Crusio, Wim E. (1993) Bi- and Multivariate Analyses of Diallel Crosses: A Tool for the Genetic Dissection of Neurobehavioral Phenotypes. Behavior Genetics 23 59-67

Bi- and Multivariate Analyses of Diallel Crosses: A Tool for the Genetic Dissection of Neurobehavioral Phenotypes

Wim E. Crusio¹

Footnote: The preparation of this article was supported in part by a NATO Science Fellowship awarded by the Netherlands Organization of Pure Research (ZWO; Den Haag, The Netherlands), a stipend from the Alexander-von-Humboldt Foundation (Bonn, Germany), and the Centre National de la Recherche Scientifique (URA 1294 CNRS; Paris, France).

Send correspondence and proofs to: Dr. Wim E. Crusio at the above address.

The genetic-correlational approach provides a very powerful tool for the analysis of causal relationships between phenotypes. It appears to be particularly appropriate for investigating the functional organization of behavior and/or of causal relationships between brain and behavior. A method for the bivariate analysis of diallel crosses that permits the estimation of correlations due to environmental effects, additive-genetic effects, and/or dominance deviations is described, together with a worked-out example stemming from a five times replicated 4 x 4 diallel cross between inbred mouse strains. The phenotypes chosen to illustrate the analysis were locomotor activity and rearing frequency in an open field. Large, positive additive-genetic and dominance correlations between these two phenotypes were obtained. This finding was replicated in another, independently-executed, diallel cross.

Keywords: diallel cross; genetic correlations; multivariate analysis; brain-behavior relationships; neurobehavioral genetics; exploratory activity; mouse.

Introduction

The ultimate aim of neurobehavioral genetics is to understand the mechanisms that underly individual differences for neural and behavioral phenotypes. We may distinguish between two different aspects of this causation: the phylogenetic and the phenogenetic ones (van Abeelen, 1979; Crusio, 1984). The former concerns the question of the adaptive value for the organism of a certain character, the latter relates to all physiological steps leading from genotype to phenotype. Quantitative-genetic methods may offer us answers to questions related to both aspects that often are impossible or very difficult to obtain otherwise: information about the genetic architecture of a trait may allow inferences about its evolutionary history (Broadhurst and Jinks, 1974), whereas the analysis of genetic correlations between physiological, neuroanatomical, neurochemical, and/or behavioral characters may provide crucial clues for the identification of functional relationships between these traits.

Genetic correlations render important information because, if different from zero, they imply that for the variables involved there exists a (set of) gene(s) simultaneously influencing both of them (pleiotropy; see also the more elaborate discussions by Carey, 1988, and Houle, 1991). In turn, this must indicate that for these characters, at least part of the physiological pathways leading from genotype to phenotype are shared and that some causal relationship between them is highly probable. It is this special property that renders the genetic-correlational approach so uniquely valuable. In the mouse, it has already been used succesfully to uncover functional relationships between neuronal characters (size of the intra- and infrapyramidal mossy fiber terminal field; hippocampal dynorphin B levels) and behaviors such as exploration in an open field (Crusio et al., 1989b; van Daal et al., 1991a and b) and two-way active-avoidance learning (Crusio, 1992b).

Among the quantitative-genetic designs that are available for the analysis of the genetic underpinnings of behavioral and neural phenotypes, the replicated diallel cross is one of the most powerful ones (Crusio, 1992a). Usually, authors analyze the genetic architecture of one or a number of phenotypes, mostly employing the methods of Hayman (1954a and b) and Jinks and Hayman (1953; see also Mather and Jinks, 1982). If more than one phenotype is being measured, bi- or multivariate analyses of the diallel cross may be fruitfully employed. Such extensions have been applied before by Sulzbach and Lynch (1984) and Fulker et al. (1972), respectively, using adaptations of Hayman's (1954a) abcd tests. However, the interpretation of these tests is not without complications, mainly because the a-item is sometimes a composite of both additive-genetic and dominance variance (Walters and Gale, 1977).

An alternative approach would be a bivariate extension of the variance/covariance analysis of Hayman (1954b) and Jinks and Hayman (1953). In the univariate case, this analysis renders estimates of E, D, H1, and H2, the environmental, the additive-genetic, and the two dominance contributions to the phenotypic variation, respectively. These parameters have unequivocal interpretations in terms of gene effects and allele frequencies (Hayman, 1954b; Mather and Jinks, 1982; see also below). Therefore, the analysis of bivariate equivalents of these parameters has distinct advantages over modifications of the abcd tests.

Although such an analysis has already been developed more than 30 years ago (Nei, 1960), it has not yet been applied by behavior geneticists (indeed, to the best of my knowledge, it has not been applied at all ever since). Moreover, up till now no worked-out example of the method has been available. Building on our previous illustration of the diallel-cross method (Crusio et al., 1984), a modified version of Nei's (1960) bivariate extension of the variance-covariance analysis is here presented. A comprehensive theoretical treatment of the method is not attempted, however. For this the reader is referred to the literature (Hayman, 1954a,b; Mather and Jinks, 1982; Nei, 1960).

Animals

The example presented here consists of a diallel cross between four inbred strains of mice: C57BL/6J//Nmg (B), DBA/2J//Nmg (D), C3H/St//Nmg (H), and CPB-K//Nmg (K). The phenotypes analyzed were locomotor activity (ambulation measured as line crossings) and the frequency of rearing-up during a 20 min observation in an open field at the age of 3 months. Since full details on breeding, and experimental procedures are not relevant in the context of this paper and can be found elsewhere (Crusio and van Abeelen, 1986), only a few points of interest are mentioned here.

From all 16 possible crosses one litter (hybrids) or two litters (inbreds) were raised simultaneously, constituting one replication or block. In all, five such replications were bred consecutively. Three male mice from each litter were observed in the open field. Thus, the entire experiment comprised 300 animals in total: 30 for each inbred strain and 15 for each F1 cross. Litter means were used as experimental units in the subsequent analyses.

Definitions and Notation

In the univariate case, we may partition the total phenotypic variance into its environmentally-induced part E and four different genetic components that have been defined by Jinks and Hayman (1953; see also Hayman, 1954b; Mather and Jinks, 1982) as follows:

where k = the number of genes involved, u_i = the frequency of the increasing allele of the ith gene, v_i = the frequency of its decreasing allele (in such a way that u_i+ v_i = 1; assuming the absence of multiple allelism), d_i = half the difference between homozygotes of the ith gene (the additive-genetic effect), and h_i = the deviation due to dominance deviations of heterozygotes for the ith gene from the midparental value. Thus, D refers to that part of the variance that is due to additive-genetic effects and H1 and H2 to variation due to dominance deviations. F is a term describing the covariation between the additive-genetic effects and dominance deviations of genes. H1 will equal H2, unless allele frequencies deviate from 0.50.

In the bivariate case, the total phenotypic covariance between two characters x and y may also be partitioned into an environmental component (E_xy) and genetic components due to additive-genetic effects (D_xy) and dominance deviations (H1_xy, H2_xy). Analogously to the univariate case, these genetic components of the phenotypic covariance are defined as follows:

(These terms are equivalent to Nei's (1960) parameters L, M1, and M2, respectively. As the present notation appears to be more consistent with the one used for the univariate analysis, I have chosen not to follow Nei's nomenclature.) Here, l is the number of genes exerting simultaneous influences on characters x and y (with l < k_x and l < k_y), d_ix and d_iy are the additive-genetic effects of the ith gene on characters x and y, respectively, and h_ix and h_iy are the respective dominance deviations due to the ith gene. Evidently, genes that have effects on only one of the characters x and y do not contribute to the genetic covariance terms. No single, direct bivariate equivalent of parameter F exists, since it already constitutes a covariance term in the univariate case. Nei (1960) defined two components to replace it: N1 describing the covariation of the additive-genetic effects of genes on character x with the dominance deviations of the same genes on character y and N2 describing the inverse relationship. As these covariation terms are not necessary for our present purpose, i.e. the estimation of genetic correlations, they are further ignored here.

Using the environmental and genetic components of the phenotypic covariance and variances, we may calculate

rE = E_xy / SQRT(E_xE_y),

rD = D_xy / SQRT(D_xD_y),

rH1 = H1_xy / SQRT(H1_xH1_y) and

rH2 = H2_xy / SQRT(H2_xH2_y)

These quantities thus provide estimates of correlations between the environmental effects (E), additive-genetic effects (D), and dominance deviations (H1 and H2), respectively, as exerted on characters x and y. Note that although H1 > H2, the same relationship need not hold for H1_xy vs. H2_xy and rH1 vs. rH2.

In what follows below, we will also use the following symbols:
n = number of parental strains (here: 4);
b = number of replications (blocks; here: 5);
d = number of nested replications of the leading diagonal (replications of the diagonal within blocks; here: 2).

Analysis

Appropriate scales were selected as described before (Crusio, 1990, and Crusio et al., 1984, following Kerbusch et al., 1981). For locomotor activity no transformation was necessary, whereas for rearing a SQRT(x) + SQRT(y transformation had to be applied. The means of the transformed litter means (averaged over reciprocals and replications), are presented for all cells of the diallel cross in Table I. A sizeable and significant, positive, phenotypic correlation, estimated as the correlation between the 100 individual littermeans, is observed for the two variables (r= 0.502, df = 98; P < 0.001). Because the analysis is completely analoguous to that of the univariate case, the present example is given for the block of means only. For further details, the reader is referred to Crusio et al. (1984).

In the univariate case, the following four statistics are used to obtain estimates of, among others, the genetic parameters D,H1, and H2: the mean of the variances of the n arrays V_r, the variance of the array means V_r, the variance of the parentals V_P, and the mean of the n array covariances of entries with their nonrecurrent parents W_r. An array consists of diallel entries having one parent in common. Because entries have been averaged over reciprocals, it is not relevant whether this is the maternal or paternal parent (see Discussion for cases where reciprocal effects may be important).

If we now convert the above variances into their analogous covariances between the appropriate entries for the variables x and y, we obtain bivariate equivalents for three of the aforementioned four statistics. From Table I we find for the first array

W1 = {(446.3 x 15.6 + ... + 326.0 x 15.9) - (446.3 + ... + 326.0) x (15.6 + ... + 15.9)/4}/3 = -3.4

Hence, the mean of the covariances of the n arrays equals

W_r,xy = {(-3.4) + 81.8 + (-13.1) + 241.6}/4 = 76.7

(Note that W_r,xy is the bivariate equivalent of the univariate V_r and not of Wr.) The bivariate equivalent to the univariate Vr is the covariance between array means, which amounts to

W_r,xy = {(392.6 x 15.7 + ... + 290.4 x 14.6) - (392.6 + ... + 290.4) x (15.7 + ... + 14.6)/4}/3 = 14.2

The equivalent of V_P is the covariance of the parental entries, for which we obtain

W_P,xy = {(446.3 x 15.6 + ... + 189.3 x 9.2) - (446.3 + ... + 189.3) x (15.6 + ... + 9.2)/4}/3 = 255.8

However, no such bivariate equivalent is readily available for the univariate W_r since it is already a covariance. Instead, we construct a new diallel table (Table II) in which the cell entries are found by calculating the differences between the entries in Table I and their non-recurrent parents. We now utilize V_r(F-P), the mean of the variances of the n arrays. Because the variance of the differences between two variables equals the sum of their respective variances minus twice their covariance, the expectation of V_r(F-P) is easily derived as being identical to that of V_r + V_P - 2W_r. In terms of genetic parameters and using the expectations presented in Table X of Crusio et al. (1984) this gives under a fixed-effects model

1/4 (D+H1+F) + (n-1) (d + 2)E/2 bn

In cases where a random-effects model is to be preferred (cf. Kuehl et al., 1968; Wearden, 1964; Wright, 1985), the expectation of the above-presented statistics will have to be modified according to Hayman (1960).

Of course, V_r(F-P) can easily be extended to the bivariate case by calculating the corresponding covariance terms between the variables x and y. This procedure deviates from the one used by Nei (1960), who calculated two cross-covariances (the covariances of entries for variable x with the corresponding non-recurrent parent entries for variable y and the other way round). However, the present method is mathematically equivalent and computationally more efficient.

From Table II we find for the first array

W_1(F-P) = {(0 x 0 + ... + 136.7 x 6.7) - (0 + ... + 136.7) x (0 + ... + 6.7)/4}/3 = 167.0

Averaged over all four arrays, we find

W_r(F-P),xy = (167.0 + ... + 87.8)/4 = 197.9

The estimate of the environmental covariance, Exy, is calculated in a way completely analogous to the univariate case (see Crusio et al., 1984), which gives in the present case 20.5. To obtain estimates for the genetic components of the covariation, we now use the following perfect-fit solutions:

D_xy = W_P,xy - E_xy/db and

H2_xy = 4(W_r,xy - W_r,xy) - 2{d(n-1)² + 2n - 2}E_xy / dn² b

(Crusio et al., 1984; Table X), together with

H1_xy = 2(W_r,xy + W_r(F-P),xy) - W_P,xy - {2d(n - 1) + n} E_xy / dnb

(See Appendix for modified perfect-fit solutions to be used with half-diallel designs, where reciprocal crosses have been omitted.) For example, for the additive-genetic component of the covariance we get

D_xy = 255.8 - 20.5/(2 x 5) = 253.8

Finally, we calculate the genetic correlations using these bivariate estimates in combination with the univariate estimates obtained before for locomotor activity and rearing (Crusio et al., 1984; Crusio, 1984). Thus, we obtain, e.g.,

r_D = 253.8 / SQRT( 12792.2 x 6.47 ) = 0.882

The results are entered in Table III. Testing the significance of these genetic correlations is problematic, however. Two different solutions are proposed here. For both, we assume that when a certain covariance deviates significantly from zero, the corresponding correlation will differ significantly from zero, too. For the first method, we estimate the different components of the total phenotypic covariance separately for each block (as was also done by Crusio et al., 1984, for the univariate components). We then calculate standard errors (here termed "empirical standard errors") from these estimates, which we use to test the deviations from zero of the components of the covariance obtained from the block of means using a t-test (with df = b - 1, which gives 4 in the present case).

For the second method, we adapt the technique proposed by Hayman (1954b; pp. 798-799 and 807), also for the univariate case. In this procedure, we estimate in the block of means the variance of (W_r- V_r). Half of that variance is used as an estimate of the sampling variation s². For each component of the variance, s² is multiplied by an appropriate factor, obtained from the inverse of the matrix of the coefficients of these components in the least squares equations (shown by Hayman, 1954b, p. 798; see also below). Its square root now provides the appropriate standard error (here termed "theoretical standard error"). In the bivariate analysis, we make use of the fact that

(W_r - V_r) = (V_P - V_r(F-P) - V_r) / 2

and calculate the variance of

(W_P,xy - W_r(F-P),xy - W_r,xy) / 2

In the present case, we find for this quantity 46.1, -35.8, -11.15, and -36.8 for arrays 1 (strain B) to 4 (strain K), respectively. Thus, the sampling variation s² equals half the variance of these four numbers, giving 755.2. Taking the standard error of D as an example, we find from Hayman's table

(n⁵ + n⁴)/n⁵ = (n + 1)/n = 5/4 = 1.25

as coefficient. The theoretical standard error of D therefore equals

SQRT( 755.2 x 1.25 ) = 30.7

The coefficients necessary to calculate the theoretical standard errors of the other components are as follows:

(n⁵ + 41n⁴ - 12n³ + 4n²)/n⁵ for H1,

36/n for H2, and

1/n for E.

We may now test the different components of the covariance using the normal distribution. For D_xy this renders

z = 253.8/30.7 = 8.3 (P < 0.001).

The results of both methods are entered in Table IV.

One final problem remains: the foregoing analysis may only be applied if the assumptions underlying it (no epistatic interactions, no multiple allelism, independent distribution of alleles between parents) are met. In the univariate case, this is tested by examining the V_r:W_r graph, which should have a slope of unity. Alternatively, we may use the V_r:V_r(F-P) graph. If the assumptions underlying the genetic model are met, this graph should have a slope of -1 (Crusio, 1985). This may be tested using a regression analysis or by using a two-way ANOVA to check whether V_r + V_r(F-P) remains constant over arrays. The extension to the bivariate case is, again, straightforward and follows the analysis outlined above. When utilising regression analyses, allowance should be made for the fact that both variables entering into the regression (V_r and V_r(F-P) viz. W_r,xy and W_r(F-P),xy) are subject to sampling error. In the present case, a method proposed by Moran (1971) is used. The regression of W_r,xy on W_r(F-P),xy has a slope of -1.4, with a standard error of 3.6, rendering

t = {(-1.4) - (-1)}/3.6 = -0.11 with

df = (b x n) - 2 = 18 (ns)

In the ANOVA of (W_r,xy + W_r(F-P),xy), we cannot test the array differences against the pooled mean squares of Blocks and Residuals, because (W_r,xy + W_r(F-P),xy) is heterogeneous over blocks (F_4,12 = 6.39; P < 0.01; cf. Crusio et al., 1984). Testing the array differences against the Block term only, renders F_3,4 = 0.63 (ns). Both the regression analysis and the ANOVA therefore indicate that serious violations of the assumptions may be safely presumed absent.

Interpretation and Discussion

Turning first to the specific results obtained here, we see from Table III that the high positive phenotypic correlation is composed of low environmental, but substantial additive-genetic and dominance correlations. Using the theoretical SEs (Table IV), we see that all genetic correlations are significant except, surprisingly, rE. Using the empirical SEs, however, rD remains significant but rH1 and rH2 become borderline cases. Of course, a t-test with just 4 degrees of freedom has only limited statistical power. Which one of the two tests is to be preferred may probably only be decided by carrying out some large-scale simulation studies. Fortunately, however, when the genetic correlation matrices are used as input for further, multivariate, analyses, the eventual significance or lack thereof of an individual correlation is not very important any more.

Although rather high, the phenotypical correlation is more moderate than the very high genetic correlation. This is due to the fact that heritabilities were rather low for rearing (especially the narrow, 0.03, but also the broad, 0.40; Crusio and van Abeelen, 1986). Genetic and environmental correlations contribute to the total phenotypical correlation proportionally to the square roots of the respective fractions that they contribute to the total phenotypical variances of characters x and y. Thus, the size of a phenotypical correlation cannot be indicative of the sizes of its components (contrary to the opinion voiced by Cheverud, 1988). This will be the case even more if genetic and environmental correlations do not all have the same sign (this is not just a theoretical possibility; cf. the results of Crusio et al., 1989a). It appears that, in the present case, the phenotypical correlation between locomotor activity and frequency of rearing-up in an open field is mainly caused by dominance deviations shared by both characters. In combination with the large additive-genetic correlation, we may take this as evidence for a common physiological regulatory mechanism that at least in part underlies individual variation for these traits. Interestingly, there exists a rather large body of data pointing in this direction (van Abeelen, 1989).

A general point of criticism of the genetic-correlational method might be, that linkage disequilibria may also lead to significant genetic correlations between traits (note that mere linkage is not a sufficient condition). Yet, such disequilibria occur only rarely in nature (Carson, 1987) and may therefore be expected to be equally rare in inbred strains. The risk of their occurrence may further be diminished by using a sample of only distantly related inbred strains, omitting strains derived from a common ancestor, and at the same time increasing the generalizability of the results obtained with that particular sample. Furthermore, because almost all characters showing continuous, non-pathological variation are polygenically regulated, the possibility of a linkage disequilibrium occuring purely by chance in a certain sample of inbred strains will be very small. On the other hand, although linkage disequilibria are the only systematic cause of non-independent allele distributions, the latter may be encountered by pure chance alone, too. In any design comprising only inbred parentals and, possibly, their F1's (that is, without any segregating generations), chance associations or dispersions of alleles over strains will have the same effects as linkage disequilibria. Fortunately, the diallel-cross analysis permits a test of the adequacy of the genetical model used. Spurious genetic correlations caused by non-independent allele-distributions, including linkage disequilibria, may be assumed absent if no serious violations of the assumptions are indicated by the analysis of W_r,xy and W_r(F-P),xy.

This possibility to test the underlying assumptions is one of the advantages the diallel cross offers over other, alternative designs. A further attractive feature is that any number of strains may be included, so the generalizability of results is much larger than when using designs assaying only 2 strains at a time. The main alternative for the diallel cross as a tool for the genetic dissection of neural and behavioral phenotypes is the estimation of genetic correlations using a battery of inbred strains (Hegmann and Possidente, 1981). Of course, the latter method does not allow the estimation of genetic correlations due to dominance or a test of the assymptions underlying the analysis. On the other hand, because no cross-breeding is necessary so that much fewer resources and effort need to be invested, we may use it as a "quick, but dirty" exploratory analysis (see Crusio, 1992a for a more elaborate comparison of the different alternative quantitative-genetic methods available).

Some more specific limitations of the diallel-cross method should also be mentioned here. To start with, by averaging over reciprocals, the analysis ignores possible reciprocal effects due to, e.g., sex linkage and pre- or postnatal maternal influences. I have shown previously that if such effects are, in fact, present, the estimates of the genetic components of the (co)variation, especially the additive-genetic ones, may be biased in an unpredictable direction and subsequently I have presented a computationally more demanding method that deals with this problem (Crusio, 1987). In combination with the present article, the extension of that method to the bivariate case is straightforward

A further point of caution relates to the fact that genetic correlations have sometimes been used to answer questions related to the phylogenetic aspect of causation. In these cases, the sign of a particular genetic correlation was interpreted in terms of selection pressures that might have been exerted on both phenotypes in the evolutionary past. Houle (1991), however, showed that the genetic correlation between two traits under directional selection may be either positive or negative at equilibrium, depending on the fitness function, the input of mutational variance, and, above all, what he termed the functional architecture of the metabolic or developmental pathways that determine the pattern of pleiotropic effects.

Finally, an important consideration in planning a diallel cross concerns the sampling problem. Disagreement exists about the adequate size of a diallel-crossing experiment. Henderson (e.g., 1989) has argued repeatedly against using smaller diallel crosses (employing 4 or 5 strains). The opposite view, with some reservations, has been expressed by Gerlai et al. (1990) and by Crusio (1992a). An indication that the problem may be less serious than sometimes feared may be the fact that the correlational estimates obtained from another diallel-cross experiment, carried out independently from the present one but using similar methods, are very close to those presented here (unpublished results; 5 times replicated 5 x 5 diallel cross with a nested replication of the diagonal, between BA//C, BALB/cJ, C57BL/6J, C57BR/cdJ, and DBA/2J, the two strains also employed in the present example belonging to different sublines; one animal per litter, giving a total of 150 subjects; see Crusio et al., 1989a, for further experimental details; phenotypical r = 0.503, df = 148; P < 0.001; rE = 0.306; rD = 0.809; rH1 = 0.792; rH2 = 0.739; using theoretical SEs: all correlations significant; using empirical SEs: only rD significant). As the sampling problem is not directly relevant to the purpose of the present article, I refer the reader to the above-mentioned publications for a more in-depth discussion.

In cases where a larger number of phenotypes are studied, the matrices of correlations obtained may be further investigated by means of multivariate techniques, such as factor analysis. In one recent example (Crusio et al., 1989a), this method was used to dissect exploratory behavior and its relationships with anatomical variation in the hippocampus in the mouse. In that particular case, environmental and genetic correlations with opposing signs caused phenotypic correlations to be low and nonsignificant. Multivariate genetic analysis thus succeeded in uncovering brain-behavior relationships where a conventional, purely phenomenological approach, would have failed. Some of the results of that study have subsequently been confirmed using other genetic research strategies (Crusio et al., 1989b, 1991), convincingly showing the potential heuristic value of the genetic-correlational approach.

In the above-outlined multivariate genetic analyses, variables were weighted by the square roots of their narrow heritabilities (h²n) or their environmentalities (e² = 1 - h²b) in the factor analyses of the matrices of additive-genetic and environmental correlations, respectively. This procedure is justified by two considerations. First, as we already have seen above, additive-genetic and environmental correlations contribute portions to the total phenotypic correlation that are proportional to these quantities (DeFries et al., 1979; for dominance correlations the appropriate quantity would be the square root of the difference between the broad and narrow heritabilities). Second, and perhaps more importantly, the precision with which these correlations can be estimated will vary with the sizes of these quantities, too. This sampling error is also responsible for the fact that sometimes correlations are obtained that are larger than |1|, which is, of course, a theoretical impossibility. Such correlations will have to be set to unity for subsequent analyses. However, empirical evidence suggests that such improbable estimates are mainly obtained when variables are involved for which the particular genetic effect is absent, that is, where formally speaking such a genetic correlation does not even exist.

Summarizing, multivariate genetic analyses by means of diallel crosses clearly provide a powerful additional tool for the neurobehavioral geneticist to dissect functional relationships between behavioral and neural phenotypes.

Acknowledgments

The experimental part of the work described here was carried out at the Dept. of Zoology, University of Nijmegen (The Netherlands) under the supervision of Dr. J.H.F. van Abeelen. The bivariate extension of the diallel-cross analysis was developed during a post-doctoral stay (1984-1986) at the Institut für Humangenetik und Anthropologie (University of Heidelberg, Germany). I am grateful to its director, Prof. Friedrich Vogel, for his generous hospitality during this period. I thank Dr. P.E. Ferreira (EMBRAPA, Brasilia, Brasil) for bringing Nei's 1960 article to my attention and Dr. J. Raaijmakers (Dept. of Mathematical Psychology, University of Nijmegen, The Netherlands) for help with the Moran regression.

Appendix: Modifications Necessary in the Analysis of a Half-Diallel Cross

In the analysis of a half-diallel cross, where reciprocal crosses have been omitted, we have to use slightly modified formulae. Based on the formulae presented by Crusio et al. (1984; Table XI) and following the notation adopted above, the expectations for the four statistics needed become:

W_P,xy = D_xy + E_xy / db,

W_r,xy = 1/4 (D_xy + H1_xy - H2_xy - F_xy) + {d(n - 1) + 1} E_xy / dn² b,

W_r,xy = 1/4 (D_xy + H1_xy - F_xy) + {d(n - 1) + 1} E_xy / dn b,

and

W_r(F-P),xy = 1/4 (D+ H1 + F) + (n - 1) (d + 1) E / dn b

Together, they provide the following perfect-fit solutions:

D_xy = W_P,xy - E_xy / dn b,

H1_xy = 2(W_r,xy + W_r(F-P),xy) - W_P,xy - {n(4d + 1) - 4d}E_xy / dn b,

and

H2_xy = 4(W_r,xy - W_r,xy) - 4{d(n - 1)2+ n - 1} E_xy / dn² b

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^a After Crusio et al. (1984)

^b After Crusio (1984)

***P<0.001;

**P<0.01;

*P<0.05;

(*)P<0.10.

^aIndependent estimates not available for each replication separately.