ABSTRACT:
The aim of this study was to determine the required sample size for estimation of the Pearson coefficient of correlation between cherry tomato variables. Two uniformity tests were set up in a protected environment in the spring/summer of 2014. The observed variables in each plant were mean fruit length, mean fruit width, mean fruit weight, number of bunches, number of fruits per bunch, number of fruits, and total weight of fruits, with calculation of the Pearson correlation matrix between them. Sixty eight sample sizes were planned for one greenhouse and 48 for another, with the initial sample size of 10 plants, and the others were obtained by adding five plants. For each planned sample size, 3000 estimates of the Pearson correlation coefficient were obtained through bootstrap resamplings with replacement. The sample size for each correlation coefficient was determined when the 95% confidence interval amplitude value was less than or equal to 0.4. Obtaining estimates of the Pearson correlation coefficient with high precision is difficult for parameters with a weak linear relation. Accordingly, a larger sample size is necessary to estimate them. Linear relations involving variables dealing with size and number of fruits per plant have less precision. To estimate the coefficient of correlation between productivity variables of cherry tomato, with a confidence interval of 95% equal to 0.4, it is necessary to sample 275 plants in a 250m² greenhouse, and 200 plants in a 200m² greenhouse.
Key words:
Solanum lycopersicum var. cerasiforme; sampling; resampling; bootstrap.
RESUMO:
O objetivo deste trabalho foi determinar o tamanho de amostra necessário para estimar o coeficiente de correlação de Pearson entre variáveis do tomate cereja. Foram instalados dois ensaios de uniformidade em ambiente protegido na primavera/verão de 2014. As variáveis observadas em cada planta foram comprimento médio de fruto, largura média de fruto, peso médio de fruto, número de cachos, número de frutos por cacho, número de frutos e peso total de frutos, sendo calculada a matriz de correlação de Pearson entre elas. Foram planejados 68 tamanhos de amostra em uma estufa e 48 em outra, com tamanho inicial composto de 10 plantas e os demais obtidos acrescentando cinco plantas. Para cada tamanho de amostra planejado foram obtidas 3000 estimativas do coeficiente de correlação de Pearson através de reamostragens “bootstrap” com reposição. O tamanho de amostra de cada coeficiente de correlação foi determinado quando o valor da amplitude do intervalo de confiança de 95% foi menor ou igual a 0,4. A obtenção das estimativas do coeficiente de correlação de Pearson com elevada precisão é difícil para caracteres com relação linear fraca e, consequentemente, maior é o tamanho amostra necessário para estimalos. As relações lineares envolvendo as variáveis relacionadas com o tamanho e o número de frutos por planta tem menor precisão. Para estimar o coeficiente de correlação entre variáveis produtivas do tomate cereja, com intervalo de confiança de 95% igual a 0,4, é necessário amostrar 275 plantas na estufa de 250m², e 200 plantas na estufa de 200m².
Palavraschave:
Solanum lycopersicum var. cerasiforme; amostragem; reamostragem; ”bootstrap”.
INTRODUCTION:
When several variables are measured in an essay, there is a possibility to study the linear relation between them. This type of information is mainly used in plant breeding and involves identifying variables that can be used in indirect selection of superior genotypes (CRUZ & REGAZZI, 1997CRUZ, C.D.; REGAZZI A.J. Modelos biométricos aplicados ao melhoramento genético. Viçosa: UFV, 1997. 390p.). In addition, knowledge of the relation between productivity parameters and total fruit yield may assist in the improvement or choice of management practices.
Pearson correlation coefficient is a dimensionless measure that determines a linear relation between two variables. Its value varies from 1, when there is a perfect negative linear relation, to +1, when there is a perfect positive linear relation. The closer this value to zero, the smaller is the degree of linear relation. From the Pearson correlation coefficient, many other statistics are calculated, such as partial correlation, direct and indirect effects between variables in track analysis, and canonical correlation (HAIR et al., 2005HAIR J.F. et al. Análise multivariada de dados. Porto Alegre: Bookman, 2005. 593p.). Thus, the precision of these statistics depends on accuracy of the estimate of Pearson’s correlation coefficient.
Sample size has a large impact on statistical significance and interpretation of a statistical result. In large samples, the coefficients of low magnitude tend to show statistical significance, even when the relation between the parameters is not important from the practical point of view. Nevertheless, when the sample size is small, the reliability of the estimates is low and may not represent the true relation between two variables (HAIR et al., 2005HAIR J.F. et al. Análise multivariada de dados. Porto Alegre: Bookman, 2005. 593p.; CARGNELUTTI FILHO et al., 2010CARGNELUTTI FILHO, A. et al. Sample size for estimating the Pearson correlation coefficient among corn characters. Pesquisa Agropecuária Brasileira, v.45, n.12, p.13631371, 2010. Available from: http://www.scielo.br/pdf/pab/v45n12/v45n12a05.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S0100 204X2010001200005.
http://www.scielo.br/pdf/pab/v45n12/v45n...
; CARGNELUTTI FILHO et al., 2011CARGNELUTTI FILHO, A. et al. Sample size to estimate the Pearson correlation coefficient among characters of Crambe abyssinica. Revista Ciência Agronômica, v.42, n.1, p.149158, 2011. Available from: http://www.scielo.br/pdf/rca/v42n1/v42n1a19.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S1806 66902011000100019.
http://www.scielo.br/pdf/rca/v42n1/v42n1...
; CARGNELUTTI FILHO et al., 2012CARGNELUTTI FILHO, A. et al. Sample size to estimate the Pearson correlation coefficient among characters of castor bean. Semina: Ciências Agrárias, v.33, p.953962, 2012. Available from: http://www.scielo.br/pdf/pab/v45n12/v45n12a05.pdf>. Accessed: Feb. 18, 2017. doi: 10.5433/1679 0359.2012v33n3p953.
http://www.scielo.br/pdf/pab/v45n12/v45n...
). Hence, we realized that an adequate sample size with acceptable accuracy is important.
The confidence interval of the Pearson correlation coefficient can be estimated from the sampling distribution transformed by Fischer (FERREIRA, 2009FERREIRA, D.F. Estatística básica. 2.ed. Lavras: Universidade Federal de Lavras, 2009. 664p.; CARGNELUTTI FILHO et al., 2011CARGNELUTTI FILHO, A. et al. Sample size to estimate the Pearson correlation coefficient among characters of Crambe abyssinica. Revista Ciência Agronômica, v.42, n.1, p.149158, 2011. Available from: http://www.scielo.br/pdf/rca/v42n1/v42n1a19.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S1806 66902011000100019.
http://www.scielo.br/pdf/rca/v42n1/v42n1...
). Another way to obtain it is by means of bootstrap confidence intervals with replacement. In this methodology, there is no need to know the variable’s probability distribution, and this approach is effective in the case of variables with an unknown or non normal distribution (CARGNELUTTI FILHO et al., 2010CARGNELUTTI FILHO, A. et al. Sample size for estimating the Pearson correlation coefficient among corn characters. Pesquisa Agropecuária Brasileira, v.45, n.12, p.13631371, 2010. Available from: http://www.scielo.br/pdf/pab/v45n12/v45n12a05.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S0100 204X2010001200005.
http://www.scielo.br/pdf/pab/v45n12/v45n...
; CARGNELUTTI FILHO et al., 2012CARGNELUTTI FILHO, A. et al. Sample size to estimate the Pearson correlation coefficient among characters of castor bean. Semina: Ciências Agrárias, v.33, p.953962, 2012. Available from: http://www.scielo.br/pdf/pab/v45n12/v45n12a05.pdf>. Accessed: Feb. 18, 2017. doi: 10.5433/1679 0359.2012v33n3p953.
http://www.scielo.br/pdf/pab/v45n12/v45n...
). This methodology is of great value because it can be used to determine the confidence interval amplitude of any variable and for any cultivated plant. The sample size required for estimation of the mean of parameters is common in the literature, but there are few studies that determine the sample size needed to estimate the Pearson correlation coefficient (CARGNELUTTI FILHO et al., 2010CARGNELUTTI FILHO, A. et al. Sample size for estimating the Pearson correlation coefficient among corn characters. Pesquisa Agropecuária Brasileira, v.45, n.12, p.13631371, 2010. Available from: http://www.scielo.br/pdf/pab/v45n12/v45n12a05.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S0100 204X2010001200005.
http://www.scielo.br/pdf/pab/v45n12/v45n...
; CARGNELUTTI FILHO et al., 2011CARGNELUTTI FILHO, A. et al. Sample size to estimate the Pearson correlation coefficient among characters of Crambe abyssinica. Revista Ciência Agronômica, v.42, n.1, p.149158, 2011. Available from: http://www.scielo.br/pdf/rca/v42n1/v42n1a19.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S1806 66902011000100019.
http://www.scielo.br/pdf/rca/v42n1/v42n1...
; CARGNELUTTI FILHO et al., 2012CARGNELUTTI FILHO, A. et al. Sample size to estimate the Pearson correlation coefficient among characters of castor bean. Semina: Ciências Agrárias, v.33, p.953962, 2012. Available from: http://www.scielo.br/pdf/pab/v45n12/v45n12a05.pdf>. Accessed: Feb. 18, 2017. doi: 10.5433/1679 0359.2012v33n3p953.
http://www.scielo.br/pdf/pab/v45n12/v45n...
). In the specific case of olive groves, studies determining a sample size for estimation of the Pearson correlation coefficient are practically nonexistent. Studies are limited to only determine a sample size for the mean of parameters (SILVA et al., 2009SILVA, G.O. da et al. Sample size for evaluation of carrot traits in agroecologic cultivation systems. Horticultura Brasileira , v.27, n.2, p.166170, 2009. Available from: http://www.scielo.br/pdf/hb/v27n2/v27n2a08.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S0102 05362009000200008.
http://www.scielo.br/pdf/hb/v27n2/v27n2a...
; SANTOS et al., 2010SANTOS, D. et al. Sample sufficiency for lettuce grown in different environments. Ciência Rural , v.40, n.4, p.800805, 2010. Available from: http://www.scielo.br/pdf/cr/v40n4/a554cr1846.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S0103 84782010000400009.
http://www.scielo.br/pdf/cr/v40n4/a554cr...
; HAESBAERT et al., 2011HAESBAERT, F.M. et al. Sample size for experiments with bean pods in different environments. Ciência Rural, v.41, n.1, p.3844, 2011. Available from: http://www.scielo.br/pdf/cr/v41n1/a834cr3400.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S0103 84782011000100007.
http://www.scielo.br/pdf/cr/v41n1/a834cr...
; SILVA et al., 2011SILVA, A.R. et al. Sample size for morphological characterization of pepper fruits. Horticultura Brasileira , v.29, n.1, p.125129, 2011. Available from: http://www.scielo.br/pdf/hb/v29n1/22.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S010205362011000100022.
http://www.scielo.br/pdf/hb/v29n1/22.pdf...
; LÚCIO et al., 2012LÚCIO A.D. et al. Sample size and plot size for growth and productivity characteristics of tomato. Horticultura Brasileira, v.30 n.4, p.660668, 2012. Available from: http://www.scielo.br/pdf/hb/v30n4/v30n4a16.pdf>. Accessed: Feb. 18, 2017.
http://www.scielo.br/pdf/hb/v30n4/v30n4a...
).
The appropriate sample size will be related to the accuracy of the estimate of Pearson’s correlation coefficient. In the case of cherry tomatoes, there are no studies in the literature that make this determination. Consequently, the aim of the present study was to determine the sample size for estimation of the Pearson correlation coefficient between productivity variables of cherry tomato.
MATERIALS AND METHODS:
Two uniformity tests were carried out in plastic greenhouses in the Plant Science Department of the Universidade Federal de Santa Maria (latitude 29°43’ S, longitude 53°43’ W and 95m altitude). Greenhouses were covered with a low density polyethylene (LDPE) film, with 150 micron thickness and an anti UV additive, placed in the northsouth direction, and the greenhouses had the following dimensions: 1) 20 × 10m (200m²), 3m right foot and 4m in the central part; 2) 25 10m (250m²), 4m right foot and 5.5m in the central part. Climate at the site of the experiments is classified as Cfa, and the soil is classified from Paleudalf (EMBRAPA, 2006EMBRAPA. Sistema brasileiro de classificação de solos. Rio de Janeiro: Centro Nacional de Pesquisa de Solos, 2006. 306p.).
A soil correction in the two greenhouses was performed 30 days before planting and served to raise pH to 6.5 and the phosphorus level to 300mg dm^{3}. Next, eight ridges were constructed in each greenhouse, with 0.20m height and 0.30m width, and spaced at 1m. These ridges were covered with a mulching black opaque LDPE film. Planting fertilization was conducted by means of 150kg ha^{1} N, 250kg ha^{1} P, and 125kg ha^{1} K for each greenhouse. The cover fertilization was performed every 23 days, with 30kg ha^{1} N, 15kg ha^{1} P, and 30kg ha^{1} K. All the procedures were carried out based on soil chemical analysis and according to recommendations of the Official Network of Soil and Plant Tissue Analysis Laboratories of the states of Rio Grande do Sul and Santa Catarina (ROLAS, 2004ROLAS (REDE OFICIAL DE LABORATÓRIOS DE ANÁLISE DE SOLO E DE TECIDO VEGETAL DOS ESTADOS DO RIO GRANDE DO SUL E DE SANTA CATARINA). Recomendações de adubações e de calagem para os estados do Rio Grande do Sul e Santa Catarina. Passo Fundo: SBCS Núcleo Regional Sul, 2004. 224p.).
The seedlings were Lily Hybrid Cherry tomatoes, and the transplant took place in the spring/summer season, on October 26, 2014, during the first flowering. Spacing between seedlings was 0.5m. Plants were vertically staked with a ribbon and driven on a double stem with elimination of lateral shoots. Every 14 days, sprays of calcium, boron, fungicides, and insecticides were administered to control pests, diseases, and physiological disturbances such as the “blossom end rot” of fruits.
All the plants in both greenhouses were evaluated, totaling 247 plants in the 200m² greenhouse and 347 in the 250m² greenhouse. All the fruits were harvested, and the following variables were determined: total weight fruit (TWF), mean fruit length (MFL), mean fruit width (MFWi), mean fruit weight (MFW), number of bunches per plant (NBP), number of fruits per plant (NFP), and the number of fruits per bunch (NFB). Variables MFWi and MFL were measured in centimeters with a caliper. MFW and TWF were measured in grams using a scale with 0.01g precision.
From the observed data, two Pearson correlation matrices were constructed, one for each for greenhouse. The correlation coefficients were tested for their significance by Student’s t test at an error probability of 5%. The third matrix (matrix of means) was compiled from the mean values of the Pearson correlation coefficients of the two matrices previously mentioned. A total of 68 sample sizes were planned for the 250m² greenhouse and 48 for the 200m² greenhouse. The initial size was 10 plants, and the others were calculated by adding five plants. For each of them, 3000 estimates of the Pearson correlation coefficient were obtained through bootstrap resampling with replacement. Later, from the 3000 estimates, we calculated the minimum value, 2.5% percentile, mean, 97.5% percentile, maximum value, and the 95% confidence interval (the difference between the 97.5% and 2.5% percentiles). The confidence interval, which represents the variation caused by the variable’s random behaviour can be represented by 1000 resamplings; the larger the number of resamplings, the more accurate is the interval (FERREIRA, 2009FERREIRA, D.F. Estatística básica. 2.ed. Lavras: Universidade Federal de Lavras, 2009. 664p.). Then, the use of the 3000 resamplings had the objective to obtain intervals with high precision.
The amplitude of the confidence interval determines variability of the correlation coefficients estimated by means of different sample sizes. The optimal sample size to estimate the Pearson correlation coefficient was determined when the amplitude value of the 95% bootstrap confidence interval was less than or equal to 0.4 (Figure 1a). We also calculated the mean of each of the 21 sample size estimates obtained for each of the two matrices. Data processing and statistical analysis were conducted using the R software (R DEVELOPMENT CORE TEAM, 2012R Development Core Team. R: a language and environment for statistical computing. Viena: R Foundation for Statistical Computing, 2016. Available from: http://www.Rproject.org>. Accessed: Feb. 18, 2017.
http://www.Rproject.org...
) and Microsoft Office Excel^{®}.
Maximum, 97.5% percentile (LS 97.5%), mean, 2.5% percentile (LI 2.5%), and the minimum for 3000 “bootstrap” estimates of the Pearson coefficient of correlation between the total weight of fruits and the mean length of fruits (A), between the mean width of a fruit and the number of fruits per plant (B), between total fruit weight and number of fruits per bunch (C), and between total fruit weight and number of fruits per plant (D) in the 250m² greenhouse.
RESULTS AND DISCUSSION:
The Pearson correlation coefficients of the 21 pairs of parameters varied between 0.019 and 0.97 in the 200m² greenhouse and between 0.083 and 0.963 in the 250m² greenhouse. Of these, 20 had statistical significance in the 200m² greenhouse and 12 in the 250m² greenhouse (Table 1). We reported that the Pearson correlation coefficients, even at low magnitude, were statistically significant, as a consequence of the large number of observations (347 in the 250m² greenhouse and 247 in the 200m² greenhouse). This finding should be interpreted with caution because a statistical test often indicates the presence of a linear relation between variables, when in fact this relation is of no practical importance (HAIR et al., 2005HAIR J.F. et al. Análise multivariada de dados. Porto Alegre: Bookman, 2005. 593p.; CARGNELUTTI FILHO et al., 2010CARGNELUTTI FILHO, A. et al. Sample size for estimating the Pearson correlation coefficient among corn characters. Pesquisa Agropecuária Brasileira, v.45, n.12, p.13631371, 2010. Available from: http://www.scielo.br/pdf/pab/v45n12/v45n12a05.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S0100 204X2010001200005.
http://www.scielo.br/pdf/pab/v45n12/v45n...
; CARGNELUTTI FILHO et al., 2011CARGNELUTTI FILHO, A. et al. Sample size to estimate the Pearson correlation coefficient among characters of Crambe abyssinica. Revista Ciência Agronômica, v.42, n.1, p.149158, 2011. Available from: http://www.scielo.br/pdf/rca/v42n1/v42n1a19.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S1806 66902011000100019.
http://www.scielo.br/pdf/rca/v42n1/v42n1...
; CARGNELUTTI FILHO et al., 2012CARGNELUTTI FILHO, A. et al. Sample size to estimate the Pearson correlation coefficient among characters of castor bean. Semina: Ciências Agrárias, v.33, p.953962, 2012. Available from: http://www.scielo.br/pdf/pab/v45n12/v45n12a05.pdf>. Accessed: Feb. 18, 2017. doi: 10.5433/1679 0359.2012v33n3p953.
http://www.scielo.br/pdf/pab/v45n12/v45n...
).
The association between fruit size variables (MFL, MFWi, and MFW) and the number of fruits per plant (NBP, NFP, and NFB) differed between the greenhouses, resulting in different Pearson correlation coefficients, both in magnitude and sign. Hence, the use of a matrix mean may not provide an adequate Pearson correlation coefficient; and consequently, the sample size calculated from it may not be reliable. Thus, we chose to determine the sample size for each of the trials separately (Table 1).
The amplitudes of the 95% confidence intervalobtained from 3000 bootstrap resamplings with replacementrevealed that the Pearson correlation coefficient estimate is more accurate for variables with a strong linear relation than for those with a weak linear relation. With the increasing sample size, the confidence interval amplitude is still high in large samples when the correlation coefficient is close to zero (Figure 1). The estimated sample sizes for the 21 correlation coefficients were inversely proportional to the magnitude of the Pearson correlation coefficient in both greenhouses (r = 0.95 in the 250m² greenhouse and r = 0.91 in the 200m^{2} greenhouse), confirming that the weaker the linear relation between variables (Pearson correlation coefficients close to zero), the larger is the sample size needed for its estimation.
A suitable sample size is necessary, especially when the correlation coefficient is close to zero, because there is great variability of magnitude and sign inversion (Table 2); this situation definitely compromises the statistical analysis. CARGNELUTTI FILHO et al. (2010CARGNELUTTI FILHO, A. et al. Sample size for estimating the Pearson correlation coefficient among corn characters. Pesquisa Agropecuária Brasileira, v.45, n.12, p.13631371, 2010. Available from: http://www.scielo.br/pdf/pab/v45n12/v45n12a05.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S0100 204X2010001200005.
http://www.scielo.br/pdf/pab/v45n12/v45n...
), after studying the linear relation between corn parameters, reported that an inadequate sample size may be linked to the discrepancies in results of scientific publications. Because of significance of the mathematical sign for interpretation of the Pearson correlation coefficient, the use of a suitable sample size for its determination is highly relevant, even more than the required sample size for mean estimation.
In the case of cherry tomatoes, we reported that the use of small samples is even more detrimental for the study of linear relations involving fruit size variables (MFL, MFWi, and MFW) and the number of fruits per plant (NBP, NFP, and NFB). For example, the estimate of the correlation between variables MFL and NBP obtained in the 250m² and 200m² greenhouses was 0.083 and 0.133, respectively. When the sample consists of only 10 plants, the amplitude of the 95% confidence interval of the 3000 bootstrap estimates with replacement ranged from 0.744 to 0.657 in the 250m² greenhouse and from 0.699 to 0.764 in the 200m² greenhouse. These results showed that at reduced sample sizes, the inferences about relations of variables can be contradictory, compromising interpretation of the results. This problem did not occur between variables TWF and NFP. In this case, the correlation coefficients obtained in the 250m² and 200m² greenhouses were 0.963 and 0.970, respectively. The 95% confidence interval of the estimates varied between 0.810 and 0.995 in the 250m² greenhouse, and between 0.864 and 0.995 in the 200m² greenhouse (Table 2).
The sample size needed to estimate the Pearson coefficient of correlation between productivity variables of cherry tomato, with a 95% confidence interval equal to 0.4, varied between 10 and 275 in the 250m² greenhouse and between 10 and 200 in the 200m² greenhouse (Table 1). Therefore, a sample size of 275 plants in the 250m² greenhouse and 200 plants in the 200m² greenhouse allows for estimating the coefficient of correlation between the cherry tomato’s productivity variables with a 95% confidence interval of at least 0.4, regardless of the relation between the parameters being studied. The use of this amplitude of the confidence interval to determine a sample size is justified because at this value, the amplitude tends to stabilize when the correlation coefficient is of low magnitude (Figure 1a).
Sample sizes determined by means of the same amplitude of a confidence interval have different precision values (this value is greater in the correlation coefficients of greater magnitude than in those with lower magnitude). Hence, it is important to highlight that if a researcher wants to estimate the Pearson correlation coefficient with the same precision, the sample size for each analysis should be different. If the same sample size is used, the precision estimation will not be the same. The better amplitude definition of the 95% confidence interval will be selected by each researcher according to the experimental precision of its estimates.
CONCLUSION:
The sample size should be larger for determining a linear relation of the variables associated with the size and number of fruits per plant, owing to the low magnitude of this correlation.
To estimate the Pearson coefficients of correlation between cherry tomato variables with a 95% confidence interval amplitude equal to 0.4, it is necessary to sample 275 plants in the 250m² greenhouse and 200 plants in the 200m^{2} greenhouse.
REFERENCES:
 CARGNELUTTI FILHO, A. et al. Sample size to estimate the Pearson correlation coefficient among characters of castor bean. Semina: Ciências Agrárias, v.33, p.953962, 2012. Available from: http://www.scielo.br/pdf/pab/v45n12/v45n12a05.pdf>. Accessed: Feb. 18, 2017. doi: 10.5433/1679 0359.2012v33n3p953.
» http://www.scielo.br/pdf/pab/v45n12/v45n12a05.pdf  CARGNELUTTI FILHO, A. et al. Sample size for estimating the Pearson correlation coefficient among corn characters. Pesquisa Agropecuária Brasileira, v.45, n.12, p.13631371, 2010. Available from: http://www.scielo.br/pdf/pab/v45n12/v45n12a05.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S0100 204X2010001200005.
» http://www.scielo.br/pdf/pab/v45n12/v45n12a05.pdf  CARGNELUTTI FILHO, A. et al. Sample size to estimate the Pearson correlation coefficient among characters of Crambe abyssinica Revista Ciência Agronômica, v.42, n.1, p.149158, 2011. Available from: http://www.scielo.br/pdf/rca/v42n1/v42n1a19.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S1806 66902011000100019.
» http://www.scielo.br/pdf/rca/v42n1/v42n1a19.pdf  CRUZ, C.D.; REGAZZI A.J. Modelos biométricos aplicados ao melhoramento genético. Viçosa: UFV, 1997. 390p.
 EMBRAPA. Sistema brasileiro de classificação de solos. Rio de Janeiro: Centro Nacional de Pesquisa de Solos, 2006. 306p.
 FERREIRA, D.F. Estatística básica. 2.ed. Lavras: Universidade Federal de Lavras, 2009. 664p.
 HAESBAERT, F.M. et al. Sample size for experiments with bean pods in different environments. Ciência Rural, v.41, n.1, p.3844, 2011. Available from: http://www.scielo.br/pdf/cr/v41n1/a834cr3400.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S0103 84782011000100007.
» http://www.scielo.br/pdf/cr/v41n1/a834cr3400.pdf  HAIR J.F. et al. Análise multivariada de dados. Porto Alegre: Bookman, 2005. 593p.
 LÚCIO A.D. et al. Sample size and plot size for growth and productivity characteristics of tomato. Horticultura Brasileira, v.30 n.4, p.660668, 2012. Available from: http://www.scielo.br/pdf/hb/v30n4/v30n4a16.pdf>. Accessed: Feb. 18, 2017.
» http://www.scielo.br/pdf/hb/v30n4/v30n4a16.pdf  R Development Core Team. R: a language and environment for statistical computing. Viena: R Foundation for Statistical Computing, 2016. Available from: http://www.Rproject.org>. Accessed: Feb. 18, 2017.
» http://www.Rproject.org  ROLAS (REDE OFICIAL DE LABORATÓRIOS DE ANÁLISE DE SOLO E DE TECIDO VEGETAL DOS ESTADOS DO RIO GRANDE DO SUL E DE SANTA CATARINA). Recomendações de adubações e de calagem para os estados do Rio Grande do Sul e Santa Catarina. Passo Fundo: SBCS Núcleo Regional Sul, 2004. 224p.
 SANTOS, D. et al. Sample sufficiency for lettuce grown in different environments. Ciência Rural , v.40, n.4, p.800805, 2010. Available from: http://www.scielo.br/pdf/cr/v40n4/a554cr1846.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S0103 84782010000400009.
» http://www.scielo.br/pdf/cr/v40n4/a554cr1846.pdf  SILVA, A.R. et al. Sample size for morphological characterization of pepper fruits. Horticultura Brasileira , v.29, n.1, p.125129, 2011. Available from: http://www.scielo.br/pdf/hb/v29n1/22.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S010205362011000100022.
» https://doi.org/10.1590/S010205362011000100022.» http://www.scielo.br/pdf/hb/v29n1/22.pdf  SILVA, G.O. da et al. Sample size for evaluation of carrot traits in agroecologic cultivation systems. Horticultura Brasileira , v.27, n.2, p.166170, 2009. Available from: http://www.scielo.br/pdf/hb/v27n2/v27n2a08.pdf>. Accessed: Feb. 18, 2017. doi: 10.1590/S0102 05362009000200008.
» http://www.scielo.br/pdf/hb/v27n2/v27n2a08.pdf

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2017
History

Received
20 Feb 2017 
Accepted
18 July 2017 
Reviewed
22 Aug 2017