An Introduction to Plant Breeding (Brown/An Introduction to Plant Breeding) || Predictions

Caligari: “CALIG_CH06” — 2007/11/5 — 16:06 — PAGE 96 — #1

6

Predictions

INTRODUCTION

Plant breeders strive to make a wide range of predic-tions which allow them to act in the most effectiveway in creating genetic variation and selecting desirablegenotypes.

There are a variety of questions that might be posed,but ones that a breeder might sensibly ask wouldinclude:

• What expression of what traits would be most suc-cessful when a cultivar is released?• What will the cultivar need to display to meet agro-

nomic requirements so as to fit ideally into the mosteffective management systems?• How stable will be the expression of the impor-

tant traits (e.g. yield) over a range of environments(especially locations and years)?• Which parents will give the best progeny for further

breeding or for commercial exploitation?• Which traits will respond most significantly to the

selection imposed?• What type and level of selection will give the optimal

response in the traits of interest?

The first question is clearly difficult to answer and isone that faces breeders all the time. The second questionis partly a matter of selection conditions, partly a matterof judging what is required and partly luck! The thirdis one on which a considerable amount of work hasbeen carried out and is, of course, one of genotype byenvironment interaction. It is an important aspect ofbreeding but despite its importance we intend to onlyspend a limited amount of time on it.

However, questions such as the last three (above)need to be answered by a combination of knowledge

of genetics, experimental design and statistics – thebetter our knowledge the more accurate should be ourpredictions!

Let us start by considering the stability of expressionover environments and its genetic determination.

Genotype × environment interactions

The performance (in other words its phenotype) of agenotype will differ in different environments. This isstraightforward and so if we give less fertilizer we gener-ally get less yield, the more we space the plants of manyspecies the more vegetative growth they make. This canclearly be handled in our selection programme by select-ing under the conditions we think are most appropriateand if necessary in several different ones. But what ismore complicated is that not all genotypes respondto the same extent or necessarily in the same way todifferences in the numerous environmental variables.

So some genotypes are more drought tolerant, somemore disease susceptible, some can withstand higher lev-els of salt than counterparts, and so on. We cannot growall the possible genotypes we are interested in underall the possible environmental conditions. We mightnote that some environmental variables are of a ‘macro’nature and fairly obvious, but there are numerous possi-ble differences in the environment that are experiencedby individual plants – ‘micro’ ones.

So winter or spring sowings give a clear set of envi-ronmental differences that we might take specificallyinto account, just as different latitudes, temperature,semi-arid and tropical climates might be important con-siderations in an international breeding programme.But differences in water availability at one end of testarea compared to the other, the row spacing produced byone piece of farm equipment compared with another,


Predictions 97

are but a few of the possible subtle differences whichmight be very important to individuals or plots or fieldsor farms.

There are a number of ways that plant breeders try totake genotype× environment interactions (G×E) intoaccount in breeding but it does mean that there a is needto carry out trials over a range of environments whichmight be simply different sites or over years/seasons orrunning trials with defined differences such as waterlevels etc. Genotype × environment at its simplest canbe examined by looking at the variance (or standarderror by taking the square root) of the phenotypes overthe range of environments and selecting for the low-est variance as being the most stable – rememberingthat we also need a good mean expression! Or a muchmore sophisticated approach is to use the mean of allthe material grown to provide a biological measure ofthe environment and compare (usually by regression−slope of the line) the individual lines, families, clones,etc. against this.

It is not appropriate to go into greater details hereabout G × E or the various possibilities to take it intoaccount. Needless to say many breeding programmeseffectively ignore G × E in an explicit way but takeit into account to a modest extent by trialling themore advanced material at different sites, by the factthat selection is carried out over a number of years,etc. A more detailed examination of analyses of mul-tiple year and multiple location trials is presented inChapter 7.

Let us now consider prediction protocols associatedwith answering which parents will give the best progenyand which traits will respond most significantly to theselection we impose?

Genetically based predictions

Plant breeders use all the genetic information (qualita-tive and quantitative) in just the same way as we usethe information from Mendelian Genetics – in otherwords to predict the properties of generations or fam-ilies that have not actually been observed. So from ananalysis of the observed variation, firstly determine howmuch of the variation is due to environmental effectsand how much due to genetic effects. Often it is desir-able to go further and separate the genetic into additive,dominance, and other genetic effects, and to determine

which direction is dominance acting in, and to whatdegree.

Let us consider one particular use of the information,how can we predict the response to selection? Before youstart a selection programme you would obviously liketo know what sort of response you might expect for anygiven input, which traits are worth targeting, whichpopulations or crosses are best to use. Is it a worth-while venture? These questions involve many aspectsof the biology of the crop, its handling in agriculture,the availability of other methods to affect crops, forexample, chemicals such as pesticides, herbicides etc.But one of the main components that will determinethe outcome is the amount and type of variation that ispresent. For example, in the extreme case of no geneticvariation the breeder is wasting time trying to selectsuperior genotypes!

If genetic variation is present, but small comparedwith that due to the environment, then progress can bemade but only very slowly, unless very large numbers arehandled – in other words much of the time the breederwill be selecting phenotypes which are ‘superior’ but asthis is mostly due to the environment it will not givea reliable indication of a ‘superior’ genotype. If on theother hand the phenotype is a good reflection of thegenotype, that is most of the variation is genetic, thenprogress will be quick, since when the breeder selects agood phenotype and uses it as a parent it will pass onthe superior attributes (via its genes) to its offspring.

One obvious question to ask is can we estimate howmuch of the variation we observe, in for example theF2 we were looking at, could be ascribed to geneticdifferences of segregating genes and how much to envi-ronmental causes? We therefore need to measure theproportion of the total observed variation, that is geneticvariation, and such a measure is called heritability.

HERITABILITY

For a modern plant breeder (or indeed a farmer with noknowledge of genetics) to make progress in an organizedprogramme of selective breeding, two conditions area must:

• There must be some phenotypic variation within thecrop. This would normally be expected, even if it weredue entirely to the effects of a variable environment


98 An Introduction to Plant Breeding

• At least some of this phenotypic variation must havea genetic basis

This relates to the concept of heritability, the propor-tion of the phenotypic variance that is genetic in origin.This proportion is called the heritability and this sectionis concerned with the ways of estimating heritability.

Values of heritability (h2) can range from zero to one.If h2 is relatively high (e.g. close to 1) there is potentialfor a breeding programme to alter the mean expressionof the character in future generations. On the otherhand, if h2 is close to zero, there will be little scope foradvancement and there would probably be little pointin trying to improve this character in a plant breedingprogramme.

There are three main ways of estimating heritability.

• Carrying out particular genetic crosses so that theresulting data can be partitioned into their geneticand environmental components• Based on the direct measurement of the degree of

resemblance between offspring and one, or both, oftheir parents. This is achieved by regression of theformer onto the latter in the absence of selection• Measuring the response of a population to given levels

of selection (this will not be discussed until we coverselection later)

The essential background theory of heritability waspresented in the previous quantitative genetics section.

Heritability is a ratio of genetic variance dividedby total phenotypic variance. In a simple additive–dominance model of quantitative inheritance the totalgenetic variance will contain dominance genetic vari-ance (denoted by VD) and additive genetic variance(denoted by VA). Dominance genetic variance is vari-ation caused by heterozygotes in the population, whileadditive genetic variance is variation between homozy-gotes in the segregating population.

Broad-sense heritability (h2b) is the total genetic vari-

ance divided by the total phenotypic variance. Thetotal genetic variance in an additive–dominance modelis simply VA + VD. The total phenotypic variance isobtained by summing the genetic variance plus theenvironmental variance.

The degree of heterozygosity within segregating pop-ulations will be related to the number of selfing gener-ations. Maximum heterozygosity will be found in the

F1 family, and will be reduced, by half, in each subse-quent selfed progeny. Similarly, the dominance geneticvariance will be dependent on the degree of heterozy-gosity in the population and will differ between filialgenerations. A more useful form of heritability for plantbreeders, therefore, is narrow-sense heritability (h2

n),which is the ratio of additive genetic variance (VA) tototal phenotypic variance.

Why should lack of resemblance between parentsand their offspring be attributable to dominance, butnot additive, components? Well, dominance effects area feature of particular genotypes; but, genotypes are‘made’ and ‘unmade’ between generations as a result ofgenetic segregation during the production of gametes.Thus, the mean dominance effect in the offspring of aparticular cross can be different from that of the par-ents, even when there is no selection. On the otherhand, when selection is applied, there may be no changeor even change in the ‘wrong’ direction. This is nottrue of additive genetic effects. The additive geneticcomponent must remain more or less constant fromone generation to the next in the absence of selection.While, if differential selection is applied, the changebetween generations must be in the direction corre-sponding to the favoured alleles. In addition, additivegenetic variance is constant between filial generationsand so narrow-sense heritability of recombinant inbredlines can be estimated from early-generation segregatingfamilies.

In the first filial generation, after hybridizationbetween two homozygous parents (F1), there is nogenetic variance between progeny and all variationobserved between F1 plants will be entirely environ-mental. The first generation for which there are bothgenetic and environmental components of phenotypicvariance is the F2. Partitioning of phenotypic varianceand the calculation of the broad-sense (and ultimatelynarrow-sense) heritabilities will be confined to thisgeneration.

Broad-sense heritability

The first step is to derive an equation for the geneticvariance of the F2 generation. The genetic variance ofthe F2 generation (without proof ) is:

σ 2F̄2= 1

2VA + 1

4VD


Predictions 99

The total phenotypic variance of any generation isthe sum of its genetic variance plus the environmentalvariance, E. So the total phenotypic variance of the F2generation can be written:

σ 2F̄2= 1

2VA + 1

4VD + σ 2

E

In terms of VA, VD and σ 2E therefore the broad-sense

heritability of the F2 generation is:

h2b =

genetic variance

total phenotypic variance

=12 VA + 1

4 VD12 VA + 1

4 VD + σ 2E

In order to estimate the broad-sense heritability of theF2 family (or indeed any other segregating family) allthat is required is an estimate of total phenotypic vari-ation, and an estimate of environmental variation. Theformer is obtained by measurements on plants withinF2 families, while the latter is estimated from mea-surements on families or plants that have a uniformgenotype (i.e. homozygous parental lines or F1 familieswhere plants are genetically identical and any variationbetween plants is due to environment).

To illustrate this, consider a simple numerical exam-ple. A field experiment with an inbreeding crop specieswas conducted which included 20 plants from Parent 1,20 plants from Parent 2 and 100 plants from the F2 fam-ily derived from selfing the F1 generation obtained byinter-crossing the two parents. These 140 plants werecompletely randomized within the experiment and atharvest the weight of seeds from each plant recorded.The variances in seed weight of the two parents were,σ 2

P̄1= 16.8 g2 and σ 2

P̄2= 18.4 g2. The phenotypic vari-

ance (which included both genetic and environmentalvariation) of the F2 was σ 2

F̄2= 56.9 kg2. Total pheno-

typic variance of the F2 generation is 12 VA+ 1

4 VD+σ 2E

and is estimated to be 56.9 k2 (the variance of the F2).It therefore follows that the broad-sense heritability, h2

bfor these data is:

h2b =

56.9− σ 2E

56.9

The problem reduces to: what is the value of theenvironmental component of the phenotypic variance,σ 2

E ? Since, by definition, both parents are completelyhomozygous inbreds, any variance displayed by either

must be attributable exclusively to the environment.The best estimate of the value ofσ 2

E is therefore the meanphenotypic variance of these two generations. Thus:

σ 2E = (16.8+ 18.4)/2 = 17.6 kg2

and

h2b =

56.9− 17.6

56.9= 0.691

In other words, 69.1% of the phenotypic variance ofthe F2 generation is estimated to be genetic in origin.

The other generation in which the phenotypic vari-ance is also entirely attributable to environmental effectsis the F1. If the phenotypic variances of all three ofthese genotypically invariate generations were avail-able, the environmental component of the phenotypicvariance of the F2 generation could be estimated asfollows:

σE =σ 2

P̄1+ 2σ 2

F̄1+ σ 2

P̄2

4

Research workers often use more elaborate formulae,but this one will serve our purpose.

Narrow-sense heritability

Often it is of more interest, for reasons already noted,to know what proportion of the total phenotypic vari-ation is traceable to additive genetic effects rather thantotal genetic effects. This ratio of additive genetic vari-ance to total phenotypic variance is called narrow-senseheritability (denoted by h2

n) and is calculated as:

h2n =

additive genetic variance

total phenotypic variance

Therefore, in terms of VA, VD and σ 2E , what is the

narrow-sense heritability of the F2 generation? Since:

h2b =

12 VA + 1

4 VD12 VA + 1

4 VD + σ 2E

it is reasonable to suppose that:

h2n =

12 VA

12 VA + 1

4 VD + σ 2E

In order to estimate the narrow-sense heritability it istherefore necessary to partition the genetic variance into



its two components (VA and VD). This is done by con-sidering the phenotypic variance of the two back crossfamilies (σ 2

B̄1and σ 2

B̄2). Without proof the expected

variances of σ 2B̄1

and σ 2B̄2

are:

σ 2B̄1= 1

4VA + 1

4VD − 1

2[�(a)×�(d)] + σ 2

E

σ 2B̄2= 1

4VA + 1

4VD + 1

2[�(a)×�(d)] + σ 2

E

The awkward expression 12 [�(a)×�(d)] disappears

when the equations are added together. Therefore:

σ 2B̄1+ σ 2

B̄2= 1

2VA + 1

2VD + 2σ 2

E

As it is also known that:

σ 2F̄2= 1

2VA + 1

4VD + σ 2

E

Provided that numerical values for σ 2F̄2

, σ 2B̄1

, σ 2B̄2

and

σ 2E can be estimated, there is sufficient information to

calculate both VA and VD, and hence the narrow-senseheritability.

To illustrate this consider the following example.A properly designed glasshouse experiment was car-ried out with pea. Progeny from the F1, F2 and bothbackcross families (B1 and B2) were arranged as sin-gle plants in a completely randomized block design andplant height recorded after flowering. The followingvariances were calculated from the recorded data.

σ 2F̄2= 358 cm2; σ 2

B̄1= 285 cm2;

σ 2B̄2= 251 cm2; σ 2

E = 155 cm2

now:

σ 2B̄1+ σ 2

B̄2− σ 2

F̄2=

(1

2VA + 1

2VD + 2σ 2

E

)

−(

1

2VA + 1

2VD + σ 2

E

)

= 1

4VD + σ 2

E

and

VD = 4(σ 2B̄1+ σ 2

B̄2− σ 2

F̄2− σ 2

E )

= 4(285+ 251− 358− 155)

= 92 cm2

Rearranging the equation for σ 2F̄2

(i.e. σ 2F̄2= 1

2 VA +14 VD + σ 2

E ) we have:

VA = 2

(σ 2

F̄2− 1

4VD − σ 2

E

)

= 2

(358−

[1

4× 92

]− 155

)

= 360 cm2

Therefore, the narrow-sense heritability for these data is:

h2n =

12 VA

12 VA + 1

4 VD + σ 2E

= 0.5× 360

0.5× 360+ 0.25× 92+ 155= 0.50

This can be derived more simply by:

h2n =

12 VA

total phenotypic variation

=12 VA

σ 2F̄2

= 0.5× 360

358= 0.50

Thus, 50% of the phenotypic variation in this F2generation of pea is genetically additive in origin.

Heritability from offspring–parentregression

In this section we will consider one other method ofestimating the narrow-sense heritability. The optionof predicting the response to selection using heri-tabilities will be discussed in the selection section(Chapter 7). However, the phenomenon does suggestanother approach to measuring the heritability of a char-acter, namely comparison of the phenotypes of offspringwith those of one or both of their parents. Close corre-spondence in the absence of selection implies that theheritability must be relatively high. On the other hand,if the phenotypes appear to vary independently of oneanother, this suggests that heritability must be low.

The foundations of this approach, which is termedoffspring–parent regression, were laid in the nineteenthcentury by Charles Darwin’s cousin Francis Galton


Predictions 101

in his study of the resemblance between fathers andsons. Therefore, the narrow-sense heritability of a met-rical character can be estimated from the regressioncoefficient of offspring phenotypes on those of theirparents.

In regression analysis, one variable is regarded as inde-pendent, while another is regarded as being potentiallydependent on it. Not surprisingly, in offspring–parentregression, the phenotype of the parent(s) correspondsto the independent variable and that of the offspringthe dependent variable.

The narrow-sense heritability of a character in the F2generation is:

h2n =

12 VA

12 VA + 1

4 VD + σ 2E

and, the regression coefficient from simple linear regres-sion is:

b = SP(x, y)/SS(x)

Therefore, there must be some relationship betweenh2

n and the regression coefficient (b). The regression rela-tionship when the offspring expression is regressed ontothe expression of one of the parents (provided withoutproof ) is:

b =14 VA

12 VA + 1

4 VD + σ 2E

and since:

h2n =

12 VA

12 VA + 1

4 VD + σ 2E

it follows that, for the regression of offspring phenotypeson the phenotypes of one of their parents:

h2n = 2× b

In short, to estimate the narrow-sense heritability(h2

n), it is necessary to perform a regression analysis ofthe mean phenotype of the offspring of individual par-ents on the phenotype of those parents. The regressioncoefficient (b) is obtained by dividing the offspring–parent covariance by the variance of the parental gener-ation. The narrow-sense heritability is then double thevalue of the regression coefficient.

When the expression of progeny are regressedonto the average performance of both parents

(the mid-parental performance) then the regressioncoefficient is (given without proof ):

b =14 VA

12 ( 1

2 VA + 14 VD + σ 2

E )

=12 VA

12 VA + 1

4 VD + σ 2E

and since:

h2n =

12 VA

12 VA + 1

4 VD + σ 2E

it follows that, for the regression of offspring phenotypeson the mean phenotypes of both their parents:

h2n = b

In short, to estimate the narrow-sense heritabilityit is necessary to perform a regression analysis of themean phenotypes of offspring on the mean phenotypesof both their parents. The regression coefficient (b),obtained by dividing the offspring-parent covarianceby the variance of the parental generation, estimates thenarrow-sense heritability directly.

Consider the following simple example. The databelow are the phenotypes of parents and their offspringfrom a number of crosses in a frost tolerant winterrapeseed breeding programme for yield (kg/ha).

Femaleparent

Maleparent

Mid-parentvalue

Offspringvalue

995 1016 1005.5 10061004 999 1001.5 10041009 996 1002.5 10081012 1014 1013.0 10101005 1014 1009.5 10131007 1004 1005.5 10071034 1014 1024.0 10241015 998 1006.5 10021017 1028 1022.5 10201003 1013 1008.0 1008



Then

Statistic Femaleparent

Maleparent

Mid-parentvalue

Regression 0.476 0.468 0.813slope (b)

se(b) 0.1632 0.1895 0.1269t8 df 2.898 2.259 6.407

From the regression of offspring on one parent:

male h2n = 2× b = 2× 0.473 = 0.946

female h2n = 2× b = 2× 0.468 = 0.936

From the regression of offspring onto the averagephenotype of both parents (mid-parent) we have:

h2n = 0.813

You will notice that the heritability using only oneparent is larger than that from both parents. It shouldbe noted that the estimation based on both parents willbe more accurate. Despite the difference, however, it isobvious that there is a high degree of additive geneticvariance for this character.

Finally, always remember that a heritability estimate,no matter which method is used to obtain it, is onlyvalid for that population, at that time, in that environ-ment! Change the environment, carry out (or allow)selection to occur, add more genotypes, sample anotherpopulation and the heritability will be different! Thisshould be clear from the descriptions and methods ofcalculating heritability but you will find many exam-ples in the literature where the basic limitations of theconcept are forgotten.

DIALLEL CROSSING DESIGNS

It has been over 130 years now since the publicationby Louis de Vilmorin that became known as Vilmorin’s

isolation principle or progeny test. He proposed thatthe only means to determine the value of an individualplant (or genotype) was to grow and evaluate its progeny.Ever since, of course, the progeny test has become wellestablished and is frequently used by plant breeders todetermine the genetic potential of parental lines. Thediallel cross is simply a more sophisticated applicationto Vilmorin’s progeny test.

The term diallel cross has been attributed to a Danishgeneticist (J. Schmidt) who first used the design inanimal breeding. The term and design came to plantbreeding and began to be used by plant scientists in themid 1950s.

The diallel cross was then described as all possiblecrosses amongst a group of parent lines. With n par-ents there would be n2 families. The n2 families orprogeny are called a complete diallel cross. If the recipro-cal crosses are not made, making n[n−1]/2 families, theresult is called a half diallel. A modified diallel is one inwhich all possible cross combinations are included butthe parental selfs (diagonal elements) are excluded. Thistype of diallel will include n2 − n families. In a partialdiallel fewer than the n[n − 1]/2 cross combinationsare completed. However, the crosses that are includedare arranged in such a way that valid statistical analysisand interpretation can be carried out.

Initially, only inbred homozygous lines were usedas parents in diallel crossing designs. Techniquesthat allow for parents to be non-inbred genotypes(i.e. heterozygous) are now available.

According to some critics of the designs ‘the diallelmating design has been used and abused more extensivelythan any other . . .’. Whether this statement is true orotherwise, there is little doubt that if the theory of dial-lel analysis is adhered to and if interpretation can becarried out in a logical manner, then the use of diallelcrossing designs can be of great benefit to plant breedersin aiding understanding of qualitative inheritance andproviding invaluable information regarding the geneticpotential of parental lines in cross combinations. Thelimitation of the design arises in terms of the sample ofparental genotypes that can be handled, which is alwayssomewhat restricted.

It will not be possible to cover the whole spec-trum of information or even indeed the types of diallelcrossing schemes that are available or to investigatethe interpretation of many examples within the spaceavailable. There are therefore several approaches tothe analysis and interpretation of diallel cross data


Predictions 103

although only two will be covered briefly in thissection:

• Analysis of general and specific combining ability.These methods are often referred to as Griffing anal-yses, after B. Griffing who published his, now famouspaper ‘Concept of general and specific combiningability in relation to diallel crossing systems’ in 1956.• Analysis of array variances and covariances, often

referred to as Hayman and Jinks analyses, after B.I.Hayman and J.L. Jinks’ paper of 1953, ‘The analysisof diallel crosses’.

Griffing’s Analysis

Griffing proposed a diallel analysis technique for deter-mining general combining ability and specific com-bining ability of a number of parental lines in crosscombination based on statistical concepts. Griffinganalyses have been used by many plant breeders andresearchers over the past 40 years and in many cases withgood success. Much of the success found in applyingGriffing analyses is the apparent ease of interpretationof results compared to other analyses available. Parentsused in diallel crosses can be homozygous or heterozy-gous, for simplicity diallel types are described here interms of homozygous (inbred) parents. Four types ofdesign analyses are available:

• Method 1. The full diallel where p parents are crossedin all possible cross combinations (including recipro-cals). Therefore with p parents the design will consistof p2 families (p2− p segregating populations or F1’sand p inbred parents).• Method 2. The half diallel where p parents are

crossed in all possible combinations, parental selfsare included but that no reciprocals are included.These types of design will contain p[p+1]/2 families([p[p − 1]/2] segregating populations of F1s and pinbred parents).• Method 3. The full diallel without parent selfs, which

consists of all cross combinations (including recipro-cals) of p parents. Method 3 differs from Method 1in that with Method 3 the inbred parents are notincluded in the diallel design.• Method 4. The half diallel, without parent selfs,

which consists of all p parents crossed in all pos-sible combinations (but with no reciprocals). The

Method 4 design differs from the Method 2 design asthe inbred parents are not including in the Method 4design.

Griffing’s Analysis allows the option to test for fixed(Model 1) or random (Model 2) effects. Fixed effectmodels are where inference is made only on the par-ents that are included in the diallel cross while randomeffects models are where inference is made regardingall possible parents from a crop species. Therefore, infixed effect models the parents used in the diallel crossare specifically chosen (i.e. because a breeder wishes tohave additional information regarding general or spe-cific combining ability of chosen lines). In randomeffects models the parental lines should be chosen com-pletely at random. If this is done then the analyses canbe interpreted to cover the eventuality that any parentsare used.

Obviously, in most cases where plant breeders areinvolved, it is often very difficult to decide whether theparental lines were chosen or identified at random. Inmany cases the parents in diallel crossing designs area sample of already commercial cultivars. In this casesome would argue that being commercial cultivars theycannot be a random sample, as by definition all com-mercial cultivars are a very narrow subset of all potentialgenotypes within a species. On the other hand, othershave argued that plant breeders are only interested ingenotypes of commercial or near-commercial standardand they can therefore quite rightly term their choice asa random sample of commercially suitable cultivars.

There are no hard and fast rules regarding fixed orrandom models, and usually there is little to be lost orgained from either argument, provided that the anal-yses are not treated as one type and interpreted asanother. For example, plant breeders and researchersoften include diverse parental genotypes as parents indiallel crossing designs (and we believe this to be anexcellent idea). However, do not choose specific parentallines which show a range of expressions for (say) yield-ing ability, cross them in a diallel design, and try to inferfrom the results what would happen if any different lineswere included.

Griffing’s Analysis requires no genetic assumptionsand has been shown by many researchers to providereliable information on the combining potential of par-ents. Once identified the ‘best’ parental lines (thosewith the highest general combining ability) can becrossed to identify optimum hybrid combinations or



to produce segregating progeny from which superiorcultivars would have high frequency.

In simplest terms, the cross between two parents(i.e. parent i× parent j) in Griffing’s Analysis wouldbe expressed as:

Xij = µ+ gi + gj + sij

where µ is the overall mean of all entries in the dialleldesign, gi is the general combining ability of the ithparent, gj is the general combining ability of the jthparent and sij is the specific combining ability betweenthe ith parent and the jth parent.

General combining ability (GCA) measures the aver-age performance of parental lines in cross combination.GCA is therefore related to the proportion of variationthat is genetically additive in nature.

Specific combining ability (SCA) is the remainingpart of the observed phenotype that is not explainedby the general combining ability of both parents thatconstituted the progeny.

Griffing’s Analysis of a diallel is by analysis of vari-ance, where the total variance of all entries is partitionedinto: general combining ability; specific combing abil-ity and error variances. In cases where reciprocalsare included, then reciprocals (or maternal effects)are also partitioned. Error variances are estimatedby replication of families. To avoid excessive repeti-tion, only Method 1 (complete diallel) and Method 2(half diallel) both including parents will be consideredfurther.

Degrees of freedom (df ), sum of squares (SS) andmean squares (MSq) from the analysis of variance forMethod 1 for the assumption of model 1 (fixed effects)are shown in Table 6.1. Also shown are the expectationsfor the mean squares (EMS).

Similar expected mean squares for Method 1,model 2 (random effects) are shown in Table 6.2.

Considering now Method 2 (the half diallel), thedegrees of freedom (df ), sum of squares (SS), meansquares (MSq) and expected mean squares (EMS) for

Table 6.1 Degrees of freedom, sum of squares and mean squares from theanalysis of variance of a full diallel including parent selfs (Method 1) assumingfixed effects. Also shown are the expectations for the mean squares.

Source df SS MSq EMS

GCA p − 1 Sg Mg σ 2 + 2p(1/(1− p))�g2i

SCA p(p − 1)/2 Ss Ms σ 2 + 2/(p(p − 1))�ij s2ijReciprocal p(p − 1)/2 Sr Mr σ 2 + 2(2/(p(p − 1)))�i<j r2

ijError (r − 1)p2 Se Me σ 2

Table 6.2 Degrees of freedom, sum of squares and mean squares from the analysisof variance of a full diallel including parent selfs (Method 1) assuming randomeffects. Also shown are the expectations for the mean squares.


GCA p − 1 Sg Mg σ 2 + 2p(1/(1− p))σ 2s + 2pσ 2

gSCA p(p − 1)/2 Ss Ms σ 2 + 2((p2 − p + 1))/p2σ 2

sReciprocal p(p − 1)/2 Sr Mr σ 2 + 2σ 2

rError (r − 1)p2 Se Me σ 2

For Method 1, where r is the number of replicates; p is the number of parents; Sg is

1/2p�i (Xi. + X.i )2 − 2/p2X 2

.. ; Ss is 1/2�ij xij (xij + xji )− 1/2p�i (X.i + Xi.)2 + 1/p2X 2

.. ;

Sr is 1/2�i<j (xij − xji )2 and Xi. is �j xij = xi1 + xi2 + xi3 + · · · , that is, sum over rows;

X.j is �i xij = x1j + x2j + x3j + · · · , that is, sum over columns and X.. is �ij xij is sum of allobservations.


Predictions 105

Table 6.3 Degrees of freedom, sum of squares and mean squares from the analysis ofvariance of a half diallel including parent selfs (Method 2) assuming fixed effects. Alsoshown are the expectations for the mean squares.


GCA p − 1 Sg Mg σ 2 + (p + 2)(1/(1− p))�g2i

SCA p(p − 1)/2 Ss Ms σ 2 + 2(p/(p − 1))�j s2ijError (r − 1){p(p + 1)/2} Se Me σ 2

Table 6.4 Degrees of freedom, sum of squares and mean squares from the analysis ofvariance of a half diallel including parent selfs (Method 2) assuming random effects. Alsoshown are the expectations for the mean squares.


GCA p − 1 Sg Mg σ 2 + σ 2s + (p + 2)σ 2

gSCA p(p − 1)/2 Ss Ms σ 2 + σ 2

sError (r − 1)[p(p + 1)/2] Se Me σ 2

Where r is number of replicates; p is number of parents; Sg is 1/(p+2){�i (Xi.+ xii )2−4/pX 2

.. }; Ss

is �i<j x2ij−1/(p+2)�i (Xi.+xii )

2+2/((p+1)(P+2)X 2.. ) and Xi. is �j xij = xi1+xi2+xi3+· · · ,

that is, sum over rows; X.. is �ij xij = is sum of all observations.

model 1 are shown in Table 6.3 and Method 2 andmodel 2 in Table 6.4.

When SCA is relatively small in comparison to GCAit should be possible to predict the performance ofspecific cross combinations based only on the val-ues obtained for GCA of parents. A relatively largeSCA/GCA variance implies the presence of dominanceand/or epistatic gene effects. It should also be noted thatif dominance × additive effects are present, the GCAcomponent will also contain some of these effects inaddition to pure additive effects.

For inbred lines, the closer that the followingequations are equal to one (i.e. as SCA becomes small orvery small compared to GCA), then greater predictabil-ity based on GCA will be possible. The ratio equationsfor each model are:

Model 1 : 2g2i /[2g2

i + s2ij ]

Model 2 : 2σ 2g /[2σ 2

g + σ 2s ]

where g2i , σ 2

g are the general combining ability mean

square and variance, respectively and sij and σ 2s are

specific combining ability mean square and variance,respectively.

The choice of Griffing method will depend on theplant breeder or researcher’s preference and on the char-acters of the crop and trial under investigation. If,for example, there is a suspicion that the particularinheritance has a maternal or cytoplasmic effect thenMethod 1 or Method 3 may be the desired choice. If,however, there is no evidence of reciprocal differencesthen Method 2 or Method 4 would be chosen. Whenthe variance components are of major importance thenit has been suggested that Method 1 will result in a moreaccurate and constant variance estimation compared tothe other methods available. Conversely, it has beenreported that the inclusion of the parental genotypesin the diallel design can cause an upward bias in theestimation of the GCA and SCA variances.

Normally the F1 generation is considered in Griffing’sAnalysis. However, as no genetic assumptions areinvolved then there are no reasons why F2 or indeedother segregating generations could not be analyzed.

Despite the attraction and simplicity of Griffing’sAnalysis several researchers have criticized the technique.



Table 6.5 Average plant height of each of the 45 F1’s and the 10 parents in a half diallel with selfs.

GlobalGlobal 328 HeliosHelios 341 352 JaguarJaguar 336 310 263 StarrStarr 329 271 293 287 93.C.393.C.3 269 308 271 312 292 WestarWestar 256 350 279 299 324 273 DNK.89.213DNK.89.213 284 313 290 266 259 270 293 CycloneCyclone 321 263 280 241 285 243 273 201 HeroHero 246 261 315 261 241 256 250 244 231 RestonReston 306 327 295 287 284 296 275 265 248 277GCA +18.4 +26.4 +9.4 +0.4 +0.4 +0.4 −6.6 −22.6 −28.4 +2.4

In open-pollinated species, where GCA is the onlyparameter of interest, then it has been suggested thatother designs such as topcross or polycross would yieldequally reliable results with less effort and that thesealternative methods provide the opportunity to testmany more parental lines. Similarly it has been arguedthat in many instances North Carolina I designs (wherea set of p parents to be tested are each inter-crossed witha set number of other parents and where each parentunder test is not necessarily crossed to the same tester)or North Carolina II designs (where a set of p parentsare crossed to a common set of n different parents andwhere each parent under test is crossed to the same set ofnon-test parental (or tester) lines) would offer a betteralternative to diallel designs and Griffing’s Analysis.

Many studies have shown that the GCA values ofparents from diallel analyses are similar to actual pheno-typic performance of the parents. It has, therefore, beenargued that it is not necessary to progeny test potentialparents in a plant breeding programme but simply to‘cross the best with the best’. Many practical plant breed-ers often add to this statement, however, ‘cross the bestwith the best, and hope for the best’, but perhaps that iswhat we would be doing anyhow.

Example of Griffing analysis of half diallelLet us consider now an example of a half diallel. A halfdiallel crossing design between ten homozygous linesof spring canola (Brassica napus) was carried out in thespring of 1992. The parental lines were: Global, Helios,Jaguar, Starr, 93.C.3.1, Westar, DNK.89.213, Cyclone,Hero and Reston. Hero and Reston are both industrial

rapeseed cultivars while the others are canola (edible)types. Crossing resulted in n[n − 1]/2 = 45 differ-ent F1 families. Over the following winter each of the45 F1 families were grown in a two replicate randomizedcomplete block design which also included the 10 par-ent selfs making a design with 55 entries (n[n + 1]/2)

and two replicates (i.e. 110 plots).Throughout the growth of this experiment a number

of different traits were recorded on each of the 110 plots.To avoid excessive repetition we will only consider oneof these characters, plant height at end of flowering.

The average plant height of each of the 45 F1s andthe 10 parents are shown in Table 6.5. The data usedwere the sum of two plant heights (cms) as two readingswere made on each of the replicate plots.

From the data the total variance (sum of squares) ispartitioned into differences between the two replicateblocks (Reps), general combining ability, specific com-bining ability and an error term (based on interactionsbetween replicates and other factors). Sum of squares(SS) and mean squares (MS) obtained are shown inTable 6.6.

The basic assumption of this experiment was that theten parental lines were chosen as representative of thewide range of B. napus cultivar types that were avail-able. We are therefore analyzing a fixed effect model andall the mean squares in the analysis are tested for signif-icance (using the ‘F’ test) against the error mean square(i.e. 1545).

From the analysis the overall replicate block effect(i.e. difference between replicate one and replicate two)was not significant. An F-value is obtained for specific


Predictions 107

Table 6.6 Degrees of freedom, sum of squares andmean squares from the analysis of variance of plant heightof a half diallel including parent selfs. In the analysis thetotal variance is partitioned into differences between thetwo replicate blocks (Reps), general combining ability,specific combining ability and an error term (based oninteractions between replicates and other factors).

Source df Sum ofsquares

Meansquares

General combining 9 108 665 12 074ability

Specific combining 45 113 497 2522ability

Replicate blocks 1 959 959Replicate Error 54 83 428 1545

Total 109 306 548 2812

combining ability by 2522/1545 = 1.63. This ‘F’ valueis compared to F-values found in statistical tables at dif-fering probability levels and with 45 and 54 degrees offreedom. When this is done, with some degree of dif-ficulty, it is found that the probability of this F valueoccurring if SCA were not significant is 95.7, there-fore specific combining ability is just significant at the5% level.

Consider now the variance ratio for general combin-ing ability. The appropriate F-value is 12 074/1545 =7.8. When this value is compared to the appropriateF-values in statistical tables with 9 and 54 degrees offreedom we find that it exceeds the appropriate expec-tation based on 99.9% confidence (i.e. approximately3.54) and so we say that general combining abilityis highly significant. This, in combination with themarginal significant of specific combining ability, sug-gests an additive–dominance model with high additiveeffects.

Now the expected mean square for specific combin-ing ability of a half diallel and fixed effects is:

σ 2 + 2(p/(p − 1))�i s2i

Therefore

2521− 1545 = 2(10/(10− 1))�i s2i

976 = 2.2�i s2i

so

�i s2i = 976/2.2 = 439

Similarly for general combining ability, the expectedmean square is:

σ 2 + (p + 2)(1/(1− p))�g2i

Therefore

12 074− 1545 = (10+ 2)(1/(1− 10))�g2i

10 529 = 1.33�g2i

so

�g2i = 10 529/1.33 = 7897

Now, from the equation above we can compare GCAand SCA effects, as noted earlier we have:

2g2i /[2g2

i + sij ] = 2× 7896.893/

[(2× 7896.893)+ 439.288]= 0.973

As this value is very close to one, it indicates thats2ij is relatively small compared to g2

i . Therefore addi-tive genetic effects predominate. This means there isa good chance that plant height at the F1 stage in aB. napus breeding programme can be predicted withgood accuracy depending on the general combiningability of chosen parental lines.

In many instances there is good agreement betweenthe general combining ability of a genotype and the phe-notypic performance of the line. If this is the case thenall that is necessary is to determine the expression of theparents and from these the expected expression of theoffspring can be estimated (compare with h2

n). There-fore in this example consider the regression of averageparental performance against the offspring (Figure 6.1).It can be clearly seen that there is relatively good agree-ment between parents and offspring. The regressionequation is offspring mean =0.7265 × parent +80.0.Therefore the narrow sense heritability of these data isapproximately 0.73, which is relatively high in that 73%of the total variation is additive genetic variance.

Hayman and Jinks’ analysis

Hayman and Jinks developed an analysis for diallelsthat has been widely used by many plant researchers toevaluate the mode of inheritance. This analysis is basedon a model that, for any one locus, i, with two alle-les, the difference between the two homozygotes is 2a.



320y = 0.73x + 80.0h2

n = 0.73300

280

260

240

220

200200 220 240 260 280

Mid-parent height (cm)

Pro

geny

hei

ght (

cm)

300 320 340

Figure 6.1 Scatter diagram of mid-parent phenotype height against average offspring progeny phenotype height from a10× 10 half diallel in Brassica napus.

The difference between the heterozygous (F1) and themid-parent value (m) is d .

To simply interpret a Hayman and Jinks’ Analysis thefollowing assumptions are made:

• Diploid segregation• Homozygous parents• No difference between reciprocal crosses• No epistasis• No multiple alleles• Genes are distributed independently between the two

parents

But these assumptions are tested in the approach.The parents and all possible F1 progenies are eval-

uated for the trait of interest. All the offspring of oneparent used in crosses is called an array. That is, in allcrosses that the particular parent was used. Seven kindsof variances and covariances are calculated including:

Vp = variance among the parent lines;

Vr = the variance among family(F1 and reciprocal)means within an array;

Vxr = variance among the means of the arrays;

V̄r = mean value of all Vr over all arrays;

Wr = the covariance between families withinthe ith array and their non-recurrent parent;

W̄r = mean value of Wr over all arrays;

σ 2E = Error variance.

From these, a number of parameters can be esti-mated, including:

VA = 4/7[Vp + W̄r + Vxr] − σ 2E

VD = 4V̄r − VA

The estimates of VA and VD indicate the amounts ofadditive variance and dominance variance among thecrosses. This estimate of VD assumes that F1 progenyare being evaluated (although other generations can beaccommodated). Obviously the frequency of heterozy-gous alleles in a population will determine the degree ofdominance variation and this will vary with successiverounds of selfing.

The most useful aspect of Hayman and Jinks’ Anal-ysis for plant breeders involves examination of varianceand covariance relationships and estimation of VA orVD. Therefore we will only cover the within array vari-ances and between array covariances, how they can helpin determining the inheritance of the character of inter-est, what the relationship of these two parameters meansin comparing different parental lines, and estimationof h2

n .Based on the assumptions (listed above) of Hayman

and Jinks analyses we have

V̄r = 1

4(VA + VD)

W̄r = 1

2VA

Consider now the relationship between Wr and Vr.If we plot Wr against Vr, the regression line must have


Predictions 109

a slope which will pass through the point (V̄r, W̄r) andwill have expected value of one only if the additive–dominance model is adequate to explain the variationobserved. It should, therefore be noted that the rela-tionship between Wr and Vr provides a test of theadditive–dominance model of gene action. If the con-tribution of many genes are not independent, thatis if there are genes interacting in their effect (epis-tasis) we would not expect the relationship betweenWr and Vr to be as described. Hence if the additive–dominance model is not adequate then the regressionof Wr against Vr will not result in a regression slopeof one.

In addition, regression of Wr against Vr will resultin a gradient which will pass through the point14 (VA + VD), 1

2 VA and which will cut the y-axisat W̄r − V̄r = 1

2 VA − 14 (VA + VD) = 1

4 (VA − VD).So we can learn something about the average dominancerelationships of the quantitative inheritance system. Ifadditive genetic variance (VA) is greater than domi-nance genetic variance (VD) then the regression linewill cut the y-axis (the Wr-axis) above zero. Similarly,the reverse will be true of VD is greater than VA.

The relative position of each array (Vr and Wr) willindicate the relative frequency of the dominant to reces-sive alleles that array parent has. Therefore, the relativeposition of the array points on the line will reflect thedirection of dominance. If an array has a scatter of pointsclose to the origin (i.e. a low Vr and Wr values) this indi-cates that the parent common to that array has a highfrequency of dominant alleles for that character of inter-est. If an array has a scatter of points at a distance fromthe origin (i.e. high Vr and Wr values) then the com-mon parent in that progeny array will have a relativelyhigh frequency of recessive alleles.

This graph (Wr/Vr) can therefore provide a greatdeal of information about the genetic situation betweenthe parents in the diallel. In plant breeding terms thefrequency of dominant (or recessive) alleles, combinedwith the average progeny performance can be usefulindicators for selection. For example, given two pos-sible parents, if one has a high frequency of recessivealleles and the other a high frequency of dominant alle-les for, say, yield. If both parents have similar generalcombining ability, then a plant breeder should choosethe recessive parent as it will be easiest to select and fixfor high yield. Similarly, selection for high yield, whichis related to a high frequency of dominant alleles will

likely have lower narrow-sense heritability compared tothe case of high recessive allele frequency.

To illustrate further, consider the example from ahalf diallel design involving 10 homozygous parents ofspring canola/rapeseed (B. napus). Although many traitshave been recorded from this trial we will again con-sider plant height which was explained as an exampleof the Griffing’s Analysis earlier. The means, over repli-cates, have therefore been shown earlier. From the arraymeans, values of within array variances (Vr) and covari-ances (Wr) were obtained. Similar Vr and Wr valueswere calculated from each of the two replicates.

Regression analysis of Vr against Wr resulted in aregression equation:

Wr = 0.60× Vr + 59.07

The analysis of regression resulted in a mean squarefor linearity of 47 473 and a mean square for depar-ture from linearity to be 9 286. From this we calculatean ‘F’ value of 5.11 which is significantly (p < 0.05)larger than would have been expected if the relationshipbetween Wr and Vr was not linear.

The standard error (seb) of the regression coefficient(b) was 0.265. From this we can calculate the Student’st as:

t = b − 1

seb= 0.400

0.265= 1.51

This did not exceed the value from t -tables with8 degrees of freedom (n − 2) and p < 0.05. There-fore b is not significantly different from a regressionslope of one and so we have a good indication that theadditive–dominance mode is adequate to explain theinheritance of plant height in B. napus.

The regression line cuts the y-axis at (Vr = 0) abovethe origin (+59.07) so we can say that additive effectsare greater than dominance effects.

A scatter diagram of the Wr and Vr values from the10 arrays (parents) is shown in Figure 6.2. From thediagram we see that the cultivars Hero, DNK.89.213,Jaguar and 93.C.3 are relatively close to the origin whileHelios, Cyclone and Westar are further from the origin.From this we can deduce that those closest to the originhave a higher frequency of dominant alleles for plantheight and those further from the origin have high-est frequency of recessive alleles for plant height. Inthe extremes, the cultivar Hero has highest relative fre-quency of dominant alleles and the cultivar Helios hashighest frequency of recessive alleles for plant height.



420

320

22093.C.3

Reston

JaguarDNK.89.213

Starr

Cyclone

Westar

Helios

Global

Hero

120

Wr

20110 160 210 260

Vr

310 360 410 Figure 6.2 Scatter diagram of Vr and Wrvalues from arrays in a 10× 10 half diallelin Brassica napus.

Moving on to the Wr + Vr and Wr − Vr from eacharray. Wr + Vr and Wr − Vr will contain all the infor-mation that Wr and Vr contain. Now if dominance ispresent then Wr + Vr will vary from array to array. Ifthere is non-allelic interaction then Wr − Vr will varyfrom array to array. If only dominance is present thenWr −Vr will not vary between arrays more than wouldbe expected by sampling variation. We can calculate thevalues of Wr+Vr and Wr−Vr from the Vr and Wr valuesobtained from each replicate and carry out a one-wayanalysis of variance on the resulting data. When this isdone we have the two analyses of variance tables:

Wr + Vr

Source df MSq F -value Significance

Between 9 17 164 4.63 (0.01 < p < 0.05)arrays

Within 10 3708arrays

Wr − Vr

Source df MSq F -value Significance

Between 9 5642 2.60 n.s.arrays

Within 10 2170arrays

Therefore values of Wr + Vr vary significantlybetween arrays (p < 0.05) so we can say that domi-nance is present. Values of Wr − Vr between differentarrays are not significantly different and therefore we cansay that there is no evidence of non-allelic interactionand that only dominance is present.

These data are from homozygous parents and F1progenies. In many cases it is difficult to obtain largequantities of F1 seed and the actual diallel analysis needsto be carried out on the F2 (or higher) generations.When this is done the same six assumptions listed atthe beginning of this section still apply. The regressionof Wr/Vr will still have an expected slope of unity ifthe additive-dominance model is adequate to explainthe inheritance, and Wr−Vr should be constant acrossarrays if no epistasis is present. Therefore, as far as theexample above is concerned, then it would make no dif-ference if F2 data were used. However, if a more detailedanalysis is to be carried out and the components VAand VD are to be estimated then some modification isneeded. The modification is not within the scope ofthis book.

Estimating h2n from Hayman and Jink’s Analysis

If the crop under investigation in a diallel crossingdesign complies with all the restraints of the Haymanand Jinks design, then it is possible to obtain accurateestimates of additive genetic variance (VA) and dom-inant genetic variance (VD) straightaway and hencedetermine the narrow-sense heritability (h2

n). The aver-age Wr value (Wr) is an estimate of 1

2 VA. The Vp value isa direct estimate of VA, and the Vxr value is an estimateof 1

4 VA. These relationships hold true irrespective of the


Predictions 111

generation (i.e. F1, F2, F3, etc.) that is analyzed. Fromthese three estimates of VA, we can produce a weightedmean where:

VA = 4/7[Vp + W̄r + Vxr]The dominance genetic variance (VD) will vary from

generation to generation. Greatest VD will be observedin the F1 generation as there is greatest frequency ofheterozygotes compared to other generations. In F1 theaverage Vr value (V̄r) is an estimate of 1

4 [VA+VD], andVD can easily be estimated by substituting the alreadycalculated VA value in to this equation. Therefore, whenanalyzing data from F1 family diallels:

D = 4V̄r − VA

From estimation of VA and VD we can now calcu-late h2

n:

h2n =

1

2VA/

[1

2VA + 1

4VD + σ 2

E

]

where, σ 2E is the replicate error term obtained from

the analysis of variance in the B. napus example shownearlier in the Griffing’s Analysis.

In F2 families VD = 14 [VA + 1

4 VD] and so VD−F2 =16V̄r − 4VA, and in F3 families, VD−F3 = 1

4 [A +1/16VD]. It should be noted that the proportion ofVD in each family is decreased each generation by [ 1

2 ]n,where n is the generation number (i.e. [ 1

2 ]1 = 1 at F1;[ 1

2 ]2 = 14 at F2; [ 1

2 ]3 = 1/16 at F3, etc.).

CROSS PREDICTION

There is one further way that it is possible to predict theresponse to selection, in the long-term, although notnecessarily the rate of response. This approach is basedon the genetics underlying the traits, was proposed byJinks and Pooni, and is currently attracting considerableattention in terms of experimental investigations and inapplying it to practical breeding. This will be covered inmore detail in the next chapter but needs mentioninghere to keep in view the options available to the breederin terms of making predictions.

If, to start with, we assume that we have an inbreedingspecies and wish to produce a final variety that is true-breeding.

What we want to know of any population or cross iswhat is the distribution of inbred lines that we predict

can be derived from it and findout what is the probabil-ity of one of these lines having a phenotype equal to, orexceeding, any target level that we set, in other words,that we would be aiming for with selection.

If we assume that the distribution of the final inbredlines that are derivable have a normal distribution as isgenerally the case in practice, then it can be described bythe mean and standard deviation. Since they are inbredlines they will have a mean of m and a standard deviationof√

VA, we can predict the properties of the distribu-tion of all inbred lines possible and hence we can obtainthe frequency (= probability) of inbreds falling intoa particular category. In other words, we can simplyuse the properties of the normal probability integral intables to say what the probability of obtaining an inbredline with expression falling in a particular category. Ifthe probability is low it will obviously be difficult toactually obtain such a line. If the probability is high itwill be easy to produce.

How do we put it into practice? If we have a set ofgenotypes for use as Parents, which ones do we crossto produce our desired new inbred lines? Do we takeA×B and C×D or A×D and C×Z etc.? We will needto decide between the crosses before we invest too muchtime and effort, otherwise we may well be spreading ourefforts over crosses that will not produce the phenotypeswe want. If we take the crosses and estimate m andVA for each, then we can estimate the probability ofobtaining our desired target values. From this we canrank the crosses on their probabilities and only thenuse the ones with the highest probabilities of producinglines with the required expression of characters deemedto be important.

In fact, the approach is even more general in that itcan be used to predict the properties of the F1 hybridsderived from the inbred lines. It can also be used topredict the probability of combination of characters,that is the probability of obtaining desirable levels ofexpression in a series of characters.

What are the drawbacks to the approach? First, weneed to estimate m and A, and this involves a certainamount of work in itself, but is fairly modest.

Second, it also assumes that the estimates we use areappropriate to the final environment that the material isto be grown. In other words, as in the case of heritabili-ties, if we carry out the experiments in one environmentat one site in one year, we are assuming that this is rep-resentative of other years and sites. We can, of course,



carry out suitable experiments to obtain estimates inmore years and sites, but this involves extra time andeffort.

The use of cross prediction techniques in selectionwill be discussed in greater detail in Chapter 7.

THINK QUESTIONS

(1) Given values for the variance of the mean of theF2, and variance of the F1, estimate h2

b and explainwhat this tells us about the genetic determinationof the trait.

VF̄2= 436.72

VF̄1= 111.72

Given below are the variances of the mean fromtwo parents (P1 and P2), the F1, F2, and bothbackcross families (B1 and B2), estimate h2

n andexplain what the value means in genetic terms.

VP̄1= 14.1 VP̄2

= 12.2

VF̄1= 13.3 VF̄2

= 40.2

VB̄1= 35.2 VB̄2

= 34.6

(2) List the six assumptions necessary for a straightforward interpretation of a Hayman and Jinks’Analysis of diallels.

Below are shown values of array means, withinarray variances (Vr) and covariances between arrayvalues and non-recurrent parents (Wr) from aHayman and Jinks Analysis of a 7 × 7 completediallel in dry pea.

Parentname

V r W r Arraymean

‘Souper’ 34.1 19.3 456‘Dleiyon’ 99.9 79.3 305‘Yielder’ 21.0 11.2 502‘Shatter’ 99.4 68.4 314‘Creamy’ 49.6 39.4 372‘SweetP’ 59.1 48.8 361‘Limer’ 61.8 49.2 393

The variate of interest is pea yield. Regres-sion of Wr against Vr resulted in the equation:Wr = 0.837 × Vr − 4.817, with standard errorof the regression slope equal to seb= 0.0878. Ananalysis of variance of Wr+Vr showed significantdifferences between arrays while a similar analy-sis of Wr − Vr showed no significant differencesbetween arrays. What can be deduced regardingthe inheritance of pea yield from the informationprovided? If you were a plant breeder interestedin developing high yielding dry pea cultivars, onwhich two parental lines would you concentrateyour breeding efforts? Briefly explain why.

(3) Below is shown an analysis of variance of plantyield from a 6 × 6 half diallel (including par-ents). The analysis of variance is from a Griffing’sAnalysis (Model 2). GCA = general combin-ing ability, SCA = specific combining ability,Error = random error obtained by replication,df = degrees of freedom and SS = sum of square.

Source df SS

GCA 5 4988SCA 15 6789Error 21 5412

Discuss the results from the analysis given thatthe 6 parents were: (1) specifically chosen and(2) chosen completely at random.

(4) It is desired to determine the narrow-sense her-itability for flowering date in spring canola(B. napus). Both parents and their offspring fromten cross combinations were grown in a properlydesigned field experiment. At harvest, yield wasrecorded for each entry and using these data theaverage phenotype of two parents (i.e. [P1+P2]/2)was considered to be the x independent variablewhile the performance of their offspring was con-sidered as the y dependant variable. A regressionanalysis is to be carried out by regression of theoffspring (y) onto the average parent (x). Thefollowing data are derived: SP(x, y) = 345.32;SS(x) = 491.41; SS(y) = 321.45. Estimate theslope of the regression (b), test if this slope is


Predictions 113

greater than zero and estimate the narrow-senseheritability from the regression equation.

How would the relationship between the regres-sion and the narrow-sense heritability differ if theregression were carried out between only the maleparent and the offspring?

(5) Four types of diallel can be analyzed usingGriffing’s Analysis. Describe these types. Fam-ilies from a 5 × 5 half diallel (including selfs)were planted in a two replicate yield trial at asingle location. The parents used in the dialleldesign were chosen to be the highest yielding linesgrown in the Pacific-Northwest region. Data foryield were analyzed using a Griffing’s Analysis ofvariance. Family means, averaged over two repli-cates, degrees of freedom and sum of squares (SS)from that analysis are shown below. Explain theresults from the Griffing’s Analysis. What differ-ences would there be in your analytical methodsif the parents used had been chosen at random.

Parent 1Parent 1 62.0 Parent 2Parent 2 71.0 69.5 Parent 3Parent 3 55.5 52.5 50.5 Parent 4Parent 4 72.5 80.5 56.5 76.5 Parent 5Parent 5 70.5 66.5 36.5 71.0 64.5

Source d.f. SS

GCA 4 6694.058SCA 10 825.676Replicates 1 8.533Error 14 317.467

Total 29 7845.733

From the same diallel data (above), withinarray variances (Vr) and between array and non-recurrent parent covariances (Wr) were calculated.The values of Vr and Wr for each parent along withthe of mean of Vr, mean of Wr, sum of squares ifVr (�V 2

r ), sum of squares of Wr(�W 2r ) and sum

of products �VrWr are shown below.

V r W r

1 98.0 102.02 66.3 53.23 161.8 207.44 50.3 65.25 71.4 83.1

Mean of Vr = 89.56; Mean of Wr =102.19; �V 2

r = 7702.01: �W 2r = 15192.05;

�VrWr = 10 535.29. From these data, testwhether the additive-dominance model is ade-quate to describe variation between the progenies.What can be determined about the importanceof additive compared to dominance genetic vari-ation in this study. From all the results (Griffingand Hayman and Jinks, above) which two par-ents would you use in your breeding programmeand why?

(6) Two genetically different homozygous lines ofcanola (B. napus L.) were crossed to produceF1 seed. Plants from the F1 family were self-pollinated to produce F2 seed. A properly designedexperiment was carried out involving both parents(P1 and P2, 10 plants each), the F1 (10 plants)and the F2 families (64 plants) and was grown inthe field. Plant height of individual plants (cm)recorded after flowering. The following are familymeans, variances and number of plants observedfor each family.

Family Mean Variance Number ofplants

P1 162 1.97 10P2 121 2.69 10F1 149 3.14 10F2 139 10.69 34

Complete a statistical test to determinewhether an additive–dominance model of inher-itance is appropriate to adequately explain theinheritance of plant height in canola. If theadditive–dominance model is inadequate, listthree factors that could cause the lack of fit ofthe model.



(7) Given values for the variance of the F2, and vari-ance of both parents (P1 and P2) and the F1,estimate h2

b and explain what this value means ingenetic terms.

σ 2F̄2= 436.72; σ 2

F̄1= 111.72

σ 2P̄1= 164.13; σ 2

P̄2= 109.33

Given the variance from two parents (P1 andP2), the F1, F2, B1 and B2 families, estimate thenarrow-sense heritability (h2

n) and explain whatthe value means in genetic terms.

σ 2P̄1= 9.5; σ 2

P̄2= 7.4

σ 2F̄1= 8.6; σ 2

F̄2= 17.7

σ 2B̄1= 14.3; σ 2

B̄2= 15.2

(8) A new oil crop (Brassica gasolinous) has been dis-covered which may have potential as a renewablebiological fuel oil substitute. This diploid speciesis tolerant to inbreeding and is self-compatible.A preliminary genetic experiment was designed toexamine the inheritance of seed yield (YIELD) andpercentage oil content (%OIL). This experimentinvolved a 4× 4 half diallel (including selfs). Thefour homozygous parental lines are represented bythe codes AAA, BBB, CCC and DDD. The halfdiallel array values (averaged over two replicates),array means, general combing ability (GCA) val-ues, mean squares from the analyses of variance(Griffing style), Vr and Wr values (Hayman andJinks’ analysis), variance of array means (Vxr) andparental variances (Vp), and the one-way analysesof variance for Vr +Wr and Vr −Wr are shownbelow for each character.

Yield %Oil

AAA 40.5 20.5BBB 38.5 29.5 20.5 25.0CCC 37.0 28.0 19.5 23.0 26.0 30.5DDD 32.5 20.5 18.5 10.0 24.5 27.5 31.0 36.0

AAA BBB CCC DDD AAA BBB CCC DDD

Array means 37.1 29.1 25.8 20.4 22.1 24.7 27.6 29.8Source df Yield %OilG.C.A. 3 796.5 180.6S.C.A. 6 90.5 30.1Replicate blocks 1 0.4 0.1Replicate error 9 51.5 5.7

V r and W r valuesYield %Oil

V r W r V r W r

AAA 46.2 171.3 15.6 50.7BBB 218.2 376.7 36.3 75.0CCC 297.7 436.7 58.3 98.0DDD 344.3 472.3 97.3 132.3

V xr and V p valuesYield %Oil

Vxr 196.9 44.4Vp 687.6 180.7

Yield %OilSource df V r +W r V r −W r V r +W r V r −W r

Between 3 8760 28.0 628 7.68Within 4 133 17.0 115 4.77


Predictions 115

Without using regression, estimate the narrow-sense heritability (h2

n) for seed yield, and explainthis value in genetic variance terms. Explain theanalyses for Yield and outline any conclusions thatcan be drawn for these data. Explain the analysisfor %Oil (percentage of seed weight that is oil) andoutline any conclusions that can be drawn fromthese data. Which one of these four genotypeswould you choose as a parent in your breed-ing programme? Explain your choice. Describeany difficulties suggested from these analyses in abreeding programme designed for selecting lineswith high yield and high percentage of oil.

(9) F1, F2, B1 and B2 families were evaluated forplant yield (kg/plot) from a cross between twohomozygous spring wheat parents. The followingvariances from each family were found:

σ 2F1= 123.7; σ 2

F2= 496.2

σ 2B1= 357.2; σ 2

B2= 324.7

Calculate the broad-sense (h2b) and narrow-

sense (h2n) heritability for plant yield. Given the

heritability estimates you have obtained, wouldyou recommend selection for yield at the F3 in awheat breeding programme, and why?

(10) Griffing has described four types of diallel cross-ing designs. Briefly outline the features of eachMethod 1, 2, 3 and 4. Why would you chooseMethod 3 over Method 1? Why would you chooseMethod 2 over Method 1?

A full diallel, including selfs was carried involv-ing five chickpea parents (assumed to be chosenas fixed parents), and all families resulting wereevaluated at the F1 stage for seed yield. Thefollowing analysis of variance for general com-bining ability (GCA), specific combining ability(SCA) and reciprocal effects (Griffing’s Analysis)was obtained:

Source d.f. MS

GCA 5 30 769SCA 10 10 934Reciprocal 10 9638Error 49 5136

Complete the analysis of variance and explainyour conclusions from the analysis. Giventhat the parents were chosen at random,how would this change the results and yourconclusions?

Plant height was also recorded on the samediallel families and an additive-dominance modelfound to be adequate to explain the genetic vari-ation in plant height. Array variances Vrs andnon-recurrent parent covariances (Wrs) were cal-culated and are shown alongside the general com-bining ability (GCA) of each of the five parents,below:

Parent V r W r GCA

1 491.4 436.8 −0.762 610.3 664.2 +12.923 302.4 234.8 −14.324 310.2 226.9 −15.775 832.7 769.4 +17.93

Without further calculations, what can bededuced about the inheritance of plant height inchickpea?

(11) A 4× 4 halfdiallel design (with selfs) was carriedout in cherry and the following fruit yields of eachpossible F1 family were observed:

Small redsSmall reds 12 Big yieldsBig yields 27 36 Jim’s delightJim’s delight 21 35 27 Jacks’ bestJack’s best 28 27 26 21

From the above data, determine the narrow-sense heritability for yield in cherry.

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An Introduction to Plant Breeding (Brown/An Introduction to Plant Breeding) || Predictions