis a vector of data of the ith bull in country
j, fjis a vector of overall means for traits of the jth
country, aijis a vector of additive genetic effects of bull
i in country j, and eijis a vector of residual effects.
Model [1] is also valid for data from countries with a
single-trait model in national genetic evaluations, with
all terms above becoming scalars. The (co)variance ma-
trix of genetic effects of the m countries and its inverse
are denoted as
G0=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
G011
G012… G01m
G022… G02m
...
?
symm.
G0mm
⎤
⎥
⎥
⎥
⎥
⎥
⎦
, [2]
and G−1
0 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
G11
0
G12
G22
0 … G1m
0 … G2m
...
0
0
?
symm.
Gmm
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
where m is the number of countries, G0jjis the original
genetic (co)variance matrix of country j, and G0jkis the
genetic covariance matrix between countries j and k.
When country j uses a single-trait model for national
evaluation, G0jjis a scalar and all corresponding off-
diagonal blocks, G0jk(j ≠ k), become vectors or scalars,
depending on the number of traits in other countries.
As usually considered in MACE, data of a bull from
different countries are assumed to be residually uncor-
related. The inverse of error (co)variance matrix of bull
i in country j is
[Var(eij)]−1= Ψij
[3]
where Ψijis the EDC matrix for bull i in country j,
converted from the reliability matrix contributed by
his daughters’ records in the jth country. The MTEDC
procedure(Liuetal.,2004a)canbeusedtoapproximate
matrix Ψ for each bull.
The Mixed Model Equations
The mixed model equations (MME) of model [1] con-
sist of equations for additive genetic effects of bulls
and fixed effects of country means. Ignoring pedigree
contributions, the equations corresponding to bull i are

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TARRE´S ET AL.
4848
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ψi1
0
Ψi2
...
0
Ψim
⎤
⎥
⎥
⎥
⎥
⎥
⎦
+ aii
⎡
⎢
⎢
⎢
⎢
⎢
⎣
G11
0
G12
G22
0 … G1m
0 … G2m
...
0
0
?
symm.
Gmm
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
[4]
⎡
⎢
⎢
⎢
⎢
⎢
⎣
a ˆi1
a ˆi2
?
a ˆim
⎤
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ψi1
0
Ψi2
...
0
Ψim
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎣
fˆ1
fˆ2
...
fˆm
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
?i1
?i2
...
?im
⎤
⎥
⎥
⎥
⎥
⎥
⎦
where aiiis the diagonal element of bull i in the inverse
of the numerator relationship matrix A, and ?ijrepre-
sents the right-hand-side (RHS) of bull i in country j.
Notethat?ijisnotdependentonacross-countrycorrela-
tions and is a function of country-specific information
only. The RHS of bull i in country j was calculated as
?ij= Ψijyij, multiplying the EDC matrix by the data
vector. Equations for the fixed effects of country means
in model [1] are
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ψf1
0
Ψf2
...
0
Ψfm
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎣
fˆ1
fˆ2
?
fˆm
⎤
⎥
⎥
⎥
⎥
⎥
⎦
[5]
+
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ψ11
0
Ψn1
0
Ψ12
…
Ψn2
...
...
0
Ψ1m
0
Ψnm
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎣
a ˆ1
?
?
a ˆn
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
∑
i
∑
i
?i1
?i2
?
?im
∑
i
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
where n is the total number of bulls with DYD,
a ˆi′ = [a ˆi1′ a ˆi2′ … a ˆim′] represents EBV of bull i in all m
nj
Ψijwith njbeing the number of countries, and Ψfj=∑
bulls with DYD in country j.
The equation system was solved using a PCG algo-
rithm and iteration on data technique (Strande ´n and
Lidauer,1999; Liuetal., 2004b).The convergencecrite-
rion, defined as the base 10 logarithm of the sum of
squares of differences in solutions between 2 consecu-
tive rounds of iteration divided by the sum of squares
of solutions in last round of iteration, was set to −10.
i−1
Journal of Dairy Science Vol. 90 No. 10, 2007
Estimation of (Co)Variance Components
An approximate REML algorithm was proposed by
Liu et al. (2004b) to estimate genetic (co)variances com-
ponents for general MT-MACE models. This algorithm
is based on the expectation maximization REML (EM-
REML) algorithm for the genetic effects only:
σ ˆ[l]
gjk= [a ˆj′A−1a ˆk+ tr(A−1Cjk)/q,[6]
where σ ˆ[l]
countries j and k at iteration l, a ˆjis the vector of EBV
at iteration l of all animals in country j (animals sorted
within trait), Cjkis the submatrix of the inverse of coef-
ficient matrix (C), corresponding to the country pair j
and k, and q is the total number of animals in the
pedigree file. For countries having multiple correlated
genetic effects, j and k represent the genetic effects
within country.
As the coefficient matrix, C, of MME is impossible to
invert for large systems, an alternative formulation of
the EM-REML formula is necessary. The numerator
relationship matrix can be decomposed as
gjkis the estimated genetic covariance between
A = TDT′,[7]
where matrix T is a lower triangular matrix and its
inverse T−1is also a lower triangular matrix with 1 in
the diagonals and the only nonzero elements, equal to
−0.5, connect an animal to its known parents; D is a
diagonal matrix for Mendelian sampling variance
(Mrode, 2005). As Mendelian sampling effects are mj=
T−1aj, the EM-REML formula [6] can be reformulated
(Fikse et al., 2003) as
σ ˆ[l]
gjk= [m ˆj′D−1m ˆk+ tr(D−1diag(T−1Cjk(T−1)′))]/q.
[8]
The first term of formula [8] is a simple summation
of Mendelian sampling estimates over all animals:
m ˆj′D−1m ˆk=∑
q
i=1
dim ˆijm ˆik,[9]
where m ˆijand m ˆikare the Mendelian sampling esti-
mates of animal i in countries (or within country trait)
j and k, respectively. For animal i, the corresponding
diagonal element of the matrix product T−1Cjk(T−1)′ is
{diag(T−1Cjk(T−1)′)}i= t′Cjk
it,[10]
where t′ = [1 − ¹⁄₂ − ¹⁄₂], and Cjk
inverse of coefficient matrix for country (subtrait) pair
i is a submatrix of the

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VALIDATION OF AN APPROXIMATE REML ALGORITHM
4849
j and k corresponding to animal i, its sire s, and its dam
d. Many of these ideas were reported by VanRaden
(2001), also accounting for inbreeding.
From the above formulas [8] and [10], it can be seen
that, in addition to diagonal elements of the inverse
of the coefficient matrix, off-diagonal elements among
animal, sire, and dam are also required for estimating
genetic(co)variancecomponents.Becauseobtainingthe
inverse of the coefficient matrix is not usually feasible
for large-scale equation systems, the MTEDC (Liu et
al., 2004a) is used to provide approximate diagonal ele-
ments of the inverse matrix. An approximation method
is presented here for calculating off-diagonal elements
of the inverse matrix corresponding to each animal,
sire, and dam trio.
Approximating (Co)Variances of Mendelian
Sampling Estimates
For each animal, the following 2 assumptions were
made. First, an animal is the only common progeny of
its parents, which is not too restrictive for livestock
species such as dairy cattle. Second, there are no envi-
ronmental covariances between any 2 members of the
trio. This assumption is easily met for model [1], which
contains only fixed and genetic effects. Under these 2
assumptions, all required elements of the submatrix
Cjk
for each animal:
i can be obtained from inverting the following matrix
Cisd=
2diG−1
[11]
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ψ*
i−sd+ diG−1
0
−1
0
−1
2diG−1
1
4diG−1
0
Ψ*
s−1+ (ds+1
4di) G−1
00
symm.
Ψ*
d−i+ (dd+1
4di) G−1
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
where Ψ*
account all sources of information, except the parental
contribution to the animal; Ψ*
of sire (dam) of animal i after accounting for all sources
of information except the contribution of animal i to its
sire (dam); and di, ds, and ddare the Mendelian sam-
pling terms respectively for the bull i, its sire s, and its
dam d (Mrode, 2005). The matrix Cisdis then inverted
for animal i:
i−sdis EDC matrix of animal i after taking into
s−i(Ψ*
d−i) is the EDC matrix
C−1
isd=
⎡
⎢
⎢
⎣
Cii
i
Cis
Css
i Cid
i Csd
Cdd
i
i
symm.
i
⎤
⎥
⎥
⎦
. [12]
Note that Cjk
submatrix of C−1
i is a 3 × 3 matrix and represents a
isd. The EDC matrices Ψ*
i−sd, Ψ*
s−i, and
Journal of Dairy Science Vol. 90 No. 10, 2007
Ψ*
be used to efficiently obtain the EDC matrices for all
animals.
d−iare expressed on an animal basis. The MTEDC can
Procedure for Estimating (Co)Variances
Across Countries
The EM-REML algorithm [8] was combined with the
PCG algorithm to estimate (co)variance parameters of
the multitrait MACE model [1]. An iterative procedure
developed for the parameter estimation comprised 4
steps:
1) Solve the equation system of the MT-MACE model
with PCG.
2) Compute the quadratic term for a given (co)vari-
ance component using formula [9].
3) Approximatepredictionerror(co)variancesofMen-
delian sampling estimates with formulas [11] and
[12] and the MTEDC approach (Liu et al., 2004a).
4) Estimate genetic (co)variances using the EM-
REML formula [8] and derive genetic correlation
estimates. Newly estimated (co)variances are used
to update covariances for the next iteration.
Repeat steps 1 through 4 until all parameters no
longer change at the fourth decimal place. The original
country-specific (co)variances can be used as starting
values.
SIMULATION OF DATA
Single-Trait MACE Scenarios
Recordsof5,000bulls,eachhavingprogenyin2coun-
tries recorded for a single trait, were simulated using
the following model:
yij= aij+ eij
[13]
where yijis the simulated DYD vector for bull i in coun-
try j, aijis a vector of additive genetic effects of bull i
in country j, and eijis a vector of residual effects. True
breeding values aijof the 5,000 bulls that were progeny
of 50 unrelated sires were obtained by adding half the
breeding value of their sire sijto a value uijincluding
the dam contribution and Mendelian sampling; that is,
representing three quarters of the total genetic vari-
ance. The genetic variances for the 2 countries were
arbitrarily chosen to be 4 and 6, respectively. The ge-
netic correlation between countries was 0.9. Residual
values eijfor the 2 traits were generated accounting for
variable amount of information summarized in yij. Its
residual variance was assumed to be heterogeneous:
var(eij) = σ2
e,j/ωij, where σ2
e,jis the residual variance for

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TARRE´S ET AL.
4850
country j chosen to result in heritabilities of 0.30 in
bothcountries;andtheweightωijwasarandomnumber
between 1 and 300 and represented approximately an
effective number of daughters of bull i in each country
j. A zero residual correlation was assumed between
countries. Two scenarios were studied: one scenario
with all bulls having data in both countries (100% com-
monbulls)andtheotherscenariowithonly10%ofbulls
beingcommon,whichrepresentsatypicalinternational
bull evaluation with 2 countries involved.
The performance of the approximate REML method
was compared using a version of the average informa-
tion REML (AI-REML) software. Although the latter
software was originally not developed for a model such
as [13] with heterogeneous residual variances, a simple
trick can avoid this limitation (Ducrocq et al., 2001).
Let vij=√ωij. Multiplying both sides of the model equa-
tion [13] by vij, one gets
y*
ij= vijyij= vijaij+ εij. [14]
Now, the residual part εijhas homogeneous variance:
var(εij) = v2
dispersion parameters of model [14] considering y*
the data and vijas a continuous covariate gives results
identical to the analysis of model [13] (Ducrocq et al.,
2001). The version of the AI-REML software from I.
Misztal and S. Tsuruta (Misztal et al., 2002) was modi-
fied by Druet to impose the constraint that genetic and
residual variances are fixed (Druet et al., 2003). This
constraint was necessary to allow convergence to be
obtained, due togenetic variances quickly goingto zero.
ijvar(eij) = σ2
e,j. The REML estimation of the
ijas
Multiple-Trait MACE Scenarios
Because the previous scenarios were rather simplis-
tic, we further extended the validation to check the
performance of the approximate EM-REML method by
simulating the actual milk production data from the
FrenchandGermannationalevaluations;thatis,using
the same number of traits, pedigree, EDC, and missing
pattern as in a real life situation. Records of animals
resembling milk production DYD of bulls included in
the Interbull 010 files from French and German na-
tional evaluations were simulated using the model in
equation[13].Theadditivegeneticeffectsweremodeled
in the same way as in national evaluation. Milk produc-
tion traits are evaluated in Germany using an RRTDM
in which the genetic effects of an animal per lactation
are modeled with a normalized orthogonal third-order
Legendre polynomial (Liu et al., 2001) and in France
withasinglerepeatabilitylactationmodel(Robert-Gra-
nie ´ et al., 1999). Thus, bulls with data in Germany
Journal of Dairy Science Vol. 90 No. 10, 2007
had 9 correlated traits; that is, 3 regression coefficients
(RRC) for the first 3 lactations; bulls with data in
France had data for a single trait.
The February 2006 Interbull bull pedigree file was
usedtogeneratetruebreedingvalues aijoftheanimals.
The complete pedigree file was read from the oldest to
the youngest, and the breeding value aijwas obtained
by adding half the breeding values of the sire and dam
and the Mendelian sampling term, representing half of
the total genetic variance. When the sire or dam was
unknown, half of the value of genetic groups plus a
value uijincluding the sire or dam contribution (i.e.,
representing one quarter of the total genetic variance)
were added. The values were drawn from a MVN(0,
G0) distribution, where G0is the genetic (co)variance
matrix. The (co)variance matrix of genetic effects of the
two countries and its inverse are defined as:
G0=
⎡
⎢
⎣
G011G012
G012G022
⎤
⎥
⎦
, and G−1
0 =
⎡
⎢
⎣
G11
G12
0 G12
0 G22
0
0
⎤
⎥
⎦
. [15]
The genetic (co)variance matrix G0used for simula-
tion is shown in Table 1.
Residual values eijwere also generated from values
drawn from an MVN(0,Ri) distribution, where Riis the
residual (co)variance matrix which is different for each
bull, i:
Ri=
⎡
⎢
⎣
Ri1 0
0 Ri2
⎤
⎥
⎦
, and R−1
i =
⎡
⎢
⎣
Ψi1 0
0 Ψi2
⎤
⎥
⎦
,[16]
where Rij= Var(eij) = [Ψij]−1is the error (co)variance
matrix of bull i in country j and it is the inverse of the
EDCmatrixΨijofbulliincountryj.Asusuallyassumed
inMACE,residualsfromdaughtersofabullfromdiffer-
ent countries are uncorrelated. Residual values eijwere
obtainedbyinvertingtheEDCmatrixfromthedatasets
for parameter estimation. Table 2 has the data and
pedigree structures for 2 scenarios of the MT-MACE
model, one scenario with full pedigree information in
which sires and dams of all bulls were known and the
other scenario with reduced pedigree information
where all dams of bulls were missing. The data and
pedigree structures were the same as those used for
parameter estimation of milk production traits from
Franceand GermanywithanMT-MACE model(Tarre ´s
et al., 2006). Only bulls born from 1985 to 2001 were in-
cluded.
Inclusion of a Year Effect
Nationalgeneticevaluationmaybebiasedforseveral
reasons. Examples are lack of information on selection