•
THE UNION OF PARTIAL DIALLEL MATING DESIGNS
AND INCOMPLETE BLOCK
~JIR01~IENTAL
DESIGNS
by
MELVIN OLE BRAATEN
A thesis submitted to the Graduate Factuty
of North Carolina State of the University
of North Carolina at Raleigh
in partial ftllfillment of the
requirements for the Degree of
Doctor of Philosophy
DEPARTI'~T
OF EXPERll\lliNTAL STATISTICS
RALEIGH
1
965
APFHOVED BY:
Chairman of Advisory Committee
•
iv
TABLE OF CONTENTS
Page
LIST OF TABLES
vi
LIST OF FIGURES •
viii
1.0
INTRODUCTION
1
2.0
REVIEW OF LITERATURE
5
2.1
General
5
7
2.2 The Partial Diallel
3.0 DESIGN
3.1
12
General
12
13
3.2 The Partial Diallel
3.3 The Blocked Partial Diallel •
17
3.3.1 Method A-Construction of a Blocked
Partial Diallel when t
=0
(mod 2) •
=1
(mod 2) •
3.3.2 Method B-Construction of a Blocked
Partial Diallel when t
3.3.3 Method C-Construction of a Blocked
Partial Diallel Using a $ymmetric
Latin Square
3.4 The Blocked Modified Diallel
18
19
20
22
3.5 The Blocked Disconnected Diallel
25
4.0 ANALYSIS OF VARIANCE
26
5.0 ESTIMATION OF a g2 AND a 2s
6.0 EFFICIENCY OF DESIGNS
34
6.1
6.2
39
General
• • • • • •
Efficiency of Designs for Estimating a 2 •
g
6.2.1
6.2.2
6.2.3
Unblocked Designs
Blocked Designs •
Critical Values of p
39
40
40
46
c
That Make
Blocking Effective
Efficiency of Designs for Estimating a 2 •
s
6.3.1 Unblocked Designs
50
v
TABLE OF CONTENTS
(continued)
Page
mocked Designs
Comparison of mocked and
Unblocked Designs •
52
52
7.0
GENERAL DISCUSSION
54
8.0
SUMMARY AND CONCLUSIONS •
64
9.0
LIST OF REFERENCES
66
APPENDICES
67
10.0
1001 A Partial Diallel is Not Necessarily
Superior to an "Experiment II"
10.2 Inversion of a Symmetric Circulant Matrix
10.3 Properties of the Mean Squares
10.3.1 The Independence of Mean Squares
100302 EXpectations of the Mean Squares
10.3.3 Variances of the Mean Squares
67
68
71
71
76
76
vi
LIST OF TABLES
Page
2.1.
Analysis of variance of a partial diallel
3.1.
Design methods applicable for blocked partial
diallels
18
A blocked partial diallel with q = 13, s = 6
and b = 3
19
4.1.
Analysis of variance of a blocked partial diallel
30
4.2.
Analysis of variance of a blocked disconnected
diallel •
32
3.2.
9
Estimators and variance of the estimators for
general combining ability variance, ~2
35
g
Estimators and variance of the estimators for
specific combining ability variance, ~2
s
36
6.1. Minimum and Maximum values of pa for which the
efficiency for ~2, Eup(i,j), = 1 for T from 48
g
to 1152 plots and asymptotic values for T
6.2.
~m
41
Segments of the spectrum of p where a partial
a
diallel with s matings per parent is superior to
all other partial diallels for estimating ~2
g
42
Efficiency for estimation of ~2 from an unblocked
g
partial diallel with i matings per parent compared
to an unblocked partial diallel with j matings,
Eup(i,j)
6.4.
44
Efficiency for estimation of ~2 of an unblocked
g
modified diallel relative to an unblocked partial
diallel with s matings per parent, Eup(*,s) •
45
Maximum and minimum values of p where the efficiency
2
a
for 0 , Ebp(i,j), ::;; 1 for T from 48 to 576 plots in
g
blocks of 12, 24 or 36 plots and asymptotic values
for T -.> m
46
vii
LIST OF TABLES
(continued)
Page
6.6. Maximum and minimum values of p where the efficiency
a
for ~2,
Ebd(i,j),
=
1
for
T
from
48 to 1152 plots
g
in blocks of 12, 24 or 36 plots and asymptotic
values for T
-.;>
47
co
Values of p
above which a blocked disconnected diallel
c
is superior to an unblocked disconnected diall.el with
the same q and s for estimation of ~2
g
6.8.
49
Range of' efficiency for estimating c1'2 of a blocked
g
partial diallel compared to a blocked disconnected
diallel for the range of p where the designs are
a
optimum
50
Asymptotic ranges of Eup(j,i), the efficiency of an
unblocked partial diallel with s = j relative to
an unblocked partial diallel with s = i, for
estimating ~2
s
6.10.
51
Range of efficiency for estimating
2
~s
of a blocked
partial diallel compared to a blocked disconnected
diallel over the range of p where the designs are
2
a
optimum for (J"
g
6.11.
53
Values of Pd above which a blocked partial diallel
is more efficient for ~2 than an unblocked partial
s
diallel
53
viii
LIST OF FIGURES
Page
=8
1.1.
Three possible designs for q
2.1.
A partial diallel with q
= 11
and s
=4 .
3.1.
A disconnected diallel with q
= 18,
s
3.2.
A partial diallel with q
= 14
and s
=5 .
3.3.
An (8x8) symmetric Latin square with the letter A
on the main diagonal
3
=3
8
and b
=3
A blocked modified diallel with q
14
16
3.4. Mating array for eight parents.
3.5.
•
21
22
= 13
and b
=3
•
24
1.0
INTRODUCTION
For the estimation of genetic variances the quantitative geneticist
is confronted with two major design problems.
One is the develollment of
a mating system Hhich allows tmbiased and efficient est1.mation of the
genetic components of variance from covariances of relatives.
The other
is the construction of environmental designs allowing efficient estimation of these vnriBnce components without introducing bias.
These
two problems are not .independent since the estlrr.ators and the variances
of the estimators are functions of batt. the mati.ng and environmental.
design parameters.
The diallel cross .• a mating system first discussed by SClll;Uftt
(1919), has been used. successfully for evaluating genetic variances.
':['11e
theory of the analysis of dl1311elE has been extensively deveJoI)cd,
and various modi.fi.cations of the dj.aI1el mating
desi~n
have been made
which lead to more efficient estimators of the general and specific
combining ability variancef;.
The mating designs primarily considered in this work include the
modified diallcl and fractions thereof
0
rrhe term modified diallel will
refer to the qql/2 possible crosses between q parents or parental inbred
lines where red.procnT crosses are not made (ql is used to denote q-l,
and in general, q.,_1
:=
q-i)
0
Incomplete ma"ting designs i-There tIle I'arents
are not mated J,n all possible combinations Hill be called partial
diallels.
A third class of designs, the diGconnected diallels, are a
subset of the partial diallels.
In the disconnected diallelG the
parents are first partitioned into groups of equal size.
'l.1:W11 the
2
parents
1J2
thin a group are mated according to either a modif:Led
part:h·.l dial1el design.
c;1'
a
Figure 1.1 gi.ves an example of each type of
mating design for eight parents.
Even for a moderate number of parents,
these designs requi.re very large block sizes if a randorrdzeCi complete
blocks envirorunental desi.gn is to De used
Methods are suggested to
partitlonthe crosses from a partial, modjfied or disconnected diallel
into groups which may be grown in incomplete blocks when size of block
is res-cricted.
The designs developed haVE! analyses whIch are not excessively
cumbersom.e; yet J they allowed estimation of the genetic components of
variance free of block effects.
The desj.gns were evaluated for
efflciency of the est:imators f'or the genetic parmneters a
2
.
and
g
0
2
,
s
the general and specific combining ability variances, respectively.
The analysis of the proposed designs was based upon the following
model:
Yijkt ;:;: 11 + gi
t-
gJ + Sij + b k + rJ, + rbk,t + eijk.t '
vfuere the phenotype of the progeny of the cross (ixj) or (jxi) grown in
the kth incomplete block of tlw .eth replicate is denoted by y,
"1 II'
~JK~
Il-
an effect common to all crosses,
gj = an effect due to the ith parent,
Sij
= an
effect due to the cross (ixj),
b
= an
effect due to the kth block or group of entries,
r
rb
k
t
kt
_. an effect due to the .tth replication,
= an
effect due to the kth block or group of entries in the tth
replication an.d
3
Figure 1.1.
1
2
3
4
Three possible designs for q
2
3
4
5
6
7
8
X
X
X
X
X
X
X
1
X
X
X
X
X
X
2
X
X
X
X
X
X
X
X
X
3
4
X
X
X
X
X
5
6
X
7
5
6
7
=8
X
2
3
4
5
6
5
6
X
3
4
X
X
X
X
X
5
8
X
X
X
X
X
X
X
X
X
Partial dia11el
2
7
X
Modified dia11e1
1
4
3
2
6
7
8
X
X
X
X
X
X
X
7
Disconnected dia11e1
4
e
ijkt
=a
random deviation or the observation on the (ixj)th cross in
the kthblock or the tth replication.
The components of this linear model; g's, s's, b's, r's, rb's and e's;
are assumed to be independent random variables with zero means and
variances;
2
~elb
2
,
g
~
~
2
s,
2
~b'
~
2
r,
2
r b and
~
~
2
e Ib; respectively.
The component
is the plot to plot variance within a block where the plots in a
replication are allotted to b blocks.
This notation is used to
emphasize the fact that the within block variance may not be constant
but related to size or block.
5
2.0
REVIEW OF LITERATURE
2.1
General
The "Method of' Diallel Crossing" or the "Method of' Complete
Intercrossing" was first introduced by Schmi.dt (1919).
Two types of
mating patterns were included under the general topic of diallel
crossing.
The f'irst case, which is frequently called testcrossing,
involved complete intermating of one group of males with another group
of females.
The second case discussed by Schmidt was the mating of' each
individual in a group with all otller individuals of that group.
If' the
members of' the group are inbred lines, this complete crossing may be
accomplished even though each individual cannot perform both as a male
and as a female.
Four dif'ferent types of diallel sets based upon the inclusion or
exclusion of' the q parents and/or the qql/2 reciprocal crosses were
described by Grif'fing (1956b).
(1)
Parents, one set of Flvs and reciprocal Fl's are included
2
[all q
(2)
The four possibilities were:
combinations].
Parents and one set of FI'S are included but reciprocal FI's
are not [q(q+l)/2 combinations].
(3)
One set of Fl's and reciprocals are included but the parents
are not [qql combinations].
(4)
One set of Fl'S but neither parents nor reciprocals are
included [qql/2 combinationsJ.
Griff'ing (1956a) used the term II modif'ied diallel" to designate those
diallel sets where the parents were not included, and it will be further
6
restricted here to mating designs which satisfy the conditions of
Griffing's Method 4.
Sprague and Tatum (1942) applied the analysis of variance technique
to the analysis of the qql/2 single crosses from a set of q lines.
They
gave the following definitions for general and specific combining
abilities and defined their variances:
l
liThe term 'general combining ability' is used to designate
1I
the average performance of a line in hybrid combinations.
liThe term 'specific combining ability' is used to designate
those cases in which certain combinations do relatively
better or worse than would be expected on the basis of the
1I
average performance of the lines involved.
The estimators of the general and specific combining ability variances
were determined by equating the mean squares to their expectations.
The mean square for within line groups was shown to be a function of
the specific combining ability variance and of the error variance.
Griffing (1956b) gave the analysis of each of the four previously
mentioned types of diallel mating designs under the assumptions:
.........,.,
(1)
that the variety and block effects are constants and
(2)
that they are random variables •
These two classes of models have been designated as Models I and II,
respectively, by Eisenhart (1947).
The progeny using Griffingls Method 3 were fitted into an
incomplete blocks design of the split-plot type by John (1963).
of the progeny from a single female are placed into a block.
1
Sprague, G. F. and L. A. Tatum. 1942.
combining ability in single crosses of corn.
923-932.
All
Sums of
General vs. specific
Amer. Soc. Agron. 34:
7
squares for general combining ability, specific combining ability and
reciprocal effects were given.
2.2
The Partial Diallel
An experimenter using the modified diallel finds that even \Ii th a
modest number of parents, the number of crosses required increases very
rapidly with q.
In the modified diallel the number of necessary crosses
is qql/2 single crosses.
Several authors have suggested methods for
reducing the total number of crosses while still maintaining a
relatively large number of parents.
Kempthorne and Curnow (1961) suggested that greater efficiency
could be attained for the estimation of the variance of general
combining ability 'by using a partial diallel which alloY,s a greater
number of parents to be evaluated.
It was also noted that the partial
diallel permits selection among crosses from a wider range of parents
and allows the estimation of the general combining ability effects of
a larger number of parents.
It must be noted that each parent would be
measured with a relatively lower precision, but larger genetic gains
may be realized because a more intensive selection pressure could be
applied.
Sometime around 1948, according to Kempthorne (1957),
Dr. G. W. Brown suggested the following procedure for the construction
of a partial diallel:
(1)
Take a large random sample of q individuals from an inbred
population, numbering them from 1 to q.
= 2k,
(2)
Obtain s crosses with each parent where q-s-l
say.
(3)
Cross line x with lines x+k+l, x+k+2, .o.,x+k+s (mod q)o
8
This procedure results in qS/2 crosses.
An example of a partial diallel
with 22 progeny from 11 parents mated 4 times each is given in Figure
2.1.
The matings in this partial diallel are chosen according to
Brown's procedure.
Figure 2.1.
A partial diallel with q
2
1
2
3
4
=
11 and s
=
4
11
5
6
7
8
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
3
4
5
6
9
10
X
7
The analysis of this design was given in detail by Kemptllorne and
Curnow (1961).
In addition to the least squares estimators of the
general combining ability effects, they worked out the analysis of
variance table together with the expectations of the mean squares
assuming Eisenhart's (1947) Model II.
The general analysis of variance
for designs generated by the procedure suggested by Brown is given in
Table 2.1.
Gilbert (1958) proposed ttJ.e following use of a (qxq) symmetric
Latin square with a single letter on the main diagonal for the
construction of partial diallels for Which q
(1)
=0
(mod 2):
Superimpose the Latin square on the (qxq) array of crosses.
9
(2)
Select s letters from the Latin square and make all crosses
corresponding to the selected letters which lie above the
main diagonal.
This procedure assures that each parent is involved in an equal number
of matings.
Furthermore, Gilbert suggested that if q ::: 0 (mod
4)
one
should use a Latin square symmetrical about both diagonals.
Table 2.1.
Analysis of variance of a partial diallel
D~grees
Source
of
Expectation of
mean squares
freedom
Reps
r
Gee
ql
Sea
qs,.j2
2
2
CJ'elb + rCJ's
Reps x crosses
r l (qs-2)/2
"c
CJ'elb
l
rq2s 2
2
2 +- CJ'
rO"
+
O"e/b
s
g
ql
c
Another procedure for developing partial mating designs was given
by Hinkelmann and Stern (1960).
The crosses to be made are determined
by numbering a sample of q parents from 1 to q.
parent x+l+pd, x
= 1,2, ••• ,q;
Parent x is mated to
p ::: 0,1,2, ••• with x+l+pd
characteristic of the design.
=q
and d is
The requirement that each parent be
mated the same number of times together with the parameter d leads to
the further requ~rement that q2
=0
combinations for a given design is
(mod d).
The total number of cross
10
q2/d
T, where
T
=
(ql-~d) = q(q+d~2)/2d
L
,
1,=0
This procedure assures that each parent is involved in s
= (q+d~2)/d
crosses.
F,yfe and Gilbert (1963) suggested two general classes of designs
which were shown in some cases to be superior for estimation of general
combining ability effects to the circulant designs;
s:.s:"
th.e Kempthorne
and Curnow (1961) and the Hinkelmann and Stern (1960) designs.
lit r i angul ar"d es~gns
.
.
requ~re
q
=
~/
2 parents.
design (1) each of the q parents is mated s
The
In their triangular
= n2n3/2
times, and in its
compliment, triangular design (2), each parent is mated s = 2n
The "factorial" designs require q
=
(3) each of the q parents is mated s
compliment, factorial design
mn parents,
=
2
times.
In factorial design
IlJ.nl times, and in its
(4), each parent is mated s = m+n-2 times.
The least squares and inverse matrices are given in terms of m and n
for all four of these designs,
Several designs which possess the property of having only two
variances for comparing the general combining ability effects are
tabled in Curnow (1963).
It was pointed out in this paper that
Kempthorne and Curnow's (1961) analysis of variance holds for all
partial diallels which have the same number of parents, q, each mated
s times.
Kempthorne and Curnow (1961) found that a partial dia1.1el with the
same number of plots was always superior to an
Ii
Experiment r
fl
design of
Comstock and Robinson (1952) for estimating both general and specific
11
combining ability variances.
Furthermore, the covariance of the
estimators of ~2 and ~2 was less negative for the partial diallel.
g
It
s
was further stated that an experiment analogous to an
II
Exper~ment
0
II
II
design of Comstock and Robinson (1952) could be shown to be inferior to
a partial diallel with q equal to the number of sires and s equal to
twice the number of dams.
In Appendix 10.1 it is shown that this
last statement is not necessarily true.
12
3.0
301
DESIm~
C~neral
The size of block required to accommodate the
to~al
number of
crosses necessary to adequately estimate the geneti.c components of
variance is generally larger than is desired.
A reduction in error
variance can be effected by fitting the crosses into an incomplete
block design, but there will be some loss of genetic information.
A
particular method of blocking is considered effective if the reduction
in the plot to plot variance adequately compensates for this loss.
The modified diallel, partial diallel and disconnected diallel
mating designs are fitted into incomplete block designs.
However, the
use of an arbitrary incomplete block environmental design will not
assure that the estimators of the variance components will be efficient.
Incomplete block designs having the expectations of all general
combining ability sum of sqQares free of the block variance,
2
~b'
can
be constructed by reqUiring that each parent is involved in an equal
number, t, of crosses within each block.
adjustment of the general
combin~ng
effects by merely subtracting the
This condition allows the
ability
SQm
Stun
of squares for block
of squares due to block contrasts.
Both the modified diallel and the partial diallel lend themselves to
fractionation into
incomple~e
blocks so that general combining ability
and block contrasts are orthogonal.
The disconnected diallel can be
readily fitted into an incomplete block design by confounding block
differences with the differences between the small partial or modified
diallels of which the disconnected diallel is formed.
An example of a
13
disconnected diallel which is composed of three small diallels is given
in Figure 3.1.
with q
=6
and s
The small diallels in this example are partial diallels
= 3.
The general design vector is (r, T/2, b, q, s, t) where:
r
= number
of replications,
T
= number
of plots in 2 replications,
b = number of block.s in each replication,
q
= number
of parental lines,
s = number of times each parent is mated in the design and
t
= number
of times each parent is mated in each block..
The experimental designs considered have the following design vectors:
Unblocked modified diallel, UMD,
(r,
qql/2, 1, q, gl' ql)
Blocked modified diallel, BI,m, (r, qql/2, b, q, ql' ql/b )1/
Unblocked partial diallel, UPD,
(r,
qs/2, 1, q, s, s)2/
Blocked partial diallel, BPD, (r, qs/2, b, q, s, S/b)1/,2/
Unblocked disconnected diallel, UDD, (r, qs/2, 1, q, s, s)3/
Blocked disconnected diallel, BDD, (r, qs/2, b, q, s, 8)3/
Restrictions:
1/
ql = 0 (mod b)
2/ 3
3/
.:s s < ql
3.:s 8.:s (q-b)/b; q = 0 (mod b).
3.2
The Partial Diallel
The following two classes of partial diallels, which include those
suggested by Hiru~elmann and Stern (1960) and Kempthorne and Curnow (1961)
14
Figure 3 .1.
1
1
2
3
4
A di sconnected diallel \vi th q = 18, s = 3 and b
2 3
X
4 5 6
X
X
X
X
X
X
X
7 8
:=
3
9 10 11 12 13 14 15 16 17 18
X
5
6
7
8
9
10
X
X
X
X
X
X
X
X
X
11
12
13
14
15
16
17
18
X
X
X
X
X
X
X
X
X
15
as special cases, were found more adaptable to incorporation into
incomplete block designs than others found in the literature.
When
the number of parents, q, is odd, a partial diallel with s matings
per parent is formed by selecting any s/2 groups of crosses from the
following ql/2 possible groups.
Group
l(t = 1,2, ••• ,ql/2) contains
the crosses;
parent i x parent i+l, i = 1,2, ••• ,q
where (i+t) is reduced mod q.
When q is even, a partial diallel with s
matings per parent is formed by selecting group l
= q/2
and any sl/2
groups of crosses from the q2/2 remaining possible groups.
Group
l(l = 1,2, ••• ,q/2) contains the crosses;
parent i x parent i+l, i
= 1,2, ••• ,q
where (i+t) is reduced mod q.
As an example, consider the UPD design (r, 35, 1, 14, 5, 5) given
in Figure 3.2.
Since q is even, group q/2
=7
and sl/2
=2
other groups
must be used to provide the desired design.
Let us choose groups 1 and
4, say.
= 1,
Repeated use of Equation 3.2 with t
desired design.
4 and 7 will yield the
This design is the special case used as an example in
Hinkelmann and Stern (1960).
When all possible qql/2 crosses among q parents have not been made,
care must be exercised during construction of the designs to assure that
the sum of squares due to general combining ability is reasonably easy
to compute.
For this reason attention will be restricted to mating
designs possessing a circulant least squares matrix for which the
inverse is relatively easy to compute.
Procedures for inverting
16
Figure 3,2.
A partial dia11e1 with q
2
1
2
3
4
5
6
7
8
9
10
11
12
13
3
4
A
5
6
7
B
8
9
A
=5
and s
10
11
C
B
B
C
B
A
lit
B
B
A
13
A
C
B
A
12
B
C
B
A
= 14
C
C
B
A
B
A
C
B
B
A
A
B
A
A
A
17
symmetric circulant rr~trices are given in Appendix 10.2. ~lrthermore,
"'r,C
"2
ttie variances of U and cr are independent of the choice of' particular
g
s
diallel mating des:l.gn if q and s are held clJnstant.
The proof of this
last statement is given in Appendix 10.3.
Some of the designs which nay be derived by procedures outlined
in this section are of no interest because their least squares matrices
are singular.
The folloWing necessary and sufficient condition for
singularity of a symmetric circulant matrix was stated without proof by
Curnow (1963).
IAI
=0
If line 1 is crossed to lines dl+l, d +l, ••• ,d s +l,
2
if and only if for some root wj
s +
'
o .
'ibis implies that q is even and every d
r
is of the form d
r
integer )/21 ,.here I 1.8 SOi.'1le fixed integer between 1 and ql.
= (q
x odd
The
subscripts on the d's in this paragraph are indices rather than
indicators of d-i.
3.3 The Blocked Partial Diallel
A partial diallel (r, qs/2, 1, q, s, s) can be fitted into an
incomplete blocks design to give a (r, qs/2, b, q, s, sib) with general
combining ability contrasts orthogonal to contrasts among the blocks
if s
=0
(mod b).
Again, the reqUirement that each of the q parental
lines occurs t times in each of the b blocks assures that the orthogonality condition is satisfied.
total of s
= bt
times.
Thus, each of the q parents is mated a
Therefore, it is necessary to develop methods
18
for partitioning the qS/2 crosses of the partial diallel into b groups
of qt/2 crosses in such a way that each parent occurs t times in every
group.
q
Wi.th the BPD the smallest attainable block size is C
=0
(mod 2) and C
Now bt
=s
=q
if q
=1
= q/2
if
(mod 2) since t must be an integer.
and both qt/2 and qS/2 must be integers.
These restrictions
lead to six possible combinations of the design parameters.
Applicable
methods for developing incomplete block designs for each of these
conditions are summarized in Table 3.1.
Table 3.1.
Design methods applicable for blocked partial diallels
Design parameters
t
q
s
b
Method a
either
even
even
either
A
even
odd
even
even
B,C
even
odd
odd
odd
B,C
aListed in order of preference.
3.3.1 Method A-Construction of a Blocked Partial Diallel when t
(mod 2)
When t
=0
=0
(mod 2) a BPD having a circulant design matrix can be
constructed by partitioning the qS/2 crosses into s/2 groups of size q
using either equation 3.1 or 3.2.
These groups of crosses are assigned
to the b blocks at random; each block is thus composed of s/2b groups.
As an example of this method of construction consider the design
(r, 39, 3, 13, 6, 2)0
Equation 3.1 is used to determine the crosses in
19
groups 1, 2 and 6.
3 respectively.
Table 3.2.
Groups 1, 2 and 6 are assigned to blocks 1, 2 and
The design is given in Table 302.
A blocked partial diallel with q = 13, s = 6 and b
Block 1
Block 2
Block 3
lx2
lx3
2x4
lx7
2x8
3x5
4Xb
3X9
4xlO
5x7
6x8
5xll
7X9
8xlO
7xl3
8xl
2x3
3x~·
4x5
5x6
6x7
7x8
8X9
=3
6x12
9xlO
lOxll
9xll
9x2
lOxl2
10x3
llxl2
1lx13
llx4
12x13
12xl
12x5
13xl
13x2
13x6
3.3.2 Method B-Construction of a Blocked Partial Diallel when t
(mod 2)
=1
It is conjectured that a BPD having a circulant design matrix can
be constructed for all values of q and s.
(mod 2) and t
=1
When q = 0 (mod 2), s = 0
(mod 2), the crosses which will be used to make up
the design can be chosen as in Method A.
The qS/2 crosses must be
partitioned into s groups of size q/2; each group must have the
property that each parent is mated once and only once in that group.
A systematic procedure has not been found for developing these groups.
20
However, the method of trial and error has been successful for the
author.
In order to construct designs with t > 1, t of the s/2 groups
with each parent mated once are assigned to each of the b blocks.
cases where s
=1
(mod 2), the addition of the group t
= q/2
For
in
equation 3.2 to the groups as developed above yields the desired
partitioning of the crosses.
As an example of this technique, a BPD (r, 18, 3, 12, 3, 1) can
be constructed by letting t
by letting t
= q/2 = 6.
= ~/2 = 5,
and since s
=1
(mod 2), also
Group 6, which is a group with t
the crosses lx?, 2x8, 3x9, 4xlO, 5xll and 6xJ.2.
= 1,
contains
Group 5 contains the
crosses lx6, lx8, 2x7, 2x9, 3x8, 3xJ.O, 4X9, 4xll, 5xlO, 5xl2, 6xll and
7x12.
The crosses in group 5 can be divided into two groups with t
One group can be composed of lx6, 2x9, 3x8, 4xll, 5xlO and 7xl2.
other contains lx8, 2x7, 3xlO, 4x9, 5xl2 and 6xll.
= 1.
The
Each of these
groups comprise a block of the desired incomplete block design.
3.3.3
Method C-Construction of a Blocked Partial Diallel Using a
SymmetriC Latin Square
Should the conjecture that it is possible to construct a BPD with
a circulant design matrix for all permissible combinations of q, sand
b be found untenable, an alternative procedure is suggested which will
divide the crosses into blocks as desired.
However, when q is large,
the design matrix may be very difficult to invert since it generally
will not be nicely patterned.
A symmetric (qxq) Latin square with a single letter on the main
diagonal is superimposed upon the mating array for q parents; q
=
°
21
(mod 2').
A subset of' s of tb.e off dIagonal letters are
s groups 01'
q/;~
to .form
C[l<:):3(.'11
crosses; eac:l1 group tms the property t :;;;; 1.
Blocks of
qS/2b plots are formed by dividing t.t~e s groups equally amonr; the 1)
blocks.
As an example of this met.hod, a BPD (r, 12, 3, 0,3, 1) is
constructed hy superimpos:i.ng th,; (Gx:8) ,~Ynimetric Latin square given
in Figure 3.3 u:pon the rooting design for eight parents
parent InUS1'; oceur once and ofl..ly
onc~~
.in each bloc:k,t.he c::rO'3ses
c.orresponding t.o a letter are ass.i.gned to one of the three blocks.
Block B contains crosses lX2, 3x:5} 4x6 B.nd ~rx8.
mock D contai.:r18
\c:ros~,eG
lx:4~ 2x6.~ )x7
mock H contains
crO:S5ei:~
lx8
Figure 3. '50
j
2x7J
and 5x8.
3x6 and 4x5o
An (8xH) sym.'lletrlc: Latin square with the letter A on the
maln diagonal
...
_""",-.-"-~~..,.,.,...,.,.n
~=.,..-~~-"-=-~,,,,,~_·.~_·
__
__
~.""""""'.""~."'",
A
B
C
D
E
F
G
II
B
A
E
J?
C
D
H
G
C
E
A
G
B
II
D
F
D
F
G
A
II
B
C
E
E
C
B
II
A
G
F
D
:1'"'
D
H
B
G
A
E
C
G
If
D
C
F
E
A
B
H
G
F
E
D
C
B
A
__
.~""
~-=.~
22
Figure 3.4.
lxl
Mating array for eight parents
lx2
lx3
lx4
lx5
lx6
lx7
lx8
2x2
2x3
2x4
2x5
2x6
2x7
2;-<:8
3x3
3x4
3x5
3x6
3x7
3x8
4x4
4x5
4x6
J.mrr
4x8
5x5
5x6
5x7
5x8
6x6
6x'7
6x8
7x7
TxB
8x8
304
The Blocked Modified Diallel
The modified diallel can be regarded as a limiting case of a
partial diallel uhere s
=
combinations ignoring order.
ql)
The q parents are mated in all possi.ble
A modified diallel (r, qQl/2, 1, q, ql'
can be acconnnodated in an incomplete block des1.gn with b blocks
of qql/2b plots in each replication.
This results in a BMD with the
design vector (r, qql/2, b, q, ql' ql/b).
By imposing the restriction
that each parent occurs in each block exactly q~/b times, the general
L
combining ability contrasts are orthogonal to the block contrasts.
This requires that the number of times eacll parent is mated in the
design must be evenly divi sible by the number of blocks;
(mod b).
The blocks of these designs have qql/2b
= qt/2
!. ~., ql
plots each.
The necessary restrictions on the design parameters are thus:
(1)
ql
= bt.
= 0
23
(2)
If q
(mod 2) )' t
::;:
0 (mod 2) since qt/2 must be an integer.
(3 )
If q : : : 0 (mod 2)y t
::
1
== 1
"
(mod 2) since ql has only odd factors.
Thus, t.he smallest attaina.ble size of block)' C, is)'
C
:::
C-
q/2
if q
=0
q
if" q
=
(mod 2) or
1 (mod 2)
Methods A, Band C discussed in Sections 3.3.1, 3.3.2 and 3.3.3
can be used to partition tl1e crosses of a modified dia11e1 into
incomplete blocks by letting s ;.: gl.
As an example using Method A the
BMD design (r, 78, 3, 13)' 12, 4) is shown in Figure 3.5.
The first
block is composed of the crosses in groups land 3 and is indicated
by the symbol
Ii
X'.
The second block contains the crosses occurring in
groups 2 and 5 and is indicated by an
indicated by an
II
II
O!I.
If' to complete the desi,gn.
J
This leaves groups 4 and 6)'
The random assignment of
the q parents to the margins of the design table will result in a
sizeable number of design choi,ces for most values of q.
Likewise,
Method B can be directly applied.
Another procedure, Method C, for dividing the crosses into be equal
size blocks is an extension of the use of a Latin square for constructing partial diallel crosses.
(mod 2) and t : : : 1 (mod 2).
on q and
t
This technique is useful is q : : : 0
It should be noted that the restrictions
do not permit q even and t even or q odd and t odd.
q is odd, a symmetri.c Latin square cannot be found.
When
Gilbert's (1958)
procedure was based on the use of a symmetric Latin square as a tool
for selecting crosses to comprise a partial d1a11e1.
Similarily, this
procedure is used to allocate the qq1/2 crosses to theb blocks.
24
Figure 3.5.
1
2
A blocked modified dial1el with q ::: 13 and b =3
2
3
4
5
6
7
8
9
10
11
12
13
X
0
X
H
0
H
H
0
H
X
0
X
X
0
X
H
0
H
H
0
H
X
0
X
0
X
II
0
H
H
0
H
X
X
0
X
H
0
H
H
0
H
X
0
X
H
0
H
H
0
X
0
X
H
0
H
H
X
0
X
H
0
H
X
0
X
H
0
X
0
X
H
X
0
X
X
0
3
4
5
6
7
8
9
10
11
12
X
To illustrate Method C, the 15 offspring from 6 parents are
divided into 5 blocks with 3 plots each.
The following symmetric (6x6)
Latin square is superimposed upon the mating array.
A
B
C
D
E
F
B
A
D
E
F
C
C
D
A
F
B
E
D
E
F
A
C
B
E
F
B
C
A
D
F
C
E
B
D
A
Now, the gIlt::: 5 blocks will each be indicated by one letter, namely
blocks B, C, D, E and F.
25
Block B contains lX2, 3x5 and 4x6.
Block
C
contains lx3, 2x6 and 4x5.
Block D contains lx4, 2x3 and 5x6.
Block E contains lx5, 2x4 and 3x6.
Block F contains lx6, 2x5 and 3x4.
3.5
The Blocked Disconnected Diallel
The construction of the BDD is straight forward.
The q parents are
first divided at random into b groups of q/b parents each.
The progeny
from matings among the parents within a group are assigned to one of
the b blocks within each replication.
If s
= (q-b)/b,
be made among the q/b parents within each block.
all matings will
However, if s is less
than (q-b)/b, the matings within each block may be made according to any
of the partial diallel designs suggested in the literature.
If all
possible matings are not made among the q/b parents in a block, the
inverse of the least squares matrix must be found within each of the b
blocks in order to determine the sum of squares for general combining
ability.
This points out the need for a careful choice of the within
block mating design.
The same design arrangement may be used within
each block, thereby simplifYing the computer programming problem for
this type of design.
An example of the BDD (r, 27, 3, 18, 3, 3) has already been given
in Figure 3.1 when each group of crosses is assigned to a block.
It
should be noted that this design can be constructed by assigning one
or more small partial diallels to each block.
earlier, if s
= (q-b)/b
As was pointed out
each small diallel would be a modified diallel.
26
4.0
ANALYSIS OF VARIANCE
Independent sets of sums of' squares are developed for each of tlle
blocked diallel designs.
Since the sums of squares developed for
general and specific combining ability are adjusted for block effects,
the estimators of
(J
2
g
and
(j
2
are free of block effects.
s
The sum of squares due to replications, blocks (unadjusted for
geneti.c effects) and replication by block interaction are defined in
the usual way.
The form of the Sllin of squares due to general combining
abi.lity differs for the various mating and environmental designs.
When
all possible qql/2 crosses have been made, the smu of squares due to
general combining ability has been given by Griffing (l956b) and others.
In the blocked modified diallel, where general combining ability is
orthogonal to blocksv and in the unblocked modified diallel the sum
of squares, S , is as follows:
g
(4.0.1)
where a dot indicates summation over the respective subscript.
The analysis of variance of partial diallels requires inversion of
a (qxq) matrix of coeffi.cients in order to determine the sum of squares
The sum of squares, S , is:
due to general combing ability.
S
g
= at A-la
-
g
2y 2
-
where G is the (qxl) vector of
• •••
(4.0.2)
rqs
!r times the total yields for the q
parental lines and the matrix A
= (a .. )
~J
is the (qxq) matrix of
27
coefficients of the general combining ability effects in the least
squares equations for general combining ability.
for all i, a ..
lJ
otherwise.
= a ji = 1
The element a"11
if cross (ixj) is sampled and a' j
1
=s
= 8.,
=0
J1
In the case where the matrix A is a symmetric circulant,
the inverse is relatively easy to find since the inverse is also a
symmetric circulant.
A (qxq) circulant matrix is characterized by
the property that element (i+l, j+l) is e~lal to element (i,j) for all
i and
j
and element (i+l, 1) is equal to element
The elements a
where r
=
ij
(i,q) for all i.
of the circulant matrix A-I may be denoted by a r
li-jl, the absolute value of (i-j).
Formulae for computation
of the elements of A-I are given in equation 10.2.5.
property of symmetry, we further have a r
= a q-r
Because of the
for all q and r.
Thus,
the work of inversion is further reduced; it is only necessary to
compute either (q+l)/2 or (q+2)/2 elements as q is either odd or even.
Since it is usually essential to use equation 4.0.2 to find the
sum of squares for a partial diallel, judicious choice of a design
matrix could possibly simplifY the computer programming and analysis
of these designs.
The sum of squares due to general combining ability in the blocked
disconnected partial is computed within each block according to
equation 4.0.1 or 4.0.2, whichever is applicable, and then is pooled
over blocks.
For the unblocked disconnected diallel, the contrasts
among groups are also general combining ability contrasts.
The expec-
tation of the mean square for groups differs from that for within
28
groups; therefore, the optimum pooling procedure is not immediately
obvious.
This topic will be discussed at greater length in Chapter
5·0.
The sum of squares due to specific combining ability adjusted for
blocks and general combining ability is given for each design.
When
possible, conditions of orthogonality are used to simplify the form
of the sum of squares.
Since blocks are orthogonal to general combining ability in the
blocked partial diallel and in the blocked modified diallel, the sum
of squares due to specific combining ability follows directly.
The
usual sum of squares ignoring blocks is adjusted by subtracting the
block sum of squares to yield the adjusted S :
s
s =1
s
E
r i<j
Y~.
1.Jk.
~y2
1 •• k.
2
- rqt
-
where Yijk • is the total of all crosses of (ixj).
S
g
Note, the
SUbscript k is given for a specific design if i and j are known.
The sum of squares due to specific combining ability in the blocked
disconnected diallel can be conveniently determined by two methods.
1
The unadjusted quadratic form -
b _2
E L
r i<j 1
~jk
1.
•
may be adjusted for the
mean, blocks and general combining ability by either of the following:
S
s
or
Ss
b
1
E E ~jk
r i<j 1 1 . ·
=-
=
1
E
r i<j
b
~ ~jk.
b
2b E y 2
- Sg
qs 1 .. k.
b
l
E ~ A- G
k-k
1
(4.0.3)
where G' is the (q/bxl) vector of ! times the total yields in the kth
-k
r
parental group (block) and A is the matrix of coefficients for the
k
particular mating pattern used in the kth group (block).
If the same
mating pattern is used within each group (block), all ~'s are
identical.
It can be readily seen in equation 4.0.3 that the specific
combining ability sum of squares is determined within groups (blocks)
and pooled over groups (blocks).
The analysis of variance tables for tlle designs discussed herein
are given in Tables 4.1 and 4.2.
Included with each table are the
formulae for computation of the sums of squares and the expectations
of the mean squares.
e
e
e
Table 4.1.
Analysis of variance of a blocked partial dialle1 8
Degrees of
freedom
Source
Sum of
b
squares
Mean
square
Elcpectation of
mean square
Reps
r1
S
r
MS
Blocks
b
f\
~.I\
CTe
Reps x blocks
r1b l
Srb
MS
f'!"2
~ 2
velb + 2 CTrb
Gea
q1
S
MS
Sea
(qS2~2bl )/2
;:,
Reps x crosses/blocks
r1(qs ~ 2b)/2
S
~en s
= qlP
i
g
....
S
e
r
rb
g
MS
MS
S
e
2
Ib
2
~
+ rO"s + 2
2
.
lb
1e
reT
2
e,ib +
reT
CT
CT
2
rqt
CTrb
2 rq2s
s + - ql
+ 2
2
CTb
2
g
CT
2
s
2
elb
CT
this table reduces to the ANOVA for a blocked modified diallel.
'When b
= Ip
this table reduces to the corresponding unblocked design.
Table Continued.
'vi
o
Table 4.1.
b
Where
(Continued)
S
ry2
2
=r
qs
"
ti. b
1
1
g
= -=-r~
S
g
=.....
1
Sa ;:;
r
rqs
q
E (Yo
1
l
- S
b
. ..
J.'
2
2
y
rqs
~
2
)
...Ly2
,
...
rqq2 •••• i-P
-
•• • •
E
= q~.J..
+h
.• o ...~ erw:::.se
rqs
2
YiJk.
0
':)
-
0
"h
u_ 2
E yrqt ~ •• K.
~
1
-
.L
Sg
r
Se =
S
2y2
U' -
iq
••••
r
+ Y
•••
A-.L....
'<"'
- S
•• k~
~
",i
?
-2y -
0 r 2
E E Y
~ ~
J. .J..
y2
.•••
rqs
I)
2 b't'V'+£.0 ...
1
rq ... 1 .. .K.
?
S
...L
••• ~
E
ro = -...
qv
S
Where
e
e
e
8 -8
E";
- S - 8s
J.jk,t
r
-b- g
i<j 1
G
rr. \
G' = (,...
\U"1' 2'···'I..Tq ),
G~ being
..L
1.r
_2..
....2
rqs'"
[total yield of progeny of parent (i)].
\.N
......
e
e
e
Table 4.2.
Analysis of variance of a blocked disconnected diallel
Degrees of
freedom
Source
Sum Of
b
squares
Mean
square
a
Expectation of
mean square
l
S
r
MS
l
~
Ml\
2
2
2..9! 2
rqs 2
0"e Ib + rO"s + 2rsCTg + 2b 0"rb + 2b O"b
Reps x blocks
rIb l
Srb
MSrb
2
as 2
0"e Ib + 2b 0"rb
Gea
q-b
S
g
MS
0"2
+ ....0"2 + r q-2b )6 0"2
elb
~ s
q-b)
g
Sca
qS2/2
S
s
MS
Reps x crosses/blocks
r (qs-2b )/2
l
S
e
MS
Reps
r
Blocks
b
8when b
= 1,
r
g
t
s
2
2
O"elb + rO"s
e
2
O"elb
this table reduces to the unblocked disconnected diallel design.
Table Continued.
\.N
rv
e
Table 4.2.
b
Where
-
e
(Continued)
~
,.,
;)
r
r
?
~
=~.EY-
qs ....1
e
Cl
e
2
?lb ,..,
~
8- -~.E";:
'-b - rqs .. .... ok.
2
b r
2'
Y·b
sg
?
""....e..E L: Y- ,
qS"1 ~
, • Kt
~
,...
~
r
Q
2
~--
.r;
r', q~,::::b) 1
1
i
;.2
b
2"\
rq \ ':l.~ '"'; 1r:
?
."
)'_
(1,
k + Y'k,i
l o.
l
.E
1
sg = ...~ ,]ro 'A..~~G.
K-l{
1 e-=k
y2
rqs
b
b q/b
?'~~
'h
S
..l.
.....
:;::
y2
rqs
.l.,
S.
y2
qs ...
t
0
b
20.r;
2
- rqs 1, v (l.K:
d.
i"
.0
r -,
.. K.
•
\,
\q~D)/';
.
..'
J.l.f
8
=
other'\ifi se
.'1
';I
Cl
"!-o
=
Cl
'h
1
s
'-',;2-
L .E
r i<j 1
~.
f'";l
.,
0=-
"t....
~
'ok.
2"
.. = \, '"':lk
,..,1
~tP
f
-.
:'"i
J
..-::
~
ri
d
g
"Ml
S :;::; Z L~~y2
L.. "",
e
'Kt
'J.;;' <j' ~
<~
<~
""'v ...
..l.
I{nere
~
lr2kJ
"•
f"'1
e
J
,::J
r
\
i k )~
-...7
~
~
"'-".
~,~
~ ":;'
~::..
0~
\~·ik oeing .;:
!"":
"i
'
r
~
~
-
-
22
y
raB
-_'
[total yield of progeny of the ith parent
in the kth block].
\.>1
\J.i
34
ESTD1ATION OF ~2 AND ~2
s
g
Estimators of general and specific combining ability variances are
obtained by equating the mean squares for general combining ability,
specific combining ability and error to their respective expectations
and then solving the resultant set of equations.
~
2
g
and
(J
2
are given in Tables
s
The est:imators for
5.1 and 5.2, respectively, for each of
the designs considered.
If the phenotypes, Yijk.t' are normally distributed, ttle sums of
squares developed for each of the designs are independent.
Hence, the
variances of the estimators of the variance components can be developed
directly from the variances of the mean squares.
The sum of squares for
I)
general combining ability is not necessarily distributed as a
variable.
X~
Nevertheless, its variance is independent of the choice of
mating design for any given combination of q and s.
The proof of
independence of the sum of squares for blocks, general combining ability
and specific combining ability together with a procedure for evaluating
the expectations and variances of the mean squares is given in Appendix
Tables 5.1 and 5.2 give the variances of the estimators of ~2
10.3.
g
2
222
2
and ~s in terms of ~g' ~s' ~b and ~elb·
It should be noted for the blocked designs that the optimum
solution is not realized since the block sum of squares is also a
function of the genetic variance components.
weights were known, better estimators of
~
If the appropriate
2
and
g
by using a weighted least squares analysis.
(J
2
could be developed
s
A similar problem arises
in the unblocked disconnected diallel design where the expectation of
e
e
e
Table 5.1.
Estimators and variances of the estimators for general combining ability variance, a 2
g
Estimator
Design
Unblocked partial
diallel
ql
ra_s(t.iS -~ )
Blocked partial
diallel
ql
rq s(MS -~ )
2
g
s
-2
g
Variance of the estimator
2
2 2
~b +ra s ) (l+Pc
S
r
2
(
~.Ib
)2 2
ql
2 2 2
[..£.- +1qS2
~s
+ 2 2 2
ras) ql
r2~s2
[--2
qS2-2bl
ql
1
+_
2~sPa
+
+
2
2
l
(q+~ s )sP
ql
(q+~ s )sP
2
ql
+
ql
Unblocked modified -!-(MS -MS )
diallel
rq2
g
s
Let s ;: ql in the unblocked partial diallel
Blocked modified
diallel
1 (
rq_ ,MS -MS )
-2
g
s
Let s ;: ql in the blocked partial dial1el
Unblocked disconnected diallel
r~s(MSg-MSS)
Blocked disconnected diallel
(q_b
r(q-2b s{MSg-MSs)b
a
ql
l
2
aJ
2
ql
2q,..sp
+
c:. a
tl
a
2
aJ
Equivalent to unblocked partial dial1el
2
2{ 0- b
2 2
-tr<rs ) {q_b)2
4r
(q_2b)2 s 2
2
[ qS 2 + q:b
+
2( q-2b )sp
[q+( q-4b )s JsP
(q-b) 2
(q-b) 2
a +
2
a
l
oJ
2· 2
2
2
2
2
+ ra )
=g
ra !(a
pa
e Ib' + ra s ) and Pc =~b 1(ae Ib
s
bIgnoring among blocks information.
VI
\Jl
e
•
e
Table 5,2,
Estimators and variances of the estimators for specific combining ability variance, ~2
s
Unblocked partial
diallel
1
-
r
(MS -MS )
1
,
s
e
\ [~Ib + O"~
r
-
(MS ~MS )
Unblocked modified
di~-I 1 el
1
r
(MB
\.l.
_
r
s
e
=;)2
+
(
~ [~!b
2
+
?
r~s)-
+
( 2
2)2
~b· ~b
]
r ( qs-2)
l
qS2
2
Blocked partial
dialJ..el
+
4
+~J
~,
r2
qS2-2bl
-MSe )
Let s
= ql
in the unblocked partial diallel
Blocked modified
diallel
1- ( MS -MS )\
r \ s
e
Let s
= ql
in the blocked partial diallel
Unblocked
di<>1 1 el
::. (MS - r.m
di3ccp~ected
.,
'1
r"
s
e
)
Equi valent to unblocked partial diallel
2
Blocked disconnected
diallel
±
(NS -MS )
r' s
e
4
r
2
[(~elb
+
N
qS2
2
s
~4
?
)+
'
r
elb
1
(qs-2b)
J
Vol
0\
37
the bet\feen groups mean square is rr;lb + rr~ + rrr; + 2rsrr~, and the
expectation of' the within groups mean square for general combining
ability is
If' the exact weights for the mean squares in an unblocked
discon~
nected diallel were known, the least squares estimators would be
r---
I
X '"
V
-
2rs
1
r
1
r
1
r
o
1
0
o
diag(V, )
~
,
,
2(rr2 + ra 2 + 2rsrr2 ).2
s
g
b
=
V
2( rr 2 + rrr2 ,2
q-b
2
s...'_ +
(q~b)2
4( rr 2 + r(l)2
V =
3
s
qS2
4
4rr
V4 = r ( qs-2b)
l
y'
4rs(q~2b)(rr2
= [MSg (between
,
i
+ rrr2 )rr2
S
fL +
2r2s[q+(q=4b)SJd-~
g
(q=b)2
,
and
groups), MSg (within groups), MSs , MSe ]
,
Since the correct weights are not known, an iterative solution
,..
for Qwou,ld 'be expected to reduce the variances of the estimf3.tWJ.
For
the weighted analysis to be appropriate, th.e qS/2 offspring should be
randarrized over all of the plots in each replication.
39
6.0
EFFICIENCY OF DESIGNS
6.1
General
The efficiencies of the estimators for general and specific
combining ability variance components were studied for the designs
developed in Chapter 3.0.
The relative efficiencies of the designs
were evaluated empirically with an IBM 1410 electronic computer.
The 16 sizes of experiments considered in the evaluation of these
designs were 48, 96, 144, 192, 240, 288, 320, 352, 384, 432, 480, 576,
720, 864, 1008 and 1152 plots.
All design comparisons were made for
two replications of the sampled crosses among the parents.
Block
sizes of C equal to 12, 24 and 36 were employed to study the effect
of increasing block size and to allow greater flexibility in the
permissible design combinations.
The frequency of matings was
permitted to take the values 3, 4, 5, 6, 8, 12 and ql' except that
the permissible values of s remained restricted by the design
conditions;
!.~.,
evaluated, T
= qs
3
~
s
~
since r
ql and T
= 2.
= rqs
2. In all cases empirically
The values of q were permitted to take
non-integer values for the modified diallel designs in order to permit
comparison with partial diallels which have the same total number of
plots, T.
The values for
222
~elb' ~b' ~s
the following four parameters.
and
2
~g
were reparamaterized to give
40
2
Pc
G:
2/( 2
(rb,Oelb + 20 s )
and
O~/Q"; Ib
Pd '"
The variances of the estimators were evaluated at 50 unequally spaeed
values of P and
a
1\.
These poInts were O.O( 001) .3( .05) 1(1) ~:(~2) 10,
where the numbers i,n parenthesis are the :i.ncremants 'between succeasive
values of' p.
a
The values of P Hnd 0d nbove which block:ing
c
\JOG
effective in reducing the variances of the estimators Were found 1.n
closed form for each combination of 'the ottier psramete:rs.
This waB
"'2
poss:ible i'or Pc because the variance of O'g 10 a monotonic increasing
quadratic function in p.
c
Likewise, the critical value of Pd' the ratio
of block variance to error variance, above which blocking was effective
for reducing the variance of
"2
(r
s
w'as found.
')
L
Efficiency of Designs for Estimating a
-~--~~----~,
Unblocked
~'he
De~
efficiency of an uriblocked modifIed
diHllel~
UMD, relatlve to
an unblocked partial diallel, UPD, 'Hi th the same ctotal number of
matings was plotted as a function of p
a
the computer.
by using a plot slibroutine on
Plots of these relati,ve efficiencies for
and 12 were plotted on the sam,a graph where s
< J-'t':
8
;:,: 3~
4
p
6, 8
Examination of
these graphs revealed that the relative eff:tciency, Eup(:i"j), of an
UPD with
S ::::.
i relative to another UPD with s "'" j, where i
monotonically decreasing function of p.
a
intersected in a very consistent pattern.
>
,j, 1. s a
Furthermore, these graphs
These points of intersection
were determined and the ranges of these intersections tabulated.
These
41
points of intersection determine the value of p
a
for which the
efficiency of an UPD with s :: i is equal to the efficiency of an UPD
with s = j; !.~., Eup(i,j)
= 1.
The asymptotic limit of p in the equation Eup( i J j) :;: ; 1. as T
a
approaches infinity is:
(6.2.1)
The minimum and maximum observed values of p for which Eup(i,j)"'o 1
a
for T from 48 to 1152 plots are given in Table 6.1 togetller with the
asymptotic value of p when T .-p..
a
Table 6.1.
~linimum
2
for (J' ,
g
00.
and Maximum values of P for which the efficiency
a
Eup(i,j), = I for T from 48 to 1152 plots and
asymptotic values for T
~~
i=3, j=4
i=4,j=6
i=6,j=8
iz:8,j;;;,12
Minimum
.68
·32
.18
.12
Maximum
·70
·35
.20
.13
Asymptotic
value
.71
·35
020
.13
The observed points of equ.al efficiency vary little from the asymptotic values, even for a small total number of plots.
The computed
efficiencies indicate that the asymptotic formula, equation 6.2.1, can
be used to compare effici.encies of any two parti.al dialle1s.
The UPD
with the lower frequency of mating is the more efficient design for all
values of p greater than the critical point determined by 6.2.1.
a
Thus,
it is possible to determine critical values of p which divide the
a
spectrum of P into intervals where a partic:u.lar UPD (2 i T/2, 1., q,
a
is more efficient than any other UPD with the same T.
6J
s)
It has already
been pointed out that these critical values are relatively insensitive
to changes in total number of plots.
Table 6.2 which is constructed
using equation 6.2.1 gives tIle portions of the spectrum where an UPD
with s as indicated is superior to all other UPD's for estimating
Segments of the spectrum of p
2
~g.
where a partial dialle1 with
a
s matings per parent is superior to all other partial
Table 6.2.
diallels for estimating ~2
g
Optimum number of matings
per parent, s
Range of Pa where the design
is optimum
-_.-=-=
3
•70'r
~ eo
4
.408
~
0707
5
.289
~
.408
6
.22)+ ~
.289
7
.183 - .224
8
.154 - .183
9
.134 - .154
10
.118 - .134
11
.105 - .118
s
1/.1616 2 - 1/JS28 3
o - 1/Jq3~
43
An unblocked disconnected diallel, UDD]I has the same efficiency
as an UPD if the among grou.ps mean square is weighted by degrees of
freedom 0 This, of course, follows immediately because the analyses
are identical for ttLe two designs.
However, if the UDD w'ere analyzed
by an iterati.ve procedure as suggested in Chapter )00]1 it should be
superior to the UPD using the same design parameterso
For large
experiments, an intragrcup analy'sis may be quite satisfactol'Yo
This
is especially true when the number of progeny in a group are quite
large.
If Table 602 were used to determine the optimum number of mati.:ngs
per parent, an UPD would be used for all cases except those with very
small values of p.
a
the designs with T
Consideri.ng this, a more detailed comparison of
= 48,
is given in Table 603.
96, 288 or 1152 and s
= 3, 4]1 5, 6, 8
or 12
The value Eup(5,3) may be found by taking the
product Eup(4,3) x ELlp(5,4).
Similarily, other untabled efficiency"
comparisons may be found 0 Triis table sheds some light on the serious=
ness of an improper choice of mating frequency 0
The efficiency of an UMD relative to an UPD with s matings per
parent for selected parameter values is given in Table
6.4.
The
values given in Table 604 may be developed from Table 6.3 by taking
appropriate products 0 These comparisons sb.ow the gains in efficiency
realized by using the appropriate mating design rather than an UMDo
When p is very small or very large these values are extreme.
a
This is
especially true when p is near zeroj the lJ'MD being considerably more
a
efficient than the UPD.
44
Table 6030
Efficiency for esti:m.ation of ()2
g from an unblocked .partial
diallel with i matings per parent compared to an unblocked
partial diallel wittl j matings.? Eup( i, j )a
0000
Pa
0.04
0008
0.16
0030
0065
L22
2.00
10000
48 plots
Eup(4,3 )
Eup(.5,4)
Eup(6,5)
Eup(*,6)b
1.86
1·72
1.61
1.38
1.30
1.23
1.22
1.16
1.0b
1.04
l.lO
1.02
1.44
1.12
1.04
1.00
1.01
.86
.81
1.03
·93
.87
·99
098
093
·9'7
.85
.88
096
.89
096
-----------_.~-------~-------
96 plots
ElJ.p(4,3 )
Eup(5,4 )
Eup( 6,5)
Eup(8,6)b
Eup(*,8)
1·94·
1. 45
1.28
1·39
1.15
1.b7
1.46
1.24
1.01
1027
1.15
1.04
1.20
1.24
1.1i+
1.07
1.02
098
·99
.93
·93
·92
1.1}+
1.06
1.02
1.79
1.35
085
.87
.81
.87
.86
.79
089
.86
.81
.94
·91
.89
1.26
1.05
1.02
.86
·99
·93
.84
.83
.94
·92
.85
·77
·79
.86
.88
.80
--------------------_.
.84
288 plots
Eup(4,3 )
Eup(5,4 )
Eup(6,5)
Eup(8,6)
Eup(12,8)b
Eup(*,12)
1.97
1.84
1.69
1.49
1.29
1.16
1. tq
1.32
1..31
1.23
1.28
1.16
1.07
1.04
1.59
1.26
1.37
1.10
1.17
1.10
·99
1)~8
094
.89
.81
.85
.86
·72
.76
.78
.70
·75
.86
.86
.88
.84
1152 plots
Eup(4,3)
Eup(5,4 )
Eup(6,5 )
Eup(8,6)
Eup(12,8)b
Eup(*,12)
2.02
1.85
1.71
1.50
1.26
1.02
1. 4 9
1.39
1.24
1.29
1.17
1005
1.17
1.18
.94
1.29
1.12
.82
1.08
L07
·95
.61
LOO
1.30
094
·93
.8.5
.84·
..50
.76
.43
1.33
1.49
1.65
2.96
1.17
.80
,72
.40
,81
.86
.78
·70
.38
a Th€: values of Eup(5,3), Eup(6,3) and Rup(6,4) can be determined from
tabulated values since Eup(i,j) = Eup(i,k) x Eup(k,j).
bEi1p (*, J) indicates the efficiency of a modified diallel relative to a
partial diallel wi.th J matings per parent. Note; '* Is not generally
an integer in these comparisons.
45
Table 6.4.
?
Efficiency for estimation of ~-g of an unblocked modified
dial1el relative to an unblocked partial diallel with s
matings per parent, Eup(*,s)
0.00
Eup(*,3 )
Eup(*,4 )
Eup(*,6)
Eup(*,3 )
Eup( *,4)
Eup(-*, 6)
Eup(*,8)
Eup(*,3 )
Eup(*,4)
Eup(*,6)
Eup(*,8)
Eup(*,12)
Eup( *,3)
Eup(*,4 )
Eup(*,6)
Eup(*,8)
Eup(*,12)
3.36
1.81
1.06
5·79
2·99
1.61
1.15
12.40
6.28
3·20
2.18
1.37
28.90
14.50
7·29
4.89
2.96
0,04
2·70
1.56
1.04
3.87
2.16
1.33
1.07
5.54
3·03
1.79
1.39
1.10
6.25
3.39
1.97
1.51
1.17
Pa
0.08
0.16
0·30
0.65
2.00
10.00
2.24
1.39
1.02
48 plots
1.68 1.21
1.17
·99
1.00
·99
.84
·97
.64
·75
.96
·58
·72
.96
2.83
1.69
1.17
1.03
96 plots
1.80 1.12
1.22
·90
.88
1.00
.94
.98
.68
.68
.78
·91
.47
·55
·72
.89
.41
·51
·70
.89
3.23
1.90
1.27
1.08
·99
288 plots
.85
1.63
.68
1.09
.65
.87
.84
·70
.89
.83
.45
.45
.51
.60
·79
·29
.33
.44
.55
.76
.24
·30
2.80
1.64
1.08
·92
.82
1152 plots
1.13
·52
.41
.76
.40
.60
.42
.58
.61
·50
.25
.25
.28
·33
.43
.15
.17
.23
.28
.40
.84-
.LJ.1
.53
·75
.12
.15
.21
.27
.38
46
6.2.2
Blocked Designs
The efficiencies of possible blocked partial diallels, BPD's,
relative to a blocked modified diallel, BMD, with the same total
nu~ber
of matings were plotted as a function of p using the plot subroutine.
a
Again, all of these graphs possessed properties similar to those
described for the UPD's.
As in the previous section, the ranges of
the points of intersection of the graphs are given in Table 6.5.
These
points, Ebp(i,j) = 1, are the values of P for which the efficiency of
a
a BPD (2, T, b, T/i, i, i/b) is equal to the efficiency of a BPD
(2, T/2, b, T/j, j, jib).
The critical value of p in the BPD's
a
compared very well with the asymptotic values given in Table 6.2,
Whatever the size of block.
Table 6.5.
Maximum and minimum values of p where the efficiency for
a
2
G , Ebp(i,j), = 1 for T from 48 to 576 plots in blocks of
g
12, 24 or 36 plots and asymptotic values for T
~
00
i=3, j=4
i=4, j=6
i=6,j=8
Minimum
·70
.33
.18
.11
Maximum
·72
·35
.20
.13
Asymptotic
value
.71
·35
.20
.13
i=8,j=l2
The efficiencies of a BMD, relative to each of the possible
blocked disconnected diallels, BDD's, were also plotted.
These
designs gave similar although slightly lower critical values than
either the BPD's or UPDts.
However, the results still were very close
to the asymptotic values.
given :i.n Table 6.6.
The critical values of Ebd(i,j) :::: 1 are
The value Ebd(i,j) is tIle eff'iciency of a EDD
j).
(2, T/2, b, T/i, i, i) relative to a BDD (2, T/2, b, T/j, j,
Table 6.6.
Maximum and minimum values of p where the efficiency t'or
a
~2, ~~d(i,j), :::: 1 for T from 48 to 1152 plots in blocks of
g
12, 24 or 36 plots and as~nptotic values for T ~ ~
i=3, ,j:=)+
1:::4, j::::6
Minimrun
.65
·31
•
Maximum
.69
·35
.19
Asymptotic
value
·71
.35
.20
i:::6,j=8
17
!
Critical Values of p That Make Blocking Effective
c
BPD's and UPD's were compared to determine the magnitude of'block
variance above which the BPD (Bl'/JD) is more efficient than the UPD (UMD).
This comparison can be made by detennining the solution of the
~2
var(rr
g
~0
from UPD) :::: var(cr; from BPD) •
BY appropriate rearrangement, the value of p c is
2
r (Q+q4 s )spL:
rQ2sPa
2
+
a
+1- +
2
2
qS2-2bl
ql
4Ql
ql
2
2
2 +l... + rq2sp a + r (Q+Q4s)sPa
2
2
QS2
Ql
4Ql
Ql
r)
Pc
::::
- 1
equation~
48
The largest value of p over all combinations of the parameters
c
considered was .05.
For all designs with 288 or more plots, the
largest value was .01.
Similarly, the magnitude of Pc above which a BDD is more efficient
than an UDD is
2
2
r Q sP
r (q+q4s)sP
2
a
+
a
+1... +
2
qs ~ 2b
2
ql
l
2
4ql
ql
r 2 {q+q4s )sp2
2 +L + rQ2sp a +
a
2
2
QS2
ql
4Ql
Ql
:2
Pc
=
~
q-2b ql \
\
Since the BDD's have b
l
- 1
fewer degrees of freedom for the general
combining ability sum of squares, the magnitude of block variance
necessary for a BDD to have a particular efficiency is usually higher
than for a BPD.
Table 6.7 gives the solution for P for selected
c
combinations of the parameters.
A comparison of particular interest is that between a BPD and a
BDD.
The efficiencies of several selected BPD's relative to BDD's
with the same r, Q, sand b parameters are given in Table 6.8.
The
ranges of these efficiencies for the optimal values of p are given in
a
Table 6.2.
The BPD's are more efficient for estimating ~2 than are
g
the BDD's for all the design comparisons where a close to optimum
frequency of mating is used.
A BPD is approximately {l+p )2 times
c
as efficient as an UPD while a BDD is approximately (l+p )2 divided
c
by the appropriate value in Table 6.8 times as efficient as an UDD.
49
Table 607.
Values of p above which a blocked disconnected dial1.el
c
is superior to an unblocked disconnected diallel v1th the
same q and s for estimation of ~2
g
-"
T,c
s=3
6=4
8=6
8=8
Pa
Pa
Pa
Pa
0
·7
10
0
·7
10
48,12
010
.06
.02
.16
.06
.01
96,12
.15
.09
.04
.18
.10
.02
288,12
.18
.11
.05
.29 .12 .03
1152,12
.19
.12
.06
.30
.13
.03
96,24
.04
.03
.01
.06
.03
288,24
.07
.05
.03
.10
1152,24
.08
.05
.03
288,36
.04·
.03
1152,36
.05
.03
-
0
·7
10
.01
.11
.03
.01
.05
.03
.19
.05
.01
.12
.06
.03
.21
.06
.01
.01
.06
.03
.01
.10
.03
.01
.02
.O~(
.04
.02
.13
.04
.02
-
0
---
07
10
.15
.03
.01
.18
.04
.01
50
Table 6.8.
Range of efficiency for estimating ~2 of a blocked partial
g
dia11e1 compared to a blocked disconnected dial1e1 for the
range of p where the designs are optimum
a
Number of matings per parent
T,C
48,12
4
3
1.05-1.10
8
6
1.12-1.18
1.19-1.28
96,12
96,24
1.04-1.05
1.07-1.08
1.10-1.14
144,24
1.04-1.06
1.07-1.10
1.13-1.18
144,36
1.02-1.03
1.04-1.05
1.06-1.08
1.16-1.22
288,24
1.06-1.07
288,36
432,36
1.10-1.12
1.15-1.18
1.11-1.14
1.17-1.21
1.17-1.22
576,36
Efficiency of Designs for Estimating
6.3.1
~
2
s
Unblocked Designs
The variance of
""2
~
s
from an UPD with T plots always is smaller for
designs with a larger number of matings per parent.
This is immediately
obvious since the number of degrees of freedom for specific combining
ability is qS2/2 = T - 2q. Let EUp(j,i) be the ratio of the variance
A2
A2
of ~ in an UPD with s = i to the variance of ~ in an UPD with s = j.
s
s
The minimum and maximum lim!ting values of EUp(!, j) as T
given in Table 6.9.
was ij2/i2j.
~
co
are
The asymptotic minimum was i1j2/i2j1 and maximum
The values of EUp(j,i) were computed for the parameter
e
e
e
Table 6.9.
Asymptotic ranges of Eup(j~i), the efficiency of an unblocked partial dia11e1 with s
2
relative to an unblocked partial dia11e1 with s = i, for estimating ~
=j
s
j
i
3
4
5
6
7
8
9
4
1.33-1.50
5
6
7
8
9
10
00
1.50-1.80
1.60-2.00
1.67-2.14
1. 71-2.25
1. 75-2.33
1. 78-2.40
2.00-3·00
1.12-1.20
1.20-1.33
1.25-1.43
1.29-1.50
1.31-1.56
1.33-1.60
1.50-2.00
1.07-1.11
1.11-1.19
1.14-1.25
1.17-1.30
1.19-1.33
1.33-1.67
1.04-1.07
1.07-1.12
1.09-1.16
1.11-1.20
1.25-1.50
1.03-1.05
1.05-1.09
1.07-1.12
1.20-1.40
1.02-1.04
1.04-1.07
1.16-1.33
1.02-1.03
1.14-1.29
\J1
I-'
52
combinations considered in the previous sections.
The asymptotic
ranges given in Table 6.9 included almost all of these observed
efficiencies.
The values which exceeded the asymptotic ranges did so
The variance of (J'"'2 using an UDD is equal
s
only by a very small amount.
to that using an UPD.
6.3.2
Blocked Designs
The variance of (J'"'2 is always greater for a BPD than for a BDD.
s
However, since the BPD is usually more efficient for estimating (J'2,
g
the magnitude of the efficiency for estimating (J'2 is given for
s
selected parameter combinations in Table 6.10.
As in the unblocked
designs, the variance of (J'"'2 in a BPD with s matings can be shown to
s
be greater than a BPD with s + 1 matings.
Thus, the BMD is the most
2
efficient blocked design for estimating (J' •
S
6.3.3
Comparison of Blocked and Unblocked Designs
The magnitude of Pd
= (J'~/(J';lb
efficient for estimating (J'
Pd
=
2
s
above which the BPD is more
than an UPD is given by:
+ 1 )
qS2
qs-2
1
(1
Values of P are given in Table 6.11 for selected combinations of
d
the parameters.
The critical values of P were not determined for the
d
disconnected diallel since only rlb degrees of freedom are lost from
error.
This implies that the reduction in error variance will more
than offset the loss in error degrees of freedom in most cases.
53
Table 6.10.
Range of efficiency for estimating (]'2 of a blocked partial
s
dia11e1 compared to a blocked disconnected dia11e1 over the
range of p where the designs are optimum for (]'2
a
g
Number of matings per parent
T,C
4
3
48,12
96,12
96,24
144,24
144,36
288,24
288,36
432,36
576,36
.875- .905
.917- .944
.875- .915
.938-.953
·917- ·937
.958-.969
.958- .9T3
.968-.981
.958- .975
·979= .987
.948- .969
.969=.981
.965- ·979
.945- .963
.972- .981
.959= ·972
Table 6.11.
8
6
·982=·989
.972= .984
.974- .982
·968-·981
Values of P above which a blocked partial dia11el is more
d
efficient for (]'2 than an unblocked partial dia11el
s
Number of matings per parent
s=3
s=4
s=5
8==6
~
~
pt,
pt,
1
T,C
0
1
10
0
48,12
96,12
96,24
288,24
288,36
.05
.12
·73
.02
.05
.34
.03
.05
.01
.01
.03
10
0
1
10
.07 .46
.11 ·71
.03 .22
.03
.04
.01
.06
.09
.02
·37
.58
.18
.22
.01
.02
.18
0
1
10
.02 .05 ·33
.04 .08 ·51
.Ol. .02 .16
.02 .04 .28
.01 .02 .16
7.0
GENERAL DISCUSSION
TIle development of designs wIlierl provide efficient and. mibi.ased
estimators of
2
and
g
0-
0-
2
involves two primar'.{ design consideratj.ons.
s
First) a mating system must be developed to provide the relatives upon
which the estimators are based.
in an environmental design.
Second, these progeny must. be grown
Since the number of progeny necessary to
adequately estim.ate components of genetic varlance Is large, complete
mating designs may be ine:fficient, and ran.domized complete blocks are
often heterogeneous and unsatisfactory.
This leads t.o the joint
problem of developing incomplete mating designs in conjunction with
incomp.lete block designs,
the
comhi.nation of which will lead to
General classes of mating designs
efficient and unbiased estimators.
are proposed and evaluated emp:Lrically
of a 2 and- ()2
g
S
0
I~or
efficiency of the estimators
Tne dedslon to use an u.nblocked. modified diallel, UMD,
or an unblocked partial
diallel.~
UPD,H:i.th s mat:Lngs can be based
entirely upon .£ l2Eiori estimates of the variance components by usi.ng
Table 6.2 as a decision rule.
~len
a randomized complete block
enviromnental design is used, e.ither tIle UPD or the unblocked
disconnected diallel, UDD, will yield equally efficient estirr.ators
prOViding both designs have the same q and s.
two designs can be based upon other criterion.
The choice between these
Ease of handling the
experimental material may be an important consideration in the choice
between these two design classes.
Also, since the efficiency of a
partial diallel design for estimating genetic variances 1.s determined
55
by q and
and not by the rooting pattern" the choice ot' a part-iel.uar
8,
A2
pa ttern in no wa:,/, effects the efficiency of cr
g
pattern uIJ.ich prov:ide8
+;11€
A,:~
and
0-
s
Rence J a .mating
0
smallest avera.ge varimlce for compar:i.ng two
treatment means can be used wIthout imJ?;.lr:lng tkle variance component
estirr~torso
~lis
last consideration was discussed at some 1engtll by
Curnow (1963).
When Lneomplete bloc}r,"'Jn:Lronrrlerrt;al des:igns a.re osed y rest.rictions
''4Ilel'-',~
kq/2
C
_.,
C
.- kq
.if q
,.~
0 (,Eod
.:;
:-
)j
if q ..- 1 (mod 2),
k. ... 1. J 2 9
k
u.
1, 2.~
0
0
0
or
.0.
The permissi.ble values of k are f.'urtlJ,errestr:ictedb,'f the relation
k '" sib if' q
c:"
0 (mod 2) or k "~ 2s/b if q "" 1. (mod 2).
On t;,he other
:().!.;,\ud, the blocked disC0Il11ected diallels, EDD'sj Ilave block sizes of
C "" qS/2b wtlere 3
.:5
s
< q/b
':EI1UB
0
the mi:n.:.tmum bloek sizes are
'l'hus, a
blocked partial diallel,
(mod 2) or if q
BPD~
> C wIlen q
=:
does not exIst if q
1 (mod 2)
not exist If tIle number ofbl0,:;ks,9
b~
0
m.~e
q
=0
Also. th::Ls: type of design does
'per re.plieat:ion is greater than
s when q :;::: 0 (mod 2) or 5/2 w.b.en q '''' l(mod 2).
possible to
> 2C when
Since it is generally
an even number of parents9 the largest attainable number
of' hlocks i.n a BPD is s.
a BDD design must be used.
If tIle desired !-l'JIlfcer of blocks exceeds 5,
However.. Lf s ,?, band s
between a BDD and a BPD is posslhle.
< q/b a choice
j
Fo:r a BDDto exist the frequency
56
of mating must be less than
q/b.
satisfied, a BPD must be used.
If this last condition is not
For very large q the block size
required for a BPD becomes prohibitively large.
Conversely, when each
parent 1s mated a large number of times, minimum block size for a BDD
is beyond reasonable limits.
Certain species may be accommodated more readily in a BDD than in
a BPD.
In addition, particularly during crossing time, record keeping
would be simplified.
Of special interest, however, is the ease with
Which missing observations can be circumvented in a BDD.
Mating
failures can be avoided by:
(1)
Substituting a group which is complete for the defective group
prior to placing the progeny on test.
(2)
Reducing the size of the experiment by omitting groups which
have missing crosses.
In the partial diallel designs, a very specific mating pattern must be
rigorously adhered to.
The loss of a single mating would result in a
nonorthogonal design even before the material is planted in the field.
If some of the material is lost at a latter stage, the blocks containing
the missing values could be eliminated from the analysis.
Of course,
the loss of information from the entire block would be appreciable, in
Which case, replacing a missing value probably
wou~d
be preferred.
If
an entire block were discarded from a BPD, a reduction in s would also
result.
In the BDD designs, s would remian constant but q would be
reduced.
While loss in efficiency' in a BDD design with one block
missing would be substantial, it would rarely be as great as in a BPD.
57
A comparison of tl'lese designs using a cost function
1!laS
not
under~
taken since experimental cost is expected to be almost directly related
to the total number of plots, whatever the mating or environmental
design.
Hence, a choice between the BPD and BDD designs can be based
upon effici.ency and convenience considerations,
! .Eriori estimates of
222
rr , rr and rr Ib when applied to the decision rule given in Chapter 6.0
g
s
e
will lead to a partial diallel design with an efficiency for rr2 '''hich
g
is optimum or nearly so, providing the
~
priori estimates are
reasonably good.
2
The loss in efficiency for estimating rrg because of inadequate ~
priori estimates can be large,
evaluated by using Table 6.3.
The magnitude of this loss can be
For example, suppose that P was assumed
a
to be ,65 when in reality it was .16,
A design with s
=4
been chosen when the correct design should have reqUired s
would have
= 8.
If the
experiment size had been 288 plots, the appropriate design would have
had an efficiency of 1.29 compared to the design with s
= 4.
efficiency was determined by evaluating the product, Eup(8,4)
x Eup(6,5) x Eup(5,4).
efficiency between 1.06 and 1.07.
Table 6,8 gives an
The loss in efficiency due to an
improper frequency of mating can be very great at times.
demonstrated for several cases in Table 6.4.
an s as high as 8 were chosen when T
=
= 1152,
.20 as efficient as an UMD if pa
the need for good
~
= R~p(8,6)
Continuing with this example, one can determine
the efficiency of a BPD compared to a BDD.
1/4.89
This
priori estimates.
This is
For instance, even though
this UPD is only
= o.
This result emphasizes
58
The validity of the asymptotic critical points determined by
equation 6.2.1 and listed in Table 6.2 is substantiated by empirical
results given in Tables 6.1, 6.5 and 6.6.
These critical points hold
very well throughout tIle range of parameters considered for all of the
designs suggested in Chapter 3.0.
In addition to the improvement which can be realized by using
partial diallels, incomplete block designs can be used Which result
in a reduction in error variance.
This allmTs more efficient estimators
to be developed prOViding the among blocks variance is sufficiently
large to offset the effects of the inevitable loss in degrees of
freedom due to blocking.
Blocking is almost always effective in the
partial diallel designs.
This is true because only specific combining
ability contrasts are confounded with blocks.
Greater differences
between the blocks are necessary, of course, for the disconnected
diallels since general combining ability contrasts are confounded
with block contrasts.
Nevertheless the magnitude of among blocks
variance 'Thich makes blocking .:1'fective was also found to be very small
in almost all cases.
A block variance greater than 2ifP of error
variance was reqUired only in cases where small block sizes were used.
In most cases this variance was considerably smaller than 2CfI~ of the
error variance.
Because the critical values of
p~
a
were very' insensitive to
blocking, the optimum desl.gn for a blocked experiment could be de-·
termined by using the decision rule given in Chapter 6.0<
Due to the
fact that blocking may result in a reduction i.n the plot to plot
59
variance,
occur
0
O;lb'
a change in the optimum mating frequency could actually
A reduction in J2,.... because of blocking will result in
Ct.
elL'
larger value for p J and
a
t~en(;e
a smaller value of
is
may be appropriate
0
The symbol (J;lb is used to denote within block variance throughout
this paper to emphasize the fact that the error y
function of block sizeo
2
()
may be a
e I'"'
0
If the experiment is unblocked this component
becomes part of the plot to plot varlance within the replication •
.
2
~
The problem of ~loint effic:iency of the eGt:imat.c:)rs of eye. and
(j'
s
g
must be considered.
for esti.mating
0
2
0
S"
:I'he modified diallel j
is best for estimating
the most effic:i,ent design
0
2 only if P", is V".:ry smalL
g
<-
The use of a partial diallel with s dete:r"!l1ined by th,e decision rule
given in Chapter 6.0 gives more efficient estimators of (J2 but at
g
.?
the expense of less efficient estimation of cr-o
s
.?
When
cr~
is large,
"2
a moderately large variance of cri.s not necessarily undesirable.
When
g
(J
2
g
and cr
2
are smally howevery even a small variance may be too largeo
s
One consoling factory howevery is that when
clg
is small, both of the
estimators are nearly optimwm if several matings are made per parent.
For instance, if p is such that s
a
=8
2
g
is optimum for cr', then
"2
0"
has
s
an efficiency of not less than 07,5, .! ..~,. , 1/1.33, from Table 6.9.
It
can be seen from tb.is table that the only really substantial reduction
in the variance of
experiment.
A?
0is
can be rea.lized by increasing the size of the
Af'ter choosing the optimtUn value of s for efficient
2
estimation of (J , the research worker will have to decide on the basis
g
of the magnitudes of variances he is willing to accept ''''hether a design
wi t,n a slightly higher frequency of mating is actually preferable
0
60
This dec:1.s.lon must be based upon the relative importance of efficiency
1")
1'or
0":"
g
rj
Hnd a C
s
Tables 603, 6.9 and 6.10 may be of some assistance while
•
consider:ing th.is problem.
is desired for
0
Generally, however J if additional precision
2 , a larger experiment should be used.
s
Of course J the
larger experiment would provide a more efficient estimator of' (J2
g
as well.
When p is small, a large number of lnatings per parent are needed
a
for efficient estimation of either of the parameters.
This large number
of matings per parent precludes using a BDD design which requires a
minimum block size of s(s+1)/2.
This presents a considerable dilemma
since with both sand q very large, a very intricate mating design is
necessary.
Mating failures are almost certain to occur.
This difficult
situation can be circumvented by using a combination design with
5(8+1)/2
matings per group in a disconnected design and then treating
each group as was done in a BPD.
The analysis for this type of design
is not g:iven but is a straight forward extension of results given in
Chapter 5.0.
each.
This procedure will allow blocks as small as s + 1 plots
AlSO, this allows for elimi.nation of lines where mating failures
have occurred.
Thus j orthogonal:ity of the
SUIll.S
of squares can be
rnainta1.ned alongwl.th a manageable analysis.
A BDD is composed of several small partial (or modified) diallel
designs; one or more of which are assigned to each block.
When more
than one are assigned to a block, all of the progeny in the block are
randomly a.ssigned to tIle plots of that block without regard to the small
•
partial diallel to which they belong.
In these cases
j
more efficient
61
estimators may be developed by appropriately weighting the am:.:mg and
within groups mean squares within eac.h of t.b.e blocks
0
Arbitrarily
weighting these mean squares by degrees of freedom is eqUivalent to
analyzing the design by the formulae for partial diallels.
increase in efficiency should not, be expected to be very
since the weights
weights
0
usir~
The
great.~
however,
degrees of freedom must be close to the correct
This 1'011mm lmmedistely by observing that the v!3riance of'
the mean square among groups is
While the variance ',dthin groups which have a £:::mall modified dial1el is
The coefficient of ~2 in 7.0.1 is 2rs and the coefficient of ~2 in
g
g
7.0.2 is greater trian or equal to 2rs/3 but less than rs.
Hence, the
numerator of equation 7.001 is greater but never more than three times
greater than the numerator of 700.2.
Thus the actual weights cannot
be grossly different from the weights using degrees of freedom.
All upnus vith fixed values of q anti
3
are equally efficient.
The
UDnus also have tbe same efficiency unless a weighted least squares
analysis is used..
If further galns in effic:i.ency are to be realized
beyond ttLose possible by varying s and
to be considered.
-by
blocking, ottier designs need
For example, it is shown in Appendix 10.1 that an
"Experiment III is superior to a partial dial1el is p is large.
a
A
62
computer comparison of these designs was not made at this time.
extension of these resul tsto a comparison of blocked diallc:l
with other designs might lead to seme interesting findings,
The
de~":i.gns
This may
be especially true for large values of p •
a
The sum of squares proposed for error :l..n the blocked designs is
not necessarily the min:i.mum variance error term.
If' t.he genet:i.e
effects./ the
g., I S and the ~"
It" do not~ i.nteract ,>lith replications. the
"1
1.J .
/
0
reps by groups (blocks)
0
Btllll
of sqiJ.e.res provides an €':Jtfmate 01' error
with the same ex:peetHtion as tb."-; defined. error tel"ln,
however ~ the error term already· I'm;:;
freedom.
is
In most designs,
large number of degrees of
The pooling of the proposed error with the rep .x group
interaction would generally have only a. small ef:t"ect on the variance
of the estimators slnce on.ly r]b
1
degrees of freedom are added.
If
the interacti.on term 1.8 pooled wit,h the error when these interactions
actually eXfst, the estimator of' Cf2 will be biased.
s
Tllis pooling of
?
sums of squares would not affect the estimator of Cf - since
g
function of only the sums of
squ~res
0"2
is a
g
0··
.for general and specific: combining
ability.
The sum of squares due to general. combining ability is not
necessar:i.ly distributed as a X
2
variable.
'I'his is readily apparent
after comparing the expectation and variance
squares.
or
any partlcu.lar sum of
Ilfle sum of squares due to general combining ability in the
2
modified diallel, however, is distributed as a X variable.
The only
other case where this 1.8 true is in the intrablock analys:i.s of a
disconnected diallel where the group of crosses which compose a block
is itself a small modified diallel.
the sum of squares is
8.
GenerallYJ the distribution of
linear function of X2 variables.
Although the
exact distribution cannot be readily deduced in all cases J the mean and
the variance of the sum of squares is known in closed form.
Since the
2
sums of squares are not distributed as X variables J the usual tests of
significance of u"'2g estimated from a partial diallel are complicated.
64
8.0
SUMMARY AND CONCLUSIONS
The modified diallel and various fractions thereof were fitted
into incomplete block environmental designs.
Three methods were
given for choosing and partitioning the crosses from the partial (or
modified) diallel into blocks.
A fourth class of designs called
disconnected diallels was developed together with a means of
partitioning the crosses into complete blocks.
The principle objective of fitting these designs into incomplete
blocks was to allow efficient and unbiased estimation of ~2 and ~2,
g
s
the general and specific combining ability variances.
This was
accomplished in the blocked partial (and modified) diallel designs by
requiring each line to occur as a parent an equal number of times in
each block.
In the blocked disconnected diallel, this objective was
met by using only an intrablock analysis.
Computational formulae for orthogonal sets of sums of squares and
the analysis of variance tables were developed for each of the suggested
designs.
Expectations of the mean squares were also determined assuming
a random model.
designs.
Estimators of
~
The variances of these
2 and
g
2 were developed for each of the
s
~
estill~tors
were given for the case
Where the yield variable was normally distributed.
FUrthermore, the
variance of the estimators was invariant under choice of mating design
for a fixed combination of the number of parents and the number of times
each was mated.
The sum of squares due to general combining ability was found to be
distributed as a x2 variable for only the modified diallel and some of
65
the disconnected diallels.
The necessary conditions that this is true
in a disconnected diallel is that the groups are actually modified
diallels and that only an 1ntrablock analysis is used.
The efficiencies of these various designs were evaluated with an
electronic computer for selected parameter values in experiments with
two replications.
Critical values of pa
=g
2~2/(~21.
e 0 + 2~2)
s were found
which partitioned the range of P into segments where a specific
a
frequency of mating gave the maximum efficiency for estimating ~2.
g
This critical value was not materially affected by size of experiment,
size of block or choice of mating design.
These proposed designs were
also evaluated for efficiency of estimation for
2
~.
s
The estimators of
~2s had efficiencies which were nearly maximum if several rnatings were
made per parent.
The magnitude of block variance which made blocking
effective for either component was evaluated and found to be relatively
small for most designs.
It can be concluded that blocking is usually effective in tIle
estimation of ~2 or ~2 even if block variance is as small as 5 or 10%.
g
s
The number of matings per parent, s, can be determined using
estimates of
~
priori
222
and ~s in the formula, Pa
~g' ~elb
and finding the appropriate value of s in Table
The blocked partial diallel is generally more efficient than the
blocked disconnected diallel for estimating ~2.
g
2
~s.
The reverse is true for
However, the mating system is rather intricate for the partial
diallel but very simple for the disconnected diallel.
The simplicity
of the latter should be seriously weiglled before a deci.sion is made to
use the more complicated design.
66
9.0
LIST OF REFERENCES
Comstock, R. E. and H. F. Robinson. 1952. Estimation of average
dominance of genes. Heterosis, pp. 494-516, Ames, Iowa State
College Press.
Curnow, R. N.
287-306.
1963.
Sampling the dia11e1 cross.
Biometrics 19:
Eisenhart, C. 194 7. The assumptions underlying the analysis of
variance. Biometrics 3:1-21.
F,yfe, J. L. and N. Gilbert.
19:278-286.
Gilbert, N. 1958.
477- 492.
Partial dia11e1 crosses.
Dia11e1 cross in plant breeding.
Biometrics
Heredity 12:
Griffing, B. 1956a. A generalized treatment of the use of dia11e1
crosses in quantitative inhertiance. Heredity 10:31-50.
Griffing, B. 1956b. Concept of general and specific combining
ability in relation to dia11el crossing systems. Aust. J.
Bio1. Sci. 9:463-493.
Hinkelmann, K. and K. Stern. 1960. Kreuzungsplane zur
Selektionszuchtung bei Waldbaumen. Silvae Genetica 9:121-123.
John, P. W. M. 1963. Analysis of dia11e1 cross experiments in a
split-plot situation. Aust. J. Biol. Sci. 16:681-687.
Kempthorne, o. 1957. An introduction to genetic statistics.
Wiley and Sons, New York.
Kempthorne, O. and R. N. Curnow.
Biometrics 17:229-250.
1961.
John
The partial dia11e1 cross.
Schmidt, Johs. 1919. Individets vaerdi sam ophav bed~mt efter den
flersidige krydsnings metode. Meddelelser fra Carlsberg
Laboratoriet 14, No. 6:1-30.
Sprague, G. F. and L. A. Tatum. 1942. General vs. specific combining
ability in single crosses of corn. J. Amer. Soc. Agron. 34:
923-932.
67
10.0 APPENDICES
10.1 A Partial Diallel is Not Necessarily
II
,
II
Superior to. an Experiment II .
The variance of the estimator of the general combining ability
variance in an Experiment II design with q males and 8/2 females
(ignoring the females mean square) is:
+ rsp)
2
4
+
R..J
s2
Similarly the variance of the estimator from a partial diallel is:
2
when
p =
for q > 2.
0, V - V OC ql (E.... + ..i..)
2
1
2 ql
qS2
q2
Hence if
p =
0, V2 < V1 .
However, the limit of V /V as
2 l
p
--...> 00
is
68
2
2r (q + q4 s )s
r
222
~s
=
for all values of q and s since ql >
S
in a partial diallel.
Hence for small p the partial diallel is superior for estimation
of cr2 while the "Elcperiment 11" is superior for large p.
g
10.2
Inversion of a Symmetric Circulant Matrix2
If A is a real symmetric circulant (qxq) matrix, the inverse of' A
may be found without much difficulty
Since the matrix A is real and
0
symmetric, the inverse of A is also real and symmetric.
Furthermore,
the charactersitic roots ~j of matrix A and l/~j of matrix Aalso real.
are
Consider the general circulant matrix
a
A
Let wj
unity •
l
o
=
= expo
{j(2~i/q)},
(j
= 1,2, ••• ,q)
be the qth roots of
'Ihen
2Kempthorne, O. and R. N. Curnow
1961. 'Ihe partial diallel cross.
Biometrics ~7:229-250. 'Ihis is an extension of the appendix Inversion
of Matrix A to the general real symmetric circulant matrix.
0
69
1
1
w.
J
A
Thus, the characteristic roots of A are
j
= 1,2, ... , q
(10.2.1)
9
,
These equations can be inverted to obtain the a s in terms of the A s
1
a
=o
q
q
L:
~
k=l
and
j ::
-1
Using the property that A
of l/~j' (j
= 1,2, ••• ,q)
a
o =1
q
q
L:
1,2, ••. , ql
(10.2.2)
is also circulant with characteristic roots
we may write
1
-
k=l ~
and
j
= 1,2, ... , q1
Since A is a real synunetric matrix, both its charactersitic roots
and its elements a
k
are real.
Substituting
~j
70
if k
ak
=
=0
if parent 1 is mated to the (k+l)th parent
if otherwise
in 10.2.1 and taking real parts,
j
=
q
= 28
A.
and
A.
a
~l
s + E a
k=l
k
cos(
J21Ck
) ,
j
q
= 1,2,···,ql
(10.2.4 )
Now, upon taking real parts of equations 10.2.3,
and
a
J
=
ql
1
1
J(q-k)
E -- cos --- - 21C
q k=l ~
q
J
j
= 1,2, ••• ,ql
For the special case of the partial diallel as given in Kempthorne and
Curnow (1961), we have
sin( q- s ).J2£
s - --~-q...
sin .J2!
,
J
= 1,2, ••• ,ql
q
(10.2.6)
A.
n
= 2s
The inverse elements of matrix A may be obtained by substitution of
equations 10.2.4 into equations 10.2.5.
The actual inversion can now
be readily accomplished with the aid of an electronic computer.
It
should be pointed out that considerable reduction in computation time
-1
can be realized by considering the synnnetry of A •
of synnnetry implies that a
r
= a q-r ,
r
= 1,2, ••• ,ql.
This requirement
71
10.3
10.3.1
Properties of the Mean Squares
The Independence of Mean Squares
The least squares equations for the blocked partial dia11e1 can be
written in partitioned form as follows:
1'1
!'~
l'X
- 2
!'~
l'X
- 4
l'
IJ.
l'y
~1
~~
x;.X2
~~
X~X4
~
r
~X
X'l
2='
X~iS.
x~X2
x~~
X~X4
x'2
b
9:
X3~
X3 X2 X3~ X3 X4 X3
rb
X3I
X'l
iF
X4~
X'4
.6:
1
~
X4 X2
X2
I
s
X'y
4y
X4~
~
X4X4
X
4
X'Y
=
2='
where Y is the (~ x 1) vector of mean yields for the ~ progeny,
~ is a (~ x r) matrix with elements ~
2 is a (~ x b) matrix with (xij,k) equal to 1 if cross i x j
X
is in block k
X4 is a (~ x q) matrix with (Xij,k) equal to 1 if i
j
= k,
=k
or if
otherwise O.
S;) identity matrix
I
is a (~ x
1
is a vector of l's
and the notation (xij,k) symbolizes the element in the ijth row and the
kth column.
The ijth row meaning the row of the least squares equation
giving the coefficients of cross i x j.
72
The genetic portion of the variance=covariance matrix of the
vector Y is
The
sums
of squares,
~ :::: y' [X (X' X )=lX' =
"0
-
S
:=
S
:=
g
S
2' 2 2
2
y' [X (Xl X f1x'
-
Y'[I
-
4 4 4
=
4
Sand S , may be
~,
=
s
g
.s.
J] Y
qs
-
:=
.s...
J]
qs
:=
Y
-
written~
y' BY
-y' AY
--
X (X'X fIx' - X (x.'x flx' +~ J] Y
2 2 2
2
4 -'"4, 4
4 qs
-
where an unsubscripted
J'
is an (~ x
¥) matrix of I' s.
=:
yl(JY'
-Ip general,
J's are matrices of l's with dimensions given by the sUbscript.
Some preliminary matrix
J' J
nm mn
results~
== m J
X'J
4 ~
:=
S
J
X J
:=
2 J
2 n
4 qn
x4''Xq
4 Jn
(XIX
4 4
)-1 J
qn
nn
qn
~
2 n
:::: 26 J
::::
126
J
qn
qn
73
(XiX )~lJ
2 2
bn
tr J
nn
tr X4 X4
= 2b J
qs
bn
=n
=:
qs
2
tr X4X4 X4X4 = tr X4X4X4X4 :; q( 6 +S) since every diagonal
4
element of X X4 iss and the element 1
occurs s ti.mes in each row.
AJ
=0
BJ
=0
CJ
=0
....,
C=I-A=B+'::""J
qs
A'A:::;. A
Since the block erfects are swept out of both general and specific
combining ability equations, the sum of squares for general combining
ability are independent of the sum of squares for specific combining
ability if and only if
CVA
Let a
= r~2s
and d
=:
2
r~
g
=0
, then
eVA
4
=::
C(a I + d \ X )A
:;
a CA + d C~X4 -
=a
CA + d CX X
4 4
4d
q
CJ
(10.3.1)
=A
- AA - X (X' X f1 X' A
=-
X (X' X (lX' +
=-
BA
2
2
2 2
2 2
2
2
R..
qs
+.s...
JA
qs
JA
However,
=0
(10.3.3)
.
Using equation 10.3.2,
CA
=-
BA
Now
= - -2q
=0
,2
,
X2 J.bq-""4
x.' + -qs JX4 X
4
.
=0
(10.3.4 )
75
Substitution of 10.3.4 and 10.3.5 into 10.3.1 yields
CVA ::: 0
Since the block effects are swept out of the general combining
ability equations,
~
and Sg are independent if and only if
BVA ::: 0
::: d BX X A
4 4
,
::: d BX X
4 4
=
4d
q
BJ
4
::: d BX X
4
::: d
[X2(X~X2flX~X4X4
-
~s
4]
JX4 X
Now substi.tute 10.3.5 to yield
BVA ::: 0
Thus
~
and Sg are independent.
Similarly Sb and Ss are independent if and only if
BVC :;; 0 •
BVC ::: a BC + d BX X C
4 4
Now
BC=B=BA-B+E...BJ
qs
:::
=
BA
::: 0
Also,
since
Hence Sb and Ss are independent.
76
10.3.2
Expectations of the Mean Squares
Under the assumption of normal:i.ty the expectation of the
Stun
of
squares due to general combining ability in the partial diallel is:
ES
g
::::
2
2
2
+ r0- ) A + reT
tr VA :::: tr [ (O-elb
s
g
::::
(0-e Ib + ro' s ) tr A + ro-g tr X4 X A
:::
222
2
2
4
2
ql (vel b + rag) + r0-g (tr X4 Xl4
2
( 2
ql 0- ~Ib + ro-s ) + rqso-g
("'.
::::
c:;
~
~X4AJ
=
4
tr J)
q
-
2rso-2
g
_. ql(eT2 + ro-C:) + rs,.) S(J'''8
'- g
elb
r.)
!')
Therefore
2
2
EMS g :: 0-e b + rJ s
I
The expectations of the rewEining mean squares for the designs
considered may also be developed by this technique.
10.3.3
Variances of the Mean Squares
Under the assumption of normality the variance of the sum of
squares due to general combining ability in the partial diallel is:
Var S
g
:::: 2 tr VAVA
:::: 2a
2
tr A + 4a(1 tr (X4 X'4 - ~.
J)
q
4 J)
+ 2d 2 tr ( X4X4I - 4 J )( X4X4I - '2
q
•
77
::;; rcr2
where
g
2( 2
••• Var
MS
g
::;;~b
222
+ r ....2 )2
V
ql
s
+
4 rq2 s (cre lE+ rcr )0"
b
s g
2
ql
The remaining variances may be developed by this method.
However,
in many cases the contrasts of which the sum of squares is composed
may all have the same expectation.
This is true in particular for the
specific combining ability and error s'wns of squares.
In this case
the usual variance ofaX 2 variate is appropriate and considerably
simpler to evaluate.