Notation used

The eight factors will be represented as A, B, H; both levels within each of these will be designated by the same letter, in lowercase and with a subscript, as a₁ and a₂ for the factor A. Interactions will be designated as A*B, A*C, A*B*C*D*E*F*G*H. In statistical models, each factor will be associated with a subscript: A, B, C, D, E, F, G, H will correspond to i, j, k, l, m, n, o, p, respectively. The repetitions will be identified with R and the last subscript that is indicated in the residual of each model used (^{Sahagún, 1998}; ^{Balzarini et al., 2008}; ^{Balzarini and Di Rienzo, 2016}).

Number of treatments and their factorial structure

Treatments will be represented by the combination of levels of two or more factors. Experimental design is a strategy of combining the structure of treatments with the structure of experimental units in such a way that alterations in responses, of at least in some subset of experimental units, can be attributed exclusively to the action of treatments, except for random variations. Thus, it is possible to compare treatment means or linear combinations of treatment means with as little noise as possible (^{Di Rienzo et al., 2008}; ^{Balzarini et al., 2016}; ^{Balzarini and Di Rienzo, 2016}). Table 1 shows the terminology, how many and what is the structure of treatments for n= 2, 3, 4, 5, 6, 7, 8.

Statistical models

To construct the statistical models used in data analysis, only the capital letters used to represent the main factors and those corresponding to their possible interactions, a specific value of n should be chosen, then the subscripts that identify them are chosen, and μ, R and ε are added. Their construction was based on the procedures described by ^{Sahagún (1998)}. For example, in a 2³ factorial experiment, if Y is the quantitative variable of interest: Y_ijkl= μ + A_i + B_j + C_k+ (AB)_ij + (AC)_ik + (BC)_jk + (ABC)_ijk + R_l + ε_ijkl.

Where: μ is the arithmetic mean of the abcr data; A_i, B_j and C_k are the effects of the i-th, j-th or k-th level of A, B or C, respectively; (AB)_ij, (AC)_ik, (BC)_jk and (ABC)_ijk are the interactions between two or three factors; R_l is the effect of the l-th repetition and ε_ijkl is the residual of the model. Table 2 shows the statistical models that will be used to analyze the data when n= 2, 3, 8.

To calculate how many interactions will be indicated in the statistical models, the combinations that are possible for the n factors taken k times are simply added, such that 2≤ k ≤8; k represents, in that order, interactions between two, three, eight factors. If n= 8: C 2 8 + C 3 8 + C 4 8 + C 5 8 + C 6 8 + C 7 8 + C 8 8 = 28 + 56 +70 +56 +28 +8 +1 = 247 interactions. To verify: 2 n - C 0 n - C 1 n = 2 8 - C 0 8 - C 1 8 = 256 -1 -8= 247 interactions.

Interactions for n= 8

Between two factors: A*B^£, A*C^£, A*D^£, A*E^£, A*F, A*G, A*H, B*C^£, B*D^£, B*E^£, B*F, B*G, B*H, C*D^£, C*E^£, C*F, C*G, C*H, D*E^£, D*F, D*G, D*H, E*F, E*G, E*H, F*G, F*H, G*H.

Between three factors: A*B*C^£, A*B*D^£, A*B*E^£, A*B*F, A*B*G, A*B*H, A*C*D^£, A*C*E^£, A*C*F, A*C*G, A*C*H, A*D*E^£, A*D*F, A*D*G, A*D*H, A*E*F, A*E*G, A*E*H, A*F*G, A*F*H, A*G*H, B*C*D^£, B*C*E^£, B*C*F, B*C*G, B*C*H, B*D*E^£, B*D*F, B*D*G, B*D*H, B*E*F, B*E*G, B*E*H, B*F*G, B*F*H, B*G*H, C*D*E^£, C*D*F, C*D*G, C*D*H, C*E*F, C*E*G, C*E*H, C*F*G, C*F*H, C*G*H, D*E*F, D*E*G, D*E*H, D*F*G, D*F*H, D*G*H, E*F*G, E*F*H, E*G*H, F*G*H.

Between four factors: A*B*C*D^£, A*B*C*E^£, A*B*C*F, A*B*C*G, A*B*C*H, A*B*D*E^£, A*B*D*F, A*B*D*G, A*B*D*H, A*B*E*F, A*B*E*G, A*B*E*H, A*B*F*G, A*B*F*H, A*B*G*H, A*C*D*E^£, A*C*D*F, A*C*D*G, A*C*D*H, A*C*E*F, A*C*E*G, A*C*E*H, A*C*F*G, A*C*F*H, A*C*G*H, A*D*E*F, A*D*E*G, A*D*E*H, A*D*F*G, A*D*F*H, A*D*G*H, A*E*F*G, A*E*F*H, A*E*G*H, A*F*G*H, B*C*D*E^£, B*C*D*F, B*C*D*G, B*C*D*H, B*C*E*F, B*C*E*G, B*C*E*H, B*C*F*G, B*C*F*H, B*C*G*H, B*D*E*F, B*D*E*G, B*D*E*H, B*D*F*G, B*D*F*H, B*D*G*H, B*E*F*G, B*E*F*H, B*E*G*H, B*F*G*H, C*D*E*F, C*D*E*G, C*D*E*H, C*D*F*G, C*D*F*H, C*D*G*H, C*E*F*G, C*E*F*H, C*E*G*H, C*F*G*H, D*E*F*G, D*E*F*H, D*E*G*H, D*F*G*H, E*F*G*H.

Between five factors: A*B*C*D*E^£, A*B*C*D*F, A*B*C*D*G, A*B*C*D*H, A*B*C*E*F, A*B*C*E*G, A*B*C*E*H, A*B*C*F*G, A*B*C*F*H, A*B*C*G*H, A*B*D*E*F, A*B*D*E*G, A*B*D*E*H, A*B*D*F*G, A*B*D*F*H, A*B*D*G*H, A*B*E*F*G, A*B*E*F*H, A*B*E*G*H, A*B*F*G*H, A*C*D*E*F, A*C*D*E*G, A*C*D*E*H, A*C*D*F*G, A*C*D*F*H, A*C*D*G*H, A*C*E*F*G, A*C*E*F*H, A*C*E*G*H, A*C*F*G*H, A*D*E*F*G, A*D*E*F*H, A*D*E*G*H, A*D*F*G*H, A*E*F*G*H, B*C*D*E*F, B*C*D*E*G, B*C*D*E*H, B*C*D*F*G, B*C*D*F*H, B*C*D*G*H, B*C*E*F*G, B*C*E*F*H, B*C*E*G*H, B*C*F*G*H, B*D*E*F*G, B*D*E*F*H, B*D*E*G*H, B*D*F*G*H, B*E*F*G*H, C*D*E*F*G, C*D*E*F*H, C*D*E*G*H, C*D*F*G*H, C*E*F*G*H, D*E*F*G*H.

Between six factors: A*B*C*D*E*F, A*B*C*D*E*G, A*B*C*D*E*H, A*B*C*D*F*G, A*B*C*D*F*H, A*B*C*D*G*H, A*B*C*E*F*G, A*B*C*E*F*H, A*B*C*E*G*H, A*B*C*F*G*H, A*B*D*E*F*G, A*B*D*E*F*H, A*B*D*E*G*H, A*B*D*F*G*H, A*B*E*F*G*H, A*C*D*E*F*G, A*C*D*E*F*H, A*C*D*E*G*H, A*C*D*F*G*H, A*C*E*F*G*H, A*D*E*F*G*H, B*C*D*E*F*G, B*C*D*E*F*H, B*C*D*E*G*H, B*C*D*F*G*H, B*C*E*F*G*H, B*D*E*F*G*H, C*D*E*F*G*H.

Between seven factors: A*B*C*D*E*F*G, A*B*C*D*E*F*H, A*B*C*D*E*G*H, A*B*C*D*F*G*H, A*B*C*E*F*G*H, A*B*D*E*F*G*H, A*C*D*E*F*G*H, B*C*D*E*F*G*H.

Between eight factors: A*B*C*D*E*F*G*H

The above interactions can also be used to construct any statistical model, such as those corresponding to the series of experiments in randomized complete blocks in arrangement of subdivided plots (^{Padilla et al., 2019}; ^{González et al., 2019}). For example, the 26 interactions that originate between factors A, B, C, D and E, and that must be fed in specifications to the model of InfoStat and InfoGen, were previously identified with the superscript £.

Some formulas for verifying sums of squares with n= 8

The formulas presented below were obtained with the procedures described by ^{Sahagún (1998)}.

Next, the second part of each formula, which corresponds to the correction factor, will be identified as FC and for convenience, the SCs of treatments 1, 2, s will be defined so that by difference the SC of the interaction of interest is calculated. The procedure is similar to that reported in ^{Padilla et al. (2019)}, for a series of experiments in randomized complete blocks in arrangement of subdivided plots.

SC for an interaction between two factors

Where: SC Treatments 1 = ∑i=1a∑j=1bYij…….2cdefghr- FC.

SC for an interaction between three factors

Where: SC Treatments 2= ∑i=1a∑j=1b∑k=1cYijk……2defghr- FC.

SC for an interaction between four factors

SC A*B*C*D = SC Treatments 3 - SC A - SC B - SC C - SC D - SC A*B - SC A*C - SC A*D - SC B*C - SC B*D - SC C*D - SC A*B*C - SC A*B*D - SC A*C*D. Where: SC Treatments 3 = ∑i=1a∑j=1b∑k=1c∑l=1dYijkl…..2efghr- FC.

SC for an interaction between five factors

SC A*B*C*D*E = SC Treatments 4 - SC A - SC B - SC C - SC D - SC E - (SC interactions between two, three and four factors). Where: SC Treatments 4 = ∑i=1a∑j=1b∑k=1c∑l=1d∑m=1eYijklm….2fghr- FC. Henceforth, the SCs for A, B, C, D, E, F, G, H will be identified as SC main factors.

SC for an interaction between six factors

SC A*B*C*D*E*F = SC Treatments 5 - SC main factors - SC interactions between two, three, four and five factors. Where: SC Treatments 5 = ∑i=1a∑j=1b∑k=1c∑l=1d∑m=1e∑n=1fYijklmn…2ghr- FC.

SC for an interaction between seven factors

SC A*B*C*D*E*F*G = SC Treatments 6 - SC main factors - SC interactions between two, three, four, five and six factors. Where: SC Treatments 6 = ∑i=1a∑j=1b∑k=1c∑l=1d∑m=1e ∑n=1f∑o=1gYijklmno..2 hr- FC.

SC for the interaction between eight factors

SC A*B*C*D*E*F*G*H = SC Treatments 7 - SC main factors - SC interactions between two, three, four, five, six and seven factors. Where: SC Treatments 7 = ∑i=1a∑j=1b∑k=1c∑l=1d∑m=1e ∑n=1f∑o=1g∑p=1hYijklmnop.2 r- FC.

Development of databases

These must have the structure of Table 3, in case the partial or total number of data captured is observed, Rep identifies repetitions, Trat defines the treatment, Ren specifies the yield and in A, B, H, the combinations between both levels of each of the eight factors are recorded.

Saving this treatment structure follows the same recommendations as for any file that contains databases, the file could be called EXP2N. IDB2.

Procedure for analyzing the data

After loading InfoStat or InfoGen, two dialog boxes are displayed on the screen, in both choose ok, their main menu will appear. In file choose the open option to load the database. InfoStat or InfoGen will display a table similar to that of Table 3.

In statistics choose analysis of variance and another dialog box will appear, where the definition of the dependent and classification variables is requested. If the objective is to analyze the 2⁸ factorial experiment, the dependent variable is rinde, the classification variables are Rep, A, B, C, D, E, F, G, H. Click accept.

Another dialog box that says analysis of variance will be displayed on the screen. Here, the user will verify that in the specifications of the terms of the model, the classification variables (Rep, A, B, H) are correct, the interactions A*B, A*C, A*B*C*D*E*F*G*H must be entered below the last main factor. In a 2³ factorial experiment, Rep, A, B, C, A*B, A*C, B*C, A*B*C should be displayed vertically in the dialog box. It is not necessary to specify either the arithmetic mean or the residual of the model. The analysis of variance will be obtained.

In the last dialog box, a comparison of treatment means with LSD Fisher, Bonferroni, Tukey, Duncan and Scheffé, among others, could be chosen. In comparisons choose: Tukey/show means according to: Rep, A, B, C, H, A*B, A*C, A*B*C*D*E*F*G*H/ presentation in descending list/significance level 0.01/accept. This procedure generates the analysis of variance and the comparison of means with the Tukey test (p= 0.01) for the eight factors and for their 247 interactions.