Heredity (1977), 38 (3), 329-338

THE CAUSES AND CONSEQUENCES OF NON-NORMALITYIN PREDICTING THE PROPERTIES OF RECOMBINANT

INBRED LINES

H. S. POONI, J. L. JINKS and M. A. CORNISHDeportment of Genetics, University of Birmingham, Birmingham 815 2TT

Received 1 .xi.76

SUMMARY

The contribution of two of the major causes of non-normality in the distribu-tion nf pure-breeding lines derivable by single seed descent from an F2,namely, linear epistasis and genotype x micro-environment interaction, areexamined theoretically by algebraic models and computer simulations. It isshown that all the parameters required for predicting the distribution of thepure-breeding lines including the sign and magnitude of the coefficients ofskewness and kurtosis which measure the deviation from normality, can beestimated from the basic set of P1, P5, F1, F2, B5 and B, generations and an F,triple test cross.

These predictions are shown to be satisfactory in practice for nine charac-ters in the generations and families derived from the cross between varieties Iand 5 of .J'Iieotiana rzutiea. Although the distributions of 80 F15 families derivedby single seed descent from this cross display non-normality which is attribut-able to epistasis and genotype x environment interaction, predictions of theirproperties which assume normality are satisfactory providing that we make acorrection to the expected means of the distributions equal to the epistaticcomponent [ij. This cnrrectinn accounts for considerably more of the dis-crepancies between observations and predictions than all other causes includingnon-normality. In practice, therefore, it appears to be the only correction thatneeds to be made for data of the kind we have analysed in which there aremoderate levels of epistasis and medium to high heritabilities.

1. INTRODUCTION

IN order to make useful predictions about the properties of the recombinantpure-breeding lines that can be derived by single seed descent from the F2of a cross between a pair of such lines, Jinks and Pooni (1976) assumed thatthe line means were approximately normally distributed even where, as inthe presence of directional epistasis, this was unlikely to be true. Neverthelessthe predictions were satisfactory for all practical purposes. In this paperwe examine two major causes of non-normality, epistasis and genotype xenvironment interactions. Using theoretical models, simulations andpractical examples we show the direction and the magnitude of the deviationsfrom normality that they produce and the consequential effects on the useful-ness of the predictions we can make in their presence.

2. EFFECT OF EPI5TA5I5

Statistically, the non-normality of a distribution is jointly measured byits third and fourth moments. The moment which is associated with theasymmetry of the distribution around its mean is termed "skewness" and

329

330 H. S. POONT, J. L. JINKS AND M. A. CORNISH

its estimate g1 (the coefficient of skewness) for a sample size of n, is calculatedas

[f(x — —

(Snedecor and Cochran, 1967). For a population of pure-breeding linesderived by single seed descent from the F2 of a cross between a pair of lines,the expected value of g1 in the presence of digenic interactions is given bythe expression:

[6dJdkiJk + 6iJSikSiJk]/[d + i]]Here, d and d represent the additive effects of alleles at jth and kth locusand jk' i8 and ks are respectively the additive x additive interaction effectsofjth and kth, jth and sth and kth and sth loci as defined on the F metric(Van der Veen, 1959).

The fourth moment, kurtosis, provides a statistical measurement of therelative peakedness (or flatness) of a distribution in terms of the projectedfrequencies in the central peak in relation to the density of the observationsfalling in its shoulders and tails. The estimate of g2, the coefficient ofkurtosis, is usually obtained from the expression

[Ef(x —x)4/Ef}/{f(x — —3;

and on our model its theoretical expectation is given by the ratio of

Zd + Ei + 6dd + 6di + 6ii+ 24Edd k(EiJSi ks + >.L jstsk + sjsk + iSJikS)

+ 24jk1 js1kt1st + 24tjkks1jttst+24Eljsljtlkslst

to {d +EiJk} , minus 3.

With the assumption that d = d over all the K loci and that jk = i forevery one of the K(K- 1) pairs of loci, the expectations of g1 and g2 simplify to

g1 = {3K(K— 1)di2 +K(K— 1)(K—2)i3}/{Kd2 +K(K— 1)i2}

and

= [({K(1+ :;(K— 1))}d4+ K(K— 1)(1 + 3(4K(K— 1)— 1)+6(K—2)

(K— 3))}i4 + (3K(K — 1)(5K—8)}d2i2)/{Kd2 + 3K(K —1)12}2]3.In these simplified forms one can see more clearly how the magnitude

and the direction of g1 and g2 directly reflect the magnitude and the directionof the epistatic component ri]. Nevertheless, a better appreciation of thesituation can be obtained by simulation. In fig. 1 are presented threesimulated distributions each for a random sample of 500 pure-breeding linesderived by single seed descent from a cross between a pair of inbreds differingat K = 10 loci. In the first sample there is no epistasis while in the secondand third samples epistasis is present at the level of Zk =--Ed, theepistasis being of the complementary type in the second sample and ofduplicate type in the third (see also the figures from Snape and Riggs, 1975).While, however, these distributions demonstrate the effect of epistasis they

1

2

RECOMBINANT INBRED LINES 331

m

FIG. 1. Distributions of Samples of 500 Inbreds from three different simulations.

are based on a heritability of one (h 1.00) and therefore do not take intoconsideration the important influence of the environment and genotype xenvironment interactions.

3. RANDOM ENVIRONMENTAL EFFECT AND G x E INTERACTION

A general effect of introducing random environmental variation on thesediscrete distributions of the genotypic scores is to produce continuity and torestore normality or at least to offset some of the non-normality producedby epistasis. The extent to which this occurs, however, will be dependent onthe proportion of the total variation in the population attributable to randomenvironmental causes. To illustrate this, theoretical estimates of g1 and g2for various levels of epistasis and heritability (h) have been tabulated intable 1. These show the expected amelioration by the random environ-mental variation of the deviations from normality.

The environmental variation displayed by different pure-breeding lineswhen exposed to micro-environmental variation, however, is not always

no epistasis

complementaryepistasis

duplicateepistasis

3

332 H. S. POONI, J. L. JINKS AND M. A. CORNISH

TABLE I

The expected values of g1 (coefficient of skewness) and g2 (coefficient of kurtosis) for various levels ofepistasis and heritability (hz) values. A random environmental variation is assumed when h is less than

10 and skewness is always positive for complementary epistasis and negative for duplicate epistasis

Heritability Level of epistasish, — —

01 025 050

Skewness

1.0 117 l53075 1•37 l•710•50 IO00•25010

Kurtosis

10 142 252 419075 108 250 403050 051 149 2690•25 —040 0•07 075010 — 123 —107 —0•81

random or homogeneous over the lines. A significant part of the environ-mental variation is often due to the differential response of different geno-types to the micro-environmental effects. In these circumstances the environ-mental variation instead of reducing the non-normality due to epistasis mayitself produce non-normality. These two causes of non-normality may,therefore, reinforce or cancel out each other's contributions to non-normalitydepending on whether they have the same or opposing signs. The directionof the genotype x environment contribution will itself be determined by thesign of the covariance between the mean and variance of the pure-breedinglines, which has the expectation 2 . 'edg1, where e5is the additive environ-mental effect and g15 the genotype x environment interaction between dand e1. Since d is always positive the sign of the covariance is essentiallydetermined by the signs of e and g1. If both have predominantly the samesign, whether positive or negative, the covariance and hence the skewnesswill be positive. If on the other hand they have predominantly oppositesigns the covariance and skewness will be negative. The covariance, how-ever, may be zero even in the presence of significant genotype x environmentinteractions. This will occur if the signs of e1and g5 are independent of oneanother. In these circumstances the interactions will produce no skewnessbut may still produce kurtosis. They will do so, for example, if the zerocovariance results from above average variances being associated withaverage means and below average variances with high and low extrememeans equally, or vice versa.

We can determine the sign and magnitude of this covariance from themeans and variances of the pure-breeding families derived by single seeddescent and test its significance as the correlation coefficient. We can alsodetermine its sign if not its precise magnitude by deriving the covarianceof mean and variance among the P1, P2 and F1 families (Perkins and Jinks,1970) or amongst the families of an F2 triple test cross (Perkins and Jinks,1971). Equally, the sign and the magnitude of sensitivity parameter

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334 H. S. POONI, J. L. JINKS AND M. A. CORNISH

S = 1(VP1_ VP2) = 2L'e5g5 in which VP1 is the variance of the parentalline with the larger score, will establish the sign and proportionate magnitudeof the genotype x environment interaction covariance.

4. EXPERIMENT AND RESULTS

The experimental material consists of:

(i) The 80 pure-breeding F11 families produced by single seed descentfrom the F2 generation of a cross between varieties 1 and 5 ofJ'*Iicotiana rustica (Pooni and Jinks, 1976).

(ii) A triple test cross set of families produced by crossing these 80inbreds to P1, P2 and the F1 of varieties 1 and 5.

(iii) The basic set of six families (P1, P2, F1, F2, B1 and B2) of the crossof varieties 1 and 5.

All the families were raised in a single experiment. All plants wereindividually randomised. Each family in sets (i) and (ii) was representedby 10 plants and the six generations in set (iii) by 120, 140, 140, 240, 160and 200 plants, respectively. Nine metrical traits were recorded; the threedevelopmental heights (H1, H2 and H3) taken at successive fortnightlyintervals after transplanting into the field; flowering time (FT) and heightat flowering (lIFT); leaf length (LL) and leaf width (LW) on the largestleaf blade; leaf spread (LS) across the plant of the two leaves involving thelargest leaf and the final height (FH) of individual plants at the end of theseason. All characters were scored in centimetres (to the nearest 05 cm)except FT which was scored in days.

TABLE 3

Dependence of Skewness and Kurtosis on epistasis ([i] /m) and environmental sensitivity Sf( VP1 + VP2)as expressed in the form of regression coefficients the estimates of which are obtained from a multiple

regression analysis

Dependent variable

Independent variable Skewness Kurtosis

Epistasis ([iJ/m) 2ll*±O.8l l59 NS±O87Environmental

sensitivityS/l-(VP1+ VP2) O77 NS±O44 O55 NS±047

NS: P>OO5. * P = OO5—OOl. ** P<OOl.

Two types of statistics can be derived from these data. The first are theestimates of the parameters which are required to predict the properties ofthe pure-breeding lines derivable from this cross by single seed descent.The second are the estimates of the parameters which describe the observedproperties of the 80 lines that have actually been derived in this way. Thefirst include the additive genetic component of variance D which we haveestimated from the triple test cross (ii) and {dJ, [iJ and S which we haveestimated from. the basic generations (iii). To facilitate some of the com-parisons we wish to make between characters we have also standardised theestimates of {i] as [i] /m and of S as S/( VP1 + VP2). The second type of

RECOMBINANT INBRED LINES 335

estimate includes the narrow heritability h, the correlation of family meansand their variance, and the coefficients of skewness and kurtosis all of whichhave been obtained from the 80 pure-breeding lines themselves (i). Theseestimates for each of the nine characters are listed in table 2.

Reference to table 2 shows an obvious relationship over the nine charac-ters between our standardised predictors of non-normality, [i] /m andS/12( VP1 + VP2), and our observed deviations from normality g1 and g2. Theresults of a multiple regression analysis in which the two predictors wereused as the independent variables and the coefficients of skewness andkurtosis as dependent variables are summarised in table 3.

In table 4 are presented the two standard predictions of Jinks and Pooni(1976) of the proportions of pure-breeding lines which are expected to havea higher score than P1 or a lower score than P2 the first prediction assumingno [i] the second allowing for the presence of [i] but in both cases assumingnormality. Also presented are the observed proportions among the 80 pure-breeding lines and the discrepancies between the observed and expectedproportions as absolute magnitudes.

5. Discussion

From the theoretical expectations in section 2 and the simulations infig. 1 it is quite clear that there is a direct and predictable relationshipbetween the occurrence, magnitude and direction of skewness as measuredby the coefficient g1 and the presence, magnitude and sign of the linearadditive x additive interaction component EiJk. There is also a relationshipthough more complex between the coefficient of kurtosis g2 and Eijk. Butwhile kurtosis in general increases with the magnitude of the epistasis thereis no simple relationship between the sign of the epistasis and that of thekurtosis. These are, however, genotypic expectations and while it is withthese that our predictions are concerned, in practice we shall usually beexamining phenotypic distributions in which random environmental variationis superimposed upon the genetic variation.

The best way of examining the effect of random environmental variationon g1 and g2 is by simulation studies of the kind summarised in table I.These show that both skewness and kurtosis are affected by decreasing theheritability and hence increasing the contribution of random micro-environmental effects. Kurtosis, however, is more sensitive to this decreaseirrespective of the level and direction of the epistasis and for moderate levelsof epistasis it reaches its smallest value at intermediate values of the herita-bility. Skewness on the other hand, is least important when the contributionof such random environmental effects is at its greatest. In fact for heritabilityvalues below 05 it is unlikely that the levels of epistasis encountered inpractice would lead to significant skewness.

The coefficient of skewness, g1, is either significant (P <0.05) or highlysignificant (P <0.01) for four of the nine characters (table 2). Furthermore,its direction and magnitude follows closely that of our standardised predictor[i] /m with a positive and significant regression of 211 08 1 (table 3). Thusthe theoretical relationship holds good in practice.

In contrast the estimates of the coefficient of kurtosis, g2, are non-significant in all the cases (table 2) even though their magnitudes and signsvary between + 069 and — 049 as would be expected from the simulations

336 H. S. POONI, J. L. JINKS AND M. A. CORNISH

TABLE 4

The observed and expected proportion of inbreds falling outside the parental range for various characters. The expectedproportions are worked out by assuming no epistasis ([d] only) and by allowing for the epistatic component ([d] and [iJ).

P1 is the larger scoring parent (Vi or V5) for the character concerned

Proportion of inbreds falling outside parental range

tSummed discrepanciesObserved between observed and

Character proportions Expected proportions expected- ____________________________________________ ____________________________-' C -' C

from [d] alone from [d] and [i]* -' -

>P1 <P > Pi <P2 > P1 <P5 with [d] with {d] and [i]H1 0•i0 O•50 0•23 0-23 007 0-52 0-40 005H2 008 064 021 0-21 004 054 056 0-14H3 018 07O 035 0-35 011 068 0-52 009FT 0-58 0-06 0•30 0-30 0-61 0-10 052 0-07HFT 030 0-20 0-27 0-27 0-36 0-19 010 007LL 0-60 031 0-43 0-43 0-63 026 O29 008LW 040 030 0-29 0-29 032 0-27 012 0.11IS 0-68 0-26 044 0-44 072 019 042 011FH 0-40 013 0l9 019 0-33 009 027 0.11

t Discrepancies between observed and expected proportions are worked out for each character bysummation of the absolute deviations between the observed and expected values.

for characters with medium h values and a moderate level of epistasis(tables 1 and 2). The sample size of 80 pure-breeding lines, however, ismuch smaller than the 500 that would be required to obtain significance foran epistatic level of 01 and a h value of 0-6 (Cornish 1976).

The largest value of g2 is obtained for the character H2 which has ahighly significant value for [i] and the largest value for [i] /m (table 2) andover all the characters there is a positive regression ofg2 on [i]/m of 1.59 087(table 3). This regression, however, is not significant which is not surprisingin that none of the g2 values is significant and we do not expect a simple linearrelationship between them (table 1).

While the estimates of g1 and g2 show the expected positive relationshipwith our standardised sensitivity parameter (table 3) neither of the regressioncoefficients is significant. Again this is not surprising in that only four of theestimates of the sensitivity parameter and none of the g2 values are significant.That the theoretical relationship holds, however, is shown conclusively byreplacing the sensitivity parameter in the multiple regression analysis by theobserved correlation between family means and the variances among the 80pure-breeding lines, whereupon both regression coefficients become signifi-cant (1.63** and 1.27* There can be no doubt, therefore,that in practice directional interactions between the genotypes and themicro-environmental variation produce both skewness and kurtosis and ourfailure to establish a significant causal relationship using our standardisedsensitivity predictor must be due to the unreliability of the estimates of thepredictor (see section 6).

Although we know that both epistasis and genotype x environmentinteraction have produced significant non-normality for more than half of thecharacters studied we have nonetheless proceeded with the prediction of the

RECOMBINANT INBRED LINES 337

proportion of pure-breeding lines that are expected to fall outside of theparental range assuming normality (table 4) in order to assess the conse-quences of the failure of this assumption. As reported in our previous paper(Jinks and Pooni, 1976) the discrepancies between observed and expectedproportions are much larger when no allowance is made for [i] than whenallowance is made for its presence. Nevertheless, even after allowing for [i]there are still residual discrepancies between the observed and expected.For example, for the first character in table 4 (H1) the summed discrepancieswhen no allowance is made for [i] are 040 and after allowing for it are 005.Hence allowing for [i] reduces the discrepancies by 0.35 and all other causesof discrepancies including non-normality account for the remaining 005.The corresponding average values over all nine characters are 024 and 009respectively. Thus even though some of these characters do not have asignificant value for [i] nonetheless the simple linear correction for [i] onaverage accounts for more of the discrepancies between observed andexpected proportions than all other causes combined. These other causesinclude non-normality and the sampling errors of our predictions and of theobserved statistics. In addition we are predicting the proportion of genotypesthat are expected to fall outside of the parental range and observing theproportion of phenotypes. The latter is a real distinction in these datawhere the heritability varies between 054 and 070 (table 2). We can, ofcourse, readily accommodate these deviations from a heritability of one inour prediction equations (Jinks and Pooni, 1976), although it is the propor-tion of genotypes we wish to predict in most circumstances. The only clearindication that non-normality may be a contributor to the residual discre-pancies is provided by character H2 (table 4) which has the largest skewness(table 2) and the largest discrepancy. The importance of the other causes,however, is equally shown by the characters H3, LL, LW, LS and FH whichalso have above average residual discrepancies but show no significantskewness.

6. CoNcLusioNs

These studies have demonstrated that directional epistasis and genotype xenvironment interactions are two major causes of non-normality in thedistribution of pure-breeding lines. This non-normality may be anticipatedand allowed for in making predictions by observing the significance anddirection of Eik and the correlation between family means and variances.Estimates of [i] obtained from the basic generations, as in the present paper,provide satisfactory predictions although they could be misleading if therewere a high degree of dispersion of the interacting alleles in the parentallines. A better prediction of the correlation between family means andvariances than the one used in the present paper is obtainable from a covari-ance analysis of (L1+L2) and (oj+o2) values from an F2 triple testcross (Perkins and Jinks, 1971). Since this design is also the best source ofthe other predictors its use would not involve any additional experimentation.

A further step in the present studies would be to use our predictors ofnon-normality to generate appropriate non-normal distributions, theprobabilities of which could then be integrated to predict the proportion ofpure-breeding lines falling outside of the parental range by having a higherscore than P1 or a lower score than P2. While this, however, may be worthy

338 H. S. POONI, J. L. JINKS AND M. A. CORNISH

of further investigation there appears to be little justification for attemptingthis except for perhaps character H2 in the present study.

Particularly impressive is the confirmation of our earlier finding (Jinks andPooni, 1976) that by using [i] to make a linear correction to the expectedmean of the distribution of pure-breeding lines we can remove most of thediscrepancies between the predicted and observed results. And while thereare residual discrepancies not even in the worst case (character H2 in table 4)are these discrepancies more than of borderline significance (P = 005—0.02).They are therefore very unlikely in practice to lead to an incorrect decisioneither in choosing between alternative crosses or in determining the samplesize required to yield a predictable proportion of pure-breeding lines whoseperformance exceeds some predetermined score with some predeterminedprobability of success.

Acknowledgments.—We are indebted to Dr M. J. Kearsey who developed the computerpackage upon which the simulations are based and for his advice on their further develop-ment.

7. REFERENCES

cORNI5H, M. A. 1976. A study of the causes of non-normality in the distribution of inbredlines derived by single seed descent. M.Sc. thesis, Dept. of Genetics, University ofBirmingham.

JINKs, J. L., AND POONI, H. s. 1976. Predicting the properties of recombinant inbred linesderived by single seed descent. Heredity, 36, 253-266.

PERKINS, j. as., AND JINKS, j. L. 1970. Detection and estimation of genotype-environmental,linkage and epistatic components of variation for a metrical trait. Heredity, 25, 157-1 77.

PERKINS, J. as., AND JINK5, j. L. 1971. Analysis of genotype x environment interaction intriple test cross data. Heredity, 26, 203-209.

POONI, H. 5., AND JINKs, J. L. 1976. The efficiency and optimal size of triple test crossdesigns for detecting epistatic variation. Heredity, 36, 215-227.

SNAPE, 3. W., AND RIGGs, T. j. 1975. Genetical consequences of single seed descent in thebreeding of self pollinating crops. Heredity, 35, 211-219.

5NEDECOR, G. W., AND cocHRAN, w. o. 1967. Statistical Methods. Iowa State UniversityPress, Ames, Iowa, U.S.A.

VAN DER VEEN, 3. H. 1959. Tests of non-allelic interaction and linkage for quantitativecharacters in generations derived from two diploid pure lines. Genetica, XXX, 201-232.