OF DIALLEL, TRIALIEL
AND QUADRALIEL CROSSES USING
~SIS
A GEImRAL GENETIC M>DEL
by
Sidney stanley Young
Institute of Statistics
Mimeograph Series No. 917
April ~974 - Raleigh
iv
'rABLE OF CONTENTS
1.
INTRODUCTION • •
1
2.
GENERAL MODEL
3
2.1
2.2
Genetic Model
Experimental Model
3
11
3.
ANALYSIS OF VARIANCE
13
4.
EXPECTED MEAN SQUARES
19
5·
T:ES T OF HYPOTHESES FOR FIXED
6.
VARIANCE COMPONENTS AND TEST OF HYPOTHESES
29
7.
DISCUSSION OF RESULTS
32
7.1 Di allel . . • • •
7.2
7.3
7.4
Triallel • . .
Quadrallel •.
General Discussion
8.
SUMMARY
9·
LIST OF REFERENCES
10.
APPENDIX • • • . • •
• • • • • •
EFFECT~)
27
32
32
35
37
41
1.
INTRODUCTION
The estimation of genetic variances is generally accomplished in
the following way, Cockerham, 1963.
Relatives are created in some
mating design and tested in some environmental design.
Expectations
of the sums of squares of a quadratic analysis of the observations
lead
to estimates of design components of variance and covariance
which can be interpreted genetically and environmentally.
The
quadratic analysis can be viewed as reSUlting from a sequential
fi tting of a progressively more complicated model, called herein
design model.
the
The components of variance of the design model are
translated into covariances of relatives.
It is the covariances of
relatives that are often interpretable in terms of components of
genetic variance.
Kempthorne, 1957, formulated a general factorial model of genetic
effects for genes at mUltiple loci in diploids.
Cockerham, 1972,
organized these effects into summary ones reflecting the ancestral
sources of the genes in the mating design.
A quadratic analysis can
be developed by successively fitting effects of this model.
In that
way, design effects are genetic effects and the procedure of
translating from design effects to genetic effects (by way of covariances of relatives) is replaced with direct attention on genetic
effects.
Eberhart, 1964, Eberhart and Gardner, 1966" and Gardner and
Eberhart, 1966, have discussed a similar genetic model for fixed
effects.
The analyses of dial1el, triallel, and quadrallel hybrids
have been considered separately by several authors, Hayman, 1954a, b,
1958a, bj 196o,; Griffing, 1950, 1956; Kempthorne, 1956, 1957; and
2
Rawlings and Cockerham, 1962a,b; to name but a few, but never before
have all three types of hybrids been analyzed in conformity with the
same general genetic model.
The purpose of this dissertation is to develop quadratic
analyses for these three types of hybrids by successive fitting of
genetic effects of a general genetic model.
The resulting analyses
can be viewed as either of fixed effects or random effects, depending
upon the experimental material utilized.
3
2.
2.1
GENERAL MODEL
Genetic Model
The factorial model of gene effects, Kempthorne, 1957, is
presented for genes at two loci in diploids.
individual genotype, loci
k, and
x
and
L indexing the alleles.
y
Consider for an
as in Figure 1 with
Using
i, j,
a for additive effects and
x
i
j
Figure 1
Diagram of two loci with indexing of positions
for dominance effects the model for the genotypic effect can be
written as
Notation
Description
Genotypic effect
=
(additive, a, and dominance,
d, effects for locus x)
+
(additive, a, ~ld dominance,
d, effects for locus y)
+ (additive x additive effects)
d
4
+ (additive x dominance effects)
+ (dominance x dominance effects)
+ (dd.~JA,{,
'lrA)
These effects can be summed over an unknown number of loci for
individuals or entries such as hybrids and indexed so that the index
is descriptive of the parental source of the genes, Cockerham, 1972.
For additive effects let
A.
indicate the summation of the additive
~
effects of genes from the i
th
parental source, and
proportion of the genes received from the i
ra.
entry under consideration
Eo .. D.. , let
~J
~J
2 .
~
parent.
Then for any
For dominance effects,
be the proportion of genotypes for loci in
0 .. (0 .. )
~J
=
~
th
~./2 the
~~
the entry with alleles from parents
and
i
j
(i) .
D..
~J
of dominance effects for these genes from parental sources
j.
EO~J'
•
= 1 and
= 20 ..
~.
~
i
and
A general model for an entry
+ Eo. .
~~
is the sum
'.1-' ~J
Jr~
as a deviation from the population mean can now be written as
G
=
'£a.A. + Eo .. D.. + (ra.A.)
~
~
~J
+ (Eo .. D.. )
~J
2
~J
~
+ (ra. A.)
~J
~
~
2
~
+ ('£a.A. )(Eo .. D.. )
~
~
~J
~J
3 + (ret.A.) 4 + . . . .
~
~
Expansions of the epistatic terms are instructive; for example,
(ra.A.)
~
~
2
=
2
Lev. (M) .. + 2E ra.~. (M) ...
.
~
~
~~
"
~<J
~
J
~J
The first summation. in the expansion is for additive x additive interaction between alleles from the same parent and the latter involves
5
alleles from different parents.
Also note that
(AA)..
~J
average of two additive x additive interaction effects:
parent
i
i
with x
with
y
genes from parent
genes from parent
j , and
y
is an
x
genes from
genes from parent
j .
Models for three types of entries, diallel, triallel, and
quadral1e1, are now presented.
First consider the entries of a dia11el
experiment in which selfs and reciprocals are omitted:
~.
~
6 ..
lJ
=~.
J
=1
+ (AD)i(ij) + (AD)j(ij) + (DD)(ij)(ij) +
Next consider progeny of a three-way cross
distinct parents:
~i
= 1,
~j
(2.1)
i x (j x k)
1
= ~k = 0ij = °ik = 2 '
Gi (jk.) = Ai + ~2 Aj + ~2 Ak + 21 Dij + 12 Dik + (AA) i i
1 ()
1 ( )
2 ( )
+ 1+
AA j j + 1+
AA kk +~
2 (
AA) ij + '2
AA ik
1
1
8 (AAA)jjj
+
1+ (AD)k(ij)
+
133
8 (AAA)kkk + 2 (AAA)iij + 2 (AAA)iik
+ (AAA)iii +
with
+
3
's
+t
(AM)kkj +
6
4"
(DD)(ik)(ik)
(AM)ijk +
+~
1
4"
.
(DD) (ij ) (ij )
(DD)(ij)(ik) + . . . .
Three locus, all-additive types of interactions are included in the
model since they are to be utilized in the analysis.
Finally, consider the model for the progeny of a four-way cross
from four distinct parents
1
=2
_
and
0ik
+
~8
(i x j) x (k x i,) , Q'i = Q'j = Q'k = Q'i,
~
1
= °ii, = u jk = °jt = 4:
((AM) ..
11J
0
+ (AM) .. k + (AAA) ..
11
+ (AM). Ok + (AAA)
JJ
IJ
11~
o. n
JJ~
+ (AM)
JJ1
0
.,
+ (AAA),.h~ + (AAA)kk' + (AAA) •• ' d
.ruu.
,.Jt'U\.,(,
6 r
()
')
()
('}
+ "8
1.. AAA ijk + CAAA ij.t + \AAA ik.t + \AAA)jk.t
7
+
fb{(AAAA)iiii
+
(AAAA)jjjj
+
(AAAA)kkkk
(AAAA)'i'k
+
(AAAA)",
+ (AAAA)""",,}
+
~1((AAAA)ii'j
~o
~
+
~
~
n
~~~~
+
(AAAA), .. ,
JJJ~
+
(AAAA)jjjk
(AAAA)jjj,t
+
(AAAA)kkki
+
+
(AAAA)kki ~n + (AAAA)kk'J~
+
(AAAA) JJ~
... k
+ (AP..AI1.).,",
+
A
(AAAA)kkkj
A
JJ~~
8
1
+
ib ((DD) (ik)(ik)
+ (DD) (iL )(iL) + (DD) (jk) (jk)
For four-way crosses,three and four-locus, all-additive
typ~of
interactions are included in the model since they are to be utilized
in the analysis.
Note that when all the effects of a particular type are added for
any model
G
= 2A
+ D + 4AA + 2AD + DD + .•••
Numerators in the
models indicate the number of distinct effects that are averaged.
When individuals are random members of a linkage equilibrium,
randomly mating population, the genetic effects are uncorrelated,
Cockerham, 1963, and the total variance can be expressed as a sum of
the variances of the effects:
9
Comparin.g this to the model of Cockerham, 1954, where variances of a
kind are summed into one term,
Total u
22222
+ 0-6 + o-~ + o~6 + u
+ ... ,
66
2
o~
G
and the translation from one representation to the other is obviow.:i.
For single crosses, G.. , and assuming uncorrelated effects, the
lJ
variance among unrelated single cross means is the total variance.
u
2
G
222
0+ 20'
2u +
A
D
M.l
2
+ o·DD
+
2
+ c..vAAA
'') ,.,.2
-r ...
III
1
The numerical subscripts refer to the number of lines invo]:ved in a
variance cOm:Ponent.
same,
~.~.,
u
2
G
2
E(M) I..I
2+
2 0-A
If we let the components within a class be the
2
E(M) ..
lJ
2 + 4 2
0-D
UAA
2
M
or
+ 2 2
0-AD +
,then
0-
2
U DD
2
2
+ 8crAM + .•• •
The variance among three-way cross means, which is not the total
variance, is
+ ....
<:'
The numerical subscripts distinguish among 1, 2, and 3 line effect:,.
Again if the cOm:Ponents are the same within a category
+ •
II'
..
.,
10
Finally, the variance among four-way crosses if;
0"
2
O"A
2
,1
G( ..J ) (k.e
- )\
1
1 2
1
3 2
+1: O"D2 +4
O"AA + 4" 0"AA + "8
4
2
1
2
+ - O"DD
32
+
3
,1
2
b4 (JDD
+
4
2
1
l6
n
C'
c.
AD"{~
+ 9
O"AM
1
16
u
+
2
AAA
,)
1
{-
+
O"Aj
"8
)3
+
2
6
16
')
1
(-
61;
cr
D:':'i"
c.
2
°AAA_.z,
.-'
If the components within a category are the same, then.
Note that when components in a category are equated, the entire;
variance, whether it be for total, single crosses, three-way crm;sfcs
or double crosses can be generated from the coefficients of
2
O"D'
and
Orgarlizing the variance components into categories reflecting r,b.€'
number of contributing lines affords a convenient way of summarizing
the kinds of effects involved in quadratic forms even thou.gh thE':
effects are viewed. as fixed effects.
Linkage affects the coefficients of the epistatic cOillponent,;,,;w:clc':n
there is control over the grandparents; for eXaillr1e, in a l'our-',1ay
cross
(i x j) x (k x .e) , without some recombination of genes witrdr,
a chromosome or reassortment of chromosomes there can
(DD) (ij ) (k.e)
ee
no
component and all of the dominance x dominance i!,t.~::­
actions would be of the
(DD) (ik) (ik)
type.
With y.·ecombir~atic:)n 'J.'~i/C'-·
reassortment, dominance x dominance interactions of the ty}'e
(DD) (ik) (i.e) and (DD) (ik) (j.e)
become possible.
.n
recombination the coefficients for triallel and quadTallel crosses
are those given in (2.1) and (2.2).
Linkage does not affect the co-
efficients of additive and dominance effects.
2.2
Experimental Model
Each type of hybrid is to be analyzed separately with the same
experimental model giving rise to three analyses of variance.
The
experimental model is
where
Y[
Jm
is the effect of replicate
m
m
[J
G[
J
is the genotypic effect of cross
e[
Jm
is the random error associated wi th cross
replicate
[J in
m.
In the case of diallel, triallel, and quadrallel,
i(jk)
m
is the overall mean
~
r
is the value of the progeny of cross [J in rep
and (ij)(k.t) ,repsectively.
[J
All line indexes,
= 1, 2,3, ... , n , have the same range, where
IT
becomes
ij ,
i, j, k, t
is the total
number of lines, except that they must be distinct for each hybri.d.
For replicates,
m
= 1, 2, 3, •.. , r .
12
It is convenient at this point to layout the notations to be
used.
A dot notation is used to indicate a summation,
~.~.,
Y(ij)(kL).
is the summation over all reps of hybrid
Y
(ij)( •• ).
is the summation of all quadrallel crosses with grand-
parental cross
i x j
over all reps.
(ij)(kL).
When parentheses are omitted,
the summation is over all hybrids with the given parental
identification regardless of how the hybrids are put together,
Y. 'k
~J
~.~.,
is the summation of all four-way crosses involving grand-
••
parents
i, j,
and
k
summed over all reps;
crosses involving lines
regardless of how the grandparents were mated,
Y. ,
~J'
i
•
is the summation of all three-way
and
j
summed over all reps.
The
summations with parentheses removed can be calculated as simple sums
of the sums with parentheses, but they are convenient for succinctly
expressing sums of squares used in the anaLyses of variance.
notation
is used to denote
ni
of combinations of
n
n-i, and
things taken
The total number of hybrids is
n
3. e3
=
/2
nnln~
en
K
The
to denote the number
K at a time.
e~ = nn/2
f or th ree-way crosses, an d
way crosses, where reciprocals are omitted.
for single crosses,
~.er.41
~
The factor of three for
three-way and four-way crosses comes from the three ways that the
same set of three or four lines can enter a cross.
13
30
A..T'fALYSES OF VARI.fu"iJ"CE
In each analysis the sums of squares for replications) treatments (hybrids)) and error are the usual least squares partitions for
a replicated experiment and are orthogonal by construction.
The
partitioning of the hybrid sum of squares follows from fitting effects
in the general model in the order
AAAy
ADy
AAAA , DD , DDy
4
2
AAAA , AAAA , AAAAy
l
2
indicating additive effects;
A) D, AA , AA , AD , AAA , AAA ,
2
l
2
2
l
and
DD ) with
4
A
D, dominance effects; repetitions of
letters, interactions; and the subscript, the number of lines involved in an interaction.
Each sum of squares in the partitioning of
the hybrid sum of squares is the additional accounted for by adding
the effect to the model.
The process of adding effects to the model
was stopped when the entire
partitioned.
hybrid sums of squares had been
Of course, it is not possible to obtain a sum of
squares for each type of effect in the model for all analyses; for
example, four-line interactions are not possible when only two-line
crosses are made.
Also some of the effects are completely con-
founded with previously fitted effects.
The analysis of variance for diallel crosses is given in Table 1.
The hybrid sum of squares is broken into two parts, additive and
The analysis of variance for triallel crosses is given in
dominance.
Table 2.
The hybrid sum of squares is broken into seven addi ti ve
parts, TA, TD, TAA , TAA , TAD )
1
2
2
TAAA~,
~nd
TAD
3
.
variance for quadrallel crosses is given in Table 3.
of squares is broken down into seven addi ti ve
QAAAy
QADy
QAAAA4' and
QDD4 •
The analysis of
The hybrid sum
parts) Q,A, QD, QAA2'
14
Table 1
Analysis of variance for progeny of a diallel cross,
selfs and reciprocals excluded
Sums of Squares
df
Source
r-1
Crosses
(2- - 1).
nnl
Additive
Dominance
2
2Y
mnl
2
2 Ey
nnl m • 'm
Replicates
n1
Total
t!l
2
E E Y2 • _ --y2
ij
rnn2
r i<j
nn3
DD =
1
rn2
1.
L: y2
L L
r i<j
i •.
..
Y. " e
1
- -rn2
LY.
i ~ •.
~
nnl
(r-1)(-·- - 1)
DE by difference
rnnl
-2- -.1
2y2
2
I: I: L Yijm mln2
i<j m
2
2
4
--rnn2
2
Yi'J.
2
2
+ rnln2 y
Error
(Crosses x
Replicates)
~
1.
DA ""
2
o
2
Table 2
Analysis of variance of three-w"ay crosses
Source
Correction
factor
Sum of Squares
df
2y2
C
1
rnnln2
d ... m
m
Replications
(r-l)
R
-
E E E
i j<k
Crosses
Additive
n
( 3C 3- 1 )
H =
nl
TA
2
y2
1
rn2 On-B)
E
i
[2Y
2 rn 3 i < j
TAAI
1
C
+Y.] 2 _
i( .. ) . . (1.).
_I_EErY . .
TD
4
nl
-
r
+ rnln2n3
Add. by Add.
One-line
i(.ig:)·
~- -
nn3
Dominance
C
nn~2
+Y . .
(J • ) •
16
1-
]2 _ _1
2 rn 2 n 3
L :2Y
i'
J
(1 • ) •
[
n4 y i( .. ). -n 2 .(1.).
-,.
Y
2
+Y
i . . . . (1.).
]2
y2
2
rnn2n3 On-B)"
L:
i
Y
] 2 ..
2
2
( 8 ) Y •...
rn2n33n-
t-'
\}1
Table 2
..
~_
(Continued)
__._--
nn3
AdEi b;.' Add
Two-line
1
rn1n3nl+
TAA2
2
~ ~
[n3Y.(ij).+YiUo).+Yj(L).J2
~<J
1
rnln2n3n4
Add by Dom
n1 n 2
Two-line
-2-
TAD2
Add by Add by Add
Three-line
nOlDS
TAAA3 =3! r E E
r i<j<k
Add by Dom
nn2 n 4
Three-line
6
TAD
3
Erro:c
(r-l)(3 C~-l)
Total
Or C~-l)
=
1
2rn3
1
~
E E [Y
i<j
Y~jk
1.
i(j.)
-
•
E E E y2
r i
j<k
[2Y i (o.).+ n2 Y .(io).J2+
E
i
i(jk)
- y
0
1
3rn4
j (1.)
EE
i<:i
J2
y2
_ _1_. E[ 2Y. r )
2rl1L i
~\.o
0
y2
ij
2
3
+
0
•
2
E
3rn3nq i
y2
0
_ Y (' ) ] 2
~o •
i ••.
•
- rn
2
2
y ••••
n n
2 3 4
-TA-TD-TAAI-TAA2-TAD2-TAAA3
0
j,k/:i
TE
... By difference
T=1:EEZ',,2
i j<k ~ ~i(jk)m - C
j,k/:i
b-,
Table 3
Ana~sis
of variance of four-way crosses
df
Source
Correction
Factor
Sum of Squares
8y2 ••.••
C - rnnln2n,
1
8 I: y2.••• m
Replications
Crosses
(r-l)
3C~-1
Additive
nl
Dominance
-2-
R _
_ C
m
nnln2 n ,
H=.!I:I:I:I: y 2
_
r 1<.) k<,t
( i j ) (kl) •
C
i, jf.k, t
i<k
QA _
2
[
rn2n,n,.
nn,
QD -
_,_7
~
J.
yi... - 16
•
n
y2 .•••• ]
4
n2
~_~ u , ( ~ ~ ~i. )(j.).
J.<J
Add. by Add.
Two-line
nn,
-2-
AA
Q 2 -
nUlnS
~6-
yL ... J
2
E r {(n 2 -7n+14)y(ij)(
) +
rn4nS (nL -7ii+I4) [_1
nln2 i<j
•••
2ngY(L)(j.).}2
Add. by Add. by Add.
Three-line
I: y2
+ !L
i
i ....
nln2
1
QAAAg • 3rn6
- n
I:
l i
y~
J. . . . .
I: I: I: y2.
_ ..,,--...;.4__
i<.:i<k iJk..
3rn4n6
6
rngn4 n 6
I:
i
Y~ ••••
32
-
+ 8y2
L: E y2
i<j
ij ••• +
y¥ .•.•
rn2ngn4n~
!::1
Table'
(Continued)
Add. by Dom.
Three-line
nnzn ..
-3-
QAD 3 • _1_
rn3
~
t t t
(ij
1<j k
t yZ
_ ___2
i<j
(ij)( .• )
rnin3
Four-line
nninZ n 7
QAAAA. . .
24
Four-line
nnin .. nS
12
QDD . . .
t t ty2
_
i<j<k ijk ••
I t y2
i<j
(i.)(j.).
t t t ty2
_-1...- t t t.~
3r i<j<k<. ijk{. 3rn6 i<j<k ijk••
L
2
rn .. nSn6
Dom. by Dom.
1
3rn3
4
rnin3'
i .• jl-k
I
Add. by Add. by Add. by
Add.
-
HIt. ).
!
r
t
i
E E E E
i<j k<t
y~ •••• +
+
4
t t y2
3rnin3 i<j
ij •••
2nl+
+ 3 rnl+nSn6
r r
i<j
r?:..
~J."
2
y2
rn3n .. nSn6
Y~ij)(k£). - QA - QD - QAA 2 - QAAA 3 - QAD 3
i,jl-k,t
_ QAAAAI+ i<k
Error
n
(r-l) (3 CI+-l)
Total
3r C.. -l
n
QE - by difference
2
1:. EEEE1: Y(ij )(kt)m - C
i<j k.d m
i,jrk,t
i<k
~
19
4.
EXPECTED MEAN SQUARES
Three methods were used in obtaining the expected mean squares
for the three analyses.
4,
Those for the diallel, Table
were
obtained by substituting the model of effects into the mean squares
and taking expectations assuming uncorrelated effects.
Table 4
Expectations of the mean squares of diallel analysis in
terms of the variance components of the general model
truncated to dominance by dominance effects
Source
E(MS )
Additive
DominaJ1Ce
Error
(J
2
e
This method was used to check some of the results for the triallel and
quadrallel analyses, but was found to be extremely tedious.
The
following method was used to obtain the expected mean squares for the
triallel aJld quadrallel
aJla~yses.
First, the covariaJlces of genetic
effects of three-way, Table 5, and four-way, Table 6, hybrid relatives
were defined and their expectations obtained in terms of components of
genetic variaJlce.
Next the expectations of the uncorrected products
and squares of sums were obtained in terms of
covariaJlces of re.lati ves.
~L
2
(J
2
r
,
2
(J
e
and the
These are given in Table '7 for the triallel
analysis and Table 8 for the quadrallel analysis.
Finally the
results of Tables 5 and 7 were substituted into Table 2 and the
Table
~
Covariances of the genotypic effects of three-way hybrid relatives and their expectations in
terms of components of genetic variance
Coefficients of Variance Component
Covariance*
Number of
lines common
0 2
A
02
D
2
°AA 1
2
(1AA2
2
(1 AD 2
2
(1 AD 3
2
(1AAAl
2
°AAA2
2
°AAA3
2
°DD2
2
°DD3
9/S
5/S
l/S
66/64
63/32
3/S
l/S
l/S
1
1/4
1/16
17/64
42/32
3/8
1/16
0
E[Gi(jk)Gi(jk)]
3
3/2
1/2
9/S
Cov2 - E[Gi(jk)Gj(ik)l
3
5/4
1/4
9/16
Cov3 - E[Gi(j_)Gi(j_)]
2
5/4
1/4
17/16
1/2
5/16
0
65/64
3002
0
1/16
0
Cov~ - E[Gi(j_)Gj(i_)]
2
1
1/4
1/2
1/2
1/4
0
16/64
24/32
0
1/16
0
Covs - E[Gi(j_>G_(ij)l
2
3/4
0
5/16
1/4
0
0
9/64
9/32
0
0
0
Cov6 - E[G_(ij)G_(ij)]
2
1/2
0
1/8
1/8
0
0
2/64
3/32
0
0
0
Cov, - E[Gi( __ )Gi( __ )l
1
1
0
1
0
0
0
64/64
0
0
0
0
Covs - E[Gi( __ )G_(i_)]
1
1/2
0
1/4
0
0
0
8/64
0
0
0
0
Covg - E[G_(i_)G_(i_>]
1
1/4
0
1/16
0
0
0
1/64
0
0
0
0
COY}
E
*Dashes indicate any lines not common in the two relatives.
I\)
o
Table 6
Covariances of the genotypic effects of four-way hybrid relatives and their expectations
in terms of canponents of genetic variance
Covarianee*
Buaher of
lines COaaGll
2
"A
Coefflcienes of yariance co.ponent
D
,,2
AAI
,,2
AA2
" iD 2
,,2
AD3
,,2
,,2
AAAI
,,2
AAA2
" iAA 3
"hAAI
aiuA2
ohAA3
ohAA~
"~D2
"~D3
"~D~
Covl - E G(lj)(kl)G(lj)(kl)
4
1
114
1/4
3/4
1/8
1/8
1116
9/16
3/8
1/64
21/64
9116
3/32
1/64
1/32
1/64
Cov2 - E G(lj)(kl)G(ij)(kll
4
1
118
1/4
3/4
1/16
1/16
1/16
9/16
3/8
1/64
21/64
9116
3/32
1/128
0
1/128
Covs - E G(lj)(k_)G(lj)(k_)
3
3/4
1/8
3/16
3/8
1/16
1/32
3/64
9/32
3/32
3/256
21/128
9164
0
1/128
1/128
0
Cov~ - E G(lj)(k_lG(ikl(j_)
3
3/4
1/16
3/16
3/8
1/32
1/64
3/64
9/32
3/32
3/256
21/128
9/64
0
1/256
0
0
Covs - E G(l_)(j_)G(l_)(j_)
2
1/2
1/16
1/8
1/8
1/32
0
1/32
3/32
0
1/128
7/128
0
0
1/256
0
0
Cov6 - E G(ij)( __ )G(lj)( __ )
2
1/2
0
1/8
1/8
0
0
1/32
3/32
0
1/128
71128
0
0
0
0
0
COV7 - ! G(lj)( __ )G(l_)(j_)
2
1/2
0
1/8
1/8
0
0
1/32
3/32
0
1/128
7/128
0
0
0
0
0
Cove - E G(l_)( __ lG(l_)( __ )
1
1/4
0
1/16
0
0
0
1/64
0
0
1/256
0
0
0
0
0
0
*D•• bes indicate any linea Dot eoa-on in the two relatives.
I\)
r"'
. +-'
J.ne ey.pec"ta~lons
CI praGue t
s "
and squares 01~.
sums ll1 t erms 0 f
of three-\vay cross l'elati ves
"'.
rrable :-
r
N.",· .. "
S tit;; 3qu&.red
or Pl"duct
'I
_L_
y1
rnnlnZ
1 y2 .
• (J • ) ,
rnlnZ
1
y
y
rnln2
i( . . ). . (i. ) .
rnz
y2
i(j.).
_1_ y2
ruz
.(ij).
_1_ y
i (j . ) .
rn2
1.
r
y
y2
i(jk).
2
y2
i ...
3rulu2
__
1_ y2.
3rn2
ij ••
L
3r
y2
ijk.
• (ij).
~
__
(12
e
COVI.
nnin2
1
2
r
nl n 2
--2-
~L
2 , U2 , er2 and the covarianees
e
r
Coefficients of Covariance
~
Cove
Covg
rn3n4
rn3n4
0
0
0
0
rn3n4
0
0
rn3n4
0
0
0
0
0
0
0
0
rn3
0
0
0
0
rn3
0
0
0
0
0
0
0
0
0
0
0
0
4rn3
-3-
4ru3
8rn3
2rn3
rU3nlf
2rn3nlf
2rn3nlf
2ru3
3
2rn3
3
4ru3
3
rn3
3
3
3
3
COV2
COV3
Cov4
COV5
COV6
'2
r
rn3
rn3
2rn3
-2-
1
r
0
2rn3
0
0
0
nln2
1
r
r
rn3
0
21: n 3
rn3
nl n 2
--2-
0
0
r
0
rn3
rn3
n2
1
r
0
rn3
0
n2
1
r
0
0
u2
0
0
r
I
1
r
0
3nlu2
1
4
')
- , , ,•. ,.__
(r\.l2+ 0 2)
r
..
...
y2
rn 1 n2
i ( .. ) .
-L.
•
r
rn3
2r
2
3u2
1
r
2r
3
1
r
2r
0
0
0
0
COV7
rn3n4
--4
rn3n4
2
6
3
3
0
0
0
0
0
0
f\)
f\)
'I'arLl.e 8
The expectations of' products and squares of sums in terms of ,./, J.L2, ()2 and the covariances
of four-way cross relatives
r
e
Coefficients of Covariance
Sum Squared
~ P
0..
•
2
(ql +0
..
roauc~
6'
__
4 __ y~....
rnnln2n3
2
2
r ) 0e
COV1
Cov2
Cov3
Cov!+
Covs
Cov6
COV7
Cove
nnln2 n 3
8
8r
16r
32rn4
64rn4
32rn4nS 8rn4nS
32rn4nS
164n4nSn6
--~--~Yir-'illD2 D 3
••••
n1 n 2 n 3
2
2r
4r
6rn4
12rn4
4rn4nS
rn4nS
4rn4nS
rn4nSn6
4
2
----- Y., .
9n2D3
6
6r
12r
12rn4
24rn4
4rn4nS
rn4nS
4rD4ns
~~
Y:~k
n~
~J""
3n3
1
r
2r
rD4
2rn4
0
0
0
0
r
r
2rn4
2rn4
rn4nS
0
0
0
4rn4
0
0
rn4nS
0
4
?
rn2u3
j
~J'"
0
..
___
1 __
r02uS
Y~i
)(j )
\.
..
02n3
1
rn~n3
Y(ij)( .• ).Y(i.)(j.),
u2 n 3
0
u2n3
2
2r
0
4rn!+
0
0
rn4nS
0
0
n3
1
r
0
rn4
0
0
0
0
0
1
1
rOO
0
0
0
0
0
.__....!i- y~ ..
rn2nS
__
I_
rn3
f
PJ)"')'
y2(~j)(k
)
•
. ..
~ Y~ij)(kl).
~-----_
0
2r
0
..
~
24
results of Tables 6 and 8 into Table 3 to give expected sums of
squares for the triallel and quadrallel analyses respectively.
Dividing by the degrees of freedom gave the expected mean squares for
the triallel analysis, Table
~
and quadrallel analysis, Table 10.
The intermediate results, the expected mean squares in terms of
covariances of relatives, are given in Appendices I and II.
These
types of results are instructive in the case of the diallel,
Kempthorne, 1957, but do not appear to be here.
A third method of calculating expected mean squares, Gaylor,
Lucas, and Anderson, 1970, using the forward solution of the
abbreviated Doolittle method\'las used to check the expected mean
squar~of
the triallel analysis.
This method would be useful for a
particular experiment where the number of lines is fixed, but it is
difficult to apply to a general analysis.
This method is of limited
utility if the number of lines is large, as an excessively large
matrix must be swept out by the abbreviated Doolittle method.
Table 9
Expectations of the mean squares of' three-way crosses in terms of components of genetic
variance
Compooents of Variance
Mean
Square
11~
a~D3
aln 3
-8-
OlAA3
al AA2
OlD2
al A2
aiAAI
alAI
{02+~02
DD2
D
}
02
A
Coefficients of Components of Variance
TA*
1
r
r 010-32)
16 (3n-8)
TD*
1
r
r(30-11)
16n3
TAA~
1
r
TAA~
1
r
TAD~
1
r
16
TAAA;
1
r
4"
1
r
#:
'rAD:!
r02
4(30-8)
rn2
8n3
r
r
3rn3
rn2(9n-20)2
r02(50-12)2
r(41n2-217n+288)
16(30-8)
3r(1010 2 -562n+784)
32 (3n-8)
r(7n-20)2
8 (3n-8)
3r04
9r03
4n3
"""16
3r(3n-10)2
16n3
r(2n-7)2
4n3
0
0
ron3
9ron2 n 3
ro02n3
4 (3n-8)
32 (3n-8)
8 (3n-8)
(3n-8)
r02n3
3rn
8 (3n-8)
3rn
8D3
0
r (3n-8)
0
3r03 (50-8)
16 (30-8)
3rnl04
rOl04
32n3
----an;-
r03
3r03
16
"""16
64 (3n-8)
r (3n-8)
16 (3n-8)
4
rn:{3n-8}
4
rn3
-4-
9r
8
r
16
TE*
..
~
Sum of squares divided by its degrees of freedom.
I\)
VI
Expectations of the mean squares of four-way crosses in terms of components of genetic
variance
Tab1::: 10
Mean
Square
QA*
QD*
Components of Variance
aZ
"ze
9r
r
32
TI
r{n Z-5n+S)
64(n 2 7n+14)
QAAA~
TI
QAD;
ill
QDDt
"iD 3
27rn4
--n-
r
r
Z
rn2DS
32(n2-7n+14)
16(n'-7n+14)
TI
0
3 Z
"AAA3+Z"AAAA3
3rn4
3rn 1 n2
9r
r
{z
16
r (3nz-25n+54) 3rn 4 n S
r(3n 3 -39n z+17n-268)
128(n 2 7n+14) 7176'(-n'2-'7~n7+71'4')
32 (n 2 -7D+14)
QAA~
QAAAA~
aiAAA 4
DD4
Z
3rn4Ds
rnltDs
(n 2
IHil -7n+T4)
-7"-+14)
Z
{ Z l
z
1Z}{zlz
"D~ADz+t6"DDz
rn3n4
-8-
-3-2-
JrnlD2DS
7
"AA z 4 AAA z+t6"AAAA z }
9rn2D3
z
8
~z
} {Z
2
lz
lz
"A+4"AA1~AAA1+64"AAAAI
rn2D3D1t
--S--
r(n z-7n+14)
16
rnlD2n4DS
...... I'
2
,_.
rnG
""i6""
rn3
64
9r
32
32
r
128
0
QE*
f\)
0\
27
5.
TESTS OF HYPO'rHESES FOR FIXED EFFECTS
Certain tests of hypotheses are available wi thout making any
assumptions about the genetic effects.
in Tables
The mean square
expectations
4, 9, and 10 in this case serve only as guides to the
types of effects that can contribute to the mean squa.res; the mean
squares actually involve quadratic functions of these types of
effects.
:F'
'rhe error mean square can be used as the der::.ominator in an
ratio testing sequential.ly up each table.
Table 11 gives lowest
order types of effects that are tested in each mean square for each
analysis; higher order effects are also tested for in each.
As we
proceed to test up the table lower order genetic effects become
involved.
The method of obtaining each analysis of variance
guarantees that quadratic forms of previously fitted effects do not
appear in SUbsequent mean squares, although similar interaction type
effects,
~.~.,
fi tting of
AD
AA
2
after the fitting of
AA
, and
AD
after the
, may appear in subsequent mean squares.
Also there
2
l
3
are two things which complicate the interpretation of the nonsignificance of a particular mean square.
First, the genetic model
effects are a sl.urunation of allelic effects and may sum to zero when
a.llelic effects are present.
Also, the qU8.dratic functions of a
particular type of genetic effect differ from mean square to mean
square so the conclusion that certain quadratic functions are zero in
one mean square does not gnarantee that the same is trCle in another
mean square.
28
~~able
Mean
Square
11
Lowest order type of effects tested for in the diallel,
triallel and quadrallel analyses for the various mean
squares. A -- indicates there is no corresponding mean
square for that analysis
DIALLEL
TRIALLEL
QUADRALLEL
A
Ai
Ai
Ai
D
D
ij
D
ij
D
ij
AAl
AA
AA2
AA
AD 2
ADi(ij)
AAA3
AAA
AD3
ADi(jk)
it
ij
ijk
AA
ij
AAA
ijk
ADi(jk)
AAAA'4
AAAA ijk £.
DD4
DD (ij) (k£.)
29
6,
VARIANCE Co.MPONENTS AND TESTS OF HYPOTHESES
By assuming the effects of the genetic model are random, and uncorrelated, and that there are cammon variances within certain
categories, genetic variance components can be estimated.
These
assumptions were made in arriving at the expectations of the mean
squares.
When the general model is truncated for each analysis to
those terms given in Table 11, the comparable variance components can
be estimated by equating mean squares to expected mean squares and
solving the resulting equations,
In the diallel it is possible to
2
2
2
2
2
2
estimate ~A and ~D; in the triallel, ~A' ~D' ~AA ' ~AA '
2
2
1 2
222
~AD '
~AAA...' and ~AD
; and in the quadrallel, ~A' ~D'
2 2
2m~
2
3 2
2
~AA '
~AAA...'
~AD '
~AAAA ' and ~DD '
Other vari ance
2--~
3
4
4
components defined and given in the tables of expected mean squares
are confounded with these estimators although not always in a simple
manner.
The diallel
~lalysis
gives a good example of simple patterns of
confounding; all one-line variance components are completely con-
~~
founded and estimated as one package,
+
variance components are estimated together,
2
+ ~DD
•
, and al.l two-line
21
2
~D + 2~AA
2
+ 2~AD
2
2
An example of a more difficult confounding pattern can be
2
2
2
in the following
DD
3
Two types of additive by dominance, three-line variance
seen in the triallel analysi s for
manner.
~~
~AD
and
~
3
components are distinguished, depending on bow the effects come to222
E(ADi ('k)} = E(ADi ('k)} = ~AD
- J_
- .J.
1 3
2
E(AD!(ik)'AD!(j~)} = ~2AD3 ' the underscore indicating whether
gether in taking expectations.
and
Let
the grandparental source of alleles is the same,
~
2
lA.D
3
,or
30
different,
0-
2
0-
3
2
the underscore emphasizes that
AD
3
I
E{AD~:C.sik)
E{ADf(.sik )} -
2
In
AD
AD~(.sik)}
,.
parental cross referenced by
i x j
confounded with
2
(JDD
and
j.
In
2
0-
AD (
~ j~,
i x k.
and
and
was the grand-
x j
the effects,
AD
2 3
) ' come from different grandparental sources,
and
i
i
and that
0-
AD
o-DD
is completely
3
The distinction, other than to show that
.
3
I
2
()AD
3
2
2
With this distinction made,
are confounded, does not appear useful so
3
(J2
assumed eqllal to
2
and both are termed
AD
is
2
GAD
.
3
3
Any variance component or sum of variance components that can be
estimated can be tested, subject to the condition that the effects in
the model are distributed normally.
The error mean square can be
used as the denominator in an F-test to test certain exact and
composite hypotheses.
Table 11,
2
(JD
=
0 ,
DD*/DE* ,
2
0
(JAD
With each analysis restricted to terms in
T~/TE* , and QDDt/Q,E* provide exact tes ts for
2
and (JDD == 0 • Co..mposi te hypotheses, testing
4
3
each mean square versus error, are possible for the linear functions
of variance components given in the tables of expected mean squares,
~.~., in the triallel
l'
lb
2
(JAD
3
TAD*~TE*
rn3
+ 16
tests the hypothesis that
3rn3 2
2
(JAD
2
+ ~ (JAP.A
2
o.
Exact tests are not generally available for testing other variance
components; however, approximate F-tests are.
Satterthwai te, 1946, suggested that a linear function of mean
squares,
(L:a MS )
i i
is approximately distributed as
x2 rl/f l
with
fl
degrees of freedom where
fl
22
= (L:a.MS.) 2
/L:(a.MS./f.)
l
l
~
~
~
(6.1)
31
and
f.
l
denotes the degrees of freedom for mean squares
MS.
l
Using
Satterthwai te I s approximation, error terms can be constructed to test
each of the components of variance.
For f'.-Xample, in the triallel
analysis
4T~ - 3TE*
has expectation
~2 + £ ~A2 ,the correct expectation of an error
e
4 D
3
term for testing the significance of
~~ in T~* and can be
used to form an approximate F-test
TAAA*
3
4T~ - 3TE*
with degrees of freedom
from (6.1).
nn n /6
l 5
and
f'
where
f'
can be obtained
32
'7.
7.1
DISCUSSION OF EESULTS
Diallel
Since truncation of the general model to additive ai1d dominance
effects corresponds exactly to the usual model for general and
specific combining ability, the partitioning of the
identical.
SlUllS
of squares is
Several tbings became apparent from examination of the
expected mean squares.
There are two types of effects, and con-
sequently variance components, single-line and two-line.
'l'he single-
line types are confounded with each other and must be estimated
jointly.
The two-line types are also completely confounded and must
be estimated in a single package.
It is the splitting of the
epistatic variance into two parts, within line and between lines, that
makes the estimation of single-line and two-line packages possible.
All single-line effects are removed with additive effects.
As one
would expect from the expression of the total genetic variance for
two-line crosses, only one-line and two-line variance components
appear in the analysis.
7.2
Triallel
Examination of the expected mean squares indj.cates that whereas
the genetic model is simple in concept and interpretation the
expected mean squares are complex.
It can be seen that the order of
fi tting of dominance and addi ti ve by addi ti ve, single-line effects is
innnaterialo
Th.is is also true
1'01'
addi ti ve by addi ti V2, two-line
effects and addi ti ve by dominance, two-line effects.
33
In considering a fixed effects model., it is possible to combine
the mean squares in the analysis presented into single-line (TA + TAA ),
1
two-line
(TD + TAA
2
+ TAD )
2
and three-line
partitions to give a new analysis.
(TA~ + TAD
3
)
In testing against error
sequentially up the resulting analysis, three-line, two-line, and
single-line effects successively come into play.
In the estimation of genetic variances mean squares of the analysis
may be combined to correspond to assumptions about the variance
components.
If it is assumed that
2
~AA
and
1
2
~AA
are identical,
2
the corresponding mean squares in the analysis can be combined to
estimate
2
~AA
Likewise if it is assumed that
.
2
~AD
and
2
~
2
AD
are
3
identical, the corresponding mean squares can be combined to estimate
2
When these mean squares are combined, weighting by the degrees
~AD'
of freedom, the coefficients of the variance components in the resulting expected mean squares remain complex.
2
2
= 0 , it then becomes possible to
2
by manipUlation of the mean squares.
~AD
and
If it is assumed that
~AAA
The point is that by assuming genetic variance components within a
category to be equal, other higher order variance components become
estimable.
This analysis can be compared to that of Rawlings and Cockerham,
1962a.
There is a simple relation between the sums of squares in the
two analyses, Table 12.
34
Relation of su.ms of squares of Rawlings and Cockerham,
1962a, to those of the general model for trial1el crosses
Table 12
Sluns of Squares
Rawlings and Cockerham
1962a
Description
Su..llS of Squares
General Model
l-line
G + °1
TA + TAA
2-line 2-alleles
S2 + ° 2
a
TD + TAA
2-1ine 3-alleles
°2
3-line 3-alleles
TAD
b
l
2
2
S3
T~
°3
TAD
3
The two analyses differ in the genetic variance components that
are estimable.
In the analysis of Rawlings and Cockerham, 1962a, the
design components of variance were expressed in terms of covariances of
relatives, and these in turn in terms of genetic variance components.
The estimation of genetic variance components was then accomplished
by equating the estimated design components of variance to their
expected values in terms of genetic components of variance, and
solving the resulting equations after suitably restricting the
genetic variance components.
When the genetic varianr:;e components
were restricted to the seven lowest order ones,
j_ t
was found that
there was a Unear dependency in the seven reSUlting equations so that
and
er
2
Q1Q1Q1
,could be estimated.
2
2
(T& '
2
2
er
,
er0'& '
cxa
In the design presented here, seven
only six genetic variance components,
er
01
35
genetic variance components can be estimat(..j,
222
uAA '
uAD '
uAM
, and
U
variance components.
Again,
L'f'
2
3
2
2
AD
3
(Y
2
'D '
2
uAA
J_
'
,but onl,y- five di.stinct kinds of
u2
AD
is as m.,med eqC1al to
then
2
one can estimate
7.3
Quadrallel
The coefficients of the genetic components of variance in the
expected mean squares are complex functions of the numbers of grandparents.
The order of fitting of
ponding mean squares.
as most reasonable.
D
and
AA
The fitting of first
2
D
affects the corresthen
AA
2
was adopted
This analysis does not offer the possibility of
combining of mean squares for the estimation of variance components as
was possible for the triallel analysis because mean squares in the
analysis are not avai lable for the two types of addi ti ve by addi ti ve
or additive by dominance effects.
It is reasonable, however, when
analyzing fixed effects, to combine the mean squares for dominance and
additive by additive, two-line effects to give a mean square corresponding to two-line effects.
Combining
mean square for three-line effects;
square for four-line effects.
Q,AA.4.
Q/l.AAA4
3
and
gi ves a
ar..d
QDD
4
, a mean
The analysis then separates one-line,
two-line effects corrected for one-line effects;
corrected for one-line and two-line effects;
thr'ee-line effects
and fouY'-lirle effects
corrected for one, two, and three-line effects.
Wi th two exceptions ttere is an exact corn;spondence between the
sums of squares for this analysis and those of Rawlings and Cockerham,
1962b, Table 13.
Reversing the order of fitting of
D
and
AA
2
36
effects gives the identical sums of squares of Ha-wlings aEd
Cockerham,1g62b,
S2
= QAA2
T
2
the order of fitting effects is
Table 13
= QD' , the prime indicating that
AA
2
, D .
Relation of SllJnS of squares of Rawlings and Cockerham,
1962b, to those of the general model for quadrallel
crosses
Description
l-line
Sum of Squares
Rawlings and Cockerham
1s;62b
Sum of Squares
GcnE.;roal Model
G
2-line 2-alleles
QD + QAA
2
QAA'
2
QD'
3-line 3-alleles
Q,AD
3
4-line 4-alleles
With a restricted genetic model, Rawlings and Cockerham were able
to estimate six genetic variance components,
CJ
2
c¥
2
ot:i
CJ
,
2
()er6 '
A corresponding variance component is estimable for
each of these in the analysis presented here.
In the analysis
37
presented here we are able to estimate a seventh variac':::e component,
2
(J'AAAA
However, Rawlings and Cockerham could have estimated a
4
corresponding variance component,
, had they not restricted
their genetic model.
7.4
General Discussion
The primary purpose of developing the analyses of variance for
diallel, triallel, and quadrallel crosses was to demonstrate how the
hybrid sum of squares would be partitioned if a uniform genetic model
was used in all three analyses.
This use of a general genetic model
for the development of the parti tioning of tb,e various hybrid sums of
squares is in contrast to previous use of design models for each of the
analyses.
The sums of squares were developed by successively fitting
a more complex genetic model so that each line in the resulting
analysis of variance is corrected for previously fitted effects.
partitioning developed can be used in three ways.
The
with no assumptions
concerning population structure) the stuns of squares can be used to
test for fixed effects.
This use would be hel.pful in analyzing crosseS
of elite lines where assumptions of random mating and of no selection
are seldom tenable.
1961,
vJi th the asslunptions given by Cockerham, 1954,
covariances of relatives can be related to genetic variance
components and the analyses presented here can be used to estimate and
test these genetic variance components.
Finally, a new set of genetic
variance components can be defined in terms of tte gF;neral genetic
model used in developing the partitioning of the hybrid
Stun
of squares
and these can be estimated, tested, and related to previously used
genetic variance components.
38
The variance components defined and used iG tLese analyses are
directly related to previously used variance components for addi ti ve
and dominance effects.
It:!.s in the epistatic variance components
that the two analyses differ; the previously defined epistatic
variance components are partitioned into variance compotwnts that
reflect the nwnber of lines contributing effectf;.
additive by additive genetic variance component,
For examp le, the
2
,
of the
o.a
standard analysis is divided into an additive by additive, one-line
component
2
(JAA
2
(J
and an additive by additive, two-line component
AA
(J
1
.
The one-line component arises from interactions of alleles
2
between loci, but between the genes contributed by one line.
The two-
line component arises from interactions of alleles between loci and
between genes of two lines.
have adapted
AA
effects from
AA
l
2
It could be argued that adapted lines
effects, giving some reason for separating
effects.
AA
l
The other epistatic components are
parti tioned similarly.
The correlations
between the addi ti ve deviations,
between the dominance deviations,
ex, and
13, of Rawlings and Cockerham,
1962a and 1962b, are directly related to the coefficients of the
genetic variance components used in expreffiing the expectations of covariances of relatives.
If .lines used in constructing hybrids are
completely inbred, summing coefficients of components of genetic
variance wi thin a category gives the corresponding correlation of
Rawlings and Cockerham when their
Ct
is multiplied by two.
1:1 trle analysis of diallel crosses the hybrid sum of squares is
partitioned into two parts, there are two covariances among relatives,
39
and with suitable restrictions there are two genetic variance
tJ2
components,
D
. h t can
,t__a·
bi~
estimated.
If'- the analysis of
triallel crosses the hybrid sum of squares is partitioned into seven
parts; there are nine covariances among relatives, and there are seven
';::J
genetic variance components,
2
tJ~
,
tJ~
,
tJ
2
D
,
iY
2
AA
,
fJ
2
AA 2
1
,
2
AD
,
tJ
2
2
; that can be estimated with sui table res tri c ti ons.
AD
tJ
If
3
variance components wi thin a category are assumed identical, then by
pooling lines in the analysis of variance there are five variance
2
(JAM '
components,
2
(JAD ' that can be estimated.
In the analysis of quadrallel cross hybrids the hybrid sum of squares
is partitioned into seven parts; there are eight covariances
among
relatives, and there are seven genetic variance components,
2
tJD
'
tJ
2
AA
,
2
AM
tJ
,
2
AD
tJ
3
2
3
with sui table restrictions.
,
2
tJAAM
...,
'
tJ
2
DD
"
,
that can be estimated
4
In tbis analysis i t is rlOt possible to
combine variance components within a category as only one variance
component within a category is estimable.
The mi.rimum number of lines necessary for a complete analysis for
each of the analyses is the minimum munber of lines necessary to construct at least one pair of u.r.re.lated hybrids.
four lines
crosses
A, B, C, D,
A x Band
For example, with
it is possible to construct unrelated single
C x D
so that a complete diallel analysis is
possible; for the triallel, six lines are needed; and for the
quadrallel, eight lines are needed.
The minimulll nUlllber of hybrids are
6, 60, and 210 for diallel, triallel, and quadrallel designs, respecti vely.
increase
Additions to the numbers of parerltal lines sampled
the number of hybrids dramatically.
Par
exarnplF~,
adding
40
only one additional line to the three designs increases the numbers
of hybrids to la, 105 and 378.
It is possible that some systematic
sUbsampling of hybrids (partial designs) in the case of the triallel
and quadrallel would be beneficial by allowing a greater sampling of
parental lines without the concomitant increase in total hybrids
required by the complete designs.
One point exemplified by these analyses is the confounding of
genotypic effects with line effects.
In the diallel analysis there
are only one and two-line type effects.
All one-line effects are
completely confounded with additive effects.
Two-line epistatic
effects are combined with dominance effects.
In the triallel analysis
there are one, two, and three-line effects and these show up in the
different mean squares of the analysis of variance,splitting mean
squares that would correspond to the usual epistatic variance
components.
In the quadrallel analysis there are one, two, three, and
four-line effects and within a category, say dominance by dominance,
the lower-line variance components,
previously fitted categories.
2
2
G"DD
2
'
G"DD
3
' are confounded with
41
8.
SUMMARY
A quadratic analysis of diallel, tri allel, aQd quadrallel hybrids
is pTovided using a general genetic model.
Sums of squares are
developed by fitting successively additive, dominance, additive by
additive, etc. effects.
In the fitting process, the standard
epistatic variance components are split into categories indexed by the
number of lines contributing alleles to the effect.
standard additive by additive variance component,
For example, the
Cf
2
ot:X
, i s split into
2
CfAA ' with numerical subscripts indexing
l
2
2
2
the number of lines contributing to the effect. Also Cf
2Cf
($X
AA
1
'1
2
(:..
2
2
2
2
+ 2(JAA ; assuming (JAA ::= (JAA ::= (JAA , then (J
- 40
AA
00
2
1
2
For the diallel analysis, the results are pssentially identical to
2
two components, Cf
AA
and
those of the standard analysis (~.. ~., Kempthorne, 1957).
Tb.ere are two
covariances of relatives, two hybrid sums of squares, and two variance
components (if the model is suitably restricted) that can be estimated.
For the triallel anaysis, the results are somewhat different from
those of the analysis of Rawlings and Cockerham, 1962a.
Both
analyses have nine covariances of relatives and seven hybrid
squares.
of
Sl,1illS
With the restrictions on the genetic model used by Rawlings
and Cockerham, 1962a, six genetic variance components can be estimated.
With the analysis presented here, seven genetic variance components
can be estimated; however, some pairs of these components correspond
to the same category of effects in the standard model,
2
correspond to
(J
AA
2
2
(J
0l0I
~'f£..,
2
(JAA
1
'
For tte quadrallel analysis, the results
are similar to the analysis of Rawlings and Cockerham, 1962b.
Both
42
analyses have eight covariances of relatives and seven hybrid sums of
squares.
With the restrictions of Rawlings and Cockerham on their
genetic model, there are six genetic variance components that can be
estimated.
Without their restrictions, seven variance components can
be estimated, which correspond to the seven variance components
estimated in the present analysis,
2
O"AAM
4
'
and
2
O"DD
4
•
222
O"A ' O"D'
O"AA
2
2
'
O"~,
2
O"AD
Genetic variance components can be tested,
3
'
although the tests usually involve linear combinations of mean squares.
Tables of expected mean squares are given and are useful in determining
confounding patterns of the genetic effects.
If the genetic effects are considered fixed, it is possible to
make certain tests of hypotheses without making any assumptions about
the genetic effects.
These tests are discussed.
43
9.
LIST OF REF'ERENCES
Cockerham, C. Clark. 1954. An extension of the concept of partitioning hereditary variance for analysis of c:u"a.Y.·iances among
relatives when epistasis is present. Genetics 39:859-882.
Cockerham, C. Clark. 1961. Implications of gc~Getic variances in
hybrid breeding program. Crop Science 1:1.+'7-52.
Cockerham, C. Clark. 1963. Estimation of genetic variances.
Symposium on statistical genetics and plant breedi.ng. NAS-NRC
983: 53-94.
Cockerham, C. Clark. 1972. Random 'Is. fixed effects in plant genetics.
Paper presented at the Seventh International Biometrics Conference.
Eberhart, S.A. 1964. Theoretical relations among single, three-way,
and double cross hybrids. Biometrics 20:522-539.
Eberhart, S.A. and C.O. Gardner. 1966.
effects. Biometrics 22:864-881.
A general model for genetic
Gardner, C.O. and S.A. Eberhart. 1966. Analysis and interpretation
of the variety cross diallel and related populations. Biometrics 22:439-452.
Gaylor, D.W., H.L. Lucas, wld R.L. ~~de~son. 1970. Calculation of
the expected mean squares by the abbreviated Doo.li ttle and square
root methods. Biometrics 26:641-655.
Griffing, B. 1950. Analysis of quantitative gene action by constant
parent regression and related techniques. Genetics 35:303-321.
Griffing, B. 1950. A generalized treatment of the use of diallel
crosses in quanti tative inheritance. Heredity 10:31-50.
Haymwl, B.1. 1954a. The analysis of variance of diallel crosses.
Biometrics 10: 235-244.
Hayman, B.1. 19541. The theory and analysis of diallel crosses.
Genetics 39:789-809·
Hayman, B. I. 1957. Interaction, heterosia and diallel crosses.
Genetics 42 :336-35~;.
Hayman, B. 1. 1958. 'l'he theory Wid analysb of diallel crosses. II.
Genetics 43 :63-85·
Hayman, B. I. 1960. 'l'he theory and analysis of diallel crosses. III.
Genetics 45: 155-1'72.
44
Kemp thorne, O. 1956.
41: 451-)-+59.
2:'he theory of the dia.llel crOSB.
Genetics
Kemp thorne, Oscar. 1957. An Introduction to Genetic Statistics.
John Wiley and Sons, Inc., New York City, New York.
Rawlings, J.O. and C. Clark Cockerham.
Crop Science 2:228-231.
1962a.
Triallel analysis.
Rawlings, J.O. and C. Clark Cockerham. 1962b. Analysis of double
cross hybrid populations. Biometrics 18:229-244.
Satterthwaite, F.E. 1946. An approximate distribution of estimates
of' variance components. Biometrics Bulletin 2:110-114.
46
Expectations of the mean squares for the triallel analysis
in terms of the covariances of relatives
Appendix I
Mean
Square
Covl
Cov2
Cov3
TA*
r
r(5n-l6)
3n-8
TD*
r
TAAr
r
-
TAA~
r
2r
n3
-2r
-
TAD;
r
-r
rn5
- rn 4
2r
TAAA~
r
2r
-2r
-2r
-4r
TAD;
Mean
Square
r
-r
-2r
r
2r
Cov4
rn3(Sn-16)
4rn3n4
2rn3 (3n-16)
3n-8
3n-8
3n-8
rnS
2rn4(2n-S)
2rn4
3n---a
3n-8
Cov€
rn3ne
TA*
r(n 2 -7n+14)
n3
rn 5
n3
Cov7
rn3n4n4
3n-8
n3
-
2r
n3
_.
3n-8
4r
n3
COVg
2rn3n4ne
rn3n4(n-16)
3n-8
3n-8
2rn4n5
- rn 4
rn4n9
- ----
n3
n3
4r
n3
TAD~
-r
- rn 4
2rn4
- rn 4
TAAA;
-r
r
4r
4r
TAD *
3
-r.
r
-2r
r
TAA~
n3
4rn4
.! (2n 2 -9n+8) rn4(n2-9n+16)
(3n-B)
2(3n-8)
r(n 2 -6n+7)
r
n3
TAA~
~ti1..n -11
2r(2n 2 -11n+16)
(3n-8)
Cov·e
3n-8
rn5
TD*
CovS
2rn4(n2-5n+8)
2rnln4n4
3n-8
3n-8
2r(n 2 -6n+6)
n3
).
Appendix II
Expectations of the mean squares for the quadrallel analysis
in terms of the covariances of relatives
Mean
Square
COVl
QA*
r
2r
r(3n-16)
2r(3n-16)
r
r(n 2 -11n+211
2rn4(n2"9n+24)
2rn4(n2-15n+46)
QD*
COV2
(n 2 -7n+14)
4rn3
(n 2 -7n+14)
2r(n3-12n?~45n -58)
(n 2 -7n+14)
4r(n 2 -11n+22)
(n 2 -7n+14)
QAA~
r
(n 2 -7n+i4)
QAAA;
r
2r
2rnlO
QAD;
r
-r
-rns
QAAAA~
r
2r
-4r
-8r
QDD~
r
-r
-4r
4r
Mean
Square
(n 2 -fn+14)
CovS
COV6
Cove
rnsne
QA*
2rnsne
QD*
rn4ns(n2-11n+42)
(n 2 -7n+14)
QAA~
4r(n 3 -10n 2 +29n-2B)
- .
(n 2 -7n+14)
--2-
rnSn6n16
2
2rn4nS
2rnsne
2
8rn4n;
(n 2 -7n+14)
r(n 4 -16n 3+93n 2 -230n+216)
2(n 2 -7n+14)
2rn4nSn6n9
(n 2 -7n+14)
8r(n2-9n+16)
rnln2n6n9
. (n 2 -7n+14) - (n 2 -7n+14)
QAAA~
- 4rn 3
-rne
- 4rn e
3r (3n-22)
QAD~
-rnS
-rns
2rns
o
QAAAA~
8r
2r
8r
-12r
QDD*
4
2r
2r
-4r
o