Introduction
The Caricaceae family has a great economic importance in the production of commercial fruits, in which Carica papaya L. stands out in the Brazilian agribusiness. In 2017, Brazil exported approximately US $ 41.35 million of papaya, mainly to the European market (^{Anuário Brasileiro da Fruticultura, 2018}). The Northeastern of Brazil accounts for 59.45% of the national papaya production, for which the state of Bahia contributes with 34.89%, followed by the states of Espírito Santo (29.43%) and Ceará (10.93%) (^{IBGE, 2017}).
In the search for a better performance of papaya cultivations, researchers have pointed out the need of experiments for the genetic improvement, and the launching of new cultivars with desirable agronomic characteristics that meet the demands of the domestic and foreign markets (^{Oliveira et al., 2010}; ^{Dantas et al., 2015}; ^{Luz et al., 2015}). Thus, important aspects related to experimental planning should be considered, in order to obtain reliable research results, such as the optimal size of the plot to be adopted (^{Cargnelutti Filho et al., 2015}).
The determination of the plot size is a relevant matter to use of the experimental design, since its optimal definition attaches a greater precision to the research. Indeed, an optimal size and shape allows of the detection of significant differences between treatments and the reduction of experimental errors (^{Donato et al., 2008}).
The estimation of the optimal plot size is one of the ways to increase the experimental accuracy, and to maximize the information obtained, which gives the opportunity to the researcher to maximize the resource utilization, as well as to gain a greater control and improve the management of the experimental area (^{Silva et al., 2012}). In this sense, the linear model of plateau response (LMPR) has been used to determine the optimum plot size of different crops, as follows: maize hybrids (^{Cargnelutti Filho et al., 2011}), in vitro conservation of passion fruit (^{Peixoto et al., 2011}), pineapple (^{Silva et al., 2012}), pineapple (^{Leonardo et al., 2014}), and sunflower (^{Sousa et al., 2015}).
The linear model of plateau response consists of two segments, in which the first one describes a descending, or ascending, line up to the point that determines the plateau. After this point, the vector assumes a constant value, determining the second segment, in which the dependent variable represents the value of the coefficient of variation, and the independent variable assumes the size values of the plots (^{Paranaiba et al., 2009}).
However, even when working with a single species, the size of the plot may vary according to the soil characteristics, genetic material used, evaluated variables, resources and management adopted in the experiment, and the method applied in the estimation (^{Donato et al., 2008}; ^{Sousa et al., 2015}).
The objective of this work was to determine the optimal size of the experimental plot for the evaluation of the agronomic characteristics and fruit quality of papaya, by the linear model of plateau response, under soil and climatic conditions of the Recôncavo Baiano.
Materials and Methods
The experiment was carried out between August 2015 and March 2017, at the experimental area of Embrapa Mandioca e Fruticultura, in the municipality of Cruz das Almas (12º40'31.62" S, 39º5'24.04"W, at 220 m altitude), in the state of Bahia, Brazil. The climate is an Af type, that is, a warm climate, according to the classification of Köppen, with the coldest month, with temperature above 18°C, and the driest month with precipitation equal to, or greater than 60 mm; the mean annual temperature is 24.5°C, the relative humidity is 82%, and the mean annual rainfall is 1,200 mm (^{Embrapa, 1993}).
The soil is classified as a cohesive yellow Alic Latosol, with a clayey texture and 0 to 3% slope, before the experimental area was cultivated with banana. The chemical characterization of the soil had characteristics described as follows. At 0-20 cm soil depths: 32 g kg^{-1} organic matter; pH (H_{2}O) 6.4; 13 mg dm^{-3} P; 0.41, 2.24, 1.28, and 1.68 cmol_{c} dm^{-3} K, Ca, Mg, and H + Al, respectively; 5.66 cmol_{c} dm^{-3} effective cation exchange capacity; and 70% base saturation (BS). At 20-40 cm soil depths, 17 g kg^{-1} organic matter; pH (H_{2}O) 6.0; 14 mg dm^{-3} P; 0.41, 2.06, 0.94, and 2.44 cmol_{c} dm^{-3} K, Ca, Mg, and H + Al, respectively; 5.66 cmol_{c} dm^{-3} effective cation exchange capacity; and 59% base saturation (BS). After the soil analysis has been performed, the correction and fertilization was proposed according to ^{Oliveira & Coelho (2009)}.
Papaya seed of the L78 lineage were obtained from the Papaya Active Germplasm Bank maintained by Embrapa. Three seed were sown in plastic bags filled with soil. Thinning was done at 15 days after plant emergence, in order to maintain one plant per plastic bag. After 40 days, plants were transplanted to the experimental area, with six plants per hole. In the beginning of flowering, another thinning was performed to keep only one plant showing hermaphrodite inflorescences.
A completely randomized design was used and a uniformity test was carried out. The area was formed by 18 rows of 24 plants each, at 3 x 2 m spacing, in which the 16 central rows with 22 plants was considered useful, totaling 352 plants and floor area of 2,112 m^{2}. Evaluations considered each plant as a basic unit (bu) in 6 m², thus making 352 basic units defined from the blank assay map, whose combinations generated 11 types of plots with different shapes and relative contributions.
Papaya agronomic characteristics and fruit quality were evaluated for the following variables: plant height (m) at six, 12, and 18 months; distance (m) from the ground surface to the insertion point of the newest leaf located in the stem apex at six, 12, and 18 months after planting; insertion height of the first fruit (m) measured from the ground surface and the insertion point of the first functional flower (able to produce fruit), at the start of fruit production; stem diameter (cm) at six, 12, and 18 months after planting, measured at 20 cm above the soil surface; precocity, measured in days, considering the date of planting and the first harvest of fruit; number of marketable fruit per plant at 14 months after planting; productivity (Mg ha^{-1}), by multiplying the number of marketable fruit per plant by the average weight of fruit per plant, considering 3 x 2 m spacing; fruit length, with a pachymeter from the fruit base to the apex; fruit diameter (cm) at the thickest part of the fruit, using a pachymeter; fruit weight (g), consisting of the weight of fruit harvested at maturity stage (¼ mature, having up to 25% of yellow bark) per plant, on an analytical balance. Fruit firmness (kg cm^{-2}) determined, with a penetrometer, from three readings taken at the central region of intact mature fruit; pulp thickness (cm), expressing the thickness of the largest pulp, after the cross-section of fruit, and measured with a pachymeter; diameter of the internal cavity of fruit (cm), measured at the central part of fruit. In case of cavities of a star shape, measurements were taken from one end to another, after selecting the ends at the greatest distance; soluble solids (^{o}Brix), obtained with a portable analog refractometer, model Brix Tester (Reichert Technologies). For the determination of the optimal plot size, the linear model of plateau response (LMPR) was used (^{Paranaiba et al., 2009}) by the equation
in which: CVi corresponds to the coefficient of variation (%) observed in the experiment; X is the size of the plot in basic units; X_{0} is the optimal plot size, represented by the point of intercession for a plateau, relatively to the abscissa; P is the coefficient of variation at the point corresponding to the plateau; β_{0} and β_{1} are the intercept and angular coefficients of the linear segment, respectively; ε is the error, associated with the CVi, considered normally and independently distributed with mean 0 and constant variance
The estimated value of is the optimal plot size (in basic units) that would be recommended for this type of experiment. Thus, the optimal plot size was estimated by the expression
The values of the coefficients of variation (CV), variance, and other necessary determinations were calculated using Microsoft Excel. Equations and model graphs were obtained with the help of the SAEG software (^{Ribeiro Júnior, 2001}), with the dependent variable (CV) and the independent variable, and plot size, in the basic unit for each variable.
Results and Discussion
The coefficients of variation ranged from 0.48 to 55.81% for fruit length and number of marketable fruit per plant in nine months, respectively (Tables 1 and 2).
Form | Dimension | Xbu | Coefficient of variation (%) | ||||||||||
F x P/F | PH6 | PH12 | PH18 | IHFF | SD6 | SD12 | SD18 | PREC | NMF9 | NMF14 | PROD | ||
Plant | 1 x 1 | 1 | 16.06 | 12.37 | 10.33 | 15.21 | 29.49 | 14.21 | 11.73 | 11.22 | 55.81 | 38.14 | 39.89 |
Row | 2 x 1 | 2 | 12.52 | 9.80 | 8.26 | 10.77 | 22.92 | 11.51 | 9.53 | 7.48 | 44.40 | 29.42 | 32.58 |
Row | 4 x 1 | 4 | 9.88 | 7.64 | 6.70 | 8.20 | 17.57 | 9.33 | 7.98 | 5.66 | 36.97 | 24.85 | 28.64 |
Row | 8 x 1 | 8 | 7.14 | 5.04 | 4.92 | 6.06 | 12.22 | 7.44 | 6.47 | 4.36 | 25.74 | 19.53 | 21.39 |
Row | 1 x 11 | 11 | 8.01 | 7.34 | 6.34 | 6.15 | 12.38 | 8.40 | 7.34 | 3.97 | 30.68 | 26.80 | 25.79 |
Row | 16 x 1 | 16 | 5.62 | 4.09 | 2.68 | 4.72 | 8.58 | 4.31 | 3.45 | 3.40 | 16.69 | 13.61 | 14.20 |
Rectangular | 2 x 11 | 22 | 7.31 | 6.16 | 5.54 | 4.27 | 11.24 | 7.77 | 6.72 | 3.37 | 27.86 | 21.28 | 22.71 |
Rectangular | 16 x 2 | 32 | 4.44 | 3.74 | 2.21 | 3.69 | 7.04 | 3.83 | 2.75 | 3.02 | 15.48 | 12.72 | 13.67 |
Rectangular | 2 x 22 | 44 | 4.88 | 4.56 | 4.81 | 3.10 | 7.13 | 6.70 | 6.01 | 1.95 | 25.68 | 19.20 | 20.73 |
Rectangular | 8 x 11 | 88 | 6.11 | 4.11 | 4.34 | 2.40 | 9.47 | 6.79 | 5.82 | 2.97 | 22.50 | 15.74 | 18.75 |
Rectangular | 16 x 11 | 176 | 5.62 | 4.00 | 2.02 | 2.40 | 8.22 | 3.68 | 2.18 | 3.19 | 12.32 | 9.56 | 10.93 |
PH6, plant height at six months; PH12, plant height at 12 months; PH18, plant height at 18 months; IHFF, insertion height of the first fruits; SD6, stem diameter at six months; SD12, stem diameter at 12 months; SD18, stem diameter at 18 months; PREC, precocity; NMF9, number of marketable fruit per plant at nine months; NMF14, number of marketable fruit per plant at 14 months; PROD, productivity; F, row; and P/F, plant per row.
Form | Dimension | Xbu | Coefficient of variation (%) | ||||||
F x P/F | FD | FL | FW | FF | PT | DIC | SS | ||
Plant | 1 x 1 | 1 | 6.63 | 6.15 | 17.14 | 19.87 | 11.53 | 9.38 | 5.50 |
Row | 2 x 1 | 2 | 4.88 | 4.37 | 12.40 | 14.14 | 8.93 | 6.73 | 3.99 |
Row | 4 x 1 | 4 | 3.46 | 3.18 | 8.89 | 10.31 | 7.07 | 4.55 | 2.81 |
Row | 8 x 1 | 8 | 2.67 | 2.60 | 6.99 | 7.85 | 5.31 | 3.16 | 1.98 |
Row | 1 x 11 | 11 | 2.66 | 2.47 | 7.34 | 7.73 | 5.45 | 2.93 | 2.12 |
Row | 16 x 1 | 16 | 2.40 | 2.03 | 5.37 | 5.45 | 4.21 | 2.68 | 1.23 |
Rectangular | 2 x 11 | 22 | 2.32 | 1.92 | 6.32 | 6.21 | 4.88 | 2.31 | 1.52 |
Rectangular | 16 x 2 | 32 | 2.03 | 1.74 | 4.32 | 4.65 | 3.68 | 1.70 | 0.94 |
Rectangular | 2 x 22 | 44 | 1.16 | 1.68 | 4.96 | 4.29 | 3.78 | 1.91 | 1.10 |
Rectangular | 8 x 11 | 88 | 1.88 | 1.38 | 4.92 | 4.68 | 4.41 | 1.30 | 1.08 |
Rectangular | 16 x 11 | 176 | 2.12 | 0.48 | 3.76 | 5.10 | 4.06 | 0.89 | 0.76 |
FD, fruit diameter; FL, fruit length; FW, fruit weight; FF, fruit firmness; PT, pulp thickness; DIC, diameter of the inner cavity; SS, soluble solids; F, row; and P/F, plant per row.
The values of the coefficients of variation generally show a decrease when the value of the plot size increases, however, it is not linear (Tables 1 and 2). This way, parcels with a larger dimension in the perpendicular direction to plant rows, irrespectively of the format, express, in general, smaller coefficients of variation, admitting that the plot shape directly influences the experimental precision. ^{Donato et al. (2008)} and ^{Santos et al. (2016)} calculated the plot size in banana and broccoli, respectively, and observed reductions of the coefficients of variation, with different rates of reduction, and increase of the plot.
The parameters β_{0} and β_{1} and CV (%) showed different values for the characteristics evaluated (Tables 3 and 4). By the LMPR, the optimal plot size is determined at the abscissa of the straight line generated by the linear model, in conjunction with the regression plateau, using the model proposed in the evaluation of rice, sunflower, and pineapple cultivation according to ^{Paranaiba et al. (2009)}, ^{Leonardo et al. (2014)}, and ^{Sousa et al. (2015)}, respectively.
Variable | If | Parameter | X_{0} | ≈ Xbu | ||
CVi | β_{0} | β_{1} | ||||
PH6 | X ≤ X_{0} | 17.39 | -1.96 | 5.75 | 6 | |
X > X_{0} | 6.12 | |||||
PH12 | X ≤ X_{0} | 13.45 | -1.51 | 5.69 | 6 | |
X > X_{0} | 4.86 | |||||
PH18 | X ≤ X_{0} | 9.41 | -0.41 | 13.70 | 14 | |
X > X_{0} | 3.79 | |||||
SD6 | X ≤ X_{0} | 32.16 | -3.79 | 5.98 | 6 | |
X > X_{0} | 9.54 | |||||
SD12 | X ≤ X_{0} | 12.99 | -0.54 | 13.37 | 14 | |
X > X_{0} | 5.75 | |||||
SD18 | X ≤ X_{0} | 10.88 | -0.45 | 13.84 | 14 | |
X > X_{0} | 4.70 | |||||
PREC | X ≤ X_{0} | 12.13 | -1.72 | 5.15 | 6 | |
X > X_{0} | 3.28 | |||||
IHFF | X ≤ X_{0} | 16.49 | -2.18 | 5.67 | 6 | |
X > X_{0} | 4.13 | |||||
NMF9 | X ≤ X_{0} | 59.53 | -5.91 | 6.32 | 7 | |
X > X_{0} | 22.12 | |||||
NMF14 | X ≤ X_{0} | 33.78 | -1.20 | 15.09 | 16 | |
X > X_{0} | 15.70 | |||||
PROD | X ≤ X_{0} | 36.90 | -1.40 | 13.93 | 14 | |
X > X_{0} | 17.36 |
PH6, plant height at six months; PH12, plant height at 12 months; PH18, plant height at 18 months; SD6, stem diameter at six months; SD12, stem diameter at 12 months; SD18, stem diameter at 18 months; PREC, precocity; IHFF, insertion height of the first fruit; NMF9, number of marketable fruit per plant at nine months; NMF14, number of marketable fruit per plant at 14 months; PROD, productivity; X, plot size in basic unit (bu); β_{0} and β_{1}, estimated parameters of the model; X_{0}, greatest plot size; Xbu, greatest plot size rounded to the nearest whole number.
Variable | If | Parameters | X_{0} | ≈ Xbu | ||
CVi | β_{0} | β_{1} | ||||
FD | X ≤ X_{0} | 7.34 | -1.01 | 5.15 | 6 | |
X > X_{0} | 2.16 | |||||
FL | X ≤ X_{0} | 6.75 | -0.93 | 5.31 | 6 | |
X > X_{0} | 1.79 | |||||
FW | X ≤ X_{0} | 18.89 | -2.61 | 5.14 | 6 | |
X > X_{0} | 5.50 | |||||
FF | X ≤ X_{0} | 21.78 | -3.00 | 5.34 | 6 | |
X > X_{0} | 5.75 | |||||
PT | X ≤ X_{0} | 12.47 | -1.41 | 5.67 | 6 | |
X > X_{0} | 4.47 | |||||
DCI | X ≤ X_{0} | 10.47 | -1.54 | 5.44 | 6 | |
X > X_{0} | 2.11 | |||||
SS | X ≤ X_{0} | 6.09 | -0.85 | 5.58 | 6 | |
X > X_{0} | 1.34 |
FD, fruit diameter; FL, fruit length; FW, fruit weight; FF, fruit firmness; PT, pulp thickness; DIC, diameter of the inner cavity; SS, soluble solids; X, plot size in basic unit (bu); β_{0} and β_{1}, estimated parameters of the model; X_{0}, greatest plot size; Xbu, greatest plot size rounded to the nearest whole number.
The estimates of the optimal plot size (X_{0}) for morphoagronomic characters and fruit quality ranged from 5.14 bu, for fruit weight, to X_{0} = 15.09 bu for the number of market fruit by plant at 14 months, corresponding to X_{0} = 7.90 bu general average, approximately 8 bu, resulting in 48 m^{2} area (Tables 3 and 4). This result differs from that found by ^{Schmildt et al. (2016)}, who carried out a field experiment with papaya, using the method of maximum modified curvature and maximum curvature of the coefficient of variation, and concluded that the optimal plot size is six plants. However, this divergence is justified by the differences between the methods used, since the modified maximum curvature method tends to estimate smaller plot sizes than other methods (^{Donato et al., 2008}, ^{2018}). In addition, there is a difference between the genetic materials and crop management applied, as well as to soil and climate conditions to which the studies were subjected (^{Boer et al., 2008}).
The determination coefficients for morphoagronomic characters of fruit quality ranged from r^{2} = 0.6011 to r^{2} = 0.8864 (Figure 1 and 2), which corresponds to the number of marketable fruit per plant at 14 months and plant height at 12 months, respectively.
It is worth noting that the variable number of marketable fruit per plant at 14 months showed a higher value of plot size (15.09 bu), for a lower value of determination coefficients (Table 3); however, this relationship is not direct in the determination of the size (r^{2} = 0.7642 and X_{0} = 13.70 bu) and precocity (r^{2} = 0.7150 and X_{0} = 5.15 bu), and it is also necessary to compare the plot sizes for plant height characters at 18 months, stem diameter at 18 months (r^{2} = 0.8158 and X_{0} = 13.84 bu), and number of marketable fruit per plan at the ninth month (r^{2} = 0.8120 and X_{0} = 6.32 bu), despite the coefficient values of determination often close that of the plot size are different. According to ^{Oliveira et al. (2011)}, the linear model of plateau response do not fit as well, since it is two straight segments to explain an exponential trend, so in this method considered the most important the point of intercession of the two straight lines, which indicates the optimum between the experimental plot size and the precision gain.
From these results, the largest plot size - 15.70 bu (~16 bu), corresponding to 96 m^{2} - should be considered for use as the optimal plot size. However, the overall mean of the evaluated characters, as adopted by ^{Peixoto et al. (2011)} and ^{Sousa et al. (2015)}, in this case of 7.90 bu (~8 bu) corresponding to 48 m^{2}.
It is important to note that the estimated plot sizes should not be observed as the maximum size of the plot, but as the minimum ones to be used, since, if there are necessary resources, it is up to the researcher to adopt any plot size above the minimum value (^{Peixoto et al., 2011}). Therefore, it is possible to use the largest parcel size found, that is, approximately 16 bu.
Conclusions
The experimental plot sizes for papaya plants, estimated using the LMPR model, vary with the evaluated characteristics.
In the soil and climatic conditions of the Recôncavo Baiano, the optimal experimental plot size for papaya cultivation is eight plants for 48 m^{2} area, using 3 m spacing between rows and 2 m between plants.