Estimates of Genetic Variance in an
F2 Maize Population
D. P. Wolf, L. A. Peternelli, and A. R. Hallauer
Maize (Zea mays L.) breeders have used several genetic-statistical models to study
the inheritance of quantitative traits. These models provide information on the importance of additive, dominance, and epistatic genetic variance for a quantitative
trait. Estimates of genetic variances are useful in understanding heterosis and determining the response to selection. The objectives of this study were to estimate
additive and dominance genetic variances and the average level of dominance for
an F2 population derived from the B73 ⴛ Mo17 hybrid and use weighted least
squares to determine the importance of digenic epistatic variances relative to additive and dominance variances. Genetic variances were estimated using Design
III and weighted least squares analyses. Both analyses determined that dominance
variance was more important than additive variance for grain yield. For other traits,
additive genetic variance was more important than dominance variance. The average level of dominance suggests either overdominant gene effects were present
for grain yield or pseudo-overdominance because of linkage disequilibrium in the
F2 population. Epistatic variances generally were not significantly different from
zero and therefore were relatively less important than additive and dominance variances. For several traits estimates of additive by additive epistatic variance decreased estimates of additive genetic variance, but generally the decrease in additive genetic variance was not significant.
From Golden Harvest Research, North Platte, Nebraska
(Wolf ), Universidade Federal De Viçosa, MG, Brazil (Peternelli), and the Department of Agronomy, Iowa State
University, Ames, IA 50011 ( Hallauer). This is a contribution of the Department of Agronomy and journal paper no. J-18459 of the Iowa Agricultural and Home Economics Experiment Station (Ames), project 3495. This
article is part of a dissertation submitted by D. P. Wolf
in partial fulfillment of the requirements for a Ph.D. degree. Address correspondence to A. R. Hallauer at the
address above or e-mail: [email protected].
2000 The American Genetic Association 91:384–391
384
Information on genetic variances, levels of
dominance, and the importance of genetic
effects have contributed to a greater understanding of the gene action involved in
the expression of heterosis. The Design III
mating design (Comstock and Robinson
1952) has primarily been used in maize F2
populations to determine the effects of
linkage on estimates of additive and dominance genetic variances and on the average level of dominance ( Hallauer and Miranda Fo 1988). Design III has shown
generally that genes controlling quantitative traits in maize F2 populations are in
the partial to complete dominance range.
There has been little evidence for genes
with overdominance controlling quantitative traits. Pseudo-overdominance, when
detected, has generally been due to linkage effects (Gardner et al. 1953; Gardner
and Lonnquist 1959; Moll et al. 1964).
Use of the Design III and other geneticstatistical models to estimate genetic variances usually assumes epistasis to be absent or of little importance. Several
studies indicate that epistasis is not a significant component of genetic variability
in maize populations (Chi et al. 1969; Eber-
hart et al. 1966; Silva and Hallauer 1975).
Other studies have shown, however, that
epistatic effects are important for specific
combinations of inbred lines ( Bauman
1959; Gorsline 1961; Lamkey et al. 1995;
Sprague et al. 1962). Specific crosses with
epistatic effects likely have unique combinations of genes contributing to heterosis. These unique combinations are restricted to the specific cross and may be
of little importance in a maize population,
and if the frequency of genetic combinations that exhibit epistatic effects are low
the variability due to epistasis may not be
detected when effects are spread throughout the population ( Hallauer and Miranda
Fo 1988).
A previous study (Wolf and Hallauer
1997) determined that epistatic effects
were significant for several traits in the
B73 ⫻ Mo17 hybrid. In the present study
it is possible to determine the importance
of epistatic genetic variance relative to additive and dominance variance for this hybrid. The objectives of our study were to
estimate additive and dominance genetic
variances and the average level of dominance for the F2 population derived from
the B73 ⫻ Mo17 cross and use weighted
least squares to determine the importance
of digenic epistatic variances relative to
additive and dominance variances.
Materials and Methods
Genetic Materials
The hybrid B73 ⫻ Mo17 was an important
and widely grown hybrid in the central
U.S. corn belt in the late 1970s and early
1980s. Inbred B73 was a selection from
Iowa Stiff Stalk Synthetic after five cycles
of half-sib recurrent selection for grain
yield (Russell 1972). Inbred Mo17 was derived by selection from the single cross of
inbred lines, CI187-2 ⫻ C103 ( Zuber 1973).
In 1991 an F2 population derived from
the B73 ⫻ Mo17 cross was grown at the
Agronomy Research Farm near Ames,
Iowa. Using the triple testcross ( TTC) mating design ( Kearsey and Jinks 1968), 100
random F2 plants (males) were crossed to
both parents ( B73 and Mo17) and the F1.
B73, Mo17, and the F1 were considered testers. Each F2 plant was selfed to form S1
progenies. For Design III analysis only data
from B73 and Mo17 testcrosses are needed.
Experimental Procedures
Testcross and S1 progeny were evaluated
in separate experiments. The 300 testcross entries were evaluated in a replications-within-sets, randomized incomplete
block design with two replications per set.
Ten sets were used, and each set included
30 entries comprised of three testcrosses
from each of 10 different F2 plants. The S1
progeny were grown in a 10 ⫻ 10 lattice
with two replications.
Both experiments were grown at the
Agronomy Research Center near Ames,
the Atomic Energy Farm in Ames, and near
Elkhart, Iowa, in 1992. In 1993, experiments were evaluated at the Agronomy
Research Center and the Ankeny Research
Farm. Each location by year combination
was treated as a different environment.
Each plot was a single row 5.49 m in length
with 0.76 m between plots. Plots were
overplanted and thinned to a stand of
57,520 plants/ha.
Sixteen traits were measured in both experiments. Days from planting to 50% anthesis and silk emergence were recorded
at the Agronomy Research Center in 1992
and 1993, and at the Atomic Energy Farm
in 1992. Silk delay was calculated as the
difference between anthesis and silk emergence. Plant and ear heights (cm) were
calculated as the average measurement of
10 competitive plants within a plot at all
environments, except Elkhart. Plant and
ear height were measured from ground
level to the collar of the flag leaf and uppermost ear node, respectively. Ten competitive plants within a plot were hand
harvested (with gleaning for dropped
ears) at all locations and ears were dried
to a uniform moisture. Data for the following traits were measured as the average of
10 primary ears or plants, ear diameter
(cm), cob diameter (cm), ear length (cm),
kernel-row number, and ears per plant.
Kernel depth was recorded as the difference between ear and cob diameter. Grain
yield was determined from all primary and
secondary ears and expressed in grams
per plant. Barren plants were expressed as
the percentage of 10 harvested plants that
did not produce an ear. Root lodging (percentage of plants leaning more than 30 degrees from vertical), stalk lodging (percentage of plants broken at or below the
primary ear node), and dropped ears (percentage of plants with dropped ears at
harvest) were based on the total number
of plants in a plot and recorded at five environments.
Statistical Analysis
Genetic Variance Components
The genetic-statistical model for Design III
was followed to derive genetic variance
components for the F2 reference population (Comstock and Robinson 1952). Analyses of variance (ANOVAs) combined
across environments were used to estimate variance components. From the Design III analysis, additive genetic (␴A2 ), ad2
ditive by environment (␴AE
), dominance
2
genetic (␴D ), and dominance by environ2
ment (␴DE
) variance components were estimated. The necessary components were
calculated as variation among males [␴M2
⫽ covariance half-sibs ⫽ (1/4)␴A2 ]; envi2
2
ronment by male [␴EM
⫽ (1/4)␴AE
]; tester
2
by male [␴MT
⫽ ␴D2 ]; and tester by male by
2
2
environment [␴TME
⫽ ␴DE
]. Estimates of
2
2
␴A and ␴D were used to estimate the aver2
age level of dominance as d̄ ⫽ ␴MT
/2␴M2 )1/2
⫽ (2␴D2 /␴A2 )1/2.
From the combined ANOVA for S1 progeny, genotypic (␴G2 ) and genotypic by en2
vironment (␴GE
) variance components
were estimated. Because an F2 population
was sampled, the expected gene frequencies of segregating loci are 0.5. At gene frequencies of 0.5 the ␴G2 of S1 progeny can
be expressed in genetic components as
␴G2 ⫽ ␴A2 ⫹ (1/4) ␴D2 . Standard errors (SE)
for all variance components were calculat-
ed using the method of Anderson and Bancroft (1952); SE ⫽ {2/C2 ⌺i [(MSi )2/(ni ⫹
2)]}1/2, where MSi ⫽ i th mean square; ni ⫽
degrees of freedom associated with the i th
mean square; and C ⫽ coefficient of the
variance component in the expected mean
square.
Variance components were considered
significantly different from zero if they
were greater than twice their standard error. If estimates are distributed normally
the 95% confidence interval will be bounded by ⫾2 standard errors of the estimate.
Estimates were considered different from
each other if their confidence intervals did
not overlap.
Because half-sib and S1 progeny were
derived from the same F2 parents, the covariance between them can be translated
into genetic variance components. Mean
products were obtained from the combined analysis of covariance between halfsib and S1 means as discussed by Matzinger and Cockerham (1963). Mean products
were multiplied by two to put them in the
same magnitude as mean squares from the
ANOVAs. Expectations for mean cross
products have the same general form as
for the mean squares (Mode and Robinson
1959), and therefore covariance components can be derived from mean products
as MXY ⫽ r␴XYE ⫹ re␴XY; MXYE ⫽ r␴XYE; ␴XY
⫽ (MXY ⫺ MXYE)/re; and ␴XYE ⫽ MXYE/r,
where MXY ⫽ mean product between halfsib ( X) and S1 ( Y) progeny; MXYE ⫽ interaction of environment by half-sib and S1
progeny mean product; ␴XY ⫽ covariance
of half-sib and S1 progeny; and ␴XYE ⫽ covariance by environment interaction. The
genetic covariance between half-sib and S1
progeny was derived by Bradshaw (1983)
and rederived by Peternelli et al. (1999)
for the special case of half-sib families obtained as the average of the three respective testcross families ( F2 ⫻ P1, F2 ⫻ P2,
and F2 ⫻ F1) as used in the present study.
For a genetic model that includes digenic
epistatic variances this covariance can be
2
expressed as ␴XY ⫽ (1/2)␴A2 ⫹ (1/4)␴AA
and
2
2
␴XYE ⫽ (1/2)␴AE ⫹ (1/4) ␴AAE. These components, however, may be biased by epistatic terms that include dominance effects (Peternelli et al. 1999). Standard
errors of components of covariance were
estimated by the following formula ( Dickerson 1969): SE ⫽ {1/C 2 ⌺i [(Mi XX)(Mi YY) ⫹
(Mi XY)2]/(nI ⫹ 2)}1/2, where C ⫽ coefficient
of the component of covariance; Mi XX and
Mi YY ⫽ mean squares for half-sib and S1
progeny; Mi XY ⫽ mean product for half-sib
and S1 progeny; and ni ⫽ degrees of freedom of i th mean product. Wolf and Hal-
Wolf et al • Genetic Variance in Maize 385
lauer (1997) used the triple testcross analysis for the same type of population, and
estimates of epistatic effects were significant for several traits. For the present
study, however, the potential bias for the
estimation of the different components of
variance will be considered either absent
or negligible to permit comparisons of the
variance component estimates from the
different populations.
Weighted Least Squares
From Design III, S1 progeny, and covariance combined analyses, there were 10
mean squares and mean products that
were translated into genetic components
of variance and error variances. Mean
squares and products were expressed in
terms of genetic components of variance
through digenic epistatic components and
error variances as follows:
From the Design III,
Males
2
2
⫽ ␴ 2e1 ⫹ rt[(1/4)␴ AE
⫹ (1/16)␴ AAE
]
2
⫹ rte[(1/4)␴ A2 ⫹ (1/16)␴ AA
];
⫹ rt[(1/4)␴
b
␴2AE
␴2D
␴2DE
␴2AA
␴2AAE
␴2DD
␴2DDE
␴2AD
␴2ADE
␴2e1
␴2e2
Design III
Males (M)
Males ⫻ environment ( E)
Males ⫻ tester ( T )
M⫻T⫻E
Error
5.00
0.00
0.00
0.00
0.00
1.00
1.00
0.00
0.00
0.00
0.00
0.00
10.00
0.00
0.00
0.00
0.00
2.00
2.00
0.00
1.25
0.00
0.00
0.00
0.00
0.25
0.25
0.00
0.00
0.00
0.00
0.00
10.00
0.00
0.00
0.00
0.00
2.00
2.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
1.00
1.00
1.00
1.00
0.00
0.00
0.00
0.00
0.00
S1 progeny
Genotypes (G)
G⫻E
Error
9.30
0.00
0.00
2.00
2.00
0.00
2.30
0.00
0.00
0.50
0.50
0.00
9.30
0.00
0.00
2.00
2.00
0.00
0.58
0.00
0.00
0.12
0.12
0.00
2.30
0.00
0.00
0.50
0.50
0.00
0.00
0.00
0.00
1.00
1.00
1.00
Mean products
Half-sib/S1
Half-sib/S1 ⫻ E
4.70
0.00
1.00
1.00
0.00
0.00
0.00
0.00
2.40
0.00
0.50
0.50
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
All traits except plant and ear heights, anthesis, silk emergence, and silk delay.
2
2
Variance components: additive genetic (␴ 2A ); dominance (␴2D ); digenic epistasis of ␴ 2A and ␴ 2D (␴ AA
, ␴ 2DD , ␴ AD
); interaction of these components by environment (␴2AE , ␴ 2DE , ␴ 2AAE , ␴ 2DDE , ␴ 2ADE ); experimental error of the design III
(␴ 2e1); and experimental error of S1 progeny (␴ 2e2).
⫽ r␴XYE ⫹ re␴XY
2
⫽ r[(1/2)␴ 2AE ⫹ (1/4)␴ AAE
]
Half-sib and S1 ⫻ environment
2
⫽ ␴ 2e1 ⫹ rt␴ EM
⫽␴
a
␴2A
2
⫹ re[(1/2)␴ A2 ⫹ (1/4)␴ AA
]; and
Males ⫻ environment
2
AE
Variance componentsb
Half-sib and S1
2
2
⫽ ␴ 2e1 ⫹ rt␴ EM
⫹ rte␴ M
2
e1
Table 1. Matrix of coefficients for mean squares and mean products in terms of genetic, genetic by
environment, and error variances for combined analysis of traits measured in five environmentsa
⫹ (1/16)␴
2
AAE
];
Male ⫻ tester
2
⫽ ␴ 2e1 ⫹ r␴ ETM
⫹ re␴ 2TM
2
2
⫽ ␴ 2e1 ⫹ r(␴ DE
⫹ ␴ DDE
)
⫹ re(␴ 2D ⫹ ␴ 2DD ); and
Male ⫻ tester ⫻ environment
2
⫽ ␴ 2e1 ⫹ r␴ ETM
2
2
⫽ ␴ 2e1 ⫹ r(␴ DE
⫹ ␴ DDE
).
From the S1 progeny,
Genotypes
2
⫽ ␴ 2e2 ⫹ r␴ GE
⫹ re␴ G2
2
2
⫽ r␴XYE ⫽ r[(1/2)␴ AE
⫹ (1/4)␴ AAE
].
2
2
2
For the mean squares, ␴AA
, ␴DD
, and ␴AD
are the digenic epistatic variance compo2
2
2
nents; ␴AAE
, ␴DDE
, and ␴ADE
are the digenic
2
epistatic by environment variances; ␴e1
is
2
the error variance of Design III; ␴e2
is the
error variance of S1 progeny; ␴XY is the covariance of half-sibs and S1 progeny; and
␴XYE is the covariance by environment.
Translation matrices of mean squares and
products into coefficients of genetic and
error variance components for the complete model are presented in Table 1.
Weighted least squares as discussed by
Nelder (1960) were used to estimate genetic variance components. The weighted
analysis can be expressed as
2
2
⫽ ␴ 2e2 ⫹ r[␴ AE
⫹ (1/4)␴ DE
⫹ ␴ 2AAE
⫹ (1/16)␴ 2DDE ⫹ (1/4)␴ 2ADE ]
⫹ re[␴ 2A ⫹ (1/4)␴ 2D ⫹ ␴ 2AA
2
⫹ (1/16)␴ DD
⫹ (1/4)␴ 2AD ]; and
Genotypes ⫻ environment
2
⫽ ␴ 2e2 ⫹ r␴ GE
2
2
⫽ ␴ 2e2 ⫹ r[␴ AE
⫹ (1/4)␴ DE
⫹ ␴ 2AAE
⫹ (1/16)␴ 2DDE ⫹ (1/4)␴ 2ADE ].
From the mean products,
386 The Journal of Heredity 2000:91(5)
B̂ ⫽ ( X⬘WX)⫺1 ( X⬘WY),
where
B̂ ⫽ column vector of estimated genetic
and error variances;
X ⫽ matrix of coefficients of the genetic
and error variances;
W ⫽ matrix with the inverse of the variances of mean squares and mean
products on the diagonal and zero
on the off diagonal; and
Y ⫽ column vector of observed mean
squares and products.
Standard errors of the parameter estimates were computed as the square root
of the associated diagonal element of the
( X⬘WX)⫺1 matrix. Variances of mean
squares and products were calculated by
the methods of Mode and Robinson
(1959). The following formula was used for
the variance of a mean square:
V(Mi ) ⫽ [2(Mi )2/dfi ⫹ 2],
where
Mi ⫽ ith mean square; and
dfi ⫽ degrees of freedom of ith mean
square.
The following formula was used for the
variance of a mean product:
V(Mi XY) ⫽ [(MiXX )(MiYY ) ⫹ (MiXY ) 2 ]
⫼ (dfi ⫹ 2),
where
Mi XX and MiYY ⫽ ith mean squares for
half-sib and S1 progeny;
Mi XY ⫽ ith mean product of halfsib and S1 progeny; and
dfi ⫽ degrees of freedom of
the ith mean product.
To estimate the genetic parameters of B̂,
different models were tested. Not all genetic and error variances, however, could
be estimated from a single model. A complete model included 10 genetic variances
and two error variances. The adequacy of
each model was tested using a chi-square
test (Mather and Jinks 1982).
X2 ⫽
冘 [(O ⫺ E) ·V],
2
where
O ⫽ observed mean square or product,
Table 2. Estimates of variance componentsa (ⴞ standard error) and average level of dominance (d ) from the Design III ANOVA across five environmentsb for
the B73 ⴛ Mo17 F 2 maize population
Trait
␴2A
Yield (g/plant)
Ear diameter (cm)d
Cob diameter (cm)d
Kernel depth (cm)d
Ear length (cm)
Kernel rows (no.)
Ears/plant (no.)d
Barren plants (%)
Root lodging (%)
Stalk lodging (%)
Dropped ears (%)
Plant height (cm)
Ear height (cm)
Anthesis (days)
Silk emergence (days)
Silk delay (days)
52.46 ⫾
2.15 ⫾
1.62 ⫾
0.89 ⫾
0.60 ⫾
1.13 ⫾
0.06 ⫾
2.91 ⫾
0.38 ⫾
6.56 ⫾
0.45 ⫾
180.01 ⫾
136.65 ⫾
3.53 ⫾
3.25 ⫾
0.40 ⫾
␴2AE
18.25
0.43
0.28
0.23
0.14
0.18
0.02
1.50
0.25
1.78
0.24
28.15
21.68
0.63
0.58
0.13
␴2D
65.68 ⫾
0.40 ⫾
0.07 ⫾
0.04 ⫾
0.60 ⫾
⫺0.02 ⫾
0.09 ⫾
6.55 ⫾
⫺0.11 ⫾
2.76 ⫾
0.67 ⫾
14.06 ⫾
13.65 ⫾
0.60 ⫾
0.72 ⫾
0.30 ⫾
27.65
0.31
0.13
0.28
0.13
0.03
0.04
2.77
0.53
2.21
0.46
4.01
3.79
0.25
0.22
0.14
155.63 ⫾
1.16 ⫾
0.24 ⫾
0.50 ⫾
0.40 ⫾
0.12 ⫾
0.00 ⫾
⫺0.24 ⫾
0.04 ⫾
1.05 ⫾
0.17 ⫾
20.68 ⫾
10.98 ⫾
0.46 ⫾
0.53 ⫾
0.07 ⫾
␴2DE
27.45
0.23
0.06
0.13
0.08
0.02
0.01
0.50
0.11
0.49
0.11
3.78
2.32
0.12
0.12
0.05
15.96 ⫾
0.14 ⫾
⫺0.05 ⫾
0.13 ⫾
0.13 ⫾
⫺0.02 ⫾
0.02 ⫾
2.70 ⫾
0.08 ⫾
⫺1.24 ⫾
0.25 ⫾
4.53 ⫾
4.35 ⫾
0.06 ⫾
0.16 ⫾
0.15 ⫾
␴2e
12.71
0.15
0.06
0.15
0.05
0.01
0.02
1.34
0.27
0.94
0.22
1.81
1.70
0.10
0.09
0.07
267.72 ⫾
3.24 ⫾
1.39 ⫾
3.14 ⫾
1.05 ⫾
0.35 ⫾
0.35 ⫾
26.79 ⫾
6.10 ⫾
23.30 ⫾
4.72 ⫾
29.28 ⫾
27.38 ⫾
1.63 ⫾
1.29 ⫾
0.91 ⫾
12.27
0.15
0.06
0.14
0.05
0.02
0.02
1.23
0.28
1.07
0.22
1.50
1.40
0.10
0.08
0.05
d
␴2D/␴2A
2.44c
1.04
0.54c
1.06
1.16
0.46c
0.32
0.41
0.47
0.57
0.88
0.48c
0.40c
0.51c
0.57c
0.61
2.97
0.54
0.15
0.56
0.67
0.10
⫺0.05
⫺0.08
0.11
0.16
0.38
0.12
0.08
0.13
0.16
0.19
Variance components: additive genetic variance (␴2A ); additive genetic by environmental variance (␴2AE ); dominance genetic variance (␴2D ); dominance genetic by environmental
variance (␴ 2DE ); and experimental error variance (␴ 2e )
b
Plant and ear heights measured in four environments, and anthesis, silk emergence, and silk delay measured in three environments.
c
Average level of dominance deviated from complete dominance at 0.01 probability level.
d
Estimates and standard errors multiplied by 100.
a
E ⫽ expected mean square or product,
and
V ⫽ inverse of the variance of the mean
square or product.
Models that included the maximum
number of parameters permitted by the
number of independent equations often
produced an X-matrix that was either singular or nearly singular. These models included two digenic epistatic terms and
gave unrealistic estimates (very large or
negative, with large standard errors). Chi
et al. (1969), Silva and Hallauer (1975), and
Wright et al. (1971) also obtained unrealistic and negative estimates as the number
of epistatic terms in the model increased.
Therefore models that included no more
than one digenic epistatic term were used.
The following six models were included
to estimate genetic and error variances:
Model
Parameters
1
2
2
2
␴ 2A , ␴ AE
, ␴ e1
, ␴ e2
2
2
2
2
2
␴ A2 , ␴ AE
, ␴ 2AA , ␴ AAE
, ␴ e1
, ␴ e2
3
2
2
2
␴ 2A , ␴ AE
, ␴ D2 , ␴ 2DE , ␴ e1
, ␴ e2
4
2
2
2
2
␴ 2A , ␴ 2AE , ␴ 2D , ␴ 2DE , ␴ AA
, ␴ AAE
, ␴ e1
, ␴ e2
5
2
2
␴ 2A , ␴ 2AE , ␴ D2 , ␴ 2DE , ␴ DD
, ␴ DDE
, ␴ 2e1 , ␴ 2e2
6
2
␴ 2A , ␴ 2AE , ␴ 2D , ␴ 2DE , ␴ AD
, ␴ 2ADE , ␴ 2e1 , ␴ 2e2 .
Heritabilities
Heritability estimates (h2) were calculated
on a progeny mean basis. Heritability of
half-sib progeny means of Design III was
calculated as
2
h2 ⫽ ␴M2 /(␴2/rte ⫹ ␴EM
/te ⫹ ␴M2 ).
Heritability of S1 progeny means was calculated as
2
/e ⫹ ␴G2 ).
h2 ⫽ ␴G2 /(␴2/re ⫹ ␴GE
Exact 90% confidence intervals for estimates of heritability were calculated, as
defined by Knapp et al. (1985).
Results
Design III
Barren plants, root lodging, and dropped
ears did not have estimates of ␴A2 significantly different from zero ( Table 2). For
other traits, estimates were generally two
to five times greater than their standard
2
errors. Estimates of ␴AE
were not different
from zero for ear and cob diameters, kernel depth, kernel-row number, root and
stalk lodging, and dropped ears. Estimates
2
of ␴AE
were larger than estimates of ␴A2 for
yield, ears per plant, and barren plants.
Estimates of dominance genetic variance (␴D2 ) were significantly different from
zero for all traits except ears per plant,
barren plants, root lodging, dropped ears,
and silk delay ( Table 2). Significant estimates of ␴D2 were generally greater than
three times their standard errors. Ear
length, barren plants, plant and ear
heights, days-to-silk emergence, and silk
2
delay had significant estimates of ␴DE
.
Estimates of ␴A2 and ␴D2 were not different from each other for ear diameter, kernel depth, and ear length, and both estimates were zero for barren plants, root
lodging, and dropped ears. For grain yield,
␴ˆ D2 was greater than ␴ˆ A2 , while the opposite
was true for the remaining traits. There-
fore the ␴ˆ D2 ·␴ˆ A2 ratio was less than one for
all traits except grain yield ( Table 2). Ratios were generally greater in magnitude
for traits in this study compared with ratios reported by Han and Hallauer (1989),
who also evaluated the F2 of the B73 ⫻
Mo17 cross.
The average level of dominance deviated from complete dominance for yield,
cob diameter, kernel-row number, plant
height, ear height, anthesis, and silk emergence ( Table 2). Of these traits, grain yield
had an average level of dominance in the
overdominant range (2.44), while the remaining traits exhibited partial dominance. Han and Hallauer (1989) reported
an average level of dominance for grain
yield of 1.28, which did not deviate from
complete dominance. The average levels
of dominance for other traits in the present study were similar to those reported
by Han and Hallauer (1989), who compared estimates in the F2 generation with
those in the same F2 generation after five
cycles of intermating. The average level of
dominance for grain yield also was greater
than estimates reported for F2 populations
by Gardner et al. (1953), Gardner and
Lonnquist (1959), Moll et al. (1964), and
Robinson et al. (1949), which ranged from
1.03 to 2.14. For other traits the level of
dominance was similar to estimates from
these studies.
S1 Progeny
Genetic variance (␴G2 ) estimates were significantly different from zero for all traits
except root lodging ( Table 3). Genetic by
2
environmental (␴GE
) variances were signif-
Wolf et al • Genetic Variance in Maize 387
Table 3. Estimates of variance componentsa (ⴞ standard error) from the S1 progeny ANOVA across five
environmentsb for the B73 ⴛ Mo17 F 2 maize population
Trait
␴2G
Yield (g/plant)
Ear diameter (cm)c
Cob diameter (cm)c
Kernel depth (cm)c
Ear length (cm)
Kernel rows (no.)
Ears/plant (no.)c
Barren plants (%)
Root lodging (%)
Stalk lodging (%)
Dropped ears (%)
Plant height (cm)
Ear height (cm)
Anthesis (days)
Silk emergence (days)
Silk delay (days)
166.65 ⫾
2.38 ⫾
1.20 ⫾
0.91 ⫾
1.04 ⫾
1.00 ⫾
0.29 ⫾
18.13 ⫾
⫺0.09 ⫾
4.30 ⫾
0.26 ⫾
210.67 ⫾
163.22 ⫾
4.95 ⫾
4.07 ⫾
0.47 ⫾
␴2GE
59.52 ⫾
0.44 ⫾
0.26 ⫾
⫺0.02 ⫾
0.07 ⫾
0.05 ⫾
0.35 ⫾
32.42 ⫾
⫺0.29 ⫾
1.12 ⫾
0.20 ⫾
8.66 ⫾
7.23 ⫾
0.98 ⫾
0.72 ⫾
0.39 ⫾
29.20
0.40
0.21
0.18
0.17
0.15
0.06
4.39
0.13
1.07
0.10
30.96
23.93
0.80
0.67
0.12
␴2e
251.02 ⫾
3.20 ⫾
1.80 ⫾
3.30 ⫾
1.12 ⫾
0.44 ⫾
0.71 ⫾
50.28 ⫾
8.76 ⫾
27.10 ⫾
3.36 ⫾
50.88 ⫾
35.04 ⫾
2.21 ⫾
2.38 ⫾
1.19 ⫾
15.89
0.18
0.10
0.15
0.06
0.02
0.06
4.53
0.42
1.40
0.18
3.47
2.48
0.23
0.22
0.11
16.55
0.21
0.12
0.20
0.07
0.03
0.05
3.32
0.58
1.79
0.22
3.78
2.60
0.19
0.20
0.10
Variance components: genetic variance (␴ 2G ); genetic by environmental variance (␴ 2GE ); and experimental error
variance (␴ 2e ).
b
Plant and ear heights measured in four environments and anthesis, silk emergence, and silk delay measured in
three environments.
c
Estimates and standard errors multiplied by 100.
a
icantly different from zero for all traits except kernel depth, ear length, root and
stalk lodging, and dropped ears. The esti2
mate of ␴GE
for ears per plant and barren
plants was larger but not significantly different from the estimate of ␴G2 . For several
traits the estimates of ␴G2 were smaller
than those reported by Han and Hallauer
(1989).
Covariance S1 and Half Sibs
Covariance of S1 and half-sibs translated
2
into ␴A2 and ␴AE
are presented in Table 4.
Estimates of ␴A2 were not different from
zero for yield and dropped ears. Additive
by environment variance was different
from zero for yield, ear diameter, ears per
plant, barren plants, days to anthesis, and
2
silk emergence. For yield, ␴AE
was signifi-
cantly greater than ␴A2 , while for ears per
plant and barren plants it was larger but
not different from ␴A2 . The estimates obtained from the covariance of S1 and half
sibs generally were not significantly smaller than estimates from Design III. For ears
per plant, barren plants, days to anthesis,
and silk delay, estimates of ␴A2 were larger
than those obtained from Design III. Over2
all, estimates of ␴A2 and ␴AE
from covariance analysis were generally within one
standard error of estimates from Design
III.
Weighted Least Squares
Across all traits the chi-square lack of fit
was generally significant for models 1 and
2, while models 3 and 4 generally provided
an adequate fit to the data ( Tables 5 and
Table 4. Estimates of additive genetic (␴2A ) and additive by environment (␴2AE ) variance components (ⴞ
standard error) from analysis of covariance between S1 and half-sib progeny across five environmentsa
for the B73 ⴛ Mo17 F 2 maize population
Trait
Yield (g/plant)
Ear diameter (cm)b
Cob diameter (cm)b
Kernel depth (cm)b
Ear length (cm)
Kernel rows (no.)
Ears/plant (no.)b
Barren plants (%)
Root lodging (%)
Stalk lodging (%)
Dropped ears (%)
Plant height (cm)
Ear height (cm)
Anthesis (days)
Silk emergence (days)
Silk delay (days)
␴2A
28.09 ⫾
1.46 ⫾
1.26 ⫾
0.50 ⫾
0.62 ⫾
0.98 ⫾
0.10 ⫾
5.51 ⫾
0.40 ⫾
4.98 ⫾
0.03 ⫾
172.87 ⫾
138.15 ⫾
4.01 ⫾
3.44 ⫾
0.50 ⫾
␴2AE
18.42
0.33
0.21
0.15
0.13
0.15
0.03
1.75
0.14
1.10
0.11
27.15
21.36
0.67
0.60
0.10
57.48 ⫾
0.40 ⫾
⫺0.06 ⫾
0.05 ⫾
0.08 ⫾
⫺0.01 ⫾
0.13 ⫾
11.70 ⫾
⫺0.17 ⫾
1.02 ⫾
0.31 ⫾
0.36 ⫾
⫺0.52 ⫾
0.49 ⫾
0.51 ⫾
⫺0.05 ⫾
14.87
0.16
0.09
0.14
0.06
0.02
0.03
2.25
0.28
1.14
0.16
2.98
2.38
0.22
0.19
0.08
Plant and ear heights measured in four environments, and anthesis, silk emergence, and silk delay measured in
three environments.
b
Estimates and standard errors were multiplied by 100.
a
388 The Journal of Heredity 2000:91(5)
6). Model 3 generally provided a good fit,
with R 2 greater than 97% and smaller standard errors of the six models for the majority of traits. Silva and Hallauer (1975)
and Wright et al. (1971) also obtained their
best results from the same model. Esti2
2
mates of ␴A2 , ␴AE
, ␴D2 , and ␴DE
from model 3
were generally similar to estimates from
Design III, and standard errors from
weighted least squares were generally less
than those from Design III. Only the esti2
mate of ␴AE
for ear length was significantly
different between the two methods, and it
was greater in Design III. Additive genetic
variance (␴ˆ A2 ) from model 3 was not different from zero for root lodging, while ␴ˆ D2
was not different from zero for ears per
plant, barren plants, root lodging, dropped
ears, and silk delay. Estimates of ␴A2 and
␴D2 were not significantly different from
each other for ear and cob diameters, kernel depth, ear length, root lodging, and
dropped ears. For the remaining traits, estimates of ␴A2 were greater than ␴D2 , except
for yield.
Inclusion of digenic epistatic variances
in models 4, 5, and 6 generally improved
the fit and increased the R 2 values compared with model 3, but the standard errors of ␴ˆ A2 and ␴ˆ D2 for models 4, 5, and 6
increased compared with model 3. The increase in standard errors as the number
of epistatic components increased is likely
unavoidable because of the high correlation between coefficients of the first-order
variance components (␴A2 and ␴D2 ) and coefficients of second-order components
2
2
2
(␴AA
, ␴AD
, ␴DD
) (Chi et al. 1969).
With model 4, several traits had decreased estimates of ␴A2 , while ␴D2 was not
affected compared with model 3 ( Tables 5
and 6). For example, the estimate of ␴A2 for
yield was decreased by 75% and for ear
length decreased by 32%. Decreases in ␴ˆ A2
resulted in nonsignificant estimates for
yield, ears per plant, barren plants, and
dropped ears in model 4. Dominance variance did not differ from zero for ears per
plant, barren plants, root lodging, dropped
ears, and silk delay. Additive genetic variance was significantly greater than ␴ˆ D2 for
cob diameter, kernel-row number, root and
stalk lodging, plant and ear heights, days
to anthesis and silk emergence, and silk
delay in model 4. Dominance variance was
significantly greater than ␴ˆ A2 for yield.
2
When significant estimates of ␴A2 , ␴AE
, ␴D2 ,
2
and ␴DE were observed in model 4, they
did not differ from corresponding estimates observed in model 3.
2
Inclusion of ␴DD
in model 5 generally
gave unrealistically large estimates of ␴D2
Table 5. Model 3a weighted least squares estimates (E) of variance components and their respective
standard errors (SE) from the combined analysis across five environments for the B73 ⴛ Mo17 F2 maize
population
Variance componentsb and standard errors
␴2A
Trait
Yield (g/plant)
Ear diameter (cm)
Cob diameter (cm)
Kernel depth (cm)
Ear length (cm)
Kernel rows (no.)
Ears/plant
Barren plants (%)
Root lodging (%)
Stalk lodging (%)
Dropped ears (%)
Plant height (cm)
Ear height (cm)
Anthesis (days)
Silk emergence (days)
Silk delay (days)
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
54.78
11.85
1.83
0.22
1.30
0.13
0.68
0.10
0.72
0.08
1.02
0.09
0.09
0.02
4.48
1.10
0.15
0.09
4.85
0.71
0.16
0.07
185.94
16.63
145.23
12.92
4.00
0.40
3.50
0.35
0.46
0.06
␴2AE
57.46
10.08
0.40
0.11
0.06
0.06
0.02
0.09
0.11
0.04
0.002
0.01
0.14
0.02
12.62
1.62
⫺0.13
0.21
1.43
0.81
0.26
0.11
6.02
1.96
4.60
1.56
0.68
0.13
0.62
0.12
0.13
0.06
␴2D
␴2DE
170.82
26.81
1.19
0.22
0.22
0.05
0.51
0.12
0.42
0.08
0.12
0.02
0.00
0.01
⫺0.26
0.50
0.00
0.11
1.01
0.48
0.19
0.11
20.73
3.78
10.97
2.31
0.46
0.11
0.53
0.12
0.07
0.04
14.93
12.19
0.15
0.14
⫺0.03
0.06
0.12
0.14
0.08
0.05
⫺0.01
0.01
0.03
0.02
3.86
1.32
0.05
0.26
⫺1.45
0.89
0.14
0.21
4.07
1.77
3.92
1.66
0.08
0.10
0.16
0.09
0.16
0.07
␴2e1
␴2e2
R2
␹2
269.15
10.99
3.20
0.13
1.38
0.06
3.14
0.12
1.12
0.04
0.34
0.01
0.33
0.01
25.13
1.13
6.17
0.24
23.73
0.94
4.87
0.19
30.37
1.41
28.56
1.31
1.61
0.09
1.30
0.07
0.93
0.05
251.87
15.20
3.21
0.19
1.93
0.10
2.96
0.17
1.07
0.07
0.47
0.02
0.79
0.04
56.29
3.11
8.24
0.45
27.00
1.53
3.30
0.19
51.87
3.39
35.96
2.38
2.31
0.18
2.41
0.18
1.28
0.09
99.4
8.2
99.8
2.2
99.2
7.7
99.8
2.8
98.8
17.5**
99.6
5.6
97.6
33.8**
97.1
41.8**
99.3
9.6*
99.8
2.3
99.6
5.1
99.3
8.5
99.0
11.4*
99.5
4.3
99.8
1.3
98.8
10.7*
*,** Chi-square lack of fit significant at the 0.05 and 0.01 probability levels, respectively.
Model 3 included ␴ 2A, ␴ 2AE, ␴ 2D, ␴ 2DE, ␴ 2e1, and ␴ 2e2.
b
Variance components: additive genetic (␴ 2A ); dominance (␴ 2D ); interaction of these components by environment
2
2
2
(␴ 2AE, ␴ DE
); experimental error of Design III (␴ e1
); and experimental error of S1 progeny (␴ e2
).
a
2
and large negative estimates of ␴DD
, which
may indicate the model was inadequate.
2
Estimates of ␴AD
generally had no effect on
estimates of ␴A2 and ␴D2 in model 6. There2
fore ␴AA
had a greater effect on estimates
2
2
of ␴A2 and ␴D2 than did either ␴DD
or ␴AD
. If
model 5 is considered inadequate, estimates of ␴D2 were generally not biased by
epistasis in the remaining models. Hallauer and Miranda Fo (1988) observed
2
that ␴AA
had the greatest bias on estimates
2
of ␴A and ␴D2 .
Estimates of digenic epistatic components from models 2, 4, 5, and 6 often were
negative, smaller than their standard errors, or unrealistically large compared
with estimates of ␴A2 and ␴D2 . These results
agree with those of Silva and Hallauer
(1975) and Wright et al. (1971) who also
observed unrealistic and negative estimates of digenic epistatic components.
Heritabilities
Heritability estimates and their 90% confidence intervals of Design III and S1 progeny are presented in Table 7. Estimates
were considered greater than zero if the
confidence interval did not overlap zero.
Estimates in both analyses were significantly greater than zero for all traits, except for root lodging of S1 progeny, which
was negative. In both analyses kernel-row
number and plant and ear heights had the
largest estimates (⬎0.91), which were significantly greater than estimates for other
traits. Estimates for S1 progeny were larger
than Design III estimates for several traits,
particularly yield. Because the expected
genetic variance of S1 progeny includes all
the ␴A2 and one-fourth of the ␴D2 of the
source population, heritabilities are expected to be larger compared with those
based on half-sib progeny, which contain
one-fourth of the ␴A2 .
Discussion
Variance Components
In both the Design III and weighted least
squares analyses, the estimate of ␴D2 was
significantly greater than the estimate of
␴A2 for grain yield. For the remaining traits,
␴ˆ A2 was greater than ␴ˆ D2 . The average level
of dominance from Design III was in the
overdominance range for grain yield and
partial to complete dominance for the remaining traits, but the average level of
dominance for grain yield may be biased
upward due to linkage. In an F2 population,
linkage disequilibrium will be at a maximum. If coupling phase linkages predominate, ␴ˆ A2 and ␴ˆ D2 will be biased upward. Repulsion phase linkages will cause a
downward bias of ␴ˆ A2 and upward bias of
␴ˆ D2 . Both types of linkage may cause an upward bias in the average level of dominance. Han and Hallauer (1989) reported
that the average level of dominance for
grain yield decreased from 1.28 to 0.95 after five generations of random mating.
Linkage did not bias estimates of ␴A2 , but
␴ˆ D2 decreased by 40% with random mating.
However, the two estimates of average level of dominance did not differ from complete dominance, indicating linkage may
have only a small bias on the average level
of dominance. The average level of dominance for yield from the present study was
2.44. This is a distinct contrast to the estimate of 1.28. If ␴ˆ D2 from the present study
is reduced by 40%, the level of dominance
is 1.89, which is still greater than the majority of estimates reported in previous
studies (Gardner et al. 1953; Gardner and
Lonnquist 1959; Moll et al. 1964). The estimated level of dominance (2.44) supports the presence of either important
overdominant gene effects or pseudooverdominance because of linkage effects
in the expression of yield.
A significant difference in the estimate
of the average level of dominance was observed by Gardner and Lonnquist (1959)
for two samples of the single cross M14 ⫻
187-2. Sample 1 had an average level of
dominance estimate of 0.59, and sample 2
had an average level of dominance estimate of 1.59; both estimates deviated from
complete dominance. Sample 1 had a larger estimate of ␴A2 , and they suggested the
environment in which sample 2 was grown
may have suppressed the estimate of ␴A2 ,
increasing the level of dominance. The
sample 1 estimate of ␴D2 was 67% of sample
2, and estimate of ␴A2 sample 2 was 18% of
that observed in sample 1.
2
The estimate of ␴AE
for yield was greater
than observed by Han and Hallauer
2
(1989). The ␴ˆ AE
· ␴ˆ A2 ratio was 1.25 compared with 0.16 in the study of Han and
2
Hallauer (1989), whereas the ␴ˆ DE
· ␴ˆ D2 ratios
were 0.10 and 0.16, respectively. The estimate of ␴D2 for yield was less affected by
environment than ␴ˆ A2 , in the present study.
The range of environments in which this
study was conducted may have decreased
Wolf et al • Genetic Variance in Maize 389
Table 6. Model 4a weighted least squares estimates (E) of variance components and their respective
standard errors (SE) from the combined analysis across five environments for the B73 ⴛ Mo17 F2 maize
population
Variance componentsb and standard errors
␴2A
Trait
Yield (g/plant)
Ear diameter (cm)
Cob diameter (cm)
Kernel depth (cm)
Ear length (cm)
Kernel rows (no.)
Ears/plant
Barren plants (%)
Root lodging (%)
Stalk lodging (%)
Dropped ears (%)
Plant height (cm)
Ear height (cm)
Anthesis (days)
Silk emergence
(days)
Silk delay (days)
␴2AE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
E
SE
13.31
24.98
1.73
0.50
1.65
0.31
0.62
0.25
0.49
0.18
1.12
0.20
⫺0.01
0.04
⫺1.75
2.46
0.74
0.24
6.78
1.80
0.10
0.20
164.97
34.66
128.98
26.81
3.10
0.81
62.40
26.21
0.38
0.29
⫺0.20
0.14
0.13
0.25
0.37
0.11
⫺0.06
0.03
⫺0.03
0.05
⫺3.38
3.76
⫺0.01
0.53
1.71
2.10
0.54
0.32
7.67
4.54
⫺3.90
3.88
0.33
0.30
E
SE
E
SE
2.98
0.73
0.44
0.14
0.61
0.27
0.06
0.14
␴2D
␴2DE
161.32 14.88
27.28 12.61
1.18
0.14
0.22
0.15
0.23 ⫺0.06
0.05
0.06
0.51
0.14
0.13
0.14
0.40
0.11
0.08
0.05
0.12 ⫺0.02
0.02
0.01
0.00
0.02
0.01
0.02
⫺0.24
2.54
0.50
1.34
0.04
0.10
0.11
0.27
1.06 ⫺1.37
0.49
0.92
0.18
0.20
0.11
0.22
20.65
4.20
3.78
1.80
10.90
3.66
2.32
1.69
0.46
0.05
0.11
0.10
0.53
0.12
0.07
0.04
␴2AA
87.38
46.23
0.17
0.76
⫺0.54
0.43
0.09
0.36
0.41
0.29
⫺0.17
0.30
0.27
0.08
16.79
5.51
⫺0.82
0.31
⫺2.83
2.43
0.09
0.26
38.02
55.22
30.99
42.63
1.77
1.36
0.16
0.09
0.13
0.07
0.97
1.18
0.03
0.22
␴2AAE
␴2e1
⫺7.44 270.00
36.48 11.56
0.03
3.21
0.41
0.14
0.43
1.43
0.21
0.06
⫺0.17
3.12
0.35
0.13
⫺0.36
1.09
0.15
0.04
0.10
0.35
0.05
0.01
0.35
0.35
0.09
0.02
33.08 27.18
7.10
1.19
⫺0.29
6.06
0.84
0.26
⫺0.49 23.59
3.06
1.00
⫺0.40
4.82
0.45
0.20
⫺2.74 30.22
6.83
1.46
6.78 29.44
5.45
1.36
0.57
1.64
0.45
0.09
0.02
0.41
0.32
0.22
1.30
0.07
0.95
0.05
␴2e2
R2
252.13
16.43
3.20
0.21
1.82
0.12
3.01
0.19
1.15
0.07
0.44
0.03
0.72
0.05
51.04
3.26
8.72
0.56
27.32
1.76
3.39
0.22
52.48
3.74
36.93
2.58
2.23
0.18
99.6
4.6
99.8
2.2
99.8
2.0
99.8
2.5
99.3
10.1**
99.9
0.6
99.8
2.0
99.8
2.8
99.9
0.9
99.9
0.8
99.6
4.3
99.3
7.9*
99.1
10.8**
99.9
0.7
99.9
0.6
99.0
8.4*
2.40
0.20
1.23
0.10
␹2
*,** Chi-square lack of fit significant at the 0.05 and 0.01 probability levels, respectively.
2
2
2
2
Model 4 included ␴2A, ␴2AE, ␴2D, ␴2DE, ␴AA
, ␴AAE
, ␴e1
, and ␴e2
.
b
Variance components: additive genetic (␴ 2A ); dominance (␴2D ); digenic epistasis of ␴ 2AA ; interaction of these com2
2
ponents by environment (␴ 2AE , ␴ 2DE , ␴ AAE
); experimental error of Design III (␴ e1
); and experimental error of S1 progeny
(␴ 2e2).
a
Table 7. Estimates of heritability (h2) with confidence intervals for half-sib progeny means from the
Design III and S1 progeny means, based on the ANOVAs across five environmentsa for the B73 ⴛ Mo17 F 2
maize population
Confidence intervalb
a
b
Confidence interval
Trait
Design III
h2
Lower
limit
Upper
limit
S1 progeny
h2
Lower
limit
Upper
limit
Yield (g/plant)
Ear diameter (cm)
Cob diameter (cm)
Kernel depth (cm)
Ear length (cm)
Kernel rows (no.)
Ears/plant (no.)
Barren plants (%)
Root lodging (%)
Stalk lodging (%)
Dropped ears (%)
Plant height (cm)
Ear height (cm)
Anthesis (days)
Silk emergence (days)
Silk delay (days)
0.44
0.75
0.85
0.59
0.65
0.94
0.40
0.30
0.24
0.56
0.30
0.94
0.93
0.83
0.83
0.50
0.27
0.67
0.80
0.46
0.54
0.93
0.22
0.10
0.01
0.43
0.08
0.93
0.91
0.77
0.77
0.32
0.58
0.81
0.89
0.69
0.73
0.96
0.55
0.48
0.43
0.67
0.47
0.96
0.95
0.87
0.88
0.63
0.81
0.84
0.83
0.74
0.89
0.95
0.66
0.59
⫺0.12
0.58
0.39
0.96
0.96
0.87
0.86
0.58
0.75
0.80
0.78
0.66
0.85
0.93
0.56
0.48
⫺0.44
0.46
0.22
0.95
0.95
0.83
0.81
0.44
0.85
0.88
0.87
0.80
0.91
0.96
0.74
0.69
0.15
0.68
0.54
0.97
0.97
0.90
0.89
0.69
Plant and ear heights measured in four environments, and anthesis, silk emergence, and silk delay measured in
three environments.
Exact 90% confidence intervals as defined by Knapp et al. (1985).
390 The Journal of Heredity 2000:91(5)
the estimate of ␴A2 , as suggested by Gardner and Lonnquist (1959). Estimates of ␴A2
and ␴D2 were 17% and 61%, respectively, of
estimates reported by Han and Hallauer
(1989), indicating both have decreased in
the present study. Additive variance for
grain yield may have been suppressed by
a large interaction with environments.
Other traits between the two studies were
less affected by environments, and average level of dominance estimates were
consistent between studies.
Weighted Least Squares
Weighted least squares analysis was conducted to determine the relative importance of epistatic variance compared with
␴A2 and ␴D2 . Triple testcross analysis indicated epistatic effects were important for
several traits (Wolf and Hallauer 1997).
2
2
2
Generally, ␴ˆ AA
, ␴ˆ DD
, and ␴ˆ AD
were not greater than twice their standard errors, negative, or unrealistic. Models that did not include digenic epistatic components often
provided an adequate fit and more precise
2
2
2
estimates. Therefore ␴AA
, ␴DD
, and ␴AD
were
2
2
less important than ␴A and ␴D for the majority of traits.
Weighted least squares analysis indicat2
ed that inclusion of ␴AA
in model 4 de2
creased estimates of ␴A for several traits.
The decrease in ␴ˆ A2 generally did not result
in nonsignificant estimates or estimates
different from those obtained with model
3. For yield, ␴ˆ A2 was not significant in model 4, as were several other traits. Therefore, although ␴ˆ A2 is biased upward if we
assume epistasis is absent, the magnitude
of bias is small. Generally ␴ˆ D2 was not biased by epistasis in models 4 and 6. Bias
observed in model 5 is likely the result of
an inadequate model. Dominance variance
was less important than ␴A2 for most traits
and may be less likely to be biased by
epistasis.
All traits had negative variance component estimates for various models, with
model 5 generally having at least two negative estimates. By definition, a variance
is always positive, but Searle (1971) indicated that there is nothing intrinsic about
the ANOVA to prevent negative estimates
from occurring. Negative estimates could
arise from an inadequate model, inadequate sampling, or inadequate experimental techniques. Searle (1971) discussed
possible solutions to negative estimates.
The best solution would be to interpret
them as zero and reestimate other components from a reduced model.
Negative estimates in the present study
were generally small and not greater than
their standard error. Negative estimates
often occurred for variance components
that were either nonsignificant or negative
when estimated in Design III or S1 progeny
experiments. Generally when a model
gave negative estimates, another model
for that trait had positive estimates with
greater precision; hence negative estimates were not a serious problem.
Dominance variance was important in
the expression of heterosis for grain yield
in the B73 ⫻ Mo17 cross. While epistasis
was less important than dominance, the
presence of significant positive epistatic
effects may have contributed to the expression of heterosis and could explain
why the B73 ⫻ Mo17 cross was an exceptional and widely grown hybrid. Epistatic
variances were not important in the F2
population of the B73 ⫻ Mo17 cross, although epistatic effects have been previously detected.
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Received June 28, 1999
Accepted March 10, 2000
Corresponding Editor: William F. Tracy
Wolf et al • Genetic Variance in Maize 391