Estimation of optimal size of plots for experiments with radiometer in beans

An experimental error can lead to rework and, consequently, to the loss of financial and human resources. One way to reduce this problem is the estimation of the optimum size of experimental plot to carry out the treatments. The objective of this study was to estimate the optimal size of plots for reflectance measurements in beans by the modified maximum curvature method and the maximum distance method. Reflectance readings were made on bean plants with the aid of the GreenSeeker ® equipment, obtaining basic units of 0.45 m2 in an area of lines 6 and 8 m in length, performing 46 combinations of experimental area. X0 was determined using the modified maximum curvature and the maximum distance method. To increase the R2, the calculations have been redone using 20 combinations of experimental area. By adopting the bigest obtained area, it was concluded that the optimum size of an experimental plot for works with reflectance in beans is 5.40 m2 and the combination that presents the best distribution is 2 lines totalling 6 m long.


INTRODUCTION
The spectral response of vegetation usually shows that plants absorb more solar energy in the visible region and the bands used for determining the vegetation indices (VI) are in the red and near infrared region (Monteiro et al., 2012).The GreenSeeker ® is an instrument that provides the normalized difference vegetation index (NDVI) via reflectance measurements, the interpretation of which can provide information in a rapid and targeted way on nutritional conditions, physiological state, stress and potential crop yields, even in cloudy days, which prevent the acquisition from satellites (Malenovský et al., 2009;Gutiérrez-Soto et al., 2011;Martin et al., 2012;Ali et al., 2015).The reflectance, percentage of light reflected by the culture, can also detect variations in leaf area of plants attacked by diseases, serving as a parameter to estimate damage to production and determine the economic damage threshold (Hikishima et al., 2010).For experiments with beans, the size of the portions differ according to the purpose of the study.To check the efficiency of the severity assessment of angular leaf spot in common bean based in healthy and diseased areas of the leaf, Parrella et al. (2013) adopted portions 8 m² (4 m long and 2 m wide).Doblinski et al. (2010) adopted a 2 m² area in the study of diffuse pollution of swine wastewater on the beans.To estimate the productivity of grains and wheat plant height using reflectance measurements Xavier et al. (2006) adopted 3.6 m² plots (3 m long by 1.2 m wide).In order to quantify the damage and the relationship between severity, reflectance and productivity in the pathosystem of Asian soybean rust, Hikishima et al. (2010) adopted as experimental unit an area of 6.75 m², or 3 lines of 5 m in length.
The economy of human and financial resources, without losing experimental precision, is considered an important factor in the design of experiments.To plan the tests and assess the magnitude of the experimental accuracy is important to determine the level of credibility of the results obtained in the research (Storck, 2011;Storck et al, 2011).The establishment of optimum plot size, in any culture, is one of the ways to increase the experimental precision and maximize the information obtained in an experiment (Silva et al., 2012), and is a recognized way to reduce experimental error, while there are several methods for its estimation based on different principles (Lorentz et al., 2012).The experimental error, which is the existing variance between experimental units that received the same treatment, is estimated by applying repetition, which is one of the principles of the trial and to avoid it is necessary to know the characteristics of the experimental area and the grown culture (Oliveira et al., 2005).Works with the right size of plots allow optimal use of resources, while also allowing the researcher greater control and management of their experiment, when performed in a smaller area (Lackey and Stein, 2014).For determining the optimum plot size through the method of maximum modified curvature and the maximum distance method a blank experiment is necessary, with the culture of interest and then the experimental area is subdivided into smaller portions, called basic units from which the data is collected independently while identifying the relative position.After the taking of the data, plots of different sizes and shapes are simulated through the sum of contiguous plots (Lorentz et al., 2012).
The objective of this study was to estimate the optimal size of plots for reflectance measurements in beans by the modified maximum curvature method and the Lorentz et al. ( 2012) called maximum distance.

MATERIALS AND METHODS
The experiment was conducted in a growing area of the State University of Londrina (UEL), in Londrina-PR, in the dry season of 2013.The cultivated beans were the IPR Andorinha (registration No. 30617, Ministry of Agriculture, Livestock and Supply), seeded with 0.45 m spacing between rows and 11 plants per meter.With the aid of the GreenSeeker ® equipment were collected reflectance values in six rows wide by 23 m long at intervals of each meter thereby obtaining 138 readings.The basic unit for this study was set at 0.45 m², obtained through the minor form: 0.45 m × 1 m (Table 1).The optimum plot size was estimated using initially the method of the modified maximum curvature proposed b y Lessman; Atkins (1963) apud Meier and Lessman (1971).In this method, the measure of variability given by coefficient of variation (CVx) and the portion size with X basic units is clarified by CVx=aX -b , where aand b are the parameters to be estimated.The optimum plot size was estimated by the expression: Where, X0 is the value of the abscissa at the point of maximum curvature, which corresponds to the optimum plot size (Meier and Lessman, 1971).
The method of maximum distance was then calculated, where its resolution of the geometry formed by ayc curve, described by CVx=aX - b , and a secant line to this curve, yr.We look for the point of the curve yc that is at the largest distance from the line yr, since the line segment along this distance is perpendicular to the line yr (Lorentz et al., 2012).The solution method presented by Lorentz et al.  2012) and that has as its solution: .The distance between the points (xcj, xcj) and (xrpj, xrpj), this distance is on line ypj, which is perpendicular to yr, and is given by

RESULTS AND DISCUSSION
By the method of the modified maximum curvature (MMC) the estimates of a and b were 2.0012 and 0.058, respectively (Figure 1), thus, the optimum plot size was the minimum measure, that is, 0.45 m² because X 0 = 0.365.The result is justified by the low coefficient of variation, with 2.423 maximum and 1.401 minimum (Table 2).In order to obtain a more representative R², we limited following Table 1.Size (X), shape and total number of plots for the determination of the optimal size in studies using reflectance.simulations to the number 20 and calculations were redone and we obtained the same value of 0.45 m² through the modified maximum curvature method (MMC), because X 0 = 0.587.In the method of maximum distance (MD), in

Conclusion
With optimum size values of plots obtained by the MMC for 46 and 20 combinations equal to 0.45 m² for MD with 20 simulations equal to 2.70 m² and MD for 46 simulations, and for variance equal to 5.40 m², it could be concluded that the highest value is the most suitable for works in beans with application of radiometer.Adopting as a criterion the lowest CV, the optimal size area is 5,40 m² with combination 2x6 that is two lines (0.90 m) wide and 6 m long.
(2012)   proposes to express the line perpendicular to the line yr as an aid to find the point sought of the yc curve.So, this line perpendicular to the line yr will be called the yp, expressed by yp=ex+f.The angular coefficient c and the linear coefficient d, both from the line yr, are fixed and can be obtained from two yr points common to the yr curve.The common point between yc e yrwhich is more to the left, given by (xcri, ycri), and the common point more to the right, given by (xcrf, ycrf), then c and d are expressed, respectively, by and or , and these expressions for d are obtained by isolating the equation for yR,having been replaced in this, the point (xcri, yrri), or the point (xcrf, ycrf).The angular coefficient e of the line yp is also fixed and may be obtained by using the condition that the yR and yP lines are perpendicular to each other.In this way, .The determination of the linear coefficient f of the line yP is part of the interactive method proposed by Lorentz et al. (

Figure 1 .
Figure 1.Regression of observed CV and estimated CV data for the variables a, b, c and d for 46 simulations.

Figure 2 .
Figure 2. Regression of observed CV and estimated CV data to obtain the variables a, b, c and d for 20 simulations.

Figure 3 .
Figure 3. Regression of observed V and estimated V data to obtain the variables a, b, c and d for 20 simulations.

Table 2 .
Standard deviation, mean and variation coefficient of reflectance data.