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CROP BREEDING, GENETICS & CYTOLOGY

Estimation of Genetic Variance in Corn from F₁ Performance with and without Pedigree Relationships among Inbred Lines

^a Dept. of Animal Science, Univ. of Nebraska, Lincoln, NE 68583-0908 USA
^b Roman L. Hruska U.S. Meat Animal Research Center, USDA, ARS, Lincoln, NE 68583-0908 USA
^c Dep. of Agronomy, Univ. of Nebraska, Lincoln, NE 68583-0908 USA
^d Pioneer Hi-Bred International, Inc., Johnston, IA 50131 USA

dvan-vleck1{at}unl.edu

	ABSTRACT

TOP NOTES ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 
Estimates of genetic variance are needed for ranking of inbred lines for selection and for prediction of response to selection. The objectives of this study were to determine whether including relationships among inbred lines affects estimates of genetic variance and whether random association among inbred lines mated together affects estimates. Genetic variance was estimated with different models with restricted maximum likelihood for eight traits from matings of inbred lines from two heterotic groups (Iowa Stiff Stalk Synthetic, SSS, and unrelated to SSS, NSSS) of corn (Zea mays L.). For each comparison relationships among one or both of the inbred lines were either considered or ignored. With relationships ignored, variance due to inbred line effects was reduced on average by 33% for SSS inbred lines and 18% for NSSS inbred lines. Estimates were also reduced for variance of SSS inbred lines by 11 to 41% when calculations were done with effects of NSSS inbred lines considered to be fixed and 6 to 31% for variance of NSSS inbred lines with SSS inbred lines considered fixed. The increase in variance with relationships among inbred lines considered indicates that potential gain from selection would be greater than predicted from estimates of variance due to line effects calculated ignoring relationships among lines. Estimates of inbred line variance within a heterotic group were usually smaller when lines in the other group were considered fixed. This result suggests that variance due to line effects can be inflated due to association of inbred lines between heterotic groups.

Abbreviations: gdu, growing degree units • MTDFREML, multiple trait derivative free restricted maximum likelihood • NSSS, unrelated to SSS • SSS, Iowa Stiff Stalk Synthetic

	INTRODUCTION

TOP NOTES ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 
SETS OF CROSSES AMONG INBRED LINES (e.g., Comstock and Robinson, 1948) are often used in corn to estimate components of genetic variance. Although with the original factorial mating design of Comstock and Robinson (1948) the parent inbred lines were assumed to be random, in applied corn breeding available inbred lines are selected and usually related within heterotic group. Thus, analyses should consider relationships among inbred lines (e.g., Bernardo, 1994). Animal geneticists regularly use genetic relationship matrices when estimating genetic components of variance. Hudson and Van Vleck (1982) and Dong and Van Vleck (1989) noted that ignoring existing relationships usually resulted in a reduction in estimates of genetic variance. The first objective of this study was to determine if estimates of genetic variances were the same when relationships among inbred lines were included or ignored.

Henderson (1973) suggested that associations between one random factor and another random factor can inflate variances of random factors such as genetic variances. Meyer (1982) and Van Vleck (1985) used models for estimating genetic variances that considered effects of previously selected older sires that have large numbers of progeny to be fixed effects and effects of young sires with only progeny for an initial progeny proof to be random effects. National dairy and beef cattle (Bos indicus and Bos taurus) genetic evaluations routinely consider herd–year–season (contemporary) effects to be fixed effects rather than random effects to account for any association of contemporary effects with effects of sires. The second objective of this study was to determine whether random association of inbred lines in crosses between inbred lines of different heterotic groups affects estimates of genetic variance. Therefore, two additional sets of analyses were done, one set with the effects of inbred lines from one heterotic group modeled as fixed effects and the second set with the effects of inbred lines from the other heterotic group modeled as fixed effects.

	Materials and methods

TOP NOTES ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 
Traits
Phenotypic measurements were recorded for eight traits (Table 1) from progeny obtained by crossing seven different sets of four inbred lines per set from SSS at random with seven sets of four NSSS inbred lines, resulting in 16 single crosses per set. The NSSS inbred lines are a mixture of different base populations. The sets were grown across twenty locations and 2 yr (1991, 1992), although no traits were measured for all plots (Table 1).

Plant and ear heights were averages of five plants per plot measured to the nearest 15.5-cm increments from the soil surface to tip of tassel or ear node, respectively. Root and stalk lodging were measured as the percentages of unlodged plants in a plot. Days to pollen shed were recorded as growing degree units (gdu) from planting until 50% of the plants were shedding. Yield was recorded as pounds of grain and converted to metric tons per hectare at 155 g H₂O kg^-1. Seedling vigor was subjectively scored from one (low) to nine (high). Number of plots, overall means, and standard deviations are reported in Table 1.

Statistical Analyses
Analyses for each trait were based on the model:

(1)

where y is the vector of observations for that trait; ß is the vector of fixed effects associated with records in y by design matrix, X (fixed effects were year by location combination, and with some models, effects of parental inbred lines of one or the other heterotic group were designated as fixed); u is the vector of random effects associated with records in y by design matrix, Z (random effects were effects of parental inbred lines not specified as fixed), with u = (u_SSS'u_NSSS')' when effects of inbred lines in both parental groups were considered random; and e is the vector of random residual effects.

The first and second moments are

(2)

and

(3)

and

(4)

where A_SSS and A_NSSS are matrices of Wright's numerator relationships among SSS and NSSS inbred lines, respectively. The variances, ²_SSS and ²_NSSS, are variances of additive genetic effects in the initial populations of SSS inbred lines and NSSS inbred lines, respectively; I_N is an identity matrix with N the number of observations, and ² is the residual variance.

In these analyses, the convention of animal breeders was followed to use the numerator relationship matrix, A, as a measure of genetic likeness among inbred lines. Plant breeders (e.g., Bernardo, 1994) typically have expressed G in terms of coefficients of coancestry, which are one-half the numerator relationships. The coefficients of coancestry were calculated from pedigrees. Coefficients of coancestry among SSS inbred lines varied from 0.148 to 0.574, with a mean of 0.363, and among NSSS inbred lines varied from 0.048 to 0.509, with a mean of 0.246. Coefficients of coancestry between SSS and NSSS inbred lines varied from 0.005 to 0.081, with a mean of 0.023. For the analyses, the SSS inbred lines were considered unrelated to the NSSS inbred lines. With a sire and dam model corresponding with fully inbred lines in heterotic groups SSS and NSSS, the sire and dam components of variance in the absence of maternal effects estimate one-half of additive genetic variance (Henderson, 1977).

Initially a component of variance for SSS x NSSS interaction was included in the model, but this component was small for all traits (average of total variance was .008) and had no effect on patterns seen from these analyses and are not reported.

For the first objective, effects of inbred lines within each heterotic group were considered random and related. This model was compared with a similar model with pedigree information excluded. For the second objective, two additional sets of analyses were done. First, calculations were done as if inbred lines in the first heterotic group (SSS) were fixed with relationships within the second heterotic group (NSSS) either considered or ignored. Second, calculations were done as if inbred lines in the second heterotic group (NSSS) were fixed with relationships within the first heterotic group (SSS) either considered or ignored. The modeling of one set of inbred line effects as fixed removes those effects from the model for the usual quadratics computed to estimate components of variance due to line effects of the other heterotic group that might inflate components of variance (e.g., Henderson, 1973, 1985). The modeling of one set of effects as fixed removes those effects from expectations of the usual quadratics computed to estimate components of variance of line effects within the other heterotic group (Van Vleck, 1985).

Estimates of variance components due to line effects were obtained with a derivative-free restricted maximum likelihood program (multiple trait derivative free restricted maximum likelihood, MTDFREML; Boldman et al., 1995). The MTDFREML package is a set of programs that use numerator relationships and a derivative-free algorithm to obtain restricted maximum likelihood estimates of variance and covariance components. These programs can be used to obtain solutions for fixed effects, breeding values and other random effects, as well as sampling variances of solutions to MME and expected values of the solutions. Fixed effects, covariates, and other random effects can be specified separately for each trait.

The matrix of numerator relationships among the SSS inbred lines and among the NSSS inbred lines was inverted with a separate program and written to files usually written by a program in the MTDFREML package that calculates elements of the inverse of the numerator relationship matrix from a list of animals with sires and dams following rules of Quaas (1976). Inbred lines in the first heterotic group were treated as first-animal genetic effects and inbred lines in the second heterotic group were treated as second-animal genetic effects (typically maternal effects in animal models). Thus, estimates of line variances for both heterotic groups were easily obtained with no reprogramming of the package.

With the derivative-free algorithm, convergence for variance and covariance component estimation occurs when the global maximum of the log likelihood function is found. The simplex (polytope) method described by Nelder and Mead (1965) is the procedure used to locate the minimum with respect to the variance components of negative two times the logarithm of the likelihood (), which corresponds with the maximum of the logarithm of the likelihood. Minus twice the likelihood, -2 = constant + log|R| + log|G| + log|C| + y'Py, with C a full-rank portion of the coefficient matrix of the mixed model equations and , where q₁ is the order of A_SSS (number of SSS inbred lines), q₂ is the order of A_NSSS, ²_SSS is the variance of effects due to SSS inbred lines, and ²_NSS is the variance of effects due to NSSS inbred lines, and y'Py is the generalized residual sum of squares (Harville, 1977; Smith and Graser, 1986; Meyer, 1989). For a single trait analysis, y'Py converges to N - rank(X). The derivative-free method basically tries different R and G (e.g., ² of R = I_N², ²_SSS of A_SSS²_SSS, ²_NSSS of A_NSSS²_NSSS) until the combination that maximizes the log of the likelihood (i.e., minimizes negative two times the log of likelihood) is found for the data vector, y. The simplex algorithm cannot guarantee convergence to a global maximum. The variance of the simplex, which is an intermediate convergence criterion, depends on the current simplex and becomes small even if convergence is to a local minimum. The program is restarted with estimates at intermediate convergence as initial values until a global maximum is found (e.g., the log likelihood does not change to third decimal place after consecutive restarts).

	Results and discussion

TOP NOTES ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 
Including or Ignoring Relationships
For the comparisons with effects of inbred lines within both heterotic groups considered random, the likelihood improved (-2logL decreased) for all eight traits after pedigree relationships were included, indicating a better fit to the data (Table 2) . The changes in -2 were generally small between including relationships and ignoring relationships.

Estimates of variance due to effects of inbred lines increased after pedigree information was included in all analyses, except for one pair of analyses for which the estimates were equal (Tables 3 and 4) . With both sets of inbred lines considered random (Table 3), the ratio of estimates of ²_SSS ignoring and considering relationships ranged from 0.51 to 0.87, with an average of 0.67 (Table 3). With effects of inbred lines of the NSSS heterotic group considered fixed and SSS inbred lines as random (Table 4), the ratios of estimates of ²_SSS ignoring and including relationships ranged from 0.59 to 0.89, with an average of 0.68.

These results show that failure to consider relationships among inbred lines may result in decreased estimates of genetic variance. These decreases are substantially greater than those observed in animal studies (e.g., Dong and Van Vleck, 1989). The average relationships in this study are much larger than those in the animal studies. In addition, the sires in the study of dairy cattle were not as inbred as the inbred lines in this study. The changes in estimates of genetic variance due to ignoring relationships will probably depend on the magnitude of the relationships. Further study would be needed to investigate cases between the extremes of unrelated inbred lines and highly related inbred lines.

Considering Lines as Fixed Effects
Estimates of ²_SSS decreased for seven of eight traits when effects of NSSS inbred lines were considered fixed rather than random in the calculations (Table 3; Columns 1 vs. 4 with relationships among SSS inbred lines ignored and Columns 2 vs. 5 with relationships considered). Similarly, variances of effects of inbred lines in the NSSS heterotic group also decreased for all except one of eight traits when SSS inbred lines were considered fixed rather than random in the calculations with relationships ignored (Table 4; Columns 1 vs. 4) and for all eight traits with relationships considered (Table 4; Columns 2 vs. 5).

These analyses demonstrate that randomly mating inbred lines in a set does not ensure that the variances will be unaffected by the mating partner probably because of the limited number of mates in a mating set. A larger number of mates would tend to average out better and worse mates more completely. For most traits the difference was small between estimates when the effects of the other heterotic group were considered fixed or random for the analysis. However, for plant height, the estimate of ²_SSS was considerably larger, by 37% (24.71 vs. 18.07), and of ²_NSS by 36% (19.42 vs. 14.32) when both sets of line effects were considered random rather than when effects of the other heterotic group were considered fixed for the calculations.

Estimates of residual variance (Table 5) for all traits and all comparisons were nearly the same whether pedigree information was included or whether effects of inbred lines in one or the other heterotic group were considered as fixed.

View this table: [in this window] [in a new window]   Table 5 Estimates of residual variances with relationships among lines in an heterotic group ignored or included in the analysis and with effects in the one or the other heterotic group considered as random or fixed effects

	Conclusions

TOP NOTES ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

	ACKNOWLEDGMENTS

 
This research was supported in part by a grant from Pioneer Hi-Bred International Inc., Johnston, IA.

	NOTES

TOP NOTES ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 
Journal Paper no. 12345 of the Nebraska Agric. Res. Div., Univ. of Nebraska, Lincoln, NE 68583-0908.

Received for publication May 10, 1999.

	REFERENCES

TOP NOTES ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 

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• Comstock R.E., Robinson H.F. The components of genetic variance in populations of biparental progenies and their use in estimating average degree of dominance. Biometrics 1948;4:245-266.
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