COMPARISON OF METHODS FOR ESTIMATING THE OPTIMUM PLOT SIZE FOR PEARL MILLET, SLENDER LEAF RATTLEBOX, AND SHOWY RATTLEBOX 1

- This study aimed to compare three methods of estimating the optimum plot size to evaluate the fresh matter productivity of pearl millet ( Pennisetum glaucum L.), slender leaf rattlebox ( Crotalaria ochroleuca ), and showy rattlebox ( Crotalaria spectabilis ). Twenty-seven uniformity trials were carried out with pearl millet, slender leaf rattlebox, and showy rattlebox cultivated alone and intercropped. Fresh matter productivity was evaluated in 972 basic experimental units (BEU) of 1 m × 1 m (36 BEU per trial). The optimum plot size was determined using the methods modified maximum curvature, linear response with plateau model, and quadratic response with plateau model. The optimum plot size differs between methods and decreases in the following order: quadratic response with plateau model (9.94 m 2 ), linear response with plateau model (7.41 m 2 ), and modified maximum curvature (3.49 m 2 ). The optimum plot size to evaluate the fresh matter productivity of pearl millet, slender leaf rattlebox, and showy rattlebox cultivated alone or intercropped is 7.41 m 2 . This size could be used as a reference for future experiments.

An important aspect when planning the experiment for a given crop is to define the optimum plot size to minimize the experimental error and, consequently, increase the precision of inferences. This size can be calculated using data from uniformity trials (blank experiments) of this same crop by different methods. Plot size has been investigated in the common cultivar of pearl millet (Pennisetum glaucum L.) (BURIN et al., 2015(BURIN et al., , 2016 and sunn hemp (Crotalaria juncea L.) (FACCO et al., 2017) using the maximum curvature of the model of the coefficient of variation (PARANAÍBA; FERREIRA; , and in C. juncea (FACCO et al., 2018) using the modified maximum curvature method (MEIER; LESSMAN, 1971).
Comparative studies with the methods modified maximum curvature (MMC) (MEIER; LESSMAN, 1971) Uniformity trials allow planning different plot sizes (X) by grouping adjacent basic experimental units (BEU) and estimating the coefficient of variation (CV (X) ) between BEU. The values of CV (X) and X can be related using the MMC, LRP, and QRP methods to determine the optimum plot size (Xo) and the coefficient of variation in the optimum plot size (CV Xo ).
Uniformity trials carried out with different soil cover species cultivated alone or intercropped and the data set analysis by different methods may provide useful information to be used as a reference in the design of experiments, aiming at higher experimental precision.
Therefore, this study aimed to compare three methods of estimating the optimum plot size to evaluate the fresh matter productivity of pearl millet (Pennisetum glaucum L.), slender leaf rattlebox (Crotalaria ochroleuca), and showy rattlebox (Crotalaria spectabilis) cultivated alone or intercropped.

MATERIAL AND METHODS
Twenty-seven uniformity trials with pearl millet (Pennisetum glaucum L.) of the cultivar BRS 1501 (M), slender leaf rattlebox (Crotalaria ochroleuca) of the common cultivar (CO), and showy rattlebox (Crotalaria spectabilis) (CS) were conducted in an experimental area located at the geographic coordinates 29°42′ S and 53°4 9′ W and 95 m of altitude. The local climate is Cfa, according to the Köppen classification, that is, a humid subtropical climate with hot summers and no dry season (ALVARES et al., 2013). The soil of the area is classified as an Arenic Dystrophic Red Argisol (SANTOS et al., 2018).
The central area of a size of 6 m × 6 m (36 m 2 ) at each uniformity trial, which had a size of 8 m × 8 m (64 m 2 ), was divided into 36 basic experimental units (BEU) of 1 m × 1 m (1 m 2 ), forming a matrix of six rows and six columns. The plants were cut close to the soil surface between January 29 and February 4, 2020, at each BEU and weighed to determine the fresh matter (FM) productivity (g m −2 ) on a digital scale (accuracy of 1 g). The pearl millet plants were at flowering and the slender leaf rattlebox and showy rattlebox plants did not reach the flowering stage at that time. The weighing was carried out immediately after cutting to minimize possible variations in plant moisture.
The FM data of the 36 BEU allowed the planning of plots with adjacent X R BEU in the row and adjacent X C BEU in the column for each uniformity trial. Plots with different sizes and/or shapes were planned as X=X R ×X C , that is, 1×1, 1×2, 1×3, 1×6, 2×1, 2×2, 2×3, 2×6, 3×1, 3×2, 3×3, 3×6, 6×1, 6×2, and 6×3. The terms Rev. Caatinga, Mossoró, v. 34, n. 2, p. 249 -256, abr. -jun., 2021 251 X R , X C , and X represent, respectively, the number of adjacent BEU in the row, the number of adjacent BEU in the column, and plot size in number of BEU. The number of plots (n) with an X BEU size (n=36/X) and the coefficient of variation (%) between plots of X BEU in size (CV (X) ) were determined for each plot size (X). The optimum plot size (Xo) was determined for each of the 27 trials using the methods modified maximum curvature ( The parameters a and b and the coefficient of determination (R 2 ) of the model were estimated for the modified maximum curvature (MMC) method (MEIER; LESSMAN, 1971). These parameters were estimated through a logarithmic transformation and linearization of , that is, , whose estimation was weighted by the degrees of freedom (DF = n − 1), associated with each plot size, according to the application of Sousa, Silva, and Assis (2016). The point corresponding to the optimum plot size (Xo) was determined algebraically using the expression . The coefficient of variation corresponding to the optimum plot size was determined by . Two segmented lines were adjusted for the linear response with plateau model (LRP) (PARANAÍBA; FERREIRA; MORAIS, 2009) and the estimations of the parameters a, b, and p and the coefficient of determination (R 2 ) were obtained. The first line is adjusted to the point corresponding to the optimum plot size (Xo), with a nonzero slope (b). The second line starts from Xo and has a zero slope, that is, it is a line parallel to the abscissa, where p is the plateau, i.e., p corresponds to CV Xo . The LRP model consisted of . The optimum plot size in the LRP model was determined by and the coefficient of variation in the optimum plot size was determined by . The adjustment of the quadratic response with plateau model (QRP) (PEIXOTO; FARIA; MORAIS, 2011) was performed using two segmented equations. Estimations were obtained for the parameters a, b, c, and p and the coefficient of determination (R 2 ). The quadratic part of the model was adjusted up to point Xo. The model becomes a straight line with a zero slope after Xo, being called a plateau, whose model is described by , where p is the plateau, i.e., p = CV Xo . Thus, the QRP model consisted of . The optimum plot size in the QRP model was determined by and the coefficient of variation in the optimum plot size was defined by . The point of union between the two segments in the LRP and QRP models corresponds to Xo in the abscissa and CV Xo in the ordinate. In the three models (MMC, LRP, and QRP), ɛ is the residual or random error of the model.
Thus, the fresh matter productivity (FM, g m −2 ) of the trial, the coefficient of variation of the trial (CV, %), and the estimates of the coefficient of determination (R 2 ), optimum plot size (Xo), and coefficient of variation in the optimum plot size (CV Xo , %) relative to the MMC, LRP, and QRP methods were obtained for each of the three uniformity trials (replications) of each of the nine compositions. The data were subjected to analysis of variance and Scott-Knott test via bootstrap with 10,000 resamples at a 5% significance level to compare the compositions. These statistical procedures are adequate to circumvent possible impacts of not meeting the assumptions of normality of errors and homogeneity of residual variances (FERREIRA, 2014). Comparisons of the means of the R 2 , Xo, and CV Xo estimates between methods (MMC versus LRP, MMC versus QRP, and LRP versus QRP) were performed using the Student ttest (bilateral) for dependent samples at a 5% significance level regardless of the composition (n = 27 uniformity trials). The results of these comparisons were represented by letters next to the means. Statistical analyses were performed using the Microsoft Office Excel ® application and the software R (R DEVELOPMENT CORE TEAM, 2020) and Sisvar (FERREIRA, 2014).

RESULTS AND DISCUSSION
The 27 uniformity trials, formed by sowing densities compositions of pearl millet (Pennisetum glaucum L.) of the cultivar BRS 1501 (M), slender leaf rattlebox (Crotalaria ochroleuca) of the common cultivar (CO), and showy rattlebox (Crotalaria spectabilis) (CS), showed a decrease in the coefficient of variation (CV (X) ) with an increase in the planned plot size (X) ( Table 1). These results indicate an improvement in experimental precision (decrease in CV (X) ) with an increase in plot size. Thus, although it is possible to evaluate the fresh matter productivity (FM) in 1-m 2 plots, as performed in this study, evaluating the precision at larger plot sizes is also important, that is, planning the experiment with the optimum plot size to ensure adequate discrimination of treatments under evaluation and reliability in inferences. It is also important to consider that smaller sizes may not represent plant development.  Table 2). The FM means from the three trials of each composition were 7325,7812,8466,8505,4511,7442,7861,7955, and 4593 g m −2 for the compositions 100% M, 75% M + 25% CO, 50% M + 50% CO, 25% M + 75% CO, 100% CO, 75% M + 25% CS, 50% M + 50% CS, 25% M + 75% CS, and 100% CS, respectively. Two groups of means were formed by the Scott-Knott test via bootstrap at a 5% significance. The group with the highest FM production was formed by compositions of pearl millet cultivated alone and intercropped. The group with the lowest FM production was formed by slender leaf rattlebox and showy rattlebox cultivated alone. FM values of 34.59, 31.35, and 33.9 Mg ha −1 were obtained by Passos et al. (2017), and Table 1. Planned plot size (X=X R ×X C ), in basic experimental units (BEU), with adjacent X R BEU in the row and adjacent X C BEU in the column, number of plots with X BEU of size (n=36/X), and coefficient of variation (%) between plots with X BEU of size [CV (X) ]. Fresh matter productivity data from three uniformity trials (repetitions) of nine compositions of pearl millet (M), slender leaf rattlebox (CO), and showy rattlebox (CS).  Table 2. Fresh matter productivity (FM), coefficient of variation (CV), coefficient of determination (R 2 ), optimum plot size (Xo, m 2 ), and coefficient of variation in the optimum plot size (CV Xo , %) as a function of maximum modified curvature (MMC) methods, linear response with plateau (LRP) model, and quadratic response with plateau (QRP) model, in three uniformity trials (repetitions) of nine sowing density compositions of pearl millet (M), slender leaf rattlebox (CO), and showy rattlebox (CS). The F-test value and respective p-value of the variance analysis via bootstrap with 10,000 resamples. (1) Each uniformity trial with an area size of 6 m × 6 m (36 m 2 ) was divided into 36 BEU of a size of 1 m × 1 m (1 m 2 ), forming a matrix of six rows and six columns.
(2) Averages that are not followed by the same letter in the column (comparison of averages of the compositions) differ at 5% significance by the Scott Knott's test via bootstrap with 10,000 resamples. (3) p-value > 0.05 means that averages of the compositions do not differ.
The means of the coefficient of determination (R 2 ), optimum plot size (Xo), and coefficient of variation in the optimum plot size (CV Xo ) did not differ between the nine compositions of sowing densities of pearl millet, slender leaf rattlebox, and showy rattlebox for the methods modified maximum curvature (MMC), linear response with plateau model (LRP), and quadratic response with plateau model (QRP) (Tables 2 and 3). Therefore, based on this finding and the lack of difference in the coefficient of variation of trials between compositions, the experimental planning regarding the plot size is similar for these crops cultivated alone or intercropped. Table 3. Averages of coefficients of determination (R 2 ), optimum plot size (Xo, m 2 ), and coefficient of variation in the optimum plot size (CV Xo , %) as a function of maximum modified curvature (MMC) methods, linear response with plateau (LRP) model, and quadratic response with plateau (QRP) model obtained from fresh matter productivity in three uniformity trials (repetitions) of nine compositions of pearl millet (M), slender leaf rattlebox (CO), and showy rattlebox (CS). 1 *Averages of R 2 , Xo, and CV Xo that are not followed by the same lowercase letter in the column (comparison of compositions within each method) differ at 5% significance by the Scott Knott's test. Averages that are not followed by the same uppercase letter in the row (comparison of methods regardless of composition, n = 27 uniformity trials) differ by the t-test of Student (bilateral) at 5% de significance.
The means of the coefficient of determination (R 2 ) did not differ between the nine compositions, varying from 0.69 to 0.89, 0.61 to 0.83, and 0.51 to 0.83 for the MMC, LRP, and QRP methods, respectively (Table 3), considering that 0.00 ≤ R 2 ≤ 1.00 and that the closer to 1.00, the better the model fits the data. In general, the R 2 values of the MMC method were higher than the values of the LRP and QRP methods within each composition. The comparison of methods showed that the R 2 value of MCC (0.81) was higher and those of LRP (0.75) and QRP (0.75) did not differ from each other, regardless of the composition of sowing densities