THEORETICAL ANALYSES OF DOUBLE-CROSS
AND THREE-WAY HYBRID POPULATIONS
by
JOHN OREN RAWLINGS
.'.4
.
: ..&: .'
C. CLARK COCKERHAM
NORTH CAROLINA STATE COLLEGE
,.
,e
Institute of Statistics
=~:,s~;~;s No. 248 :~"\~~~
'f
iv
1.0 TABLE OF CONTENTS
Page
. 2.0 LIST C1P TABLES
•
· ..
•
vi
3.0 INTRODUCTION • •
4.0 REVIEW OF LITERATURE •
I
.1
•
4.1 Correlations between Relatives •
4.2 Analysis of Single-Cross Hybr~ds
5.0 ANALYSIS OF VARlANCE FOR DOUBLE-CROSS HYBRIDS
2
•
•
•
•
•
•
•
•
· '. .
10
5.1 Introduction
.• •
5.2 Expectations of.Quadratic Forms in Terms of
Covariances..
•
••
•
5.3 Genetic Expectations of Covariances
•
• •
5.4 Partitioning of Sums of Squares •
••
•
•
5.5 Expectations of Sums of Squares in Terms of
Covariances • • •
•. •
5.6 Model of Linear Effects
•••
5.7 Covariances as Functions of the Components of Variance
5.8 Expectations of Mean Squares in Terms of COmponents
of Variance and Estimation of Components of Variance
5.9 Components of Variance as Functions of Covarianc~s
between Hybrid Relatives.
••••
5.10 Genetic Interpretations of the Components of
Variance and Tests of Significance
•
5.11 Estimation of Genetic Parameters. •
•
5.12 Discussion of the Analysis of Double-Crosses.
6.0 ANALYSIS OF VARIANCE FOR THREE-WAY HYBRIDS.
•
•
6.1 Introduotion..
• •
•
•
6.2 Expectation of Quadratic Forms in Terms of
Covariances • •
• •
•
• •
6.3 Genetic Expectations of Covariances
• •
•
6.4 Partitioning of Sums of Squares. • •
•
6.5 Expectations of Sums of Squares in Terms of
.Covariances. • _ •
••••
6.6 Model of Linear Effects..
•
•
6.7 Covariances as Functions of the Components of Variance
6.8 Expectations of Mean Squares in Terms of Components
of Variance and Estimation of Components of Variance
6.9 Components of Variance as Functions of Covariances
•
between Hybrid Relatives..
•
6.10 Genetic Interpretations of the Components of
Variance and Tests of Significance •
• •
• ,. •
•
6.11 Estimation of Genetic Parameters
• •
6.12 Discussion of the Analysis of Three-Way Hybrids.
2
6
10
11
13
14
19
25
28
28
32
35
42
45
•
48
•
48
48
51
52
•
•
57
57
63
64
•
66
•
67
73
76
v
1.0 TABLE OF CONTENTS(continued)
Page
7.0 GENERAL DISCUSSION
•
•
8.0 SUMMARY AND CONCLUSIONS.
9.0 LIST OF REFERENCES
.
•
•
. .
•
•
•
•
•
•
79
•
•
82
•
•
•
•
84
vi
2.0 LIST OF TABLES
Page
1.
2.
Correlations between additive deviations, p, and dominance
deviations, q, of double-cross hybrid relatives.
13
Description of sums pertinent to the analysis of
double-cross hybrids.
16
3.
Analysis of variance of double-cross hybrids
•
17
4.
Expectations of qncorrected sums of squares in terms of
the covariances between double-cross hybrid relatives.
20
Expectations of mean squares in terms of the covariances
between hybrid relatives for the analysis of double-cross
hybrids.
••
•
24
Covariances between double-cross hybrid relatives expressed
as functions of the components of variance •
•
28
Expectations of mean squares with Model A for the analysis
of double-crosses.
29
for the analys#is
•
•
30
5.
6.
7.
8.
9.
10.
11.
12.
13.
·0
14.
It
Expectations of mean squares with Model
of double-crosses.
• •
B
Components of variance as functions of the covariances
between double-cross hybrid relatives for Model A •
33
Components of variance as functions of the covariances _
between double-cross hybrid relatives for Model B-.
34
Components of variance in the analysis of double-crosses
as functions of the components of genetic variance up
to and including the three-factor epistatic components
36
•
F-tests of hypotheses that various components of variance
are zero in the analysis of double-crosses •
39
Correlations between additive deviations, p, and dominance
deviations, q, of three-way hybrid relatives.
50
Description of sums pertinent to the analysis of three-way
hybrids •
•
•
o·
..
..
15.
Analysis of variance of three-way hybrids
16.
Expectations of uncorrected sums of squares in terms of the
covariances between three-way hybrid relatives •
••
"
•
54
.55
58
vii
2.0 LIST OF TABLES(continued)
P~e
17.
18.
19.
20.
21.
22.
·e
Expectations of mean squares in terms of the covariances
between hybrid relatives for the analysis of three-way
hybrids
•
60
Covariances bet~een three-way hybrid relatives expressed
as functions of the components of variance
63
Expectations of mean squares for the analysis of three-way
hybrids • • • • • • • • • ~ • •
65
Components of variance as functions of the covariances
between three-way hybrid relatives
•
68
•
Components of variance in the analysis of three-way crosses
as functions of the components of genetic variance up to
and including the three-factor epistatic components
F-tests of hypotheses that various components of
are zero in the analysis of three-way crosses
69
va~iance
•
•
71
3.0 INTRODUCTION
The use of statistical concepts to charact.erize the variation in
quantitative characteristics has beco~e widely accepted as an approach
to studying the inheritance of such characters.
variation in quantitative characters, that
is,
The nature of the
the absence of discrete
classes, precludes the-use of classical genetic methods of investigating
the inheritance.
A knowledge of the mode of inheritance of characters
is essential to the development of efficient breeding programs for
.improvement of organisms.
As a rule, however, precise estimation of the
genetic parameters which prove most useful is difficult and generally
requires very elaborate and extensive experiments.
It would be desirai1e
if these estimates could be obtained as a matter of course in a breeding
program without extensive special effort.
The evaluation of potential hybrids constitutes a large part of
most corn breeding programs.
The analysis of variance of all possible
single-cross hybrids from a set of pareats has been shown to provide
estimates of some of the genetic parameters.
It is the purpose oftnis
dissertation to develop analyses of variance for all possible double-cross
hybrids and three-way hybrids from a set of p lines and to determine the
genetic information available from such analyses.
2
4.0
4.1
REVIEW OF LITERATURE
Correlations between Relatives
The first general treatment of the theoretical genetic correlations
between relatives was given by Fisher (1918).
The heritable variation
for a locus was partitioned into additive variance and dominance variance.
Non-additivity of the effects between loci was defined as epistacy.
The
correlations for parent-offspring, full-sib, uncle-nephew, cousin, and
double-first-cousins were treated for random mating populations with
any degree of dominance but no epistacy.
an arbitrary number of alleles.
The general solution considerea
The effects of two-locus epistatic
deviations on the covariances of the above relatives were presented.
Assortative mating and coupling effects were studied.
Wright (1921) proposed the use of the method of path coefficients
for the study of the genetic correlations between relatives in the case
of no dominance or epistasis.
Dominance was considered for some special
cases but, primarily, only the additive effects of the genes were considered.
The method of path coefficients did not provide a general
treatment of non-additive gene effects so that the covariances between
dominance deviations were not encountered by this method.
The parent-
offspring and full-sib correlations were given by Wright (1935) for a
special model in which the value of a secondary character is dependent
upon the deviation from an optimum of a primary character.
The cases of
either no dominance or complete dominance at all loci were considered.
3
Mal'cot
1
(Kempthorne, 1954, 1957) gave a general treatment of the
covariance between relatives for the case of random mating for a single
locus with arbitrary dominance among an arbitrary number of alleles in
.
1948.
The only restriction appliad "by Mal'cot was that of no inbreeding •
The covariance between two individuals was given as
~. 2 AA,.2
( 2 ) 0"A + '1"1' (Tn
2
2
where (TA is the additive genetic variance, (Tn is the dominance variance
and' and "
are measures of the relationship between the two individuals
for the locus under consideration.
'and"
can be used as above only
in the particular case when x , y can be identified as genes contrib. s
s
uted to individuals X and Y, respectively, on one chromosome and xd '
Yd as genes contributed to X and Y on the other chromosome
with no
relationship between the two c.hromosomes received by X or by Y, i.e.,
when the sire genes, x s ' and the dam genes, xd ' are distinct sets.
where P(x s = ys ) is the probability that genes x and yare identical by descent. The more general
"
s
s
result for the covariance between X = (Ax A ) and Y = (A A )
s xd
Ys Yd
was given as
Then ~
= P(xs = Ys )
= P(xd = Yd)'
and "
(4. L 1)
«
Anderson and Kempthorne (1954) presented a factorial gene model
which included epistatic parameters for an arbitrary number of loci with
gene frequency of one-half.
1Ma 1'"ecot, G., 1948
.•
Masson et Cie.
The model was applied to populations
Les mathematiques de l'heredite.
Paris,
4
obtained by successive selfing of a heterozygous individual and to populations arising trom crossing two inbred 11nes with subsequent crossing
and s8l£1ng.
.
Cockerham (1954) partitioned the hereditary variance of diploid
populations into additive, dominance, additive x additive, additive x
dominance, dominance x dominance and higher order epistatic components
for the case of two alleles at each of an arbitrary number of loci with
no linkage.
Correlations between the hereditary deviations were consid-
ered for (1) relatives from random mating populations, (2) relatives from
a self-fertilizing population and (3) offspring from randomly mated
inbred parents.
The correlations between the epistatic deviations are
products of the one-factor additive and one-factor dominance correlations
for cases one and three above and for case two when gene frequency is
one-half and the relatives are in the same generation.
Hence, the covar-
iance between the hereditary values of two relatives can be expressed
for the above situations as
(4.1.2)
where p and q are the correlations between additive and dominance deviations, respectively, A is the number of times additive appears in the
component, D is the number of times dominance appears and n is the
number of loci.
Kempthorne (1954, 1955a, 1957) presented the same type of analysis
but extended it to include an arbitrary number of alleles for a random
mating population but with no inbreeding.
The covariance of any two
5
individuals was expressed by an extension of Ma1~cotfs notation as
(¥)~ (~,,)SO"~ D
.
.
(4.1.3)
r s
where A ~ppears r times and D appears stimes.
• and ~' are as defined
previously if the sire and dam genes are distinct sets.
If sire and dam
genes are not distinct sets,~,and •• ' are defined as the coef2
2
ficients of O"A and CT , respectively, as given in expression (4.1.1).
D
Horner and Kempthorne (1955) determined the correlation between
relatives in random mating populations for which the gene effects were
the same for all loci assuming only two alleles per locus and gene frequeney of one-half with no linkage.
Some of the classical epistatic
models were considered.
Kempthorne' (1955b) considered the correlations between relatives
derived from a random mating population by some regular system of inbreeding.
The covariance between full-sibs after n generations of
inbreeding and the covariance between a parent in the nth generation
and offspring in the (n + l)th generation were derived for the particular
case of full-sib inbreeding.
The effects of linkage in studies of correlations between relatives
have generally been assumed absent.
Cockerham (1956) studied the effects
of linkage on the covariances between relatives in random mating populations which are in linkage equilibrium assuming absence of position
effects.
Ancestral covariances were found to be unaffected by linkage,
whereas, the covariances not involving ancestral relationships were
affected.
The effect of linkage in nonancestral relationships was to in-
crease the covariance but only in the coefficients of the epistatic components and not in the coefficients of the additive and dominance components.
6
4.2
Analysis of Single-Cross Hybrids
The analysis of diallel crosses or the analysis of all possible
single-crosses from a set of lines has been used to obtain information
concerning quantitative inheritance.
The methods of analysis have been
varied.
Hull (1946) discussed the interpretation of the constant parent
regression analysis of all possible single-crosses from a set of
homozygous lines.
Epistacy was assumed to be absent.
The regression
of the constant parent regression coefficients on the phenotypic values
of the constant parent, b , was interpreted as an indication of the
2
degree of dominance.
Griffing (1950) described in detail the constant parent regression
analysis of quantitative genetic data.
W~th
nO,epistacy, b
inversely with h, the degree of dominance, i.e., b
2
= - h4d'
2
varied
However,
with dominance and a multiplicative type epistacy at two loci, the
direction of slope and degree of curvilinearity of b
was shown to
2
be dependent upon the particular combination of degree of dominance,
h, and the degree of epistacy,e.
For h= 0, i.e., no dominance, and
e)O, the trend of the constant parent regression values are opposite
to that for dominance alone.
Burdick (1957) presented a constant parent regression model for
multiplicative type epistasis with the assumption of no dominance.
With
this model, b varied directly with e, a measure of the degree of contri2
bution of multiplicative effects, in contrast to the dominance model with
no epistacy in which b varies inversely with the degree of dominance.
2
The sign of e with this model may be either positive or negative. Hence,
7
if multiplicative type epistasis were the cause of heterosis" b
2
could
be either positive or negative depending on the sign of e.
Sprague and Tatum (1942) presented an application of the analysis of
variance technique to the analysis of the n(n-l)12 single-crosses from
a set of n lines for the estimation of variance due to general combining
ability and variance due to specific combining ability.
The expectation
of the mean square for within line groups involved only variance due to
specific combining ability and an error variance.
Yates (1947) presented an analysis of variance of all possible reciprocal crosses from a set of lines for the following cases:
(1) se1f-
sterility, (2) no self-sterility and (3) self-sterility with incompatibility within groups of lines.
Additivity of parental effects was
assumed but a test for the departure from this additivity was available.
Griffing (1950) presented a
par~~~~o~in~~of
the variance among
Fl'S within a constant parent group into a basic gene variance" a dominance variance and a covariance between the basic gene effect and the
dominance effect.
It was assumed that epistasis was either absent or
that the proper transformation had been applied to the data to make
between locus effects and environmental effects additive.
Hayman (1954a, 1954b) included dominance in the model for the
analysis of the dial1el set of all n
2
matings with the following.assump-
tions:
(1)
Diploid segregation
(2)
No difference between reciprocals
(3)
Independent action of non-allelic genes
(4)
No multiple allelism
(5)
Homozygous parents
8
(6)
Genes independently distributed between the parents.
Methods for detecting epistacy and for estimating the degree of dominance
were presented.
Dickinson and Jinks (1956) extended this method to include heterozygous parental lines.
A method was presented for estimating the degree
of heterozygosity or the inbreeding coefficient and whether dominant
or recessive genes were in excess.
Griffing (1956) suggested the use of the term ''modified diallel"
to designate the diallel set where the parents are not included.
The
analysis of the modified diallel from homozygous parents with and without reciprocal crosses was generalized to include an arbitrary number
of alleles at an arbitrary number of loci with any magnitude of additive genetic, dominance and epistatic effects.
The assumptions and
conditions necessary for valid inductive inferences about the parental
population were presented as follows:
(1)
The set of inbreds is a random sample from a population
of inbred lines derived from the parent population by
means of an inbreeding system free from forces which
change gene frequency.
(2)
The parent population is under equilibrium conditions
with random mating.
(3)
It is necessary to use the ''modified diallel" to obtain
unbiased estimates of population parameters.
The variances of general and specific combining ability were given
explicitly in terms of additive, dominance and epistatic components of
variance.
The variance of general combining ability was shown to be equal
to the parent-offspring covariance in a random mating population in
equilibrium.
9
Kempthorne (1956) removed the assumptions of no epistacy and no
multiple alleles as given by Hayman (1954b) and presented the generalized
expectations of the sums of squares of the analyses given by Yates (1947)
and Hayman (1954a).
The parents were assumed to be homozygous lines
obtained without selection from a random mating population at equilibrium.
It was illustrated that only in the absence of epistacy could
information be obtained from the diallel analysis about the original
population.
Arguments were presented to show that previous analyses
in which the parents were not a random sample from a random mating
population at equilibrium are not valid.
Matzinger and Kempthorne (1956) extended the analysis of the
'~odified
diallel" to include an arbitrary degree of inbreeding for the
parents of the diallel and to obtain estimates of the interactions of
the genotypic components of variance and environments.
Kempthore (1957) extended the diallel analysis presented earlier
(Kempthorne, 1956) to include arbitrary'inbreeding of the parental
lines~
10
5.0
ANALYSIS OF VARIANCE FOR DOUBLE,-CROSS HYBRIDS
5.1
Introduction
The term double-cross hybrid is used to denote the progeny of the
cross of two F hybrids in which none of
l
t~e
four parents or lines are
the same and may be symbolized by (A x B)(C x D).
crosses, there are 3 pC different double-,cross
4
Omitting reciprocal
~ybr~ds
possible from
C ~,P! I'; p! :::; 1.2 .. ,P2,Pl'P.
p n
n. Pn'
The notat.ion Piis used herein to denote ,(p - i), e'. g., PI = (p - 1),
a set of p grandparents or lines, where
There are pC
4
different combinations of four 'grandparents, each combi-
nation giving rise to three different
pairing, which will be
~n
referre~
the single-cross patents.
d01ible,.,~rosses by
changing the
to as ordering, of 'the grandparents
For example, the three orders or double-
crosses for four lines A, B, C, and D are (A x B)(C x D), (A x C)(B x D),
and (A x D)(B x C).
The analysis of variance presented is of the complete set of 3 pC
double-cross hybrids from a set of p lines.
It is assumed that the p
lines constitute a random sample of all possible lines produced from
some random mating population by some system of inbreeding to an arbitrary degree without selection of any form.
The only restrictions on
the system of inbreeding are that the inbreeding coefficient can be
determined at any particular stage and that all p lines are equally
inbred.
4
•
11
5.2 Expectations of Quadratic Forms in Terms of Covariances
The phenotype of the double-cross (A x B)(C x D) grown in the mth
replication will be denoted in the following derivations by Y(AB) (CD)m
or, in general, Y(ij) (kl)m'
The paired subscripts within the paren-
theses indicate the parents of each'of the parental single-crosses.
The model for
~he
Y(ij) (kl)m
analysis of variance in such an experiment is
= r+ r m + G(~j)(kl) + e(ij)(kl)m
(5.2.1)
where
r = a contribution common to all entries
r
m
G(ij) (kl)
=a
=a
p
contribution particular to replication m,
contribution particular to double-cross
(i x j)(k x 1), and
e(ij) (kl)m
= a random deviation of the observation on the
double-cross (1 x J)(k x 1) grown in replication m.
It is assumed that e(ij)(kl)m's, rm's and G(ij) (kl) 's are independent
222
random variables with zero means and variances C1' , C1' and C1' , respectively,
e
r
y
and that the e's, r's and G's are uncorrelated with each other. A
constant number of individuals in each double-cross progeny in each
replication is assumed.
The expectations of the quadratic forms which will be encountered
222
can be defined in terms of U , C1' , cr , and the covariances between the
r
e
r
effects of the particular double-cross hybrids, i.e., the G(ij) (kl) 's.
The expectation of the square of the phenotype of a double-cross grown
2
2
2
in replication m will contain ru + crr + cre plus the covariance of the
effect of a double-cross with itself, i.e., cov(G(ij) (kl) G(ij)kl»'
The
expectation of the product of the phenotypes of two different double-crosses
in the same replication will contain
f2
+ cr; plus the covariance of the
12
e'ffectso'fthe: )twQ --4oul;rle-CTps~e,~, i. e,_, C,ov(G (ij) (kl) G(mn)'(op».
~pe~ta:.t'i..Qn
of:
th~ prp4u(::~, o~ ~:he phe~()~yp$& q~-th(\\
.same or
The
,dif.f~;r,ent
~oub:le-~o:sl:r~s>,-grown·in'different 1:'ept:i)c~~,i:ons/w;)U-,c~nta~~,,....~ 'plus,
t,he
The covariances between the effects of double-crosses are defined
as listed in Table 1.
Cov
Cov
It will be noted that:
l
= the
covariance of the effect of a double-cross with itself,
2
= the
covariance between the effects of two double-cross
hybrids with all four lines common but in different orders,
Cov
3
= the
covariance between the effects of two double-cross hybrids
with only three lines common and in the same order,
Cov
4
= the
covariance between the effects of two double-cross
hybrids with only three lines common but in different orders,
Cov
s = the
covariance between the effects of two double-cross hybrids
with two lines common and with the two common lines in
different parental single-crosses in each case,
Cov
6
= the
covariance between the effects of two double-cross hybrids
with two lines common and with both of the common lines in
the same parental single-cross in each case,
Cov
7
= the
covariance between the effects of two double-cross hybrids
with two lines common and with the common lines in the same
parental single-cross in one double-cross and in different
parental single-crosses in the other double-cross and
Cova
= the
covariance between the effects of two double-cross hybrids
with only one line common.
The covariance between the effects of two double-cross hybrids with no
lines common is zero.
13
Table 1-
Correlations between additive deviations, p, and dominance
a
deviations, q, of double-cross hybrid relatives
Number
of lines
in common
Covariance
Correlations
Inbreeding = F
F
P
9
P
=1
9
l
= Cov
[G (ij) (kl) G(ij) (kl) 1
4
F/2
F'1./ 4
1/2
1/4
2
= Cov
[G(ij) (kl)G(ik) (jl) 1
4
F/2
2
F /8
1/2
1/8
3
= Cov
[G(ij)(kl)G(ij)(km)l
3
3F/8
2
F /8
3/8
1/8
4
= Cov
[G(ij) (kl)G(ik) (jm) 1
3
3F/8
2
F /l6
3/8
1/16
5
= Cov
[G(ij)(kl)G(im)(kn)l
2
F/4
2
F /l6
1/4
1/16
6
= Cov
[G (ij) (kl) G(ij) (mn) 1
2
F/4
0
1/4
0
7
= Cov
[G(ij)(kl)G(ik)(mn)l
2
F/4
0
1/4
0
Cov
8
= Cov
[G(ij) (kl)G(im) (no)l
1
F/S
0
1/8
0
0
0
0
0
0
Cov
Cov
Cov
Cov
Cov
Cov
Cov
Cov [G(ij)(kl)G(mn)(op)l
a,rom Cockerham, C.
c.
5.3
(1957) (see footnote 2, Section 5.3).
Genetic Expectations of Covariances
In order to express the covariances between effects of relatives
as defined in Section 5.2 in terms of the genetic parameters as defined
by Cockerham (1954) or Kempthorne (1954), it is necessary to impose the
assumption that the p parents are a random sample of lines which would
result by inbreeding some equilibrium random mating population without
any type of selection.
In addition it is necessary to assume no linkage
and equal inbreeding for all parents.
Cockerham (1954) indicates that the
assumption of no linkage is necessary only if the parents are partially
inbred (O<F<l).
For completely inbred parents, the assumption of linkage
14
equilibrium in lieu of no linkage is adequate.
With these assumptions,
the covariances between the phenotypes of relatives, if the environments
are not correlated, can be expressed as
n
1:
A,D=O
AD 2
p q
~AD(Cockerham,
1954).
(5.3.1)
l~A-H>~n
2
Cockerham derived the correlations between the additive and dominance deviations, p and q in expression (5.3.1), for various hybrid
relatives.
The correlations for double-cross hybrid relatives are given
in Table 1.
5.4
Partitioning of Sums of Squares
The following notation will be used to indicate the individual observat ions and sums which are pertinent to the partitioning of the sums of
squares:
Y(ij) (kl)m = the phenotypic value of double-cross (i x j)(k x 1)
in the mth replication; m = l,2, ••• ,r; i,j,k,l
= 1,2, ••• ,p where i
+ j + k + 1.
r
Y(ij)(kl). = m:1Y(ij) (kl)m = sum of double-cross (i x j)(k x 1)
qver r replications.
~
= 1: Y(ij)(kl).= sum of the three orders or double-crosses
o
involving the combination of four lines i,j,k and 1.
2cockerham, C. C. 1957, Coefficients of the additive and dominance
components of variance between hybrid relatives. (unpublished).
15
Y
(ij)(k.) •
P3
=
E
Y(ij)(kl).
l+i,j,k
= sum
of all double-crosses with
parental single-cross (i x j) and grandparent k.
3
Y
ijk .•
=E
Y(ij)(k.).
o
= sum
of all double-crosses with lines i,j
and k as grandparents.
Y
(ij)( .. ).
1 P2
=- E
.
Y
2 k+i'
,J
(iJ)(k.).
=
Y(ij)(kl).
= sum of all
double-crosses with parental single-cross (i x j).
C
2
P2 2
Y
E
(t.)(j.).
k,l+i,j
k<l
E Y(ik) (jl).
o
= sum
of all
double-crosses with grandparental lines i and j in
different parental single-crosses.
Y
ij •••
1
P2
=Y (ij)( .. ). +Y (i.).(j.). =- .E.
Y
2 it"i, j ijk ••
=
P~C2
3
E
k,l+i,j
E
o
Y
(ij)(kl).
= sum
of all double-crosses
k~l
with line i and j as grandparents
Y
i ....
=.!
c:
p
El Y
3 j+i
ij ...
=
Pl 3
3
E
E Y(ij)(kl).
j,k,l+i o
j<k<l
= sum of all double-
crosses with line i as one grandparent.
16
Y
1 P
=-EY
4 i i ....
Other pertinent facts
Table 2.
=
pC4
E
i,1,k,1
i<j<k<l
3
E Y(ij)(kl).
o
= grand
total.
regarding·,thesens.utnilations;.~aJ:'e
. presented in~.Table
Description of sums pertinent to the analysis of double-cross
hybrids
One possible method of partitioning the sums of squares from all
possible double-cross hybrids grown in r replications in one environment
is presented in Table 3.
The partitions of the sums of squares into
2.
-
e
Table 3.
e
Analysis of variance of double-cross hybrids
Source
d.f.
Correction factor
Sums of squares
8y2
..
·C = rpP P P
I 2 3
...
I
M.S.
C
r
Replications
(r-l)
8 1:: y2
R = m=l··· ..m
- C
PP I P2P3
C
1? ~:
2
1:: Y(ij)(kl).
i<j<k<l o
-_ C
H=
r
1::
Hybrids
3
C
P 4
- I
2
I-line general
PI
G=
~
R*
3
-H*
y2
i....
4P
1
i
---C
rP 2P3P4
P4
G*
-........
C
2-line general
PP 3 /2
P 2 2
2 1:: Yij •••
i<j
S =
3rp P
4 S
6P 2P3
3P3
C --G
- P4PS
Ps
S*
p C3 .
1:: y2
3-line general
PP l PS /6
4P 3
2P
3P4
U = i<j<k ijk ••
--C--G--S
S
P6
P6
· P6
3rP6
U*
.....
--.J
e
e
e
Table 3(cont1nued)
Source
d.f.
Sums of squares
p
4-line general
PP 1P2P7 /24
C4
1:
y2 ·
w == i<j<k<1 ijkl.
3r
.-
C
2-line order
PP 3 /2
C
C
P 3
3-line order
PP 2P4/ 3
1:
V== i<j<k
Error
PP1P4P5/12
(r-l)(3 pC4 - 1)
3r pC4 - 1
P 2 2
p
3
1: y2
0
(1j)(k.).
1:
y2
i<j<k ijk ••
p
-r
M==1:
.m
C
1:
i<j<k<l
T*
2P
-
3
2
-T
P
3
V*
C
4
1:
y2
i(j<k<l ijkl.
3r
- T - V
X*
E*
E=M-R-H
P 4
3rP 1P2
C
3
3 2
1:
~ Y(ij) (kl) •
X == i<j<k<1 o
C4
2 1: Yij ...
i<j
1: y(1.)(j.).
1<j
.
r p l,P
2
3rp
r
Total
+
C
2
rP 3
p
4-line order
p2
2 1: y2
1<j (ij) ( •• ) •
rpiP2"
w*
C - G - .S' - U
p 2
T ==
M.S.
3
1: y2
0
(ij)(k1)m - C
f-&
00
19
variance due to replications, hybrids, and error are the least squares
partit~susing Model
(5.2.1) and are orthogonal.
The partitioning
of the hybrid sums of squares was arrived at by largely a trial and
error procedure.
It is, however, an orthogonal partitioning.
The proof
of orthogonality is long and involved and is not presented in this thesis.
Mean squares are indicated by an asterisk on the letter symbolizing the
corresponding sum of squares.
5.5
Expectations of Sums of Squares in Terms of Covariances
The expectations of the sums of squares can be derived in terms of
2
u , ,,2, er2 and the covariances between the effec:ts of relatives as defined
're
in Section 5.2. These expectations are presented in Table 4 for the uncorrected sums of squares.
The derivation of the expectation of the"
one-line general sum of squares,
will
b~
presented for
il1ustra~ion.
2
terms of Y(ij) (kl)m so that Yi •••• will con;ain, a1fogether)
2 222
.
r Pl~2P3/4 squares or ~roducts of these terms. Of these,rPlP2P)/2
,
222
will be squares and have expecta~ion ~ + ere + err + Cov l • ~ach of- the
rPlP2P3/2 terms will be involved in products with (r-l)~iJte terms in
other replications so that r(r-l)PlP2P3/2 products will have expectatioq
2
2
~ + Cov • There will ber PlP2P3 product~,between double-crosses with
l
four lines ¢ommon but in
be between
differen~ orde~.
double-c~osses
in the -same
Of these,
rPlP~P3
of them will
r~plicationwithexpectation
2 2 ·
+ err + Cov 2 and r(r-l)Pl P2P3 of them will be between
~
different replications with expectation f 2 + Cov •
2
dou~le~ctosse~
in
e
Table 4.
e
e
Expectations of uncorrected sums of squares in terms of the covariances between double-cross
hybrid relatives
Sums of squares
r: p2
.
+ ci)
r
2
CT
e
Coefficients of covariances
Cov
Cov
Cov
Cov
1
2 Cov3
4
S
,Cov 6
Cov
Covs
7
y2
.....
3 pC4
3r C
p 4
2
1
r
2r
~i y~1. ••••
C
P 2 2 '
I: Y
i<j ij •••
3rP2P3
2
I.:
Y. jk
i<j<k 1.
3rP3
'-'Srp
,4
4rP4PS
rP4PS
' 6'P~4
2rpP4PS
rpP4PS
2
4rP4PS 2rP4PSP6
"
rP 1P2P3
C
P3
4rP4
••
12 C
p 4
P
rp
2rp
3rpP4
IS pC4
C
p 2
rpP
l
2
rpP l
rpP 1P4 2rpp.i~4
12 C
p 4
C
p 3
r~C'3
',p,
2rpP4PS
IrPP4PSP6
2
rpP 1P4PS rpPIP4PS rpPi~4~S
3
12
3
0
2r pC3 tpit. p,Cj , 2rP4 pC3
0
0
0
0
2r
0
0
0
0
C
P 4
2
Y
i<j<:k<l ijkl.
3 r
I:
3 pC4
rC
p.4
pC4
C
p 4
0
.
0
"
,
N
o
e
e
e
Table 4(continued)
.•.
Sums of squares
( rp.2 + crr
.2)
cr,2
e
..-
Cov l
Coefficients of covariances
Cov
Cov
Cov
Cov4
2
S
3
C
P 2
2 r. '. y2
1<3;, (ij)( .• ).
6 pC 4
rP2P3
. C
2
PI: y2
i<j (L)(j.).
rP2P3
pC3
r.
i<J<k
3 '2
r. Y(ij)(k.).
0
rp3
pC4
r.
i<j<k<l
12 pC
4
r
pC 2 r pC 2
0
2rP4 pC 2
0
0
,C
r C r pC 2rP4 pC 2rP4 pC rp PS pC
2
2 4
2
p 2.
2
P 2
.'
Cov
6
'
, rP4PS
C2,
.
p
2
Cov
7
;Co~8
0
0
0
0
0
I
12 pC 4
3 pC) 3r pC)
0
3 pC4
.3 '-C
p. 4 3r pC4
0
3rP4 pC 3
0
0
0
0
0
0
0
0
0
0
3
2
r. y (ij)(k1).
0
r
-
P
I
....
N
22
2
Of the 3r PlP2P3P4/2 products between
double~crosses
with three
lines common and in the same order, 3rPlP2P3P4/2 will be between
double~
crosses in the same replication with expectation p.2 + CT~ + Cov
3r(r~l)PlP2P3P4/2
with expectation
and
3
will be between double-crosses in different replications
~
2
+ Cov3 • There will be twice as many products involv-
ing hybrids with three lines common but in different order.
3rPlP2P3P4 of the products will have expectation
3r(r-l)PlP2P3P4 of them will have expectation f
2
~
2
Hence,
+ CTr2 + Cov4 and
+ Cov4 •
2
Of the r PlP2P3P4PS products between double-crosses which have two
lines common but in different parental single-crosses in both hybrids,
rPlP2P3P4PS will be between hybrids in the same replication with the expec.
2
2
+ CTr + CovS and r(r-l)Pl P2P3P4PS will be between hybrids in
different replications with expectation ~2 + Cov • There will be oneS
tation~
fourth as many products between hybrids with two lines common but
in
the
same parental sil!'gle-cross; hence, rPIP2P3P4PS/4 will have ,expectation
2
2
2
+ Cov 6
and r~-1~PlP2P3P4PS/4 will have expec~ation,... + COV 6 •
The number of products between· hybrids with two lines common but in
f'A
+ CTr
different order will equal the number with the two
co~on
lines in
different parental single-crosses; hence, rPlP2P3P4PS products will have
2
2
expectation fA' + CTr + Cov7 and r(r-l)PlP2P3P4PS will have expectation
2·
fA + COV7 •
2
Of the r PlP2P3P4PSP6/4 products between double-crosses with only
one line common, rPlP2P3P4PSP6/4 will be between double-crosses in ttte
same replication and r(r-l)PlP2P3P4PSP6/4 will be between double-crosses
222
in different replication with expectations f + CTr + CovS and f + CovS'
respectively.
23
There will be no products with expectation
22
f2
or ~ + a r since all
double-crosses in the sum yi •••• have at least one line common.
In summary,
The expectations of the sums of squares as partitioned in Table 3
may be derived as simple linear functions,of the expectations presentedin
Table 4, i.e., E(E aix i ) = E a i E(x i ).
If we define ! as the (9 x 1) vector of sum of squaTes as listed
in Table 4, M as the (9 x 10) matrix of coefficients in Taule 4, and
£'
= ( rf2
+ a~,a:,covl,cov2,cov3,cov4,coV5,cov6,cov7,COV8)' then
E(!) =
~.
Further, if we define S' = (G, S, U, W, T,
-p
V~
to the (7 x 9) transformation matrix such that
X) from Table 3 and N equal
!p = N!
(i.e., N is the
matrix of coefficients which transforms the uncorrected sums of squares,
!, into the partitioned sums of squares,
E(!P)
S ), then
-p
= NE(!) = (NM)Q
Each expectation will contain, in addition to the covariances, the degrees
222
of freedom times O"e' The terms tJ- and O"r drop out of the expectations of
all the hybrid sums of squares.
The expected mean squares are then found
by dividing the expected sums of squares by the degrees of freedom.
2
The
mean square expectations are shown in Table 5 in terms of a and eight
e
functions of the covariances.
e
e
Table S.
Expectations of mean squares in terms of the covariances between hybrid relatives for the
analysts (:of ,od~"bJ.,~r,c~(iSS hybrids
~~~~_._~~~~_.
M.S.
e
2 Cov
(J'
l
e
+
2Cov
2
Cov
1
- Cov
2
Cov
C()effIe{e.:tl~s~~~f~c()varJ~ances~-~
3
+ 2Cov'4
Cov
3
4CovS + Cov 6
+ 4Cov 7
cov S' + "CdV 6',
, ,
- 2Cov "
7
0
rPSPS/2
0
0
2
r(p -2lpi92)/6
0
- 3rP6P9 '
0
3r(3p-'Z2)
0
-l2r
- Cov
4
C
oVa
G*
1
r
0
S*
1
r
0
2rP
u*
1
r
0
rPlO
0
-rp
w*
1
r
0
-4r
0
2r
T*
1
0
r
' '9,
2rP4
0
rP4P5'!~'
0
v*
1
0
r
0
rP 7
0
-rp
0
x*
1
0
r
0
-4r
0
2r
0
E*
1
0
0
0
0
0
0
0
r(3p-l6)
7
a
rpsP61>16/ 2 .
S
too,)
~
25
5.6
MOdel of Linear Effects
The development of a model of linear effects is not essential for
the estimation of genetic parameters.
The results of Section 5.3
(the genetic expectations of the covariances) and Section 5.5 (expectations of the sums of squares in terms of covariances) can be used
directly to estimate the components of genetic variance.
The develop-
ment of a model of linear effects is necessary, however, for the development of tests of significance of various genetic hypotheses.
The effect model presented in this section is the join result of
intuitive reasoning as to what would constitute a logical model and an
attempt to make the linear model compatible with the previously defined
covariances at least in their more apparent features.
It is known, for
example, on the basis of genetic expectations of the covariances that
Cov
6
= Cov 7 >Cova
(Table 1).
It would be most desirable that these
covariances expressed in terms of the components of variance would
follow the same pattern.
With these and other conSiderations, the
following is proposed as one possible model of linear effects:
Y(ij) (kl)m = ~ + r m + G(ij) (kl) + e(ij) (kl)m
= p. +
r m + gi + 8j +
~
(5.6.1)
+ 8 1 + Sij + sik + 8 U
+ Sjk + Sjl + skl + t ik + t u + t jk + t jl + u ijk
+ u ijl + u ikl + u jkl + v(ij)k + v(ij)l + v(kl)i
+ v(kl)j + wijkl + x(ij) (kl) + e(ij) (kl)m
where Y(ij) (kl)m'
tA' r m,
Section 5.2 and where
G(ij) (kl) and e(ij) (kl)m are as defined in
(5.6.2)
26
gi = a contribution particular to line i,
Sij
=a
contribution particular to line i and line j
appearing~'together
Sij + t
ij
as a single-cross parent,
= a contribution particular to line i and line j
appearing in different parental single-crosses,
u
ijk
= a contribution particular to lines i, j and k
appearing together in any order.
v(ij)k = a contribution particular to lines i, j and k
appearing in the order (ixj)k,
w
= a contribution particular to lines i, j, k and 1
ijk1
appearing as the four grandparents in any order.
X(ij)(k1) = a contribution particular to lines i, j, k and 1
appearing together in the order (ixj) (kx1).
The terms gi' Sij' u ijk ' and wijk1 will be referred to as one-,
two-, three-, and four-line general effects, respectively, and t ,
ij
v(ij)k' and X(ij) (k1) as two-, three-, and four-line order effects,
respectively.
It is assumed that effects gi' Sij' t ij , u ijk ' and
w
are independent random variables with means zero and variances
ijk1
2 2
222
0'g , 0's , O't' (ju and 0'w, respectively, and that they are uncorrelated with
each other.
The effects v(ij)k and x(ij)(kl) may be treated under two different
assumptions.
In Model A, the effects v(ij)k and X(ij) (kl) are assumed
to be random variables with means zero and variances (j2 and .0'2, respec..,
v
tively.
x
In Model B, the three 3-1ine order effects involving the same
set of three lines and the three 4-line order effects involving the same
four lines are assumed to sum to zero within each set;
27
that is,
V(ij)k + v(ik)j + v(jk)i
=0
for all combinations of three lines
and
X(ij)(k1) + X(ik) (jl) + X(il) (jk)
=0
for all combinations of four
lines.
The variances of the effects v(ij)k and X(ij) (k1) are defined for MOdel B
as
222
1/2 E(v(ij)k + v(ik)j + v(jk)i)
= CTv2
and
222
2
2
= CTx
1/2 E(X(ij) (k1) + x(ik)(jl) + X(il) (jk)) = 0-x
ijk1
or
2 2 2
E(v(ij)k) = 2/3 CTv ' E(v(ij)k v(ik)j) = -1/3 o-~
2
2
E(x(ij)(k1)) = 2/3 o-x' and E(X(ij) (kl) x(ik)(jl))
= -1/3
2
x
CT •
The covariances between three-line order effects not involving the same
combination of three lines and the covariances between four-line order
effects not involving the same combination of four lines are all assumed
to be zero.
It should be noted that the two-line order effect, t
, could be
ij
handled under similar alternative models by defining two effects, t(ij)
and t (i) (j)' for example,· such that t (ij) + 2t (i) (j) = O.
No arrange-
ment of this system, however, was found which was compatible with the
genetic expectations of the covariances.
2a
Covariances as Functions of the Components of Variance
5.7
The covariances as defined in Section 5.2 may be expressed in terms
of the components of variance as defined in Section 5.6 by taking the
expectations with respect to the model of linear effects.
These results
= Cov7 ~ Cova
in both Table 1
are presented in Table 6 for Model A.
and Table 6.
Cov6
Hence, at least in this major respect, the l,inear model of
effects is compatible with the genetic expectations of the covariances.
Covariances between double-cross hybrid relatives
expressed as functions of the components of variance
Coefficients of components of variance
2
Covariance
2
2
2
2
2
0'2
'o-CT
CT
CT
CT
CT
CT
g
w
s
t
u
x
v
Table 6.
cov
Cov
2
Cov
3
Cov
Cov
Cov
Cov
l
4
5
6
7
Cov
a
5.a
4
6
4
4
4
1
1
4
6
2
4
0
1
0
3
3
2
1
1
0
0
3
3
1
1
0
0
0
2
1
1
0
0
0
0
2
1
0
0
0
0
0
2
1
0
0
0
0
0
1
0
0
0
0
0
0
Expectations of Mean Squares in Terms of Components of
Variance and Estimation of Components of Variance
The expectations of the mean squares in terms of the components of
variance are presented for Model A and MOdel B in Table 7 and Table
respectively.
a,
the derivation of the expectations of the mean squares
are straight forward and are not presented here.
For the completely
29
Table 7.
M.S.
Expectations of mean squares with Model A for the analysis
of double-crosses
Coefficients of components of variance
2 (T2 (T2
2
(T2
(T2
(T2
(T2
(T2
(Te x
(Tt
g
w
s
r
u
v
R*
1
0
0
0
0
0
G*
1
r
3rP4
2rP3P4
3r
9rP4 9rP3P4/2 rP2P3P4/2
S*
1
r
2rpS
2rP 4PS/3
3r
6rpS
3rP4PS/2
0
0
u*
1
r
rP6
0
3r
3rP6
0
0
0
w*
1
r
0
0
3r
0
0
0
0
T*
1
r
2rP2
rP1P2/3
0
0
0
0
0
v*
1
r
rP 3
0
0
0
0
0
0
x*
1
r
0
0
0
0
0
0
0
E*
1
0
0
0
0
0
0
0
0
0
0
3 pC
4
0
random model, Model A, the expectation of the mean squares can be derived
directly by simple substitution of the results in Table 6, where the
covariances are expressed as functions of the components, in the results
in Table 5, where the expectations of mean squares are given in terms of
covariances.
It is apparent from the expectations of the mean squares that
2
exact tests of significance are available only for the hypotheses (Tx = 0,
2
2
.
2!
2'
2'
2t
(Tw
= O.and
(T
=
0
for
Model
A
and
(T.=
0,
(J'
=
0,
(T
=
0
and
(J'
=0
·
v
x
w
v
u
in Model B.
Tests of significance of the remaining components are
complex and are approximations.
The appropriate F-tests and their
interpretations are presented in Section 5.10.
30
Expectations of mean squares with Model B for the analysis
of double-crosses
Coefficients of components of 'variance
...
2' 2'
2'
2'
2'
2'
2'
2'
,cr:.2'
cre crx
cr
crv
crw
cru
crt
crr
g
s
Table 8.
M.S.
.
R*
1
0
0
0
0
0
G*
1
0
0
2rP3P4
3r
9rP4
S*
1
0
0
3r
6rp
U*
1
0
0
2rP P /3
4 S
0
w*
1
0
0
0
3r
0
T*
1
r
r~1P:2J3
0
v*
1
r
2r P
2
rp3
0
X*
1
r
0
E*
1
0
0
0
3 pC
4
0
0
9rP3P4/2 rP2P3P4/2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3r
S 3rP4PS/2
0
3rP6
The formulae for the estimation of the various components of
.
2
variance for the two models are as follows where the estimate of cr is
2
denoted by s.
A prime is used on the component to indicate an estimate
for Model B.
Model A:
s2
x
= (X*
- E*)/r
2
s w = (W* - X*)/3r
2
Sv
s
s
= (v*
- x*)/rP 3
2
P6
= [u* - w* - ~(v* - x*)]/3rP 6
u
P
3
2
t
2P 2
= 3 [T* - -lfft
P3 _
P
+ - l X*l/rp P
P3
1 2
~l
Model B:
2'
2
s x == s x
2'
s w == (W* - E*)/3r
2'
2
2'
= St
s v == s v
2'
Su = (U* - W*)/ 3rP 6
St
S
2
2'
2
s == s S
2'
2
s g == s g
2
2
Only c:rw
and c:r'u require different estimators in the two models.
Further discussion of the above components will be delayed until after
the expectations of these components are given in terms of genetic
parameters.
Since the analysis as presented is an orthogonal partiti?ning of the
sums of squares, the mean squares are uncorrelated.
If the observations
are normally distributed, then each mean square is a sum of squared
linear funtions of NID variates and is distributed as 1l2c:r~ with f
degrees of freedom.
Hence, the variances of the mean squares are equal
4
2c:ri
to T
2(MS
and are estimated by
i
i
i
)21
£i +
~
(Anderson and Bsnoro£t, 1952) where
32
£i indicates degrees of freedom for the ith mean square.
The variance of
the estimated components of variance may then be computed by applying the
rules for finding the variance of a linear function.
That is, the
but since the mean squares are uncorrelated,
For example,
arid
2:
y*2
V(Su2) ... ~2~2- f + 2
9r P6u .
"l
5.9
w*2
+fw .+ 2
p 6..
v*2
+ 2( f
P3
v
+2+
x*2
f
. x
~]
. + .2
...
Components of Variance as Func·tj,ons of Covariances
between Hybrid Relatives
The expectations of the
the components
re~pectively,
mea~
squares have been derived in terms of
~f v~riance forMode~s
and in
ter~
of the
A and
B,
covari~nces
Table 7 and Table 8,
between hybrid relatives,
From these"results~ tqe components of variance for the two
Table 5.
models mar be expressed as functions of the covariances between hybrid
relatives.
Model A:
Let. -e.
S'
= (G*
!' =
- E*, S* - E*, U* - E*, W* - E*, T* - E*, V* - E*, X* - E*),
(0";, 0";,
O"~,
0";' 0"=,
O"~,
0":>,
33
= the
A
7 x 7 matrix of coefficients of the components of variance
in the expected mean squares of Table 7, excluding R* and E*
2
2
mean squares apd the columns for CTe and CTr ,
B = the '7 x 7 matrix of the coefficients of the functions of
covariances given in Table S, excluding E* mean square and.
2
the column for
CT
e
, and
Qi = (Cov l
+ 2 COV 2' cov l - COV 2' Cov3 + 2 COV 4' Cov 3 - COV 4'
4 covS + Cov 6 + 4 Cov7' CovS + Cov 6 - 2 COV7' Cova).
E(~)
Then, from Table 7,
~
= B~
or
~
= A-1 BQf'
=~
and, from Table S,
E(~)
= BQf.
Hence,
This solution expresses the components of
variance in Model A as functions of the covariances between relatives
(Table 9).
Table 9.
Components of variance as functions of the covariances
between double-cross hybrid relatives for :ttode1 A
Coefficients of covariances
. Cov
Cov
Cov
Cov
Cov
6
4
S
3
7
2
Cov
0
0
0
0
1
0
0
0
-1/3
4/3
-2
0
0
0
1
1
-2
0
0
0
0
1
-1
0
-2
3
0
0
1
-1
-1
-1
2
0
0
1
0
-4
2
0
4
-4
1
-1
-4
4
·2
2
-4
0
Component of
variance
Cov
2
g
0
0
0
0
0
0
CT
2
CT
s
2
t
2
CT
u
CT
2
CT
v
2
CT
w
2
CT
x
l
Cov
a
34
Model B:
2'
Let g' = (O'x '
2'
O'v '
2'
2'
O't' O'w'
2'
2'
2!
eTs ' erg') and
O'u '
D = the 7 x 7 matrix of coefficients of the components of variance
in the expected mean squares in Table 8, excluding.R* and E*
mean squares and the columns for
Dg
Hence,
g
Then, E(§'e) =
0';.
Dg.
D-1 ~f'
This solution is given in Table 10.
2
2' 2. 2' 2
2'
It is apparent from these results that O'g = O'g, O's = O's ' O't = O't'
2
2'
2
2'
2
2'
2
2'
0' = 0' and 0' = 0' but 0' + CT and 0' + CT.
v
v
x
x
u
u
w
w Hence, changing the definition of the three- and four-line order effects from random to fi~ed,
2
2
i.e~, MOdel A to MOdel B, changes the interpretations of only 0' and 0' •
=
B~
or
CT~ and
=
u
W
Table 10.
Components of var~ance as functions of the covariances
between double-cross hybrid relatives for Model B
Coefficients of covariances
Component of
Cov
Cov
Cov
Cov
Cov
Cov
Cov
.Cav
variance
2
3
4
5
6
7
8
l
I
2'
O'
g
2'
O'
s
2'
O'
t
2'
0'
u
2'
0'
V
2'
0'
w
2'
x
0'
0
0
0
0
0
0
0
1
0
0
0
0
0
-1/3
4/3
-2
0
0
0
0
1
-2
0
0
0
1/3
2/3
-4/3
-1/3
-4/3
3
0
0
1
-1
-1
-1
2
0
1/3
2/3
-4/3
-8/3
8/3
2/3
8/3
-4
1
-1
-4
4
2
2
-4
0
1
5.10 Genetic Interpretations of the Components of Variance
and Tests of Significance
The covariances between double-cross hybrid relatives can be expressed
in terms of the genetic components of variance by function (5.3.1) where
the values of p and q in the function are given in Table 1 for each of the
covariances.
The genetic interpretations of the components of variance
can then be derived by applying (5.3.1) to the results of Section 5.5
(Table 9, Table 10).
For example,
2
2!
n
O'g = O'g =:CovS = I:
A,D=O
2
(!)A(O.)D
S
O'AD
l~A~n
F 2
F2 2
= i 0'10 + 64 0'20 +
F3 2
ill 0'30 + •..
and
The expressions of the components of variance in terms of one-, two-,
and three-factor genetic components of variance are presented in
Table 11.
The genetic interpretations of the effect components of variance
222
are apparent from Table 11. The genetic expectations of O's' O'u' O'v'
2
2
0'w and 0'x involve only epistatic components of genetic variance.
2
2'
CT = CT is a function of the dominance x dominance epistatic component
x
x
of genetic variance and all three-factor or higher epistatic components
except the all-additive types.
Hence, a test of the hypothesis that
e
e
e
Table 11.
Components of variance in the analysis of double-crosses as functions of the components of
genet~c variance up to and including the three-factor epistatic components
Coefficients of components of genetic variance
Component
2
2
2
2
.2 '
.2 .
.2
2
.2
0"10
0"01
0"20
0"11
0"02
0"30
0"21
0"12
0"03
F/8
0
F2 /64
0
0
F3 /512
0
0
0
= 0"2'
s
0
0
F2 /64
0
0
3F3 /256
0
0
0
0"2 = 0"2'
0
F2 /16
0
F3 /64
F4 /256
F4 /256
F5 /1024
F6 /4096
0"2
u
0
0
0
F3 /128
0
3F3 /25n
5F 4 /1024
F5 /2048
0
0"2'
0
0
0
3
F /96
F~1384
3F3 /256
4
5F /768
5F5 /3072
6
F /2048
0"2 = 0"2'
0
0
0
F3 /128
4
F /128
0
5F 4 /1024
7F 5 /2048
3F6 /2048
0"2w
0
0
0
0
F4 /128
0
F4 /256
F5 /256
3F 6 /2048
~'
w
0
0
0
0
F4 /96
0
F4 /192
5F5 /768
6
F /256
= 0"2'
x
0
0
0
0
4
F /128
0
F4 /256
F5 /128
15F 6 /2048
0"2 = 0"2'
g
g
0"2
s
t
t
u
v
0"2
x
v
0
I.rJ
Q\
37
a
2
x
=0
is a test of the hypothesis that this particular function of the
2
epistatic variances equals zero. Similarly, the tests that aw = o and
2'
('J"
= 0 are tests that a slightly different function of the same epiw
2
2'
static variances equal zero. The. functions for a and a place relaw
w
tively less emphasis on the higher-factor components than the function
2
x
for a •
2
2'
The test of the hypothesis av • av = 0 is a test that a function
involving additive x dominance, dominance x dominance, and all higherfactor epistatic variances except the all-additive types is equal to
2'
zero. a is a function of all epiStatic components of variance except
u
2'
additive x additive epistatic variance and a test of au = 0 is a test
of these components.
2
The test of au
=0
is a test of additive x domi-
nance epistatic variance and all higher-factor epistatic variances
2'
except the all-dominance types. The test of a2 • a = 0 is a test of
s
s
only the all-additive types of epistasis. The test of at2 • at2' = 0 is
a test of dominance variance and all epistatic variances involving
dominance in their nomenclature and the test of a2 = ('J"2' = 0 is a test
g
g
of additive genetic variance plus the all-additive types of epistatic
variance.
Thus, testing the hypotheses that the various effect components of
variance are zero will provide considerable information on the gene
However, a' indicated in Section 5.2, simple F-tests are possible only for the hypotheses concerning ('J"2, a2, and a2 in
w
x
v
2' 2' 2'
2'
Model A and ('J" , a , a , and a in Model B unless some of these comx
v
w
u
ponents can be assumed equal to zero, either on the basis of the F-test
action involved.
or from other genetic information.
If some can be assumed equal to zero,
then exact F-tests will be available for more of the remaining components
of variance.
38
. ~,Sa.tterthwaite (1946) suggested that a linear function of mean '
2
2,
squares, 1:(a MS ), is approximately distributed as 'X,; fI /f'; with f'
i i
degrees of freedom where
(5.10.1)
The degrees of freedom for mean square MS
is denoted by fie With the use
i
of Satterthwaite's approximation, two different approximate tests have
been proposed for the cases where no simple error term is available.
One
procedure is to construct an error term with the proper expectation as
linear function of mean squares and assume it is distributed asX2fI2 ffo
with f' degrees of freedom.
Satterthwaite, however, has indicated that
caution should be exercised when using this approximation if the linear
function of mean squares involves some negative a • Cochran (1951) sugi
gested an F-test whereby negative a can be avoided by adding all
i
negative terms to both numerator and denominator of the F-ratio involving the constructed error term.
This ratio is then assumed to follow
the F-distribution with fi and fi degrees of freedom.
Anderson and
Bancroft (1952) indicated that Cochran's test is more powerful than the
first test but whether or not it is enough better to offset the need fot
computing two estimated degrees of freedom instead of one is doubtful.
The exact and approximate F-tests are presented in Table 12 for the hypotheses that the various components of variance equal zero.
F is used
to indicate an F-test to avoid confusion with F used', to indicate the
inbreeding coefficient.
jt indicates an approximate test formed by con-
structing an appropriate error term and
Fft
indicates an approximate test
using Cochran's suggestion.
A much simpler system of tests, which are
~alid
F-tests,
~re
availa-
ble but they will not generally yield as much information concerning
Table 12.
F-tests of hypotheses that various components of variance are zero in the analysis of
double-crosses
Cotn:Ponent
2 21
ux ' Ux
2
U
w
U
21
w
2
U ,
v
e
e
e
21
. F_-~esL . _
.
a
_._.. __.
~egrees_oJ_freedom
F •
-
X*/E*
PP1P4PS/12, (r-l)(3 pC4 - 1)
F •
W*/X*
PP 1P2P7 /24, PP 1P4PS/ 12
F •
W*/E*
PP 1P2P7 /24, (r-l)(3 pC4 - 1)
v
F • V*/X*
PP 2P4/ 3 , PP 1P4PS /12
2
Fl.
PP 1PS/6, fl
U
U
u
11*
P6 .
w* + - (V* - X*)
P3
P
11* +..§. X*
F"·
.
P3
f"
W*+!!V*
P3
U
21
U
2 21
uti ut
l'
ftt
2
F • U*/W*
PP 1PS /6, PP 1P2P7 /24
j
PP /2, fl
3
I.
-::--_--=T*
_ _-
Pl
2P 2
-V*--X*
P3
P3
W
\D
e
e
e
Table l2(continued)
ComRQp.e:J:1f:~ ._.~ __ ~
. _.1!'",,-tes_t
~Degrees
of freedoma
P
2
O"e
21
0"t
T* +..1 x*
_ " P3 "
2P2
F"=
f"
l'
fit
2
~v*'
P3
2
0" ,
S
21
0"
S
S*
-I
F == 2P 5
'P 6
P4
u* - -
. P-6
P
- '
F"
g
21
g
.
FI
, P 1P 2
T* -
P6 . '
P 1P3· "
- ..""
2P4PS· ' 2P 4'S
~6
0"
+.
,
'P 1P3
v* +
P~3
PP3/2, fl
x*
4p P '.
-u*+
2
W*
2l?4PS
S* + ..! W* + ' 4 S v*
- == 2PS
0" ,
'4P 4PS
. 2P4PS"
= 3P
-
3
.T*+,..
P 1P 2
P 2P3
,"
G*"'
." 3P ,.
2
--U*
P6·
·
s*
Ps "
·x*
fIt fll
l' 2
".P1' fl
P P
2 3
+ ---:PSP6
W*
3p
+ --1 u*
-F"·- - '6
G*
'.--S S"'"
3P3
,---
a
" 2P3
-:I- --.PP
fIt
. l'
fll
2
w*
S 6
.
fl and f" indicate approximate degrees of freedom as suggested by Satterthwaite (l946)(see equation
So1001)
~
o
41
gene action as the tests indicated in Table 12.
The error mean square
for the experiment is a valid error term for testing each of the entry
mean squares.
The hypotheses being tested, however, are the composite
hypotheses that all genetic components of variance appearing in the
expectation of the respective mean squares are equal to zero.
Since the
genetic expectations of the mean squares are independent of the effect
model, the interpretations of the F-tests using the common error term
The tests X*/E*, W* IE* I V* IF.* and
are the same for both models.
ra:r IE*
have essentially the same interpretation as the F-tests of the corre222
2
sponding components of variance 0'x , 0'w, 0'v , and O't' respectively.
F =U*/E* is, however, a test of all the components of epistatic variance except the additive x additive component.
F. S*/E*
is a test of
dominance variance and all epistatic components and F • O*/E* is a test
of all components of genetic variance.
The ability to test only the
all-additive components of epistatic variance and to test the additive
genetic variance plus only the all-additive epistatic components is lost
with this procedure of testing.
The most powerful test of dominance and epistasis is
with P(P1P2P3 - 8) and
(r-~3
p. C4 - 1) degrees of freedom.
The simplest
overall test of epistasis is
_ . 8(U + W+ V + X) /pP2PS(P + 1)
F •
; E*
with PP2PS(P + 1)/8 and (r-l)(3 PC4 - 1) degrees of freedom. This test,
however, excludes the effects of two-factor additive x additive epistasis.
In order to include these effects in an all-epistatic test, the test would
have to be a complex test.
42
5.11 Estimation!of Genetic Parameters
Equating the estimates of the components qf variance to their
expectations in terms of components of genetic variance gives a set of
2
2
2
'
n(n+3)
seven equati9ns in
2
unknowns, ~lO' ~Ol' ~20' ••• , where n is the
~ociinvolved
number of
(Table 9).
.
I
It is necessary
to
assume a re-
stricted genetic model in order that the number of unknowns be reduced
to an estimable number.
We may assume (1) additive and dominance effects
. .
222
2
but complete independenc~ between loci, i.e., ~G= ~10 + ~Ol' where ~G =
total genetic variance, (2) additive and dominance effects and only
2
-2
2
2
2
2
dual-factor epistatic effects, i.e., ~G = ~10 + ~Ol + ~20 + ~ll + ~02'
or (3) additive and dominance effects, dual-factor epistatic effects and
the 3-factor additive x additive x additive epistatic effects, i.e.,
222
2
2
·22
~G =~lO + ~Ol + ~20 + Q: U + ~02 + ~30· Unless our assumed genetic
.
model includes
-2
2
~12' ~03
or 4-factor or higher epistatic parameters,
there are only six independent equations since E(~2) = E(~2) =:I kE(~2° )
...
w
x
w
for (1), (2), and (3) above. Since the expectations of the mean squares
are independent of the effect model, the same
estima~es
will be obtained
using either effect Model A or B•. The results from Model A will be used
to derive the estimators for the six genetic variances
2
2
2
222
2
~G
(110 + ~Ol + ~2'0 + ~ll + ~02 + ~30'
=:I
which assumes no epistatic effects involving more than two loci other
than additive x additive x additive effects.
The seven equations in six
unknowns to be solved are shown in Table 11 by
del~ting
the last three
columns and substituting estimates for the components and for the
genetic parameters.
43
Two estimates of cr:
2
Sw
=
2
are available from
F4
ill
2
s02 and
2
Sx
4
F
2
= 128 s02
Hence, cr~2 may be estimated by using either a weighted or unweighted mean
of the two components.
F
-1
=
Using an unweighted mean,
8/3F
-81.3F
8/3F
0
-32/F2
24/F2
-32/.r2: . 32/F 2
3
l28/F
0
-32/F2
8/F
-8/F
0
0
0
l6/F
0
64/F2
0
0
0
0
0
0
0
... 0
0
0
0
256/3F3,..:· - 256 /3F
2
-1
I1 = FQ or G = F Il'
3
12~/F.4
0..
and the estimation equations are as follows:
-128/F
3
256/3F
3
44
2
s20
s2 + s2
32 [2 2 s2.+ s2 _ ( w
x)]
Ss - u
v
2.
.;:I
256s
[ 2
s 2. . 30 '3F3
u
2
2
Using a weighted mean .of Sw and Sx yields a similar set of
2
2
equations with the
.' weighted mean substitued for (s w + sIX)/2. The variances for the estimates of genetic variances may be found by applying
the rules for variances of linear functions, i.e.,
:
2
V(Eaix i ) = Eai V(x )
1E Eaia j Cov(xix j )
i
i<j
+
For example,
It must be remembered that although the individual mean squares are
uncorrelated, many of the functions of mean squares estimating the components of variance are correlated.
Henc~,
the covariances between
estimates of variance components must be considered in deriving the variances of the estimates of the genetic components of variance.
45
5.12 Discussion of the Analysis of Double-Crosses
The
of variance for double-crosses is parallel in many
4
respects to the analysis of a p factorial experiment. The four factors
ana~ysis
in this case are the four positions a line may take in the pedigree of
a double-cross and the p "levels" are the p parental lines.
This de-
generat.s into a partial factorial because of the restriction that a
particular line can occur only once in a particular double-eross.
There
are only 3 pC 4 combinations instead of p4 as there would be in a complete factorial. The effects of the factors are not distinguishable in
the double-cross analysis so that the one-line general sum of squares is
comparable to the sum of squares due to the pooled main effects in a
factorial.
Similarly" the two-" three- and four;;.line general sums of
squares are comparable to the sums of squares due to the pooled two,:,,,
three- and four-factor interaetion effects.
The order sums of squares
do not have parallels in a factorial analysis.
They are new features
which depend upon the order in which the four lines are combined.
There is lack of complete balance in the double-cross analysis in
that a particular line
gree.
c~not
appear more than once in a particular pedi-
There is,, however, balance in the sense that every line appears an
equal number of ttmes with every other line.
There is also a certain
symmetry about the way the lines are combined in the hybrid.
This
balance and symmetry have simplified the development of a model of linear
effects which is compatible with the genetic expectations of' the covar!ances between hybrid relatives.
Considerable
i~formation
concerning gene action can be obtained from
the analysis of double-cross hybrids under the assumptions as stated.
Direct '-tests are available for different fractions of the genetic
46
variance including several all epistatic fractions.
A qualitative com-
p&rison of the relative importance of the different types of genetic
variance may be obtained from these tests.
F-tests are available for the
detection of dominance plus epistatic variance and for the detection of
all components of epistatic variance except the additive x additive component.A complex function is necessary in order to test all components
of epistatic variance with a single F-test.
A quantitative estimate of
additive genetic variance, dominance variance and four epistatic components of genetic variance can be obtained as shown in Section 5.11 if
all other epistatic components are assumed'to be equal to zero.
bias, if this assumption
~s
The
not valid, is not likely to be large, i f the
fc;~r epistatic parameters estimated. are the three two-factor
components
..
,
.'
,and the
~hree-factor
co~fficients
;.
additive x additive x additive component, since the
of the higher order components of epistatic variance in most
cases decrease in size raptdly as, order increases.
The-relatively small
size of the coefficients of the genetic variances in Table 9 may cause
some difficulty in 'obtaining enough precision to detect two-factor epistasis, even if it is an important part of total genetic variance.
The variances due to the interaction of the various hybrid effects
with replications have been assumed to be equal in this analysis and have
been pooled into a single error sum of squares.
It is possible and may
be desirable, if there are reasons to believe inequality exists, to partition the error sum
o~
squares into interaction sums of squares corre-
sponding to the partitioning of the hybrid sums of
The analysis of variance for double-cross
for r replications in one environment.
sq~ares.
hybr~ds
as presented was
This can readily be extended to
include tests over years and locations with a more complex model.
It
47
would be desirable in the extended analysis to partition the sums of
squares due to the interaction of hybrids with years, locations and years
x locations into the sums of squares due to the interaction of each of
the separate effects defined in the linear model with years, locations
and years x locations.
These
in~eraction
components may be different.
If so, they will have different biological interpretations as they do in
the analysis of single-cross hybrids (Matzinger, et al., 1959).
A serious limitation on the practical usefulness of this analysis
is the rapidity with Which the number of double-crosses increases as p
increases.
The degrees of freedom indicate that p = 8 is a minimum
number of parents which can be used for a complete analysis since there
will not be any degrees of freedom with Which to estimate W* if
p~7.
Thus, 210 entries, p = 8, are required for a complete analysis.
If
p = 9, there are 378 double-crosses and if p = 10, there are 630 doublecrosses.
From a practical standpoint, p= 8 or p = 9 are probably
upper bounds on the number of parents for the complete analysis of
double-crosses.
48
6.0 ANALYSIS OF VARIANCE FOR THREE-WAY HYBRIDS
6.1
Introduction
The term three-way hybrid is used herein to denote the hybrid
progeny resulting from the cross of a line with the F of a cross of
l
two other lines and may be symbolized by A(B x C). Omitting reciprocal
crosses, there are 3 pC
set of p lines.
different three-way hybrids possible from a
3
Anyone of the three parents in a particular combination
of three lines may appear as the single parent in the hybrid so there
are three order hybrids for each combination of lines,
For example, the
three orders or three-way hybrids for the lines A, B, and Care A(B x C),
B(A x C) andC(A xB).
The analysis of
varia~cepresented is
of the complete set of all
three-way hybrids from a set of p random lines.
The p lines are, as
in the double-cross analysis, assumed to be a random sample of lines
derived from a random mating population.
It i. assumed that all lines
have an equal but arbitrary degree of inbreeding.
6.2 Expectation of Quadratic Forms in Terms of Covariances
The phenotype of the three-way hybrid A(B x C) grown in themth
replication in one environment will be denoted in the 'following derivations by YA(BC)m or, in general, Yi(jk)m.
parenth~s denote
The subscripts within the
the lines which form the single-cross parent and the
single subscript before the parentheses denotes the single parent in
the three-way cross.
The model for the
analysi~
of variance is
(6.2.1)
49
where
fA = a contribution common to all entries"
r
m
=a
ei(jk)m = a
Gi(jk)
and
=a
contribution particular to repfication m"
contribution particular to three-way
~ross
i(j x k),
random deviation of the observation on the three-way
cross i(j x k) grown in replication m.
It is assumed that the ei(jk)m's" rm's and Gi(jk)'s are independent
~
2 respecrandom var i a b1 es wi t·h zero means and var.Lances
t1-2 , 0'2 an d 0'"
e r
y
tively" and that the e's" res and G's are not correlated with each
other.
A constant number of individuals in each hybrid progeny in each
replication is assumed.
The expectations of the quadratic forms which will be encountered
2 O'r2 an d the covariancesetween
b
can be defined in terms .ofp2 , . O'e"
the
effects of the three-way hybrids" i.e." the Gi(jk) IS.
The expectation
of the square of the phenotype of a hybrid grown in replication m will
contain
222
~
+ O'r + O'e plus the covariance of the effect of the three-way
hybrid with itself.
The expectation of the product of the phenotypes of
two different hybrids grown in the same replication will 'contain
,.i + C1'~
plus the covariance of the effects of the two three-way hybrids.
The
expectation of the product of the phenotypes of the same or different
hybrids grown in di,fferent replicatioos will contain t-"2 plus the covariance of the effects of the two hybrids involved.
The covariances between the effects of three-way crosses are defined
as listed in Table 13.
Cov
l
It will be noted that:
= the covariance of the effect of a three-way cross with
itself,
SFrom Cockerham, C. C. (1957) (see footnote 2, Section 5.3).
Cov • the covariance between the effects of two three-way hybrids
2
with all three lines common but in different orders,
Cov • the covariance between the effects of two three-way
3
hybrids with two lines common and with the same line
as the single parent in both hybrids,
Cov4 • the covariance between the effects of two three-way
hybrids with two lines common and with a different line
as the single
p~rent
in the two hybrids,
51
cov
5
• the covariance between the effects of two three-way
hybrids with two lines common and with one of the lines
the single parent in only one of the hybrids,
Cov
6
= the
covariance between the effects of two three-way
hybrids with two lines common and with both the lines
as a
Cov
7
= the
si~gle-cross
in both hybrids,
covariance between the effects of two three-way
hybrids with one line common as the single parent
in both hybrids,
Cov
a = the
covariance between the effects of two three-way
hybrids with one line common but with the common line
as the single parent in only one hybrid and
Cov
9
= the
covariance between the effects of two three-way
hybrids with one line common in the single-cross
parent in both hybrids.
The covariance between the effects of two three-way hybrids with no lines
common is zero.
6.3
Genetic Expectations of Covariances
In order to express the covariances between three-way hybrid relatives in terms of the genetic parameters as defined by Cockerham (1954),
or Kempthorne (1954), it is necessary to impose the same assumptions as
given in Section 5.3 for the covariances of related double-cross hybrids.
With these assumptions, the covariances between the effects of relatives,
if the environments are not correlated, can be expressed by equation
5.3.1.
The correlations between the additive and dominance deviations,
_( . 0'..:.
52
p and q, respectively, in equation 5.3.1, were derived by Cockerham
(see footnote 2, Section 5.3) for various hybrid relatives and are presented in Table 13 for three-way
hybrid:~relatives.
6.4 Partitioning of Sums of Squares
The notation for the observations and various sUms which will be
encountered
in partitioning
~i(jk)m
= the
the sums of squares are defined as follows:
phenotypic observation on three-way cross
i(j x k) grown in replication m. ;m
i, j,
k
= 1,2, ••• ,
r
Yi(jk)~
= m:l
Yi(jk)m
= sum
p; i
= l,2, ••• ,r;
+ j +k.
over replicatipns of the three-way
hybrid i(j x k).
3
= E Yi(jk).
o
= sum
of the three orders of hybrids involving
the combination of lines i, j, and k.
P2
Y
i (j •) •
= k+i,
E
j
Yi(jk).
= sum
of all hybrids in which line i appears
as the single parent and line j appears in the singlecross parent.
Y• (ij).
=
P2
E
Yk(ij). = sum,. of all hybrids in which lines i and
k+i,j
j appear in ~he single-cross parent.
P2
Y
ij ••
= Yi(j)
.••
+ Yj(i)
(ij)
. • • + Y•
•
hybrids in Which lines i and
=k+i,j
E
j
Yijk •• sum of all
appear in any order.
53
Y
1
i( ••).
=-
C
Pl 2
E Yi(jk)
j,k+i
•
Pl
E
~
Y
2 j+i ,i(j.).
= sum
of all hybrids in
j<k
which line i appears as the single parent.
G
PI
Y
• (i.).
=
E
j+i
Pl,2
Y
• (ij).
=
2
E Yj(ik).
E
j,k+i o
= sum of
all hybrids
j<k
in which line i appears in the single-cross parent.
Pl
Y
i •••
=¥
i( •• ).
= sum of
Y
....
+Y
.(i.).
=1
E
2 j+i
Y
ij ••
all hybrids in which line i appears.
1 p
PC3
=-3 Ei Yi... = i<j<k
E
Y
ijk •
= g~and
total of all
observations.
Other pertinent facts regarding these sums are given in Table 14.
One possible partitioning of the sums of squares from all possible
three-way hybrids from a set of p parents grown in r replications in
one environment is shown in Table 15.
The sums of squares for repli-
cations, hybrids, and error are the ordinary least squares partitions
for these effects using Model (6.2.1) and are orthogonal.
The parti-
tioning of the hybrid sum of squares is the result of largely trial and
error procedures.
The partitions are, however, an orthogonal set.
The
proof of orthogonality is long and involved and is not presented in
this thesis.
This system of partitioning appears logical and closely parallels
the double-cross partitioning with two exceptions.
I
First, it is possible
e
54
Table 14.
Description of sums pertinent to the analysis of three-way
hybrids
Number of
such sums
Yi(jk)m
1
rpPlP2
2
Yi(jk):
r
PP l P2
2
Yo •
Yijk •
3r
PP l P2
Y
Yi(j.).
rP 2
PP l
2Y
Y.(ij).
rP 2
PP l
2
Y
Yij ••
6
~~l
y
•
0
•
ill _
·...
•
0
••
·...
3Y
Yi ( •• ).
rP l P2
2
P
Y
Y• (1.).
rP l P2
P
2Y
Yi ...
3rP l P2
2
P
3Y
....
rpP 1P2
2
1
Y
3rP 2
•
• •••
2
Y
..
Total of all
such sums
equals
Number of
observations/sum
Sum
·...
,
·...
• •••
• •••
in the three-way partitioning to identify two different types of oneline sums, i.e., Yi ( •• ). and Y.(i.).' giving rise to one-line general
and one-line order sums of squares. This was not possible in the
double-cross analysis.
Secondly, in the three-way analysis the two-line
order 'sums Yi(j.).and Yj(i.). are distinct sums and give rise to an
Table 15.
e
e
e
Analysis of variance of three-way hybrids
Source
Correction factor
Replications
Hybrids
Sums of squares
d.f.
1
(r-l)
(3 pC3 - 1)
2y2
•• ••
C - rpP P
l 2
PI
2-line general
PP 3 /2
R*
- C
A*
~
y2
3P l
i i •••
G ==
- -C
31'P 2P3
P3
C
P 2 2
i~j Yij.. 3P 2 . 2P3
S=
.
--C--G
3rP4
P4
P4
2
I-line general
C
r
2 E y2
. mal •••m.
R=
- C
PP 1P2
C .
P 3 3 2
E
: Yi(jk?
A == i<.1<k
r
M.S.
G*
S*
C
p3
2
Yi ok
i<j<k J .
U ==
- C - G- S
3r
1::
3-line
genera~
PP l P5 /6
u*
IJ1
IJ1
e
e
e
Table 15(cont1nued)
d.f.
Source
SU$S of squares
,P
2
I-line order
PI
2
~ 1'1(.~).
H=
+~1'2
P
.(f~)
i
rpP2
2-line order
/2
1' (j.).
+
1'i(j.).
VS.
1
p 2
I: [1'
+2
T • i<3 iO.).
1'j(i.)]
VS.
, ,
Y.
j(J..) •
..,'
Yj(:t;:):
~
2rPl
PP3
D•
Error
PP 2P4 /3
pc~
3
I: 1'2
V. i<Jsko
I:
i(jk).
i<j4
M-
p 3
I:
f<:j<k
3
I:
l
3rP
l
...e... H
- 2P
T*
l
2
p 2
I: [1'1 (j.). + 1'j(i.).]
3P
2 H
1<J
_ --
"2rp3
1'2
1jk.
3r
r
- H- T - D
2P3
D*
v*
E*
P 3
0 -1)
p, 3
rP
2
,I: 1'1j ••
i<1
0
2
(r-l) (3 0 -1) E - M - R - A
(3r
i<1
3
0
Total
P 2
o
P 2
I: 1'.(ij)
+
H*
3rpP 2
P 2 2
p 2 2 + I: 1'j(i.).
I: 1'iO .). _ i<1
",
i<j
rp
!O '
P 3
I:
3-line order
o
o,
•• P I P 212
'.
••.
'.
o
0
l' • (1j).
2
2 I: 1'i
i
M.S.
r
I:
om-I
2
I:
i(jk)m - 0
VI.
0\
57
additional source of variance, the
these two sums.
varian~e
of the difference between
In the double-cross partitioning, Y(i.)(i.). == Y(j.)(i.).
so that this source of variance does not exist.
6.5 Expectations of Sums of Squares in Terms of Covariances
.The expectations of the sums of squares in Table 15 can be derived
222
in terms of I·
u , CTr , CTe and
. the covariances between the effects of relatives as defined in Section 6.2 in the same manner as illustrated in
the analysis of double-crosses.
The expectations for the uncorrected
sums of squares are presented in Table 16.
The expectations of sums, of squares as partitioned in Section 6.4
may be derived from the results presented in Table 16 by applying the
rules for the expectation of linear functions.
If we let ! beCthe
(10 x 1) column vector of uncorrected sums of squares in Table 16, £
222
be the (11 x 1) column vector of (ru
+ CTr , CTe and the nine covariI·
ances, and M be the (10 x 11) matrix of coefficients in Table 16, then
E(~ ==
MQ.
Letting S be the (7 x 1) column vector of partitioned hybrid
-p
sums of squares, i.e.,
~
== (G, S, U, H, T, D, V), and N be the trans-
formation matrix such that -p
S == N§., then E(S
) == NE(!> == NM£. The
-p
2
coefficient of CT in the expectation of each of the sums of squares is
e
the degrees of freedom for the sum of squares. The expectations of the
mean squares are then found by dividing by degrees of freedom.
The ex-
pectations of the mean squares are presented in Table 17 in terms,. . of····.···2
CT
e
and seven functions of the covariances •
. 6.6 Model of Linear Effects
It is important to recognize that the development of a model of
linear effects is not essential for the estimation of genetic parameters.
Table 16.
sums. of
: Squares
2y2
•• ••
rpP 1P2
e
e
e
Expeet4tions of uneorreetedsums of squares in terms of the eovarianc;:es between three-way
hybrid relatives
2:
2
(t;p +'O"i)
O"e
PP 1P2
1
2
2
Coeffieients-of
eoval:':tanees ._...... ..._..'
Cov
Cov
Cov4
:Co~5
Cov6
cov
7
2
3
'--~~~-
Cov
l
r
\2r
_~
2rP 3
2rP3
4rP 3
rP 3
4q>P
-3 3
4rpP
SrpP3
-3 3
rP 3P 4
2
:Cov
Covg
2rP 3P4
2rP 3P4
s
P 2
21: Yi •••
i
3PP 1P2
2
P
3PP 1P2
PP 1
2
-2
P 3 2
1: Yijk.
i<i<k
3r
PP 1P2
PP 1'2
2
6
P 2
21: Yi ( .• ).
i
rp" P'-2
l
PP 1P2
3rP 1P2
C
P 2 y2
1:
ij ••
i<i
3rP2
rp2rp
~P1
2
rpP
1
3
-:--...3
3
2rpP
rpP 1P3 rpP 1P3 2rpP 1P3 r pP1P3
_. ""3' - 3
3
6
~PP3P4
2rpP 3P4 2rpP 3P4
633
o
o
o
C
--C-
rPP 1P2 rpP 1P2
6
3
o
o
o
o
o
o
o
o
rp P P
3 4
o
o
o
rpP 3P4
2
P
rp
o
2rpP 3
o
o
PP 1P2
P
rp
rp
,rpP 3
o
2rpP 3
2'
P 2
1:Y'.'(i.l.'
i
rPiP2
rpP3
o
VI
ClO
e
e
e
Table 16(continued)
2
Sums of squares
C
p2
£
1<j
rP 2
+ CTr )
2
CT
COV
PP I P2
PP l
rpP
2
2
2
PP I P2
PP I
rpP
rpP
PP l P2
PP l
"2
(rp
2
Y.(ij).
····2
e
I
l
Coeffic1ents of covar1ances
Cov
Cov3
Cov4 CovS
2
o
o
Cov6
rpP 1P3
2
Cov
Cova
Covg
o
o
o
7
o
o
o
o
o
o
o
o
C
P 2
2
£ [Y
+ y2
I
1<1. 1(j.). . j(1.).
rP 2
C
p 2
£ [Y
+Y
1<j . 10.) ...
2rP2
C
p 3
£
1<j<k
3
£ y
0
r
j(1.).
]2
I
l
o
rpP l
---r ---r
rpP 1P3
r pPI P3 rpP I P3
2
2
o
o
o
o
o
o
o
o
o
o
o
o
2
1(jk).
PP I P2
2
PP l P2 rpP 1P2
2
2
o
VI
\0
e
e
~
e
,
Table 17.
M.S.
Expectations of mean squares in terms of the covariances between hybrid relatives for the
analysis of three~way hybrids.
Coefficients of covariances
2· ,
.
2Cov + 2Cov
.. . Cov + Cov
Cov + 4Cov Cov ~ 2Cov
a 7
3
4 Cov ;- Cov4
O"e Cov + 2Cov Cov ~ Cov
7
a
3
4
1
2 + 4Cov + Cov
3
l
2
~ 4Cov + 2Cov
+
Cov
+
4Cov
6
6
S
9
9
S
G*
1
r
0
r(2p-9)/3
S*
1
r
0
U*
1
r
0
H*
1
0
r
0
3rp /2
3
T*
1
0
r
0
D*
1
0
r
v*
1
0
E*
1
0
rP 7 /3
0
0
rP4P9/6
0
0
~r (2p-ll)
0
0
0
13
0
r
0
rp /6
3
0
rP3P413
-3r/2
r(2p~S)/6
0
~rP4/3
0
r(2p-9) 12
~r/2
0
-rp
r
0
~3r/2
~r/2
0
r
0
0
0
0
0
~r
0
4
0'\
o
61
The results of the previous five sections can be used directly to estimate the components of genetic -variance.
A model of linear effects is
necessary, however, in order to develop tests of significance of vario,us
genetic hypotheses. - As in the double-cross analysis" the model of
linear effects which is presented is only one of the possible models
which could be used.
The model as presented closely parallels the model
of linear- effects for the double-crosses
The effect gi as defined
here~
(5.6~2)
with few exceptions.
has approximately the same'genetic inter-
pretation as the effect gi for the double-cross model but does not have
the same statistical definition.
Additional effects, hi and di(j)'
have been added to the-model of linear effects for three-way crosses to
account for the extra sources of variation described in Section 6.4.
The following model of linear effects appears reasonable for the
analysis of three-way hybrids:
(6.6.1)
Yi(jk)m = ~+ r m + Gi(jk) + ei(jk)m
=
fA + r m + 2g i + gj +
~
+ hi + Sij + sik + s jk
+ t ij + t ik + di(j) + diCk) + u ijk + vi(jk)
(6.6.2)
+ ei(jk)m
where Yi(jk)m' f,r m, Gi(jk) and ei(jk)m,~re defined in Section 6.2
and where
gi = a basic contribution particular to line i appearing
as anyone of
gi + hi • an additional
theparentalline'~,
con~ribution
particular tQ line i appearing
as the single parent,
Sij = a contribution particular to linesi and j appearing
togethe~in
the three-way hybrid,
62
t ij - an additional
cont~ibution particular
to line i and
line j appearing together with one or the other as
the single parent,
di(j) -an additional contribution particular to line i and
line j appearing together with line i as tthe single
parent,
and
vi(jk) - an additional contribution particular to line's i, J,
and k appearing in the order i(j x k).
It is assumed that ~he ei(jk)m's are uncorrelated with mean zero
and variance u2 and that there {is, a constant number of individuals in
e
each plot.
The terms gi' hi' Sij' t ij , and u ijk are assumed to be random
2 2 2 2
2
variables with zero means and variances O'g' Uh ' Us' Ut and uu ' respectively. The use of gi + hi to represent the additional effect due to
line i being the single pa2:ent is synonymous with defining an effect hi* to
represent this contribution where hi* -gi + hi. ,However, hi* and gi are
2
* gi) - Cov (gi + hi' gi) - ug and
then correlated with Cov (hi'
2
2
Var(h*
i ) - Var (gi + hi) - ug + Uh • Hence, the use of gi + hi gives
effects which are uncorrelated.
The terms di(j) and vi(jk) may be "
treated either as completely random variables with means zero and vari2
ances u and u2 , respectively, or with the restrictions that
d
v
di(j) • - dj(i) for all ij combinations and
vi(jk) + vj(ik) + vk(ij) • o for all ijk combinations, respectively.
63
It was illustrated in the double-cross analysis that the choice of the
model,.inso far as random or fixed effects were concerned, had no affect
on the information obtained fro,m the analysis concerning genetic parameters.
Thus, for the analysis of three-way hybrids, only one model will
be used; that model assuming all effects to be completely random and
independent variables.
6.7
Covariances as Functions of the Components of Variance
The covariances as defined in Section 6.2 can be expressed in terms
of the effect components of variance (Section 6.6) by taking the expectations of the respective quadratic. forms in terms of the model of
linear effects.
These results are presented in Table lS.
Table lS.
Covariances between three-way hybrid relatives
expressed as functions of the components of variance
Coefficients of components of variance
:).2
Covariance
2
2
2
2
(J'2
(J'2
0'0'
0"
O"g
O"h
O"d
;::t
u
v
S
Cov
1
Cov
2
Cov
3
Cov
4
Cov
S
Cov
6
Cov
7
Cov
S
Cov
9
6
1
3
::2
2
1
1
5
0
3
'.1
0
1
0
5
1
1
1
1
0
0
4
0
1
1
0
0
0
3
·0
1
0
0
0
0
2
0
1
0
0
0
0
4
1
0
0
0
0
0
2
0
0
0
0
0
0
1
0
0
0
0
0
0
64
It would be desirable for the linear effects model to be compatible
with the genetic expectations of the covariances as defined in Section
i··"
However, by the genetic expectations, Cov 6 = Cov and Cov 2 = Cov •
S
3
These equalities are not satisfied by the model of linear effects. No
6.2.
way was found to reconcile these differences with a linear model.
It
appears that a non-linear model would be necessary to completely describe
the situation in the analysis of three-way hybrids.
6.S
Expectations of Mean Squares in Terms of Components
of Variance and Estimation of Components of Variance
The expectations of the mean squares in terms of the effect components of variance are presented in Table 19.
The detailed derivations
of the expectations of the mean squares are not presented here.
They
may be derived either by the usual method of taking expectations of
quadratic forms or by substituting the results of Table lS in Table 17.
In most of the following derivations, including the results presented in
·22
Table 19, the components of variance as and ah are grouped so as to
present a variance due to average one-line effect, 0'g2 , and a variance
222
due to one-line order effect, O'h* = O'h + O'S. This makes the derivations
s~pler,
since these variances coincide with particular sources of vari-
ation in the analysis Qf variance.
The estimators for the effect components of variance can be determined by inspection from Table 19.
mate of
2 denoted by s 2 :
0'
s2 = (V* - E*)/r
v
2·
s u = (U* - V*)/3r
They are as follows with the esti-
65
2
2P4
Ss = [S* - u* - ---(T* - V*)]/3rp
Pl
4
· pp~
3P 2
2
2
2
= s + s =3(H* - -2- T* - --D* +---& V*)/rpp and
h* h
g
2Pl
2P3
P1P3
2 .
2
P
2P 3
P3.
3P2P~
P2 . .
s = 2(G* .- - 3 H* - --S* + --- T* + --- D* + ..A U* - -::=-V*)/lSrp
p
g
2p
p4
4P 1
4p
P4
2Pl
. 2 3
s
Expectations of mean squares for the analysis of three-way
hybrids
Coefficients of components of variance
2
2
2 0'2
2
2
2
2
2
O'h*=
0'
O'
O'
O'
O'
O'
O"g
e v
d
t
s
r
u
2· 2
O' + 0'
g
h
Table 19.
M.S.
R*
1
0
0
0
G*
1
r
4rP3/3
8rp3 13
S*
1
r
2rP4/3
4rP4/3
0
3r
3rp4
0
0
U*
1
r
0
0
0
3r
0
0
0
H*
1
r
rp/3
rpP /3
2
0
0
0
0
T*
1
r
2rPl/3
0
0
0
0
0
D*
1
r
rp 13
1
rP 3
0
0
0
0
0
0
V*
1
r
0
0
0
0
0
0
0
E*
1
0
0
0
0
0
0
0
0
r(5p-9)/3
0
0
r P2P3/ 6 3r
0
0
6rP 3 5rP2P3/2
3 pC 3
0
66
In lieu of estimating a g2, which has an extraordinarily complex estimator,
222
2
we may estimate ah* + l5ag • ah. + l6ag as follows:
2
2
2P3
P2
s + 16s = 6(G* - S* + - u*) Irp p
g
g
,
P4
P4
2 3
This is a much simpler esttm&tor and, consequently, has a smaller
variance.
The
vari~ces
of these estimators can be approximated as shown in
I
Section 5.8 in the double-cross analysis.
6.9 Components of Variance as Functions
of Covariances between Hybrid Relatives
The components of variance can be expressed as functions of the
covariances between hybrid relatives by use of the expectations in
Table 17 and Table 19.
In matrix notation, E(S-e ) •
AL
where
st •
-e
(G* - E*, S* - E*, U* - E*, H* - E*, T* - E*, D* - E*,
V* - E*),
A • the (7 x 7) matrix of coefficients of the components of
variance for the expectations of the mean squares in Table 19
omitting mean squares R* and E* and the columns for a~ and
ar2 and
I
t
2 at'
2 a 2*1 au"
2 as'
2 a 2) • Al so.. .\IID(S)
BC.~
• (2
av ' ad'
-e • .:.:f wnere
h
g
B • the (7 x 7) matrix of coefficients of the functions of
covariances in the expectations of the mean squares in
Table 17 omitting mean square E* and the column for a~ and
67
£f = (Cov1 + 2Cov2,
Cov - Cov , 2Cov3 + 2Cov4 + 4Cov + cov 6,
1
2
S
Cov - Cov , Cov + Cov - 4Cov +2Cov 6, cov + 4Cov
3
4
S
7
3
S
4
:+ 4Cov9 , Cov7 - 2CovS + COV9 )·
Then Al. •
-1
or ! = A
B~
-1
B~.
!
",1
Note that! = A E(!e) orA
is the transformation matrix giving the
estimators of the components of variance.
Hence, A-1 may be written
from the estimation equations in Section 6.S.
A-1
=1r
Thus,
0
0
0
o
o
1
0
0
:;0::
o
o
-lIp 3
0
0
0
o
3/2P1
-1/ 2P 3 ·~R.4J~i~3
0\
0
0
0
0
1/3
o
1/3P4
2/ SP2P3 -4/SP 2P4
-1/3P4
-3/2P1P2 -9/2PP3
o
0
::2t~~3P4 "'!.~t~'P~2
-2/3P 1
3/P 1P3
0
-1/3
0
2/3P 1
1/1Op1P2 3/1OpP3 -1/5P 1P3
A rearrangement of the product matrix (A-1 B) is given in Table 20.
6.10 Genetic Interpretations of the Components
of Variance and Tests of Significance
The covariances between three-way hybrid relatives can be expressed
in terms of genetic components of variance from Table 13 by expression
(S.3.1).
The genetic interpretations can then bep1aced on the compo-
nents of variance from the relationships shown in Table 20.
These are
presented in Table 21 up to and including the three-factor epistatic
components of variance.
The interpretations for various tests of significance are apparent
from Table 21.
2
The component a is; a function of additive x dominance
v
epistatic yariance and all higher order epistatic components except the
68
Components of variance as functions of the covariances
between three-way hybrid relatives
Coefficients of covariances
Component of
Cov
Cov Cov
O
Cov Cov
Cov
variance
2 CO"3
4
S C V6
8
l
7
Table 20.
(J'2
g
2
O'h*
Cov
9
0
0
0
0
0
0
0
2/5
1/5
Q
q
q
0..
.~
Q
1
-2
1
2
(J's
0
0
0
0
4/3
-1/3
0
-4/3
-2/3
2
(J't
0
0
0
1
-2
1
0
0
0
2
(J'd
0
0
1
-1
0
0
-1
2
-1
0
1
0
-1
-2
0
0
2
1
1
-1
-2
1
2
-1
1
-2
1
(J'2
u
(J'2
v
all-additive types.
Hence, the test of the hypothesis that (J'~
==
0 is a
test that this particular combination of epistatic components of variance
2
2
is. equal to zero. The tests of the hypotheses (J'u .. 0 and (J'd .. 0 are
tests that the additive x dominance and all higher epistatic components
of variance except the all-dominance types are equal to zero.
The test
2
u
of (J' • 0 gives relatively more weight to the three-factor all-additive
2
epistatic component than the test of (J'd .. O.
The test of the hypothesis ~. 0 is a test of all components of
r
variance except additive genetic variance.
The test of
c,2s
== 0 is a
test of the all-additive types epistatic variances.
.
2
The tests of the hypotheses that (J'h*
.
II
2
0 and C1'g .. 0 are tests of
additive genetic variance and the all-additive epistatic components of
variance with the latter giving considerably more weight, relatively,
to the additive component of genetic variance.
e
e
""
-
.
-
Components of variance in the analysis of three-way cros$es as functions of the components
of genetic variance up to and including the three-factor epistatic components
~--"" "-.Coefflcientsof components of genetic variance
Component
22222
2
222
°'io
0"01
0"20
0"11
0"02
0"30
0"21
0"12
0"03
Table 21.
2
'/8
0
9,2/ 320
0
0
17,3/ 2560
0
0
0
O"h*
'/8
0
9,2/64
0
0
49,3/ 512
0
0
0
O"s
2'
0
0
7'~ /96
0
0
3
llF /256
0
0
0
crt
2
0
2
O"d
0
ff
2
0"2
O"g
2
..
,2/32
,3/8
0
0
0
0
0
0
,2/4
,4/16
9,3/ 256
,4/ 16
,5/32
,6/64
,3/32
0
3,3/128
9,4/ 256
,5/ 128
0
·0
,3/32
0
5,3/ 64
9,4/ 256
,5/ 128
0
0
,3/32
0
13,4/ 256
13,5/128
"
u
v
,4/8
3,6/ 32
0\
\0
70
Hence, considerable information can be obtained concerning gene
action from the tests of the hypotheses that the various components of
variance equal zero.
It is evident from Table 19 that simple F-tests
22.
.
2
are available only for the hypotheses that ~ = 0, ~ = 0 and ad
O.
v
u
The remaining tests make use of Satterthwaite's approximation as ex-
=
plained in Section 5.10.
The exact and approximate F-tests for the
various hypotheses are give.n in Table 22.
tests
F'
and
F"
The difference between the
is explained in Section S.lQ.
F is
used to indicate
an F-test in order to avoid confusion with the F used to indicate the
degree of inbreeding.
Another system of F-tests is available in which all the tests are
exact tests but which will not generally yield as much information concerning gene action as the tests indicated in Table 22.
As in the
double-cross analysis, the error mean square is a valid error term for
testing all of the hybrid mean squares.
The hypotheses being tested,
however, are the composite hypotheses that all genetic components of
variance appearing in the respective mean squares are equal to zero.
As a result, the interpretations of the tests will be considerably
different from those in Table 22.
The test
2 .
F =,*/E*
is the same test
.
as for the hypothesis ~v • Q. F. U*/E* and F = D*/E* are, however,
tests of all epistatic components of variance except the two-factor
If
additive x additive component.
F.
T*/E* is a test of aU components
of genetic variance except additive genetic variance.
F = S*/E*
is a
test of the same hypothesis with relatively ~9re weight given to the
aU additive types of epistasis.
o~;all
F = H*/E* and F = G*/E* are both tests
components of genetic variance with the latter giving relatively
more weight., to the additive genetic variance component.
It is apparent
e
Table 22.
e
e
...
F-tests of hypotheses that various components of variance are zero in the analysis of
three-way crosses
Degrees of freedoma
F-test .
Component
a:v
2
F = V*/E*
PP 2P4 /3 1 (r-1)(3 pC3 - 1)
eT
2
u
F == U*/V*
PP 1P5 /6 1 PP 2P4 /3
2
eTd
i · D*/V*
P1P 2/2 1 PP 2P413
2
eTt
F'. ii,..
P
.
.
T*
.2p
3P3
2
eTs
PP3/2If'
1
-D*+--!V*
F •
3P3
s*
2P4'~
U* + -
P1
('1'* -
PP 3 /2 1 f'
V*)
2p
F "=
s* +~V*
....
P1
-2-"::'
-::...-0..
U*+ P4 T*
Pl
fll 1 fll
1
2
,..,.
....
e
e
e
..
,
Table 22(continued)
222
O"h* = O"h + O"g
a
Degrees of freedom
F-test
COII1ponent
-1
F
=
",..-.;H:...*
_
3P 2 PP2
..l!.'l* +-D*
--V*
2p
2P 3
P1P3
PI' fl
PP
H*+~V*
i" - . - -
fIt
l'
P 1P 3
fIt
2
-.
3p
~1:* + --! D*
2P 1 .
2P 3
2
F1 =
O"g
-.".
..;;;;G-*
2P 3
P2
P3
. P3 . 3P 2
P2
--U'*+-V*
--H*+-S*--'l*--D*
2p
P4
4P 1
4p
P4
2Pl
_
F II =
p
3p
P
G* +--11:* +--!D* +...lU*
4p
4p
P
P
.1
.
-
2p
-
P
4
..2 H* + -..2 S* + --! V*
2p
a
P4
-
PI' fl
fIt fll
l' 2
2P 1
fl and fIt indicate the approxtmate degrees of freedom as suggested by Satterthwaite (1946) (see
equation,S. 10. i)
.....
N
73
that this system will not generally yield as much information.
For
example, the ability to test only the all-additive types of epistasis
and the ability to test additive genetic variance plus only the alladditive types epistasis are lost.
The most powerful test of dominance and epistasis is
F ==
with (p + l)(p
2
2 (8
+. T + D + U + V) /(p +
lHp2 - 4p
+
2)
E*
- 4p + 2)/2 and (r - 1)( 3 pC
3
- 1) degrees of freedom.
The simplest most powerful test for epistasis is
- _ 2(D + U + V)/(p 3 - 5p 2.+ 4p + 2)
F E*
with (p3 - 5pZ + 4p + 2)/2 and (r-l)(3 pC
- 1) degrees of freedom.
3
This,test, however, does not include the two-factor additive x additive
epistatic deviations.
In order to include these deviations in a single
aU-epistatic test, the test must be complex and an approximation.
6.11 Estimation of Genetic Parameters
There are two approaches available for estimation of the genetic
components of variance.
One approach is to fit by least squares proce-
dures progressively more complex models to the estimated components of
variance until no significant deviation from the model is obtained.
The
,components of genetic variance are then estimated as the regression
coefficients in the particular models.
Anoth~r
method of estimating the genetic parameters is to equate
the estimates of the components of variance to their expec:t,ations in
terms of the genetic par.ameters and
solVe~
since the number of genetic parameters
n(n
+ 3)/2,
the set of' equations.
invol~ed
However,
in the expectations is
where n is the. number of loci, a restricted genetic model
74
must be assumed in order to reduc$ the number of parameters to an
estimable number.
In the three-way analysis there are
s~ven
equations
but a dependency exists among them such that. there are only six independent equations.
An
i ndependent set
Hence, only six genetic parameters may be estimated.
0
-"" b"nau:.
,. A·d.l-eng.' silHan.
2.. d:'~~u.
2
f s i x equat i ons can -be f orme~y
This set will allow estimation of the parameters for the model
2
2
2
2
222
O'G == 0'10 + 0'01 + CJ'20 + CJ'11 + CJ'oa' + CJ'30'
2
where CJ'G is total genetic variance, which assumes no epistatic effects
involving more than two loci other than additive x additive x':additive
-1
effects. For this genetic model, 1 == FQ or Q == F L where
2
2 2
2 2 2
2 2
(sh* + 15S g, sh*' ss' St' sd + su' s v ),
2
2
2
2
2
2
Q' == (s10' sOl).
s2()' su' s02' s30)'
l'
==
arid
2F
F/8
0
F ==
from Table 21.
O~
0
2
9F /16
2
9F /64
2
7F /96
p2/ 32
0
0
0
0
0
p3/8
0
4
F /16
0
0
0
2
F /4
0
0
0
::F 3 /16
0
0
0
p3/32
4
F /8
3
25F /128
3
49F /512
3
llF//256
3
9F /256
3
13F /128
0
75
Then
0.30/F
-1.89/F
F
-1
2
1. 73/F2
=
4.76/F
-1.19/F
.-2.93/F
3
4
3
3.19/F
-8. 47;/F
30.17/F2
-45.35/F
-27.61/F
-76.13/F
19.03/F
2
3
4
46.85/F3
53.65/F
2
2
110. 12/F3
-27.53/F
4
-67.76/F3
and the estimation equations are as follows:
5.11.
0
4/F
o
0
2
-7/F2
-2/F
o
0
0
0
16/F
0
_4/F 4
0
0
3
o
76
6.12 Discussion of the Analysis of Three-Way Crosses
The analysis of variance for three-way hybrids may be compared to
the analysis of a p
3
factorial experiment.
The three factors are the
three positions a line may take in the pedigree of the hybrid and the
p levels are the parental lines.
This is not a complete factorial
because of the restriction that a particular line cannot appear more
There are only 3 pC combinations
3
3
instead of p as there would be in a complete factorial. The effects
than once in a particular hybrid.
of the factors, i.e., the positions of the lines in the pedigrees 3 are
not distinguishable so that the one-line general sum of squares is
comparable to the sum of squares due to the pooled main effects of the
factors in a factorial.
Similarly, the two- and three-line general sums
"
of squares are comparable to the sums of squares due to the pooled twoand three-factor interaction effects,
re~pectively.
The order sums of
There is lack of complete balance in the three-way analysis, since
a particular line may not appear in a pedigree with itself.
There is,
however, balance in the sense that each line appears an equal number of
times with every other line.
Because of the asymmetry of the three-way
hybrids no linear model has been found as yet which is compatible with
t
the covariances.
The inability to reconcile the model and the covari-
ances is apparently due to the inability to find a linear function the
square of which will reconcile itself with all possible quadratic forms.
Qualitative estimation of the importance of various functions of
the components of genetic variance, including several all-epistatic
fU,nctions,. is available from the F-tests in the three-way analysis of
77
variance.
Tests for deviations from an additive model and from an
additive plus dominance plus additive x additive model are available.
No simple test for deviations from an additive plus dominance model was
found.
Quantitative estimates of additive genetic variance, dominance
variance, the two-factor epistatic variances and the three-factor alladditive epistatic variance component are obtainable if all other components of epistatic variance are assumed equal to zero.
There will be
some bias in the estimates if this assumption is not valid.
This bias
may under some conditions be quite large, if epistatic variance is important, since some of the coefficients of the higher epistatic
compo~
nents in the genetic expectations of the effect components of variance
are of about the same magnitude as the coefficients of the estimated
components.
The variances due to the interaction of the various hybrid effects
with replications have been assumed to be equal in the three-way analysis
of variance and have been pooled into a single error sum of squares.
It
may be desirable under some conditions to partition the error sum of
squares into the individual
inter~ction
sums of squares.
The analysis as presented i. for r replications grown in only one
environment.
This can be extended to include tests over years and
locations by incorporating these effects in the model of linear effects.
.
-
It would be desirable in the extended analysis to partition the hybrids x
environments sum of squares into the sums of squares due to the interactions of each of the hybrid effects defined in the linear model with
environments.
The degrees of freedom in the analysis of variance indicate that
the complete analysis can be computed only if
p~6.
For p • 5, there are
78
no degrees of freedom available for estimation of the three-line general
mean square.
Hence, the complete three-way hybrid analysis can be com-
puted with only sixty hybrids.
This size analysis, however, will not
have enough degrees of freedom to give good estimates of the parameters.
For example, the largest number of degrees of freedom among the hybrid
mean squares will be ten.
Considering the size of the variance generally
associated with estimates of components of variance, it would be desirable to use some larger number of lines if the purpose of the study is
primarily that of estimation of the components of genetic variance •
•
J.
79
7.0 General Discussion
The primary purpose of developing the analyses of variance for
double-crosses and three-way crosses was to determine what genetic
information could be obtained from some of the designs commonly used in
practical breeding programs.
of corn breeding programs.
routine evaluation
progr~
Evaluation of hybrids is a routine part
It is not likely that the' hyqrids in any
would satisfy all ofth. assumptions necessary ,
for the genetic analysis, partic\Jlarly the assumption regarding the
source of the lines.
The analysesphave shQWD., however, that it is
"
possible to obtaiti'considerable genetic information from double-crosses
and three-way crosses and possibly, with some preliminary planning, as a
by-product of a practical breeding program.
Both of the analyses presented allow estimation of six components
of genetic variance.
One of the limitations to the use of double-cross
and three-way cross analyses for estimation is the rapidity with which
the number of hybrids increases as the number of lines increases.
The
sample of lines from the population must be kept small in order to handle
the resulting hybrids.
The use of the complete double-cross and three-
way analyses probably is not the most efficient method of estimation.
There is some indication that a
syste~tic
subset of the crosses in each
",
case, with an increase in the number of parental lines, may give more
precision to some of the estimates than the complete analyses.
More
precision may also be obtained by using an incomplete analysis on several
sets of hybrids from less than the minimum number of lines required for
the complete analysis.
For example, the use of five different sets of
all possible double-crosses from six lines gives a total number of
hybrids only slightly more than would one set of p
III
8
lin~s
necessary
80
if the complete analysis is to be used.
The five sets of p= 6 lines
would provide a better sample of the original population than eight lines
and, at the same time, the smaller blocks, 45 entries versus 210, will
allow more control of environmental variation.
Total information might also be increased by including singlecrosses and computing the three covariance analyses between the three
sets of hybrids,that is, between the-single-crosses, the three-way
crosses and the double-crosses.
The information contained in the covar-
iance analyses remains to be elaborated.
The theoretical analyses of variance for double-cross hybrids and
three-way hybrids in themselves enhance the understanding of the genetics
involved.
It is apparent from the double-cross analysis that, if the
additive effects are the only genetic effects of importance, the order
in which four lines are
combin~d
has no effect on the performance of the
hybrid, since the one-line general sum of squares is the only one which
involves additive genetic variance.
.
With three-way hybrids, ordering is
.
important even if all the genetic variance is due to additive effects as
is indicated by the presence of additive genetic variance in the twoline order sum of squares.
Further, it is evident from the analyses that
the generally accepted method of predicting hybrid performance, i.e.,
using the mean of the non-parental single-crosses, is the appropriate
method for both double-cross hybrids and three-way hybrids if epistasis
is not important.
These, of course, are not new concepts but they are
exemplified by the analyses.
A problem which warrants investigation in connection with the
double-cross and three-way cross analyses is the development of prediction formulae for use in selection of the better hybrids from
81.
three-way or double-cross hybrid evaluation trials.
The usual method of
selecting in an evaluation study ieto simply pick the hybrids on the
basis of their. own performance.
This, however,
the information contained in the analysis.
•
doesr~not
make use of all
For example, it does not take
into consideration the information contained in the rest of the test concerning the average performance of the lines in any particular hybrid.
The same applies to all of the other effects defined in the model.
A
better selection scheme would be to base selection on a predicted phenotypic performance using a weighting function of the estimates of the
effects obtained from the entire analysis.
82
8.0 Summary and Conclusions
Statistical concepts provide methods for the estimation of genetic
parameters which are essential for the designing of efficient breeding
programs.
•
It would be desirable to be able to estimate these parameters
from designs commonly used in breeding programs.
Analyses of variance are developed for all-possible double-cross
hybrids and for all-possible three-way hybrids from a set of lines which
have arbitrary but equal inbreeding to determine what genetic information
is available from such analyses.
The set of lines are assumed to be a
random sample derived from a random mating population.
The hybrid sum of squares is partitioned into seven orthogonal
sums of squares in both analyses.
The expectations of the mean squares
are presented in terms of the covariances between hybrid relatives and
in terms of the effects defined in·· the models of linear effects for
double-cross hybrids and for three-way hybrids.
The genetic expecta~ions of the components of yariance are derived
from the covariances between hybrid relatives by means of which the
genetic interpretations of various F-tests are determined.
Several
F-tests are presented in each analysis whitch provide information relative
to various genetic hypotheses, many of which involve only epistatic
components of genetic variance.
Relativeiy powerful F-tests are pre-
sented in each analysis for dominance plus epistatic variance and for
deviations from an additive plus dominance plus additive x additive
model.
Each analysis provides information
wh~ch
allows the estimation of
additive genetic variancel dominance variance, the three two-factor
epistatic components of variance and the three-factor all-additive
83
epistatic component.
The estimators for these components are presented
for each analysis and the method of computing their variances is .'.,
indicated.
Limitations on the usefulness of these designs for estimation
purposes are discussed and
s~e
possible modifications to improve
efficiency are indicated.
It is suggested that an analysis of a system-
atic subset of hybrids may provide more efficient estimation.
It is concluded that considerable information relative to gene
action can be obtained from the analysis of variance of double-cross
and three-way cross hybrids through inferences from theF-tests and
from the estimates of the genetic components of variance.