Theory and analysis of partial diallel crosses. Parents and F 2 generations

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1 Theory and analysis of partial diallel crosses. Parents and F generations José Marcelo Soriano Viana * Cosme Damião Cruz and Antonio Américo Cardoso Departamento de Biologia Geral Universidade Federal de Viçosa Viçosa Minas Gerais Brasil. Departamento de Fitotecnia Universidade Federal de Viçosa Viçosa Minas Gerais Brasil. *Author for correspondence. ABSTRACT. This paper presents theory and analysis of partial diallels using data from the parents and F generations obtained from crossings among two groups of parents. Theory and analysis are based on Hayman's 95 and 958 suggestions. The diallel analysis allows the characterization of the polygenic systems under analysis through the estimation of genetic and non-genetic components of variation in relation to each group of parents or to each parent depending on the component and on the genetic parameter. The genetic parameters are the mean products of the allelic gene frequencies mean degrees of dominance mean proportions between dominant and recessive genes in the parents direction of dominance and heritability. A diallel analysis for the common bean is included. Key words: diallel analysis partial diallel biometrical genetics. RESUMO. Teoria e análise de dialelos parciais. Pais e gerações F. Este trabalho apresenta teoria e análise de dialelos parciais usando dados dos pais e das gerações F. A metodologia está fundamentada nos métodos propostos por Hayman em 95 e 958. A análise dialélica possibilita a caracterização dos sistemas poligênicos em estudo por meio da estimação de componentes genéticos e não genéticos de variação e de parâmetros genéticos em relação a cada grupo de pais ou a cada pai dependendo do componente de variação ou parâmetro de natureza genética. Os parâmetros genéticos são: médias dos produtos das freqüências de alelos graus médios de dominância proporções médias de genes dominantes e recessivos nos pais direção de dominância e herdabilidade. É apresentada uma análise dialélica com feijoeiro. Palavras-chaves: análise dialélica dialelo parcial genética biométrica. The main aim of the diallel analysis is the study of the genetic control of quantitative traits which is essential for planning and carrying out breeding programs. Several methods for analysis were proposed in which the parents may be pure lines or open pollinated varieties among others (Jins and Hayman 953; Hayman 95; Dicinson and Jins 956; Griffing 956; Gardner and Eberhart 966). Information about general and specific combining ability heterosis reciprocal effects maternal and paternal effects may also be obtained (Par and Davis 976; Cocerham and Weir 977; Foolad and Bassiri 983; Nienhuis and Singh 986; Yanchu 996). Hayman (958) presented the theory and analysis of diallel crosses based on the genotypic and phenotypic values of the parents F hybrids and F generations. Although the diallel analysis using data from homozygous parents and their F hybrids allows estimation of genetic parameters unbiased by linage effects (Mather and Jins 97) and the best assessment of dominance in the polygenic systems under study the analysis using the information from the parents and the F generations may be an excellent alternative when dealing with a species from which it is difficult to obtain a large number of F hybrids. The methodology developed by Hayman (958) is not applicable to partial diallels which involve two groups of parents. This paper presents the theory and analysis of the partial diallel crosses based on Hayman's proposals (95 958) taing into consideration the data from the parents and their corresponding F generations. Theory The theory presented here is a generalization of the methodology of Hayman (95 958). If gene frequencies are the same for the two parent groups results are equal to those presented by this author. Let A and a be the alleles of one of the genes (locus a) that control a quantitative trait in a diploid Acta Scientiarum Maringá v. 3 n. p

2 68 Viana et al. species in which A increases and a decreases the trait expression. Let N homozygous parents (N 6) be divided into two groups one with n parents and the other with n' parents (n + n' N n and n' 3). The groups with N n and n' parents define three polygenic systems to which the following assumptions are attached: a) Mendelian inheritance; b) no reciprocal effects; c) no interaction between non-allelic genes; d) only two allelic forms; and e) no correlation in the distribution of non-allelic genes in the parents. In the F generation obtained from the cross between the rth (r... n) and sth (s... n') parents the genotypic frequencies of AA Aa and aa are: P(AA) P(Aa) F rs P (AA) rs + F rs (Aa) P rs (Aa) P rs P(aa) F rs P (aa) rs + (Aa) P rs P(AA)rs P(Aa)rs and P(aa)rs are the probabilities of the hybrid from the rth and sth parents to be AA Aa and aa respectively given by: P(AA) rs ( + θ ra + θsa + θra θsa ) P(Aa) rs ( θ ra θsa ) P(aa) rs ( θ ra θsa + θra θsa ) For all loci in the polygenic system under analysis the variable θ assumes the value if the parent is homozygous for the gene that decreases trait expression or if it is homozygous in relation to the allele that increases trait expression. Thus the genotypic mean of the F population obtained from the cross between the rth and sth parents is: g rs m + θ + θ + θ θ + p s ) + [da ( ra sa ) ha ( ra sa )] hrs (p r where: m m ; a p m + d θ is the genotypic value of the r a ra rth parent; p m + d θ is the genotypic value of the sth s a sa parent and hrs h a ( ra sa) is their specific heterosis. θ θ The parameter ma is the mean of the genotypic values of the homozygotes d a is the difference between the genotypic value of the homozygote of largest expression and m a h a is the difference between the genotypic value of the heterozygote and m a. Means of the rth and sth arrays (a group of F generations with a common initial parent) are: g r E(g rs ) m ) (p r + + hr g s E(g rs ) m ) (p s + + hs where: m E(p r ) m + d is the genotypic a wa mean of the n parents of a group; m E(p s ) m + da w is the genotypic a mean of the n' parents of the other group; E(hrs ) h a ( w a θra ) is the varietal hr heterosis of the rth parent and hs s E(hrs ) fixed h a ( w a θsa ) is the varietal heterosis of the sth parent. The parameters wa ua va and w'a u'a v'a are respectively the expected values of θ ra and θ sa. The parameters ua and va and u'a and v'a are the frequencies of the alleles A and a in the groups with n and n' parents respectively. The mean of the F generations is: E (g rs ) m L (m + m ) + h where: h E(hrs ) h a ( wa w a ) is the average heterosis. A discussion about the information provided by the specific varietal and mean heteroses and by the contrast m m L 0 is presented by Viana et al. (999). Genetic components of variation and genetic parameters. The following second degree statistics can be used to estimate the genetic components of variation and consequently the genetic parameters: () Variance of the genotypic means of the n parents V 0 () V(p da ( wa ) u a va da r ) D () Acta Scientiarum Maringá v. 3 n. p

3 Partial diallel with parents and F generations 69 () Variance of the genotypic means of the n' parents V 0 () V(p d w ' a ) u ' a v' a da s ) a ( D () (3) Variance of the genotypic means of the N parents V0 V(p d w t ) a ( a ) D p D () + q D () + pqd (3) where: p pw qw ' a p n / N q n ' t m + d a θta (t... N) wa a + / N and D (3) da ( wa w ' a ) () Covariance between genotypic mean of F generation of the rth parent and the mean of the non-recurrent parent (covariance in the rth array) W0(r) Wr Cov(g rs p s ) D () 8 Fr Fr d θ w ' a ha ra ( a ) (5) Variance of the genotypic means of the F generations of the rth parent (variance in the rth array) V (r)l Vr V(g rs ) D () Fr + H 8 6 () H () ha ( w ' a ) u ' a v ' a ha (6) Covariance in the sth array W0(s) Ws Cov(g rs p r ) D () 8 Fs Fs d θ wa a ha sa ( ) (7) Variance in the sth array V (s)l Vs V(g rs ) D () Fs + H 8 6 ( ) H ( ) ha ( wa ) ua va ha (8) Covariance between genotypic mean of F generation of the rth parent and the mean of the non-recurrent parent array W0(r)L V(g rs g s ) D () Fr F () + H () H r fi d F () E(Fr ) d w ' 8 u ' a v ' a ha wa ( a ) a (u a va ) d a ha Hr ha ( wa θra )( w ' a ) (9) Covariance between genotypic mean of F generation of the sth parent and the mean of the non-recurrent parent array W0(s)L V(g rs g r ) D () Fs F () + H ( ) H s F () E(Fs ) d w ' w 8 u va (u' a v ' a ha a ( a ) a a ) d a ha Hs ha ( w ' a θsa )( wa ) (0) Variance of the genotypic means of the F generations V(g rs ) E(H ) V F D () + D () F () F () + H ( ) + H () 6 H E( Hr ) E( Hs ) ha ( wa )( w ' a ) 6 u ha a va u ' a v ' a () Variance of the genotypic values of individuals of the F population obtained from the cross between the rth and sth parents + + V(rs)L { P(AA) F rs(da ) P(Aa) F rs(ha ) P(aa) F rs( da ) [ da ( θ ra + θsa ) + ha ( θraθsa )] } D (rs) + H 8 (rs) D (rs) da ( θra θsa ) H (rs) ha ( θra θsa ) () Mean of the total genotypic variances of the F generations of the rth parent V (r)l E( V(rs)L ) Dr + H 8 r Dr E( D (rs) ) da ( w ' a θra ) Hr E( H (rs) ) ha ( w ' a θra ) (3) Mean of the total genotypic variances of the F generations of the sth parent V ) V (s)l E( (rs)l Ds + H 8 s Ds E( D (rs) ) da ( wa θsa ) Hs E( H (rs) ) ha ( wa θsa ) () Mean of the total genotypic variances of the F generations of the parents of the diallel V L E( V(rs)L ) D (3) + H 8 (3) Acta Scientiarum Maringá v. 3 n. p

4 630 Viana et al. H (3) E( H (rs) ) E( Hr ) E( Hs ) ha ( wa w ' a ) D ( 3) E( D (rs) ) E(Dr ) E(Ds ) The sum [(/)D (rs) + (/8)H (rs) ] can be estimated but the components D (rs) and H (rs) cannot because there are not enough equations. Similarly [(/)Dr + (/8)Hr] and [(/)Ds + (/8)Hs] can be estimated but the components Dr Ds Hr and Hs cannot. In the absence of dominance the quadratic components D(rs) Dr and Ds are estimable and supply information which cannot be obtained from the linear components h rs h r and h s because they are all zero. The additive component D (rs) is nil when the rth and sth parents have the same genotype (θ ra θ sa for every a). Its value is largest when the parents are carriers of distinct alleles for all the segregant genes in the polygenic system (θ ra θ sa for every a). The D component of a parent is minimum when it carries the most frequent genes in the group of parents to which it doesn t belong. Its value is largest when the parent is carrier of the less frequent genes in the group of parents to which it does not belong. The mean of the products of the allele frequencies in the groups with n and n' parents are respectively H /H () and H /H (). The average degree of dominance in the polygenic systems defined by the groups with n and n' parents are respectively H ( ) / D ( ) and H () / D (). If the estimate of V L is taen into account the dominance component H (3) may be estimated. This allows the estimation of a third average degree of dominance which is valid for the polygenic system under study and not for a given group of parents since given that ha h and da d H (3) / D (3) h / d. The proportions between numbers of dominant and recessive genes in the groups with n and n' parents are respectively: D() H () + F () D() H ( ) + F () and D() H () F () D() H ( ) F () The direction of dominance is h D () D () / H D(3. Detailed discussion about ) the genetic components of variation and genetic parameters are presented by Viana et al. (999). When observations of the individuals in each F generation are available heritabilities in the broad (B) and narrow (R) senses can be defined. At the individual level the broad sense heritability in the F generation obtained from the cross between the rth and sth parents is: V(rs)L h B V(rs)L The mean broad sense heritability at the individual level in the F generations of the rth parent is: V(r)L h B V(r)L The mean broad sense heritability at the individual level in the F generations of the sth parent is: V(s)L h B V(s)L The mean heritabilities at the individual level in the F generations of the diallel are: VL h B VL (/ ) D (3) h R VL E0 is the non-genetic component of the variance of the phenotypic values of individuals in a single population. The broad sense heritability at F individual level is the square of the correlation between the phenotypic and the genotypic values of same F plant. The narrow sense heritability is the square of the correlation between the phenotypic and the additive genetic values of same F plant (Falconer and MacKay 996). The later should be used to evaluate the efficiency of the mass selection. The relationship between variance and covariance in the arrays. If the additive-dominant model is adequate there is a functional relation between W r and V r and between W s and V s and the regression coefficients of Wr on Vr (Wr β 0 + β V r ) and of W s on V s (W s β 0 + β V s ) are equal to unit as the differences W r V r (/)D() (/6)H() and Ws Vs (/)D() (/6)H () are constants. The intercepts of the two regressions are respectively β 0 E( Wr Vr ) D () H 6 () β 0 E( Ws Vs ) D () H 6 ( ) which do not give an indication of the average degree of dominance. The straight lines W r (/)[D () (/)H()] + Vr and Ws (/)[D() (/)H()] + Vs are limited by the parabolas W r V 0() V r and W s V 0() V s respectively. The absence of a functional relation between variance and covariance in the parental arrays in each group indicates that there is no Acta Scientiarum Maringá v. 3 n. p

5 Partial diallel with parents and F generations 63 dominance in the polygenic system under study (the coordinates of the points are ((/)D () (/)D () ) for the arrays of the group with n parents and ((/)D () (/)D () ) for the arrays of the group with n' parents). If the regression coefficients are different from the additive-dominant model is inadequate. If the regression coefficient of Wt on V t (W t β 0 + β V t ) is equal to unity the genes are equally frequent in the two groups of parents. The regression has a coefficient equal to because given that wa w a ' D() D() D(3) D F() F() F and H() H() H where D F and H are parameters of the Hayman method. Then W r V r W s V s W t V t (/)D (/6)H. The straight line Wt (/)[D (/)H ] + V t is limited by the parabola W t DV t. The relationship between the sum of the variance and covariance in the arrays and the genotypic value of the common parent. The analysis of the regressions of p r on (W r + V r ) (p r β 0 + β (W r + V r )) and of p s on (W s + V s ) (ps β 0 + β (Ws + Vs)) supply information about the presence of dominance in the polygenic system under study and also whether dominance is predominantly uni or bi-directional. The regression coefficients are respectively d w w ' ) a h a ( a )( a β d a h a ( wa )( w ' a ) d w w ' ) a h a ( a )( a β d a h a ( wa ) ( w ' a ) When β > 0 the deviations due to dominance are predominantly negative contributing to a decrease in the trait expression. Positive unidirectional dominance maes β < 0. If in each group of parents there is no functional relation between p and W + V (β 0) and the points on the graph of p versus (W + V) are randomly distributed dominance is bi-directional. When there is no dominance in the polygenic systems the points on the graphs of p r over (W r + V r ) and of p s over (W s + V s ) are (p r (3/)D () ) and (p s (3/)D () ) respectively. Non-genetic components of variation. Let yt and yrs be respectively the mean phenotypic values of a parent and of the F individuals obtained from the cross between the rth and sth parents. Thus y t p t + e t (e t ~ N(0 E) independents) y rs g rs + e rs (e rs ~ N(0 E") independents) p t and g rs are genotypic values and e t and e rs are non-genetic effects. Allowing e t and e rs be independent of the genotypic values and also between themselves then: V (y r ) V 0() + E V (y s ) V 0() + E V (y t ) V0 + E Cov(y rs y s ) Wr Cov(y rs y r ) Ws V(y rs ) Vr + E'' V(y rs ) Vs + E'' Cov(y rs y. s ) W0(r)L + E'' / n Cov(y rs y r. ) W0(s)L + E'' / n' And E {[ mˆ L ( mˆ + mˆ ' )] } [ ml ( m + m ' )] + [(N/ )E + E"] nn' ( /6)h + [(N/)E + E"] nn' The error mean square of the analysis of variance of the data of the F generations divided by the number of replications (b) is an estimator of E". An estimator of E is the error mean square of the analysis of variance of the data of the parents divided by b. If the means of the parents and F generations have the same precision then E and E'' can be estimated by the error mean square of the analysis of variance of the parents and F generations data divided by b. Let yrsij be the phenotypic value of the ith individual (i... l) of the F generation obtained from the cross between the rth and sth parents in the jth replication (j... b) (y rsij g rsij + e rsij (e rsij ~ N(0 E 0 ) independents). Considering the jth replication: l ( yrs. j E( Vˆ ) (rs)l j) E{ [ y ]} (l ) rsij i l l V(rs)Lj + E( e le ) (l ) rsij rs. j i V (rs)lj Thus Acta Scientiarum Maringá v. 3 n. p

6 63 Viana et al. b E ( Vˆ (rs)l j ) V b (rs)l j An estimator of the environmental variance among individuals (E 0 ) is the mean variance of the phenotypic values of individuals in the same parent population (mean of N or bn variances for the analyses considering one bloc and the b blocs respectively). Analysis Estimation of the genetic and non-genetic components of variation. If individual measurements are not taen in each F generation the estimation of the genetic and non-genetic components may be carried out by fitting the general linear model Y Xβ + ε (ε ~ N(Φ Cov(ε ))) where: Y' [ V 0() V$ 0( ) V$ 0 W 0( r ) W$ 0 ( r n V $ ( r ) L... V$ r n ) L W$ 0( s )... W$ 0( s n') V $ ( s ) L... V$ s n ') L W$ 0( r ) L... W$ 0( r n) L W $ 0( s ) L... W$ 0 s n ') L [ $ L ( $ ' $ 0)] + ml E$ E$ "] n W0() rl ( / ) D( ) [( n+ ) / 6n] Fr ( / 6n) Fr' + r' r' r ( / 6) H ( ) ( / 6) Hr + E"/n n' W0() sl ( / ) D() [( n' + ) / 6n'] Fs ( / 6n') Fs' + s' s' s ( / 6) H () ( / 6) H s + E"/n' β' [ D() D( ) D() 3 Fr LFr n Fs L Fs n' H () H ( ) Hr LHr n Hs L Hs n' h E E"] The estimation method utilized by Hayman (95 958) was the weighted least squares (Cov(ε) σ D where D is a diagonal matrix) since the means of the parents F hybrids and/or F generations were associated to different number of observations. The elements of D were defined in these papers. If the diallel table means have same precision the ordinary least squares method (Cov(ε) σ I) should be used. The maximum lielihood method (Cov(ε) V where V is a variancecovariance matrix) is an alternative. A simple iterative process for maximum lielihood estimation is presented by Hayman (960). Under normality assumptions the maximum lielihood estimator reduces to the ordinary least squares estimator when V σ I (Searle 97). Application Table shows the grain yield per plant of nine lines of common bean (Phaseolus vulgaris) six as parents of group (n 6) and three as parents of group (n' 3) and 8 F generations obtained from the partial diallel among the two parent groups. Means refer to the observed values of one of four blocs of an experiment carried out during the winter of 993 at the Federal University of Viçosa Viçosa state of Minas Gerais Brazil. In the case of the other blocs the additive-dominant model is not adequate to explain the observed results. Table shows the results of the regression analyses of Wr on Vr and of Ws on Vs. Tests show absence of functional relation between variance and covariance in the arrays of the two groups. In this case there is no dominance in the polygenic systems under study. Result is confirmed by the analysis of variance of the diallel table (Viana et al. 000). Table. Grain yield in grams of 9 lines of common bean and 8 derived F generations Parent BAT-30 () FT-8-835() Batatinha (3) Ricopardo 896 () Ouro Negro () Antioquia 8 (3) DOR. () RAB 9 (5) Ouro (6) Table. Summary of the regression analyses of W r on V r and of W s on V s estimates of regression coefficients and level of significance of the test of the hypothesis H 0 : β in relation to grain yield of common bean plants in grams Mean Square Source of Variation Degrees of Freedom Group Group Regression 0.9 ns ns.65x0 Error Regression Coefficient 0.5* Using t statistic with four and one degrees of freedom for error in analyses considering groups and respectively; With four and one degrees of freedom in the analyses for groups and respectively; ns: not significant (F less than one); *: 0.0 < P < 0.05; +: P > 0.0 The diallel analysis of the parents and F hybrids (Viana et al. 999) showed that grain yield depended largely on dominance effects. The results of the analysis presented here show that one generation of selfing made these effects negligible. The estimation of the genetic and non-genetic components of variation was carried out by adjusting the additive model using the ordinary least squares method. The estimates of V(rs)L were used in the calculations. The error mean squares from the analyses of variance considering only the parents and only the F generations were.3 and.60 respectively. The former is an estimate of the E component and the Acta Scientiarum Maringá v. 3 n. p

7 Partial diallel with parents and F generations 633 later is an estimate of E". The estimate of E0 was obtained as the weighted mean of the nine variances among plants in the same parental population (.86). Table 3 shows the estimates of the genetic and non-genetic components of variation. There is variability in group but there is little or no genetic variability in group. Thus the allelic frequencies of the non-fixed genes in the polygenic system defined by the parents of group are near one and zero. Table 3. Estimates of the additives and non-genetic (E E" and E 0 ) components of variation in relation to grain yield of common bean plants in grams Parameter Estimate Standard Error Probability D () D () D () D () D (3) D () D () D (3) D (3) D (3) D (33) D () D () D (3) D (5) D (5) D (53) D (6) D (6) D (63) E E" E D r D r D r D r D r D r D s D s D s D (3) Values not shown correspond to negative estimates given as equal to zero; Based on t test with 9 degrees of freedom Analysis of the means of the two groups shows that the genes that increase grain yield have a greater frequency in group. In this group Ricopardo 896 Ouro Negro and DOR. have similar genotypes to those of parents in group and carry the most frequent genes in this group which on the whole decrease grain yield. Crosses between these parents or between them and the parents of group should produce F generations with little variability for selection purposes. This is also true for crosses between the parents of group. In group BAT- 30 has the highest mean and is as expected the carrier of the more frequent genes in group. In group the parents Ricopardo 896 and Antioquia 8 produce the highest yield. Antioquia 8 should not have many of the more frequent genes in group which as already stated decrease grain yield. Therefore these two parents may be crossed with one or more of the parents of group. During the breeding process the objective should be to fix in a line the desirable genes of each parent. As already mentioned crosses involving Ricopardo 896 will produce F generations with reduced variability which may complicate the selection of lines with better performance than the best parent. In the F generations obtained from these crosses the breeder can deal with a smaller number of plants and families. Crosses between Antioquia 8 RAB 9 and Ouro or involving a parent from group should produce segregant generations with larger variability. This may favor the selection of a genotype superior to the best parent carrier of desirable genes from both parents. The breeder however will have to deal with many plants and families from these generations. The narrow sense heritabilities values at the individual level in the F generations of the diallel (Table ) show that mass selection are inefficient to increase grain yield in most of the segregant generations obtained from the cross between any two lines. The exceptions are the F generations derived from the crosses RAB 9 x FT and Ouro x Batatinha. Even in these cases it is adequate that selection for grain yield be performed on family means preferably in advanced generations with high degree of homozygosis. Table. Estimates of narrow sense heritabilities at individual level in the F generations of the diallel for grain yield of common bean plants in grams Parent BAT-30 FT Batatinha Mean Ricopardo Ouro Negro Antioquia DOR RAB Ouro Mean Values not shown correspond to negative estimates given as equal to zero References COCKERHAM C.C.; WEIR B.S. Quadratic analyses of reciprocal crosses. Biometrics Washington v. 33 no. p DICKINSON A.G.; JINKS J.L. A Generalised analysis of diallel crosses. Genetics Baltimore v. no. p FALCONER D.S.; MACKAY T.F.C. Introduction to quantitative Genetics.. ed. England: Longman 996. Acta Scientiarum Maringá v. 3 n. p

8 63 Viana et al. FOOLAD M.R.; BASSIRI A. Estimates of combining ability reciprocal effects and heterosis for yield and yield components in a common bean diallel cross. J. Agricult. Sci. New Yor v. 00 no GARDNER C.O.; EBERHART S.A. Analysis and interpretation of the variety cross diallel and related populations. Biometrics Washington v. no. 3 p GRIFFING B. Concept of general and specific combining ability in relation to diallel crossing system. Austr. J. Biol. Sci. Melbourne v. 9 no. p HAYMAN B.I. The theory and analysis of diallel crosses. Genetics Baltimore v. 39 no. 6 p HAYMAN B.I. Maximum lielihood estimation of genetic components of variation. Biometrics Washington v. 6 no. 3 p HAYMAN B.I. The theory and analysis of diallel crosses. II. Genetics Baltimore v. 3 no. p JINKS J.L.; HAYMAN B.I. The analysis of diallel crosses. Maize Genetics Cooperation Newsletter Columbia v. 7 p MATHER K.; JINKS J. L. Biometrical Genetics..ed. Ithaca New Yor: Cornell University Press 97. NIENHUIS J.; SINGH S.P. Combining ability and relationships among yield yield components and architectural traits in dry bean. Crop Science Madison v. 6 no. p PARK H.G.; DAVIS D.W. Inheritance of interlocular cavitation in a six-parent diallel cross in snap beans (Phaseolus vulgaris L.). J. Am. Soc. Horticult. Sci. Alexandria v. 0 no. p SEARLE S.R. Linear Models. New Yor: John Wiley & Sons 97. VIANA J.M.S. et al. Theory and analysis of partial diallel crosses. Genetics and Molecular Biology Ribeirão Preto v. no. p VIANA J.M.S. et al. Analysis of variance of partial diallel tables. Genetics and Molecular Biology Ribeirão Preto v. 3 no. p YANCHUK A.D. General and specific combining ability from disconnected partial diallels of coastal douglas-fir. Silvae Genetica Franfurt v. 5 no. p Received on August Accepted on March Acta Scientiarum Maringá v. 3 n. p