Euphytica 88 : 55-60,1996 .

55© 1996 Kluwer Academic Publishers. Printed in the Netherlands .

Estimation of genetic variance components and heritabilities for cut-floweryield in gerbera using least squares and maximum likelihood methods

James Harding, Hongzhan Huang & Thomas ByrneDepartment of Environmental Horticulture, University of California, Davis, CA 95616, USA

Received 3 August 1994 ; accepted 16 August 1995

Key words : additive variance, heritability, selection, least squares, maximum likelihood, gerbera, Gerbera hybrida

Summary

Genetic variance components and heritability were estimated for cut-flower yield of gerbera in the Davis populationusing ordinary least squares and maximum likelihood methods . The overall estimate of narrow-sense heritabilityis 0.33 based on least squares (LS) and 0 .31 based on maximum likelihood (ML) . The results of the study indicatethat (1) ML and LS provide very similar results if sample size is large enough, suggesting both are useful for plantbreeding programs ; (2) about one third of the variation in gerbera cut-flower yield is additive, implying selectionin cut-flower will be successful ; and (3) although additive variation gradually decreased, heritability remained near0.27 suggesting there is still potential variation in the population for further selection .

Introduction

Heritabilities have been traditionally estimated by ordi-nary least squares (LS) methods (see e.g. Becker,1984). However, with developments in statistical pro-cedures and computer technology, maximum likeli-hood (ML) methodology can now be used (Meyer,1990) .

It is generally assumed that a cross-breeding popu-lation is under Hardy-Weiberg equilibrium when esti-mating heritabilities (Falconer, 1989) . Breeding popu-lations usually do not follow these conditions becauseselection occurs in such populations .

Many long-term selection experiments have beenpracticed in animals and plants (Wright, 1977) . How-ever, such selection in flower crops has not been report-ed in the literature . A long-term selection experimentfor cut-flower yield has been conducted in the Davispopulation of gerbera (Gerbera hybrida, Compositae)for 16 generations . Variance components and heritabil-ities have been reported (Harding et al ., 1981, 1985 ; Yuet al ., 1992). The purpose of this study is to explore thequantitative structure of cut-flower yield in gerbera by(1) estimating variance components and heritabilitiesfor 16 generations using least squares and maximumlikelihood methods; (2) comparing LS and ML meth-

ods; (3) analyzing the changes in the variance com-ponents that have occurred during the 16 generationsand discussing the potential for further selection in thispopulation .

Materials and methods

Genetic materials used for this study are all the plantsfrom generations one through 16 plus the original par-ents of the Davis population of gerbera . There wereabout 400 plants per generation . In most generations,about 40 plants were selected based on their perfor-mance and used as parents to produce the next genera-tion (see Harding et al ., 1991 for details). Plants weregrown as an annual greenhouse crop at the Universityof California, Davis, details of growing procedure canbe seen in Harding et al . (1981) .

Factorial mating (NCII) designs (Comstock &Robinson, 1948) were used in this population exceptfor generations one through four, and nine . For gen-erations one through four, each parent was crossed toa mass collection of pollen obtained from all parents .The progenies, therefore, can be classified by theirmaternal parent into half-sib families . In generationnine, however, the mating followed a circular design .

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Fifty parents were randomly numbered from 1 to 50 .Each parent was crossed to three other parents with thenearest following numbers .

Cut-flower yield of gerbera is defined as the num-ber of flowers harvested from a plant in a six-monthperiod from September to February. Flowers were har-vested three times a week as pollen appeared; recordswere kept for each plant . For generations 0 (the originalparents) through seven, the yield was collected for sixmonths (181 days), but it was recorded for 24 weeks(168 days) for generations eight through sixteen . Con-sequently, yield in later generations was multiplied by181/168 to standardize all data to 181 days for estimat-ing mean yield in each generation .

Half-sib one way classified model

The statistical model for each of generations onethrough four is

Yij = p +Pi +Eij

where Yi j is the cut-flower yield of plant j under mater-nal parent i, Pi the effect of maternal parent i, and Eijthe residual effect .

Additive genetic components of variances and nar-row sense heritability can be estimated from the vari-ance of P (a P) and the residual variance a 2El

as = 4aP

2

2

2aph = aP + CE

2h2 = as

2ph

and

where as is the additive variance, 0";h the total phe-notypic variance, and h 2 the narrow-sense heritability .The non-additive variance a,2, a which includes domi-nant and epistatic variances can be obtained by

2

2

2ana = aph - as

For generation nine, half-sib family analysis was basedon the full-sib family means . The phenotype variancewas estimated by

aph = aP + kaE

where k is full-sib family size.

NCII factorial model

In NCII mating designs, parents are divided into twosets . Matings between these two sets of parents form atwo way classified design

Yijk =p +Ai +Bj +(AB)ij +Eijk

where Yi j k is the cut-flower yield of k-th plant underparents i and j, Ai the effect of set A parent i, Bj theeffect of set B parent j, (AB)ij the interaction effect,and Eij k the residual effect . From this model, bothadditive and dominance components of variance canbe obtained,

as = 2(aA + aB)

0';h - aA + CB + aAB +a2

2ad = 4a

2AB

and2

h 2 = as

ap2h

2

2H2 =

oa+ada 2ph

where as and as are additive and dominance compo-nents of variance respectively, ap h is the total phe-notypic variance, and h2 and H2 are the narrow- andbroad-sense heritabilities, respectively. The residualvariance a2 is obtained by subtracting additive anddominance variances from the total phenotypic vari-ance.

The data analyses were done using SAS programs(SAS Institute, 1988) on an IBM PC computer . Sincefamily size is not equal in most cases, the SAS GeneralLinear Model was used to calculate the mean squaresand expected mean squares .

Maximmum likelihood methods

The following additive genetic model (Henderson,1984) is used to estimate variance components andnarrow-sense heritability,

y =Xb +Za +e

where y is an N x 1 vector of observations for cut-floweryield, N = 6200, b is a p x 1 vector of fixed effects, i .e .

Table 1 . Mean yield (Y), variance components and heritabil-ities estimated by LS

Gen Y

oph

oa

va

h2

H2

year effects, p = 17, a is an Nx 1 vector of additivegenetic effects, e is an Nx 1 residual vector, and X andZ are incidence matrices .

This model is referred to as an animal model in theanimal breeding literature (Henderson, 1984) . Ran-dom vectors a and e are uncorrelated with zero means,and

Var(a) = Ava

Var(e) = Ide

where A is the numerator relationship matrix, ~aand ve are the additive genetic and residual variancesrespectively, then

E(y) = Xb

Var(y) = ZAZ'va + Ive

This model partitions the phenotypic variance intoadditive and non-additive components and, therefore,only narrow-sense heritability can be estimated .

Derivative free restricted maximum likelihood(DFREML) (Graser et al ., 1987) was used to estimatethe variance components . This method involves matrixabsorption by Gaussian elimination (Smith & Gras-er, 1986) and an iterative procedure to maximize the

Fig . 1 . Heritability of cut-flower yield based on cumulative data .

log likelihood with respect to the variance ratio r (r =a2/c ) . The data analysis for this study was carried outwith a program from Meyer (1988) on a VAX comput-er.

Results

Mean-cut flower yield was calculated for each gener-ation (Table 1). It increased from 15 .3 in generationone to 28 .3 in generation 16 . LS estimates of variancecomponents, and narrow- and broad-sense heritabili-ties were calculated for each generation and are shownin Table 1 . The estimated narrow-sense heritabilityvaries from - 0.40 in generation seven to 1 .25 in gen-eration three .

In addition, DFREML (ML) estimates of variancecomponents were obtained . Additive genetic varianceis 28 .54, non-additive variance 65 .13 and phenotypicvariance 93 .66. Therefore, the narrow-sense heritabil-ity is 0 .31 . In order to study the change of heritabilityover generations, for each generation, variance com-ponents were calculated using data cumulative to thatgeneration . Cumulated generation data is used becauseDFREML estimates are based on the entire pedigree .These estimates are listed in Table 2 and illustrated inFig. 1 .

To compare the methods of LS and DFREML, her-itability estimates from least squares methods werecumulated over generations in the same way cumu-

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1 15 .3 43 .87 8 .86 0 .202 17 .3 75.33 61 .55 0.823 17.1 56.21 70.23 - 1 .25 -4 18 .4 38.18 7 .62 - 0.20 -5 20.0 60.52 22.85 23.87 0 .38 0 .776 18.6 74.11 43.79 41 .94 0.59 1 .167 30.8 101 .38 -40.45 144 .45 -0.40 1 .038 13 .5 33.29 14 .91 5 .27 0.45 0 .619 20.4 69.61 15 .98 0.23 -10 28.9 97.37 20.48 9 .43 0.21 0 .3111 24.2 81 .20 30.53 -1 .58 0.38 0 .3612 24.5 100.62 36.22 -1 .37 0.36 0 .3513 30.2 116.70 29.34 5 .19 0.25 0 .3014 23.9 75 .91 24 .02 -5.55 0 .32 0 .2415 26.3 72.83 27.88 10.44 0.38 0 .5316 28.3 99.36 16 .32 -25.02 0.16 -0.09

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150

125

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750.EU 50toU

0 25

0

-25

-50

0.60

0 .50

0 .40

0 .30U

0.20

0.10

0

Non.additive

0 AddIOv.

0

0 1 2 3 4 5 6 7 8 91011121314151617

Generation

Fig . 2. Additive variance components (+) and non-additive variancecomponents (o) estimated by LS for each generation .

0 •

Ph

•

enotypic

+

.

+

Non-addidv

•

0 0

0

0

0Additive

0.00 ............

0 1 2 3 4 5 6 7 8 91011121314151617

Generation

Fig. 3 . Phenotypic (iii), additive (o) and non-additve (+) coefficientsof variation estimated by LS for each generation .

Table 2 . Variance components and heritabilitiesestimated by DFREML based on cumulative data

Table 3 . Cumulative weighted average nar-row-sense heritabilities (by LS)

•

0

•

0 +

•

0

lative DFREML estimates were obtained (Table 3 andFig. 1) . The weighted averaged narrow-sense heritabil-ity for all sixteen generations is 0 .33 .

Regression coefficients of variance components mean or generation increased. To remove scale effects,(LS) on generation number and on mean yield of gen- coefficients of variation (CV) for phenotypic valueseration were estimated to study the changes of these in each generation were calculated . Additive and non-components over generations (Table 4 and Fig . 2) . additive CVs were estimated as a ratio of the standardThe phenotypic variance component increased as the

deviation over phenotypic mean. Figure 3 illustrates

Gen . o 22 ae h2

1 11 .29 28 .85 0.282 33 .04 26 .86 0.553 35 .98 23 .67 0.604 26 .95 26 .45 0.515 15 .74 37 .98 0.29

6 20 .32 37.39 0.357 22 .12 46.77 0 .328 22 .33 41 .72 0.359 24 .97 39 .59 0.39

10 23 .23 47 .82 0.3311 24 .27 50.54 0.32

12 30 .12 53 .14 0.3613 27 .46 63 .19 0 .30

14 27 .44 63 .26 0 .30

15 28 .75 61 .91 0 .3216 28 .54 65 .13 0 .31

Gen . Size Cumulative

weighted average

1 168 0 .202 186 0 .53

3 196 0 .78

4 245 0.605 252 0.55

6 263 0.567 299 0.38

8 397 0.399 393 0 .37

10 384 0.3511 552 0.35

12 552 0.3713 735 0 .35

14 510 0.3515 469 0 .35

16 571 0 .33

Table 4. Regression coefficients (b) of variance components ongeneration and mean yield per generation

the decreased tendencies of these CVs across genera-tions .

Discussion

Changes of variance components

Mean cut-flow yield in the Davis population of gerberaincreased from generation to generation . As the meanincreased, phenotypic variance also increased (Table4). However, the phenotypic coefficient of variationdecreased (Fig . 3). This suggests that the increase ofphenotypic variance was caused by scale effects, andthe actual variation decreased . This decrease shouldbe the function of genetic effects . In fact, both addi-tive variance and additive CV estimated by LS showeda tendency to decrease as the number of generationsincreased (Figs 2 and 3) ; the association between CVand generation was statistically significant . This indi-cates a decrease in additive genetic variation . Non-additive CV, on the other hand, remained unchanged(Fig. 3), although the variance increased. The non-additive effect includes dominance and environmentaleffects. Dominance variance over generations showeda decreasing tendency. Therefore, scale effects causedby the increase of phenotypic variance must haveresulted from environmental effects.

Changes of variance components will affect thechange of heritability . Figure 3 suggests the rate ofdecrease in additive variation was greater than thedecrease in phenotypic variation ; this indicates adecreasing tendency in narrow-sense heritability .

Estimation of heritability

Estimates of narrow-sense heritability based on LSvary from generation to generation . In early genera-tions two and three, it appears to be very high (> 0 .80)and then decreases significantly after generation threeand remains near 0 .30 with decreasing tendency in latergenerations eight through 16.

This pattern may be caused by many factors, suchas recombination, natural or artificial selection, changeof environmental effects, inbreeding and sampling.Recombination in early generations added genetic vari-ation because the original parents were from differentsources, and heritability may have increased. Afterseveral generations of random mating, the populationappears to have reached linkage equilibrium . Afterselection for several generations, major genes mayhave become fixed . Combining the two forces couldlead to the decrease in heritability . However, cut-flower yield as a quantitative trait is controlled bymany minor genes with large environmental effects .These polygenes are not likely to be fixed soon ; theyform the potential variation for selection . With long-term selection, inbreeding increases gradually (Huanget al ., 1994) and this will affect the variance . There-fore, heritability remains at an intermediate level witha decreasing tendency. Environmental effects are sub-ject to change and consequently cause fluctuations ofheritability estimates . For example, in generation sev-en, the management of the greenhouses changed andestimated heritability was - 0.34. The fluctuation in theestimate could also be caused by sampling error .

The same pattern appeared in estimates based onML (Fig . 1). In generation 3, the heritability reachedthe maximum, decreased and tended to plateau by gen-eration 5 .

Comparison between LS and ML estimates

Estimates of heritability between ML and LS methodsusing cumulative data (Fig . 1) are similar and stablefrom generation seven to 16 . ML estimates tended toplateau two generations earlier than LS estimates illus-trating that ML estimates are more efficient than LSestimates. Although LS estimates fluctuate, the cumu-lative averages are stable and close to the ML estimatesby generation 7 . As sample size increases, cumulativeLS estimates are closer to ML estimates due to the factthat LS estimators are statistically unbiased .

Ordinary LS estimates of genetic components aregenerally subject to selection bias (Robertson, 1977) .

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Generation Mean yieldb p-value b p-value

(Mean Yield) 0.82 0.003aph 3 .04 0.015 3.89 0.0001

LS ad -6.91 0 .063 1 .76 0.537LS oa -0.63 0 .653 - 1 .65 0.158LS ana 3.66 0.057 5.53 0.0001

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The initial motivation to develop ML to estimate genet-ic parameters was a concern about this bias due toselection (Henderson, 1949) . ML with Henderson'smodel utilizes the entire pedigree and accounts for allthe relationships between individuals, therefore, elim-inates selection bias. However, results from this studysuggest that if the sample size is large enough, ML andLS provide similar results .

Comparison with other studies

The overall estimate of narrow-sense heritability ofcut-flower yield for 16 generations in the Davis popu-lation of gerbera is 0 .33 based on least square methodsand 0.31 based on the restricted maximum likelihoodmethod. Estimates from other populations are : 0.32(Muceniece et al ., 1978), 0 .30 (DeLeo et al ., 1978),and 0.50 (Wricke et al ., 1971) . The results of thesestudies indicate that about one third of the variation ingerbera cut-flower yield is additive . This implies thatselection for cut-flower yield will be successful, as hasbeen demonstrated by the increases of cut-flower yieldin this population . The analysis of changes of variancecomponents and heritability suggested a gradual reduc-tion in genetic variation . However, heritability remainsnear 0 .27, suggesting that cut-flower yield will contin-ue to respond to selection after 16 generations, but therate of increase will decline .

References

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