Algebraic procedures

The sums of squares of crosses will be split into GCA and SCA; then, the interaction crosses x environments (C x A) will be fractionated into GCA x A and SCA x A. It will be considered t= (ps/2) = 20 crosses (p=8, s=5), where p and s are the number of parents and times each of these participates in the partial diallel, respectively. Totals or arithmetic means can be used in calculations (^{Kempthorne and Curnow, 1961}; ^{Dhillon, 1978}; ^{Martínez, 1983}; ^{Mastache et al., 1998}).

Statistical models

One environment: Y_ijk= μ + C_ij + R_k + E_ijk, multiple environments: Y_ijkl = µ + A_k + C_ij + (AC)_ijk + R_l(k) + E_ijkl.

Genetic models

One environment: Y_ijk= µ + [g_i + g_j + S_ij] + R_k + E_ijk, Multiple environments: Y_ijkl = µ + A_k +[g_i + g_j+ S_ij] + [(gl)_ik + (gl)_jk +(Sl)_ijk] + R_l(k) + E_ijkl. Where: μ is the population arithmetic mean; g_i, g_j are the effects of GCA in parents; S_ij are the effects of SCA in crosses; A_k is the contribution of the k-th environment; (gl)_ik, (gl)_jk are the interaction effects GCA x A; (Sl)_ijk is the interaction SCA x A; R_l, R_l(k) are the effects of repeats and repeats nested within environments; E_ijk, E_ijkl are the residuals of the models (^{Dhillon, 1978}; ^{Jasso et al., 2022}), in which ^{Martínez (1983}) and ^{Mastache et al. (1998}) imposed the restriction n_ij= 1 or n_ij= 0 whether or not the cross i_xj is included, respectively. ^{Jasso et al. (2022)} also introduced the restriction i< j, as well as i_xj ≠ of crosses that complete, along with the partial diallel, ^{Griffing’s (1956}) method 4.

General variance analysis

The design of the crosses and the Anova for one trial was shown by ^{Jasso et al. (2022}). For the Anova of the series of experiments, the following stages will be applied: Stage 1. Construct Table 1; generate the Anova for each trial and obtain the sums of squares of repetitions within environments and the combined error (^{Gomez and Gomez, 1984}).

Stage 2. Calculate degrees of freedom (DF)

DF of the Total= [ar(ps/2)] - 1= 3(4)(20) - 1= 239; DF Environments (A)= a -1= 2; DF Rep(A)= a(r-1)= 3(3)= 9. For verification: DF Trat 1= DF A + DF Rep(A)= ar - 1, then: DF Rep(A)= DF Trat 1 - DF A= (ar - 1) - (a - 1)= ar -1 - a + 1= a(r-1)= 9; DF Crosses (C)= (ps/2) - 1= [(8) (5)/2] - 1= 19; DF C x A= [(ps/2) - 1] (a-1)= 19(2)= 38; DF Error= DF Total - DF A - DF Rep (A) - DF C - DF C x A= 239 - 2 - 9 - 19 - 38= 171. Also: DF Error= a(r-1) [(ps/2) - 1]= 3(3) (19)= 171. For verification: DF Error= {[a(ps/2)r] - 1} - (a -1) - a(r-1)- [(ps/2) - 1] - {[(ps/2) - 1] (a-1)}= [a(ps/2)r] - 1 - a + 1 - ar + a - (ps/2) + 1 - (aps/2) + a + (ps/2) -1= (apsr/2) - ar - a(ps/2) + a= a[(psr)/2) - r - (ps/2) + 1 ]= a(r-1) [(ps/2) - 1].

Stage 3. Obtain sums of squares (SS)

SS A _(k=3) =

S C _(ij=20) =

To calculate SS C x A, one first defines SS Trat 1 = SS A + SS C + SS C x A,

where:

SS Trat 1_(ijk=60) =

SS C x A _{(ijk= 60)}= SS Trat 1 - SS A - SS C= 374.38 - 260.94 - 62.35= 51.09

SS Rep (A)=

=SS RepA₁ +SS RepA₂ +SS RepA₃= 1.765+6.34+37.52= 45.625

Table A x Rep allows verifying the previous calculation (Table 2) and the procedure shown below is very useful to calculate the contribution of any interaction, in this and in other factorial experiments (^{Padilla et al., 2019}; ^{González et al., 2019}; ^{Pérez et al., 2021}).

SS TRAT 2_(kl=12)= SS A + SS Rep (A)=

SS A =

SS Rep (A)= SS Trat 2 - SS A = 308.35 - 262.57= 45.78

SS Error=

SSErrorA₁ + SSErrorA₂ + SSErrorA₃ = 46.32 + 61.53 + 73.88= 181.73

SS Total= SS A + SS Rep (A) + SS C + SS CxA + SS Error= 601.75

Also: SS Total_(ijkl=240)=

Stage 4. Break down DF and SS of crosses into GCA and SCA

DF GCA= p - 1= 7; DF SCA= DF Crosses - DF GCA= 19 - 7= 12. Also: DF SCA= [(ps/2) - 1] - (p - 1) = (ps/2) - 1 - p +1= (ps - 2p)/2= p (s-2)/2= 8(3)/2= 12

Similarly to the case of individual trials, in the series of experiments:

SS GCA =

where: Q_i =

The correction factor for crosses is

a^ij or A^-1

is the inverse of the matrix

A_pxp = [sI + N] = [_aij], Q_j

forms a column vector (H) and 𝑔 𝑖 generates another column vector (G), in the latter its solution is: G = A^-1 H (^{Jasso et al., 2022}). Q_i values are calculated using the data in Table 3.

Q₁= (Y_13..- CFQ) + (Y_14..- CFQ) + (Y_15.. - CFQ) + (Y_16.. - CFQ) + (Y_17.. - CFQ); CFQ= 2(1786)/40= 89.3. Q₁= (90.48 - 89.3) + (100.4 - 89.3) + (89.4 - 89.3) + (95.92 - 89.3) + (93.08 - 89.3)= 22.78. Likewise: Q₂= 16.54; Q₃= -6.58; Q₄= 14.66; Q₅= -14.86; Q₆= -10.58; Q₇= -11.5; Q₈= -10.46.

The sum over Q_i is zero and the GCA estimators for parents ( g i ) are calculated as:

With this information, one carries out the breakdown of the SS of crosses into

[0.552 (22.78) + 0.492(16.54) +,…,+ (-0.1752) (-10.46)] = 39.529

SS SCA= SS Crosses - SS GCA = 62.36 - 39.529 = 22.831

Stage 5. Split DF and SS of interaction crosses x environments (CxA)

DF GCA x A= (p-1)(a-1)= 7(2)= 14; DF SCA x A= DF C x A - DF GCA x A= 38 - 14= 24. Also: DF SCA x A= [(ps/2) - 1] (a-1) - (p-1)(a-1)= (aps/2) - a - (ps/2) + 1 - pa + p + a - 1= (aps/2) - (ps/2) - pa + p= p(as - s - 2a + 2p )/2 = p(a - 1)(s - 2 )/2= 8(2)(3)/2= 24. To split SS (C x A), Q_ik and (gl)_ik must first be calculated as:

For environment k= 1, the correction factor, CFA₁= 2(Y_..1.)/ps= 2(546.8)/40= 27.34. Q₁₁= (Y_131. - CFA₁) + (Y_141. - CFA₁) + (Y_151. - CFA₁) + (Y_161. - CFA₁) + (Y_171. - CFA₁)= (27.48 - 27.34) + (29.8 - 27.34) + (26.88 - 27.34) + (29.84 - 27.34) + (29.32 - 27.34)= 6.62, Q₂₁= 4.94; Q₃₁= 2.14; Q₄₁= 5.18; Q₅₁= -7.06; Q₆₁= -2.22; Q₇₁= -5.7; Q₈₁= -3.9;

For environment k=2, the correction factor, CFA₂= 2(Y_..2.)/ps= 2(526.48)/40= 26.324. Q₁₂= (Y_132. - CFA₂) + (Y_142. - CFA₂) + (Y_152. - CFA₂) + (Y_162. - CFA₂) + (Y_172. - CFA₂)= (23.92 - 26.324) + (28.68 - 26.324) + (27.84 - 26.324) + (27.96 - 26.324) + (26.28 - 26.324)= 3.06, Q₂₂= 6.54; Q₃₂= -8.34; Q₄₂= 2.46; Q₅₂= -3.38; Q₆₂= -0.34; Q₇₂= 3.66; Q₈₂= -3.66;

For environment k= 3, the correction factor, CFA₃= 2(Y_..3.)/ps= 2(712.72)/40= 35.636. Q₁₃= (Y_133. - CFA₃) + (Y_143. - CFA₃) + (Y_153. - CFA₃) + (Y_163. - CFA₃) + (Y_173. - CFA₃)= (39.08 - 35.636) + (41.92 - 35.636) + (34.68 - 35.636) + (38.12 - 35.636) + (37.48 - 35.636)= 13.1, Q₂₃= 5.06; Q₃₃= -0.38; Q₄₃= 7.02; Q₅₃= -4.42; Q₆₃= -8.02; Q₇₃= -9.46; Q₈₃= -2.9;

then:

For trial 1

For trial 2

For trial 3

Considering the above estimators, the following is calculated

^{Dhillon (1978}) developed the formulas for estimating

SS SCAxA=

Also, by statistical inference from the previous calculations, it deduced that:

SS SCA x A =

(SS CA₁ + SS CA₂ + SS CA₃) - SS C - SS GCA x A = (SS Trat 1 - SS A - SS C) - SS GCA x A= SS CxA - SS GCA x A.

Thus, by approximation of the results that the above will generate, it must be obtained that: SS SCA x A= 31.897.

Stage 6. Obtain mean squares (MS)

MS A= SS A / (a-1)= 260.94/2= 130.47; MS Rep(A)= SS Rep(A)/a(r-1)= 45.625/9= 5.07; MS C= SS C/[(ps/2) - 1]= 62.35/19= 3.28; MS GCA= SS GCA/(p-1)= 39.529/7= 5.647; MS SCA= SS SCA /[p(s-2)/2]= 22.831/12= 1.90; MS C x A= SS C x A/(a-1)[(ps/2) - 1]= 51.09/38 = 1.34; MS GCA x A = SS GCA x A/(p-1)(a-1) = 19.925/14 = 1.37; MS SCA x A = SS SCA x A/[p(a-1)(s-2)]/2 = 31.89/24 = 1.32; MS Error = SS Error/{a(r-1)[(ps/2) - 1]}= 181.746/171= 1.06.

Stage 7. Calculate F values

In this section, reference should be made to the type of model selected (fixed, random, mixed) and based on this define the correct GLs and MSs to test the statistical hypotheses of interest (^{Sahagún, 1998}). The F tests (Table 4) were proposed by ^{Martínez (1983}). F (GCA)= [MS GCA /(MS SCA + MS GCA x A - MS SCA x A)]= 2.94; F (SCA)= MS SCA/MS SCA x A= 1.44; F (GCA x A)= MS GCA x W/MS SCA x A= 1.01; F (SCA x A)= MS SCA x A/MS Error= 1.24.

Stage 8. Concentrate the calculations previously obtained in the Anova format

Application of the Opstat Software

The software is freely available at https://14.139.232.166/opstat. This analyzes a partial diallel cross based on a circulating sample (^{Kempthorne and Curnow, 1961}). To create the database on its web page, the crosses are ordered in the rows and the repetitions in the columns, the latter delimited by spaces (*.prn) or by tabs (*.txt); between rows there are no additional spaces. If this is done from Microsoft Excel, the file will have an extension *.prn or *.txt. The labels on the first row will correspond to crosses, R1, R2, ..., Rn, the latter are repetitions. Each cross could be identified as 1x3, 1x4, 1x5, ..., 6x8. When pasting the data on the web page, the labels are not included. If the parents are included, they are captured below the last cross, as 1x1, 2x2, pxp (^{Sheoran et al., 1998}).

To send the data to the Opstat platform, submit is selected; the software will request number of parents, number of repetitions, and sample size. Press Analyse. The output, for a single trial, includes (^{Sheoran et al., 1998}; ^{Jasso et al., 2022}): a) Anova for a randomized complete block design, b) arithmetic mean and standard error for each cross, c) g_i estimators for each parent, d) Anova with the breakdown of the effects between crosses in GCA and SCA, e) estimates of variances of GCA, SCA, additive, dominant and average and, f) standard error to compare g_i values, and g) estimation of narrow-sense heritability.

Analyses of variance obtained with Opstat

To generate the series of experiments, the analysis of variance for each environment and the corresponding to the table crosses x environments must be performed.

To split the SS (C x A), the following is calculated

SS SCA x A =

(SS SCAA₁ + SS SCAA₂ + SS SCAA₃) - SS SCA= (18.892 + 11.458 + 24.599) - 22.696= 32.253.

By subtraction, one obtains:

SS GCA x A= SS C x A - SS SCA x A= 51.09 - 32.253= 18.837.

Both calculations are approximate to those obtained with other methods.

The first three sums of squares for SCA are obtained from individual analyses of variance (A₁, A₂, A₃; Tables 5, 6, 7); when considering the fourth sum of squares (Table 8), it must be corrected by multiplying it by r= 4, because the table of values crosses x environments was built by adding over the four repetitions that are in each environment.

In this last table, repetitions=environments, error= C x A, Total= Trat 1= SS A + SS C + SS C x A; each source of variation must be multiplied by 4 (number of repetitions) and F values must be obtained by applying the procedures described by ^{Martínez (1991}) or ^{Dhillon (1978}), among others (Table 4).

The procedures described for a single environment were also referenced by ^{Singh and Chauhdary (1985}); the artifices implemented in the present study to be analyzed with Opstat can also be applied to the analysis of a partial diallel cross corresponding to the methodologies proposed by ^{Fyfe and Gilbert (1963}); ^{Federer (1967}); ^{Narain et al. (1974}), as suggested by ^{Dhillon (1978}); ^{Martínez (1991}).