Revista Mexicana Ciencias Agrícolas   volume 12   number 6   August 14 - September 27, 2021

DOI: https://doi.org/10.29312/remexca.v12i6.3013

Article

Design of 2n factorial experiments for their analysis with Infostat and Infogen

Delfina de Jesús Pérez López

J. Ramón Pascual Franco Martínez

Francisco Gutiérrez Rodríguez

Jesús Hernández Ávila

Artemio Balbuena Melgarejo

Andrés González Huerta§

Center for Research and Advanced Studies in Plant Breeding-Faculty of Agricultural Sciences-Autonomous University of the State of Mexico. ‘El Cerrillo’ University Campus, Toluca, State of Mexico, Mexico. AP. 435. Tel. 7222965518, ext. 148. (djperezl@uaemex.mx; jrfrancom@uaemex.mx; fgutierrezr@uaemex.mx; jhernandeza@uaemex.mx; abalbuenam@uaemex.mx).

§Corresponding author: agonzalezh@uaemex.mx.

Abstract

The design and analysis of data from factorial experiments using statistical packages saves time, simplifies calculations, and allows discussion of the individual or joint contribution of several factors. This study indicates the procedures for analyzing data with the InfoStat and InfoGen statistical packages in 2n factorial experiments, with n= 2, 3, 4, 5, 6, 7 and 8 factors of productivity. In addition, the design of its treatment structure, possible interactions, statistical models for a randomize complete block design and some formulas for calculating sums of squares for n= 8 are shown. The relationships between the linear models are established and also the procedures that must be applied in both statistical packages to obtain an analysis of variance and the comparison of treatment means if the general statistical model is used, the instructions to estimate the effects for each of the eight main factors and for their possible interactions with the Tukey test (p= 0.01) are also indicated. The design of this type of experiments was made to be analyzed with academic test versions, which are free for six months in both statistical packages. The user could purchase a license for one year to install it on two personal computers paying, for each software, only USD$50.00. Both statistical packages are very friendly and can be used to analyze any experimental design if the appropriate statistical model is indicated.

Keywords: construction of factorial treatments, exploratory experimental design, randomized complete blocks, statistical models.

Reception date: June 2021

Acceptance date: August 2021

Introduction

The sequential and iterative nature of the response surface methodology requires that in each phase significant factors be identified, model adaptation is made, and the direction in which the optimal experimental conditions are determined. To identify the factors that have the greatest influence on the response and to examine the goodness of fit of the statistical model, the analysis of variance should be used (Jiménez, 2015).

In this context, the design and analysis of factorial experiments in randomized complete blocks has been used as a very valuable tool in agricultural sciences, and in other branches of science and technology (Sahagún, 1998; Medina and López, 2011; Jiménez, 2015; Álvarez et al., 2018), in addition to determining the contribution of each of the factors studied, it allows simultaneously knowing what are the effects of their interactions, in each trial and through years, localities or their combinations in time and/or space (Montgomery, 2010; Gomez and Gomez, 1984; Di Rienzo et al., 2008; Balzarini and Di Rienzo, 2016).

The 2n experiments are families of tests with n factors and two levels within each of them that generally represent very wide intervals of variability in exploratory studies, when there is no or when there is little published information and consequently, it is desirable to redesign their research with a more important subset (Gomez and Gomez, 1984; Montgomery, 2010; Medina and López, 2011; Jiménez, 2015).

The evaluation of two promising crosses formed with the most outstanding lines, the introduction to a new region of cash crops, of plant species in the process of domestication or of animals of various breeds, as well as the generation, validation, application and transfer of technology could represent a desirable case for the design and analysis of this type of research (Torres et al., 2011; Reynoso et al., 2014; Torres et al., 2017; González et al., 2019). The determination of optimal doses when organic and inorganic fertilizers are applied, in different population densities, on several planting dates, or different localities and years, could have as a prerequisite the analysis of data from exploratory trials (Rodríguez et al., 2015; Quiroz et al., 2017; Padilla et al., 2019; González et al., 2019).

The 2n factorial experiments are difficult to analyze with a desktop calculator when n increases, due to an increase in the number of interactions to be estimated, when n= 6, 7 and 8, there are 57, 120 and 247 interactions, respectively. With the use of statistical packages, their design and analysis save time and simplify calculations, especially in series of experiments or in arrangements of experimental units that are designed to facilitate their handling in the field and to obtain different levels of precision in the estimation of the factors that make them up (Gomez and Gomez, 1984; Martínez, 1988; Padilla et al., 2019; González et al., 2019).

The first versions of InfoStat and InfoGen appeared before 2008 and 2016, respectively, both were created by programmers and analysts from the University of Córdoba, in Argentina and their academic versions can be downloaded for free from their websites.

Both programs are very friendly and can be used to analyze any type of experiment if the correct statistical model is introduced, because these can now enhance its use; through R-software (Sahagún, 1998; Balzarini et al., 2008; Di Rienzo et al., 2008; Balzarini and Di Rienzo, 2016). As a reference in the use of the SAS (Statistical Analysis System, 1989); InfoStat and InfoGen; Padilla et al. (2019); González et al. (2019) analyzed data on the production in green pod of three broad bean cultivars (Vicia faba L.), planted in two Mexican localities, with three population densities and three fertilization formulas: in the first publication, the codes to analyze and validate a series of experiments in randomized complete blocks in arrangement of subdivided plots were presented and in the second publication, the program to estimate mutually orthogonal contrasts in this type of trials was described.

In the above context, the main objective of this study was to design the procedures for analyzing 2n factorial experiments in randomized complete blocks with r repetitions using InfoStat and InfoGen, to generate an analysis of variance and a comparison of treatment means with the Tukey test.

Materials and methods

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.

Table 1. Number of factors, codes to represent them, number and structure of treatments.

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: Yijkl= μ + Ai + Bj + Ck+ (AB)ij + (AC)ik + (BC)jk + (ABC)ijk + Rl + εijkl.

Where: μ is the arithmetic mean of the abcr data; Ai, Bj and Ck 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; Rl is the effect of the l-th repetition and εijkl 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.

Table 2. Statistical models used in data analysis with InfoStat and InfoGen.

&= the possible interactions for n= 5, 6, 7 and 8 are shown below.

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:  = 28 + 56 +70 +56 +28 +8 +1 = 247 interactions. To verify:

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.