Theor Appl Genet (1987) 73:371-378
© Springer-Verlag 1987
Testcross selection theory *
N.M. Cowen **
Department of Agronomy, University of Missouri, Columbia, MO, USA
Received June 20, 1986; Accepted September 15, 1986
Communicated by A. R. Hallauer
Summary. The purpose of this paper is to extend the
theoretical basis for testcross selection theory from
models assuming two alleles per locus to a model
which is general for number and frequency of alleles.
The expectations of genetic variances expressed among
and within testcross families is presented for both
inbred and population testers. Predicted change due to
selection in testcross, non-inbred and selfed population
performance with testcross selection are derived. Expected changes in testcross heterosis and inbreeding
depression in the population are also derived. Approximate confidence intervals for predicted selection
response are developed and appropriate sets of progeny
to evaluate in order to estimate parameters of interest
are identified.
Key words: Heterosis - Inbreeding - Prediction Genetic variance - Genetic theory
ical basis for the expected genetic variances among and within
testcrosses, and predicted response to selection requires the
assumption of two alleles per locus (Hallauer and Miranda
1981; Rawlings and Thompson 1962). This assumption is
overly restrictive, and application of current theoretical models
to populations with multiple alleles may yield inaccurate
results.
The purpose of this paper is to extend the theoretical model for testcross selection from two alleles per
locus to a model which is general for number and
frequency of alleles.
The population
The population that is considered is assumed to be in
linkage equilibrium, with no linkage of genes affecting
the trait, and no epistasis. Thus, all variances and
quadratic components defined for the population will
be the sum of the values for the individual loci, and
consideration can be given to a single locus. The structure of the noninbred population at a single locus is
Introduction
L.i L.j Pi Pj Ai Aj ,
Testcross progeny of individuals or lines from breeding
populations are commonly used to identify lines with
superior hybrid performance in combination with the
tester or combining ability (Hallauer and Miranda
1981).
Recurrent selection of a population using a tester is a
common procedure for improving the combining ability of the
population (Hallauer and Miranda 1981; Homer et al. 1973;
Russell et al. 1973; Homer et al. 1976). Currently, the theoret-
and the structure at a level of inbreeding where the
inbreeding coefficient = F is
* Contribution from the Missouri Agricultural Experiment
Station. Journal Series No. 10063
** Current address: United Agriseeds, P.O. Box 4011, Champaign, IL 61820, USA
F
L.. Pi Ai Ai + (1 -
F)
L.i L.j Pi Pj Ai Aj ,
where Pi is the frequency of allele Ai in the population.
The structure of the fully inbred (F = 1) population is
L. Pi Ai Ai' Assuming a model which allows dominance,
but no epistasis, the genotypic value for Ai A j is
G ij = It
+ (J(i + (J(j + (iij ,
where It is the population mean genotypic value, (J(i and
(J(j are the additive effects of alleles Ai and Aj fit by
372
least squares, and bij is the dominance deviation. By
definition
Li Pi lXi = Lj Pj IXj =
0.
squared mean genotypic value; i.e.,
L Pi (lXi + IXk + bik)2- IX~
i
= L Pi IXI + 2 L Pi lXi IXk + 2 L Pi lXi bik + IX~
All genotypic values in the remaining discussion will
be coded by substracting J1: G ij = G~ - J1.
+ 2 L Pi IXk c5ik + L Pi b?k - IX~
i
=
i
L Pi IX I + 2 L Pi lXi c5ik + L Pi b?k
i
i
=
Genetic variances among and within progenies
Consider an inbred tester with genotype Ak A k. The
additive effect of allele A k is defined as the deviation
of the mean of all testcrosses from the mean of the
population:
IXK =
L Pi G ik ·
Any dominance deviation is defined as the difference
between the genotypic value and the predicted value
based on the additive effects of the alleles.
The gametic output of an individual Ai Aj is
-}(Ai +Aj ).
-}ax + 2 Cov(AD
T)
+ V (D T ) .
If we consider testcrosses of individuals from a generation with inbreeding level F, the total variability may
be subdivided into portions expressed among and
within testcross families. The variance among testcross
families may be obtained as the mean of the squared,
coded, family genotypic means minus the squared
mean genotypic value, i.e.
F
L Pi (lXi + IXk + bik )2
+(1- F) L L Pi Pj [-} (21Xk + lXi + IXj + bik + bjk)]2- IX~
i
j
= F L Pi IXI + 2 F L Pi lXi bik + F L Pi brk i
+ (I - F) IX~ + -} (1 - F)
The genetic structure of the testcross of this individuals
+ -} (I - F) L Pi b?k
IS
(I - F) IX'
i
L Pi IX I + (1 i
F)
L Pi lXi bik
i
i
ax +(1 + F) Cov(AD
-} (Ai A k + Aj A k) .
= t (I + F)
The coded genotypic value of the testcross is thus
The variance expressed within testcrosses is the difference between the total variance and the variance
expressed among testcross families. The coefficients of
these three variances or quadratic components expressed among and within testcross families for several
levels of inbreeding are given in Table 1. The coefficient of the three components among families are all
linear functions of 1 + F, while the coefficients for
within families are all linear function of 1 - F.
Now consider a population rather than an inbred
line as a tester. The tester may not be in equilibrium,
but I will assume that the gametic output of the tester
is identical to the gametic output of the tester
population at equilibrium. The gametic output of the
tester population is assumed to be
-} [(lXi + IXk + bik ) + (lXj + IXk + c5jk)].
The variance expressed among and within testcrosses
are linear functions of the additive genetic variance in
the population
ax = 2 L Pi IXI;
i
the covariance of additive and testcross dominance
deviations
Cov(AD T )
= L Pi lXi bik ;
i
and the variance of the testcross dominance deviations
T)
+ -} (1 + F) V (D T )
.
LPkAb
k
The total variance expressed among testcross individuals is constant with inbreeding in the population
because the gametic output of the population does not
change. The genetic structure of population testcross is
LPiAiAk'
where Pk is the frequency of allele Ak in the tester.
Given an individual Ai Aj , the genetic structure of its
testcross is
-} [~ Pk (Ai Ak + Aj Ak) ] .
i
The variance expressed among individuals is obtained
as the mean of the squared genotypic values minus the
The coded genotypic value of an individual Ai A k is
373
Table 1. Variances among and within testcross families using an inbred tester
Generation of testcross
Individual
Line
So
SI
S2
S3
SJ
S4
Ss
Sn
Soo
Sl
S2
SJ
S3
S4
Ss
S6
Sn+l
Soo
Among
Within
ax
01
Cov(AD T) V(D T)
ax
Cov(AD T) V(D T)
0.25
0.375
0.438
0.469
0.484
0.492
(l
(I +F n)
I
1.5
1.75
1.875
1.938
1.969
(l + F n)
(I
2
0.25
0.125
0.063
0.031
0.016
0.008
t(l-F
t(I-F n)
0
I
0.5
0.25
0.125
0.063
0.031
(1- F n)
0
t
I
I2'
0.5
0.75
0.875
0.938
0.969
0.984
t(l+F
t(l + F n)
I
where
0.5
0.25
0.125
0.063
0.031
0.016
t(l-F
t(lFn)
0
and the variance of the testcross dominance deviations
ak= L: Pi G ik
i
and 0ik is a testcross dominance deviation. The mean
additive effect of alleles in the tester are defined as:
ii. =
rz.
L: Pk ak ;
k
L: Pk 0ik;
L: L: Pi Pk (ai + ak + Oik) = L: Pi (ai + rz.ii. + £5;J
J;J = rz..
ii..
i
k
and the mean testcross dominance deviation of allele
Ak as:
b k=
J:k
L:i L:k Pi Pk Ai Ak;
Ak;
with coded, mean genotypic value of
the mean testcross dominance deviation of allele Ai as:
bi. =
The genetic structure of the population testcross is
L: Pi 0ik'
k
i
The total variance expressed among individuals in the
population testcross is
L: L: Pi Pk
pdai
(ai + ak + Oik)2- iirz22
i
k
It can be shown that
£5;. = O.
L:i L:k Pi Pk 0ik = L:i Pi J;.
The variances expressed among and within testcross
families are linear functions of six populations variances or quadratic components. The components are:
the population additive genetic variance
al
a.~ = 2 L: Pi aL
i
the covariance of population additive effects and mean
testcross dominance deviations
.
Cov(A b) = 'V
~
b·1.,
I.,
L...J p.1I a·1J bi
the variance of the mean testcross dominance deviations
vV (b) == L: Pi Jf.;
R;
i
the additive genetic variance within the tester
alTc == 2 L: Pk aa;
a~;
k
the covariance of tester additive effects and mean
testcross dominance deviations
Cov(A
Cov (A bhc =
L: Pk ak J:k;
k
+
L: L: Pi Pk Ork i
ii,z
rz?
k
I
2 + talTC
I
2 +2
- 2 Cov(Abhc+
-2
= tax
b)
- a.
rz~
=
2'aA
+2'aATC+
Cov (A 0)
+ Cov(AOhc+ V(O)
V(O)-
Using arguments similar to those presented for an
inbred tester, the variance expressed among testcross
families of individuals with inbreeding coefficient F is
t
t
:t(l
+F)Cov(Ab)+t(l
+F)V(b).
(1 +F)ax+(I
+ F) al + (1 +
F) Cov (A b) + (1 +
F) V (b) .
It is obvious, since the variance within families is the
difference between total variance and the among families variance, that the variance within testcross families
is a linear function of all six variances or quadratic
components, and is equal to
i
t
:t (l
(I - F) ax + taXTC
+(1b)
aXTC +
(1 - F) Cov(A
Cov (A J)
+ 2 Cov(A bhc +
+ V (0) - tt (l
(I +
+ F) V (b) - rz?
+
ii?
The only real simplification in these linear functions
results when the tester population is identical to the
population that is being improved. Under these conditions;
aXTC
= ax;
alTc=
V (b)
Cov(Abhc
= 0; •
Cov(A b)= Cov(A
bhc=
= 0; ii.=O;
rz.=o;
374
and V(c5) = a5
u5 for the population. The variance expressed among testcross families reduces to t (1 + F) ui,
ai,
and the variance expressed within families reduces to
[1 - (1
(l + F)] ai
ui + a5.
u5· These are the expectations of
the variance expressed among and within half sib
families (Hallauer and Miranda 1981). The coefficients
ui, Cov (A 15)
b) and V (b)
(15) expressed among testcross
for ai,
families when the tester is a population are identical to
those for ai,
ui, Cov
Cov(ADT)
(ADT) and V
V(D
(D T ), respectively,
I.
shown in Table 1.
The additive genetic variance used here is that of
the equilibrium population. The proportion of ai
ui
expressed among testcrosses depends on the inbreeding
level of the progeny testcrossed, and is the same for
both inbred and noninbred testers. Both Cov(AD T)
and Cov (A £5)
b) can, by definition, be negative. Hence,
the total variability expressed among testcross families
may not increase with inbreeding in the population.
The variability among testcross families may be zero in
two instances. First the trivial case when all comui
ponents are zero; and second when Cov (ADT ) = - t ai
- t V
V(D
(D T) or Cov(A £5)
b) = - aj,"ul- t V(b).
V(J). These are
the lower limits for these two covariances. Combinations of population and tester where this situation
occurs will not be used for selection because there will
be no genetic variability among testcrosses at any
inbreeding level of the population.
t
t
with genotypic value:
(Xi + (Xk+
bik·
(Xle+ bile'
The gametic output of the population is:
L,PIAI .
LPIAI.
I
The genetic structure of the So progeny of this Sn
individual when recombined with the population is:
L, PIAiAI .
LPIAiA
I
If these So progeny are self-pollinated to the Sn their
genetic structure is:
2F n
(Ai Ai +
"
L.
PI Al AI) +
(1 - F n)
2
'T PI Ai AI .
"
The genetic structure of the testcross progeny of these
Sn individuals is:
2F
n
Alek +
Ai A
+
+(( 1I
n "
F
2'T
Alek
PI Al A
~Fn
Ale + ~
P1AIAle) =t (Ai Ale + tPIAIA+
~ PI Al Ale) ;
~Fn)) (Ai
(AiAk+
~PIAIAk)=t(AiAk+
with genotypic value:
+
((Xi + ~PI(XI+2(Xk+bik+
~PI~k).
~Pl(XI+2(Xle+bile+ ~Pl~le)'
t((Xi+
The product of the genotypic value of the testcross
family means of the Sn individual and its Sn progeny
following intermating is:
Predicted selection response
«(Xi
((Xi + (Xk
(Xle + bik) (t (Xi + t t~ PI (XI
(Xl + (Xk + t bik + t t~ PI ~k) .
Predicted selection response can be obtained as the
products of the selection differential and the regression
of progeny testcross mean performance in cycle t + 1
I
on parental testcross mean performance in cycle t
(Hallauer and Miranda 1981). The individuals or lines
that are testcrossed are related through an So individual.
The regression function involved is the ratio of the
covariance of testcross family means to the variance of
testcross means in cycle 1. If intermating takes place
exclusively among selfed progeny of selected individuals then the regression of testcross progeny means is
on mid parent value. The denominator of the regression equation is thus the variance of mid-parent values
or t the variance of a testcross mean in cycle 1.
Assuming no environmental covariance between
relatives, then the covariance of testcross family means
is obtained in two parts. First, consider an Sn individual whose genotype is Ai Ai; the gametic output of this
individual is Ai' With the inbred tester described above
the testcross of this individual is genetically:
Second, consider an Sn individual which is genetically
Ai Ajj ; the gametic output of this individual is
AiA
Ai Ale;
k;
Aj).
t(Ai + A
j ).
The genetic structure of the testcross progeny of this
individual with the inbred tester is:
t (AiA
(Ai Alek + A
Ajj Ak);
);
with genotypic value:
((Xi + (Xj + 22(Xk
(Xk + bile
bik + bbjk).
t «(Xi
jk ).
The So progeny from the cross of this individual with
the population has genetic structure:
t
L.
L PI (Ai A, ++AjA
Aj AI);
I);
I
resulting in an Sn progeny which has genetic structure:
~2 [~(AA+AA+2
2
J J
'T'" PI Al AI) .
1
1
375
The genetic structure of the testcross of this Sn progeny
is:
~
+~ [~n~n (AiAk+AjAk+2~PIAIAk)
[
(Ai At + Aj At + 2
PIAl At)
t
(i-
+ (l
(I - Fn)
F n) (t Ai At
Ak + t A
Ajj Ak
At +
l;
~ PI AIAJ Ak)]
At) ;
with genotypic value:
where X STC and X are the mean of the selected
testcross families and the grand mean, respectively,
Var (}{Tc)
O{Tc) is the phenotypic variance of a testcross
family mean and all other tenns
terms are defined above.
tenn Var(XTc)s"
The term
Var(XTC)s. is equal to:
2
aE
RL
2
aGE
L
~+ UGE +UG2 .
-+--+aG'
2
'
u~ is the error variance from the analysis of
where a~
variance, ubE
a6E is the variance of interaction effects of
genotypes with environments, u~ the genetic variance
expressed among testcross families of Sn individuals
given in Table I, and Rand L are the numbers of
replications and locations used in evaluation of testcross families, respectively.
Another situation encountered in recurrent selection
with a tester is where an So individual is self pollinated,
and a single SI
S\ progeny is crossed to the tester with
remnant S\
SI seed being used to recombine selected
families (Dr. K. Lamkey, USDA-ARS, Iowa State
University, personal communication). Under these circumstances the selection response equation takes the
fonn:
form:
aa
.The product of the genotypic values of the testcrosses
of this Sn individual and its Sn progeny following
intennating
intermating is:
(Xj + 22IXk
(Xt + l5ik
1/S «(Xi
(lXi + IXj
jk )
it + I5jt
. ((Xi
(Xj + 4IXk
4 (Xt + l5it
I5jt + 2 ~ PI
(XI + 2 ~ PI ~k)
~t) .
(lXi + IXj
~k + I5jk
PIIXI
The covariance of Sn testcross family means with the
testcross family means of their Sn progeny is thus:
[(lXi + (Xt
IXk + l5ik
F n L Pi [«(Xi
it )
i
t
[t ax + Cov(AD
Cov (ADT) +
V(D T)]]
[t[iul+
+tV(D
X
_ -) [t
( STCX
)
,
I
2 Var(XTC s,
'2
+ (1 - F n) L ~ Pi Pj 1/S
(Xj + 2IXk
2 (Xt + ~t
liS [«(Xi
[(lXi + IXj
~k + 15jk)
I5jk)
J
1
. ((Xi
+ (Xj
(Xk + l5ik
l5ik + I5I5jk
- (X,
(lXi +
IXj + 44IXk
jk + 2 ~ PI ~k)] -IX~
t5?kt
~ L Pi (Xf+ FnL Pi (Xi l5it + Fn(X,+ F2n L
Pi I5t
=~LPilXf+FnLPilXil5ik+FnlX~+
LPi
2 i i i
=
2 i i i
2
(l-Fn)~
(1
- Fn ) ~ (Xi 15ik
) ~
1l-F
- F nn )",
+ ( --4- 4 - ~ Pi (Xi
lXi + --2- ~ Pi lXi
t5 ik
7
7
IX~+ ~
Fnn
- F n) (X, + ((1-4
1 F )
+ (l
(1-
+
1 + Fn 2 1 + Fn
=
---UA+--=-S-aA+--2S
2
~ Pi 15?k
- (X,
I5lk-lX~
Cov (AD)
(AD T ) +--+ 1 + F n V(D T) •
T --4T·
4
The covariance of testcross family means with a population as a tester is:
1 + Fn 1 + Fn 2 1 + F n
- S - UA
aA + - 2 - Cov (A 15) + - 4 - V
(15) .
V(15).
The fonn
form of the selection response equation with an
inbred tester for testcrosses of Sn individuals is:
_
Var(XTc)s, is
where all tenns
terms are defined as above and Var(XTds,
the phenotypic variance of the mean of a testcross
SI individual to the
family resulting from crossing an Sf
tester.
Comparison of the predicted selection response,
where a single St
S\ individual is crossed to the tester and
intennating, with the
remnant SI seed is used in intermating,
response predicted when numerous individuals are
sampled within the SI
St line and remnant SI seed
intennating, is of interest to breeders.
is used in intermating,
The numerators of the two response equations are
identical. The denominator of the equation for the
situation where several SI individuals are crossed to
the tester is smaller than the denominator where a
S\ individual within the line is crossed to the
single SI
tester. The genetic component of the phenotypic variance is 1.5 times as large for the latter situation as it is
former. Thus, given the same selection differenfor the fonner.
tial, which mayor may not occur, predicted response
will be larger where several individuals are sampled
St line. Similar results are obtained with a
within the S\
Cov(A 15) and V(15)
V(~) subpopulation as a tester, with Cov(Ae5)
stituting for Cov(ADT) and V(D T) in the prediction
fonnulae.
formulae.
n
+F
(I- +2F-nn )) V(DTT)
1 [1+Fn
2 +
+ (I+F
_ 21[I
-4
- UA
+ (1 +
+ F nn)) Cov(AD TT)) +
aA
STC - X)
(X STC
1
t
t Var
(XTC)s.
Var(XTC)s"
l)]j
376
Other covariances of interest
The covariance of testcross mean performance with Sl
progeny mean performance in the following cycle is:
Not only are breeders interested in predicting response
to selection or the change in the mean of testcrosses
from cycle to cycle, but they may also be interested in
the relationships between testcross performance and
the performance of either non-inbred or inbred progenies within the population. Consider the situation
where an So individual is crossed to an inbred tester
and is also self-pollinated. Breeders are interested in
the genetic relationships between the testcross family
and either the So individual or the corresponding Sl
line both in the same cycle of selection and in the following cycle. Because the reference population used to
define all the effects is the non-inbred, equilibrium
population, these relationships can be defined (see
Smith 1986 for an alternate derivation). Two additional
quadratic components need first to be defined, the first
of these is the covariance of additive and homozygous
dominance deviations (Cockerham 1983):
COy (AD H ) =
L. ri (Xi t5
Cov(SI+1, TC I) = l/16
i
L. Pi t5ik t5
In both instances the covariance of progenies within
cycles is exactly twice the covariance of progenies
separated by a cycle of recombination. Similar quadratic terms can be defined with a population as a
tester leading to a similar set of equations for the
covariances of these progenies. The quadratic components are COy (A <5), which replaces COy (AD T) in the
equations, and
which substitutes for Coy (D H D T) in the equations.
jj .
Predicted changes in heterosis and inbreeding depression
The covariance of So and testcross mean performance within the same cycle is
t L. L. Pi Pj «(Xi + (Xj + t5ij)
j
i
. «(Xi + (Xj + 2 (Xk + t5ik + t5jk) = t 0"1 + COy (AD T)
where the superscript denotes the cycle. The covariance
of Sl and testcross mean performance within the same
cycle is:
Prior to specifying exact equations for the changes in
heterosis in testcross (the deviation of the mean of all
testcross families from the population mean) and inbreeding depression within the population, the predicted changes in the noninbred and selfed population
need to be specified. The predicted change in the noninbred population from testcross selection is:
- [COy (Sb+ 1, TC I)]
STC
(X
- X)
Var (TCh.
t
t L. L. Pi Pj «(Xi + (Xj + i (t5ii + t5jj + 2t5ij»
i
j
. «(Xi + (Xj + 2 (Xk + t5ik + t5jk ) -
t (Xk L. Pi t5
jj
= (XSTC - X)
i
= L. Pi (XT +
i
+i
=
i
i
i
COy (Sf, TCI) =
t (Xk L. Pi dii
COy (D H (5) = '"
~ p.1 15··
11 (5.1.,•
jj ;
the second is the covariance of homozygous and testcross dominance deviations:
COy (Sb, TC) =
j
. «(Xi + (Xj + 2 (Xk + t5ik + t5jk) -
i
Cov(D H D T) =
L. L. Pi Pj
i L. Pi (Xi t5 + t (Xk L. Pi t5 + L. Pi (Xi t5ik
jj
i
jj
i
L.i Pi t5ik t5ii - t (Xk L.i Pi t5
i
•
j
[i 0"1 + t COY (ADT)]
I
2
O"E
2
O"OE
I
2
)
1,
2(RL +T+'4O"A + Cov(ADT)+2" V(DTJ)
jj
t 0"1 + i COy (AD H ) + COy (ADT) + i COy (DH D T) .
for an inbred tester, and
The covariance of testcross mean performance with So
progeny mean performance in the following cycle is:
COy (Sb+ 1, TC I) =
i L. L. Pi Pj «(Xi + (Xj)
i
j
for a population tester, where all terms are defined as
previously. The predicted change from testcross selection in the population selfed is:
377
[Cov(Sj+I,TCt)]
X118 Cov(AD H) + i-COV(AD
tCOV(AD T) + 1I8Cov(D
- [Cov(SI+I,
TC t)] -(X
- [to"i+
Hax + 1/8
1/8 Cov(DHD
HD T)]
STC
I (2
(X
- X)
I V TC
-= ( STC - X)
2
)
"22" are
ar ( )so
_2 ~
+ 1. a 2 + Cov (AD ) + 1. V (D )
;~ + aOE
a~E
2 RL
L +tax+Cov(ADT)+i-V(DT)
4 A
T
2
T
for an inbred tester, and
-
- [t ax + 1/8 Cov(AD H) + i- Cov(A <5") + 1/8 Cov(D H<5")]
-
(X STC - X)
1
2
(2
2
aE
aOE
- -
)
RL +T+taX+Cov(Ab)+i-V(b)
for a population tester. The predicted testcross heterosis for cycle t + I can be obtained as the observed heterosis in
cycle t plus the difference between the predicted changes in testcross and So performance with an inbred tester:
[(118
[(1/8
I
_
2
Cov (AD T) + t~ V(DT»-(t
V(D T»- (~ax
))]
ax + ti- Cov(AD
ax + tt Cov(ADT»]
abE 1
a5E
I)
"2
+T+4ax+Cov(ADT)+"2V(DT)
2I (a~
RL +T+4ax+Cov(ADT)+2"V(DT)
change in testcross performance is estimated by the
product of the observedse1ectional differential and the
heritability expressed on an entry-mean basis estimated
from the analysis of testcross performance. Table 2 is a
standard analysis of variance table for a randomized
complete block design with T testcrosses, L locations,
and R replicates per location. Based on the analysis of
variance in Table 2 predicted selection response in testcross performance is equal to:
(X STC - X]
[X
"2 Var(T )soo
=
h VV(Dr)
(D T) -- 1/8
118 aA)
a A) i-I Var(TCh
C ''
- (Xk
~k +
+(4
XSo)t is the expected difference between
where (X TC - Xso)t
testcross and populations means in cycle t and is equal
V(D T) > 2
~k' Testcross heterosis will increase if V(Dr)
to (Xk.
and otherwise will decrease since· the term within the
bracket is strictly nonnegative. The equation for a
IX., COY
Cov (A J),
<5"), and
population tester is similar with tX.,
V(<5") substituting for (Xk,
~k, Cov(ADT) and V(DT) respecVel)
tively.
The predicted change in inbreeding depression
within the population can be obtained as the observed
inbreeding depression plus the difference in the predicted changes in So and SI
S\ performance; i.e.
ax
[t(MS 3 - MS 4)/LR)]
~ [±(MS
(X STC - X)
t MS /LR
±
3
-X-) [(-;}ai+±Cov(ADT»-(;}aX+1/8Cov(ADH)+i-COV(ADT+1/8Cov(DHDr)]
[(:\-ax+tCov(ADT»-(~ax+1I8Cov(ADH)+tCov(ADT+1I8Cov(DHDT)]
- -X-) (X(XSo
s, t + STC
aE
a
OE 1 22
aE
aOE
1
aA + Cov(ADT)+1: V(DT)
"21 RL
+T+4aA+COV(ADr)+2"V(DT)
+T+4
(2
1
I '~
"
2
src
STC - X]
»"22"[XVar(TCh
=
(Cov (ADH)
Cov (D HDr»
DT
= 1:2" ~ Pi (jii
bii - 118
1/8 (Cov(AD
H) + Cov(D
)
~'--=----=-~'--=-----=--
"":1
-:-'1
Xs,)t is the expected difference in mean
where (X so - XS,)t
SI generation in cycle t, and is
performance of So and Sl
t L Pi (jii·
equal to ±
bii ·
Li
Inbreeding depression in the population will be less
severe in cycle t + 1I than in cycle t if Coy
Cov (AD H)
D T) is positive. Both terms are covariances
+ Cov(D H Dr)
and thus may be negative.
Estimation
The mean differences (Xso - Xs,)t and (X TC
rc - XSo)t can
be estimated directly from the observe means for the
corresponding generations in cycle 1. The means, of
course, should both be expressed on the same basis, i.e.
on an individual plant or plot basis. The variances of
the differences are simply the sum of the variances of
the corresponding generations means. The predicted
o
~
where (X src
STC - X) is the observed selection differential
and MS 3 and MS4 are the mean squares for testcrosses
and testcrosses x location, respectively.
The estimation of predicted changes in the noninbred and selfed populations can proceed once the So
individual plant mean data has been adjusted to the
corresponding plot values, assuming a perfect stand.
SI, and testcross data from cycle t all
With So, St.
expressed on a plot basis the covariances of So and
testcross, and S\ and testcross means may be estimated.
Attention should be paid, in the estimation of these
two covariances, to the assumption of no environmental covariances of relatives (see Casler 1982). The
predicted response to testcross selection in the non-
378
Table 2. Analysis of variance for a randomized complete block design replicated at L locations,
including the expectations of mean squares (E (MS» for Testcross, Testcrosses x Location and Error
Source of variation
df
SS
MS
E(Ms)a
Locations
Replicate (locations)
Testcrosses
Testcrosses x locations
Error
L-I
L(R-I)
T-I
(T-l) (L-I)
(T-I)(R-I)L
SSt
SS2
SS3
SS4
SSs
MS 1
MS 2
MS 3
MS 4
MS s
oi + R (TtL + R L
(T~ + R
(T~
a (T~ is the error variance, (TfL is the variance component for testcrosses x location and
genetic comparent of variance expressed among families
inbred population is estimated as:
=----=
STC
[X - X]
MS 3 /LR
X
Cov (~)
0"
~
where X STC - X is the observed selection differential
and Cov (Sb, TC I ) is the estimated covariance of So and
testcross means in cycle 1. The predicted response to
testcross selection in the population selfed is estimated
as:
~
[X STC - X]
MS 3 /LR
X
Cov (stTC)
........------where Cov(Sl, TC
I"
I
) is the estimated covariance of SI
and testcross family means in cycle 1. There two
formulae are used for both inbred and population
testers. All other predicted changes are functions of the
estimates described above.
Approximate confidence intervals on selection response
Approximate confidence intervals for regressions, or heritabilities which are linear functions of ratios of mean squares with
known distribution, have been defined (Knapp et al. 1985).
An appropriate (I-IX) confidence interval on the regression
function used in predicting selection response is obtained as:
1- [(M3/~) FI-al2:df3.df.l-l
1- [(M3/~) Fal2:df3,df.]-1
and
which are the lower and upper confidence limits. In order to
obtain an approximate I - IX confidence interval on selection
response, the further assumption, that the selection differential
is a constant, needs to be made. This, of course, is not strictly
true. The variance of the selection differential can be shown to
be different from zero. However, there is no population parameter measured with greater precision than the selection
differential with the single exception of the grand mean.
Given the assumption that the selection differential is a constant, an appropriate I - IX confidence interval on predicted
selection response is obtained by multiplying the upper and
lower confidence limits by the selection differential.
Conclusions
The proportion of the additive genetic variance expressed among testcross families or involved in covariances of testcross means is independent of the choice of
(T(;
(TtL
(T(;
is the
tester, but is dependent on the inbreeding level in the
population prior to crossing to the tester. The tester
which maximizes the response to selection maximizes
the variance expressed among testcrosses, i.e. maximizes Cov (AD T ) + tV (D T ) or Cov (A£5) + t V(15).
These are independent of (XK or iX., hence, the tester
which maximizes the selection response may not express the greatest heterosis. An inbred tester will not be
inherently better unless Cov (AD T) + tV (D T ) is consistently greater than Cov (A£5) + tV (£5). As indicated
previously testcross heterosis will increase over cycles
only if V(D T) > 2 a1 or V (£5) > 2 a1 which is not likely
to be consistently true. Inbreeding depression will be
less severe with cycles of selection if either Cov (D HH)
+ Cov(DHD T) or Cov(AD H) + Cov(D H£5) is positive,
a situation which may not occur, since all of these
quadratic components may be negative. The actual
values for population parameters and size of confidence intervals are best examined in real rather than
hypothetical populations. This will be dealt with in an
article currently in preparation.
References
Casler MD (1982) Genotype x environment interestion bias to
parent-offspring regression heritability estimates. Crop Sci
22:540-542
Cockerham CC (1983) Covariances of relatives from self
fertilization. Crop Sci 23: 1177-1180
Hallauer AR, Miranda JB, Fo (1981) Quantitative genetics in
maize breeding. Iowa State University Press, Ames, Iowa
Homer ES, Cundy HW, Lutrick MC, Chapman WH (1973)
Comparison of three methods of recurrent selection in
maize. Crop Sci 13:485-488
Horner ES, Lutrick MC, Chapman WH, Martin FG (1976)
Effect of recurrent selection for combining ability with a
single-cross tester in maize. Science 16: 5- 8
Knapp SJ, Stroup WW, Ross WM (1985) Exact confidence
intervals for heritability on a progeny mean basis. Crop Sci
25(1): 192-194
Rawlings JO, Thompson DL (1962) Performance level as criterion for the choice of maize testers. Crop Sci 2: 217 - 220
Russell WA, Eberhart SA, Vega UH, 0 (1973) Recurrent
selection for specific combining ability for yield in two
maize populations. Crop Sci 13:257- 261
Smith OS (1986) Covariance between line per se and testcross
performance. Crop Sci 26: 540- 543