Design of the crosses

K must first be calculated, which must be an integer. Thus: k= (p + 1 - s) 2 . The parents, randomized and numbered consecutively, will generate the crosses: Parent 1 x parent k + 1, k + 2, k + 3, ..., k + s. Parent 2 x parent k + 2, k + 3, k + 4, ..., k + 1 + s. Parent p x parent k + p, k + p + 1, ..., k + p - 1 + s. With eight corn lines, s= 5 and k= 2, the 20 crosses that will be sampled are: Line 1 x lines 2+1, 2+2, 2+3, 2+4, 2+5 = 1x3, 1x4, 1x5, 1x6, 1x7. Line 2 x lines 2+2, 2+3, 2+4, 2+5, 2+6 = 2x4, 2x5, 2x6, 2x7, 2x8. Line 3 x lines 2+3, 2+4, 2+5, 2+6, 2+7 = 3x5, 3x6, 3x7, 3x8. Line 4 x lines 2+4, 2+5, 2+6, 2+7, 2+8 = 4x6, 4x7, 4x8. Line 5 x lines 2+5, 2+6, 2+7, 2+8, 2+9 = 5x7, 5x8. Line 6 x lines 2+6, 2+7, 2+8, 2+9, 2+10 = 6x8. Line 7 would cross with males 2+7, 2+8, 2+9, 2+10 and 2+11, as they are greater than p, multiples of 8 are applied: 7 x (9-8) = 7 x 1; 7 x (10 -8) = 7 x 2; 7 x (11-8) = 7 x 3; 7 x (12-8) = 7 x 4; 7 x (13-8) = 7 x 5. These are eliminated because they are RCs, and the crosses originated with line 8 (Table 1).

General analysis of variance (Anova)

The statistical model for a randomized complete block experimental design is: Y_ij = μ + τ_i + β_j + ε_ij. Where: μ is the arithmetic mean of the tr data, τ_i is the effect of the i-th treatment, β_j is the contribution of the j-th repetition, and ε_ij is the experimental error or residual of the model.

Stages to obtain a general Anava

E1. Concentrate the data in a table: the rows (subscripts ij) will represent crosses and the columns (subscript k) repetitions, calculate totals and arithmetic means (Table 2).

E2. Define the format of the general Anova (Table 3). Concentrate the above calculations in the format of the general Anova.

Calculate degrees of freedom (DF)

In this section, t is the number of crosses sampled, so t=ps/2. DF of the total = (ps/2) r-1= 20(4)- 1= 79; DF Rep= r - 1= 4 - 1= 3; DF crosses= (ps/2) - 1= 20 - 1= 19; DF of the error= [(ps/2)- 1] (r- 1)= DF total - DF Rep - DF crosses = 57.

Estimate sums of squares (SS)

Here the restriction i< j is introduced. Additionally, ij ≠ 12, 18, 23, 34, 45, 56, 67 and 78, which are the crosses not sampled in the partial diallelic, in the order of female and male. Total SS = ∑i=1p∑j=1p∑k=1rYijk2-Y…2psr2 = (7.56²+6.62²+, +6.83²)-[(543.53)²/80] = 86.9. SS Rep = ∑k=1rY..k2(ps2)-Y…2(psr2) -Y…2(psr2)= 137.792+,…,143.042 20-543.53220(4) = 17.07. SS crosses = ∑i=1p∑j=1pYij2r  -Y…2(psr2)   -Y…2(psr2) = 29.172+27.982+26.502+,…,+26.482 4-543.53220(4) -543.53220(4) = 22.439. SS error = SS total - SS Rep - SS crosses = 86.898 - 17.07 - 22.439 = 47.384.

Determine mean squares (MS)

MS Rep= SS Rep/r -1= 17.07/3= 5.691; MS crosses = SS crosses /[(ps/2)-1]= 22.439/19 = 1.181; MS error = SS error / (r-1) [(ps/2)-1] = 47.384/57 = 0.8312

Get calculated F values

F Rep = MS Rep/MS error = 5.691/0.8312 = 6.846. F Treat = MS crosses/MS error = 1.181/ 0.8312 = 1.42.

Definition of the circulant matrix A = [sI + N]

A_8x8= 5011111005011111105011111105011111105011111105011111105001111105. The sum in each row (r) or column (c) is:

The inverse of A was obtained with a matrix calculator (https://matrixcalc.org.es). As follows:

A^-1 _8x8 =0.2380.027-0.029-0.044-0.047-0.044-0.0290.0270.0270.2380.027-0.029-0.044-0.047-0.044-0.029-0.0290.0270.2380.027-0.029-0.044-0.047-0.044-0.044-0.0290.0270.2380.027-0.029-0.044-0.047-0.047-0.044-0.0290.0270.2380.027-0.029-0.044-0.044-0.047-0.044-0.0290.0270.2380.027-0.029-0.029-0.044-0.047-0.044-0.0290.0270.2380.0270.027-0.029-0.044-0.047-0.044-0.0290.0270.238  The sum in each

row (r) or column (c) is: ∑r=0p-1ar=∑c=0p-1ac=12s=12(5)= 0.1

When only a desktop calculator is available, the methodologies of ^{Kempthorne and Curnow (1961)}; ^{Singh and Chaudhary (1985)}; ^{Martinez (1991)} could be applied. To obtain the elements of the first row of its inverse, ^{Martínez (1991)} calculated its characteristic roots as: λj = ∑l=1pb1lCosj[2π(l-1)p] , j= 1, 2, 3, …, p. In this, λp = ∑l=1pb1l=2s. λ₁= b11cos⁡2π1-18+b12cos⁡2π2-18+b13cos⁡2π3-18+b14cos⁡2π4-18+b15cos⁡2π5-18+ b16cos⁡2π6-18+b17cos⁡2π7-18+b18cos⁡2π8-18

The crosses 1x2 and 1x8 were not sampled (b₁₂ and b₁₈ = 0) and will be eliminated from this formula; in b₁₁, its coefficient is s=5 and in the remaining bs is 1, after regrouping its components: λ₁ = 5cos⁡0+cos(⁡π2)+cos(⁡3π4)+cos(⁡π)+cos(⁡5π4)+. = 5 - 0.00000367 - 0.7071 - 1 - 0.7071 + 0.000011 = 2.5858. λ₂ = 5cos⁡0+cos(⁡π)+cos(⁡3π2)+cos(⁡2π)+cos(⁡5π2)+cos(⁡3π) . = 5 - 1 + 0.000011 + 1 -0.0000183 - 1 = 4. Similarly: λ₃= 5.4141; λ₄= 6; λ₅= 5.414; λ₆= 4; λ₇= 2.5857; λ₈= 2s =10.

^{Kempthorne and Curnow (1961)} estimated the λs as: λ_j = s-Sin (n- 2)jπnSin jπ= n sSinjπn- Sin(n- s)(jπn)Sin(jπn)). Where j=1, 2, …, n-1; λ_n = 2s, and s is the number of crosses sampled per parent. Λ₁= 5Sinπ8-Sin(8-5)(π 8)Sin(π8) = 5Sin0.3927-Sin(1.1781)Sin(0.3927) =2.5857. Λ ₂ = 5Sin 2π8-Sin(8-5) (2π 8)Sin (2π8) = 5Sin 0.7854-Sin (2.3562)Sin (0.7854) = 4.

The remaining λs are identical to those calculated with the method of ^{Martínez (1991)}. With these, the elements of the first row of the inverse of matrix A are obtained. a^o= 1n1λ1+1λ2+,…,+1λ8= (18)12.5857+14.0+15.4142+16.0+15.4142+14+12.5857+110.0= 0.2387. The other elements of the first row are calculated as follows: a^j = (1n)∑l=1n1λl  Cosjn-ln(2π) . In this, j=1, 2, …, n-1. a¹=18[1λ1Cos18-12π8+1λ2Cos18-22π8+1λ3Cos18-32π8+1λ4Cos18-42π8+1λ5Cos18-52π8+1λ6Cos18-62π8+1λ7Cos18-72π8+1λ8Cos18-82π8=18[1λ1Cos7π4+1λ2Cos3π2+1λ3Cos5π4+1λ4Cos (π)+1λ5Cos3π4+1λ6Cosπ2+1λ7Cosπ4+1λ8Cos0=18(0.2734+0.00000275-0.1306-0.1666666-0.1306-0.00000092+0.27347+0.1= 0.2198= 0.027..

This is the value of row 1, column 2

The other elements are calculated similarly, as the matrix is symmetric and circulant, to obtain those of the next row, those of the previous one simply moves one column to the right. This procedure is repeated until all the rows in the inverse of the matrix are completed.

Calculation of the corrected sum of the sampled crosses (Q_i)

The correction factor (CFQ) is calculated with the same restrictions as for SS crosses: CFQ = 2∑i=1p∑j=1pY-ij.ps=2(135.877)8(5)=2∑i=1t∑j=1pYij.rps=2Y…rps= 2(543.53)48(5)= 6.794 t ha^-1. The corrected means of the 20 crosses are calculated as the difference between the arithmetic mean of each of them and CFQ. For the cross 1x3 = 7.292 - 6.794 = 0.498. The remaining values are shown in Table 4.

The sum of Q_is is zero and these are calculated as: Q₁= 1x3 + 1x4 + 1x5 + 1x6 + 1x7= 0.498 + 0.201 - 0.168 + 0.068 - 0.716= - 0.118. Q₂= 2x4 + 2x5 + 2x6 + 2x7 + 2x8 = - 0.361 + 0.248 + 0.968 + 0.123 + 1.063= 2.041. Q₃= 0.140; Q₄= -1.057; Q₅= -0.781; Q₆= 0.36 ; Q₇= -2.168; Q₈= 1.585.

Estimation of the effects of general combining ability (GCA)

The estimation of the GCA for each parent (g_i) is done with matrix algebra. A G = H, its solution is: G = A^-1 H. Where: A^-1 is the inverse of matrix A; H is a column vector formed by the values of the corrected sums of the sampled crosses (Q_i), and G is another column vector composed of the estimates of g_i.

Thus: G _8x1 = A^-1 _8x8 H _8x1

The value of g₁ is obtained as the sum of the products of the first row of the inverse of matrix A with the only column of matrix H; that is: g₁ = [(0.238) (-0.118) + (0.027) (2.041) + (-0.029) (0.114) +, ..., + (0.027) (1.585)] = 0.1977. Also: g₂= 0.5853; g₃= 0.1028; g₄= -0.3137; g₅= -0.3005; g₆=-0.1071; g₇= -0.4879; g₈= 0.3236.

Estimation of S_ij values

The effects of the specific combining ability (S_ij) for each single cross are estimated as: S_ij = Ȳ_ij - μ - g_i - g_j. Where: μ is the arithmetic mean of the (ps/2) r data, Ȳ_ij is the average value of the cross between the parents i, j, g_i, g_j are the estimates of the general combining ability of lines i, j. The restrictions are: ∑i=1pgi = 0 and ∑j=1p-1Sij= 0. For crosses that involve parent 1, one will have: S₁₃= Ȳ₁₃ - µ - g₁ - g₃= 7.292 - 6.794 - 0.1977 - 0.1028 = 0.1974_. Similarly: S₁₄= 0.3172; S₁₅= -0.066; S₁₆= -0.0223; S₁₇= -0.4265.

Calculation of SS GCA

The sum of squares between treatments is divided into GCA and SCA; if the value of two of these three sources of variation is known in the diallelic table, the third is calculated by difference. SS GCA = rΣG_i’ H_i. Where: r is the number of repetitions, G_i’ is the transpose of the column vector 8x1 formed by the values of g_i and H_i is the column vector integrated with the values of Q_i., it is multiplied by r because arithmetic means were used in the calculations. SS GCA = 4[(0.1977) (-0.118) + (0.5853) (2.041) +, + (0.3236) (1.585) = 13.137. SS SCA = SS Crosses - SS GCA = 22.439 - 13.1370 = 9.3019.

From Table 5, it can be concluded that only between repetitions and between effects of the general combining ability, there were significant differences (p= 0.05 and p= 0.01, respectively). These results indicate that the blocking in the experimental area, established perpendicular to the gradient of environmental heterogeneity, was efficient. This helped to decrease the experimental error. Regarding the grain yield of the 20 crosses sampled, in these, the additive genetic effects were of greater importance than the non-additive ones (dominance and epistasis), so that this group of eight corn parents could be used efficiently in the formation of synthetic varieties. In the segregating generations (F₂ or higher), from random mating between this group of inbred lines, plants could be selected to obtain new and better varieties of free pollination.

Estimation of components of variance and heritability. σ² _e = mean square of the error = 0.8312. σ² _s = (MS of SCA - MS of the error)/r = (0.7751 - 0.8312)/4 = - 0.014. σ² _g = (p-1) (MS GCA - MS SCA)/rs(p-2) = 7(1.8767 - 0.7751)/4(5)(6)= 0.0642. The average variance for calculating the differences between two estimated g_i values is calculated as: AV ( gi-gj ) = 2pa°p-1-12sp-1σs2+σe2r = 28(0.238)7-12(5)(7)-0.014+0.83124 = 2(0.272 - 0.014286) (0.1938)= 0.0998. And its standard error is: EE (g_i - g_j) = AV (gi - gj) 0.0998 = 0.3159.

The equivalence between the previously calculated variances and the additive and dominance variances is established when it is assumed that the inbreeding coefficient is equal to one, because the lines used in the partial diallelic are inbred (S₇). So: σ² _A = 2 σ² _g = 2(0.0642) = 0.1284; σ² _D = σ² _s = -0.014. The total genetic variance, σ² _G, is estimated as: σ² _G = σ² _A + σ² _D = 0.1284 - 0.014 = 0.1144. Broad-sense heritability, H², is estimated as: H² = 100 (σ² _G / σ² _F) = 100 (0.1144/0.3222) = 35.5%, where σ² _F is the phenotypic variance. σ² _F = 2 σ² _g + σ² _s + (σ² _e /r) = 0.1284 - 0.014 + 0.2078 = 0.3222. Narrow-sense heritability, h², is estimated as: h² = 100 (σ² _A / σ² _F) = 100 (0.1284/0.3222) = 39.85%.

The negative estimate of the variance of the specific combining ability (non-additive genetic effects) suggests that there was an underestimation in broad-sense heritability (35.5%) compared to the estimate of narrow-sense heritability (39.85%). The value of h² suggests that the total phenotypic variability measured in the 20 single crosses of corn is related to additive genes that determine grain yield. In this context, it is assumed that 39.8% of the total phenotypic variability estimated in this quantitative variable is attributed to the differences that exist between the inbred parents, whose predominant effects are additive.