~SIS OF DIALLEL, TRIALIELAND QUADRALIEL CROSSES USING

A GEImRAL GENETIC M>DEL

by

Sidney stanley Young

Institute of StatisticsMimeograph Series No. 917April ~974 - Raleigh

1.

2.

'rABLE OF CONTENTS

INTRODUCTION • •

GENERAL MODEL

iv

1

3

2.12.2

Genetic ModelExperimental Model

311

3. ANALYSIS OF VARIANCE

4. EXPECTED MEAN SQUARES

13

19

5·

6.

T:ES T OF HYPOTHESES FOR FIXED EFFECT~)

VARIANCE COMPONENTS AND TEST OF HYPOTHESES

27

29

7. DISCUSSION OF RESULTS

7.1 Di allel . . • • •7.2 Triallel • . .7.3 Quadrallel •.7.4 General Discussion

32

32323537

8.

9·

10.

SUMMARY • • • • • •

LIST OF REFERENCES

APPENDIX •••.••

41

1. INTRODUCTION

The estimation of genetic variances is generally accomplished in

the following way, Cockerham, 1963. Relatives are created in some

mating design and tested in some environmental design. Expectations

of the sums of squares of a quadratic analysis of the observations

lead to estimates of design components of variance and covariance

which can be interpreted genetically and environmentally. The

quadratic analysis can be viewed as reSUlting from a sequential

fi tting of a progressively more complicated model, called herein the

design model. 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 co­

variances of relatives) is replaced with direct attention on genetic

effects. Eberhart, 1964, Eberhart and Gardner, 1966" and Gardner and

Eberhart, 1966, have discussed a similar genetic model for fixed

effects. The analyses of dial1el, triallel, and quadrallel hybrids

have been considered separately by several authors, Hayman, 1954a, b,

1958a, bj 196o,; Griffing, 1950, 1956; Kempthorne, 1956, 1957; and

Rawlings and Cockerham, 1962a,b; to name but a few, but never before

have all three types of hybrids been analyzed in conformity with the

same general genetic model.

The purpose of this dissertation is to develop quadratic

analyses for these three types of hybrids by successive fitting of

genetic effects of a general genetic model. The resulting analyses

can be viewed as either of fixed effects or random effects, depending

upon the experimental material utilized.

2

3

2. GENERAL MODEL

2.1 Genetic Model

The factorial model of gene effects, Kempthorne, 1957, is

presented for genes at two loci in diploids. Consider for an

individual genotype, loci x and y as in Figure 1 with i, j,

k, and L indexing the alleles. Using a for additive effects and d

x

i

j

Figure 1 Diagram of two loci with indexing of positions

for dominance effects the model for the genotypic effect can be

written as

Notation Description

Genotypic effect =

(additive, a, and dominance,d, effects for locus x)

+ (additive, a, ~ld dominance,d, effects for locus y)

+ (additive x additive effects)

4

+ (dd. 'lrA)~JA,{,

+ (additive x dominance effects)

+ (dominance x dominance effects)

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 ~./2 the~

proportion of the genes received from the ith

parent. Then for any

entry under consideration ra. = 2 .~

For dominance effects,

be the proportion of genotypes for loci inEo ..D.. , let 0.. (0 .. )~J ~J ~J ~~

the entry with alleles from parents i and j (i) . D..~J

is the sum

of dominance effects for these genes from parental sources i and

j. EO~J' = 1 and ~. = 20 .. + Eo. . A general model for an entry• ~ ~~ '.1-' ~J

Jr~

as a deviation from the population mean can now be written as

2G = '£a.A. + Eo ..D.. + (ra.A.) + ('£a.A. )(Eo ..D.. )

~ ~ ~J ~J ~ ~ ~ ~ ~J ~J

2 3 4+ (Eo ..D.. ) + (ra. A.) + (ret.A.) + ....~J ~J ~ ~ ~ ~

Expansions of the epistatic terms are instructive; for example,

2 2(ra.A.) = Lev. (M) .. + 2E ra.~. (M) ...~ ~ . ~ ~~ " ~ J ~J

~ ~<J

The first summation. in the expansion is for additive x additive inter-

action between alleles from the same parent and the latter involves

alleles from different parents. Also note that (AA).. is an~J

5

average of two additive x additive interaction effects: x genes from

parent i with y genes from parent j, and y genes from parent

i with x genes from parent j .

Models for three types of entries, diallel, triallel, and

quadral1e1, are now presented. First consider the entries of a dia11el

experiment in which selfs and reciprocals are omitted: ~. =~. 6..~ J lJ

= 1

+ (AD)i(ij) + (AD)j(ij) + (DD)(ij)(ij) + (2.1)

Next consider progeny of a three-way cross i x (j x k) with

1distinct parents: ~i = 1, ~j = ~k = 0ij = °ik = 2 '

G A + ~ A + ~ A + 1 D + 1 D + (AA)i (jk.) = i 2 j 2 k 2 ij 2 ik ii

1 () 1 ( ) +~ () 2 ( )+ 1+ AA jj + 1+ AA kk 2 AA ij + '2 AA ik

1 1+ 1+ (AD)k(ij) + (AAA)iii + 8 (AAA)jjj

133+ 8 (AAA)kkk + 2 (AAA)iij + 2 (AAA)iik

3 6 1 .+ 's (AM)kkj + 4" (AM)ijk + 4" (DD) (ij ) (ij )

+t (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.

Finally, consider the model for the progeny of a four-way cross

from four distinct parents (i x j) x (k x i,) , Q'i = Q'j = Q'k = Q'i,

1 _ ~ 1= 2 and 0ik = °ii, = u jk = °jt = 4:

+ ~8 ((AM) .. 0 + (AM) .. k + (AAA) .. IJ + (AM) 0 .,

11J 11 11~ JJ1

+ (AM). Ok + (AAA) o. n + (AAA),.h~ + (AAA)kk' + (AAA) ••'dJ J J J ~ .ruu. ,.Jt'U\.,(,

6 r() ') () ('}+ "8 1.. AAA ijk + CAAA ij.t + \AAA ik.t + \AAA)jk.t

+ fb{(AAAA)iiii + (AAAA)jjjj + (AAAA)kkkk

+ (AAAA)""",,}

+ ~1((AAAA)ii'j + (AAAA)'i'k + (AAAA)", n + (AAAA), .. ,~o ~ ~ ~ ~~~~ JJJ~

+ (AAAA)jjjk + (AAAA)jjj,t + (AAAA)kkki + (AAAA)kkkj

+ (AAAA)kki n + (AAAA)kk' A + (AAAA) ... k + (AP..AI1.).,", A~ J~ JJ~ JJ~~

7

8

1+ ib ((DD) (ik)(ik) + (DD) (iL )(iL) + (DD) (jk) (jk)

For four-way crosses,three and four-locus, all-additive typ~of

interactions are included in the model since they are to be utilized

in the analysis.

Note that when all the effects of a particular type are added for

any model G = 2A + D + 4AA + 2AD + DD + .••• Numerators in the

models indicate the number of distinct effects that are averaged.

When individuals are random members of a linkage equilibrium,

randomly mating population, the genetic effects are uncorrelated,

Cockerham, 1963, and the total variance can be expressed as a sum of

the variances of the effects:

9

Comparin.g this to the model of Cockerham, 1954, where variances of a

kind are summed into one term,

2Total u

G22222

o~ + 0-6 + o-~ + o~6 + u66 + ... ,

and the translation from one representation to the other is obviow.:i.

For single crosses, G.. , and assuming uncorrelated effects, thelJ

variance among unrelated single cross means is the total variance.

2uG

2222u

A+ 0- + 20' +

D M.l2 ') 2+ o· + ' ,.,.DD c..vAAA

2 1-r ... III .,

The numerical subscripts refer to the number of lines invo]:ved in a

variance cOm:Ponent. If we let the components within a class be the

same, ~.~.,2E(M) ..II

2E(M) ..lJ

or 20- ,thenM

2

2uG

2 2+ 2 + 4 2 + 2 2 2 8 20-A 0-D UAA 0-AD + UDD + crAM + .•• •

The variance among three-way cross means, which is not the total

variance, is

+ .... <:'

The numerical subscripts distinguish among 1, 2, and 3 line effect:,.

Again if the cOm:Ponents are the same within a category

+ • II' ..

10

Finally, the variance among four-way crosses if;

2 2 +1: 2 1 2 3 2 1 n 1 ,) 1 ')c. {- (-

0" O"A O"D +4 O"AA + 4" 0" + "8 C' + "8 O"Aj + 61; crG( .. ) (k.e \ 4 AA2 AD" )3 D:':'i",1 J - ) 1 {~ c.

1 2 ,1 2 1 2 9 2 6 2+- O"DD + b4 (JDD + l6 O"AM + 16 uAAA

+16 °AAA_.z,32

3 4 1 2 .-'

If the components within a category are the same, then.

Note that when components in a category are equated, the entire;

variance, whether it be for total, single crosses, three-way crm;sfcs

or double crosses can be generated from the coefficients of and

2O"D' Orgarlizing the variance components into categories reflecting r,b.€'

number of contributing lines affords a convenient way of summarizing

the kinds of effects involved in quadratic forms even thou.gh thE':

effects are viewed. as fixed effects.

Linkage affects the coefficients of the epistatic cOillponent,;,,;w:clc':n

there is control over the grandparents; for eXaillr1e, in a l'our-',1ay

cross (i x j) x (k x .e) , without some recombination of genes witrdr,

a chromosome or reassortment of chromosomes there can ee no

(DD) (ij ) (k.e) component and all of the dominance x dominance i!,t.~::­

actions would be of the (DD) (ik) (ik) type. With y.·ecombir~atic:)n 'J.'~i/C'-·

reassortment, dominance x dominance interactions of the ty}'e

(DD) (ik) (i.e) and (DD) (ik) (j.e) become possible.

recombination the coefficients for triallel and quadTallel crosses

.n

are those given in (2.1) and (2.2). Linkage does not affect the co-

efficients of additive and dominance effects.

2.2 Experimental Model

Each type of hybrid is to be analyzed separately with the same

experimental model giving rise to three analyses of variance. The

experimental model is

where

Y[ Jm is the value of the progeny of cross [J in rep m

~ is the overall mean

r is the effect of replicate mm

G[ J is the genotypic effect of cross [J

e[ Jm is the random error associated wi th cross [J

replicate m.

in

In the case of diallel, triallel, and quadrallel, [J becomes ij ,

i(jk) and (ij)(k.t) ,repsectively. All line indexes, i, j, k, t

= 1, 2,3, ... , n , have the same range, where IT is the total

number of lines, except that they must be distinct for each hybri.d.

For replicates, m = 1, 2, 3, •.. , r .

12

It is convenient at this point to layout the notations to be

used. A dot notation is used to indicate a summation, ~.~.,

Y(ij)(kL). is the summation over all reps of hybrid (ij)(kL).

Y is the summation of all quadrallel crosses with grand-(ij)( •• ).

parental cross i x j over all reps. 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 is the summation of all four-way crosses involving grand-~J ••

parents i, j, and k regardless of how the grandparents were mated,

summed over all reps; Y. ,~J' •

is the summation of all three-way

crosses involving lines i and j summed over all reps. The

summations with parentheses removed can be calculated as simple sums

of the sums with parentheses, but they are convenient for succinctly

expressing sums of squares used in the anaLyses of variance. The

notation ni is used to denote n-i, and en to denote the numberK

of combinations of n things taken K at a time.

The total number of hybrids is e~ = nn/2

3 en /2 f th d ~.er.41. 3 =nnln~ or ree-way crosses, an ~

for single crosses,

way crosses, where reciprocals are omitted. The factor of three for

three-way and four-way crosses comes from the three ways that the

same set of three or four lines can enter a cross.

13

30 A..T'fALYSES OF VARI.fu"iJ"CE

In each analysis the sums of squares for replications) treat­

ments (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 A) D, AAl

, AA2

, AD2

, AAAl

, AAA2

,

AAAy ADy AAAAl , AAAA2, AAAAy AAAA4, DD2, DDy and DD4) with A

indicating additive effects; D, dominance effects; repetitions of

letters, interactions; and the subscript, the number of lines in-

volved 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 hybrid sums of squares had been

partitioned. Of course, it is not possible to obtain a sum of

squares for each type of effect in the model for all analyses; for

example, four-line interactions are not possible when only two-line

crosses are made. Also some of the effects are completely con-

founded with previously fitted effects.

The analysis of variance for diallel crosses is given in Table 1.

The hybrid sum of squares is broken into two parts, additive and

dominance. The analysis of variance for triallel crosses is given in

Table 2. The hybrid sum of squares is broken into seven addi tive

parts, TA, TD, TAA1, TAA2, TAD2) TAAA~, ~nd TAD3

. The analysis of

variance for quadrallel crosses is given in Table 3. The hybrid sum

of squares is broken down into seven addi tive parts) Q,A, QD, QAA2'

QAAAy QADy QAAAA4' and QDD4 •

Table 1 Analysis of variance for progeny of a diallel cross,selfs and reciprocals excluded

14

Source df Sums of Squares

Ey2 2Replicates r-1 2 2Y o ~ t!l

nnl m • 'm mnl

nnl1. E E 2 2Crosses (- - 1) Yij •

_ --y22 .r i<j rnn2

Additive n1 1 L:y2 4 2DA "" --- Y. " e

rn2 i •. rnn2

Dominance nn31. L

2 1 2DD = L Yi' - -- LY.

2 r i<j J. rn2 i ~ •.

+ 2 y2rnln2 .. ~

nnlError (r-1)(-·- - 1) DE by difference

(Crosses x 2

Replicates)

22y2

rnnl I: I: L Yijm -Total -2- -.1 i<j mln2m

Table 2 Analysis of variance of three-w"ay crosses

Source

Correctionfactor

df

1C

Sum of Squares

2y2

rnnln2

Replications (r-l) R

d ...mm

nn~2

- C

Crossesn

( 3C 3- 1 ) H =

E E E y2i j<k i(.ig:)·

~--r

- C

Additive nl TA1

rn2 On-B)E [2Y +Y.] 2 _ 16 2i i( .. ) .. (1.). 1- -,. Y

Dominancenn3

2TD

_I_EErY .. +Y .. ]2 __12 rn 3 i < j 1 (J • ) • J (1 • ) • 2 rn 2n 3

L :2Y +Y ]2i' i. ... (1.).

+ 4rnln2n3

y 2

Add. by Add.One-line

nl TAAI2

rnn2n3 On-B)"] 2 2 2[ y -n 2Y .. (8)YL: n4 i( .. ). .(1.). rn2n33n- •...

i

t-'\}1

Table 2 (Continued)

~_..__._--

AdEi b;.' AddTwo-line

nn3

2 TAA21

rn1n3nl+ ~ ~ [n3Y.(ij).+YiUo).+Yj(L).J2~<J

1rnln2n3n4 E

i[2Y i (o.).+ n2 Y.(io).J2+ 2 y2

Add by Dom n1 n 2 1 E E [ Y - y J2 _ _1_. E[2Y. r ) _ Y (' ) ] 2Two-line -2- TAD2 = 2rn3i<j i(j.) 0 j (1.) 0 2rl1L

3i ~\.o 0 • ~o •

2 y 2 2 2TAAA3 =3! r E E Y~jk 1

EE y2 + E - y ••••Add by Add by Add nOlDS- 3rn4 ij 0 • 3rn3nq i i ••. rn

2n

3n

4Three-line6 r i<j<k 1. • i<:i

Add by DomThree-line

Erro:c

nn2 n 4

3

(r-l)(3 C~-l)

TAD

TE

1~ E E E y2 -TA-TD-TAAI-TAA2-TAD2-TAAA3r i j<k i(jk) 0

j,k/:i... By difference

Total Or C~-l) T=1:EEZ',,2i j<k ~ ~i(jk)m - C

j,k/:i

b-,

Table 3 Ana~sis of variance of four-way crosses

Source df Sum of Squares

CorrectionFactor 1

8y2 ••.••C - rnnln2n,

Replications (r-l)

8 I: y2.•••mR _ m _ C

nnln2n ,

Additive nl

nn,Dominance -2-

Add. by Add. nn,

Two-line -2- [_1 E r {(n2 -7n+14)y(ij)( ) +nln2 i<j •••

- n I: y~ + 8y2l i J. ....

QD - _,_7 ~_~ u, ( ~ ~ ~i. )(j.).J.<J

!L yL ... Jnln2

I: y2 +i i ....

4n2

y2 .•••• ]~ yi ... - 16J. • n

2QAA

2 - rn4nS (nL -7ii+I4)

2ngY(L)(j.).}2

H=.!I:I:I:I: y 2 _r 1<.) k<,t (ij) (kl) • C

i, jf.k, ti<k

QA _ 2 [rn2n,n,.

3C~-1Crosses

Add. by Add. by Add.Three-line

nUlnS~6-

1QAAAg • 3rn6

I: I: I: y2. _ ..,,--...;.4__i<.:i<k iJk.. 3rn4n6

L: E y2i<j ij ••• +

6rngn4 n 6

32 y¥ .•.•I: Y~ •••• - rn2ngn4n~i

!::1

I t y2 + 4 t t y2i<j (i.)(j.). 3rnin3 i<j ij •••

Table'

Add. by Dom.Three-line

(Continued)

nnzn ..-3-

QAD 3 • _1_ t t t ~rn3 1<j k (ij HIt. ).

i .• jl-kI t yZ _ ___2i<j (ij)( .• ) rnin3

1- 3rn3

t t ty2 _ 4i<j<k ijk •• rnin3'

Add. by Add. by Add. byAdd.Four-line

nninZ n 7

24

QAAAA... L t t t ty2 _-1...- t t t.~3r i<j<k<. ijk{. 3rn6 i<j<k ijk••

2nl+ r r r?:+ 3 ..rnl+nSn6 i<j ~J."

2rn .. nSn6

2t y~ •••• + rn3n ..nSn6i

y2

Dom. by Dom.Four-line

Error

nnin .. nS

12

n(r-l) (3 CI+-l)

QDD... ! E E E Er i<j k<t

i,jl-k,t_ QAAAAI+ i<k

QE - by difference

Y~ij)(k£). - QA - QD - QAA 2 - QAAA 3 - QAD 3

Totaln

3r C.. -l 1:. EEEE1:i<j k.d mi,jrk,t

i<k

2Y(ij )(kt)m - C

~

19

4. EXPECTED MEAN SQUARES

Three methods were used in obtaining the expected mean squares

for the three analyses. Those for the diallel, Table 4, were

obtained by substituting the model of effects into the mean squares

and taking expectations assuming uncorrelated effects.

Table 4

Source

Additive

DominaJ1Ce

Error

Expectations of the mean squares of diallel analysis interms of the variance components of the general modeltruncated to dominance by dominance effects

E(MS )

2(J

e

This method was used to check some of the results for the triallel and

quadrallel analyses, but was found to be extremely tedious. The

following method was used to obtain the expected mean squares for the

triallel aJld quadrallel aJla~yses. First, the covariaJlces of genetic

effects of three-way, Table 5, and four-way, Table 6, hybrid relatives

were defined and their expectations obtained in terms of components of

genetic variaJlce. Next the expectations of the uncorrected products

of obtained in terms of2 2 2

and theand squares sums were ~L (J , (Jr e

covariaJlces of re.latives. These are given in Table '7 for the triallel

analysis and Table 8 for the quadrallel analysis. Finally the

results of Tables 5 and 7 were substituted into Table 2 and the

Table ~ Covariances of the genotypic effects of three-way hybrid relatives and their expectations interms of components of genetic variance

Coefficients of Variance ComponentNumber of

Covariance* lines common 0 2 0 2 2 2 2 2 2 2 2 2 2A D °AA1 (1AA2 (1 AD 2 (1 AD 3 (1AAAl °AAA2 °AAA3 °DD2 °DD3

COY} E E[Gi(jk)Gi(jk)] 3 3/2 1/2 9/S 9/S 5/S l/S 66/64 63/32 3/S l/S l/S

Cov2 - E[Gi(jk)Gj(ik)l 3 5/4 1/4 9/16 1 1/4 1/16 17/64 42/32 3/8 1/16 0

Cov3 - E[Gi(j_)Gi(j_)] 2 5/4 1/4 17/16 1/2 5/16 0 65/64 3002 0 1/16 0

Cov~ - E[Gi(j_)Gj(i_)] 2 1 1/4 1/2 1/2 1/4 0 16/64 24/32 0 1/16 0

Covs - E[Gi(j_>G_(ij)l 2 3/4 0 5/16 1/4 0 0 9/64 9/32 0 0 0

Cov6 - E[G_(ij)G_(ij)] 2 1/2 0 1/8 1/8 0 0 2/64 3/32 0 0 0

Cov, - E[Gi( __ )Gi( __ )l 1 1 0 1 0 0 0 64/64 0 0 0 0

Covs - E[Gi( __ )G_(i_)] 1 1/2 0 1/4 0 0 0 8/64 0 0 0 0

Covg - E[G_(i_)G_(i_>] 1 1/4 0 1/16 0 0 0 1/64 0 0 0 0

*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 expectationsin terms of canponents of genetic variance

Buaher ofCoefflcienes of yariance co.ponent

2 ,,2 ,,2 ,,2" iD2

,,2 ,,2 ,,2" iAA3 "~D2 "~D3 "~D~Covarianee* lines COaaGll "A D AAI AA2 AD3 AAAI AAA2 "hAAI aiuA2 ohAA3 ohAA~

Covl - E G(lj)(kl)G(lj)(kl) 4 1 114 1/4 3/4 1/8 1/8 1116 9/16 3/8 1/64 21/64 9116 3/32 1/64 1/32 1/64

Cov2 - E G(lj)(kl)G(ij)(kll 4 1 118 1/4 3/4 1/16 1/16 1/16 9/16 3/8 1/64 21/64 9116 3/32 1/128 0 1/128

Covs - E G(lj)(k_)G(lj)(k_) 3 3/4 1/8 3/16 3/8 1/16 1/32 3/64 9/32 3/32 3/256 21/128 9164 0 1/128 1/128 0

Cov~ - E G(lj)(k_lG(ikl(j_) 3 3/4 1/16 3/16 3/8 1/32 1/64 3/64 9/32 3/32 3/256 21/128 9/64 0 1/256 0 0

Covs - E G(l_)(j_)G(l_)(j_) 2 1/2 1/16 1/8 1/8 1/32 0 1/32 3/32 0 1/128 7/128 0 0 1/256 0 0

Cov6 - E G(ij)(__)G(lj)(__ ) 2 1/2 0 1/8 1/8 0 0 1/32 3/32 0 1/128 71128 0 0 0 0 0

COV7 - ! G(lj)( __ )G(l_)(j_) 2 1/2 0 1/8 1/8 0 0 1/32 3/32 0 1/128 7/128 0 0 0 0 0

Cove - E G(l_)(__ lG(l_)( __ ) 1 1/4 0 1/16 0 0 0 1/64 0 0 1/256 0 0 0 0 0 0

*D••bes indicate any linea Dot eoa-on in the two relatives.

I\)r"'

rrable :- "'. . +-' r • t" ~. t f 2 2 2and the covarianeesJ.ne ey.pec"ta~lons CI praGue s and squares 01 sums ll1 erms 0 ~L , U

r, er

of three-\vay cross l'elativese

Coefficients of CovarianceS tit;; 3qu&.red

N.",· .. " -,,,•. ,.__ ~__ ~

or Pl"duct (r\.l2+0 2) (12 COVI COV2 COV3 Cov4 COV5 COV6 COV7 Cove Covgr e .

'I nnin2 1 r rn3 rn3n4_L_ y1

4 2 '2 r rn3 rn3 2rn3 -2- --- rn3n4 rn3n4rnnlnZ .. 4

') nl n 2 rn3n4... y2 --2- 1 r 0 2rn3 0 0 0 0 0

rn 1n2 i ( .. ) . 2

1 y2 . 1nln2 r r rn3 0 21:n 3 rn3 0 0 rn3n4rnlnZ • (J • ) ,

nl n 21 y y --2- 0 0 r 0 rn3 rn3 0 0 rn3n4 0

rnln2 i( .. ). . (i. ) .

-L. y2 n2 1 r 0 rn3 0 0 0 0 0 0rnz i(j.).

_1_ y2 n2 1 r 0 0 0 0 rn3 0 0 0ruz .(ij).

_1_ y y• (ij). u2 0 0 r 0 rn3 0 0 0 0 0

rn2 i (j . ) .

1. y2 I 1 r 0 0 0 0 0 0 0 0r i(jk).

4rn3 4ru3 8rn3 2rn3 rU3nlf 2rn3nlf 2rn3nlf2 y2 3nlu2 -3-1 r 2r 3 3 3 6 3 3

3rulu2 i ...2 2ru3 2rn3 4ru3 rn3

__1_ y2.3u2 1 r 2r 3 3 3 3 0 0 0

3rn2 ij ••

L y2 3 1 r 2r 0 0 0 0 0 0 03r ijk.

f\)f\)

'I'arLl.e 8 The expectations of' products and squares of sums in terms of ,./, J.L2, ()2 and the covariancesof four-way cross relatives r e

Coefficients of CovarianceSum Squared 2 2 2~ P • .. (ql +0 ) 0 COV1 Cov2 Cov3 Cov!+ Covs Cov6 COV7 Cove

0.. roauc~ r e

6'__4__ y~.... nnln2 n 3 8 8r 16r 32rn4 64rn4 32rn4nS 8rn4nS 32rn4nS 164n4nSn6rnnln2n3

4 ?--~--~Yi- n1 n 2n 3 2 2r 4r 6rn4 12rn4 4rn4nS rn4nS 4rn4nS rn4nSn6

r-'illD2 D 3 ••••

4 2----- Y., . 9n2D3 6 6r 12r 12rn4 24rn4 4rn4nS rn4nS 4rD4ns 0rn2u3 ~J'"

~~ Y:~k 3n3 1 r 2r rD4 2rn4 0 0 0 0j ..n~ ~J""

___1__ Y~i )(j ) 02n3 1 r r 2rn4 2rn4 rn4nS 0 0 0r02uS \. ..

rn~n3 Y(ij)( .• ).Y(i.)(j.), u2 n 3 0 0 2r 0 4rn4 0 0 rn4nS 0

.__....!i- y~ .. f u2n3 2 2r 0 4rn!+ 0 0 rn4nS 0 0rn2nS PJ)"')'

__I_ y2(~j)(k ) n3 1 r 0 rn4 0 0 0 0 0rn3 • . ..

~ Y~ij)(kl). 1 1 rOO 0 0 0 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 and II. These

types of results are instructive in the case of the diallel,

Kempthorne, 1957, but do not appear to be here.

A third method of calculating expected mean squares, Gaylor,

Lucas, and Anderson, 1970, using the forward solution of the

abbreviated Doolittle method\'las used to check the expected mean

squar~of the triallel analysis. This method would be useful for a

particular experiment where the number of lines is fixed, but it is

difficult to apply to a general analysis. This method is of limited

utility if the number of lines is large, as an excessively large

matrix must be swept out by the abbreviated Doolittle method.

Table 9 Expectations of the mean squares of' three-way crosses in terms of components of geneticvariance

Compooents of Variance

MeanSquare

a~D311~ -8- aln

3 OlAA3 OlD2 alAA2 alA2 aiAAI alAI {02+~02 } 0 2D DD2 A

Coefficients of Components of Variance

3r(3n-10)2 r(2n-7)2 rn3

16n3 4n30 0 -4-

3r03 (50-8) ron3 9ron2 n 3 ro02n3

16 (30-8) 4 (3n-8) 32 (3n-8) 8 (3n-8)

3rnl04 rOl04

32n3 ----an;-3r03

"""16

r 010-32) 3rn3 r(41n2-217n+288)TA* 1 r 16 (3n-8) (3n-8) 16(30-8)

r(30-11) 3r04 9r03TD* 1 r 16n3 4n3 """16

TAA~r02 3rn r02n3

1 r 4(30-8) 8 (3n-8) r (3n-8)

TAA~rn2 3rn 01 r8n3 8D3

TAD~r r03

1 r 16 0 16

TAAA; 1 r 9rr 4" 8

#: r'rAD:! 1 r 16

TE*..~ Sum of squares divided by its degrees of freedom.

3r(1010 2-562n+784)32 (3n-8)

r(7n-20)28 (3n-8)

rn2(9n-20)2

64 (3n-8)

r02(50-12)2

16 (3n-8)r (3n-8)

4

rn:{3n-8}

4

I\)VI

Tab1::: 10 Expectations of the mean squares of four-way crosses in terms of components of geneticvariance

MeanSquare "ze a Z

DD4 aiAAA 4 "iD 3

Components of Variance

{z 3 Z } {Z ~z 7 Z"AAA3+Z"AAAA3 "AAz 4 AAAz+t6"AAAAz }

{ Z l z 1Z}{zlz lz lz"D~ADz+t6"DDz "A+4"AA1~AAA1+64"AAAAI

QA* rTI

9r32

3rn4

1627rn4

--n-9rn2D3

-3-2-rn3n4-8-

rn2D3D1t--S--

QD*r (3nz-25n+54) 3rn4n S128(n2 7n+14) 7176'(-n'2-'7~n7+71'4')

r(3n 3-39nz+17n-268)32 (n2 -7D+14)

z3rn4Ds

8 (n 2 -7"-+14)

ZrnltDs

IHil2 -7n+T4)r(n z-7n+14)

16

QAA~

QAAA~

QAD;

QAAAA~

QDDt

QE*

3rn 1n2 Z

r{n Z-5n+S) rn2DS

64(n2 7n+14) 32(n2-7n+14) 16(n'-7n+14)

r 9r rnG

TI TI ""i6""

r rn3ill 0

64r 9r

32 32r 0

128

JrnlD2DS rnlD2n4DS...... I' 2 ,_.

f\)0\

27

5. TESTS OF HYPO'rHESES FOR FIXED EFFECTS

Certain tests of hypotheses are available wi thout making any

assumptions about the genetic effects. The mean square expectations

in Tables 4, 9, and 10 in this case serve only as guides to the

types of effects that can contribute to the mean squa.res; the mean

squares actually involve quadratic functions of these types of

effects. 'rhe error mean square can be used as the der::.ominator in an

:F' ratio testing sequential.ly up each table. Table 11 gives lowest

order types of effects that are tested in each mean square for each

analysis; higher order effects are also tested for in each. As we

proceed to test up the table lower order genetic effects become

involved. The method of obtaining each analysis of variance

guarantees that quadratic forms of previously fitted effects do not

appear in SUbsequent mean squares, although similar interaction type

effects, ~.~., AA2

after the fitting of AAl

, and AD3

after the

fi tting of AD2

, may appear in subsequent mean squares. Also there

are two things which complicate the interpretation of the non­

significance of a particular mean square. First, the genetic model

effects are a sl.urunation of allelic effects and may sum to zero when

a.llelic effects are present. Also, the qU8.dratic functions of a

particular type of genetic effect differ from mean square to mean

square so the conclusion that certain quadratic functions are zero in

one mean square does not gnarantee that the same is trCle in another

mean square.

~~able 11 Lowest order type of effects tested for in the diallel,triallel and quadrallel analyses for the various meansquares. A -- indicates there is no corresponding meansquare for that analysis

28

MeanSquare DIALLEL TRIALLEL QUADRALLEL

A Ai Ai Ai

D Dij

Dij

Dij

AAl AAit

AA2 AAij

AAij

AD 2 ADi(ij)

AAA3 AAAijk

AAAijk

AD3 ADi(jk) ADi(jk)

AAAA'4 AAAAijk£.

DD4 DD (ij) (k£.)

29

6, VARIANCE Co.MPONENTS AND TESTS OF HYPOTHESES

By assuming the effects of the genetic model are random, and un-

correlated, and that there are cammon variances within certain

categories, genetic variance components can be estimated. These

assumptions were made in arriving at the expectations of the mean

squares. When the general model is truncated for each analysis to

those terms given in Table 11, the comparable variance components can

be estimated by equating mean squares to expected mean squares and

solving the resulting equations, In the diallel it is possible to

2~AA '

2

2~AA '

2 1 2~A' ~D'

2 2 2 2estimate ~A and ~D; in the triallel, ~A' ~D'

222~AD ' ~AAA...' and ~AD ; and in the quadrallel,

2 2 2m~ 2 3 2 2~AA ' ~AAA...' ~AD ' ~AAAA ' and ~DD

4' Other vari ance

2--~ 3 4components defined and given in the tables of expected mean squares

are confounded with these estimators although not always in a simple

manner.

The diallel ~lalysis gives a good example of simple patterns of

confounding; all one-line variance components are completely con-

variance components are estimated together,

founded and estimated as one package,

2~ in the following

DD3three-line variance

and2~AD

3dominance,

~~ + ~~ , and al.l two-line

21 2 2~D + 2~AA + 2~AD

2 2An example of a more difficult confounding pattern can be

Two types of additive by

the triallel analysi s for

2+ ~DD •

2seen in

manner.

components are distinguished, depending on bow the effects come to­

222gether in taking expectations. Let E(ADi ('k)} = E(ADi ('k)} = ~AD

- J_ - .J. 1 32

and E(AD!(ik)'AD!(j~)} = ~2AD3 ' the underscore indicating whether2

the grandparental source of alleles is the same, ~ ,orlA.D3

30

is

that can be

sources,

is completely

the effects,

was the grand-

2o-DD

3than to show that

i x j

the underscore emphasizes that

2and both are termed GAD .

3variance components

2(J

2AD

3component or sum of

20- AD . The distinction, other

I 3are confounded, does not appear useful so

2In 0- AD

2 3AD ( )' come from different grandparental~ j~,

i x k. With this distinction made,

and

and

Any variance

i x j

confounded with

2 2(JDD and ()AD

3 3assumed eqllal to

2 2different, 0- AD In 0- AD

2 3 I 3E{ADf(.sik )} - E{AD~:C.sik) ,. AD~(.sik)} and that

parental cross referenced by i and j.

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. With each analysis restricted to terms in

Table 11, DD*/DE* , T~/TE* , and QDDt/Q,E* provide exact tes ts for

2 20 and 2

== 0 • Co..mposi te hypotheses, testing(JD = 0 , (JAD (JDD3 4

each mean square versus error, are possible for the linear functions

of variance components given in the tables of expected mean squares,

o .

~.~., in the triallel TAD*~TE* tests the hypothesis that

l' 2 rn3 2 3rn3 2lb (JAD

3+ 16 (JAD

2+~ (JAP.A

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, (L:ai MSi ) is approximately distributed as x2rl/f l with fl

degrees of freedom where

222fl = (L:a.MS.) /L:(a.MS./f.)

~ ~ ~ l l(6.1)

31

and f.l

denotes the degrees of freedom for mean squares MS.l

Using

Satterthwai te I s approximation, error terms can be constructed to test

each of the components of variance. For f'.-Xample, in the triallel

analysis

4T~ - 3TE*

has expectation

term for testing and can bein

expectation of an error

*T~

~2 + £ ~A2 ,the correcte 4 D3

the significance of ~~

used to form an approximate F-test

TAAA*3

4T~ - 3TE*

with degrees of freedom nn l n5/6 and f' where f' can be obtained

from (6.1).

32

'7. DISCUSSION OF EESULTS

7.1 Diallel

Since truncation of the general model to additive ai1d dominance

effects corresponds exactly to the usual model for general and

specific combining ability, the partitioning of the SlUllS 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. 'l'he single­

line types are confounded with each other and must be estimated

jointly. The two-line types are also completely confounded and must

be estimated in a single package. It is the splitting of the

epistatic variance into two parts, within line and between lines, that

makes the estimation of single-line and two-line packages possible.

All single-line effects are removed with additive effects. As one

would expect from the expression of the total genetic variance for

two-line crosses, only one-line and two-line variance components

appear in the analysis.

7.2 Triallel

Examination of the expected mean squares indj.cates that whereas

the genetic model is simple in concept and interpretation the

expected mean squares are complex. It can be seen that the order of

fi tting of dominance and addi tive by addi tive, single-line effects is

innnaterialo Th.is is also true 1'01' addi tive by addi tiV2, two-line

effects and addi tive by dominance, two-line effects.

33

In considering a fixed effects model., it is possible to combine

the mean squares in the analysis presented into single-line (TA + TAA1

),

two-line (TD + TAA2 + TAD2 ) and three-line (TA~ + TAD3

)

partitions to give a new analysis. In testing against error

sequentially up the resulting analysis, three-line, two-line, and

single-line effects successively come into play.

In the estimation of genetic variances mean squares of the analysis

may be combined to correspond to assumptions about the variance

2components. If it is assumed that ~AA and

1the corresponding mean squares in the analysis

2~AA are identical,

2can be combined to

identical, the corresponding mean squares can be

andestimate2

~AA . Likewise if it is assumed that 2~AD

2combined

2~ areAD

3to estimate

2~AD' When these mean squares are combined, weighting by the degrees

of freedom, the coefficients of the variance components in the result-

ing expected mean squares remain complex. If it is assumed that

2 2~AD and ~AAA = 0 , it then becomes possible to

2by manipUlation of the mean squares.

The point is that by assuming genetic variance components within a

category to be equal, other higher order variance components become

estimable.

This analysis can be compared to that of Rawlings and Cockerham,

1962a. There is a simple relation between the sums of squares in the

two analyses, Table 12.

Table 12

34

Relation of su.ms of squares of Rawlings and Cockerham,1962a, to those of the general model for trial1el crosses

Description

l-line

2-line 2-alleles

2-1ine 3-alleles

3-line 3-alleles

Sluns of SquaresRawlings and Cockerham Su..llS of Squares

1962a General Model

G + °1 TA + TAAl

S2 + ° TD + TAA22a

°2 TAD2b

S3 T~

°3 TAD3

The two analyses differ in the genetic variance components that

are estimable. In the analysis of Rawlings and Cockerham, 1962a, the

design components of variance were expressed in terms of covariances of

relatives, and these in turn in terms of genetic variance components.

The estimation of genetic variance components was then accomplished

by equating the estimated design components of variance to their

expected values in terms of genetic components of variance, and

solving the resulting equations after suitably restricting the

genetic variance components. When the genetic varianr:;e components

were restricted to the seven lowest order ones, j_ t was found that

there was a Unear dependency in the seven reSUlting equations so that

In the design presented here, seven

only six genetic variance components,

and 2er ,could be estimated.Q1Q1Q1

2er

01

2(T& '

2er ,cxa

2er0'& '

35

genetic variance components can be estimat(..j,

then

2uAA '

J_kinds of

2(Y

'D '

eqC1al tois as m.,med

2U ,but onl,y- five di.stinctAD

3'f' 2L u

AD2

, and

Again,

222uAA ' uAD ' uAM

2 2 3variance components.

one can estimate

7.3 Quadrallel

The coefficients of the genetic components of variance in the

expected mean squares are complex functions of the numbers of grand-

parents. The order of fitting of D and AA2

affects the corres­

ponding mean squares. The fitting of first D then AA2

was adopted

as most reasonable. This analysis does not offer the possibility of

combining of mean squares for the estimation of variance components as

was possible for the triallel analysis because mean squares in the

analysis are not avai lable for the two types of addi tive by addi tive

or additive by dominance effects. It is reasonable, however, when

analyzing fixed effects, to combine the mean squares for dominance and

additive by additive, two-line effects to give a mean square corres-

ponding to two-line effects. Combining Q,AA.4.3

ar..d gives a

mean square for three-line effects; Q/l.AAA4 and QDD4 , a mean

square for four-line effects. The analysis then separates one-line,

two-line effects corrected for one-line effects; thr'ee-line effects

corrected for one-line and two-line effects; and fouY'-lirle effects

corrected for one, two, and three-line effects.

Wi th two exceptions ttere is an exact corn;spondence between the

sums of squares for this analysis and those of Rawlings and Cockerham,

1962b, Table 13. Reversing the order of fitting of D and AA2

36

effects gives the identical sums of squares of Ha-wlings aEd

Cockerham,1g62b, S2 = QAA2 T2 = QD' , the prime indicating that

the order of fitting effects is AA2

, D .

Table 13 Relation of SllJnS of squares of Rawlings and Cockerham,1962b, to those of the general model for quadrallelcrosses

Description

l-line

2-line 2-alleles

3-line 3-alleles

4-line 4-alleles

Sum of SquaresRawlings and Cockerham

1s;62b

G

Sum of SquaresGcnE.;roal Model

QD + QAA2

QAA'2

QD'

Q,AD3

With a restricted genetic model, Rawlings and Cockerham were able

to estimate six genetic variance components, 2CJ

c¥

2CJ ,

ot:i

2()er6 '

A corresponding variance component is estimable for

each of these in the analysis presented here. In the analysis

37

presented here we are able to estimate a seventh variac':::e component,

However, Rawlings and Cockerham could have estimated a2(J'AAAA

4corresponding variance component, , had they not restricted

their genetic model.

7.4 General Discussion

The primary purpose of developing the analyses of variance for

diallel, triallel, and quadrallel crosses was to demonstrate how the

hybrid sum of squares would be partitioned if a uniform genetic model

was used in all three analyses. This use of a general genetic model

for the development of the parti tioning of tb,e various hybrid sums of

squares is in contrast to previous use of design models for each of the

analyses. The sums of squares were developed by successively fitting

a more complex genetic model so that each line in the resulting

analysis of variance is corrected for previously fitted effects. The

partitioning developed can be used in three ways. with no assumptions

concerning population structure) the stuns of squares can be used to

test for fixed effects. This use would be hel.pful in analyzing crosseS

of elite lines where assumptions of random mating and of no selection

are seldom tenable. vJi th the asslunptions given by Cockerham, 1954,

1961, covariances of relatives can be related to genetic variance

components and the analyses presented here can be used to estimate and

test these genetic variance components. Finally, a new set of genetic

variance components can be defined in terms of tte gF;neral genetic

model used in developing the partitioning of the hybrid Stun of squares

and these can be estimated, tested, and related to previously used

genetic variance components.

38

The variance components defined and used iG tLese analyses are

directly related to previously used variance components for addi tive

and dominance effects. It:!.s in the epistatic variance components

that the two analyses differ; the previously defined epistatic

variance components are partitioned into variance compotwnts that

additive by additive genetic variance component,

reflect the nwnber of lines contributing effectf;. For examp le, the

2(J , of the

o.a

standard analysis is divided into an additive by additive, one-line

and an additive by additive, two-line component

component arises from interactions of alleles

The two-loci, but between the genes contributed by one line.

2component (JAA

1The one-line2

(JAA .2

between

line component arises from interactions of alleles between loci and

between genes of two lines. It could be argued that adapted lines

have adapted AAl effects, giving some reason for separating AAl

effects from AA2 effects. The other epistatic components are

parti tioned similarly.

The correlations between the addi tive deviations, ex, and

between the dominance deviations, 13, of Rawlings and Cockerham,

1962a and 1962b, are directly related to the coefficients of the

genetic variance components used in expreffiing the expectations of co-

variances of relatives. If .lines used in constructing hybrids are

completely inbred, summing coefficients of components of genetic

variance wi thin a category gives the corresponding correlation of

Rawlings and Cockerham when their Ct is multiplied by two.

1:1 trle analysis of diallel crosses the hybrid sum of squares is

partitioned into two parts, there are two covariances among relatives,

39

and with suitable restrictions there are two genetic variance

components, 2 . h ttJD

,t__a· can bi~ estimated. If'- the analysis of

triallel crosses the hybrid sum of squares is partitioned into seven

parts; there are nine covariances among relatives, and there are seven

';::J 2 2 2 2genetic variance components, tJ~ , tJD

, iYAA, fJAA

, tJAD

,2 2 1 2 2

tJ~, tJ

AD; that can be estimated with sui table res tri c ti ons. If

3variance components wi thin a category are assumed identical, then by

pooling lines in the analysis of variance there are five variance

components,2

(JAM '2

(JAD ' that can be estimated.

In the analysis of quadrallel cross hybrids the hybrid sum of squares

is partitioned into seven parts; there are eight covariances

relatives, and there are seven genetic variance components,

among

2 2 2 2 2 2tJD ' tJAA , tJ

AM, tJ

AD,

tJAAM ' tJDD

, that can be estimated2 3 3 "

4...,

with sui table restrictions. In tbis analysis it is rlOt possible to

combine variance components within a category as only one variance

component within a category is estimable.

The mi.rimum number of lines necessary for a complete analysis for

each of the analyses is the minimum munber of lines necessary to con-

struct at least one pair of u.r.re.lated hybrids. For example, with

four lines A, B, C, D, it is possible to construct unrelated single

crosses A x Band C x D so that a complete diallel analysis is

possible; for the triallel, six lines are needed; and for the

quadrallel, eight lines are needed. The minimulll nUlllber of hybrids are

6, 60, and 210 for diallel, triallel, and quadrallel designs, re-

spectively. Additions to the numbers of parerltal lines sampled

increase the number of hybrids dramatically. Par exarnplF~, adding

40

only one additional line to the three designs increases the numbers

of hybrids to la, 105 and 378. It is possible that some systematic

sUbsampling of hybrids (partial designs) in the case of the triallel

and quadrallel would be beneficial by allowing a greater sampling of

parental lines without the concomitant increase in total hybrids

required by the complete designs.

One point exemplified by these analyses is the confounding of

genotypic effects with line effects. In the diallel analysis there

are only one and two-line type effects. All one-line effects are

completely confounded with additive effects. Two-line epistatic

effects are combined with dominance effects. In the triallel analysis

there are one, two, and three-line effects and these show up in the

different mean squares of the analysis of variance,splitting mean

squares that would correspond to the usual epistatic variance

components. In the quadrallel analysis there are one, two, three, and

four-line effects and within a category, say dominance by dominance,

the lower-line variance components,

previously fitted categories.

2G"DD '

2

2G"DD ' are confounded with

3

41

8. SUMMARY

A quadratic analysis of diallel, tri allel, aQd quadrallel hybrids

is pTovided using a general genetic model. Sums of squares are

developed by fitting successively additive, dominance, additive by

additive, etc. effects. In the fitting process, the standard

epistatic variance components are split into categories indexed by the

number of lines contributing alleles to the effect.

standard additive by additive variance component,

For example, the

2Cf ,is split into

ot:X

two components,2

and2

with numerical subscriptsCf CfAA ' indexingAAl 2

2 2the number of lines contributing to the effect. Also Cf 2CfAA($X

2 2 2 2 '1 2 1(:..

+ 2(JAA ; assuming (JAA::=

(JAA::=

(JAA , then (J - 402 1 2 00 AA

For the diallel analysis, the results are pssentially identical to

those of the standard analysis (~.. ~., Kempthorne, 1957). Tb.ere are two

covariances of relatives, two hybrid sums of squares, and two variance

components (if the model is suitably restricted) that can be estimated.

For the triallel anaysis, the results are somewhat different from

those of the analysis of Rawlings and Cockerham, 1962a. Both

analyses have nine covariances of relatives and seven hybrid Sl,1illS of

squares. With the restrictions on the genetic model used by Rawlings

and Cockerham, 1962a, six genetic variance components can be estimated.

With the analysis presented here, seven genetic variance components

can be estimated; however, some pairs of these components correspond

to the same category of effects in the standard model, ~'f£..,

2 2(J correspond to (J For tte quadrallel analysis, the

AA2

0l0I

are similar to the analysis of Rawlings and Cockerham, 1962b.

2(JAA '

1results

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 2O"~, O"AD

3'

be tested,

222O"A ' O"D' O"AA '

2Genetic variance components can2

O"DD •4

tests usually involve linear combinations of mean squares.the

and2O"AAM '4although

Tables of expected mean squares are given and are useful in determining

confounding patterns of the genetic effects.

If the genetic effects are considered fixed, it is possible to

make certain tests of hypotheses without making any assumptions about

the genetic effects. These tests are discussed.

43

9. LIST OF REF'ERENCES

Cockerham, C. Clark. 1954. An extension of the concept of partition­ing hereditary variance for analysis of c:u"a.Y.·iances amongrelatives when epistasis is present. Genetics 39:859-882.

Cockerham, C. Clark. 1961. Implications of gc~Getic variances inhybrid breeding program. Crop Science 1:1.+'7-52.

Cockerham, C. Clark. 1963. Estimation of genetic variances.Symposium on statistical genetics and plant breedi.ng. NAS-NRC983: 53-94.

Cockerham, C. Clark. 1972. Random 'Is. fixed effects in plant genetics.Paper presented at the Seventh International Biometrics Conference.

Eberhart, S.A. 1964. Theoretical relations among single, three-way,and double cross hybrids. Biometrics 20:522-539.

Eberhart, S.A. and C.O. Gardner. 1966. A general model for geneticeffects. Biometrics 22:864-881.

Gardner, C.O. and S.A. Eberhart. 1966. Analysis and interpretationof the variety cross diallel and related populations. Bio­metrics 22:439-452.

Gaylor, D.W., H.L. Lucas, wld R.L. ~~de~son. 1970. Calculation ofthe expected mean squares by the abbreviated Doo.li ttle and squareroot methods. Biometrics 26:641-655.

Griffing, B. 1950. Analysis of quantitative gene action by constantparent regression and related techniques. Genetics 35:303-321.

Griffing, B. 1950. A generalized treatment of the use of diallelcrosses in quanti tative inheritance. Heredity 10:31-50.

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44

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Appendix I

46

Expectations of the mean squares for the triallel analysisin terms of the covariances of relatives

MeanCovl Cov2 Cov3 Cov4 CovSSquare

r(5n-l6) rn3(Sn-16) 4rn3n4 2rn3 (3n-16)TA* r 3n-8 3n-8 3n-8 3n-8

rnSr(n 2-7n+14) ~ti1..n-11 ).TD* r rn 5 -

n3 n3 n3

2rn4 2rn4(2n-S) 2r(2n 2-11n+16) 4rn4TAAr r - 3n---a 3n-8 (3n-8) 3n-8

TAA~2r -2r 2r 4rrn3 - _.

n3 n3

TAD; r -r rn5 - rn 4 2r

TAAA~ r 2r -2r -2r -4r

TAD; -2r r 2rr -r

MeanCov€Square Cov7 Cov·e COVg

rn3ne rn3n4n4 2rn3n4ne rn3n4(n-16)TA*

3n-8 3n-8 3n-8 3n-8

rn5 2rn4n5 rn4n9TD* - rn 4n3 - ----n3 n3

TAA~ .! (2n 2-9n+8) rn4(n2-9n+16) 2rn4(n2-5n+8) 2rnln4n4(3n-B) 2(3n-8) 3n-8 3n-8

TAA~r(n 2-6n+7) 4r 2r(n 2-6n+6)r

n3 n3 n3

TAD~ -r - rn 4 2rn4 - rn 4

TAAA; -r r 4r 4r

*TAD 3 -r. r -2r r

Appendix II Expectations of the mean squares for the quadrallel analysisin terms of the covariances of relatives

r(n 4-16n 3+93n 2-230n+216)2(n 2 -7n+14)

8r(n2-9n+16) rnln2n6n9. (n 2 -7n+14) - (n 2 -7n+14)

MeanSquare COVl COV2

QA* r 2r

QD* r r(n 2 -11n+211(n 2 -7n+14)

4rn3QAA~ r (n 2-7n+i4)

QAAA; r 2r

QAD; r -r

QAAAA~ r 2r

QDD~ r -r

MeanSquare CovS

QA* 2rnsne

rn4ns(n2-11n+42)QD*

(n 2-7n+14)

QAA~4r(n 3-10n 2+29n-2B)

- . (n 2 -7n+14)

QAAA~ - 4rn 3

QAD~ -rnS

QAAAA~ 8r

QDD* 2r4

r(3n-16)

2rn4(n2"9n+24)

(n 2 -7n+14)

2r(n3-12n?~45n -58)(n 2 -fn+14)

-4r

-4r

COV6

rnsne--2- 2

2rn4nS

-rne

-rns

2r

2r

2r(3n-16)

2rn4(n2-15n+46)

(n 2 -7n+14)

4r(n 2-11n+22)(n 2 -7n+14)

2rnlO

-rns

-8r

4r

2rnsne

8rn4n;

(n 2 -7n+14)

- 4rn e

2rns

8r

-4r

Cove

rnSn6n16

22rn4nSn6n9

(n 2 -7n+14)

3r (3n-22)

o

-12r

o