Introduction

Crotalaria species, such as C. juncea, are used as cover plants in crop rotation systems with high production of fresh matter (^{Chaudhary, 2016}) and nitrogen supply to the subsequent crops, positively influencing plant growth and productivity (^{Diniz et al., 2017}; ^{Elsaid & Silva, 2017}). Other species of crotalaria, such as C. spectabilis, C. breviflora, and C. ochroleuca, can reduce the incidence or pest infestation, diseases and nematodes (^{Deberdt et al., 2015}; ^{Braz et al., 2016}; ^{Reigada et al., 2016}).

Although crotalaria species are of agronomic importance, their genetic improvement is still incipient (^{Bhandari et al., 2016}). In plant breeding programs, it is important to know the linear relationships of traits, mainly when the simultaneous selection of traits is desired, or when the main trait has low heritability, or is difficult to measure (^{Cruz et al., 2012}). The linear relationships between traits can be evaluated with the Pearson’s linear correlation coefficients (r), in the range of -1 ≤ r ≤ 1, in which the intensity of the linear correlation is larger when r is closer to |1| (^{Ferreira, 2009}).

Complementary studies can be performed from the correlation coefficients for the definition of cause and effect relationships, and indirect selection of plants (^{Cruz et al., 2012}). In this sense, if a given correlation matrix is estimated from an insufficient sample size, it is likely that the diagnosis of the multicollinearity by the different indicators will be biased or questionable. In addition, complementary analyses of a correlation matrix - such as partial correlation analysis, path analysis, and canonical correlation analysis - could generate biased coefficients. Also, the principal components analysis from a correlation matrix could generate biased eigenvalues and eigenvectors. Finally, any other statistical procedure, besides those mentioned, performed from an estimated correlation matrix with low precision can generate unreliable results. Therefore, if the sample size for the estimation of the correlations is insufficient, all subsequent analyses may be biased, or not compatible, with the behavior at the population level.

Given the importance of knowing the linear relations between traits, it is necessary to define the sample size to be used for the estimation of correlation coefficients. In this sense, ^{Cargnelutti Filho et al. (2010)} and ^{Toebe et al. (2015)} defined the sample size for the estimation of r in single, triple, and double corn hybrids. ^{Toebe et al. (2015)} verified that the sample size varies among corn hybrids, crops, and pairs of traits, and that a larger sample size is required to estimate the correlation coefficients between weakly correlated traits and vice-versa, in agreement with that established in studies by ^{Bonett & Wright (2000)} and ^{Olivoto et al. (2018}). The sample size to estimate the Pearson’s correlation coefficients was also performed at precision levels in other agricultural crops, such as crambe (Crambe abyssinica) (^{Cargnelutti Filho et al., 2011}), castor bean (Ricinus communis) (^{Cargnelutti Filho et al., 2012}), and cherry tomato (Solanum lycopersicum 'Cerasiforme') (^{Sari et al., 2017}). A study was recently developed to evaluate the influence of sample size and magnitude of correlation on the confidence interval width for Pearson’s correlation coefficients, with real and simulated data (^{Olivoto et al., 2018}). According to ^{Kozak et al. (2012)}, it should be noted that if the correlation coefficient is estimated from a small sample size, the confidence interval for the population correlation will be very wide, and the interpretations will have little precision.

Studies of sample size for crotalaria species have been already carried out for the estimation of the mean and coefficient of variation (^{Toebe et al., 2017}, ²⁰¹⁸), but we did not find in the literature, studies on sample size for the estimation of correlation coefficients in this genus. It is likely that the sample size varies between species of crotalaria, and between pairs of traits of certain species.

The objective of this work was to determine the sample size necessary to estimate the Pearson’s linear correlation coefficients for four species of crotalaria at precision levels.

Materials and Methods

Four uniformity trials - blank experiments, that is, without treatments - were carried out in the season of 2014/2015, in the experimental area of Universidade Federal do Pampa, campus Itaqui, located in the municipality of Itaqui (29º 09' 25" S, 56º 33' 16" W, at 74 m altitude), in the state of Rio Grande do Sul, Brazil. According to the classification of Köppen-Geiger, the climate of the region is Cfa type, humid subtropical with hot summers, and without a defined dry season (^{Wrege et al., 2012}); its soil is classified as a Plintossolo Háplico (^{Santos et al., 2013}), i.e., a Haplic Plinthosol. Each one of the four species of crotalaria - C. juncea, C. spectabilis, C. breviflora and C. ochroleuca - was allocated in a uniformity trial area of 65.61 m² (8.1 m length × 8.1 m width), treated with fertilizer at 25 kg ha^-1 N, 100 kg ha^-1 P₂O₅, and 100 kg ha^-1 K₂O.

The four species were sown in October 2014, with 0.45 m spacing between rows and 27, 33, 33, and 44 seed m of the row, respectively, for C. juncea, C. spectabilis, C. breviflora, and C. ochroleuca. The other cultural treatments were carried out in a uniform way within the sample area. In the period from March to June 2015, successive harvests of pods were held randomly, in accordance with the productive cycle of each species. From each species, 1,000 pods were collected and, in each pod, the following traits were evaluated: mass of pod with seed (MPWS), mass of pod without seed (MPWOS), length of pod (LP), width of pod (WP), height of pod (HP), number of seed per pod (NSP), mass of seed per pod (MSP = MPWS - MPWOS) and mass of one hundred seed (MHS = MSP × 100/NSP). More details of the conduction of this experiment were described by ^{Toebe et al. (2018)}.

Pearson’s linear correlation coefficient (r) was calculated for each species of crotalaria in the 28 pairs of traits, and the significance of r was checked out by Student’s t-test, at 5% probability. The sample size was obtained via resampling with the replacement technique, which is considered adequate for conditions in which the distribution of the data is not known (^{Ferreira, 2009}). In this sense, 199 sample sizes were planned, that is, the smallest sample size of 10 pods, and the other sample sizes obtained with the addition of five pods, in such a way that the planned sample sizes were n = 10, 15, 20, ..., 1,000 pods. For each planned sample size of each species, 10,000 resamples with replacements were obtained and, in each resample, r of each of the 28 pairs of traits were estimated. Based on the 10,000 estimates, the percentile 2.5^th, the mean, and the percentile 97.5^th were determined. The amplitude of the 95% confidence interval was calculated (CI_95%) by the difference between the percentile 97.5^th and the percentile 2.5^th.

To determine the sample size (number of pods) required for the r estimation from each of the 28 pairs of traits, in each species, CI_95% of r was initially set as equal to 0.10 (higher precision), 0.20, 0.30, and 0.40 (lower precision). The optimal sample size (n) was considered as the minimum number of pods from which CI_95% of r was less or equal to the limit for each precision level (0.10, 0.20, 0.30 or 0.40), as previously described by ^{Cargnelutti Filho et al. (2010)}, ^{Toebe et al. (2015)}, and ^{Olivoto et al. (2018)}. Statistical analyses were performed with the aid of the program R (^{R Core Team, 2018}) and the Microsoft Office Excel.

Results and Discussion

Only for two (LP×MHS and WP×HP) of the 28 pairs of traits in C. juncea, the correlations were not significant (Table 1). In C. spectabilis, C. breviflora, and C. ochroleuca all trait pairs showed significant correlations. Thus, out of the 112 evaluated correlation (4 species × 28 pairs of traits), 110 were significant at 5% probability. It is important to observe the practical significance, since the high original sample size (1,000 pods) causes low-magnitude correlations to become significant. As highlighted by ^{Hair Jr. et al. (2009)}, the practical significance indicates whether the result is useful or not to achieve the research objectives. In this sense, ^{Mukaka (2012)} emphasizes that the misuse of correlation is common among researchers. According to ^{Kozak (2008)} and ^{Kozak et al. (2012)}, very small correlation coefficients can be statistically significant, when a large sample size is used and vice-versa. According to these authors, significance merely suggests the presence of a nonzero population correlation coefficient, not necessarily an important correlation.

Adopting the classification of the correlation coefficient with practical magnitude proposed by ^{Hinkle et al. (2003)}, in all species of crotalaria, the correlation between MPWS×MSP was very high (0.90 to 1.00) (Table 1). The very high correlation between these two variables is expected, since MSP is obtained from the difference between MPWS and MPWOS, that is, the smaller the MPWOS interference, the greater the association between MSP and MPWS. In C. spectabilis, a very high correlation was also observed between NSP×MSP. A high and positive correlation (0.70 to 0.90) was found between the following trait pairs: MPWS×NSP and NSP×MSP in C. juncea; MPWS×MPWOS, MPWS×WP, MPWS×NSP, MPWOS×LP, and MPWOS×WP, in C. spectabilis; MPWS×MPWOS, MPWS×NSP, and NSP×MSP, in C. breviflora; and between MPWS×MPWOS, MPWOS×LP, and NSP×MSP, in C. ochroleuca (Figures 1 and 2).

Figure 1. Matrix with a histogram frequency (in diagonal) and dispersion graphs between mass of pod with seed (MPWS, g), mass of pods without seed (MPWOS, g), length of pod (LP, mm), width of pod (WP, mm), height of pod (HP, mm), number of seed per pod (NSP, unity), mass of seed per pod (MSP, g), and mass of 100 seed (MHS, g), in 1,000 pods of Crotalaria juncea (A) and Crotalaria spectabilis (B). 

Figure 2. Matrix with a histogram frequency (in diagonal) and dispersion graphs between mass of pod with seed (MPWS, g), mass of pods without seed (MPWOS, g), length of pod (LP, mm), width of pod (WP, mm), height of pod (HP, mm), number of seed per pod (NSP, unity), mass of seed per pod (MSP, g), and mass of 100 seed (MHS, g), in 1,000 pods of Crotalaria breviflora (A) and Crotalaria ochroleuca (B). 

Correlations considered negligible from a practical point of view (-0.30 ≤ r ≤ 0.30) were obtained for the following trait pairs: MPWOS×MHS, LP×MHS, WP×HP, WP×NSP, HP×MHS, and NSP×MHS, in C. juncea; for MPWS×MHS, MPWOS×MHS, LP×MHS, WP×MHS, HP×MHS, and NSP×MHS, in C. spectabilis; for MPWOS×MHS, LP×MHS, WP×MHS, HP×NSP, HP×MHS, and NSP×MHS, in C. breviflora; and for WP×NSP, WP×MHS, HP×NSP, HP×MHS, and NSP×MHS, in C. ochroleuca (Table 1). In general, MHS showed the lowest values of correlation with the other traits. The other pairs of traits showed low or moderate positive correlations (0.30 to 0.70). Additionally, Pearson’s linear correlation coefficients between species, based on the 28 correlation values between pairs of traits, were high to very high (0.842 ≤ r ≤ 0.922), indicating that, in general, the studied crotalaria species have similar association patterns.

Depending on the pair of traits considered, the sample size for the estimation of the Pearson’s linear correlation coefficient with the highest precision, established in this study (CI_95% of 0.10), ranged as follows: from 10 to more than 1,000 pods in C. juncea; from 45 to more than 1,000 pods in C. spectabilis; from 25 to more than 1,000 pods in C. breviflora; and from 50 to more than 1,000 pods in C. ochroleuca (Table 2). For all species, the smallest sample size at this level of precision was verified for the correlation between MPWS×MSP. As previously mentioned, this pair of traits was the only one to show a very high correlation (Table 1), according to the classification of ^{Hinkle et al. (2003)}, in all species of crotalaria, as expected, since MSP is obtained from the difference between MPWS and MPWOS. These results indicate that high correlations can be estimated with precision from smaller sample sizes.

Considering an intermediate precision in the Pearson’s linear correlation coefficient estimation (CI_95% of 0.20), the sample size ranged from 10 to 440 pods in C. juncea, from 15 to 415 pods in C. spectabilis, from 10 to 425 pods in C. breviflora, and from 20 to 380 pods in C. ochroleuca, depending on the pair of traits considered (Table 2). In general, a larger magnitude of correlations was found for MPWS×MPWOS, MPWS×NSP, MPWS×MSP, and NSP×MSP; in at least three species, these correlations were considered high or very high (Table 1). Accordingly, in general, these pairs of traits required the smallest sample size for the estimation of correlations (Table 2). However, in at least three species, the correlations between MPWS×MHS, LP×MHS, WP×MHS, HP×MHS, and NSP×MHS were considered negligible. In these pairs of traits, in general, a larger sample size was required to estimate the correlations. The use of 440 pods would allow of the estimation of correlations with 0.20 as the maximum CI_95%, independently of the species and pair of traits considered. Thus, if, for instance, an experiment with five treatments and four replicates is carried out with 20 plots, the evaluations should be performed for 22 pods per plot to estimate the correlation at this precision level. That is, the evaluation of 22 pods per plot would allow to adequately estimate the correlation of all pairs of traits, irrespectively of the crotalaria species used, with an executable number of measurements from a practical point of view.

As previously mentioned, in all species, the correlation between MPWS×MSP was very high, and the correlation between NSP×MHS was negligible from a practical point of view (Table 1). In this sense, it is possible to verify the difference of the confidence interval of the correlation coefficients for these two pairs of traits in all the species (Figure 3 A-H). Also, it can be verified that the sample size required to estimate the linear correlations decreases as the correlation strength increases (Figure 4). In this sense, ^{Olivoto et al. (2018)} verified that the Pearson’s confidence interval width is inversely proportional to the strength of the association between traits. The inverse relationship of strength of association between traits and sample sizes needed to estimate the correlations was also observed in studies applied to maize (^{Cargnelutti Filho et al., 2010}; ^{Toebe et al., 2015}), crambe (^{Cargnelutti Filho et al., 2011}), castor bean (^{Cargnelutti Filho et al., 2012}) and cherry tomato (^{Sari et al., 2017}).

Figure 3. Percentile 2.5^th, mean, and percentile 97.5^th of 10,000 estimates of Pearson’s linear correlation coefficients, as follows: between mass of pod with seed and mass of seed per pod (MPWS×MSP) in Crotalaria juncea (A), Crotalaria spectabilis (C); Crotalaria breviflora (E); Crotalaria ochroleuca (G); and between number of seed per pod and mass of 100 seed (NSP×MHS) in C. juncea (B), C. spectabilis (D), C. breviflora (F), and C. ochroleuca (H). 

Figure 4. Relationships between the correlation magnitude and the sample size recommended for Crotalaria juncea, Crotalaria spectabilis, Crotalaria breviflora, and Crotalaria ochroleuca), based on the 28 correlations of the trait pairs of each species (Table 1) and the corresponding sample size for CI_95% of 0.20 (Table 2). 

Considering CI_95% of 0.30, the sample size ranged from 10 to 200 pods in C. juncea, from 10 to 185 pods in C. spectabilis, from 10 to 190 pods in C. breviflora, and from 10 to 180 pods in C. ochroleuca, depending on the pair of traits considered (Table 3). Considering CI_95% of 0.40, the sample size ranged from 10 to 110 pods in C. juncea, from 10 to 105 pods in C. spectabilis, from 10 to 110 pods in C. breviflora, and from 10 to 105 pods in C. ochroleuca, depending on the pair of traits considered. In general, a greater variability of sample size was observed between the pairs of traits than between species for a given pair of traits (Tables 2 and 3).

Conclusions