Published in Crop Sci. 43:1930-1937 (2003).
© 2003 Crop Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
REVIEW & INTERPRETATION
Clarification and Reevaluation of Population-Based Diallel Analyses
Gardner and Eberhart Analyses II and III Revisited
Leigh W. Murray*,a,
Ian M. Rayb,
Haiying Donga and
Armando Segovia-Lermab
a University Statistics Center MSC 3CQ, New Mexico State Univ., P.O. Box 30001, Las Cruces, NM 88003-8001
b Dept. of Agronomy and Horticulture MSC 3Q, New Mexico State Univ., P.O. Box 30003, Las Cruces, NM 88003-8003
* Corresponding author (estatu10{at}nmsuvm1.nmsu.edu).
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ABSTRACT
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Plant breeders and geneticists often use diallel mating designs to obtain genetic information about a trait of interest from a fixed or randomly chosen set of parental lines. Diallel analyses of broad-based populations have frequently been conducted by means of three analyses presented by Gardner and Eberhart in 1966. The original paper of Gardner and Eberhart used sequential model fitting to obtain estimates of effects and corresponding sums of squares. This approach, although having a long history, suffers from shortcomings which have led to confusion about what hypotheses the analyses actually test. The objectives of this paper were to delineate clearly all models implicitly required to perform Gardner and Eberhart Analyses II and III, and to present explicit formulas for effects in terms of the population means which are fundamental and unambiguous. While developing formulas of effects, we discovered a typographic error associated with variety effects in the original example of Analysis II. Our results also indicate that Analyses II and III effect formulas are rather nonintuitive both biologically and genetically, and incorporate multipliers that are functions of the number of parents. Another specific result shows that the varietal effects obtained in Gardner and Eberhart's Analysis III are "unconstrained" estimates, while those from the Analysis II are estimates constrained by the assumption of "no heterosis." These results have implications for the use and interpretation of such effects. A SAS computer program for analyzing diallels among broad-based populations according to Gardner and Eberhart's Analyses II and III is also reported.
Abbreviations: GE1, GE2, and GE3, Gardner and Eberhart Analyses I, II, and III, respectively • GCA, general combining ability • SCA, specific combining ability
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INTRODUCTION
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PLANT BREEDERS AND GENETICISTS frequently use diallel mating designs to obtain genetic information for a fixed or randomly selected chosen set of parental lines. Diallel designs and analyses have been developed for parents that range from inbred lines to genetically broad-based varieties (Griffing, 1956; Gardner and Eberhart, 1966). Computer programs have been developed for Griffing's diallel analyses (Burow and Coors, 1994; Magari and Kang, 1994; Zhang and Kang, 1997) but none is widely available for Gardner and Eberhart's analyses.
Diallel analyses of broad-based populations (Crossa et al., 1990; Gerrish, 1983; Mungoma and Pollak, 1988; Ouendeba et al., 1996; Widstrom and Snook, 1998) have generally been conducted according to Gardner and Eberhart (1966) Analysis I, II, or III (hereafter referred to as GE1, GE2, and GE3, respectively). While developing a SAS IML/MACRO program (Dong, 1999; SAS Institute, 1989a, 1990) to perform Gardner and Eberhart's original implementation for analyzing data from a diallel among genetically broad-based alfalfa (Medicago sativa L.) germplasms, we identified several models that were not explicitly provided in the original paper (Gardner and Eberhart, 1966), but which are necessary for the analyses. Hence, our first objective was to delineate clearly all models that are implicitly required for the GE2 and GE3 analyses.
Original implementation of GE2 and GE3 also involved sequential model-fitting. This approach is conditional upon assumptions that may not always be correct, and is sensitive to nonorthogonality between sets of effects. Given these limitations, our second objective was to use full-rank, cell-means models, in conjunction with the General Linear Hypothesis method, as an alternative approach to estimate effects and test hypotheses for GE2 and GE3. As part of this objective, we determined explicit formulas for all effects in GE2 and GE3 in terms of parental and cross sample means. Formulas for hypotheses being tested were defined in terms of population means. Our results indicate that several of the GE2 and GE3 formulas, which are the result of the sequential model-fitting process, seem to be rather nonintuitive biologically and genetically. We also identified other issues with GE2 and GE3, with respect to assumptions being made and an apparent typographic error in the Gardner and Eberhart (1966) paper. These points merit discussion and potentially reevaluation.
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Review of Gardner and Eberhart's Analyses I, II, and III (GE1, GE2, and GE3)
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To set the context for this discussion, we first provide a brief review of GE1, GE2, and GE3 as described by Gardner and Eberhart (1966). GE1 requires the evaluation of n parents and their n(n - 1)/2 crosses and inbred progeny developed from parents and crosses. This approach allows information to be obtained on additive and dominance gene action, in addition to heterosis and inbreeding depression. The concepts of GE1 were subsequently extended to include information on additive x additive epistatic effects (Eberhart and Gardner, 1966). The high resource demands imposed by this approach limit its practical utility in applied breeding programs, and it will not be discussed further. In GE2, n parents and their n(n - 1)/2 crosses are evaluated, and variation among all populations (entries) is partitioned into varieties (parents) and midparent heterosis. In hybrid breeding programs that rely on heterosis for cultivar development, GE2 has been reported to maximize the information on parental (variety) performance and the expression of heterosis of their crosses. Additive and dominance parameters cannot be estimated separately in GE2 because they are confounded within the "variety" parameter. This analysis further partitions heterosis into average, variety, and specific heterosis. GE3 evaluates parents, parents versus crosses, and crosses, and provides estimates of both variety and general combining ability (GCA) effects. Estimation of GCA effects in GE3 is identical to that of Griffing (1956) Model I, method 4. Average heterosis and specific combining ability (SCA) can be estimated by either GE2 or GE3.
GE2 has been reported to be superior to GE3 because the single mean square for heterosis and its three partitions—average heterosis, parental heterosis, and specific heterosis—are all due to dominance and differences in allelic frequencies between any two populations, assuming a restricted genetic model of additive and dominance effects only (Gardner and Eberhart, 1966; Ouendeba et al., 1996). These reports also indicate that the single mean square for parents (i.e., varieties) in GE2 contains all variation due to additive effects and some dominance. Two mean squares in GE3 (parents and GCA) contain variation due to additive effects and some dominance. To illustrate the application of GE2 and GE3, Gardner and Eberhart (1966) used a subset of data derived from six maize (Zea mays L.) variety crosses (Lonnquist and Gardner, 1961).
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Clarification of Models Associated with Analysis II (GE2)
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For objective 1, we begin by presenting a more detailed review of Analysis II. The following GE2 models 1, 2, 3, and 4 are explicitly given, as numbered, by Gardner and Eberhart (1966). Also included are Gardner and Eberhart's definitions of effects.
where Yjj' is the mean of a parent when j = j' and of a cross when j 
j'; µv is "the mean of all parental varieties included;" vj is "the variety effect when parental varieties are included in the analysis," with the restriction
hjj' is a "heterosis parameter" which is "due to differences in gene frequencies in varieties j and j' and to dominance," with hjj' = 
+ hj + hj' + sjj'; is "the average heterosis contributed by the particular set of varieties used in the crosses;" hj is "the average heterosis contributed by variety j in its crosses measured as a deviation from average heterosis," with the restriction
sjj' is "the specific heterosis that occurs when variety j is mated to variety j'," with the restrictions
= 0 if j = j' (i.e., for a variety), and = 1 if j j' (i.e., for a cross).
Note that the heterosis effect, hjj', does not appear directly in any of the four explicit GE2 models but is specified as a linear combination of average, varietal, and specific heterosis effects. Clearly, hjj' can be estimated by this relationship. However, obtaining a standard error is more difficult because covariances between the constituent effects must be known, in addition to the effect variances. An alternative approach is to estimate the heterosis effect and its standard error directly from fitting an implicit model, which we denote as Model H:
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where
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These five models (1, 2, 3, 4, and H) are fitted with the given restrictions and use all data (i.e., data from both varieties and crosses) to obtain estimates and standard errors for means and effects. In addition, model sums of squares (denoted SS1, SS2, SS3, SS4, and SSH) are obtained and used to construct the analysis of variance table summarized in Table 1
. Thus, GE2 partitions the overall population sum of squares into varieties and heterosis. Heterosis is then further partitioned into average heterosis, varietal heterosis, and specific heterosis. The first four columns in Table 1 contain the information from the first analysis of variance table reported by Gardner and Eberhart (1966). The last column, "Alternative Form of Sum of Squares" is the result of the present work and gives additional information on the calculation of the GE2 sums of squares.
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Table 1. Analysis of variance table for Gardner and Eberhart Analysis II (GE2): partitioning of the overall population sum of squares.
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Clarification of Models Associated with Analysis III (GE3)
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Continuing with objective 1, the models used in GE3 are not as clearly delineated in Gardner and Eberhart (1966) as those in GE2. Models 1 and 2 from GE2 are used again in GE3, with one important difference that will be discussed later. Another (unnumbered) model is given, which is analogous to Model 4 in GE2 and is here denoted as Model S (again with definitions from Gardner and Eberhart, 1966):
where Cjj' is the mean of the "variety cross" from parents j and j'; µc is the "mean of all crosses in the diallel set;" gj is the "variety effect in crosses" for the jth variety, with the restriction that
sjj' is the same as in GE2, with the same restrictions,
but is now called specific combining ability.
In addition, two other models are implicitly required, analogous to Models 3 and H in GE2 and denoted here as Models G and X, respectively:
where xjj' is the cross effect when variety j is mated to variety j', with xjj' = gj + gj' + sjj' and
Thus, in GE3, the sum of squares for populations is partitioned into varieties, crosses, and varieties versus crosses (Table 2) . Crosses are further partitioned into GCA (gj) and SCA (sjj'). As before, the Models 1, 2, G, S, and X are fitted to obtain effect estimates and standard errors, as well as model sums of squares. The first four columns of Table 2 essentially reflect the second analysis of variance table reported by Gardner and Eberhart (1966). As with the models for GE3, the sums of squares for GE3 were not as clearly delineated as those for GE2. The last column of Table 2, "Alternative Form of Sum of Squares," is provided to clarify how the sums of squares are calculated.
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Table 2. Analysis of variance table for Gardner and Eberhart Analysis III (GE3): partitioning of the overall population sum of squares.
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As in GE2, Model 2 is fitted with data from both varieties and crosses in GE3. The result of this is that, as Gardner and Eberhart noted, the GE3 effect and sum of squares for "varieties versus crosses" are the same as those for GE2's "average heterosis." In contrast, the GE3 Model 1 is fitted using the data only from parents (varieties), whereas in GE2, Model 1 is fitted using all data. To emphasize this difference, we denote the GE3 variety sum of squares from Model 1 as SS1v. The consequence of this difference will be discussed later in this report. Finally, the GE3 Models G, S, and X are fitted using only the cross data. Another equivalence between the two analyses is that the effect estimates and sum of squares for "SCA" of GE3 are equivalent to those of "specific heterosis" for GE2, as noted by Gardner and Eberhart (1966).
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Implementation of Analysis II and III: Sequential Model-Fitting versus the General Linear Hypothesis Approach
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Original implementation of the GE2 and GE3 analyses involved sequential model-fitting. This approach can be very tedious, but given the limited computing resources available in the 1960s was the best strategy available at the time. In sequential model-fitting, a series of models is fitted in a specific order so that at each stage of the process new effects are added to the model. The entire model is then refitted. Estimates are then obtained for all effects currently in the model, together with the sum of squares associated with the newly added effects. More specifically, the refitting process at each stage involves reducing a nonfull rank design matrix to full-column rank by the imposition of conditions on the effects being fitted (usually that a set of effects sum to zero in a certain way). The reduced, now full-rank, model is then fitted using least squares (i.e., estimates are obtained from solving the normal equations from the reduced model). The model sum of squares of the previous stage is then subtracted from the model sum of squares of the current stage, the difference representing the sum of squares associated with the new effects. Note that any one stage, one takes both the estimates of the newly added effects and their associated sum of squares for that specific model. Also note that a critical assumption of the sequential model-fitting process is that effects are fitted under the assumption that all subsequent effects not yet fitted are, in fact, zero. Thus the sequential estimates of the effects and their corresponding sum of squares are conditional on the assumption that these remaining effects are zero. In many cases this is not a tenable assumption. If all sets of effects are mutually orthogonal to one another, then the order in which effects are added to the model will not matter. As we shall see later, the nonorthogonality between sets of effects causes some problems with respect to GE2.
An early alternative to the sequential model-fitting method was the partial method of fitting models. As this is only peripherally related to the analyses proposed by Gardner and Eberhart (1966), we will only discuss a few important points. First, the "partial" estimates and sums of squares for a specific set of effects are unconditional, that is, they are obtained under the (frequently more reasonable) assumption that other proposed effects may be nonzero. Note that, when all sets of effects are orthogonal to each other, the sequential and partial methods will produce the same results. When computing ability is quite limited, as it was prior to the early 1970s, the sequential method is easier to implement computationally. This may account, in part, for the early popularity of sequential methods.
As advancements have occurred in both computing resources and in the theory of linear models, the conditionality of estimates and sums of squares has made the sequential method less popular in comparison to the partial method. There are also many matrix routines currently available, which make partial model-fitting much easier than before. In addition, from the standpoint of linear models theory, we have the development of full-rank models used in conjunction with the General Linear Hypothesis approach to estimate effects and to test hypotheses (Hocking, 1996). In this approach one specifies a full-rank model based on a set of fundamental parameters (here, the population means). Next, one defines linear (additive) combinations of those means which are to be estimated or tested. Then the estimates or sums of squares are obtained directly. Note that testing single degree-of-freedom contrasts is just a special case of the General Linear Hypothesis method, which, for example, is used by Proc GLM and MIXED (SAS Institute, 1989b, 1992) to generate sums of squares and least squares means for fixed effects. The results of this approach are analogous to those from "partial-model fitting" in that estimates and sums of squares are unconditional. However, the estimates are clearly defined in terms of population means.
Full-rank models, in conjunction with the General Linear Hypothesis method, can be used for two practical purposes. The first, used primarily in this paper, is to determine explicit formulas for effects and tests for existing numerical methods like GE2 and GE3. The second is to develop new analyses by first deciding what are reasonable definitions of effects in terms of population means, and then developing estimates and tests for those effects. This second use should be very applicable to complicated genetics models because the researcher can decide a priori exactly how concepts like GCA and SCA should be defined in terms of parent and cross means.
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Application of the General Linear Hypothesis Approach to Analyses II and III
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To attain objective 2, we used the General Linear Hypothesis approach with full-rank, cell-means models (Hocking, 1996) to obtain theoretical results on what is being estimated and tested in GE2 and GE3. Our results about effect estimates and tests of hypothesis for GE2 and GE3 are defined in terms of means. Therefore, Table 3
provides notation for sample and population means for varieties, crosses, the mean of all varieties (parents), the mean of all crosses, and the mean of crosses with parent j. The formulas for effect estimates and tests of hypotheses were obtained by fitting the full-rank, cell-means versions of Models 1, 2, 3, 4, H, G, S, and X with their given restrictions, by hand, and confirming them numerically. The null hypotheses being tested by various sums of squares were that the expected values of the corresponding effects were equal to zero. Formulas for sums of squares were obtained by means of the General Linear Hypothesis approach and were verified numerically. The effect estimates and hypotheses tested in GE2 and GE3 are shown in Tables 4 and 5
, respectively.
Several points should be emphasized about Gardner and Eberhart's original numerical results and our formulaic and numerical results. First, our numerical results match all original GE2 and GE3 estimates and mean squares, except for the variety effects in GE2 (Tables 1, 2, and 6)
. Note that Gardner and Eberhart's (1966) results have the same estimates for the variety effects in GE2 and GE3 (Table 6), but different mean squares (Tables 1 and 2). This cannot be correct because estimates and their respective mean square (or sum of squares) must match. Also, note that if the original GE2 (or GE3) variety effects are squared and summed, the result is the variety sum of squares for GE3, not GE2. Thus, the variety effects reported in Gardner and Eberhart (1966) are correct for GE3, but are incorrect for GE2. The corrected GE2 variety effect estimates are given in Table 6. To emphasize that GE2 and GE3 variety effects are different, we have denoted the GE2 effects as v*j and kept vj for GE3 variety effects in Tables 1, 2, 4, 5, and 6.
We believe that this mistake is just a typographic error that could have occurred during the manuscript preparation process. Alternatively, it may have resulted from picking the correct variety sum of squares at one step in the sequential model-fitting process and inadvertently picking the effect estimates from a later step. If all sets of effects (variety, average heterosis, variety heterosis, and specific heterosis) were mutually orthogonal, then the variety estimates would not change as the models are fitted sequentially. However, it turns out that variety effects and variety-heterosis effects are not orthogonal in GE2 (data not shown). Therefore, when variety-heterosis effects are entered into the model, the variety effect estimates change from the v*j estimates we report in Table 6 to the original Gardner and Eberhart estimates of vj.
Given that the GE2 and GE3 variety effect estimates are numerically different, what exactly is the formulaic difference? Both analyses fit Model 1, but do so in different ways. GE3 fits Model 1 using only the variety (i.e., parent) data. Because no other effects are in the model at this stage in the sequential model-fitting process, the single condition is imposed that the variety effects sum to zero. Consequently, the variety effect parameters and estimates are exactly what one would want them to be, the difference between a variety mean and the mean of all varieties (Table 5). On the other hand, GE2 fits Model 1 using data from both varieties and crosses. In the original Gardner and Eberhart implementation, this version of Model 1 has a more complicated design matrix than that of GE3 and must be reduced to full rank differently, i.e., by the imposition of extra conditions, in addition to the condition that the variety effects sum to zero. In this case the extra conditions are those of "no heterosis," that is, the GE2 variety effects are estimating µjj - µv subject to the constraint of µjj' = 1/2 (µjj + µj'j'). We have verified this formulaically using statistical theory for constrained linear models (Hocking, 1996). This is in agreement with our previous statement that sequential model-fitting methods assume that all subsequent effects (e.g., variety-heterosis effects) not yet fitted are actually zero. This assumption has a direct impact on variety and variety-heterosis effects in GE2 because they are not orthogonal to each other. Thus, the presence or absence of one set of effects from the model impacts the numerical estimates of the other set of effects. This difference between GE2 and GE3 variety effects has been alluded to by Hallauer and Miranda (1988), who indicate that the GE2 variety mean square includes information not only on the performance of varieties themselves, but also on variety crosses.
As a practical concern, a comparison of the numerical estimates for the two variety effects (Table 6) shows that they are not necessarily in agreement. The low variety effect (variety 3) is the same in rank for the two analyses. The second and third lowest (varieties 5 and 1) have switched rankings, however, as have the top two (varieties 4 and 6). Furthermore, the effect for variety 6 is over twice as large for GE2 as GE3. If it is known that there is no mid-parent heterosis, then using the variety effects and test of hypothesis of GE2 makes sense. However, given that significant heterosis has been reported in most studies that utilized these methods (Crossa et al., 1990; Gerrish, 1983; Mungoma and Pollak, 1988; Widstrom and Snook, 1998), GE3 seems a better choice for estimating variety effects. The constraint placed on the variety effect in GE2 does not negate the usefulness of this approach but rather emphasizes that its utility lies in its ability to detect and partition variation for heterosis.
Interpreting the two variety effects from the standpoint of genetics, we note that the estimates of the GE2 variety effect (Table 4) contains both variety effects (
jj - v) and GCA effects (jc - c). From a breeding perspective, this confounding reduces the ability to determine a parent's usefulness in a recurrent selection program and/or in making hybrid performance predictions on parental per se effects. Another consequence of having a constrained estimator (GE2) instead of an unconstrained one (GE3) is that the variance of GE2 variety effect estimates will always be smaller than that for GE3 by a factor of 4/(n + 2). For example, in a completely randomized design or randomized complete block design with r replications, the variance of the estimate of v*j is
while that for the estimate of vj is
From the practical standpoint of calculating effects and tests of hypotheses for either GE2 or GE3, the formulas of the effect estimates in terms of parent and cross sample means (Tables 4 and 5) can be used to generate coefficients of contrasts of the population means. Estimates of effects (with standard errors and t tests) and their sums of squares with corresponding F tests can be obtained by, respectively, ESTIMATE and CONTRAST statements in SAS Proc GLM or Proc MIXED (SAS Institute, 1989b, 1992). For a population diallel cross experiment (with parent number n > 3), the coefficients of ESTIMATE or CONTRAST statements for GE2 and GE3 effects are listed in Tables 7 and 8
, respectively. For example, v*1 (the GE2 effect for Parent 1) can be estimated from the following relationship extracted from Table 7 for n parents:
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Table 7. Coefficients for population means µjj' in a one-way arrangement for estimates and contrasts in Gardner and Eberhart's Analysis II (GE2) for effects numbered (i) for Parent (i) or (i, i') for Cross with Parents (i) and (i'), with i = 1,..., n and i' = 1,..., n.
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Table 8. Coefficients for population means µjj' in a one-way arrangement for estimates and contrasts in Gardner and Eberhart's Analysis III (GE3) for effects numbered (i) for Parent (i) or (i, i') for Cross with Parents (i) and (i'), i = 1,..., n, i' = 1,..., n.
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The corresponding SAS ESTIMATE statement for the case of n = 4 parents (e.g., 11, 22, ...) and their crosses (e.g., 12...) in the indicated population order is- *COMMENT on POP order: 11 12 13 14 22 23 24 33 34 44;
- ESTIMATE ‘V1 * ’ POP 12 4 4 4 -4 -4 -4 -4 -4 -4/DIVISOR=24;
For analyzing any n - population (n > 3) diallel cross experiment using either Proc GLM or Proc MIXED, for each of GE2 or GE3, there will be five CONTRAST statements and (n2 + n + 1) ESTIMATE statements, each with (n(n + 1)/2) coefficients. Entering these coefficients can constitute a large amount of work. Therefore, a general SAS MACRO has been implemented in IML (SAS Institute, 1989a, 1990) as part of the third author's masters project (Dong, 1999) and is available from the corresponding author.
Finally, we note that the effect estimates resulting from either GE2 or GE3 analyses contain multipliers like 4/(n + 2), (n - 1)/(n - 2), and n/(n - 2), which are unintended artifacts of the sequential model-fitting technique (Tables 4 and 5). These multipliers may cause problems in comparing results from diallel experiments with differing numbers of parents (n), because effect estimates depend on n in nontrivial ways. Thus, we raise the question, are these coefficients biologically meaningful or can they be eliminated from the formulas? For example, consider a modified version of GE3 (Table 5) with the same definitions for "Varieties," "Varieties versus Crosses," and "Crosses," but with the (n - 1)/(n - 2) and n/(n - 2) weighting coefficients removed from "GCA" and "SCA." The modified GCA effect is then just the unweighted deviation between parent j's crosses and the overall cross mean. The modified SCA effect, without the weighting factors, now resembles a two-way interaction effect. Similarly, the (n - 1)/(n - 2) coefficient can be removed from the "variety heterosis" effect of GE2 (Table 4), leaving an unweighted measure of deviation of the average midpoint heterosis for parent j from the average heterosis over all parents.
Other researchers may disagree with these specific modifications, but we hope that our results will at least initiate a discussion about these types of analyses. Our recommendation is that future approaches should clearly delineate meaningful quantities like "heterosis effects," "variety effects," and "GCA" directly from variety and cross means (µjj and µjj') and then use an analysis which will estimate and test those quantities.
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ACKNOWLEDGMENTS
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Research supported by the New Mexico Agricultural Experiment Station. We thank the reviewers of this manuscript for their in-depth comments and suggestions. We also thank the International Biometric Society for permission to reprint portions of Tables 1 through 4 as reported in Gardner and Eberhart (1966).
Received for publication November 1, 2001.
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TOP
ABSTRACT
INTRODUCTION
