E S T I M A T I O N OF C O M P O N E N T S OF V A R I A N C E A N D C O V A R I A N C E
BY S Y M M E T R I C D I F F E R E N C E S S Q U A R E D
AND MINIMUM NORM QUADRATIC UNBIASED ESTIMATION:
A C O M P A R ISON i
J. w. Keele 2 and W. R. Harvey 3
The Ohio State University, Columbus 43210
ABSTRACT
Standard errors were obtained of four different estimators of components of variance
and covariance applied to five specific data set designs. The designs were for estimating the
genetic and environmental components of variance and covariance of a maternal model.
The four estimators were symmetric differences squared (SDS), minimum norm quadratic
unbiased estimation (MINQUE) with all prior values for the components of variance and
covariance set to zero except for the environmental variance (MINQUE(O)), MINQUE with
all prior values for the components of variance and covariance set to 1 (MINQUE(1)), and
minimum variance quadratic unbiased estimation (MIVQUE). The standard errors of SDS,
MINQUE(O), MINQUE(1), and MIVQUE estimates of variances due to direct and maternal
additive effects of genes, covariance between direct and maternal additive effects of genes,
variance due to environmental effects common to maternal half- and full-siblings and variance due to temporary environmental effects were computed for data sets consisting of 200
sets of Thompson's A or B design (1976), or 200 sets of Eisen's 1, 2, or 3 design (1967).
Efficiency was computed as standard error of MIVQUE divided by standard error of SDS,
MINQUE(O), or M1NQUE(1). Symmetric differences squared weighted by the inverse of the
environmental variance-covariance matrix (WSDS) is equivalent to MINQUE(O). Over all
parameter sets, components estimated, and designs, the efficiency of MINQUE(1) ranged
from .80 to .99, the efficiency of SDS ranged from .30 to .95 and the efficiency of MINQUE(O) ranged from .33 to .99. On the average, MINQUE(O) was more efficient than SDS
and MINQUE(1) was more efficient than MINQUE(O). The efficiency of MINQUE(1) was
insensitive to differences between the true components of variance and covariance and their
prior values.
(Key Words: Variance, Covariance, Maternal Effects, Genetic Differences, Computer
Techniques.)
J. Anim. Sci. 1989.67:348-356
Introduction
Traditionally, variances and covariances were
estimated by ANOVA and regression. One dis-
1Salaries and research support provided by state
and federal funds appropriated to the Ohio Agric.
Res. and Development Center, The Ohio State Univ.
Journal Article No. 125-88.
2Present address: Roman L. Hruska Meat Anim.
Res. Center, ARS, USDA, Clay Center, NE 68933.
a Dairy Sci. Dept., The Ohio State Univ.
Received April 28, 1988.
Accepted August 16, 1988.
advantage of these methods is that they do not
account for all relationships among animals.
Minimum norm quadratic unbiased estimation
(MINQUE) (Rao, 1971) and restricted maximum likelihood (REML) (Patterson and
Thompson, 1971) can be used to account for
all relationships (Henderson, 1985). Accounting for all relationships is expected to result in
an estimate that is less biased by selection and
more precise than estimates obtained by traditional methods. However, REML and MINQUE
require substantially more computer time and
memory than ANOVA or regression. The precision and accuracy of MINQUE depend on
348
ESTIMATION OF COMPONENTS OF COVARIANCE
how close the prior values for the c o m p o n e n t s
of variance and covariance are to their true
values.
Grimes and Harvey (1980) developed symmetric differences squared (SDS) in an a t t e m p t
to estimate variances and covariances accounting for all relationships with a reasonable
a m o u n t of c o m p u t e r t i m e and m e m o r y . In light
of Bruckner and Slanger (1986), prior to this
study, i n f o r m a t i o n on the precision o f SDS
relative to that of M I N Q U E was n o t k n o w n .
T h e objectives of this paper were to 1) prove
t h a t SDS weighted by t h e inverse o f t h e env i r o n m e n t a l variance-covariance m a t r i x (WSDS)
is equivalent to M I N Q U E with all prior values
for t h e c o m p o n e n t s o f variance and covariance
e x c e p t the environmental variance set to 0
( M I N Q U E ( 0 ) ) and 2) c o m p a r e t h e precision o f
SDS, M I N Q U E ( 0 ) and M I N Q U E w i t h all prior
values for the c o m p o n e n t s o f variance and covariance set to o n e (MINQUE(1)).
Methodology
Model. The m o d e l is a single-trait version of
t h e multiple-trait animal m o d e l described by
Quaas and Pollack (1980) for weaning weight
and post-weaning gain. A n additive genetic
m o d e l w i t h maternal effects is assumed. However, models involving d o m i n a n c e and epistasis
could be a c c o m m o d a t e d . The m o d e l o f Quaas
and Pollack (1980) can be written for one trait
as y = X/3 + ZdU d + Z m u m + Zpup + e, where
y is a vector of p h e n o t y p e s , /3 is a v e c t o r of
fixed effects, u d is a r a n d o m vector representing direct breeding values, u m is a r a n d o m vector representing maternal breeding values, Up is
a r a n d o m vector representing p e r m a n e n t maternal e n v i r o n m e n t a l effects, e is a r a n d o m vector
representing e n v i r o n m e n t a l effects peculiar to
each observation, X is k n o w n incidence relating
p h e n o t y p e s to fixed effects, and Z d ( Z m , Zp) is
a k n o w n incidence m a t r i x relating p h e n o t y p e s
to elements of Ud(Um, up). u d and u m usually
will have m o r e elements than y, so that Z d and
Z m have more c o l u m n s t h a n rows. C o l u m n s o f
Z d corresponding to animals w i t h o u t phenotypes are c o m p o s e d o f zeros. Columns of Z m
corresponding to males and virgin females are
c o m p o s e d of zeros. The variance-covariance
m a t r i x among p h e n o t y p e s is V = OIV1 +
O2V2 + . . . § OsVs, where V1 = ZdAZ'd, V2 =
ZmAZ~I + Z d A Z m , V3 = Z m A Z m , V4 = ZpZ~,
V5 = In, O1 is the variance of direct breeding
values, O2 is the covariance b e t w e e n direct and
maternal breeding values, O3 is the variance of
349
maternal breeding values, 0 4 is the variance
of p e r m a n e n t maternal e n v i r o n m e n t a l effects,
and Os is t h e variance of e n v i r o n m e n t a l effects
peculiar to a single observation. Let O = [@l
@2 03 @4 O s ] ' . F o r simplicity, we assume a
r a n d o m m o d e l so that X = I n (an n x 1 vector
of ones) and/3 = bt (an overall mean).
Symmetric Differences Squared. To obtain
an estimate o f @ by s y m m e t r i c differences
squared, first c o m p u t e the vector o f all r =
n(n -- 1)/2 possible squared differences (Y)
a m o n g observations. Second, the expected
value o f Y is calculated. T h e e x p e c t e d value
of Y is a linear f u n c t i o n of t h e true O. The
squared differences are set equal to their expectations and the resulting equations are
solved by o r d i n a r y least squares to obtain estimates of O. The kth e l e m e n t o f Y is (Yi -- yj)2,
where Yi and yj are the ith and jth e l e m e n t s of
y andi = 1,2, 3,...,n1 ; j = i + 1, i + 2,
i + 3.....
n ; k = 1, 2, 3, . . . , r with k =
(i - 1)n + j - i(i + 1)/2. Express Y as a v e c t o r
of squares (HY1) minus 2 times a vector o f
crossproducts (Y2), as f o l l o w s HY1 - 2YR. T h e
iTM element of Y1 is y~ and the k th element o f
Y2 is YiYj- H is an r x n incidence m a t r i x containing zeros and ones that specifies t h e elements of Y1 to be i n c o r p o r a t e d into Y. H is
1 1 0 0 0 0 .
O0
1 0 1 0 0 0 .
O0
1 0 0 1 0 0 .
O0
1 0 0 0 1 0 .
O0
1 0 0 0 0 1
....
O0
1 0 0 0 0 0 .
O1
0 1 1 0 0 0 .
O0
0 1 0 1 0 0 .
O0
0 0 0 0 0 0 .
11
350
KEELE AND HARVEY
Each row of H contains exactly 2 ones and
n - 2 zeros, and each c o l u m n of H contains
n - 1 ones and (n - 1)(n - 2)/2 zeros. Each
pair of c o l u m n s has a one in the same row o n l y
one time. Therefore, H'H = (n - 2)I n + l n l n ,
H1 n = 21 r, and H ' I r = (n -- 1)1 n. Let the
expected value of Y equal XO, where X = HX1
- 2X2. The element in the i th row and pth
(p = 1, 2 . . . . .
5) c o l u m n of X1 is the ith
diagonal e l e m e n t of Vp, and the d e m e n t in the
k th row and pth c o l u m n of X2 is the e l e m e n t in
the i th row and jth c o l u m n of Vp. O b t a i n SDS
estimates of O b y solving X ' X O = X'Y.
Weighted SDS (Christian, 1980) estimates
of O can be o b t a i n e d b y solving X ' W - I X O =
X ' W - I Y , where W is proportional to the variance-covariance matrix among Y due to
e n v i r o n m e n t a l effects. W = HH' + 21r, The
inverse of W is (1/2)(1 r -- ( 1 / n ) H H ' +
( 2 / n 2 ) l r l r ) , which can be proved b y showing
that WW - 1 = I r or w - l w = I r.
MINQUE. M i n i m u m n o r m quadratic unbiased estimates of O (Rao, 1971) can be obtained b y solving CO = t, where C is a 5 x 5
matrix in which the element in the pth row and
the qth c o l u m n is tr(PVpPVq), and the pth
row of t is y'PVpPy, where tr(.) is the sum of
the diagonal elements of the square matrix in
parentheses. P = V - 1 - V - 1 1 n ( l n V - : l n ) - :
l n V - : . V = wlV1 + w2V2 + 9 9 9 + wsVs. T h e
weightings: w = [WlW2 9 . 9 w s ] ' are prior
values for the true parameters: O. (p, q) = 1, 2,
....
5. MINQUE is MINQUE(0) if all t h e elements of w are set to zero except ws, which is
set to one. For MINQUE(0), the elements of C
and t simplify to Cpq = tr(MVpMVq), and
tp = y'MVpMy, where M = I n - ( 1 / n ) l n l ; : .
MINQUE is MINQUE(1) if all the elements of
w are set to one. MINQUE is m i n i m u m variance
quadratic unbiased estimation (MIVQUE) if
and o n l y if w is proportional to O and normality is assumed.
Equivalence
of WSDS and MINQUE(O).
We will show that WSDS is
MINQUE(0) b y showing that C
t = X ' W - I y . To show that
expand X ' W - l X through W - :
to o b t a i n
equivalent to
= X ' W - I X and
C = X'W-lX,
and rearrange
(x'xx, + 2x~x:)
-
(2/n)(X~H + X'I)(H'X2 + X l )
I
+ (1/n2)(X'l I n + 2X~ l r ) ( l n X l + 21'rX2).
Since: the element in the pth row and qth
c o l u m n of X'IX1 + 2X~X2 is tr(VpVq), the pth
c o l u m n of H'X2 + X l is V p l n , and the pth
element of 21~X2 + l n X 1 is l n V p l n , then:
the d e m e n t in the pth row and qth c o l u m n of
X ' W - - l X is tr(VpV a) -- ( 2 / n ) l n V o V a l n +
2
'
t
--.
.
--(1/n)lnVplnlnVqln,
which :s t r ( [ I n -( 1 / n ) 1 n 1 n] Vp [I n - ( 1 / n ) l n l n] Vq ), which is
t
--1
tr(MVpMVq), or Cpq. Therefore, C = X W X.
To show that t = X ' W - : Y , e x p a n d X ' W - : Y
through W - 1 to o b t a i n
(XtlY1 + 2X~Y2)
-
-
(2/n)(X~H + X'I)(H'Y2 + Y1)
t
t
,
+ (1/n2)(X'l I n + 2X2 l r ) ( l n Y 1 + 21rY2).
Since: the pth element of XrlY1 + 2X~Y2 is
y'Vpy, the pth c o l u m n of H'X2 + Xl is V p l n ,
the pth element of 21~X 2 + 1"X 1 is l n V p l n,
H'Y2 + YI = y y ' l n , and l n, Y 1 + 21'rY2 =
l n y y ' l n , then: the pth e l e m e n t of X ' W - - : Y
t
t
t
2
t
is yrVpy -- ( 2 / n ) l n V p y y I n + ( 1 / n ) l n V p •
l n l n y y l n , which is y ' [ I n - ( 1 / n ) l n l " ] Vp •
[I n - ( 1 / n ) l n l n ] y , which is y'MVpMy, or tp.
Therefore, t = X ' W - : Y .
Study of the Standard Errors of tbe Estimators. The sampling variances of SDS, MINQUE(0), MINQUE(1), and MIVQUE were
c o m p u t e d for five different mating designs and
for 16 different sets of values for the components of variance and covarinace. The sampling
variance of an estimator (SDS, MINQUE(0),
MINQUE(1), or MIVQUE) of Op was computed using 2tr(QpVQpV) (Searle, 1958),
where Qp is a symmetric m a t r i x such that the
estimate of @p = y ' Q p y . This expression
assumes normality. The standard error of an
estimate of @p was c o m p u t e d as the square
root of its sampling variance. The efficiency of
SDS, MINQUE(0), or MINQUE(1) was computed as the standard error of MIVQUE divided
b y the standard error of SDS, MINQUE(0), or
MINQUE(1).
Five mating designs were used. Two designs
d e n o t e d T A and TB were described in detail b y
T h o m p s o n (1976). Three designs denoted E l ,
E2, and E3 were described b y Eisen (1967).
For the T h o m p s o n (1976) designs, a r a n d o m
sire is m a t e d to two r a n d o m dams to produce a
male and a female offspring f r o m each mating.
For TA, a female progeny from one dam and a
male progeny from the other d a m are bred to
r a n d o m mates to produce two progeny each.
For TB, the female progeny f r o m b o t h dams
are bred to r a n d o m males to produce two progeny each. These mating schemes result in eight
progeny with phenotypes.
ESTIMATION OF COMPONENTS OF COVARIANCE
For the Eisen (1967) designs, t w o sires ($I
and $2) are m a t e d to f o u r dams each. These
eight dams are d e n o t e d D1, D2 . . . .
, D8. $1 is
m a t e d to D1, D : , DT, and Ds; and $2 is m a t e d
to D3, D4, Ds, and D 6. The offspring o f DI, D : ,
9 . . , D8 are O1, 02, . . . , O8, respectively.
Parents belong to three unrelated families. F o r
E l , S1 and $2 are full-sibs; D1, D2, D3 and D4
are full-sibs; and Ds, D6, DT, and Ds are halfsibs. For E2, $1, D1, and D2 are full-sibs; $2,
D3, and D4 are full-sibs; and Ds, D6, DT, and
Ds are full-sibs. E3 is the same as E2, with t h e
e x c e p t i o n that Ds, D6, Dr, and D8 are halfsibs instead o f full-sibs. This resulted in 18
individuals with p h e n o t y p e s for each Eisen
design.
The sampling variances were c o m p u t e d
assuming that O was estimated f r o m 200 replicates o f a design. Animals in different replicates
were assumed to be unrelated. This resulted in
1,600 animals for t h e T h o m p s o n designs and
3,600 animals for the Eisen designs. The o n l y
fixed effect considered was the overall average
of 1,600 ( T h o m p s o n ) or 3,600 (Eisen) animals.
This was d o n e because SDS is o n l y defined
w h e n this is the case (a r a n d o m model). T h e
block-diagonality of V was exploited w h e n
c o m p u t i n g t h e sampling variances. C o m p u t a tional details can be f o u n d in Keele (1986).
3 51
The 16 different sets o f values for O used in
the simulation s t u d y in Table 1 are expressed
in multiples of O s . In t h e scale o f Table 1, Os
was o n e for all 16 sets o f values.
Results and Discussion
The average, m i n i m u m , and m a x i m u m standard errors for M I V Q U E over the sixteen
p a r a m e t e r sets b y c o m p o n e n t and design are
given in Table 2. Standard errors o f M I V Q U E
were m u c h smaller for Eisen designs than for
T h o m p s o n designs, as w o u l d be e x p e c t e d because there were m o r e animals for Eisen designs
(3,600) t h a n for T h o m p s o n designs (1,600).
There also were large differences in standard
errors o f M I V Q U E b e t w e e n T h o m p s o n A and B
designs. On the o t h e r hand, differences in standard errors o f M I V Q U E a m o n g Eisen designs
were small. In general, t h e differences b e t w e e n
m i n i m u m and m a x i m u m standard errors o f
M I V Q U E were small, indicating that the relative p r o p o r t i o n s o f t h e different c o m p o n e n t s o f
variance and covariance have little effect on t h e
standard error of M I V Q U E .
The efficiency of SDS, M I N Q U E ( 0 ) , or
M I N Q U E ( 1 ) has a lower limit of zero and an
upper limit of 1. As t h e efficiency for a m e t h o d
o t h e r t h a n M I V Q U E approaches 1, t h e pre-
TABLE 1. VALUES FOR THE COMPONENTS OF VARIANCE AND COVARIANCE
FOR EACH PARAMETER SET EXPRESSED IN MULTIPLES
OF THE ENVI RONMENTAL VARIANCE
,,.,
Parameter
set
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Parametera
O1
02
03
.25
.28
-.076
-.028
.25
.28
.34
.38
.31
-.133
.63
.42
.63
.38
.76
.79
.52
1.15
.65
.85
1.69
1.62
3.25
3.75
7.50
-.133
-.054
-.054
-.237
-.155
-.115
-.065
-.359
-.369
-.229
-.229
-1.250
-.750
.31
.76
.38
.79
.52
1.15
.65
1.69
.85
3.25
1.62
3.75
7.50
.42
.76
.51
.53
1.72
.77
2.17
2.82
2.82
5.41
5.41
6.25
12.50
a| is the variance due to average direct effects of genes. 02 is the covariance of direct with maternal average
effects of genes. 03 is the variance of average maternal effects of genes. 04 is the variance of environmental
effects common to members of the same litter.
352
KEELE AND HARVEY
cision o f t h a t m e t h o d a p p r o a c h e s t h e precision
o f M I V Q U E . Because M I V Q U E d o e s n ' t exist in
practice, an e f f i c i e n c y close to 1 is i n t e r p r e t e d
to indicate t h a t a s u b s t a n t i a l l y m o r e precise
m e t h o d does n o t exist.
Average, m i n i m u m , and m a x i m u m efficiencies over t h e 16 p a r a m e t e r sets b y c o m p o n e n t
e s t i m a t e d , m e t h o d , and design are given in
Table 3. M a x i m u m efficiencies f o r MINQUE(O)
and M I N Q U E ( 1 ) were usually similar and
m i n i m u m efficiencies were usually l o w e r for
MINQUE(O) t h a n for M I N Q U E ( 1 ) , so t h a t t h e
average e f f i c i e n c y was l o w e r f o r MINQUE(O)
t h a n for M1NQUE(1). In general, t h e efficiency
o f M I N Q U E ( 1 ) always was q u i t e high w i t h very
little d i f f e r e n c e b e t w e e n m i n i m u m and maxi-
TABLE 2. AVERAGE, MINIMUM, AND MAXIMUM STANDARD ERROR OFMIVQUE a,
AND THE PARAMETER SET WITH THE SMALLEST OR LARGEST
STANDARD ERROR FOR MIVQUE OVER THE
16 PARAMETER SETS BY COMPONENT
OF VARIANCE OR COVARIANCE
ESTIMATED AND DESIGN
Minimum
Maximum
Component b
Average
Set c
Value
Set
Value
Thompson A
|
|
|
|
05
12.5
11.4
21.1
14.5
6.5
4
12
12
11
4
11.6
9.4
19.2
13.9
5.8
6
5
6
6
13
13.0
12.4
21.9
15.4
6.9
Thompson B
Ol
02
|
04
05
22.1
12.1
14.2
10.1
11.3
5
9
12
11
5
19.1
11.O
13.4
9.0
10.1
4
8
6
6
4
25.1
13.1
15.1
11.1
12.5
Eisen 1
|
02
|
|
|
5.1
4.0
6.6
4.8
3.0
4
12
12
12
12
4.2
3.2
6.1
4.4
2.1
13
13
7
5
5
5.8
4.5
7.1
5.2
3.8
Eisen 2
OI
02
|
O,~
05
5.6
3.9
5.8
4.3
3.1
4
4
10
11
4
4.6
3.0
5.4
4.2
2.3
13
13
3
6
13
6.4
4.6
6.2
4.5
3.7
Eisen 3
|
|
0~
04
05
5.1
3.9
6.6
4.9
3.1
4
4
12
12
4
4.3
3.2
6.0
4.4
2.2
13
13
7
5
5
5.8
4.3
7.2
5.4
3.9
aMIVQUE is minimum variance quadratic unbiased estimation (Rao, 1971).
b|
is the variance of direct breeding values, 02 is the covariance between direct and maternal breeding
values, 03 is the variance of maternal breeding values, 04 is the variance of permanent maternal environmental
effects, and O s is the variance of environmental effects peculiar to a single observation.
Cset is the parameter set with largest or smallest standard error for MIVQUE.
ESTIMATION O F COMPONENTS OF COVARIANCE
mum efficiency. Average, minimum, and maximum efficiencies usually were lower for SDS
than for MINQUE(O) or MINQUE(1).
Parameter
sets that
yielded
the minimum
and maximum
efficiencies of SDS,
MINQUE(0),
and MINQUE(1)
a r e g i v e n in
T a b l e 4. I n g e n e r a l , p a r a m e t e r s e t s t h a t y i e l d e d
3 53
minimum efficiencies for SDS or MINQUE(0)
contained true values for the components of
variance other than the environmental variance
t h a t w e r e l a r g e r t h a n t h e e n v i r o n m e n t a l variance. Conversely, parameter sets that yielded
maximum efficiencies for SDS and MINQUE(O)
contained true values for the components of
T A B L E 3. A V E R A G E (Ave), MINIMUM (Min), AND MAXIMUM (Max) EFFICIENCY a
F O R 16 P A R A M E T E R SETS BY COMPONENT OF VARIANCE
O R COVARIANCE ESTIMATED, METHOD, AND DESIGN
SDS c
MINQUE(0) d
MINQUE(1) e
Component b
Ave
Min
Max
Ave
Min
Max
Ave
Min
Max
Thompson A
0~
02
| 3
| 4
|
.62
.58
.62
.69
.59
.51
.44
.52
.63
.48
.68
.67
.68
.72
.67
.86
.80
.83
.89
.86
.71
.61
.69
.80
.71
.96
.95
.95
.97
.97
.95
.92
.93
.94
.95
.91
.87
.87
.90
.91
.99
.97
.97
.97
.99
Thompson B
|
02
03
| 4
|
.82
.90
.84
.94
.82
.80
.88
.77
.93
.80
.84
.91
.89
.95
.84
.97
.97
.90
.97
.97
.93
.95
.80
.95
.93
.99
.99
.97
.99
.99
.93
.94
.93
.94
.93
.87
.92
.88
.92
.89
.97
.97
.97
.97
.97
Eisen 1
O1
|
| 3
| 6
|
.47
.64
.59
.65
.48
.35
.52
.54
.62
.30
.56
.72
.64
.69
.65
.78
.81
.80
.65
.55
.59
.64
.68
.52
.33
.92
.94
.92
.82
.80
.89
.89
.86
.90
.88
.81
.80
.80
.86
.83
.96
.95
.93
.96
.95
Eisen 2
| 1
|
0~
| 4
@s
.55
.70
.69
.78
.53
.41
.57
.64
.75
.36
.66
.78
.73
.81
.69
.80
.83
.85
.75
.73
.63
.67
.76
.94
.95
.94
.88
.91
.90
.92
.92
.94
.90
.85
.85
.88
.91
.86
.96
.97
.95
.97
.96
.61
.82
.82
.87
.57
.50
.72
.73
.80
.41
.70
.88
.90
.91
.72
.83
.89
.86
.72
.60
.91
.91
.90
.94
.90
.86
.82
.86
.91
.87
.97
.96
.95
.97
.96
E~en 3
01
0~
0~
| 4
|
.66
.52
.68
.95
.77
.97
.75
.59
.40
.95
~87
.83
aEfficiency is the standard error of t h e m i n i m u m variance quadratic unbiased estimator divided by the standard error o f SDS, MINQUE(O), OR MINQUE(1).
b|
is the variance of direct breeding values, |
is t h e covariance between direct and maternal breeding
values, 03 is the variance of maternal breeding values, | 4 is t h e variance of p e r m a n e n t maternal environmental
effects, and O 5 is the variance of environmental effects peculiar to a single observation.
C
.
.
9
SDS is s y m m e t r i c differences squared (Grimes and Harvey, 1980).
dMINQUE(O) is m i n i m u m n o r m quadratic unbiased estimator with all prior values except the environmental
variance set to zero (Rao, 1971).
eMINQUE(1) is m i n i m u m n o r m quadratic unbiased e s t i m a t o r with all prior values set to one (Rao, 1971).
354
KEELE AND HARVEY
variance other t h a n the environmental variance
t h a t w e r e s m a l l e r t h a n t h e e n v i r o n m e n t a l varia n c e . T h e p a r a m e t e r sets c a n b e r o u g h l y d i v i d e d
i n t o t w o c a t e g o r i e s , t h o s e w i t h t r u e values f o r
t h e c o m p o n e n t s o f v a r i a n c e o t h e r t h a n t h e env i r o n m e n t a l v a r i a n c e t h a t are s m a l l e r t h a n t h e
e n v i r o n m e n t a l v a r i a n c e ( p a r a m e t e r s e t s 1 to 7)
and those with true values for the c o m p o n e n t s
of variance other t h a n the environmental
v a r i a n c e t h a t are larger t h a n t h e e n v i r o n m e n t a l
v a r i a n c e ( p a r a m e t e r s e t s 8 t o 16) (Table 1).
P a r a m e t e r s e t s 8 t o 16 y i e l d e d m i n i m u m effi-
TABLE 4. PARAMETER SETS a WITH MINIMUM (Min) AND MAXIMUM (Max) EFFICIENCY b
BY COMPONENT OF VARIANCE OR COVARIANCE ESTIMATED,
METHOD, AND DESIGN
SDS d
MINQUE(O) e
MINQUE(1) f
Component c
Min
Max
Min
Max
Min
Max
Thompson A
|
|
|
|
|
13
13
13
13
13
4
1
1
2
4
16
13
13
13
16
1
1
1
1
1
1
1
1
1
13
9
9
9
9
9
Thompson B
|
02
03
04
05
1
8
13
8
5
12
4
2
1
12
9
8
13
10
9
1
1
1
1
1
1
8
13
8
1
16
6
6
6
16
Eisen 1
O1
|
|
04
05
13
13
13
13
16
4
1
3
3
1
16
16
16
16
16
1
1
1
1
1
1
13
11
1
16
9
6
9
9
9
Eisen 2
|
02
03
04
05
13
13
15
1
13
4
4
2
13
7
16
13
16
16
13
1
1
1
1
1
1
13
11
11
13
9
9
6
6
9
Eisen 3
|
|
03
|
|
13
13
16
16
13
4
4
1
1
2
16
13
16
16
13
1
1
1
1
1
1
13
13
1
1
9
6
9
9
9
aparameter sets are defined in Table 1.
bEfficiency is the standard error of the minimum variance quadratic unbiased estimator divided by the standard error of SDS, MINQUE(O), OR MINQUE(1).
c|
is the variance of direct breeding values, 02 is the covariance between direct and maternal breeding
values, | is the variance of maternal breeding values, 04 is the variance of permanent maternal environmental
effects, and 05 is the variance of environmental effects peculiar to a single observation.
dSDS is symmetric differences squared (Grimes and Harvey, 1980).
eMINQUE(O) is minimum norm quadratic unbiased estimator with all prior values except the environmental
variance set to zero (Rao, 1971).
fMINQUE(1) is minimum norm quadratic unbiased estimator with all prior values set to one (Rao, 1971).
ESTIMATION OF COMPONENTS OF COVARIANCE
ciency for all but 3 of the 25 component x
design combinations for SDS, and for all 25
component x design combinations for MINQUE(0). Parameter sets 1 to 7 yielded maximum efficiency for all but 3 of the component
• design combinations for SDS, and for all of
the component • design combinations for
MINQUE(0). This is consistent with the expectation that MINQUE(0) or SDS increases in
efficiency as the values for the components of
variance and covariance other than the error
variance approach zero.
One would expect the efficiency of MINQUE(1) to be greatest for the parameter sets
with values for the components of variance and
covariance closest to one. Parameter set 9
yielded maximum efficiency for MINQUE(1)
for 16 o f the component • design combinations. The true values for the components of
variance and covariance were closer to one for
parameter set 9 than for any other parameter
set (Table 1). However, parameter set 6 or 16
yielded maximum efficiency for MINQUE(1)
for the remaining 9 component x design combinations. Neither parameter set 6 nor 16 had
true values for the components of variance and
covariance that were near one. Parameter sets 1,
8, 11, 13, or 16 yielded minimum efficiency for
MINQUE(1). Consistent with expectation, all
of these parameter sets had true values for the
components of variance and covariance that
were quite different from one (Table 1). The
small differences between minimum and maximum efficiencies for MINQUE(1) indicate that
the efficiency of MINQUE(1) is relatively insensitive to changes in the values of the true
components of variance and covariance to
values quite different from one.
Other Competitive Methods. Bruckner and
Slanger (1986) showed that SDS was more precise than analysis of variance procedures (Deese
and Koger, 1967; Hohenboken and Brinks,
1971; Koch, 1972; Cantet et al., 1984) for estimating the same components of variance and
covariance of this paper. Using SDS as a reference method between the study of Bruckner
and Slanger and the current study, we conclude
that MINQUE(0) and MINQUE(1) may be
considerably better than analysis of variance
procedures. However, analysis of variance
procedures require fewer computations than
SDS, MINQUE(O), or MINQUE(1).
Restricted maximum likelihood estimates
for most situations would be more precise than
SDS or MINQUE(O). REML is unlikely to be
355
substantially more precise than MINQUE(1)
because REML can't be more precise than
MIVQUE, and MINQUE(1) is almost as precise as MIVQUE. However, REML is less biased
b y selection than MINQUE(1) (Sorenson
and Kennedy, 1984). Symmetric differences
squared and MINQUE(O) offer a computational
advantage over REML; this advantage probably
is not worth the sacrifice in precision unless
estimation by REML is impossible or prior information indicates that the components of
variance and covariance other than the environmental variance are small relative to the environmental variance.
Henderson's simple method ("diagonal MIVQUE") (Hudson and Van Vleck, 1982) and
pseudo-expectation approach (Schaeffer, 1986)
both have about the same computational requirements as SDS or MINQUE(O). Dempfle
et al. (1983) evaluated diagonal MIVQUE and
found that it performed fairly well with a dairy
cattle population when the only components of
variance that were assumed to exist were the
variance due to direct breeding values and variance due to environmental effects and when the
only relationships assumed to exist were
paternal-half-sibs. Likewise, Schaeffer (1986)
showed that pseudo-expectation was almost as
precise as REML by simulation making the
same assumptions as Dempfle et al. (1983). The
precision and accuracy of diagonal MIVQUE
and pseudo-expectation approach with the
existence of maternal genetic effects and all
relationships assumed to exist have not been
studied.
MINQUE(1) offers a computational advantage over REML using the iterative MINQUE
algorithm because MINQUE(1) requires the
same computations as one iterate of iterative
MINQUE. REML by the expectation maximization (EM) (Dempster et al., 1977) algorithm
or the derivative-free algorithm described by
Graser et al. (1987) requires fewer computations per iterate than MINQUE(1). The EM
algorithm o f REML requires many iterates to
converge. The derivative-free algorithm of
Graser et al. would be cumbersome for models
more complicated than they assumed (the only
random effects in their model were Ud and e).
Meyer (1986) described reparameterizations of
certain nested models that improve the convergence of REML, but it is not clear whether
the same kind of reparameterizations exist for
models in which all relationships are considered. Schaeffer (1979) has suggested some
356
KEELE AND HARVEY
t e c h n i q u e s for r e d u c i n g t h e n u m b e r o f i t e r a t e s
r e q u i r e d to o b t a i n R E M L estimates b y t h e EM
algorithm.
Conclusions. S y m m e t r i c d i f f e r e n c e s s q u a r e d
w e i g h t e d b y t h e inverse o f t h e e n v i r o n m e n t a l
variance-covariance m a t r i x is equivalent t o
minimum norm quadratic unbiased estimation
w i t h all prior values f o r t h e variances and covariances equal t o zero e x c e p t t h e e n v i r o n m e n tal variance.
O n t h e average, M I N Q U E ( 0 ) was m o r e efficient t h a n SDS, and M I N Q U E ( 1 ) was m o r e
efficient t h a n M I N Q U E ( 0 ) . M I N Q U E ( 1 ) is less
sensitive t o changes in t h e values o f t h e t r u e
c o m p o n e n t s o f variance and covariance t o
values d i f f e r e n t f r o m t h e prior values t h a n is
M I N Q U E ( 0 ) . In practice, it is u n l i k e l y t h a t t h e
s t a n d a r d error o f R E M L w o u l d be m u c h less
than the standard error of MINQUE(1). However, R E M L is less biased b y s e l e c t i o n t h a n
M I N Q U E ( 1 ) . Because t h e e f f i c i e n c y o f MINQ U E ( 0 ) is relatively high w h e n t h e c o m p o n e n t s o f variance o t h e r t h a n t h e e n v i r o n m e n t a l
variance are small relative t o t h e e n v i r o n m e n t a l
variance, M I N Q U E ( 0 ) w o u l d b e a useful
m e t h o d for e s t i m a t i n g c o m p o n e n t s o f variance
and covariance f o r c h a r a c t e r s w i t h low heritability such as m o s t traits related t o r e p r o d u c t i o n in livestock.
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