- Email: [email protected]
Multitrait Animal Model with Genetic Groups1
Multitrait Animal Model with Genetic Groups1 Y. DA and M. GROSSMAN2 Department of Animal Sciences University of Illinois Urbana 61801 MIXED MODEL EQUATIONS
ABSTRACT
Genetic groups of unknown parents are extended to the multitrait animal model. Computationally feasible mixed model equations are obtained. A strategy to include genetic groups for missing data is proposed. Canonical and triangular transformations can be applied to the multitrait animal model with groups if the transformations can be applied to the same model without groups. Formulations for REML estimation with groups are derived, and the results are almost as feasible as REML estimation without groups. A numerical example is given to illustrate computations of REML formulations. (Key words: multitrait, animal model, genetic group, REML)
Assumptions
For the unitrait animal model, the major assumption for genetic groups is that genetic values have nonzero means that represent different levels of genetic merit of base animals (10, 16). Other assumptions include autosomal inheritance and yach unknown parent having only one progeny. For the multitrait animal model, we shall assume 1) additive genetic values of each trait have a nonzero mean, defined as a linear function of fixed group effects of unknown parents of identified animals, to account for different genetic levels of each trait for base animals; and 2) each trait has same additive relationship between the animal and groups of unknown parents, a reasonable assumption if each trait is subjected to the same selection pressure and controlled by autosomal loci.
INTRODUCTION
The animal model with genetic groups of unknown parents (10, 16) accounts for different genetic merits of base animals, defined as unknown parents of identified animals, by assigning the unknown parents to different genetic groups. This grouping strategy has been applied to an animal model with multiple factors (17) and to one with maternal effects (15). The multitrait animal model with genetic groups and related issues remain undiscussed. The purpose of this paper is to extend genetic groups of unknown parents to the multitrait animal model. The extension includes obtaining mixed model equations that are computationally feasible, proving that canonical and triangular transformations are applicable, and deriving appropriate REML fonnulations.
Received October I, 1990. Accepted February 25, 199I. lSupported in part by the Illinois Agricultural Experiment Station, Hatch Project 35-367. 2Reprint requests. 1991 J Dairy Sci 74:3183--3195
Model and Mixed Model Equations
The animal model with genetic groups can be described in several ways (1, 10, 16, 17). Following the approach described by Da et al. (1), we write the multitrait animal model with genetic groups as
y
= Xb
+ Za +
ZA~
+ e
[lJ
where y = N x 1 vector of observations on t traits; X and Z are incidence matrices; Am =qt X nt matrix of additive genetic relationships that relates each of t traits on q animals with n genetic groups of unknown parents; b =pt x 1 vector of fixed effects (other than group effects); a = qt x 1 vector of random additive genetic effects of known animals; g = nt x 1 vector of fixed group effects; e = N x 1 vector of random residuals; and where N =number of observations, p = number of levels in b, n = number of group effects, and q = number of additive genetic effects. Also, N = qt, assuming one record per trait and no missing data.
3183
3184
DA AND GROSSMAN
Special attention is given to Am in Model [1]. The structure of Am depends on how data are ordered: animals in traits or traits in animals. This study uses animal in traits, i.e., data are arranged as
Usual assumptions for first and second moments of Model [1] are
= Xb + = ZGZ'
+R
= var(a) = Go
0 A,
E(y)
var(y)
ZA~
where
where Yi = column vector of observations on animals for trait i; i = 1 to t. By assumption 2 and by [2], Am can be expressed as
G and R
[4]
= var(e)
= Ro 0 Iq, for one record per trait and no missing data; [5]
and where [3]
A
where It = identity matrix of order t, 0 denotes the Kronecker product (13), and Ag = q x n matrix of additive genetic relationships between known animals and groups of their unknown parents (1).
Iq Go
Ro
= q x q matrix of additive relationships between identified animals, = identity matrix of order q,
= txt covariance
matrix of additive genetic values for the t traits, and = txt covariance matrix of residuals for the t traits.
The mixed model equations for Model [1] are
X'R-1Z Z'R-1Z + G-l ~Z'R-IZ or Cg
= v.
[7]
Equation [6] is not computationally feasible because Am is difficult to compute, i.e., computation of one element in Am might require a search through the entire pedigree. As shown for unitrait animal models, computationally feasible mixed model equations can be obtained (10, 16, 17). Analogous to unitrait animal models with genetic groups, we define O=A+A~
[8]
and then apply with QP transformation (11) to obtain mixed model equations that are computationally feasible and yield solutions defined by [8]. The QP transformation of [7] has the form
wa
=
[9]
where W = (p-l)'C(p-l), = (M), t = (p-l),v.
a
Journal of Dairy Science Vol. 74, No.9. 1991
[10]
3185
MULTITRAlT ANIMAL MODEL WIlli GROUPS
To obtain mixed model equations whose solution vector yields 0, we find a P such that
(6', 0', In', and then substitute Pinto [9]. The P matrix is P
=
[
IPt 0
o
a=
~qt ~m]
OInt
where Ipt, I qt , and I nt are identity matrices with orders pt, qt, and nt. The resulting mixed model equations are
In [11], we let
A_l = T.
[ _A~-l
[12]
Equation [11] is computationally feasible because the T matrix can be constructed easily using a method (10, 16) similar to Henderson's method to construct A-l (3). If there are no missing data and all traits have the same model matrix, then R in [11] has the structure of [5], and X and Z can be written as [13] [14]
X = It ® Xo Z = It ® Zo where Xo and Zo are incidence matrices for a trait. Based on [5], [12], [13J, and [14J, Equation [11] can be written as l
RO ®
~Zo
o ® X~y
6
o
RQl ® Z'azo + Gal ® T uu ~f ® Tug -G-oI ® T gu Go ® T gg
0 ][]
~
(R
=
a ® Z~y
(R [
II
']
.
0 [15]
For missing data, [5], [13], and [14] do not hold, and mixed model equations cannot be written as [15]. Equation [11] still applies in principle, but its construction is not as efficient as the strategy proposed by Henderson and Quaas (7). Their strategy was to set to zero elements corresponding to missing data in y, X, Z, and R, to keep G unchanged, and then to replace R-I in the mixed model equations by the zero-type generalized inverse of R (5), denoted by R-. We use this strategy for missing data under the multitrait model with genetic groups. In addition, we set to zero elements corresponding to missing data in ZAm, and keep Am unchanged. Note that Journal of Dairy Science Vol. 74, No.9, 1991
3186
DA AND GROSSMAN
when a row of Zm is set to zero, the same row in ZAm is set to zero automatically. It is unnecessary, therefore, to change Am for the purpose of setting to zero elements in ZAm. This is fortunate because the strategies mentioned probably could not be used if Am had to be changed for the purpose of setting to zero certain rows in ZA m. Using the described strategies, the general mixed model equations for the multirrait animal model with genetic groups and missing data are X'R-X X'R-Z Z'R-X Z'R-Z + G [
o
-G- l
c/ 0
T uu
o 0 T gu
c/ 0
-G o
Tug
]
[&] = 0 ~
Gal 0 Tgg
Z'R-y ] [X'R-Y
0
[16]
Note that in [11], [15], and [16], (} is the prediction of a nonestimable function, U = a + A~, (1, 15). However, contrasts of elements of u, such as ui - Uj, generally are estimable. For no missing data. estimability needs be examined only for one rrait, whereas for missing data, estimability may need be examined for each trait separately. CANONICAL AND TRIANGULAR TRANSFORMATIONS
a* = Q:a
A canonical transformation can be used to transform a multitrait problem into a unitrait problem if the multitrait model has no missing data and has only two random factors, including residuals (8, 9, 12, 14). For a multirrait model with missing data, a triangular transformation can be used to simplify the structure of the mixed model equations if missing data have a specific pattern (9, 12). It can be shown that canonical and triangular transformations can be applied to the animal model with genetic groups if the transformations are applicable to the same model without groups. Let the multitrait animal model without genetic groups be
y = Xb + Za + e
[17]
where terms in the model are as defined previously. Let Qy be the transformation matrix for canonical or triangular transformation. Then a canonical or triangular transformation for [17] can be performed if the following holds:
e*
= Q~e
and where, in general, Qb and ~ each has an order different from that for Qy. Equation [18] shows that incidence matrices and the transformation must satisfy the relationships
Q~ Q~Z
= XQ~ = ZQ:
[19] [20]
Suppose a canonical or triangular transformation can be applied to the model without groups, i.e., [19] and [20] hold, then the transformation for Model [17] can be applied to Model [1]:
Q~Y
= Xb*
+ Za* + ZQ~A~ + e* [21]
To complete the transformation of [21], it is necessary to prove
Q~Y = Q~(Xb) + Q~(Za) + Q~e
=
=
X(Q~b) + Z(Q~a) + Q~e Xb* + Za* + e*
where b* = Q'b b
Journal of Dairy Science Vol. 74, No.9, 1991
[22] [18]
where Qg is a matrix conformable for the multiplication of Qgg. With the assumption of animals ordered in traits, and one record per animal, transforma-
MUL1TI'RAIT ANIMAL MODEL
tion matrices in [18] and [21] can be written as
Q~ = M' ® Iq Q~ = M' ® I p = M' ® Iq
wrm
3187
GROUPS
where
Substituting [24] into [21], the transformed model is
Q:
[23)
where M is a txt matrix to diagonalize Ro for a triangular transfonnation or to diagonalize Ro and Go for a canonical transfonnation. Using [3] and [23], then [22] is proved by showing
(M' ® Iq)(I t ® Ag) = M' ® Ag = (I, ® Ag)(M' ® In> = AmQ~ [24]
Q~Y = Xb* + Za* + ZAmg* + e* where
g* = Q~. Therefore, canonical and triangular transformations can be applied to a multitrait model with genetic groups, and the only conditions required are those for a model without genetic groups.
RESTRICTED MAXIMUM LIKELIHOOD ESTIMATION REML Equations
We shall refer to equations whose solutions are REML estimates of parameters as "REML equations". The REML equations will be derived using the computationally feasible system of [16]. The REML estimates of variance-covariance components for Model [1], using mixed model Equations [6], are obtained by solving the following REML equations (4, 6): A'Qi,;4 e'Pije
= tr(QijG) = tr(PijR) -
tr(QijC aa) tr(PijFC-F')
[25] [26]
where C aa CQij Pij F ~
= var(A - a) = covariance matrix of prediction = a generalized inverse of C of [6], = G-IG~G-l = (G(/ ® A-l)G\GOI ® A-I), g = R-R~R-, = (X, Z, ZAm>,
errors of A,
[27] [28]
~
(29] [30]
= Y - Xb - ZAtd - Zll
and where
G~ = partial R~ = partial
derivative of G with respect to element ij of Go. gij [31] derivative of R with respect to element ij of
Ro,
rij' [32]
The computation of Qij [28] can be simplified, and this simplified fonnula will play a major role in deriving computationally feasible REML equations: [33] Journal of Daily Science Vol. 74, No.9, 1991
3188
DA AND GROSSMAN
where S~ is a txt matrix given by
S~
~~' = ufo,~ (~,~'
=j
[34]
ifi:;t:j
[35]
if i
=
and where
~ = column i of GOl .
[36]
To derive REM!.. equations for multitrait REM!.. estimation with genetic groups, replace d and eaa in [25] and C- in [26] with results from [16], and require that resulting equations must not need the computation of Am. For the left side of [25], we express 11 in terms of 0 and ~ using [8], [37] Substituting [37] and [33] into the left side of [25] yields 11'Qij3
= [0 - (It = O'QijO =
~ Ag)~],Qij[O
- (It ~ Ag)~ 2f'(lt ~ A~Qij)O + f'(lt ~ A~QijIt ~ Ag)~
O'(S~ ~ A-l)O - 2f'(S~ ~ A~-I)O + f'(S~ ~ A~-IAg)~ [38]
= (0',
4')Sij(O', 4')'
[39]
where
=
Sij
[s~ T~ @
o~
SIJ
T
SU
S~ ~ Tug Sijo ~ T gg
1
[40]
For the right side of [25], we express eaa in terms of a generalized inverse of the coefficient matrix of [16], denoted by W-. Using the method to derive prediction error variance in (1), we have
var
6 ]
[
0 - u
Then by [27], [37], and [41],
= W- =
i
[Wbb Wbu Wbg WU wu wug b
Wsb
eaa
u
]
wsu wgg
[41]
can be expressed as
eaa = var(d
- a) = var[(O - u) - Am
Journal of Dairy Science Vol. 74, No.9, 1991
[42]
3189
MULTITRAIT ANIMAL MODEL WI1H GROUPS
Substituting [42] and [33] into tr(QijC3a) of [25] yields tr(QijC3a) = tr(S~ ® A-1(WuU - WUgA~- AmWgu + AmWggA~)]
= tr(sg
® A-IWuu) - 2tr(Sg ®
A~-lWUS) + tr(sg ® A~-lAgW~
= tr(SijW22)
[43] (44]
where Sij is given by [40], and W 22 is a submatrix of W- given by
_ [wu wug ] . u
W22 -
Wgu
wgg
[45]
For the left side of [26], using solutions from [16], we express
e
=y
- xI) - ZOo
e of
(30] as [46]
FOf the right side of [26], using the relationship between C and W given in [10], we express FC-F' in terms of W from [16] as FC-F'
= (X,
Z)Wll(X,
zy
[47]
where
Wll -
Wbb Woo ] [ WUb
wuu
.
[48]
In [47], C- is replaced by a submatrix of W-, and Am in F of [29] is no longer needed. Summarizing (34], (43], (46], and [47], REML equations are obtained as
(0', W>Sij(O', W>' e'Pije
= tr(QijG) = tr(PijR) -
tr(SijW22)
[49]
tr[Pij(X, Z)Wll(X, Z)1
[50]
e
where Sij is defined by [40], W22 by [45], by [46], and WIl by [48]. Equations [49] and [50] are general; they apply with or without missing data and to a full-rank or a singular W. Also, Equations [49] and [50] are almost as feasible as equations fOf multitrait REML without groups given by Henderson (6). iteration AlgOrithms
To solve [49] and [50], two iteration algorithms can be used (4, 6). The first algorithm is obtained by writing G ([4]) and R ([5]) as a linear combination of 1/2(t + l)t variance and covariance components: G R
= G;lgU + G;2g12 + '" • +• = RUfH R12f12 + '"
+ G~gtt
• + Rtftt
[51] [52]
Journal of Daily Science Vol. 74, No.9, 1991
3190
DA AND GROSSMAN
where G~ is defined by [31] and R~ by [32], i and j = 1"", t. Substituting [51] into [49] and [52] into [50] yields
(0', r>SiiO',
r>'
+ tr(S ij W 22) = tr[QiiG;lgll + G;2g12 + .. , + G:gtJ]
[53]
• +• • @'Pij@ + tr[Pij(X, Z)W ll (X, Z)1 = tr[Pij(Rllrll R l2 r l2 + . ,. + RtftJ]·
[54]
Then, REML estimates are obtained by solving [53] for all gij and [54] for all rij' Let
=
1/2(t + l)t + l)t + l)t Va = 1/2(t + l)t qe = 1/2(t + l)t t e = 1/2(t + l)t He = l/2(t + l)t V e = l/2(t + l)t
qa
ta Ha
= 1/2(t = 1/2(t
x 1 vector of (0', r>S~(O', r>' x 1 vector of tr(S ij W2..) x 1/2(t + l)t matrix of tr(QijG~) x 1 vector of gij x 1 vector of matrix of @'Pr @ x 1 vector of tr[Pij(X, Z)Wh(X, Z)1 x 1/2(t + I)t matrix of tr(PijR~) x I vector of rij
so that REML estimates are
~a = H;~\qa +
~e
= H~\qe
ta>
[55]
+ le)
[56]
Note that coefficients for all gij in [53] or in H a of [55] can be simplified as ifi=jandk=l
[57]
ifi=jandk:;t:l ifi:;t:jandk:;t:l
[58] [59]
where, for example, gik is element ik of GOI . Assuming no missing data and one record per animal, then coefficients for the rij in [54] or He of [56] also can be simplified
tr(PijR~) = q(rik)2
= 2qrikrjl = 2q(rikrjl + rilrjli::)
if i = j and k = I
[60]
if i = j and k :;t: I ifi:;t:jandk:;t:l
[61] [62]
where r ik is element ik of R01. Again, assuming no missing data, P ij can be simplified as
P IJ.· -- Eij0
iO>
'01'
Iq,
[63]
where if i Journal of Dairy Science Vol. 74, No.9, 1991
=j
[64]
3191
MULTITRAIT ANIMAL MODEL WI1lI GROUPS
if i
*j
[65]
and r io
= column
i of R-0 1
[66)
Using [57] through [62], therefore, computations of the t(t + 1) matrices of order qt in [53] and [54]. or in [55] and [56]. are replaced by simple scalar operations. The second algorithm by Henderson (4) has a formulation similar to that for unitrait REML estimation proposed by Henderson (2):
~ij = [l\~A-l!ij + tr(A-IC ij)]lq iij
= [@~@j
[67]
+ tr(Bij)]lq
[68]
where
av,
Cij = cov[(ij (3· - 3y1 Bjj = submatrix of Fe-F' corresponding to traits i and j.
For REML estimation using system [16), Equations [67] and [68] can be expressed as gij = [(O~, ~)T(O;. t~' + tr(TW~)]/q
~ij
= [e~~j
+ tr(Bij)]/q
where OJ, ~j, and @j are subvectors of 0, g and e for trait i. Bij is now a submatrix of (X, Z)Wll(X., Z)' corresponding to traitS i and j, W~2 is a submatrix of W 22 corresponding to traits i and j, and has the form IJ
and where Wiju, W':l, and Wr are submatrices of WUU, WUg • wgg corresponding to traits i and j. NUMERICAL EXAMPLE
A hypothetical data set will be used to illustrate the computation of REML estimates using formulations derived in this paper. The data set is designed to have three traits (t = 3). five animals (q = 5), two fixed effects (p =2), one record per animal, and no missing data (N = 3 x 5 = 15). The pedigree from Da et at (1) will be used. where two groups of unknown parents (n = 2) were defined. Observations on the three traits are
Journal of Daby Science Vol. 74, No.9, 1991
3192
DA AND GROSSMAN
Yl = (34, 37, 35, 41, 38)', Y2 = (11, 12, 8, 10, 13)', Y3 = (5, 3, 4, 6, 8)'
The X and Z matrices for each of the three traits are
Xo= Zo
1 1 1 0 0 [0 0 0 1 1
= identity
J'
matrix of order 5.
The T-matrix [12] is the same as that of Da et al. (1): 3
1 2 -1
2 Tuu = A-I =
1
11
2
6"
-1
-1
2
2
0
-3
0
3
0
-3
3
0
0
3
7
-1
0
0
-3
0
0
0
-3
2
4
2
4
Tug = (-1/2, -1/2, 1/3, 0, -2/3)', Tgg = 5/6. Note that Tug and Tgg are obtained after restricting solution of group 2 to zero (1). The starting values for Go and Ro are chosen to be
Go
=
Ro=
[gil g21 g31
[r
ll
r2I r31
g12 g22 g32
g13 g23 g33
r12 r22 r32
rl3 r23 r33
]
=[
~O189 .0567 -1.4075
] [.no! =
.7651 -.0775
.0567 .1286 -.0268
-1.4073 -.0268 .9823
.7651 4.3139 -.1043
-.0775 -.1043 .0438
J J
The inverse of Go is gil
gl2
g21
g22
g23
[ g31
g32
g33
g13]
=
[10,004.6301-1432.1896 14,294,1396] -1432.1896 212.8423 -2046.0311 . 14,294.1396-2046.0311 20,423.8104
Solutions to the mixed model equations, which can be constructed by [15] or [16] because there are no missing data, are Journal of Dairy Science Vol. 74. No.9.
1991
3193
MULTITRAIT ANIMAL MODEL WITII GROUPS
b = (37.67, 41.50, 7.31, 8.46, 2.46, 5.69)' fi 1 = (-3.73, -1.01, -2.27, -.50, -3.50)' fi2 = (3.00, 3.07, 3.00, 1.54, 4.54)' fi3 = (2.51, .61, 1.49, .31, 2.31)' g = (-4.74, 6.07, 3.12)' The REML estimates are computed using [55] and [56]. From [34J through [36], we compute the Sij as o 10004,6 ] -1432.2 [10004.6 -1432.2
I1 So -
[
=
12 _
So -
100,092,623.9 -14,328,527.6 2,051,167.2 [ 143,007,579.2 -20,471,918.6 -14,328,527.6
10,004.6 -1432.2] -1432.2 212.8 [ 14,294.1
-2046.0
-28,657,055,2 =
14294.1]
14294.1
-4,180,575.7
[ -40,941,702.5
[
143,007,579.2 -20,471 ,918.6 204,322,425.8
212.8 -1432.2 10,004.6 -1432.2
4,180,575.7 -609,661.1 -5,972,702.1
]
-2046.0 ] 14,292.1
-40,941,702.5 ] -5,972,702.1 -58,492,507.0
13 S22 . imil'ar manner. and S0' 0' S23 0' an d S33 0 are computed mas
Substituting S~ into [38] or [39] yields the left side of [53]: qa
=
Oi'Q11 ll, a'Q1211, Il'Q1311, 1l'~2a, Il'Q2311, §'Q33a)'
=
(3.1433, -.7624, 13.598, .2046, -1.9839, 14.8833)'
Substituting S~, into [43] or [44] yields ta of [55]; ta = [tr(Qll caa), tr(QI2caa), tr(Q13caa), tr(Q22caa), tr(Q23caa), tr(Q33caa)]' = (40871.8, -11708.7, 116804.0, 877.5, -16727.5, 83449.9)'.
The computation of H a in [55] is illustrated by computing three elements in H a. By [57], Journal of Daily Science Vol. 74, No.9, 1991
3194
DA AND GROSSMAN
element (1, 1) of H a is
By [58], element (1. 2) of H a is tr(Q12G ;1)
= 2q(g]2g 11) = 10(-1432.1896)(10004.6301) = -143285276.2.
By [59], element (2, 3) of H a is tr(Q12G;~
= 2q(gllg23
+ g13g21)
= 10(10004.6301}(-2046.031l)
= -409417025.5.
+ (14294.1396)(-1432.1896}J
Then, REML estimates for genetic variance and covariance components are
ta
=
~lb t12, t13, ~22, ~23, ~33)'
= u;-l(qa
+ ta>
= (2.0194, .0626, -1.4071, .1284, -.0309..9818)'
For residual variance-covariance components, all Pij can be computed by [63] through (66], and He by (60J through (62]. Then, using (46] and (47], qe and te in (56] are qe
Ie
= (2.2348,
.1903, 4.1924, .358, .466, 2.0246)'
= (4870.6, -1377.2, 15390.8, 98.6, -2187.6, 12212.4)'
and REML estimates for residual variance-covariance components are ge = (i'll, i 12 • i'13' i'22' i'23' i'33)' = H;l(qe + tJ = (.2226, -.1545, -.154, 4.7009, 5185, .1436)' CONCLUSIONS
Mixed model equations for a multitrait animal model with genetic groups can be constructed almost as easily as without groups, and missing data cause no additional difficulty. Canonical and triangular transformations can be applied to the multitrait animal model and the only conditions are those required by these transfonnations under the same model without groups. Formulations for REML estimation with groups can be obtained and made almost as feasible as those without groups. These results imply that applications of a multitrait Journal of Dairy Science Vol. 74, No.9. 199]
animal model can include genetic groups with little additional computation. ACKNOWLEDGMENT
The authors wish to thank the reviewers and R L. Fernando for their helpful comments and suggestions. REFERENCES
] Da. Y., M
Grossman, llDd I. MiszIal. 1989. Prediction variaDcc and restricted maximum likelihood estimation for animal model with relationship grouping. J. Dairy Sci 72:2125. enol"
MUL'ITI'RAIT ANIMAL MODEL WI'IH GROUPS
2 Henderson, C. R. 1973. Sire evaluation and genetic
Coop~
trends. Page 10 ill Proc. of the Anim. Breeding Genet. Symp. in Honor of Dr. lay L. Lu5h. Am. Soc. Anim. Sci. and Am. Dairy Sci. Assoc., Ownpaign, n... 3 Henderson, C. R. 1976. A simple method !oJ' computiD& the inverse of a oumerator-relationship matrix used in prediction of breeding values. Biometrics 32: 69. 4 Henderson, C. R. 1984. ANOVA, MIVQUE, REML, and ML algorithms for estimation of variances and covariances. Page 257 in Statistics: an appraisal. Proc. SOth Anniversary Cont. H. A. David and H. T. David, ed. Iowa Sr.ate Univ. Press, Ames. S Henderson, C. R. 1984. Page 362 in Applications of linear models in animal breeding. Univ. Guelph, Guelph, ON. 6 Henderson, C. R 1986. Recent developments in variance and covariance estimation. 1. AninI. Sci. 208: 216. 7 Henderson, C. R., and R. L. Quaas. 1976. Multiple trait evaluation usiD& relatives' records. 1. AninI. Sci. 43:1188. 8 Meyer, K. 1985. Maximum likelihood estimation of variance components for a mnItivariate mixed model with equal design matricell. Biometrics 41:153. 9 Pollak, E. 1. 1984. Transformation for systematically missing data to facilitate multiple trait evaluations. Page 55 in Genetic Research. Report to Eastern AI
NY.
3195
Inc., Dep. Anim. Sci., Cornell Univ., Ithaca,
10 Quaas. R. L. 1988. Additive geuetic model with groups and relationships. J. Dairy Sci. 71:1338. II Quaas. R. L., and B. J. Pollak. 1981. Modified cquatioIls for sire models with groups. 1. Dairy Sci. 64:
1868. 12 Scbadfer, L. R., and B. W. Kennedy. 1986. ComputiDg solutions to the mixed model equations. Pmc. 3rd World Coogr. Genet. Appli. Livest. Prod. Lincoln, NB XII:391. 13 Searle, S. R. 1982. Page 263 in Matrix algebra useful for statistics. lohn Wiley &: Sons. Inc., New York,
NY. 14 Thompson, R. 1976. Estimation of quantitative genetic parameters. Page 639 in Pmc. Int. Con!. Quantitative Genet. B. Pollak, O. Kempthome, and T. B. Bailey, ed. Iowa State Univ. Press, Ames. IS Van Vleck, L D. 1990. Breeding value prediction with maternal genetic group effects. 1. Anim. Sci. 68:
3998. 16 Westell, R. A., R. L. Quaas, and L. D. Van Vleck. 1988. Genetic groups in an animal model. 1. Dairy Sci. 71:1310. 17 Wiggans, G. R., I. Misztal, and L. D. Van Vleck. 1988. Animal model evaluation of Ayrshire milk yield with all lactations, herd-sire interaction and groups based on unknown parents. J. Dairy Sci. 71:1319.
Jownal of Dairy Science Vol. 74, No.9, 1991
ABSTRACT
Genetic groups of unknown parents are extended to the multitrait animal model. Computationally feasible mixed model equations are obtained. A strategy to include genetic groups for missing data is proposed. Canonical and triangular transformations can be applied to the multitrait animal model with groups if the transformations can be applied to the same model without groups. Formulations for REML estimation with groups are derived, and the results are almost as feasible as REML estimation without groups. A numerical example is given to illustrate computations of REML formulations. (Key words: multitrait, animal model, genetic group, REML)
Assumptions
For the unitrait animal model, the major assumption for genetic groups is that genetic values have nonzero means that represent different levels of genetic merit of base animals (10, 16). Other assumptions include autosomal inheritance and yach unknown parent having only one progeny. For the multitrait animal model, we shall assume 1) additive genetic values of each trait have a nonzero mean, defined as a linear function of fixed group effects of unknown parents of identified animals, to account for different genetic levels of each trait for base animals; and 2) each trait has same additive relationship between the animal and groups of unknown parents, a reasonable assumption if each trait is subjected to the same selection pressure and controlled by autosomal loci.
INTRODUCTION
The animal model with genetic groups of unknown parents (10, 16) accounts for different genetic merits of base animals, defined as unknown parents of identified animals, by assigning the unknown parents to different genetic groups. This grouping strategy has been applied to an animal model with multiple factors (17) and to one with maternal effects (15). The multitrait animal model with genetic groups and related issues remain undiscussed. The purpose of this paper is to extend genetic groups of unknown parents to the multitrait animal model. The extension includes obtaining mixed model equations that are computationally feasible, proving that canonical and triangular transformations are applicable, and deriving appropriate REML fonnulations.
Received October I, 1990. Accepted February 25, 199I. lSupported in part by the Illinois Agricultural Experiment Station, Hatch Project 35-367. 2Reprint requests. 1991 J Dairy Sci 74:3183--3195
Model and Mixed Model Equations
The animal model with genetic groups can be described in several ways (1, 10, 16, 17). Following the approach described by Da et al. (1), we write the multitrait animal model with genetic groups as
y
= Xb
+ Za +
ZA~
+ e
[lJ
where y = N x 1 vector of observations on t traits; X and Z are incidence matrices; Am =qt X nt matrix of additive genetic relationships that relates each of t traits on q animals with n genetic groups of unknown parents; b =pt x 1 vector of fixed effects (other than group effects); a = qt x 1 vector of random additive genetic effects of known animals; g = nt x 1 vector of fixed group effects; e = N x 1 vector of random residuals; and where N =number of observations, p = number of levels in b, n = number of group effects, and q = number of additive genetic effects. Also, N = qt, assuming one record per trait and no missing data.
3183
3184
DA AND GROSSMAN
Special attention is given to Am in Model [1]. The structure of Am depends on how data are ordered: animals in traits or traits in animals. This study uses animal in traits, i.e., data are arranged as
Usual assumptions for first and second moments of Model [1] are
= Xb + = ZGZ'
+R
= var(a) = Go
0 A,
E(y)
var(y)
ZA~
where
where Yi = column vector of observations on animals for trait i; i = 1 to t. By assumption 2 and by [2], Am can be expressed as
G and R
[4]
= var(e)
= Ro 0 Iq, for one record per trait and no missing data; [5]
and where [3]
A
where It = identity matrix of order t, 0 denotes the Kronecker product (13), and Ag = q x n matrix of additive genetic relationships between known animals and groups of their unknown parents (1).
Iq Go
Ro
= q x q matrix of additive relationships between identified animals, = identity matrix of order q,
= txt covariance
matrix of additive genetic values for the t traits, and = txt covariance matrix of residuals for the t traits.
The mixed model equations for Model [1] are
X'R-1Z Z'R-1Z + G-l ~Z'R-IZ or Cg
= v.
[7]
Equation [6] is not computationally feasible because Am is difficult to compute, i.e., computation of one element in Am might require a search through the entire pedigree. As shown for unitrait animal models, computationally feasible mixed model equations can be obtained (10, 16, 17). Analogous to unitrait animal models with genetic groups, we define O=A+A~
[8]
and then apply with QP transformation (11) to obtain mixed model equations that are computationally feasible and yield solutions defined by [8]. The QP transformation of [7] has the form
wa
=
[9]
where W = (p-l)'C(p-l), = (M), t = (p-l),v.
a
Journal of Dairy Science Vol. 74, No.9. 1991
[10]
3185
MULTITRAlT ANIMAL MODEL WIlli GROUPS
To obtain mixed model equations whose solution vector yields 0, we find a P such that
(6', 0', In', and then substitute Pinto [9]. The P matrix is P
=
[
IPt 0
o
a=
~qt ~m]
OInt
where Ipt, I qt , and I nt are identity matrices with orders pt, qt, and nt. The resulting mixed model equations are
In [11], we let
A_l = T.
[ _A~-l
[12]
Equation [11] is computationally feasible because the T matrix can be constructed easily using a method (10, 16) similar to Henderson's method to construct A-l (3). If there are no missing data and all traits have the same model matrix, then R in [11] has the structure of [5], and X and Z can be written as [13] [14]
X = It ® Xo Z = It ® Zo where Xo and Zo are incidence matrices for a trait. Based on [5], [12], [13J, and [14J, Equation [11] can be written as l
RO ®
~Zo
o ® X~y
6
o
RQl ® Z'azo + Gal ® T uu ~f ® Tug -G-oI ® T gu Go ® T gg
0 ][]
~
(R
=
a ® Z~y
(R [
II
']
.
0 [15]
For missing data, [5], [13], and [14] do not hold, and mixed model equations cannot be written as [15]. Equation [11] still applies in principle, but its construction is not as efficient as the strategy proposed by Henderson and Quaas (7). Their strategy was to set to zero elements corresponding to missing data in y, X, Z, and R, to keep G unchanged, and then to replace R-I in the mixed model equations by the zero-type generalized inverse of R (5), denoted by R-. We use this strategy for missing data under the multitrait model with genetic groups. In addition, we set to zero elements corresponding to missing data in ZAm, and keep Am unchanged. Note that Journal of Dairy Science Vol. 74, No.9, 1991
3186
DA AND GROSSMAN
when a row of Zm is set to zero, the same row in ZAm is set to zero automatically. It is unnecessary, therefore, to change Am for the purpose of setting to zero elements in ZAm. This is fortunate because the strategies mentioned probably could not be used if Am had to be changed for the purpose of setting to zero certain rows in ZA m. Using the described strategies, the general mixed model equations for the multirrait animal model with genetic groups and missing data are X'R-X X'R-Z Z'R-X Z'R-Z + G [
o
-G- l
c/ 0
T uu
o 0 T gu
c/ 0
-G o
Tug
]
[&] = 0 ~
Gal 0 Tgg
Z'R-y ] [X'R-Y
0
[16]
Note that in [11], [15], and [16], (} is the prediction of a nonestimable function, U = a + A~, (1, 15). However, contrasts of elements of u, such as ui - Uj, generally are estimable. For no missing data. estimability needs be examined only for one rrait, whereas for missing data, estimability may need be examined for each trait separately. CANONICAL AND TRIANGULAR TRANSFORMATIONS
a* = Q:a
A canonical transformation can be used to transform a multitrait problem into a unitrait problem if the multitrait model has no missing data and has only two random factors, including residuals (8, 9, 12, 14). For a multirrait model with missing data, a triangular transformation can be used to simplify the structure of the mixed model equations if missing data have a specific pattern (9, 12). It can be shown that canonical and triangular transformations can be applied to the animal model with genetic groups if the transformations are applicable to the same model without groups. Let the multitrait animal model without genetic groups be
y = Xb + Za + e
[17]
where terms in the model are as defined previously. Let Qy be the transformation matrix for canonical or triangular transformation. Then a canonical or triangular transformation for [17] can be performed if the following holds:
e*
= Q~e
and where, in general, Qb and ~ each has an order different from that for Qy. Equation [18] shows that incidence matrices and the transformation must satisfy the relationships
Q~ Q~Z
= XQ~ = ZQ:
[19] [20]
Suppose a canonical or triangular transformation can be applied to the model without groups, i.e., [19] and [20] hold, then the transformation for Model [17] can be applied to Model [1]:
Q~Y
= Xb*
+ Za* + ZQ~A~ + e* [21]
To complete the transformation of [21], it is necessary to prove
Q~Y = Q~(Xb) + Q~(Za) + Q~e
=
=
X(Q~b) + Z(Q~a) + Q~e Xb* + Za* + e*
where b* = Q'b b
Journal of Dairy Science Vol. 74, No.9, 1991
[22] [18]
where Qg is a matrix conformable for the multiplication of Qgg. With the assumption of animals ordered in traits, and one record per animal, transforma-
MUL1TI'RAIT ANIMAL MODEL
tion matrices in [18] and [21] can be written as
Q~ = M' ® Iq Q~ = M' ® I p = M' ® Iq
wrm
3187
GROUPS
where
Substituting [24] into [21], the transformed model is
Q:
[23)
where M is a txt matrix to diagonalize Ro for a triangular transfonnation or to diagonalize Ro and Go for a canonical transfonnation. Using [3] and [23], then [22] is proved by showing
(M' ® Iq)(I t ® Ag) = M' ® Ag = (I, ® Ag)(M' ® In> = AmQ~ [24]
Q~Y = Xb* + Za* + ZAmg* + e* where
g* = Q~. Therefore, canonical and triangular transformations can be applied to a multitrait model with genetic groups, and the only conditions required are those for a model without genetic groups.
RESTRICTED MAXIMUM LIKELIHOOD ESTIMATION REML Equations
We shall refer to equations whose solutions are REML estimates of parameters as "REML equations". The REML equations will be derived using the computationally feasible system of [16]. The REML estimates of variance-covariance components for Model [1], using mixed model Equations [6], are obtained by solving the following REML equations (4, 6): A'Qi,;4 e'Pije
= tr(QijG) = tr(PijR) -
tr(QijC aa) tr(PijFC-F')
[25] [26]
where C aa CQij Pij F ~
= var(A - a) = covariance matrix of prediction = a generalized inverse of C of [6], = G-IG~G-l = (G(/ ® A-l)G\GOI ® A-I), g = R-R~R-, = (X, Z, ZAm>,
errors of A,
[27] [28]
~
(29] [30]
= Y - Xb - ZAtd - Zll
and where
G~ = partial R~ = partial
derivative of G with respect to element ij of Go. gij [31] derivative of R with respect to element ij of
Ro,
rij' [32]
The computation of Qij [28] can be simplified, and this simplified fonnula will play a major role in deriving computationally feasible REML equations: [33] Journal of Daily Science Vol. 74, No.9, 1991
3188
DA AND GROSSMAN
where S~ is a txt matrix given by
S~
~~' = ufo,~ (~,~'
=j
[34]
ifi:;t:j
[35]
if i
=
and where
~ = column i of GOl .
[36]
To derive REM!.. equations for multitrait REM!.. estimation with genetic groups, replace d and eaa in [25] and C- in [26] with results from [16], and require that resulting equations must not need the computation of Am. For the left side of [25], we express 11 in terms of 0 and ~ using [8], [37] Substituting [37] and [33] into the left side of [25] yields 11'Qij3
= [0 - (It = O'QijO =
~ Ag)~],Qij[O
- (It ~ Ag)~ 2f'(lt ~ A~Qij)O + f'(lt ~ A~QijIt ~ Ag)~
O'(S~ ~ A-l)O - 2f'(S~ ~ A~-I)O + f'(S~ ~ A~-IAg)~ [38]
= (0',
4')Sij(O', 4')'
[39]
where
=
Sij
[s~ T~ @
o~
SIJ
T
SU
S~ ~ Tug Sijo ~ T gg
1
[40]
For the right side of [25], we express eaa in terms of a generalized inverse of the coefficient matrix of [16], denoted by W-. Using the method to derive prediction error variance in (1), we have
var
6 ]
[
0 - u
Then by [27], [37], and [41],
= W- =
i
[Wbb Wbu Wbg WU wu wug b
Wsb
eaa
u
]
wsu wgg
[41]
can be expressed as
eaa = var(d
- a) = var[(O - u) - Am
Journal of Dairy Science Vol. 74, No.9, 1991
[42]
3189
MULTITRAIT ANIMAL MODEL WI1H GROUPS
Substituting [42] and [33] into tr(QijC3a) of [25] yields tr(QijC3a) = tr(S~ ® A-1(WuU - WUgA~- AmWgu + AmWggA~)]
= tr(sg
® A-IWuu) - 2tr(Sg ®
A~-lWUS) + tr(sg ® A~-lAgW~
= tr(SijW22)
[43] (44]
where Sij is given by [40], and W 22 is a submatrix of W- given by
_ [wu wug ] . u
W22 -
Wgu
wgg
[45]
For the left side of [26], using solutions from [16], we express
e
=y
- xI) - ZOo
e of
(30] as [46]
FOf the right side of [26], using the relationship between C and W given in [10], we express FC-F' in terms of W from [16] as FC-F'
= (X,
Z)Wll(X,
zy
[47]
where
Wll -
Wbb Woo ] [ WUb
wuu
.
[48]
In [47], C- is replaced by a submatrix of W-, and Am in F of [29] is no longer needed. Summarizing (34], (43], (46], and [47], REML equations are obtained as
(0', W>Sij(O', W>' e'Pije
= tr(QijG) = tr(PijR) -
tr(SijW22)
[49]
tr[Pij(X, Z)Wll(X, Z)1
[50]
e
where Sij is defined by [40], W22 by [45], by [46], and WIl by [48]. Equations [49] and [50] are general; they apply with or without missing data and to a full-rank or a singular W. Also, Equations [49] and [50] are almost as feasible as equations fOf multitrait REML without groups given by Henderson (6). iteration AlgOrithms
To solve [49] and [50], two iteration algorithms can be used (4, 6). The first algorithm is obtained by writing G ([4]) and R ([5]) as a linear combination of 1/2(t + l)t variance and covariance components: G R
= G;lgU + G;2g12 + '" • +• = RUfH R12f12 + '"
+ G~gtt
• + Rtftt
[51] [52]
Journal of Daily Science Vol. 74, No.9, 1991
3190
DA AND GROSSMAN
where G~ is defined by [31] and R~ by [32], i and j = 1"", t. Substituting [51] into [49] and [52] into [50] yields
(0', r>SiiO',
r>'
+ tr(S ij W 22) = tr[QiiG;lgll + G;2g12 + .. , + G:gtJ]
[53]
• +• • @'Pij@ + tr[Pij(X, Z)W ll (X, Z)1 = tr[Pij(Rllrll R l2 r l2 + . ,. + RtftJ]·
[54]
Then, REML estimates are obtained by solving [53] for all gij and [54] for all rij' Let
=
1/2(t + l)t + l)t + l)t Va = 1/2(t + l)t qe = 1/2(t + l)t t e = 1/2(t + l)t He = l/2(t + l)t V e = l/2(t + l)t
qa
ta Ha
= 1/2(t = 1/2(t
x 1 vector of (0', r>S~(O', r>' x 1 vector of tr(S ij W2..) x 1/2(t + l)t matrix of tr(QijG~) x 1 vector of gij x 1 vector of matrix of @'Pr @ x 1 vector of tr[Pij(X, Z)Wh(X, Z)1 x 1/2(t + I)t matrix of tr(PijR~) x I vector of rij
so that REML estimates are
~a = H;~\qa +
~e
= H~\qe
ta>
[55]
+ le)
[56]
Note that coefficients for all gij in [53] or in H a of [55] can be simplified as ifi=jandk=l
[57]
ifi=jandk:;t:l ifi:;t:jandk:;t:l
[58] [59]
where, for example, gik is element ik of GOI . Assuming no missing data and one record per animal, then coefficients for the rij in [54] or He of [56] also can be simplified
tr(PijR~) = q(rik)2
= 2qrikrjl = 2q(rikrjl + rilrjli::)
if i = j and k = I
[60]
if i = j and k :;t: I ifi:;t:jandk:;t:l
[61] [62]
where r ik is element ik of R01. Again, assuming no missing data, P ij can be simplified as
P IJ.· -- Eij0
iO>
'01'
Iq,
[63]
where if i Journal of Dairy Science Vol. 74, No.9, 1991
=j
[64]
3191
MULTITRAIT ANIMAL MODEL WI1lI GROUPS
if i
*j
[65]
and r io
= column
i of R-0 1
[66)
Using [57] through [62], therefore, computations of the t(t + 1) matrices of order qt in [53] and [54]. or in [55] and [56]. are replaced by simple scalar operations. The second algorithm by Henderson (4) has a formulation similar to that for unitrait REML estimation proposed by Henderson (2):
~ij = [l\~A-l!ij + tr(A-IC ij)]lq iij
= [@~@j
[67]
+ tr(Bij)]lq
[68]
where
av,
Cij = cov[(ij (3· - 3y1 Bjj = submatrix of Fe-F' corresponding to traits i and j.
For REML estimation using system [16), Equations [67] and [68] can be expressed as gij = [(O~, ~)T(O;. t~' + tr(TW~)]/q
~ij
= [e~~j
+ tr(Bij)]/q
where OJ, ~j, and @j are subvectors of 0, g and e for trait i. Bij is now a submatrix of (X, Z)Wll(X., Z)' corresponding to traitS i and j, W~2 is a submatrix of W 22 corresponding to traits i and j, and has the form IJ
and where Wiju, W':l, and Wr are submatrices of WUU, WUg • wgg corresponding to traits i and j. NUMERICAL EXAMPLE
A hypothetical data set will be used to illustrate the computation of REML estimates using formulations derived in this paper. The data set is designed to have three traits (t = 3). five animals (q = 5), two fixed effects (p =2), one record per animal, and no missing data (N = 3 x 5 = 15). The pedigree from Da et at (1) will be used. where two groups of unknown parents (n = 2) were defined. Observations on the three traits are
Journal of Daby Science Vol. 74, No.9, 1991
3192
DA AND GROSSMAN
Yl = (34, 37, 35, 41, 38)', Y2 = (11, 12, 8, 10, 13)', Y3 = (5, 3, 4, 6, 8)'
The X and Z matrices for each of the three traits are
Xo= Zo
1 1 1 0 0 [0 0 0 1 1
= identity
J'
matrix of order 5.
The T-matrix [12] is the same as that of Da et al. (1): 3
1 2 -1
2 Tuu = A-I =
1
11
2
6"
-1
-1
2
2
0
-3
0
3
0
-3
3
0
0
3
7
-1
0
0
-3
0
0
0
-3
2
4
2
4
Tug = (-1/2, -1/2, 1/3, 0, -2/3)', Tgg = 5/6. Note that Tug and Tgg are obtained after restricting solution of group 2 to zero (1). The starting values for Go and Ro are chosen to be
Go
=
Ro=
[gil g21 g31
[r
ll
r2I r31
g12 g22 g32
g13 g23 g33
r12 r22 r32
rl3 r23 r33
]
=[
~O189 .0567 -1.4075
] [.no! =
.7651 -.0775
.0567 .1286 -.0268
-1.4073 -.0268 .9823
.7651 4.3139 -.1043
-.0775 -.1043 .0438
J J
The inverse of Go is gil
gl2
g21
g22
g23
[ g31
g32
g33
g13]
=
[10,004.6301-1432.1896 14,294,1396] -1432.1896 212.8423 -2046.0311 . 14,294.1396-2046.0311 20,423.8104
Solutions to the mixed model equations, which can be constructed by [15] or [16] because there are no missing data, are Journal of Dairy Science Vol. 74. No.9.
1991
3193
MULTITRAIT ANIMAL MODEL WITII GROUPS
b = (37.67, 41.50, 7.31, 8.46, 2.46, 5.69)' fi 1 = (-3.73, -1.01, -2.27, -.50, -3.50)' fi2 = (3.00, 3.07, 3.00, 1.54, 4.54)' fi3 = (2.51, .61, 1.49, .31, 2.31)' g = (-4.74, 6.07, 3.12)' The REML estimates are computed using [55] and [56]. From [34J through [36], we compute the Sij as o 10004,6 ] -1432.2 [10004.6 -1432.2
I1 So -
[
=
12 _
So -
100,092,623.9 -14,328,527.6 2,051,167.2 [ 143,007,579.2 -20,471,918.6 -14,328,527.6
10,004.6 -1432.2] -1432.2 212.8 [ 14,294.1
-2046.0
-28,657,055,2 =
14294.1]
14294.1
-4,180,575.7
[ -40,941,702.5
[
143,007,579.2 -20,471 ,918.6 204,322,425.8
212.8 -1432.2 10,004.6 -1432.2
4,180,575.7 -609,661.1 -5,972,702.1
]
-2046.0 ] 14,292.1
-40,941,702.5 ] -5,972,702.1 -58,492,507.0
13 S22 . imil'ar manner. and S0' 0' S23 0' an d S33 0 are computed mas
Substituting S~ into [38] or [39] yields the left side of [53]: qa
=
Oi'Q11 ll, a'Q1211, Il'Q1311, 1l'~2a, Il'Q2311, §'Q33a)'
=
(3.1433, -.7624, 13.598, .2046, -1.9839, 14.8833)'
Substituting S~, into [43] or [44] yields ta of [55]; ta = [tr(Qll caa), tr(QI2caa), tr(Q13caa), tr(Q22caa), tr(Q23caa), tr(Q33caa)]' = (40871.8, -11708.7, 116804.0, 877.5, -16727.5, 83449.9)'.
The computation of H a in [55] is illustrated by computing three elements in H a. By [57], Journal of Daily Science Vol. 74, No.9, 1991
3194
DA AND GROSSMAN
element (1, 1) of H a is
By [58], element (1. 2) of H a is tr(Q12G ;1)
= 2q(g]2g 11) = 10(-1432.1896)(10004.6301) = -143285276.2.
By [59], element (2, 3) of H a is tr(Q12G;~
= 2q(gllg23
+ g13g21)
= 10(10004.6301}(-2046.031l)
= -409417025.5.
+ (14294.1396)(-1432.1896}J
Then, REML estimates for genetic variance and covariance components are
ta
=
~lb t12, t13, ~22, ~23, ~33)'
= u;-l(qa
+ ta>
= (2.0194, .0626, -1.4071, .1284, -.0309..9818)'
For residual variance-covariance components, all Pij can be computed by [63] through (66], and He by (60J through (62]. Then, using (46] and (47], qe and te in (56] are qe
Ie
= (2.2348,
.1903, 4.1924, .358, .466, 2.0246)'
= (4870.6, -1377.2, 15390.8, 98.6, -2187.6, 12212.4)'
and REML estimates for residual variance-covariance components are ge = (i'll, i 12 • i'13' i'22' i'23' i'33)' = H;l(qe + tJ = (.2226, -.1545, -.154, 4.7009, 5185, .1436)' CONCLUSIONS
Mixed model equations for a multitrait animal model with genetic groups can be constructed almost as easily as without groups, and missing data cause no additional difficulty. Canonical and triangular transformations can be applied to the multitrait animal model and the only conditions are those required by these transfonnations under the same model without groups. Formulations for REML estimation with groups can be obtained and made almost as feasible as those without groups. These results imply that applications of a multitrait Journal of Dairy Science Vol. 74, No.9. 199]
animal model can include genetic groups with little additional computation. ACKNOWLEDGMENT
The authors wish to thank the reviewers and R L. Fernando for their helpful comments and suggestions. REFERENCES
] Da. Y., M
Grossman, llDd I. MiszIal. 1989. Prediction variaDcc and restricted maximum likelihood estimation for animal model with relationship grouping. J. Dairy Sci 72:2125. enol"
MUL'ITI'RAIT ANIMAL MODEL WI'IH GROUPS
2 Henderson, C. R. 1973. Sire evaluation and genetic
Coop~
trends. Page 10 ill Proc. of the Anim. Breeding Genet. Symp. in Honor of Dr. lay L. Lu5h. Am. Soc. Anim. Sci. and Am. Dairy Sci. Assoc., Ownpaign, n... 3 Henderson, C. R. 1976. A simple method !oJ' computiD& the inverse of a oumerator-relationship matrix used in prediction of breeding values. Biometrics 32: 69. 4 Henderson, C. R. 1984. ANOVA, MIVQUE, REML, and ML algorithms for estimation of variances and covariances. Page 257 in Statistics: an appraisal. Proc. SOth Anniversary Cont. H. A. David and H. T. David, ed. Iowa Sr.ate Univ. Press, Ames. S Henderson, C. R. 1984. Page 362 in Applications of linear models in animal breeding. Univ. Guelph, Guelph, ON. 6 Henderson, C. R 1986. Recent developments in variance and covariance estimation. 1. AninI. Sci. 208: 216. 7 Henderson, C. R., and R. L. Quaas. 1976. Multiple trait evaluation usiD& relatives' records. 1. AninI. Sci. 43:1188. 8 Meyer, K. 1985. Maximum likelihood estimation of variance components for a mnItivariate mixed model with equal design matricell. Biometrics 41:153. 9 Pollak, E. 1. 1984. Transformation for systematically missing data to facilitate multiple trait evaluations. Page 55 in Genetic Research. Report to Eastern AI
NY.
3195
Inc., Dep. Anim. Sci., Cornell Univ., Ithaca,
10 Quaas. R. L. 1988. Additive geuetic model with groups and relationships. J. Dairy Sci. 71:1338. II Quaas. R. L., and B. J. Pollak. 1981. Modified cquatioIls for sire models with groups. 1. Dairy Sci. 64:
1868. 12 Scbadfer, L. R., and B. W. Kennedy. 1986. ComputiDg solutions to the mixed model equations. Pmc. 3rd World Coogr. Genet. Appli. Livest. Prod. Lincoln, NB XII:391. 13 Searle, S. R. 1982. Page 263 in Matrix algebra useful for statistics. lohn Wiley &: Sons. Inc., New York,
NY. 14 Thompson, R. 1976. Estimation of quantitative genetic parameters. Page 639 in Pmc. Int. Con!. Quantitative Genet. B. Pollak, O. Kempthome, and T. B. Bailey, ed. Iowa State Univ. Press, Ames. IS Van Vleck, L D. 1990. Breeding value prediction with maternal genetic group effects. 1. Anim. Sci. 68:
3998. 16 Westell, R. A., R. L. Quaas, and L. D. Van Vleck. 1988. Genetic groups in an animal model. 1. Dairy Sci. 71:1310. 17 Wiggans, G. R., I. Misztal, and L. D. Van Vleck. 1988. Animal model evaluation of Ayrshire milk yield with all lactations, herd-sire interaction and groups based on unknown parents. J. Dairy Sci. 71:1319.
Jownal of Dairy Science Vol. 74, No.9, 1991