Experimental dimensions and precision in trials with millet and slender leaf rattlebox 1

- The objective of this study was to determine the optimal plot size to evaluate fresh matter in millet ( Pennisetum glaucum L.) and slender leaf rattlebox ( Crotalaria ochroleuca ), in scenarios formed by combinations of numbers of treatments, numbers of replicates, and levels of precision. Fifteen uniformity trials with millet and slender leaf rattlebox, in single cropping or intercropping, were carried out. Fresh matter was evaluated in 540 basic experimental units (BEU) of 1 m × 1 m (15 trials × 36 BEU per trial). The soil heterogeneity index of Smith (1938) was estimated. Plot size was determined by the method of Hatheway (1961) in scenarios formed by combinations of i treatments (i = 5, 10, 15 and 20), r replicates (r = 3, 4, 5, 6, 7 and 8), and d precision levels (d = 2%, 4%, 6%, 8%, 10%, 12%, 14%, 16%, 18% and 20%). To evaluate the fresh matter of millet and slender leaf rattlebox, in single or intercropping, in experiments in completely randomized or randomized block designs, with 5 to 20 treatments and with ﬁ ve replicates, plots with 10 m² of usable area are suf ﬁ cient for differences between treatments of 10% of the overall mean of the experiment to be considered signi ﬁ cant at 0.05 probability level.

Plot size has been investigated in millet (Pennisetum glaucum L.), cv.'Comum' (BURIN et al., 2015(BURIN et al., , 2016) ) and in sunn hemp (Crotalaria juncea L.) (FACCO et al., 2017) through the maximum curvature of the coeffi cient of variation model (PARANAÍBA; FERREIRA; MORAIS, 2009) and also in C. juncea (FACCO et al., 2018) through the modifi ed maximum curvature method (MEIER; LESSMAN, 1971).It is assumed that the intercropping, commonly used with soil cover plants, can generate distinct experimental planning patterns and, furthermore, that the use of the methodologies of Smith (1938) and Hatheway (1961), in another millet cultivar and in another sunn hemp species, can aggregate important information for the planning of experiments with these two soil cover plants.
Thus, the objective of this study was to determine the optimal plot size to evaluate the fresh matter of millet (Pennisetum glaucum L.) and slender leaf rattlebox (Crotalaria ochroleuca) in scenarios formed by combinations of numbers of treatments, numbers of replicates and levels of experimental precision.
In each uniformity trial, the central area with size of 6 m × 6 m (36 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.On January 29 and 30, 2020, in the flowering of millet, in each BEU, the plants were cut near the soil surface and their fresh matter (FM) was immediately weighed, expressed in g m -2 .Weighing was performed immediately after cutting in order to minimize possible variations in plant moisture.
For each plot size (X), the following parameters were determined: n -number of plots with X BEU in size (n=36/X); M (X) -mean of plots with X BEU in size; V (X) -variance between plots with X BEU in size; CV (X) -coeffi cient of variation (in %) between the plots with X BEU in size; and VU (X) -variance per BEU between the plots with X BEU in size [VU (X) =V (X) /X 2 ].
The parameters V1 (estimate of variance per BEU between the plots with size of one BEU) and b (estimate of soil heterogeneity index) and the coeffi cient of determination (r 2 ) of the function VU (X) =V1/X b , of Smith (1938), were estimated.These parameters were estimated by logarithmic transformation and linearization of the function VU (X) =V1/ X b , that is, logVU (X) = logV1 -b logX, 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 observed values of dependent variables [VU (X) ] and independent variables (X) and the function VU (X) =V1/X b (SMITH, 1938) were plotted.
For each experimental plan, the optimal plot size (Xo), in number of BEU (approximated to the next integer), was calculated using the expression (HATHEWAY, 1961).In this expression, b is the estimate of the soil heterogeneity index (in this study, for each composition, the mean of b of the three uniformity trials was considered); t 1 is the critical value of Student's t-distribution for the signifi cance level of the test (type I error) of α=5% (bilateral test at 5%), with DF degrees of freedom; t 2 is the critical value of Student's t-distribution, corresponding to 2(1-P) (bilateral test), where P is the probability of obtaining signifi cant results, that is, the power of the test (P=0.80, in this study), with DF degrees of freedom; CV is the estimate of the coeffi cient of variation between the plots with size of one BEU (in this study, for each composition, the mean of CV of the three uniformity trials was considered), in percentage; r is the number of replicates and d is the difference between treatment means to be detected as signifi cant at 0.05 probability level, expressed as a percentage of the overall mean of the experiment (precision).The degrees of freedom (DF) to obtain the critical values (tabulate) of the Student's tdistribution were obtained by the expressions DF=(i)(r-1), for the completely randomized design, and DF=(i-1)(r-1), for the randomized complete block design, where i is the number of treatments and r is the number of replicates.The values o f t 1 a nd t 2 in this study were obtained with the Microsoft Offi ce Excel ® application, using the functions t 1 =INVT(0.05;DF)and t 2 =INVT(0.40;DF),respectively.Statistical analyses were performed with Microsoft Offi ce Excel ® .
Table 1 -Planned plot size (X=X R ×X C ), in basic experimental units (BEU), with X R BEU adjacent in row and X C BEU adjacent in column; number of plots with X BEU in size (n=36/X); mean of plots with X BEU in size [M (X) ], in g; and coefficient of variation (in %) between the plots with X BEU in size [CV (X) ].Fresh matter data for sowing densities of millet (M) and slender leaf rattlebox (SLR) Experimental dimensions and precision in trials with millet and slender leaf rattlebox (1) Each uniformity trial of size 6 m × 6 m (36 m 2 ) was divided into 36 BEU of 1 m × 1 m (1 m 2 ), forming an matrix of six rows and six columns The soil heterogeneity index (b) of Smith (1938), among the 15 uniformity trials, ranged from 0.6587 to 1.7891 (Figure 1).The means of b of the three trials of each composition were 1.0330, 1.4709, 0.9183, 0.9535 and 1.1444 for the compositions of 100% M, 75% M + 25% SLR, 50% M + 50% SLR, 25% M + 75% SLR and 100% SLR, respectively.According to Smith (1938), the index value describes, in addition to soil heterogeneity, other variations, such as those related to plant production, climatic conditions, management and experimental data collection.The presence of these sources of variability tend to increase the value of the soil heterogeneity index (b).The values close to the unit indicate high soil heterogeneity or low correlation between adjacent plots.According to Lin and Binns (1986), when b > 0.7, plot size should be increased, when b < 0.2, the number of replicates should be increased and, in cases of 0.2 ≤ b ≤ 0.7, the researcher should investigate the best combination between plot size and number of replicates.Therefore, the high values of b and the similarity between the compositions suggest that experiments with millet and slender leaf rattlebox in single cropping or intercropping, should place greater emphasis on the use of larger plots.
In the 15 uniformity trials, there were reductions in the coefficient of variation [CV (X) ] and in the variance per BEU between the plots [VU (X) ], with the increase in the planned plot size (X) (Table 1 and Figure 1).Then, it can be inferred that there is improvement in experimental precision (decrease in CV (X) and VU (X) ) with the increase in plot size.In practice, as demonstrated in this study, it is possible to evaluate the fresh matter (FM) in plots of 1 m 2 .However, smaller plots may not represent the development of plants in single cropping and intercropping.Conversely, larger sizes would make it possible to evaluate the plants in the central area of the plot (usable area) and disregard the borders, thus reducing the interference of plants of the adjacent plots, that is, the inter-plot competition.Thus, it is important to determine the optimal plot size to ensure adequate discrimination of treatments under evaluation and reliability in the inferences.

Continuation table 1
There were marked reductions in variance per BEU [VU (X) ] with plots of up to four BEU in size (4 m²), intermediate reductions with plots between four and ten BEU, and stabilization trend with plots larger than ten BEU (Figure 1).In species with potential for soil cover, such as: turnip (CARGNELUTTI FILHO et al., 2014a); velvet bean (CARGNELUTTI FILHO et al., 2014b); fl ax (CARGNELUTTI FILHO et al., 2018); and black oats with common vetch (CARGNELUTTI FILHO et al., 2020), the pattern was similar.Therefore, to evaluate the fresh matter of millet and slender leaf rattlebox, in single cropping or intercropping, a plot of up to ten BEU (10 m 2 ) is suggested because the gain in experimental precision (decrease in VU (X) ) with progressive increases in plot size, from ten BEU, was not signifi cant.This value of 10 m 2 is relatively higher than the optimal plot size required to evaluate the fresh matter of millet, cv.'Comum', which was 4.46 m 2 in three evaluation times (BURIN et al., 2015) and 4.97 m 2 , for the three times of sowing and cuts (BURIN et al., 2016).It was also higher than the sizes of 2.04 m 2 (FACCO et al., 2017) and 1.98 m 2 (FACCO et al., 2018) to evaluate the fresh matter of sunn hemp.The differences between the environments, millet cultivars and sunn hemp species and also the methodologies used to determine plot size contribute to explaining the different results from those obtained in this study.
In the methodology of Hatheway (1961), based on fi xed values of the soil heterogeneity index (b) of Smith (1938) and coeffi cient of variation (CV), it is possible to determine different optimal plot sizes (Xo), as a function of the number of treatments (i), number of replicates (r) and precision (d) (Tables 2 and 3).The results obtained using this methodology allow the researcher to investigate within his/ her availability of experimental area, number of treatments to be evaluated and desired precision, which combination of plot size and number of replicates is more appropriate.
With fi xed values of i and r, the Xo increased with the increment in precision (d) (Tables 2 and 3).For example, to evaluate FM in an experiment with millet in single cropping (100% M), conducted in a completely randomized design (CRD), with fi ve treatments and three replicates, aiming that in 80% of the experiments Experimental dimensions and precision in trials with millet and slender leaf rattlebox    (power=0.80)differences between treatments of d=20% of the overall mean of the experiment (lower precision) are detected as signifi cant at 5% probability level, the plot size should be three BEU (3 m 2 ) (Table 2).Plots of 10 m 2 would make it possible to improve precision, that is, to obtain d=10%.To further increase precision, that is, to obtain d=2%, a plot with 214 BEU (214 m 2 ) would be necessary.Obviously, the experimental precision is higher, but conducting fi eld experiment with a plot of 214 m 2 is impractical.Therefore, high experimental precisions (low percentages of d) are diffi cult to be achieved in practice, due to the need for large plot size, as already pointed out by Cargnelutti Filho et al. (2014a, 2014b, 2018, 2020).A similar pattern was observed in the compositions of 75% M + 25% SLR, 50% M + 50% SLR, 25% M + 75% SLR and 100% SLR (Tables 2 and 3).
With fi xed values of i and d, the Xo decreased with the increment in r.Also, with fi xed values of r and d, there was a reduction in Xo with the increase in i (Tables 2 and 3).The higher the number of treatments and the number of replicates, the greater the number of degrees of freedom of the error and, consequently, the lower the estimate of the residual variance (mean square of the error), that is, the greater the experimental precision.
The results of this study serve as a reference for the defi nition of plot size and the number of replicates in experiments to evaluate the fresh matter of millet and slender leaf rattlebox, in single cropping or intercropping, in experiments conducted in CRD and RCBD.The use of plots of 10 m 2 is recommended due to the practical feasibility in the fi eld and the stabilization of precision from this size.Additionally, it is an intermediate size, that is, slightly larger than the sizes determined for millet (Pennisetum glaucum L.), cv.'Comum' (BURIN et al., 2015(BURIN et al., , 2016) ) and for sunn hemp (Crotalatia juncea) (FACCO et al., 2017(FACCO et al., , 2018) )

CONCLUSIONS
In experiments to evaluate the fresh matter of millet and slender leaf rattlebox, in single cropping or intercropping, in completely randomized or randomized complete block designs, with 5 to 20 treatments and with fi ve replications, plots of 10 m² of usable area are suffi cient for differences between treatments of 10% of the overall mean of the experiment to be considered signifi cant at 0.05 probability level.

Table 2 -
Optimal plot size, in m 2 , for completely randomized design, in combinations of i treatments, r replicates and d precision levels, for fresh matter at sowing densities of millet and slender leaf rattlebox

Table 3 -
Optimal plot size, in m 2 , for randomized complete block design, in combinations of i treatments, r replicates and d precision levels, for fresh matter at sowing densities of millet and slender leaf rattlebox