ANALYSIS OF DJALLEL, TRrALLEL
AND QUADRALLEL CROSSES USING
A GENERAL GENETIC IDDEL
by
Sidney stanley Young
Institute of Statistics
Mimeograph Series No. 917
April ~974 - Raleigh
iv
TABLE OF CONTENTS
Pag',::
1.
INTRODUCTION • •
1
2.
GENERAL MODEL
3
Genetic Model
Experimental Model
3
11
3.
ANALYSIS OF VARIANCE
13
4.
EXPECTED MEAN SQUARES
19
TEST OF HYTOTRESES FOR FIXED EFFECTS
6.
VARIANCE COMPONENTS AND TEST OF HYPOTHESES •
29
7·
DISCUSSION OF RESULTS
32
7.1
7.2
7.3
7.4
Diallel. • • •
Triallel.
Quadrallel • .
General Discussion
32
32
35
37
8.
SUM:MAR.Y
• • • • • •
41
9·
LIST OF REFERENCES •
43
10.
APPENDIX. • • . •
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, trial1el, and quadra1lel hybrids
have been considered separately by several authors, Hayman, 1954a, b,
1958a, b; 1960; 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, dependicg
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
t
x
and
indexing the alleles.
y
Consider for an
as in Figure 1 with
Using
i, j,
a for additive effects and
x
¥..
i
k
j
Diagram of two loci with indexing of positions
Figure 1
for dominance effects the model for the genotypic effect can be
written as
Notation
Description
Genotypic effect
(a. + a , +
J.
J
d .. )
J.J
=
(additive, a, and dominance,
d, effects for locus x)
+ (additive, a, and dominance,
d, effects for locus y)
+ (additive x additive effects)
d
4
+ (additive x dominance effects)
+ (dominance x dominance effects)
+ (dd .. Lon)
~J~
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
entry under consideration
U .. D.. ,let
~J
~J
~
=2 .
parent.
Then for any
For dominance effects,
be the proportion of genotypes for loci in
5.. (5 .. )
~J
ra.
th
a./2
the
~
~~
the entry with alleles from parents
i
and
j
(i) .
D..
~J
of dominance effects for these genes from parental sources
j.
~6~J'
...
=1
and
= 25 ~~
..
a.
~
+ Z5 ..•
'l:.~J
is the sum
i
and
A general model for an entry
Ji~
as a deviation from the population mean can now be written as
G
= retiA.~
+ ~5 .. D.. + (ret.A.)
~J
+ (~5 .. D.. )
~J
~J
~J
~
~
2
+ (La.A. )(1:5 .. D.. )
~
~
~J
~J
2
34
+ (ra.A.) + (La.A.) + . . . .
~ ~
~ ~
Expansions of the epistatic terms are instructive; for example,
(ra.A.)
~
~
2
2
=!'a'. (M) .. + 21: Eo'.a. (M) ..•
. ~
~~
.. ~ J
~J
~
~<J
The first summation. in the expansion is for addi ti ve x addi ti ve interaction between alleles from the same parent and the latter involves
5
alleles from different parents.
Also note that
(AA)ij
average of two additive x additive interaction effects:
parent
i
i
with
with
x
y
genes from parent
genes from parent
j , and
y
is
fu~
x
genes from
genes from parent
j .
Models for three types of entries, diallel, triallel, and
quadrallel, are now presented.
First consider the entries of a diallel
experiment in which selfs and reciprocals are omitted:
o..
1.J
01.=01.
~
J
=1
G..
~J
= A;
•
+ A. + D.. + (AA) .. + (AA) .. + 2(AA) ..
J
lJ
II
JJ
lJ
+ (AD)i (i j) + (AD) j (i j) + (DD) (i j )( i j ) + .•• •
Next consider progeny of a three-way cross
distinct parents:
G
i (jk.)
= A.
1.
01.
1.
+
=1 ,
~ A
2
j
+
==
~ A
2
k
i x (j x k)
1
°ik - 2 '
+ 1 D
+ ~ D
+ (AA)
"2 ij
2 ik
ii
+
112
2
4 (AA)jj + 4 (AA)kk + 2 (AA)ij + 2 (AA)ik
+
4
+
2 1 1
(AA)jk +"2 (AD)i(ij) +"2 (AD)i(ik)
133
8 (AAA)kkk
+
2 (AAA)iij
+"2 (AAA)iik
(2.1)
with
+
~
(DD) (ik) (ik) +
~
(DD) (ij ) (ik) + ... •
Three locus, all-additive types of interactions are included in the
model since they are to be utilized in the analysis.
}I'inall,y, c:onsider the model for
1
2
and
8ik
=
8i ,t
=
8 jk
progeny of a four-way :-;rc;"
(i x j) x (k x ,t) , O'i = 0' j = Cl'k = Cl',t
from four distinct parents
=
th(~
=
8
jL
=
3
1
4:
+ -8 ((AM) ... + (AM) .. k + (AAA) ..
llJ
II
h
ll~
+ (AAA) ...
JJl
'7
+
81
((
,
()
('
AD)i(ik)
+ AD i(i,e) + AD)j (jk) + ,AD)j (.ji,)
+
~l
((AAAA) ....
~o
2222
I
"
+ (AAAA) .... + (AAAA)kkkk
JJJJ
+ (AAAA),ettt}
+ (AAAA)jjjk + (AAAA)jjjt + (AAAA)kkki + (~!i)kkkj
+ (AAAA)k"
n
Kl~
+ (AAAA)kk' n + (AAAA). '; k + (APA.£J..) ,. ~
J~
JJ-
n
JJ~~
8
1
+
lb [(DD)(ik)(ik)
+
lb ((DD) (ik)(jJ,)
2
+ (DD)(it)(it) + (DD)(jk)(jk)
+ (DD) (it) (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 equilibrilun,
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:
('.
j
Comparing this to tbe model of Cockerham, 1954, where varia.'1ces of a
kind are summed into one term,
Total u
2
G
22222
+ U + u~ + u
o
ao + u 06 + ..• ,
o~
and the translation from one representation to the other is obvious.
For single crosses, G.. , and assuming uncorrelated effects, the
J.J
variance among unrelated single cross means is the total variance.
The numerical subscripts refer to the number of lines involved in a
variance component.
same,
~.~.,
2
(JG
If we let the components within a class be the
2
2
E(AA) .. ,
E (AA) ..
J.J.
2 +
2 (JA
2 +
(JD
J.J
4(J2
AA
2
(JAA
or
2
(JAA
+ 2 2 +
(JAD
2
+
(JDD
' then
2
1
8(JAM
2
+ ••• •
The variance among three-way cross means, which is not the total
variance, is
+.28
2
(JAA
+
2
L
+ 1
(J2
8 AD
2
"8
+ ••
(J2
AD
+
3
~
1:
8
2
(Jr,D
~
:::>
The mllllerical subscripts distinguish among 1, 2, and 3 line
effecL-~.
Again if the components are the same within a category
=~ 2
2
(JA
+
1: 2
2
(JD
+
9
'4
2
(JAA
+
3 2
4'
(JAD
+ 1
4'
2
27 2
DD + 8 (JAM + .•.
(I
2
0
10
Finally, the variance among four-way crosses is
1 2
2
.1 2
(JA + 4" (JD +
(JAA
4"
.1
+-
32
2
DD
O'
3
1
2
+ 64 (JAAAA
If the
corr~onents
+
1
2
1
IJ
b4
DD
21
1
+
4
3
4"
(J 2
AA
+
2
1
2
+Ib (JAM
2
+ 64
1 (J L
"8 AlJ
2
+ 9
16
3
+
(TAM
2
6
°AAAA
+
3
2
36 2
2
,~
+
C'AD
"8
1
+ 6L~ (JAAAA
2
.1
6
16
u
0
MA_A4
+ 64 cf
')
1
cr
f
t4
_.
T"j:',
.L~J2
2
AAA
3
+ •...
within a category are the same, then
+ .... •
Note that when components in a category are equated) the entirco
variance, whether it be for total, single crosses) three-way
(::ross·c~s
or double crosses can be generated from the coefficients 01'
2
C'D'
d.nd
Organizing the variance components into categories reflecting tb.e
number of contributing lines affords a convenient way of summarizing
the kinds of effects involved in quadratic forms even tho-ugh th,,;
effects are viewed as fixed effects.
Linkage affects the coefficients of the epistatic component-s'Nklen
there is control over the grandparents; for example, in a four-'.Nay
cross
(i x j) x (k x
iJ ,
without some recombination of ger~e8 'iii thi n
a chromosome or reassortment of chromosomes there can be no
(DD)(ij)(k.£,)
component and 8.11 of the dominance x dominance i"t~c'~­
actions would be of tile
(DD) (ik) (ik)
type.
With recombirlatiu r
reassortment, dominance x dominance interactions of the
(DD)(ik)(i.£,) and (DD)(ik)(j.e)
become possible.
ty};E:
With free
arl:]! C~·
11
recombination the coefficients for triallel and quadrallel crosses
are those given in
(2.1)
and
(2.2).
Linkage does not affect the co-
efficients of addi ti ve 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
m
is the effect of replicate
m
G[ 1
is the genotypic effect of cross
e[ Jm
is the random error associated with cross [J
replicate
=
1, 2,
[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)(~) ,repsectively.
[J
All line indexes,
becomes
ij,
i, j, k, t
3, ... , n , have the same range, where n is the total
number of lines, except that they must be distinct for each hybrid.
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)(kt).
is the summation over all reps of hybrid
Y
is the summation of all quadrallel crosses with grand-
(ij)( •. ).
parental cross
i x j
over all reps.
(ij)(kt).
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
lJ .•
parents
~.~.,
is the summation of all four-way crosses involving grandi, j,
and
k
summed over all reps;
crosses involving lines
regardless of how the grandparents were mated,
Y..
lJ ..
i
is the stunmation of all three-way
and
j
summed over all reps.
The
summations with parentheses removed can be calculated as simple
SQ~S
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
of combinations of
n
n-i , and
things taken
The total number of hybrids is
3.C~ = nnln~2
The
en to denote the number
K
K at a time.
e~:=: nn/2
for three-way crosses, and
way crosses, where reciprocals are ami tted.
for single crosses,
nn n 2n /8 for fourl
3
The factor of three for
3.C
4
=
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
3.
ANALYSES OF VARIANCE
In each analysis the S"wms 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
AA.A;;'
AD , AAAA l , AAAA2 ,
3
~,
indicating additive effects;
A, D, AA , AA , AD , AAA , AAA ,
l
2
2
2
l
AAAA 4, DD2' DD ,
D,
and
3
domin~lce
DD4, wi th
A
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
vari~lce
for diallel crosses is given in Table 1-
The hybrid sum of squares is broken into two parts, additive
dominance.
Table 2.
The analysis of variance for triallel crosses is given in
The hybrid sum of squares is broken into seven
parts, TA, TD, TAA , TAA , TAD , TAAAy
2
1
2
a.l1d TAD
3
.
variance for quadrallel crosses is given in Table 3.
of squares is broken down into seven additive
QAAAy
~ld
QADy
QAAAA4' and
QDD4 •
addi ti ve
The analysis of
The hybrid sum
parts, QA, QD, QAA2'
14
Table 1
Analysis of variance for progeny of a diallt:d cross,
selfs and reciprocals excluded
Replicates
Sums of Squares
df
Source
2
r-1
nnl m . 'm
nnl
Crosses
1.
( - - 1)
r
2
Additive
Dominance
2
y2
_ _2_ y2
ij'
rnn2
>:: >::
'<'
1
2Y •••
---mUl
J
1
mz
DA
=
DD
=1
>:: >::
r i<j
4
-mnz
y~j
1.
nnl
Error
(Crosses x
Replicates)
Total
(r-1) (_.-
2
mnl
-2- -.1
- 1)
DE by difference
_
-l- LY~
mZ;....
1 ••
=able 2
of variance of
An~lysis
df
Source
Correctlon
factor
crosses
tr~ee-way
Sum of Squares
2y 2
c
1
....
rooloZ
2
2IT ••• m
m
Replications
(r-l)
R
- C
f1..IlJ:'1
2
2: 2: 2:
i j<k
Crosses
n
( 3C 3-1)
H
Additive
°1
TA
Domioance
nn3
-2-
TD
.i,kli
1
rn2 (3n-8)
One-line
2:
i
____ L L [Y
2 rn 3 i < j
4
[2Y.
+Y.
1( .. ) • . (1.).
.
J2
+y
i (J • ) •
j (i .) .
]2 _
1
2rn2n3
16
-,' Y
,-
'\'
~
i
+y
:~. v
..•.1.
2
• •
•
(i
)
••
12
j
y2
rnl02n3
2
°1
- c
l'
+
Add. by Add.
y2
i elK)'
TAA 1 = rnn2n3(3n 8)
2:
i
r
l
Y
.
J Z' . rn 2 n 3 2( 3n _ 8)Y 2••••
n 4 v• i ( .. ) . -n2 . (i . .
) .
t-'
\J1
(Continued)
TabJe 2
. -.. -
-----~._
«~"'~-'-'"-'
-
======:================================-======-==
-
nn3
Add by Add
Two-line
1
TAA 2
2
~ ~
rnln3n4
[n3 Y .(ij).+Y!U.).+YH!.).J2
~<J
1
rnln2n3n4
Add by Dc:"
n1 n 2
Two-line
2
Add by Add by Add
Three-line
Add by Dom
Three-line
.1_
2rn3
TADZ
r
6
nn2 n 4
r
3
n
( r - 1 X3 C 3 - 1 )
"!'ot~J
Or C~-l)
'
~
~ r r r y2
TAD
Error
i<j <k
=
TE
i
j<k
[2Y i ( .. ).+ n2 Y .(L).J2+
[ Y.
i(].).
TAAA3 =31 r r r Y?jk
OOIOS
"
l: 1:
i<j
L.
i
-
1
•
i(jk).
3rn4
Y
j(L).
L..E
i<j
] 2 -
2
Yi j
_1:_
2rnri
L.[ 2Y .
3i
..
+
3
y2
2
~( ..
2
L.
rn3n4 i
,
).
2
Y1
...
.
Y (.
-
\~.
).
J2
2
2
Y ••••
rn 2n 3n 4
-TA-TD-TAAI-TAA2-TAD2-TAAA3
j,k,ti
By difference
T .. I:Z221: y2
i
j<k m -i(jk)m - C
j,k#
------_._-_. - - - - - - - -
----------- ----------------
~
Table 3
Ana~sis
of variance of four-way crosses
df
Source
Correction
Factor
Sum of SCJ.uares
1
8y2 .•.••
C .,
rnn1n2n3
(r-l)
R ..
8 .E y2•••• m
Replications
Crosses
3C~-1
m
nn1 n 2 n 3
-
C
H=!.E.E.E.E y 2
_
r i<j k<t
(ij) (kl) •
C
i,jfk,t
i<k
2
Additive
n1
QA .,
Dominance
nn3
-2-
QD" _1_2
rn2n3n4
[.E y2
_ !6 y2. . ••• ]
i
i· . ..
n
~_--,-.n[ ~~~i.)(j.).
4
n2
J.<J
Add. by Add.
Two-line
nn3
-2-
2
1
2
2
- ill ~ \ . . . .
+
J.
nn1 n S
-6-
yL ... J
QAA 2 .. -- __ (_L,,_--,-,,-, [n1 n 2 ~<~ {(n -7n+14)y(ij)( .. ).+
2
2 n 3 Y (1. ) (j • ) . }
Add. by Add. by Add.
Three-line
.E y2
+ 1.L
i
i ....
n1 n 2
1
QAAA 3 .. 3rn6
.E.E.Ey2
i<j<k ijk •. -
6
rn3n4n6
4
3rn4n6
32
.E yi •.•• i
l:
8y
2
l: y2.
+
iJ ••.
i<j
y? ...
rn2n3n4n~
~
Tabl.e 3
(Continued)
Add. by DOJa,
nD2. u 4
Three-line
~3~-
QAD 3 -
_1_
TD3
~1J
.. )(k. )
•
r r!
i<j):.
-
1
3rn3
1: 1: n2
_
i<j<k ijk ••
4
rUIuS'
i,jl-k
2
I : Y~ij)( .. )- rninS
~
I I y2
i<j
(L)(j.).
+
4
1: 1: y2
3rDlo3 i<j
ij •••
1<J
Add. by Add. by Add. by
Add.
Four-line
nnlo2 D 7
24
1
2
1
2
QAAAAl, .. ! I: I: 1:Y • , - - 1: 1: l:Y
3r i<j<k<. iJki:. 3rn6 i<j<k ijk..
2
rnltDSn6
Dom. by Dom.
Four-line
nnl°:.DS
(r-l) (3 C~-l)
Total
3r CIt-l
D
!
QDD 4
_
QE
by difference
12
Error
!
i
~
l: Z l: L:
i<j k4
i,jl-k,i
_ QAAAAl, i<k
y~ •••• +
+
20l,
E Ey2
..
3rDl,DSo6 i<j ~J'"
2
y2
rnSn4DSD6
Y~ij)(k£). - QA - QD -
QAA2 -
QAAAS -
QADS
r
'.J: .. L: l: l: l: L:
i<;j k.d m
y2;
(ij ) (k..t)m
- C
i, jf.k,t
i<k
b,
19
4.
EXPECTED MEAN SQUARES
Three methods were used in obtaining the expected mean squares
for the three analyses.
4, were
Those for the diallel, Table
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
Additive
Dominance
E(MS )
2
2
2
+ cr2 ) + rn (cr2
(cr + 2cr
+
2cr
AD
2 A
DD
e
AA2
D
2
2
2
2
2
2
2
)
cre + l' (cr + 2crAA + 2crAD + 'cr
DD
D
2
2
cr
2
+
l'
:+
cr2
)
AA
l
2
Error
cr
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 and quadrallel analyses.
First, the covariances of genetic
effects of three-way, Table 5, and four-way, 'rable 6, hybrid relatives
were defined and their eXIJectations obtained in terms of components of
genetic variance.
Next the expectati ons of the uncorrected products
and squares of sums were obtained in terms of
covariances of relatives.
IJ.
2
,
2
Cf
l'
,
cr
2
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 5
Covariances of the genotypic effects of three-way hybrid relatives and their expectations in
terms of components of genetic variance
Coefficients of Variance Component
Number of
lines common
Covariance'"
2
O"AAA3
C1
63/32
3/8
1/8
17/64
42/32
3/8
1/16
0
0"2
A
0"2
D
2
O"AAl
2
O"AA2
2
O" AD 2
C1 2
C1 2
C1
9/8
5/8
1/8
66/64
1
1/4
1/16
AD3
AAA1
2
AAA2
2
DD2
C1 2
DD3
1/8
E[Gi(jk)Gi(jk)]
3
3/2
1/2
9/8
Cov2 • E[Gi(jk)Gj(ik)]
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
30J32
0
1/16
0
COV4 • E[Gi(j_)Gj(i_)]
2
1
1/4
1/2
1/2
1/4
0
16/64
24/32
0
1/16
0
E[Gi(j_)G_(ij)]
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
= E[Gi( __ )G i
( __ )]
1
1
0
1
0
0
0
64/64
0
0
0
0
Cova • E[Gi( __ )G_(i_)]
1
1/2
0
1/4
0
0
0
8/64
0
0
0
0
1
1/4
0
1/16
0
0
0
1/64
0
0
0
0
Covl
CavS
Cov7
COVg
=
=
=
E[G_(i_)G_(i_)]
*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 components of genetic variance
Covariance*
Nu.ber of
lines COllman
Coefficients of variance component
2
OA
02
D
oZ
AAI
oZ
AAZ
Cl
Dz
°iD3
oiAAI
oiAAz
oiAA]
oiAAAI
ohAAz
oiAAA]
ohAA.
o~Dz
21/64
9/16
3/32
o~D]
o~D.
1/64
1/32
1/64
1/128
0
1/128
0
Covi • E G(1j)(kl)G(lj)(kl)
&
1
1/4
1/4
3/4
1/8
1/8
1/16
9/16
3/8
1/64
Covz - E G(lj)(kl)G(lj)(kl)
4
1
1/8
1/4
3/4
1/16
1/16
1/16
9/16
3/8
1/64
21/64
9116
3/32
COY] • E G(1j)(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
Cov• • E G(1j)(k_)G(1k)(j_)
3
3/4
1/16
3/16
3/8
1/32
1/64
3/64
9/32
3/32
3/256
21/128
9164
0
1/256
0
0
COYs • E G(l_)(j_)G(i_)(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
Cove· E G(1j)( __ )G(lj)( __ )
2
1/2
0
1/8
1/8
0
0
1/32
3/32
0
1/128
7/128
0
0
0
0
0
COV7· E G(lj)( __ )G(1_)(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· Z G(1_)( __ )G(1_)( __ )
1
1/4
0
1/16
0
0
0
1/64
0
0
1/256
0
0
0
0
0
0
*Dashes indicate any lines not common in the two relatives.
I\)
.....
The expectatioDs of products and squares of sums in terms of
of three-way cross relatives
Table 7
:::U:i:~
('
,..
Sc;:uare-d
rtcduct
-----~_.
___
1_ y2
t"no!nZ
t>. '" •
~,
v2
, i ( .. ) .
L
rOIn?
- -1y -?
rnln2
.(L).
1
Y
rOl02 Y i (..). . (L ) .
-.L
roZ
y2
i(j.),
_1_ y2
r02
.(ij).
1
y
y , 1.C
r02
J.) • • (ij) •
1.
r
v2
·i(jk),
•.__
.'.
~
.. ·_·C. _._". _ _ _
(r1l 2 +o 2 )
r
02
COy,
001 n 2
---4-
1
2
r
°1°2
--2-
.
Coefficients of Covariance
Cov3
Cov4
CovS
Cov6
Cov7
Cova
Cov9
rn3
I
r
rn 3
r03
2rn3
-2-
r0304
--4-
r0304
r0304
1
r
0
2r03
0
0
0
0
0
°1°2
1
r
r
r03
0
2rn3
r03
0
0
r03n4
°1°2
--2-
0
0
r
0
r03
r03
0
0
r0304
0
°2
1
r
0
rn3
0
0
0
0
0
0
°2
1
r
0
0
0
0
rn3
0
0
0
°2
0
0
r
0
r03
0
0
0
0
0
1
1
r
0
0
0
0
0
0
0
0
4r03
4r03
-3-
8r03
2r03
-3-
r0304
2r0304
2rn3n4
2r03
-3-
4rn3
rn3
3
3
0
0
r0304
3°1°2
2
1
__
1_ y2
3r02
ij. ,
3n2
1
r
2r
L
3
1
r
2r
y2
ijk.
2
2
cr and the covariances
e
r'
,7
Cov2
e
2
y2
31'o1n2 i .•.
3r
~L2J
r
2r
3
2r03
3
0
3
0
2
6
3
3
0
0
0
0
0
0
I\)
I\)
'laoJ.-e b
The expectations of products and squares of sums in terms of ,..l2, f-l2, (j2 and the covariances
' t"lves
r
e
of f our-way cross reia
Coefficients of Covariance
Sum. Square;j
or 1 roduc ':
64 _
r-nr.., 1 n'~ :15
y~
~
'i2
r~nln2~.13 "·i
4
(ru 2 +a 2 ) 0 2
________________________r=-_-=e=--
••
--
1
- ij .. _
yL
3rn3 -ij!<"
I
v2
rn2nS - ( L ) ( j . ) .
].
y
rU2nj
-(ij)( .• ).
4
--rnznZJ
,r 2.
i
"(ij)( .• ).
.)
-_..::- Y""
(ij)(k.).
rD3
1
r
COV2
Cov3
COVIt
Covs
COV6
COY?
Cova
_
nnln2 n 3
8
8r
16r
32rn4
64rn4
32rnlj. TI S 8rnltnS
32rnl+nS
164nl+115n6
TI1TI2n3
2
2r
4r
6rnlt
12rnlt
4rnl+nS
rnl+nS
4rnl+nS
rnl+nSn6
902i13
6
6r
12r
12rnl+
24rnl+
4rnltnS
rnl+nS
4rnl+nS
3u3
1
r
2r
rnl+
2rnlt
0
0
0
0
n2 n 3
1
r
r
2rnlt
2rn4
rnltnS
0
0
0
n2 n 3
0
4rn4
0
0
rnl+nS
0
n2 n 3
2
2r
0
4ru4
0
0
rnl+nS
0
0
n3
1
r
0
rnll
0
0
0
0
0
1
r
0
0
0
0
0
0
0
?
y~
rn2!:3
Covl
2
Y(ij)(kl).
y
(L)(j.).
0
2r
0
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 wld II.
These
types of results a.re 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 Nas used to check the expected mean
squarES 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 Dooli ttle method.
Expectations of the mean squares of three-way crosses in terms of components of genetic
variance
Table 9
Components of Variance
Mean
Square
2
°e
(J~D3
aiD
. __
8
3
oiD 2
OiAA3
OiAA2
aiA2
a2
OiAAI
AAI
{0~+lta~D2}
02
A
Coefficients of Components of Variance
;p~*
1
r
r(11n-32)
16(3n-S)
TD*
1
r
r(3n-ll)
16n3
HAt
1
r
TAA~
1
r
rn2
TAD~
TAlA:';
"
TAn"
4 (3n-8)
rnZ
1
r
1
r
1
"
8n3
r
3rn3
rn2(5n-12)2
r(41n 2 -217n+288)
16 (3n-8)
3r(101n 2 -562n+784)
32(3n-8)
r (7n-20) 2
8 (3n-8)
3rn4
- 4n 3
9rn3
16
3r (3n-l0) 2
16n3
r(2n-7)2
4n3
0
rnn3
9rnn2n3
rnn2n3
4 (3n-8)
32 (3n-8)
8(3n-8)
rU2D3
3rn
8(3n-8)
3rn
8n3
16
0
r
9r
"4
rn2(9n-20)2
(3n-8 )
r (3n-8)
0
3rn3 (5n-8)
16(3n-8)
3rnln4
rnln ...
3zn3
8ti"3
rn3
3rn3
~
16
64 (3n-8)
16 (3n-8)
0
r(3n-8)
--4-
rni3n-S)
4
rn3
-4-
"8
r
Ib
'l'E"
" Sum of squares divided by its degrees of freedom.
I\)
\.11
Tabl'
M'~au
Squa.re
QA*
QD*
QAA~
QAAA~
QAD~
QAAAAI;
QDD~
Expectations of the mean
variance
, 1-:'
.LV
squ~ces
of four-way crosses in terms of components of genetic
Components of Variance
02
02
e
criAAA4
DL4
9r
r
32
3:<
r(3n2-ZSn+S4) 3rn 4 "s
lZ8(n Z-7r.+14) '1'6'(-n"2--"7-n+7'1T4 ')
r(n 2 -Sn+S)
64(n z ·-7n+14)
r
32
r:
128
r
32
r
°iD3
{2
32
l{2
3 2
7 2
°AAA3+ZoAAAA3
°AA 2 +t°AAA 2 +16oAAAA2}
3rn4
Z7rn4
9rn2n 3
---rG
-3-Z·
-3-Z-
r(3n 3 -39n 2+17n-Z6S)
3Z(n Z -7n+14)
3rn4ns
2
2
2
rnttnS
S (n Z -7n+14)
16(n z -7n+14)
3rr.l n 2
rn2nS
3rnln2nS
rnln2n4nS
32 (nZ-7n+14)
16 (n L 7n+14)
16(n Z-7n+14)
32(n Z-7n+14)
rn6
9rn6
16
32
9r
32
0
{ 2 1 2
1
2
°D+ZOAD2~DD2
rn3n4
-S-
1
2 1 2
1
2
-cr
2
(oA+4oAAl~AAAl+~AAAA.
r02n3n4
--S--
r(n 2 -7n+14)
16
rn3
64
9r
32
0
ill
QE*
I\)
0\
27
5.
1
TESTS OF HYPCTHESES :FOR :FIXED Eli FEI,;TS
Certain tests of hypotheses are available wi thout making any
assumptiom; about the genetic effects.
The mean square
expectations
in Tables J+, 9, and 10 in -chis case serve only as guides to the
types of effects that can contribute to the mean squares; the mean
squares actually involve quadratic functions of these types of
effects.
F
The error mean square can be used as the denominator in an
ratio testing sequentially up each table.
Table IJ. 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
, wid
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 summation of allelic effects and may sum to zero when
allelic effects are present.
Also, the quadratic functions of a
particular type of genetic effect differ from mean square to mean
square so the conclusion that certain quadratic funct.ions are zero in
one mean square does not gll-arantee that 'the
mean square.
S8.mt'~
is true in another
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 ther'e is no corresponding mean
square for that analysis
DIALLEL
TRIALLEL
QUADRALLEL
A
Ai
Ai
A.1.
J)
D
ij
D
ij
D
ij
AAI
AA
AA2
AA
AD2
ADi(ij)
AAA3
AAA
AD3
ADi(jk)
it
ij
ijk
AAij
MA.1.)'k
ADi(jk)
A.AAJ,4
AAAAijkR.
DD 4
DD(ij) (U)
29
6,
VARIANCE COMPONENTS AlIJD TES'rS OF :rr.{POTHESES
By assuming the effects of the genetic model are random, and uncorrelated, and that there are common 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,
2
G and
A
estimate
In the diallel it is possible to
2
2
G ; in the triallel,
D
2
G ,
A
G ,
D
222
' GAAA ' and GAD ; and in the quadrallel,
22
2 3
2
3 2
2
GAA ' GAAA-. ' GAD ' GAAAA ' and GDD . Other ·rariance
2
~
3
4
4
components defined and given in the tables of expected mean squares
GAD
u
__
are confounded with these estimators although not always in a simple
manner.
The diallel analysis gives a good example of simple patterns of
confounding; all one-line variance components are completely confounded and estimated as one package,
2
GA
+
2
' an.d all two-li ne
AA
(J
2
1
2
2
G + 2G
+ 2G
D
AA2
AD 2
An example of a more difficult confounding pattern can be
variance components are estimated together,
o
seen in the triallel analysis for
manner.
2
and
2
in the following
AD":J
DD3
Two types of additive by dominance, three-line variance
G
G
components are distinguished, depending on how the effects come together in taking expectations.
and
E[ADo~JJ.k)* AD.!. (j~)} = u:'
AD3
Let
2 2 2
E(AD. ('k)·} = E(AD. ('k)} = (J°AD
~ J_
~.J..
1
' the underscore indicating whether
the grandparental source of alleles is the same,
G
2
lAD3
,or
3
30
2
(J
2AD3
do; ff'er"'nt
- v,
~
In
2
(J
1AD
the underscore emphasizes that
3
2
E[AD. ('k)} - E[AD. ('k) .• AD, (ok'}
and that i x j
J
~\J )
2
parental cross referenced by i and j . In (J
~ ~
was the grand-
~
the effects,
AD
2 3
i x j
AD.~ ('k)
J_,
and
AD;!;. (.sik )
and
i x k.
confounded with
2
(JDD
3
and
2
0'AD
come from different grandparental sources,
,
With this distinction made,
2
(JDD
is completely
3
(J2
The distinction, otber than to show that
lAD3
are confounded, does not appeal' useful so
is
3
2
2
(J
and both are termed OM .
AD
2 3
3
Any variance component or sum of variance components that can be
assumed equal to
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
(JAD
0
3
With each analysis restricted to terms in
TAD)/TE* , and Q,DDt/QE* provide exact tes ts for
2
and (JDD = 0 • Composite hypotheses, testing
4
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
TAD*clTE*
tests the hypothesis that
3rn
2
3 (J
+ --.16
AAA.
o.
2
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,
(.Ea MS )
i i
is approximately dis tributed as
2
X (J2I f '
with
fl
degrees of freedom where
(6.1)
31
and
f.1
denotes the degrees of freedom for mean squares
MS.
1
Using
Satterthwaite's approximation, error terms can be constructed to test
each of the components of variance.
For p-xample, in the triallel
analysis
4T~ - 3TE*
has expectation
2
r
2
+ - ~
e
4 AD
~
,the correct expectation of an error
3
~~ in T~
-)\-
term for testing the significance of
and can be
used to form an approximate F-test
4T~ - 3TE*
with degrees of freedom
from (6.1).
nn n /6
l 5
and
f'
where
f'
can be obtained
32
7.
DISCUSSION OF RESULTS
7.1 Diallel
Since truncation of the general model to add.iti 'fE; and dominance
effects corresponds exactly to the usual model for general and
specific combining ability, the partitioning of the sums of squares is
identical.
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.
The 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 Tri allel
Examination of the expected mean sq"LlareS indjcates that whereas
the genetic model is simple in concept and interpretation the
expected mea.YJ. squares are complex.
It can be seen that the order of
fi tting of dominance and addi ti "ve by addi ti ve, single-Hne effects is
innnaterial.
This is also true for addi ti ve by addi ti V2, two-line
effects and additive by dominance, two-Jine 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.
sequentially up the resulting
(~+
TAD )
3
In testing against error
three-line, two-line, and
~lalysis,
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.
2
If it is assumed that
and
~AA
2
are identical,
~AA
2
1
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.
If it is assumed that
~~D and ~~ = 0 , it then becomes possible to
2
by manipulation of the mean squares.
1
n
3
T.AM - TE*} •
2
The point is that by assll.rning genetic variance components wi thin 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
Table 12
Relation of sums of square[; of Rawlings and Cocke:rham,
1962a, to those of the general model for ~riallel crosses
Sruns of Squares
Rawlings and Cockerham
1962a
Description
l-line
G + °1
2-line 2-alleles
S2 + ° 2
2-line 3-alleles
°2
3-line 3-alleles
of Squares
General Model
S1..L.lllS
TA + TAA
l
TD + TAA
a
TAD
b
2
2
TAAA
3
S3
TAD
°3
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.
\Alhen the genetic variance components
were restricted to the seven lowest order ones, it wa[: found that
there was a li near dependerlcy in the seven reSUlting equation.s so that
only six genetic variance components,
2
IT
01
and
IT
2
Q1Q1Q1
,could be estimated.
2
c"o '
2
IT0'0
'
In the design presented here, seven
35
2
AA
genetic variance components can be estimatL-'Cl,
(J
1
'
222
2
'
(JAD '
(JAM , and (JAD ,but only five distinct kinds of
2
2
~
3
variance components. Again, if (J~D2 is asswmed equal to
then
(JAA
one can estimate
7.3
2
(JDD
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.
and
D
AA..
The fitting of first
2
D
affects the corresthen
AA
was adopted
2
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 available for the two
or addi ti 'Ie by dominance effects.
t~Tes
of additive by additive
It is reasonab.le, however, when
analyzing fixed effects, to combine the mean squares for dominance and
addi tive by addi ti ve, 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.
QAA~
..J
QAAAA
4
and
gives 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
effe(~ts;
thr'ee-line effects
and fom'-line effects
corrected for one, two, and trrree-line effec.ts.
Wi th two exceptions t.Lere is an exact correspondEnce between the
sums of squares for this analysi s and those of Ra'iVlings and Cocke:cham,
1962b, Table 13.
Reversing the order of fitting of
D
and
AA')
Co
36
effects gives the identical sums of squares of' Ha-wlings and
Cockerham, 1962b,
S2 =
Q),A2
QD' , the prime indicating that
the order of fitting effects is
Table 13
AA
2
, D •
Relation of sums of squares of Hawlings and Cockerham,
1962b, to those of' the general model for quadrallel
crosses
Description
I-line
Sum of Squares
Rawlings and Cockerham
19S2b
Smn of Squares
Gener"al Model
G
QA
QD + QM
2-line 2-alleles
2
Q),A'
2
QD'
3-line 3-alleles
QAD
3
4-line 4-alleles
Q,AAAA4
Wi th a restricted genetic model, RawUngs and Cockerham were able
2
to estimate six genetic variance components,
eve '
(J
A corresponding variance compoGent
each of these in the analysis pres"3nted here.
lS
estimable for
In the analysis
37
presented here we are able to estimate a seventh yariance component,
2
(JAAAA
However, Hawlings and Cockerham could have estimated a
4
corresponding variance component,
2
(J
,had they not restricted
01010101
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 partitioning of the 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.
parti tioning developed can be used in three ways.
The
With no assumptions
concerning population structure,the sums of squares can be used to
test for fixed effects.
This use would be helpful in analyzing crosses
of elite lines where assumptions of random mating and of no selection
are seldom tenable.
With the assumptions given by Cockerham, 1954,
1961, covariances of relatives can be related to genetic variance
components aJld 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 the ger.eral genetic
model used in developing the partitioning of the hybrid sum of squares
and these can be estimated, tested, and related to previously used
genetic variance components.
38
The 'farianee components defined and used iG these a.r1alyses are
directly related to previously used variance components for additive
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 components that
reflect the number of lines contributing effects.
additive by additive genetic variance component,
For examp le, the
2
()
, of the
ao
standard analysis is divided into an addi ti ve by addi ti ve, one-line
component
2
and an additive by additive, two-line component
()AA
1
2
The one-line component arises from interactions of alleles
()AA
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 additive deviations,
between the dominance deviations,
(X,
and
0, of Rawlings and Cockerham,
1962a and 1962b, are directly related to the coefficients of the
genetic variance components used in expreming the expectati.ons of covariances of relatives.
If .lines used in constructing hybrids are
completely inbred, summing coefficients of compot'_ents of genetic
variance wi thin a category gives the corresponding correlation of
Rawlings and Cockerham when their
1:1
ex
is multi.plied by two.
tt"e analysis of diallel crosses the hybrid .sum of squares is
partitioned into two parts, there are two covariances among relatives,
39
and with sui table restrictions there are two genetic variance
2 ,'vhat
t(JD
can be estimated.
components,
Ie 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
2
(JA
genetic variance components,
2
(JD
,
2
(JAA
,
.'
1
2
(JAM
2
(JAD
,
3
2
(JAA
,
,:)
2
(JAD
,
2
; that can be estimated with sui table restrictions.
J..
If
3
variance components within a category are assumed identical, then by
pooling lines in the analy'Sis of variance there are five variance
2
(JAM '
2
(JD '
components,
2
(TAD' 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
relatives, and there are seven genetic variance components,
2
(JD '
2
(JAA
,
2
(JAAA
,
2
(JAD
2
3
3
with sui table restrictions.
2
(JAAAA '
,
4
2
(JDD
,
among
2
(JA '
that can be estimated
4
In this analysis i t is not possible to
combine variance components within a category as only one variance
component within a category is estimable.
The
mi~imum
number of lines necessary for a complete analysis for
each of the analyses is the minimum number of lines necessary to construct at least one pair of unrelated hybrids.
four lines
crosses
A, B, C, D,
A x Band
For example, with
it is possible to construct unrelated single
C x D
80
that a complete diallel analysis is
possible; for the triallel, six lines are needed; and for the
quadrallel, eight lines are needed.
Tt.e minimum number of hybrids are
6, 60, and 210 for diallel, triallel, and quadrallel designs, respecti vely.
increase
Addi tions to tlle numbers of parental. lines sampled
the number of hybrids dramatically.
Fa!' examp lp., adding
40
only one additional line to the three designs increases the numbers
of hybrids to 10, 105 and 378.
It is possible that some systematic
subs amp ling 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,
2
~DD
3
previously fitted categories.
' are confounded with
41
8.
SUMMARY
A quadratic analysi oS of diallel, triallel, and quadrallel hybrids
is provided using a general genetic model.
Swns of squares are
developed by fitting successively addi ti ve, dominance, addi ti ve by
additive, etc. effects.
In the fitting process, the standard
epistatic variance components are split into categories indexed by the
nwnber of lines contributing alleles to the effect.
standard additive by additive variance component,
2
two components, cr
AA
For example, the
cr
2
CiCX
, is split into
2
cr
, with numerical SUbscripts indexing
AA
1
2
the number of lines contributing to the effect.
2
and
222
2
cr
~ cr
= crAA ' then cr
AA
AA
eta
2
2
l
For the diallel analysis, the results are t~ssent:ially identical to
; assuming
+ 2crAA
those of the standard analysis (~:~., Kempthorrw, 1957).
There are two
covariar::.ces of relatives, two hybrid swns of squares, arId 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 sums of
squares.
With the restrictions on the genetic model used by Rawlings
and Cockerham, 1962a, six genetic variance compo:'1ents can be es timated.
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
cr
AA 2
correspond to
cr
2
~.£..,
2
crAA
1
'
For the quadrallel analysis, the results
CiOl
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
(J'AAAA
4
' and
2
(J'DD
4
•
222
(J'A ' (J'D'
(J'AA
2
2
'
~'
[J
2
(J'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.
Lls'r O£il REFERENCES
CockeI'ham, C. Clark. .l95~·. An extension of the concept of parti tioning hereditary variance for analysis of cO"Ia..r-iances among
relatives when epistasis is present. Genetics 39:859-882.
Cockerham, C. Clark. 1961. Implications of gCG.-:;tic variances in
hybrid breeding program. Crop Science 1:4'7- j2.
C
Cockerham, C. Clark. 1963. Estimation of genetic variances.
SymposiUlll on statistical genetics and plant breeding. NAS-NRC
983: 53-94.
Cockerham, C. Clark. 1972. Random vs. 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. a.YJ.d S. A. Eberhart. 1966. Analysis and interpretation
of the variety cross dialle1 and related populations. Biometrics 22:439-452.
Gaylor .. D.W., H.L. Lucas, and R.L. Ande~son. 1970. Calculation of
the expected mean squares by the abbreviated Doolittle 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.
Hayman, B.l. 1954a. The analysis of variance of diallel crosses.
Biometrics 10:235-244.
Hayman, B. I. 1954b. The theory and analysis of diallel crosses.
Genetics 39:789-809·
Hayman, B. I. 1957. l.nte~action.,
Genetics 42:336-355.
hete~osis
and diallel crosses.
Hayman, B. I. 19:)8. The theory and analysis of diallel crosses. II.
Genetics 43 :63-85.
Hayman, B. 1. 1960. 'The theory and analysis of diallel crosses. III.
Genetics 45: 155-1'{2.
44
Kempthorne, O. 1956.
41:451-459·
The theory of the diallel cross.
Genetics
Kempthorne, Oscar. 1957. An Introduction. to Genetic Statistics.
Jot1n Wiley and Sons, Inc., New York City, New York.
Rawlings, J. O. and C. Clark Cockerham.
Crop Science 2:228-231.
1962a.
Trial1el 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
in terms of the covariances of relatives
Appendix I
Mean
Square
ana~sis
Covl
COV2
Cov3
TA*
I'
r(5n-16)
3n-8
rn3(5n-16)
3n-8
4rn3nl+
3n-8
2rn3 (3n-16)
--3n-8
TD*
r
rn S
-
TAA!
I'
-
r(n 2 -7n+14)
°3
2r(2n 2 -11n+16)
(3n-8)
TAA~
I'
f..!.
°3
-2r
Covl+
rn S
n3
2rnl+(2n-5)
3n-8
2rnl+
3n-8
-
Covs
~U3n-lll
n3
4rnl+
3n- 8
2r
n3
- 4r
n3
TAD;
I'
-r
rnS
-rnl+
2r
TAAA~
I'
2r
-2r
-2r
-4r
-I'
-2r
I'
2r
I'
TAD;
Mean
Square
TA*
Cov6
rn3ne
30-8
Cov,
rn3nl+nl+
3n- 8
rns
TD'"
°3
TAA!
TAA~
~(2nL9n+82
(3n-8)
r (n 2 -60+7)
°3
-rnl+
Cove
2rn3nl+Oe
3n-8
2rn4nS
rn3nl+(n-16)
3n-8
rn4n9
n3
2rnl+(n2-5n+8)
rOI+ (n2-9n+16)
Covg
- --n3
r
~.!.
n3
2rnlnl+nl+
3n-8
2r(n 2 -6n+6)
n3
2(3n-8)
3n-8
TAD!
-r
-rnl+
2rnl+
-rnl+
TAAA;
-r
r
4r
4r
-r
r
-2r
r
'I<
l'AD 3
----
Appendix II
Mean
Square
Covl
QA'"
r
Expectations of the mean squares for the quadrallel analysis
in terms of the covariances of relatives
Cov2
COva
2r
r(3n-16)
2r (3n-16)
r (n L 11n+2ll
(n 2 -7n+14)
2rn4(n2"9n+24)
2rn4(n2-15n+46)
QD'"
r
QAA~
r
(n 2 -7n+14)
QAAA~
r
2r
QAD;
r
-r
QAAAAe
r
2r
-4r
-8r
QDDe
r
-r
-4r
4r
4rna
Mean
Square
QA'"
QD,r,
QAA~
(n 2 -7n+14)
2r(na-12n?~45n -58)
(n 2 -7n+14)
(n 2 -7n+14)
4r(n 2 -11n+22)
(n 2 -7n+14)
-rns
Covs
2rnsne
rn4ns(n2-11n+42)
COV6
rnsne
--2- 2
2rn4nS
(n 2 -7n+14)
2
+29n-28)
2 -7n+14)
- 4r(n a(n-10n
COV7
2rnsne
2
8rn4nS
(n 2 -7n+145
-
r(n 4 -16n 3 +93n 2 -230n+216)
2(n2-7n+14)
-
Cove
rnSnSn16
2
2rn4nSn6n9
(n 2 -7n+14)
rnln2n6n9
(n 2 -7n+14)
~ill2-9n+16)
(n 2 -7n+14)
QAAA!
- 4rn a
-rne
- 4rn e
3r (3n-22)
QADt
-rns
-rns
2rnS
0
QAAAAe
8r
2r
8r
-12r
QDDe
2r
2r
-4r
0