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Pesquisa Agropecuária Brasileira

Print version ISSN 0100-204XOn-line version ISSN 1678-3921

Pesq. agropec. bras. vol.54  Brasília  2019  Epub Oct 21, 2019

https://doi.org/10.1590/s1678-3921.pab2019.v54.01027 

STATISTICS

Sample size for the estimation of Pearson’s linear correlation in crotalaria species

Tamanho de amostra para estimar coeficientes de correlação linear de Pearson em espécies de crotalária

Marcos Toebe(1)  * 
http://orcid.org/0000-0003-2033-1467

Letícia Nunes Machado(2) 
http://orcid.org/0000-0003-2497-0645

Francieli de Lima Tartaglia(3) 
http://orcid.org/0000-0003-0489-9554

Juliana Oliveira de Carvalho(4) 
http://orcid.org/0000-0002-6754-1464

Cirineu Tolfo Bandeira(3) 
http://orcid.org/0000-0001-5019-5959

Alberto Cargnelutti Filho(3) 
http://orcid.org/0000-0002-8608-9960

(1) Universidade Federal de Santa Maria, Departamento de Ciências Agronômicas e Ambientais, Linha 7 de Setembro, s/no, BR-386, Km 40, CEP 98400-000 Frederico Westphalen, RS, Brazil. E-mail: m.toebe@gmail.com

(2) Universidade Federal do Pampa, Rua Luiz Joaquim de Sá Britto, s/no, Promorar, CEP 97650-000 Itaqui, RS, Brazil. E-mail: leticia.nunes.machado@gmail.com

(3) Universidade Federal de Santa Maria, Departamento de Fitotecnia, Avenida Roraima, no 1.000, Camobi, CEP 97105-900 Santa Maria, RS, Brazil. E-mail: francielitartaglia@hotmail.com, cirineu_bandeira@hotmail.com, alberto.cargnelutti.filho@gmail.com

(4) Universidade Federal do Rio Grande do Sul, Departamento de Biociências, Avenida Bento Gonçalves, no 9.500, CEP 91501-970 Porto Alegre, RS, Brazil. E-mail: ju.olic@gmail.com


Abstract:

The objective of this work was to determine the necessary sample size to estimate Pearson’s linear correlation coefficients of four species of crotalaria at precision levels. The experiment was carried out with Crotalaria juncea, Crotalaria spectabilis, Crotalaria breviflora, and Crotalaria ochroleuca, during the 2014/2015 crop year. Eight crotalaria traits were evaluated in 1,000 randomly collected pods per species. For each species, the correlation coefficients were estimated for the 28 pairs of traits, and the sample size necessary to estimate the correlation coefficients was determined at four precision levels [0.10, 0.20, 0.30, and 0.40 amplitudes of the 95% (CI95%) confidence interval] by resampling with replacement. The sample size varies between crotalaria species and, especially, between pairs of traits, as a function of the magnitude of the correlation coefficient. At a certain precision level, the smallest sample size is required to estimate the correlation coefficients between highly correlated traits and vice-versa. To estimate the correlation coefficients with CI95% of 0.20, 10 to 440 pods are required, depending on the species, pairs of traits, and magnitude of the correlation coefficient.

Index terms: Crotalaria; linear relationships; resampling; sample precision

Resumo:

O objetivo deste trabalho foi determinar o tamanho de amostra necessário para estimar os coeficientes de correlação linear de Pearson em quatro espécies de crotalária, em níveis de precisão. O experimento foi realizado com Crotalaria juncea, Crotalaria spectabilis, Crotalaria breviflora e Crotalaria ochroleuca, no ano agrícola 2014/2015. Oito características da crotalária foram avaliadas em 1.000 vagens coletadas aleatoriamente por espécie. Para cada espécie, estimaram-se os coeficientes de correlação para os 28 pares de características e determinou-se o tamanho de amostra necessário para a estimação dos coeficientes de correlação, em quatro níveis de precisão [amplitudes do intervalo de confiança de 95% (CI95%) de 0,10, 0,20, 0,30 e 0,40] por reamostragem com reposição. O tamanho de amostra varia entre as espécies de crotalária e, principalmente, entre os pares de características, em função da magnitude do coeficiente de correlação. Em determinado nível de precisão, o menor tamanho de amostra é necessário para a estimação de coeficientes de correlação de alta magnitude e vice-versa. Para estimar coeficientes de correlação com CI95% de 0,20, são necessárias de 10 a 440 vagens, a depender da espécie, dos pares de características e da magnitude do coeficiente de correlação.

Termos para indexação: Crotalaria; relações lineares; reamostragem; precisão amostral

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 2.5th, the mean, and the percentile 97.5th were determined. The amplitude of the 95% confidence interval was calculated (CI95%) by the difference between the percentile 97.5th and the percentile 2.5th.

To determine the sample size (number of pods) required for the r estimation from each of the 28 pairs of traits, in each species, CI95% 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 CI95% 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.

Table 1. Estimates of Pearson’s linear correlation coefficients in 28 pairs of traits measured in 1,000 pods of crotalaria species - Crotalaria juncea, Crotalaria spectabilis, Crotalaria breviflora, and Crotalaria ochroleuca - in the crop year 2014/2015. 

Trait pair(1) C. juncea C. spectabilis C. breviflora C. ochroleuca Mean
MPWS×MPWOS 0.646* 0.892* 0.762* 0.885* 0.796
MPWS×LP 0.509* 0.690* 0.566* 0.691* 0.614
MPWS×WP 0.481* 0.736* 0.583* 0.599* 0.600
MPWS×HP 0.356* 0.472* 0.492* 0.432* 0.438
MPWS×NSP 0.773* 0.834* 0.779* 0.654* 0.760
MPWS×MSP 0.984* 0.927* 0.942* 0.915* 0.942
MPWS×MHS 0.497* 0.294* 0.442* 0.555* 0.447
MPWOS×LP 0.685* 0.701* 0.699* 0.705* 0.697
MPWOS×WP 0.571* 0.760* 0.649* 0.692* 0.668
MPWOS×HP 0.390* 0.449* 0.578* 0.483* 0.475
MPWOS×NSP 0.390* 0.586* 0.372* 0.395* 0.436
MPWOS×MSP 0.501* 0.657* 0.499* 0.621* 0.569
MPWOS×MHS 0.265* 0.220* 0.300* 0.436* 0.305
LP×WP 0.454* 0.686* 0.494* 0.590* 0.556
LP×HP 0.395* 0.304* 0.412* 0.519* 0.407
LP×NSP 0.435* 0.558* 0.332* 0.412* 0.434
LP×MSP 0.418* 0.569* 0.394* 0.551* 0.483
LP×MHS 0.043 ns 0.067* 0.167* 0.312* 0.147
WP×HP 0.052 ns 0.386* 0.522* 0.489* 0.362
WP×NSP 0.254* 0.572* 0.331* 0.267* 0.356
WP×MSP 0.413* 0.597* 0.442* 0.406* 0.464
WP×MHS 0.314* 0.110* 0.265* 0.274* 0.241
HP×NSP 0.319* 0.366* 0.292* 0.182* 0.290
HP×MSP 0.313* 0.415* 0.358* 0.308* 0.348
HP×MHS 0.073* 0.147* 0.170* 0.240* 0.157
NSP×MSP 0.787* 0.904* 0.848* 0.759* 0.825
NSP×MHS -0.105* -0.118* -0.093* -0.106* -0.105
MSP×MHS 0.502* 0.308* 0.435* 0.556* 0.450
Correlation between species calculated with the correlation of the 28 pairs of traits
CJ CS CB CO
C. juncea (CJ) - 0.842 0.893 0.858
C. spectabilis (CS) 0.842 - 0.884 0.820
C. breviflora (CB) 0.893 0.884 - 0.922
C. ochroleuca (CO) 0.858 0.820 0.922 -

(1)MPWS, mass of pods with seed; MPWOS, mass of pods without seed; LP, length of pod; WP, width of pod; HP, height of pod; NSP, number of seed per pod; MSP, mass of seed per pod; MHS, mass of 100 seed. *Significant by the t-test, at 5% probability. nsNonsignificant.

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 (CI95% 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.

Table 2. Sample size (number of pods) for the estimation of Pearson’s linear correlation coefficients in 28 pairs of traits, in Crotalaria juncea, Crotalaria spectabilis, Crotalaria breviflora, and Crotalaria ochroleuca), with 95% confidence interval (CI95%) of 0.10 and 0.20. 

Trait pair(1) CI95% of 0.10 CI95% of 0.20
C. juncea C. spectabilis C. breviflora C. ochroleuca C. juncea C. spectabilis C. breviflora C. ochroleuca
MPWS×MPWOS 545 90 295 75 145 30 80 25
MPWS×LP 915 470 630 470 235 120 165 120
MPWS×WP 865 335 670 675 220 90 170 175
MPWS×HP >1,000 >1,000 810 955 270 275 205 240
MPWS×NSP 265 175 250 550 70 50 70 140
MPWS×MSP 10 45 25 50 10 15 10 20
MPWS×MHS 845 >1,000 >1,000 >1,000 215 355 305 260
MPWOS×LP 440 465 330 395 115 120 90 110
MPWOS×WP 605 310 460 430 150 85 120 110
MPWOS×HP >1,000 995 615 915 270 245 160 225
MPWOS×NSP >1,000 780 >1,000 >1,000 290 200 280 275
MPWOS×MSP 915 675 935 575 240 175 240 150
MPWOS×MHS >1,000 >1,000 >1,000 >1,000 375 395 320 310
LP×WP 885 480 945 600 230 125 240 155
LP×HP >1,000 >1,000 930 815 335 340 240 210
LP×NSP >1,000 800 >1,000 >1,000 280 210 315 255
LP×MSP >1,000 770 >1,000 800 305 200 265 210
LP×MHS >1,000 >1,000 >1,000 >1,000 370 355 330 345
WP×HP >1,000 >1,000 755 875 440 320 195 220
WP×NSP >1,000 735 >1,000 >1,000 350 195 335 365
WP×MSP >1,000 675 >1,000 >1,000 260 175 265 300
WP×MHS >1,000 >1,000 >1,000 >1,000 265 340 270 330
HP×NSP >1,000 >1,000 >1,000 >1,000 315 325 330 370
HP×MSP >1,000 >1,000 >1,000 >1,000 285 310 280 315
HP×MHS >1,000 >1,000 >1,000 >1,000 315 350 365 360
NSP×MSP 250 85 155 305 70 30 45 80
NSP×MHS >1,000 >1,000 >1,000 >1,000 370 415 425 380
MSP×MHS 790 >1,000 >1,000 >1,000 205 390 335 265

(1)MPWS, mass of pods with seed; MPWOS, mass of pods without seed; LP, length of pod; WP, width of pod; HP, height of pod; NSP, number of seed per pod; MSP, mass of seed per pod; MHS, mass of 100 seed.

Considering an intermediate precision in the Pearson’s linear correlation coefficient estimation (CI95% 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 CI95%, 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 2.5th, mean, and percentile 97.5th 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 CI95% of 0.20 (Table 2). 

Considering CI95% 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 CI95% 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).

Table 3. Sample size (number of pods) for the estimation of Pearson’s linear correlation coefficients in 28 pairs of traits, in Crotalaria juncea, Crotalaria spectabilis, Crotalaria breviflora, and Crotalaria ochroleuca, with 95% confidence interval (CI95%) of 0.30 and 0.40. 

Trait pair(1) CI95% of 0.30 CI95% of 0.40
C. juncea C. spectabilis C. breviflora C. ochroleuca C. juncea C. spectabilis C. breviflora C. ochroleuca
MPWS×MPWOS 70 15 40 15 40 10 25 10
MPWS×LP 110 60 75 60 60 35 45 35
MPWS×WP 100 45 75 80 55 25 45 50
MPWS×HP 125 120 95 110 70 70 55 65
MPWS×NSP 35 25 35 65 20 15 25 40
MPWS×MSP 10 10 10 10 10 10 10 10
MPWS×MHS 95 160 135 120 55 90 75 70
MPWOS×LP 55 55 40 50 35 35 25 30
MPWOS×WP 70 40 60 50 40 25 35 30
MPWOS×HP 120 115 75 105 70 65 45 60
MPWOS×NSP 135 90 130 130 75 55 75 70
MPWOS×MSP 105 85 110 65 60 45 65 40
MPWOS×MHS 175 185 150 135 95 105 85 80
LP×WP 105 60 110 70 60 35 65 40
LP×HP 155 155 110 95 85 90 65 55
LP×NSP 125 95 140 120 75 55 85 70
LP×MSP 135 85 115 95 80 55 70 55
LP×MHS 160 160 150 155 95 95 85 95
WP×HP 200 140 90 100 110 80 50 60
WP×NSP 155 90 150 165 85 55 90 95
WP×MSP 120 80 120 135 70 45 70 80
WP×MHS 115 150 125 150 70 90 70 85
HP×NSP 140 145 150 165 80 85 85 100
HP×MSP 130 140 135 140 70 80 75 80
HP×MHS 150 160 160 160 85 95 95 95
NSP×MSP 35 15 25 40 20 10 15 25
NSP×MHS 160 185 190 180 95 100 110 105
MSP×MHS 95 175 155 120 55 95 90 70

(1)MPWS, mass of pods with seed; MPWOS, mass of pods without seed; LP, length of pod; WP, width of pod; HP, height of pod; NSP, number of seed per pod; MSP, mass of seed per pod; and MHS, mass of 100 seed.

Conclusions

  1. The sample size varies between crotalaria species and, especially, between pairs of traits as a function of the magnitude of the correlation coefficient.

  2. Smaller sample sizes are required to estimate the correlation coefficients between highly correlated traits.

  3. To estimate the correlation coefficients with CI95% of 0.20, 10 to 440 pods are required, depending on the species, pairs of traits, and magnitude of the correlation coefficient.

Acknowledgments

To Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), the Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul (Fapergs), the Programa de Educação Tutorial do Ministério da Educação, and to Fundação Universidade Federal do Pampa (Unipampa), for scholarships and financial support; to the scholarship students and volunteers, for their help in the conduction of the experiment and data collection, and to the company Piraí Sementes, for granting seed for research purposes.

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Received: September 05, 2018; Accepted: August 05, 2019

*Corresponding author m.toebe@gmail.com

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