ABSTRACT
Evaluating the quality of scions is extremely important and it can be done by characteristics of shoots and roots. This experiment evaluated height of the aerial part, stem diameter, number of leaves, petiole length and length of roots of papaya seedlings. Analyses were performed from a blank trial with 240 seedlings of "Golden Pecíolo Curto". The determination of the optimum plot size was done by applying the methods of maximum curvature, maximum curvature of coefficient of variation and a new proposed method, which incorporates the bootstrap resampling simulation to the maximum curvature method. According to the results obtained, five is the optimal number of seedlings of papaya "Golden Pecíolo Curto" per plot. The proposed method of bootstrap simulation with replacement provides optimal plot sizes equal or higher than the maximum curvature method and provides same plot size than maximum curvature method of the coefficient of variation.
Key words:
Carica papaya L; experimental design; experimental precision
RESUMO
Em experimentos com mudas de mamoeiro é de extrema importância que se avalie a qualidade das mudas o que é feito por características da parte aérea e radicular. Foram avaliados altura da parte aérea, diâmetro do caule, número de folhas, comprimento do pecíolo e comprimento da maior raiz em plântulas de mamoeiro com objetivo de estimar o tamanho ótimo de parcelas. As análises foram feitas a partir de ensaio em branco com 240 plântulas de 'Golden Pecíolo Curto'. A determinação do tamanho ótimo de parcela foi feita aplicandose os métodos de máxima curvatura, da máxima curvatura do coeficiente de variação e um método proposto, que incorpora a simulação bootstrap de reamostragem ao método da máxima curvatura. O número ótimo de plantas por parcela para avaliação de mudas de mamoeiro 'Golden Pecíolo Curto' é de cinco. O método proposto de simulação bootstrap com reposição proporciona tamanhos ótimos de parcela iguais ou superiores ao método da máxima curvatura e, proporciona mesmo tamanho de parcela do método da máxima curvatura do coeficiente de variação.
Palavraschave:
Carica papaya L.; precisão experimental; planejamento experimental
INTRODUCTION
In every experiment, one of the main objectives is to reduce the error. In general, the experimental unit should be chosen to minimize the experimental error, which is the measure of variation that exists among observations of experimental units equally treated throughout the experiment (Steel ., 1997Steel RGD, Torrie JH & Dickey DA (1997) Principles and procedures of statistics: a biometrical approach. 3ª ed. New York, MacGrawHill Book Companies. 666p.; Storck ., 2011Storck L, Garcia DC, Lopes SJ & Estefanel V (2011) Experimentação vegetal. 3ª ed. Santa Maria, UFSM. 198p.).
Although one considers that the larger the plot size, the lower the experimental error and, consequently, the greater the accuracy of the experiment, this relationship is not linear (Smith, 1938Smith HF (1938) An empirical law describing heterogeneity in the yields of agricultural crops. Journal of Agricultural Science, 28:0123.; Paranaíba ., 2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.; Barbin, 2013Barbin D (2013) Planejamento e análise estatística de experimentos agronômicos. Londrina, Macenas. 213p.). The increase in the size of the plot initially leads to a decrease of the experimental error up to some extent, from which the precision gain is very small (Paranaíba ., 2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.; Storck ., 2011Storck L, Garcia DC, Lopes SJ & Estefanel V (2011) Experimentação vegetal. 3ª ed. Santa Maria, UFSM. 198p.).
An adequate experimental design involves the determination of the plot size and it will also depend on the crop, number of treatments, and environmental conditions of each experiment (Federer, 1977Federer WT (1977) Experimental design: theory and application. 3ªed. Nova York, Oxford & IBH Publishing. 593p.; Storck ., 2011Storck L, Garcia DC, Lopes SJ & Estefanel V (2011) Experimentação vegetal. 3ª ed. Santa Maria, UFSM. 198p.).
Several methods have been reported in the literature for estimation of the size of the plot. The most commonly used method is the modified maximum curvature, according to Meier & Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ). Another method widely used in the last years is the method of maximum curvature of the coefficient of variation (Paranaíba ., 2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.) which reduces the calculations to determine the optimum plot size as the great advantage over the previous methods. Even so, the blank trial is still needed, and in this experiment, the plants need to be set in rows, and evaluated in the exact sequence in which they are found, in order to estimate the coefficient of spatial autocorrelation of the first order.
Recently, Santos . (2012Santos D, Haesbaert FM, Lúcio AD, Storck L & Cargnelutti Filho A (2012) Tamanho ótimo de parcela para a cultura do feijãovagem. Revista Ciência Agronômica, 43:119128.) and Storck . (2014Storck L, Lúcio AD, Krause W, Araújo DV & Silva CA (2014) Scaling the number of plants per plot and number of plots per genotype of yellow passion fruit plants. Acta Scientiarum. Agronomy, 36:7378.) incorporated the bootstrap simulation to the method of maximum curvature of coefficient of variation proposed by Paranaíba . (2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.) and, Brito . (2014Brito MCM, HumadaGonzález GG, Morais AR & Moreira JM (2014) Avaliação do desempenho do algoritmo de reamostragem bootstrap na verificação da estimação do tamanho ótimo da parcela. Revista da Estatística UFOP, 3:255259.) to the linear response plateau method. However, the incorporation of simulation to the Meier & Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ) method was not found in the literature.
To determine the optimum plot size in a simpler manner is expected since the formation of clusters would be made from the simulations. The issue on flaws in the final stand (Brum . 2016Brum B, Brandelero FD, Vargas TO, Storck L & Zanini PPG (2016) Tamanho ótimo de parcela para avaliação da massa e diâmetro de cabeças de brócolis. Ciência Rural, 46:447463.), which is common in experiments involving seedlings, could also be contoured from the use of bootstrap simulation.
In field experiments involving papaya crops, several useful plot sizes are found, arbitrarily set since there are no studies reporting which plot size should be used. There are reports of the use of only one plant per plot (Pratissoli ., 2007Pratissoli D, Almeida GD, Jesus Júnior WC, Vicentini VB, Holtz AM & Cocheto JG (2007) Fertilizante organomineral e argila silicatada como indutores de resistência à varíola do mamoeiro. Idesia, 25:6367.; Melo , 2009Melo WB, Pereira RF, Silva MFD, Diniz PF, Gomes RCP, Santos JGR & Andrade R (2009) Variação da produção do mamoeiro Havaí em função de diferentes dosagens e de intervalos de aplicação de biofertilizante. Revista de Biologia e Ciências da Terra, 9:5459.) to 20 plants per plot (Vivas ., 2011Vivas M, Silveira SF, Terra CEP & Pereira MG (2011) Testers for combining ability and selection of papaya hybrids resistant to fungal diseases. Crop Breeding and Applied Biotechnology , 11:3642. ). In trials with production of papaya seedlings in nursery, useful sizes of plots follow the same arbitrariness, also due to the lack of studies on plot designs. Evaluation of experiments with plots sizes of four (Melo ., 2007Melo AS, Costa CX, Brito MEB, Viégas PRA & Silva Junior CD (2007) Produção de mudas de mamoeiro em diferentes substratos e doses de fósforo. Revista Brasileira de Ciências Agrárias, 2:257261.), six (Sá , 2013Sá FVS, Brito MEB, Melo AS, Antônio Neto P, Fernandes POD & Ferreira IB (2013) Produção de mudas de mamoeiro irrigadas com água salina. Revista Brasileira de Engenharia Agrícola e Ambiental, 17:10471054.), 10 (Paixão ., 2012Paixão MVS, Schmildt ER, Mattiello HN, Ferreguetti GA & Alexandre RS (2012) Frações orgânicas e mineral da produção de mudas de mamoeiro. Revista Brasileira de Fruticultura, 34:11051112.; Mengarda ., 2014Mengarda LHG, Lopes JC & Buffon RB (2014) Emergência e vigor de mudas de genótipos de mamoeiro em função da irradiância. Pesquisa Agropecuária Tropical, 44:325333.) and 12 (Serrano ., 2010Serrano LAL, Cattaneo LF & Ferreguetti GA (2010) Adubo de liberação lenta na produção de mudas de mamoeiro. Revista Brasileira de Fruticultura , 32:874883.) seedlings per plot are reported.
The objective of this study was to comparatively determine the optimal size of plots for evaluation of papaya seedlings by the method of maximum curvature of Meier & Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ), by the method of maximum curvature of the coefficient of variation according to Paranaíba . (2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.) and by a new method that incorporates the bootstrap simulation to the method of Meier & Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ).
MATERIAL AND METHODS
The data used in this study were obtained from a greenhouse at the Experimental Farm of CEUNES/UFES in São Mateus, state of Espírito Santo, between parallels 18°40'19.6" South latitude and 39°51'23.7" West longitude. The climate according to Köppen classification is Aw (tropical humid), with rains in summer and dry winter.
The optimal size of plots was determined using papaya seedlings (Carica papaya L.) cv. Golden Pecíolo Curto, whose seeds were obtained from the Caliman Agrícola S. A. company. The blank test was carried out using three black polyethylene trays containing 10x14 tubes of 50 cm^{3}. The trays were allocated together to provide 14 rows of 30 tubes, totaling 420 tubes. All 420 tubes were sown in summer with a single seed, being utilized for evaluating only the seedlings in the eight central rows, corresponding to 240 seedlings. The tubes were filled with Bioplant^{(r)} substrate, adding the slow release fertilizer Basacot mini 3M^{(r)} at a dose of 10 g dm^{}³ substrate (Paixão ., 2012Paixão MVS, Schmildt ER, Mattiello HN, Ferreguetti GA & Alexandre RS (2012) Frações orgânicas e mineral da produção de mudas de mamoeiro. Revista Brasileira de Fruticultura, 34:11051112.).
The characters evaluated 30 days after sowing were as follow: SH: seedling height  determined with the aid of a centimeter graduated ruler, by measuring the base of the stem to the apex of the last leaf; SD: stem diameter  obtained with a digital caliper (mm) measured in the middle region of the stem; NL: number of leaves  counting of full grown leaves; PL: petiole length  obtained by measuring with centimeter graduated ruler from the connection point in the plant to the insertion point on the leaf; and LLR: length of the longest root  determined by measuring from the base of the seedling to its end, with a centimeter graduated ruler.
By using the five characters, it was determined the optimum size of the plot using the following methods: maximum curvature method, according to Meier & Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ); method of maximum curvature of the coefficient of variation according to Paranaíba . (2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.); maximum curvature method according to Meier & Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ) using bootstrap simulation, which is a proposal made in this work.
To determine the optimum plot size by the method of maximum curvature of Meier & Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ), 240 seedlings from the blank trial were structured in basic experimental units (BEU), where each BEU was composed of a seedling. The BEU were grouped using the exact dividing seedling number of the total number of seedlings from the blank trial, ranging from 1 BEU to 60 BEU, providing 12 clusters. For each specific cluster, all the possibilities of clustering composition were evaluated, characterizing different compositions (Table 1).
For each X_{i} BEU, it was calculated: , mean of the plots with Xi UEB in size; , variance among plots with Xi UEB in size; , coefficient of variation among plots with Xi UEB in size; and , variance per BEU among plots of Xi BEU in size. From the cluster of 12 data of Xi and the constants and the regression coefficient were estimated by log transformation of the function weighing it by degrees of freedom associated to the number of applicable plots with X_{i} UEB in size for each size of the designed plot in the uniformity test (Steel ., 1997Steel RGD, Torrie JH & Dickey DA (1997) Principles and procedures of statistics: a biometrical approach. 3ª ed. New York, MacGrawHill Book Companies. 666p.). Similarly, Smith's (1938Smith HF (1938) An empirical law describing heterogeneity in the yields of agricultural crops. Journal of Agricultural Science, 28:0123.) heterogeneity index (b) was estimated from the relationship between and Xi. By using the values of and the optimal size of the plot was calculated given by .
For calculations of the optimal size of the plot by the method of maximum curvature of the coefficient of variation according to Paranaíba et al. (2008Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.), the 240 seedlings of the blank trial received sequential numeration from 1 to 240, in which the five characteristics were measured in these appropriately identified seedlings. From the values of those characteristics, it was determined the sample mean (m), the sample variation (s2) and the estimate of the coefficient of the spatial autocorrelation of the first order (), in which , and where xi is the value observed in the plant i. It was also determined the coefficient of variation, which is given by where Xi indicates the number of BUE. Finally, it was determined the plot optimal size, given by
The proposed method is based on the maximum curvature method, according to Meier & Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ) with the proposed modification of clustering the Xi BEU (Table 1) by bootstrap simulation with replacement (Efron, 1979Efron B (1979) Bootstrap methods: another look at the jackknife. The annals of statistics, 7:0126.; Martinez & Louzada Neto, 2001Martinez EZ & Louzada Neto F (2001) Estimação intervalar via bootstrap. Revista de Matemática e Estatística, 19:217251.).
For the simulations, 12 sample sizes were designed (1, 2, 3, 4, 5, 6, 10, 12, 15, 30 and 60 UEB) for each character. Then, for each designed sample size of each characteristic, 2,000 simulations were performed by resampling with replacement. For each simulated sample, the mean was estimated. Thus, for each sample size of each characteristic, 2,000 mean estimates were obtained (Ferreira, 2009Ferreira DF (2009) Estatística básica. 2ª ed. Lavras, UFLA. 664p.) and from these, a coefficient of variation was obtained for each designed sample size, which we named
From the set of 12 data of Xi and , the constant and the regression coefficient were estimated by log transformation of the function By using the values and , the optimum plot size was calculated given by . The heterogeneity index (b) was calculated by log transformation of the function according to Smith (1938Smith HF (1938) An empirical law describing heterogeneity in the yields of agricultural crops. Journal of Agricultural Science, 28:0123.).
Performance of each of the three methods was demonstrated graphically by the relationship between the coefficients of variation and the number of BEU and the presentation of the optimum plot size. Data were analyzed using the computational resources of the R software (R Development Core Team, 2014R Development Core Team (2014) R: A language and environment for statistical computing. Disponível em: <http://www.Rproject.org/>. Acessado em: 4 de dezembro de 2014.). Because it is a discrete random variable, the optimum plot size was presented by a full number, rounding up to the higher entire number.
Procedure for determining and from bootstrap simulation in software R is described in scrip below for the designed plots with one and two seedlings from the "golden pecíolo curto" file for the characteristic seedling height (SH). For other designed plot sizes (3; 4; 5; 6; 10; 12; 15; 20; 30; 60), it proceeds in a similar manner.
X<read.table("e:\\data\\golden peciolo curto.txt",header=T) # data import
R = 2000 # number of resamplings
boot.means = numeric(R)
for (i in 1:R) { boot.sample = sample(X$AP, 1, replace=T)
boot.means[i] = mean(boot.sample) }
m1<mean(boot.means) #mean
d1<sd(boot.means) # standard deviation
cv1 =(d1*100)/m1 # coefficient of variation
v1<d1^2 # variance
vu1<v1/1^2 # variance per beu
R = 2000 # number of resamplings
boot.means = numeric(R)
for (i in 1:R) { boot.sample = sample(X$AP, 2, replace=T)
boot.means[i] = mean(boot.sample) }
m2<mean(boot.means) #mean
d2<sd(boot.means) # standard deviation
cv2 =(d2*100)/m2 # coeficient of variation
v2<(2*d2)^2 # variance
vu2<v2/2^2 # variance per beu
RESULTS AND DISCUSSION
The results of the analysis of the coefficients of variation in function of the different plot sizes, measured by the number of basic experimental units (BEU) from 240 seedlings for height, stem diameter, number of leaves, petiole length and length of the longest root in papaya seedlings (Carica papaya L.) cv. Golden Pecíolo Curto are shown in Figures 1 to 5, respectively. For all 15 adjusted curves (three methods x five characters), a decrease in the coefficient of variation decreased as the size of the plot increased, a result expected from the statistical point of view (Barbin, 2013Barbin D (2013) Planejamento e análise estatística de experimentos agronômicos. Londrina, Macenas. 213p.). It is noteworthy that, although the method of maximum curvature of the coefficient of variation according to Paranaíba . (2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.) does not require BEU clustering to estimate the optimum plot size, the method allows the determination of the coefficient of variation according to the different sizes of the plots, and therefore it also provides a graphic representation and adjustment of a power model nonlinear regression.
It is observed that the optimum plot size estimated was four seedlings per plot when using seedling height (Figure 1) and stem diameter (Figure 2) and five seedlings per plot when using number of leaves (Figure 3), petiole length (Figure 4), and length of the longest root (Figure 5). Thus, it is pointed the number of five seedlings per plot as optimum size for 'Golden Pecíolo Curto'. Different estimates of sample sizes for different characters of the same plants were also detected in the production of coffee seedlings "Catuaí Amarelo" (Firmino ., 2012Firmino RA, Cogo FD, Almeida SLS, Campos KA & Morais AR (2012) Tamanho ótimo de parcela para experimentos com mudas de café Catuai Amarelo 2SL. Revista Tecnologia e Ciência Agropecuária, 6:0912.) and coffee "Rubi" (Cipriano ., 2012Cipriano PE, Cogo FD, Campos KA & Almeida SLS (2012) Suficiência amostral para mudas de cafeeiro cv. Rubi. Revista Agrogeoambiental, 4:6166.). In papaya, with the lack of scientific results on optimum plot size in the production of seedlings, we verified great variation since experiments in which plots were used with four seedlings (Melo ., 2007Melo AS, Costa CX, Brito MEB, Viégas PRA & Silva Junior CD (2007) Produção de mudas de mamoeiro em diferentes substratos e doses de fósforo. Revista Brasileira de Ciências Agrárias, 2:257261.) up experiments whose parcels contained 12 seedlings (Serrano ., 2010Serrano LAL, Cattaneo LF & Ferreguetti GA (2010) Adubo de liberação lenta na produção de mudas de mamoeiro. Revista Brasileira de Fruticultura , 32:874883.). The use of appropriate plot size in the experiments is crucial for reducing experimental error and a consequent increase in experimental precision (Catapatti ., 2008Catapatti TR, Gonçalves MC, Silva Neto MR & Sobroza R (2008) Tamanho de amostra e número de repetições para avaliação de caracteres agronômicos em milhopipoca. Ciência e Agrotecnologia, 32:855862.). Therefore, the researcher who is using more plants in the plot than the recommended may be spending more than necessary for his or her experimentation with technical, physical or financial resources.
Relationship between coefficient of variation (CV) and size of the designed plot, in UEB and estimates of the optimum size of the plot (X_{0}) by three methods using the trait height of seedlings of papaya (Carica papaya L.) cv. Golden Pecíolo Curto in a trial for uniformity with 240 BEU. Methods: A  maximum curvature according to Meier and Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ); maximum curvature of coefficient of variation according to Paranaíba . (2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.); C  maximum curvature according to Meier and Lessman (1971) using bootstrap simulation.
Relationship between coefficient of variation (CV) and size of the designed plot, in BEU and estimates of the optimum size of the plot (X_{0}) by three methods using the trait stem diameter of papaya seedlings (Carica papaya L.) cv. Golden Pecíolo Curto in a trial for uniformity with 240 UEB. Methods: A  maximum curvature according to Meier and Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ); maximum curvature of coefficient of variation according to Paranaíba . (2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.); C  maximum curvature according to Meier and Lessman (1971) using bootstrap simulation.
Relationship between coefficient of variation (CV) and size of the designed plot, in BEU and estimates of the optimum size of the plot (X_{0}) by three methods using the trait number of leaves papaya tree plantlet (Carica papaya L.) cv. Golden Pecíolo Curto in a trial for uniformity with 240 UEB. Methods: A  maximum curvature according to Meier and Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ); maximum curvature of coefficient of variation according to Paranaíba . (2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.); C  maximum curvature according to Meier and Lessman (1971) using bootstrap simulation.
Relationship between coefficient of variation (CV) and size of the designed plot, in BEU and estimates of the optimum size of the plot (X_{0}) by three methods using the trait length of petiole of papaya seedlings (Carica papaya L.) cv. Golden Pecíolo Curto in a trial for uniformity with 240 BEU. Methods: A  maximum curvature according to Meier and Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ); maximum curvature of coefficient of variation according to Paranaíba . (2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.); C  maximum curvature according to Meier and Lessman (1971) using bootstrap simulation.
Relationship between coefficient of variation (CV) and size of designed plot, in BEU and estimates of the optimum plot size (X_{0}) by means of three methods, using the trait length of the longest root of papaya tree plantlet (Carica papaya L.) cv. Golden Pecíolo Curto in an uniformity trial with 240 BEU. Methods: A  maximum curvature according to Meier and Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ); maximum curvature of coefficient of variation according to Paranaíba . (2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.); C  maximum curvature according to Meier and Lessman (1971) using bootstrap simulation.
When the three methods for determining the sample size are compared, it can be seen that the coefficient values of and and the optimum plot size are similar among the methods of maximum curvature of coefficient of variation and the maximum curvature method with bootstrap simulation to SH ( figures 1B, 1C), SD (figures 2B, 2C), NL (figures 3B, 3C), PL (figures 4B, 4C) and LLR (figures 5B, 5C), and the values of the coefficient get closer to 0.5. Thus, it is clear that the bootstrap simulation with replacement leads to similar results to the method of maximum curvature of coefficient of variation presented by Paranaíba . (2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.), and with the advantage of not needing to identify the sequence of plots in the uniformity test since the bootstrap simulation the drawing is at random.
The means of the characters are presented in Table 2, where it is observed that the mean of the 2,000 estimates by bootstrap present values very close to the real values presented in the methods of Meier & Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ) and Paranaíba . (2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.). This is because the resampling is done thousands of times and the bootstrap technique with replacement allows the same probability of drawing to all values of the sample (Ferreira, 2009Ferreira DF (2009) Estatística básica. 2ª ed. Lavras, UFLA. 664p.).
The values of the coefficient of variation in the evaluated characters also present similarity among each other in the comparison of three methodologies. It is noteworthy that, for the calculation of this statistic by the method proposed by Paranaíba . (2009Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.), the numerator of the equation contains the spatial autocorrelation coefficient of the first order (), which ranges from 1 to +1. Algebraically, when the CV method proposed by Paranaíba et al. (2009)Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268. will have a value close to the CV of the method by Meier & Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ), as it can be observed for the five characters (Table 2). The autocorrelation close to zero indicates random distribution between the seedlings, which is what happened to the five evaluated characters, which can be explained by the fact that each seedling is contained in a different tube.
It can be seen in Table 2 that the heterogeneity index of Smith (1938Smith HF (1938) An empirical law describing heterogeneity in the yields of agricultural crops. Journal of Agricultural Science, 28:0123.), (b) is as twice as the value of the coefficient (Figures 1A, 2A, 3A, 4A, 5A) estimated in the equation that determines the optimum plot size by using the modified maximum curvature method (Meier & Lessmam, 1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ). This relationship, also was reported by Lorentz . (2012Lorentz LH, Erichsen R & Lúcio AD (2012) Proposta de método para estimação de tamanho de parcela para culturas agrícolas. Revista Ceres, 59:772780.), and it can also be verified in the proposed method; however, the values are close to 0.500 and the b values are close to 1.000. Considering that Smith's b values (1938) range from zero to one and that values closer to one indicate heterogeneity in crop environment, it is clear that the method proposed by bootstrap simulation is valuing the maximum of heterogeneity. Santos . (2012Santos D, Haesbaert FM, Lúcio AD, Storck L & Cargnelutti Filho A (2012) Tamanho ótimo de parcela para a cultura do feijãovagem. Revista Ciência Agronômica, 43:119128.) report that when heterogeneity is large, the plots are less related to each other and in this case, they should be larger to obtain the same degree of experimental precision. Thus, it is expected that the proposed method will present optimal plot size equal to or greater than the method for maximum curvature modified by Meier & Lessman (1971Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650. ) and this is interesting from the practical point of view since that method sometimes determines optimal size of plots smaller than a plant (Leite ., 2006Leite MSO, Peternelli LA & Barbosa MHP (2006) Effects of plot size on the estimation of genetic parameters in sugarcane families. Crop Breeding and Applied Biotechnology, 6:4046.), which is a criticism of the method.
Taking as an example the tray model used in this experiment (10x14 = 140 tubes) for further studies, a tray would be enough to allocate 28 plots of 5 seedlings.
CONCLUSIONS
The optimum number of plants per plot for evaluation of 'Golden Pecíolo Curto' papaya seedlings is five.
The method proposed by bootstrap simulation with replacement provides optimum sizes of plots equal to or higher than the method of maximum curvature. It also provides the same size of plot than the method of maximum curvature of the coefficient of variation.
ACKNOWLEDGEMENTS
The authors thank CAPES for granting scholarships and to Caliman Agrícola S.A., for supplying seeds for the study.
REFERENCES
 Barbin D (2013) Planejamento e análise estatística de experimentos agronômicos. Londrina, Macenas. 213p.
 Brito MCM, HumadaGonzález GG, Morais AR & Moreira JM (2014) Avaliação do desempenho do algoritmo de reamostragem bootstrap na verificação da estimação do tamanho ótimo da parcela. Revista da Estatística UFOP, 3:255259.
 Brum B, Brandelero FD, Vargas TO, Storck L & Zanini PPG (2016) Tamanho ótimo de parcela para avaliação da massa e diâmetro de cabeças de brócolis. Ciência Rural, 46:447463.
 Catapatti TR, Gonçalves MC, Silva Neto MR & Sobroza R (2008) Tamanho de amostra e número de repetições para avaliação de caracteres agronômicos em milhopipoca. Ciência e Agrotecnologia, 32:855862.
 Cipriano PE, Cogo FD, Campos KA & Almeida SLS (2012) Suficiência amostral para mudas de cafeeiro cv. Rubi. Revista Agrogeoambiental, 4:6166.
 Efron B (1979) Bootstrap methods: another look at the jackknife. The annals of statistics, 7:0126.
 Federer WT (1977) Experimental design: theory and application. 3ªed. Nova York, Oxford & IBH Publishing. 593p.
 Ferreira DF (2009) Estatística básica. 2ª ed. Lavras, UFLA. 664p.
 Firmino RA, Cogo FD, Almeida SLS, Campos KA & Morais AR (2012) Tamanho ótimo de parcela para experimentos com mudas de café Catuai Amarelo 2SL. Revista Tecnologia e Ciência Agropecuária, 6:0912.
 Leite MSO, Peternelli LA & Barbosa MHP (2006) Effects of plot size on the estimation of genetic parameters in sugarcane families. Crop Breeding and Applied Biotechnology, 6:4046.
 Lorentz LH, Erichsen R & Lúcio AD (2012) Proposta de método para estimação de tamanho de parcela para culturas agrícolas. Revista Ceres, 59:772780.
 Martinez EZ & Louzada Neto F (2001) Estimação intervalar via bootstrap. Revista de Matemática e Estatística, 19:217251.
 Meier VD & Lessman KJ (1971) Estimation of optimum Field plot shape and size for testing yield in Crambe abyssinica Hochst. Crop Science, 11:648650.
 Melo AS, Costa CX, Brito MEB, Viégas PRA & Silva Junior CD (2007) Produção de mudas de mamoeiro em diferentes substratos e doses de fósforo. Revista Brasileira de Ciências Agrárias, 2:257261.
 Melo WB, Pereira RF, Silva MFD, Diniz PF, Gomes RCP, Santos JGR & Andrade R (2009) Variação da produção do mamoeiro Havaí em função de diferentes dosagens e de intervalos de aplicação de biofertilizante. Revista de Biologia e Ciências da Terra, 9:5459.
 Mengarda LHG, Lopes JC & Buffon RB (2014) Emergência e vigor de mudas de genótipos de mamoeiro em função da irradiância. Pesquisa Agropecuária Tropical, 44:325333.
 Paixão MVS, Schmildt ER, Mattiello HN, Ferreguetti GA & Alexandre RS (2012) Frações orgânicas e mineral da produção de mudas de mamoeiro. Revista Brasileira de Fruticultura, 34:11051112.
 Paranaíba PF, Ferreira DF & Morais AR (2009) Tamanho ótimo de parcelas experimentais: proposição de métodos de estimação. Revista Brasileira de Biometria, 27:255268.
 Pratissoli D, Almeida GD, Jesus Júnior WC, Vicentini VB, Holtz AM & Cocheto JG (2007) Fertilizante organomineral e argila silicatada como indutores de resistência à varíola do mamoeiro. Idesia, 25:6367.
 R Development Core Team (2014) R: A language and environment for statistical computing. Disponível em: <http://www.Rproject.org/>. Acessado em: 4 de dezembro de 2014.
 Sá FVS, Brito MEB, Melo AS, Antônio Neto P, Fernandes POD & Ferreira IB (2013) Produção de mudas de mamoeiro irrigadas com água salina. Revista Brasileira de Engenharia Agrícola e Ambiental, 17:10471054.
 Santos D, Haesbaert FM, Lúcio AD, Storck L & Cargnelutti Filho A (2012) Tamanho ótimo de parcela para a cultura do feijãovagem. Revista Ciência Agronômica, 43:119128.
 Serrano LAL, Cattaneo LF & Ferreguetti GA (2010) Adubo de liberação lenta na produção de mudas de mamoeiro. Revista Brasileira de Fruticultura , 32:874883.
 Smith HF (1938) An empirical law describing heterogeneity in the yields of agricultural crops. Journal of Agricultural Science, 28:0123.
 Steel RGD, Torrie JH & Dickey DA (1997) Principles and procedures of statistics: a biometrical approach. 3ª ed. New York, MacGrawHill Book Companies. 666p.
 Storck L, Garcia DC, Lopes SJ & Estefanel V (2011) Experimentação vegetal. 3ª ed. Santa Maria, UFSM. 198p.
 Storck L, Lúcio AD, Krause W, Araújo DV & Silva CA (2014) Scaling the number of plants per plot and number of plots per genotype of yellow passion fruit plants. Acta Scientiarum. Agronomy, 36:7378.
 Vivas M, Silveira SF, Terra CEP & Pereira MG (2011) Testers for combining ability and selection of papaya hybrids resistant to fungal diseases. Crop Breeding and Applied Biotechnology , 11:3642.
Publication Dates

Publication in this collection
JulAug 2016
History

Received
06 Apr 2015 
Accepted
11 Apr 2016